Article

A Statistical Analysis of Cryptocurrencies

Stephen Chan 1 Jeffrey Chu 1 Saralees Nadarajah 1 and Joerg Osterrieder 2

1 School of Mathematics University of Manchester Manchester M13 9PL UKstephenchanmanchesteracuk (SC) jeffreychumanchesteracuk (JC)

2 School of Engineering Zurich University of Applied Sciences 8401 Winterthur Switzerlandjoergosterriederzhawch

Correspondence saraleesnadarjahmanchesteracuk Tel +44-161-275-5912

Academic Editor Charles S TapieroReceived 7 April 2017 Accepted 27 May 2017 Published 31 May 2017

Abstract We analyze statistical properties of the largest cryptocurrencies (determined by marketcapitalization) of which Bitcoin is the most prominent example We characterize their exchangerates versus the US Dollar by fitting parametric distributions to them It is shown that returnsare clearly non-normal however no single distribution fits well jointly to all the cryptocurrenciesanalysed We find that for the most popular currencies such as Bitcoin and Litecoin the generalizedhyperbolic distribution gives the best fit while for the smaller cryptocurrencies the normal inverseGaussian distribution generalized t distribution and Laplace distribution give good fits The resultsare important for investment and risk management purposes

Keywords exchange rate distributions blockchain Bitcoin

JEL Classification C1

1 Introduction

Bitcoin the first decentralized cryptocurrency has gained a large following from the mediaacademics and the finance industry since its inception in 2009 Built upon blockchain technology it hasestablished itself as the leader of cryptocurrencies and shows no signs of slowing down Instead ofbeing based on traditional trust the currency is based on cryptographic proof which provides manyadvantages over traditional payment methods (such as Visa and Mastercard) including high liquiditylower transaction costs and anonymity to name just a few

Indeed the global interest in Bitcoin has spiked once again in recent months for example the UKgovernment is considering paying out research grants in Bitcoin an increasing number of IT companiesare stockpiling Bitcoin to defend against ransomware growing numbers in China are buying intoBitcoin and seeing it as an investment opportunity Perhaps most significantly the Chair of the Board ofGovernors of the US Federal Reserve has been encouraging central bankers to study new innovationsin the financial industry In particular the Chair expressed a need to learn more about financialinnovations including Bitcoin Blockchain and distributed ledger technologies With this recent surgein interest we believe that now is the time to start studying Bitcoin (and other major cryptocurrencies)as key pieces of financial technology

Since 2009 numerous cryptocurrencies have been developed with as of February 2017 720 inexistence Bitcoin is the largest and most popular representing over 81 of the total market ofcryptocurrencies (CoinMarketCap 2017) However as the statistics show many have not garnered thesame level of interest The combined market capitalization of all cryptocurrencies is approximatelyUSD $19 billion (as of February 2017) with the top 15 currencies representing over 97 of the marketand seven of these accounting for 90 of the total market capitalization In our analysis we focus on

J Risk Financial Manag 2017 10 12 doi103390jrfm10020012 wwwmdpicomjournaljrfm

J Risk Financial Manag 2017 10 12 2 of 23

these seven cryptocurrencies which fall into the category of having existed for more than two yearsand are within the top 15 currencies by market capitalization These are Bitcoin Ripple LitecoinMonero Dash MaidSafeCoin and Dogecoin

There exists much research on Bitcoinmdashthe most popular cryptocurrency We briefly discuss theanalysis and results of some of the most popular studies Hencic and Gourieroux (2014) model andpredict the exchange rate of Bitcoin versus the US Dollar using a noncausal autoregressive processwith Cauchy errors Their results show that the daily BitcoinUSD exchange rate shows local trendswhich could indicate periods of speculative behaviour from online trading Sapuric and Kokkinaki (2014)investigate the volatility of Bitcoin using data from July 2010 to April 2014 by comparing it to thevolatility of the exchange rates of major global currencies Their analysis indicates that the exchange rateof Bitcoin has high annualised volatility however it can be considered more stable when transactionvolume is taken into consideration Briere et al (2015) use weekly data from 2010 to 2013 to analysediversified investment portfolios and find that Bitcoin is extremely volatile and shows large averagereturns Perhaps surprisingly the results indicate that Bitcoin offers little correlation with other assetsalthough it can help to diversify investment portfolios In Kristoufek (2015) the influencing factors ofthe price of Bitcoin are investigated and applied to the Chinese Bitcoin market Short and long termlinks are found and Bitcoin is shown to exhibit the properties of both standard financial assets but alsospeculative assets which fuel further discussion on whether Bitcoin should be classed as a currencyasset or an investment vehicle Chu et al (2015) give the first statistical analysis of the exchange rateof Bitcoin They fit fifteen of the most common distributions used in finance to the log returns ofthe exchange rate of Bitcoin versus the US Dollar Using data from 2011 to 2014 they show that thegeneralized hyperbolic distribution gives the best fit

There are thousands of papers published on exchange rates Hence it is impossible providea review of all papers Here we mention seven of the most recent papers Corlu and Corlu (2015)compare the performance of the generalized lambda distribution against other flexible distributionssuch as the skewed t distribution unbounded Johnson family of distributions and the normal inverseGaussian distribution in capturing the skewness and peakedness of the returns of exchange ratesThey conclude that for the Value-at-Risk and Expected Shortfall the generalised lambda distributiongives a similar performance and in general it can be used as an alternative for fitting the heavytail behaviour in financial data Nadarajah et al (2015) revisit the study of exchange rate returns inCorlu and Corlu (2015) and show that the Studentrsquos t distribution can give a similar performance tothose of the distributions tested in Corlu and Corlu (2015) Bruneau and Moran (2017) investigatethe effect of exchange rate fluctuations on labour market adjustments in Canadian manufacturingindustries Dai et al (2017) examine the role of exchange rates on economic growth in east Asiancountries Parlapiano et al (2017) examine exchange rate risk exposure on the value of European firmsSchroeder (2017) investigates the macroeconomic performance in developing countries with respect toexchange rates Seyyedi (2017) provides an analysis of the interactive linkages between gold prices oilprices and the exchange rate in India

One of the aspects we are looking at is the volatility of cryptocurrencies There are numerousdefinitions and methods for computing volatility in the literature For example in Sapuric and Kokkinaki(2014) volatility is defined as the annualised volatility (or the standard deviation) representing the dailyvolatility of an exchange rate This is calculated by multiplying the standard deviation of the exchangerate by the square root of the number of trading days per year Briere et al (2015) use annualised returnsin their analysis and also compute the annualised volatility Other types of volatility in the literature canbe broadly categorised as future historical forecast and implied volatility (Natenberg 2007) In theremainder of this paper the term volatility is defined as the spread of the daily exchange rates and logreturns of the exchange rates of the cryptocurrencies over the time period considered

The paper is organised as follows In Section 2 we give an overview of the data used in ouranalysis including descriptions and sources Section 3 examines the statistical properties of thecryptocurrencies by fitting a wide range of parametric distributions to the data Section 4 provides

J Risk Financial Manag 2017 10 12 3 of 23

a discussion of our results Finally Section 5 concludes and summarizes our findings Throughout thepaper 0000 should not be interpreted as an absolute zero It only means that the first three decimalplaces are equal to zero

2 Data

The data used in this paper are the historical global price indices of cryptocurrencies andwere obtained from the BNC2 database from Quandl The global price indices were used as theyrepresent a weighted average of the price of the respective cryptocurrencies using prices frommultiple exchanges For our analysis we choose to use daily data from 23 June 2014 until theend of February 2017 A start date of June 2014 was deliberately chosen so that we can analyze sevenof the top fifteen cryptocurrencies ranked by their market capitalization as of February 2017mdashseeCoinMarketCap (2017) for the current rankings of cryptocurrencies by market capitalization The sevencryptocurrencies chosen to be part of our analysis are Bitcoin Dash LiteCoin MaidSafeCoin MoneroDogeCoin and Ripple It should be noted that we omitted Ethereum arguably the second largestcryptocurrency at present and Ethereum Classic as those two only started trading in 2015 and 2016respectively Other notable cryptocurrencies such as Agur and NEM were also omitted due to the lackof data We believe that our choice of cryptocurrencies covers the most prominent currencies andindeed they represent 90 of the market capitalization as of February 2017 (CoinMarketCap 2017)In the following we provide a brief introduction of the seven cryptocurrencies chosen

Bitcoin is undoubtedly the most popular and prominent cryptocurrency It was the first realisationof the idea of a new type of money mentioned over two decades ago that ldquouses cryptographyto control its creation and transactions rather than a central authorityrdquo (Bitcoin Project 2017)This decentralization means that the Bitcoin network is controlled and owned by all of its usersand as all users must adhere to the same set of rules there is a great incentive to maintain thedecentralized nature of the network Bitcoin uses blockchain technology which keeps a record ofevery single transaction and the processing and authentication of transactions are carried out by thenetwork of users (Bitcoin Project 2017) Although the decentralized nature offers many advantagessuch as being free from government control and regulation critics often argue that apart from its usersthere is nobody overlooking the whole system and that the value of Bitcoin is unfounded In return forcontributing their computing power to the network to carry out some of the tasks mentioned abovealso known as ldquominingrdquo users are rewarded with Bitcoins These properties set Bitcoin apart fromtraditional currencies which are controlled and backed by a central bank or governing body

Dash (formerly known as Darkcoin and XCoin) is a ldquoprivacy-centric digital currency with instanttransactionsrdquo (The Dash Network 2017) Although it is based upon Bitcoinrsquos foundations and sharessimilar properties Dashrsquos network is two-tiered improving upon that of Bitcoinrsquos In contrastwith Bitcoin Dash is overseen by a decentralized network of serversmdashknown as ldquoMasternodesrdquo(The Dash Network 2017) which alleviates the need for a third party governing body and allows forfunctions such as financial privacy and instant transactions On the other hand users or ldquominersrdquo in thenetwork provide the computing power for basic functions such as sending and receiving currency andthe prevention of double spending The advantage of utilizing Masternodes is that transactions can beconfirmed almost in real time (compared with the Bitcoin network) because Masternodes are separatefrom miners and the two have non-overlapping functions (The Dash Network 2017) Dash utilizes theX11 chained proof-of-work hashing algorithm which helps to distribute the processing evenly acrossthe network while maintaining a similar coin distribution to Bitcoin Using eleven different hashesincreases security and reduces the uncertainty in Dash Dash operates ldquoDecentralized Governanceby Blockchainrdquo (The Dash Network 2017) which allows owners of Masternodes to make decisionsand provides a method for the platform to fund its own development

LiteCoin (LTC) was created in 2011 by Charles Lee with support from the Bitcoin communityBased on the same peer-to-peer protocol used by Bitcoin it is often cited as Bitcoinrsquos leading rivalas it features improvements over the current implementation of Bitcoin It has two main features

J Risk Financial Manag 2017 10 12 4 of 23

which distinguish it from Bitcoin its use of scrypt as a proof-of-work algorithm and a significantlyfaster confirmation time for transactions The former enables standard computational hardware toverify transactions and reduces the incentive to use specially designed hardware while the latterreduces transaction confirmation times to minutes rather than hours and is particularly attractive intime-critical situations (LiteCoin Project 2017)

MaidSafeCoin is a digital currency which powers the peer-to-peer Secure Access For Everyone(SAFE) network which combines the computing power of all its users and can be thought of asa ldquocrowd-sourced internetrdquo (MaidSafe 2017a) Each MadeSafe coin has a unique identity and thereexists a hard upper limit of 43 billion coins as opposed to Bitcoinrsquos 21 million As the currency isused to pay for services on the SAFE network the currency will be recycled meaning that in theorythe amount of MaidSafe coins will never be exhausted The process of generating new currencyis similar to other cryptocurrencies and in the case of the SAFE network it is known as ldquofarmingrdquo(MaidSafe 2017b) Users contribute their computing power and storage space to the network and arerewarded with coins when the network accesses data from their store (MaidSafe 2017b)

Monero (XMR) is a ldquosecure private untraceable currencyrdquo (Monero 2017) centred arounddecentralization and scalability that was launched in April 2014 The currency itself is completelydonation-based community driven and based entirely on proof-of-work Whilst transactions in thenetwork are private by default users can set their level of privacy allowing as much or as littleaccess to their transactions as they wish Although it employs a proof-of-work algorithm Monerois more similar to LiteCoin in that mining of the currency can be done by any modern computerand is not restricted to specially designed hardware It arguably holds some advantages over otherBitcoin-based cryptocurrencies such as having a dynamic block size (overcoming the problem ofscalability) and being a disinflationary currency meaning that there will always exist an incentive toproduce the Monero currency (Monero 2017)

Dogecoin (Dogecoin 2017) originated from a popular internet meme in December 2013 Createdby an Australian brand and marketing specialist and a programmer in Portland Oregon it initiallystarted off as a joke currency but quickly gained traction It is a variation on Litecoin running on thecryptographic scrypt enabling similar advantages over Bitcoin such as faster transaction processingtimes Part of the attraction of Dogecoin is its light-hearted culture and lower barriers to entry toinvesting in or acquiring cryptocurrencies One of the most popular uses for Dogecoin is the tipping ofothers on the internet who create or share interesting content and can be thought of as the next levelup from a ldquolikerdquo on social media or an ldquoupvoterdquo on internet forums This in part has arisen from thefact that it has now become too expensive to tip using Bitcoin

Ripple was originally developed in 2012 and is the first global real-time gross settlement network(RTGS) which ldquoenables banks to send real-time international payments across networksrdquo (Ripple 2017)The Ripple network is a blockchain network which incorporates a payment system and a currencysystem known as XRP which is not based on proof-of-work like Monero and Dash A unique propertyof Ripple is that XRP is not compulsory for transactions on the network although it is encouragedas a bridge currency for more competitive cross border payments (Ripple 2017) The Ripple protocolis currently used by companies such as UBS Santander and Standard Chartered and increasinglybeing used by the financial services industry as technology in settlements Compared with Bitcoinit has advantages such as greater control over the system as it is not subject to the price volatility of theunderlying currencies and it has a more secure distributed authentication process

Summary statistics of the exchange rates and log returns of the exchange rates of the sevencryptocurrencies are given in Tables 1 and 2 respectively In Table 1 the summary statistics for theraw exchange rates of the cryptocurrencies we see a simple reflection of the ldquoworthrdquo or value of eachcurrency It can clearly be seen that the exchange rate of Dogecoin is the least significant and at thetime of writing (February 2017) the exchange rate is approximately $00002 USD to one DogecoinThis supports the evidence that Dogecoin is primarily used as a currency for online tipping ratherthan as a currency for standard payments It has the lowest minimum first quartile median mean

J Risk Financial Manag 2017 10 12 5 of 23

third quartile and maximum values In contrast being the most popular cryptocurrency Bitcoinhas the largest minimum first quartile median mean third quartile and maximum values whichshow its greater significance and higher ldquovaluerdquo to those with a vested interest in cryptocurrenciesThe exchange rates of all seven currencies are positively skewed with Litecoin Monero and Ripplebeing the most skewed In terms of kurtosis MaidSafeCoin shows less peakedness than that of thenormal distribution Bitcoin Dash and Dogecoin show levels similar to the normal distributionLitecoin Monero and Ripple have significantly greater peakedness than the normal distributionThe exchange rates of Dogecoin MaidSafeCoin and Ripple have the smallest variances and standarddeviations indicating that their low volatility can perhaps be explained by the low values of theexchange rates coupled with the fact that their range and interquartile ranges are very limitedOn the other hand Bitcoin Dash and Litecoinrsquos exchange rates show the greatest variance andstandard deviation

Table 1 Summary statistics of daily exchange rates of Bitcoin Dash Dogecoin Litecoin MaidSafeCoinMonero Ripple and Euro versus the US Dollar from 23 June 2014 until 28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum 192700 1178 0000 1269 0012 0235 0003 0626Q1 273600 2577 0000 3091 0020 0491 0006 0736

Median 415200 3623 0000 3662 0029 0811 0007 0779Mean 447400 5385 0000 3659 0046 2355 0008 0830

Q3 593000 7921 0000 4021 0074 1970 0008 0856Maximum 1140000 17560 0000 9793 0152 17590 0028 1207Skewness 0841 1006 0417 1363 0849 2108 2543 1127Kurtosis 3096 2992 3175 6621 2503 6526 10693 3067

SD 193241 3583 0000 1433 0032 3397 0004 0142Variance 37342159 12838 0000 2053 0001 11543 0000 0020

CV 0432 0665 0294 0392 0695 1443 0471 0171Range 946938 16385 0000 8524 0140 17358 0025 0581IQR 319400 5344 0000 0930 0054 1479 0002 0119

Table 2 Summary statistics of daily log returns of the exchange rates of Bitcoin Dash DogecoinLitecoin MaidSafeCoin Monero Ripple and the Euro versus the US Dollar from 23 June 2014 until28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum minus0159 minus0580 minus0385 minus0278 minus0404 minus0560 minus0299 minus0046Q1 minus0011 minus0019 minus0009 minus0010 minus0026 minus0026 minus0014 minus0004

Median minus0001 0003 0002 0000 minus0001 0002 0002 0Mean minus0001 minus0001 0000 0001 minus0002 minus0001 minus0000 minus000004

Q3 0008 0020 0015 0009 0023 0028 0017 0003Maximum 0205 0411 0188 0433 0241 0277 0288 0038Skewness 0758 minus1487 minus2506 0756 minus0478 minus1414 minus0401 minus0145Kurtosis 11568 26805 24434 22385 8520 13954 13818 2662

SD 0028 0051 0042 0042 0054 0062 0046 0006Variance 0001 0003 0002 0002 0003 0004 0002 000004

CV minus47976 minus84519 89782 45619 minus21499 minus54548 minus96585 minus143498Range 0364 0991 0573 0711 0645 0837 0587 0085IQR 0019 0039 0023 0019 0049 0054 0030 0007

Table 2 gives the summary statistics for the log returns of the exchange rates of the sevencryptocurrencies Here the log returns show some slightly different results Dash MaidSafeCoinand Monero have the lowest minimum values while Dash and Litecoin have the largest maximumsThe means and medians of the log returns of all seven currencies are similar and almost equal to zeroOnly the log returns of Bitcoin and Litecoin are positively skewed all others are negatively skewedwith Dogecoin being the most significant Log returns of all seven currencies have a peakednesssignificantly greater than that of the normal distribution with the most peaked being those of Dash

J Risk Financial Manag 2017 10 12 6 of 23

Dogecoin and Litecoin Noteworthy as much has been discussed about the volatility of Bitcoin returnsthe log returns of Bitcoin have the lowest variance and standard deviation of the seven cryptocurrenciesThose with the highest variation are Dash and perhaps unexpectedly MaidSafeCoin and Monero

Also shown in Tables 1 and 2 are the summary statistics of the exchange rates of the Euro and thesummary statistics of its log returns For the exchange rates the summary statistics for the Euro appearmuch smaller than those of Bitcoin but comparable to those of other cryptocurrencies For the logreturns of the exchange rates the summary statistics for the Euro generally appear smaller comparedto all the cryptocurrencies An exception is the coefficient of variation The magnitude of this statisticfor the Euro appears largest compared to all the cryptocurrencies

Fitting of a statistical distribution usually assumes that the data are independent andidentically distributed (ie randomness) have no serial correlation and have no heteroskedasticityWe tested for randomness using the difference sign and rank tests We tested for no serial correlationusing Durbin and Watson (1950 1951 1971)rsquos method We tested for no heteroskedasticity usingBreusch and Pagan (1979)rsquos test These tests showed that the log returns of the exchange rates of theseven cryptocurrencies can be assumed to be approximately independent and identically distributedhave no serial correlation and have no heteroskedasticity

3 Distributions Fitted

Having briefly examined the summary statistics for both the exchange rates and the log returns ofexchange rates of the seven cryptocurrencies we provide a visual representation of the distribution ofthe log returns Figure 1 shows the histograms of the daily log returns of the exchange rate (versus theUS Dollar) for all seven cryptocurrencies From the plots we find that the log returns in the cases ofall seven cryptocurrencies show significant deviation from the normal distribution Next we proceedto fit the parametric distributions to the data

Bitcoin

Log returns

Fre

quency

minus02 minus01 00 01 02

0300

Dash

Log returns

Fre

quency

minus06 minus02 00 02 04

0300

Dogecoin

Log returns

Fre

quency

minus04 minus02 00 01 02

0300

Litecoin

Log returns

Fre

quency

minus02 00 02 04

0300

MaidSafecoin

Log returns

Fre

quency

minus04 minus02 00 02

0300

Monero

Log returns

Fre

quency

minus06 minus04 minus02 00 02

0300

Ripples

Log returns

Fre

quency

minus03 minus01 01 02 03

0300

Figure 1 Histograms of daily log returns of the exchange rates of the seven cryptocurrencies versusthe US Dollar from 23 June 2014 until 28 February 2017

J Risk Financial Manag 2017 10 12 7 of 23

Let X denote a continuous random variable representing the log returns of the exchange rate ofthe cryptocurrency of interest Let f (x) denote the probability density function (pdf) of X Let F(x)denote the cumulative distribution function (cdf) of X We suppose X follows one of eight possibledistributions the most popular parametric distributions used in finance They are specified as follows

bull the Studentrsquos t distribution (Gosset 1908) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 and ν gt 0 where K(ν) =radic

νB (ν2 12) and B(middot middot)denotes the beta function defined by

B(a b) =int 1

0taminus1(1minus t)bminus1dt

bull the Laplace distribution (Laplace 1774) with

f (x) =1

2σexp

(minus| xminus micro |

σ

)for minusinfin lt x lt infin minusinfin lt micro lt infin and σ gt 0

bull the skew t distribution (Azzalini and Capitanio 2003) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

+2K2(ν)λ(xminus micro)

σ2 2F1

(12

1 + ν

2

32

minusλ2(xminus micro)2

σ2ν

)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin σ gt 0 and ν gt 0 where 2F1(a b c x) denotes theGauss hypergeometric function defined by

2F1 (a b c x) =infin

sumk=0

(a)k (b)k(c)k

xk

k

where (e)k = e(e + 1) middot middot middot (e + kminus 1) denotes the ascending factorialbull the generalized t distribution (McDonald and Newey 1988) with

f (x) =τ

2σν1νB (ν 1τ)

[1 +

1ν

∣∣∣∣ xminus micro

σ

∣∣∣∣τ]minus(ν+1τ)

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 ν gt 0 and τ gt 0bull the skewed Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =K (ν)

σ

1 +

1ν

[xminus micro

2σα

]2minus ν+1

2

if x le micro

1 +

1ν

[xminus micro

2σ (1minus α)

]2minus ν+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 and ν gt 0

J Risk Financial Manag 2017 10 12 8 of 23

bull the asymmetric Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =1σ

α

αlowastK (ν1)

1 +

1ν1

[xminus micro

2σαlowast

]2minus ν1+1

2

if x le micro

1minus α

1minus αlowastK (ν2)

1 +

1ν2

[xminus micro

2σ (1minus αlowast)

]2minus ν2+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 ν1 gt 0 and ν2 gt 0 where

αlowast =αK (ν1)

αK (ν1) + (1minus α)K (ν2)

bull the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ αradic

2πKminus12 (δγ)exp [β(xminus micro)]

[δ2 + (xminus micro)2

]minus1Kminus1

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2 and Kν(middot) denotesthe modified Bessel function of the second kind of order ν defined by

Kν(x) =

πcsc(πν)

2[Iminusν(x)minus Iν(x)] if ν 6isin Z

limmicrorarrν

Kmicro(x) if ν isin Z

where Iν(middot) denotes the modified Bessel function of the first kind of order ν defined by

Iν(x) =infin

sumk=0

1Γ(k + ν + 1)k

( x2

)2k+ν

where Γ(middot) denotes the gamma function defined by

Γ(a) =int infin

0taminus1 exp(minust)dt

bull the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ α12minusλ

radic2πKλ (δγ)

exp [β(xminus micro)][δ2 + (xminus micro)2

]λminus12Kλminus12

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2

Many of these distributions are nested the skew t distribution for λ = 0 is the Studentrsquos tdistribution the generalized t distribution for τ = 2 is the Studentrsquos t distribution the skewed Studentrsquost distribution for α = 12 is the Studentrsquos t distribution the asymmetric Studentrsquos t distribution forν1 = ν2 is the skewed Studentrsquos t distribution the generalized hyperbolic distribution for λ = minus12 isthe normal inverse Gaussian distribution and so on

All but one of the distributions (the Laplace distribution) are heavy tailed Heavy tails are commonin financial data The Studentrsquos t distribution is perhaps the simplest of the heavy tailed distributionsIt does not allow for asymmetry The skew t distribution due to Azzalini and Capitanio (2003) isan asymmetric generalization of the Studentrsquos t distribution The generalized t distribution due toMcDonald and Newey (1988) has two parameters controlling its heavy tails adding more flexibility

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 3 of 23

a discussion of our results Finally Section 5 concludes and summarizes our findings Throughout thepaper 0000 should not be interpreted as an absolute zero It only means that the first three decimalplaces are equal to zero

2 Data

The data used in this paper are the historical global price indices of cryptocurrencies andwere obtained from the BNC2 database from Quandl The global price indices were used as theyrepresent a weighted average of the price of the respective cryptocurrencies using prices frommultiple exchanges For our analysis we choose to use daily data from 23 June 2014 until theend of February 2017 A start date of June 2014 was deliberately chosen so that we can analyze sevenof the top fifteen cryptocurrencies ranked by their market capitalization as of February 2017mdashseeCoinMarketCap (2017) for the current rankings of cryptocurrencies by market capitalization The sevencryptocurrencies chosen to be part of our analysis are Bitcoin Dash LiteCoin MaidSafeCoin MoneroDogeCoin and Ripple It should be noted that we omitted Ethereum arguably the second largestcryptocurrency at present and Ethereum Classic as those two only started trading in 2015 and 2016respectively Other notable cryptocurrencies such as Agur and NEM were also omitted due to the lackof data We believe that our choice of cryptocurrencies covers the most prominent currencies andindeed they represent 90 of the market capitalization as of February 2017 (CoinMarketCap 2017)In the following we provide a brief introduction of the seven cryptocurrencies chosen

Bitcoin is undoubtedly the most popular and prominent cryptocurrency It was the first realisationof the idea of a new type of money mentioned over two decades ago that ldquouses cryptographyto control its creation and transactions rather than a central authorityrdquo (Bitcoin Project 2017)This decentralization means that the Bitcoin network is controlled and owned by all of its usersand as all users must adhere to the same set of rules there is a great incentive to maintain thedecentralized nature of the network Bitcoin uses blockchain technology which keeps a record ofevery single transaction and the processing and authentication of transactions are carried out by thenetwork of users (Bitcoin Project 2017) Although the decentralized nature offers many advantagessuch as being free from government control and regulation critics often argue that apart from its usersthere is nobody overlooking the whole system and that the value of Bitcoin is unfounded In return forcontributing their computing power to the network to carry out some of the tasks mentioned abovealso known as ldquominingrdquo users are rewarded with Bitcoins These properties set Bitcoin apart fromtraditional currencies which are controlled and backed by a central bank or governing body

Dash (formerly known as Darkcoin and XCoin) is a ldquoprivacy-centric digital currency with instanttransactionsrdquo (The Dash Network 2017) Although it is based upon Bitcoinrsquos foundations and sharessimilar properties Dashrsquos network is two-tiered improving upon that of Bitcoinrsquos In contrastwith Bitcoin Dash is overseen by a decentralized network of serversmdashknown as ldquoMasternodesrdquo(The Dash Network 2017) which alleviates the need for a third party governing body and allows forfunctions such as financial privacy and instant transactions On the other hand users or ldquominersrdquo in thenetwork provide the computing power for basic functions such as sending and receiving currency andthe prevention of double spending The advantage of utilizing Masternodes is that transactions can beconfirmed almost in real time (compared with the Bitcoin network) because Masternodes are separatefrom miners and the two have non-overlapping functions (The Dash Network 2017) Dash utilizes theX11 chained proof-of-work hashing algorithm which helps to distribute the processing evenly acrossthe network while maintaining a similar coin distribution to Bitcoin Using eleven different hashesincreases security and reduces the uncertainty in Dash Dash operates ldquoDecentralized Governanceby Blockchainrdquo (The Dash Network 2017) which allows owners of Masternodes to make decisionsand provides a method for the platform to fund its own development

LiteCoin (LTC) was created in 2011 by Charles Lee with support from the Bitcoin communityBased on the same peer-to-peer protocol used by Bitcoin it is often cited as Bitcoinrsquos leading rivalas it features improvements over the current implementation of Bitcoin It has two main features

J Risk Financial Manag 2017 10 12 4 of 23

which distinguish it from Bitcoin its use of scrypt as a proof-of-work algorithm and a significantlyfaster confirmation time for transactions The former enables standard computational hardware toverify transactions and reduces the incentive to use specially designed hardware while the latterreduces transaction confirmation times to minutes rather than hours and is particularly attractive intime-critical situations (LiteCoin Project 2017)

MaidSafeCoin is a digital currency which powers the peer-to-peer Secure Access For Everyone(SAFE) network which combines the computing power of all its users and can be thought of asa ldquocrowd-sourced internetrdquo (MaidSafe 2017a) Each MadeSafe coin has a unique identity and thereexists a hard upper limit of 43 billion coins as opposed to Bitcoinrsquos 21 million As the currency isused to pay for services on the SAFE network the currency will be recycled meaning that in theorythe amount of MaidSafe coins will never be exhausted The process of generating new currencyis similar to other cryptocurrencies and in the case of the SAFE network it is known as ldquofarmingrdquo(MaidSafe 2017b) Users contribute their computing power and storage space to the network and arerewarded with coins when the network accesses data from their store (MaidSafe 2017b)

Monero (XMR) is a ldquosecure private untraceable currencyrdquo (Monero 2017) centred arounddecentralization and scalability that was launched in April 2014 The currency itself is completelydonation-based community driven and based entirely on proof-of-work Whilst transactions in thenetwork are private by default users can set their level of privacy allowing as much or as littleaccess to their transactions as they wish Although it employs a proof-of-work algorithm Monerois more similar to LiteCoin in that mining of the currency can be done by any modern computerand is not restricted to specially designed hardware It arguably holds some advantages over otherBitcoin-based cryptocurrencies such as having a dynamic block size (overcoming the problem ofscalability) and being a disinflationary currency meaning that there will always exist an incentive toproduce the Monero currency (Monero 2017)

Dogecoin (Dogecoin 2017) originated from a popular internet meme in December 2013 Createdby an Australian brand and marketing specialist and a programmer in Portland Oregon it initiallystarted off as a joke currency but quickly gained traction It is a variation on Litecoin running on thecryptographic scrypt enabling similar advantages over Bitcoin such as faster transaction processingtimes Part of the attraction of Dogecoin is its light-hearted culture and lower barriers to entry toinvesting in or acquiring cryptocurrencies One of the most popular uses for Dogecoin is the tipping ofothers on the internet who create or share interesting content and can be thought of as the next levelup from a ldquolikerdquo on social media or an ldquoupvoterdquo on internet forums This in part has arisen from thefact that it has now become too expensive to tip using Bitcoin

Ripple was originally developed in 2012 and is the first global real-time gross settlement network(RTGS) which ldquoenables banks to send real-time international payments across networksrdquo (Ripple 2017)The Ripple network is a blockchain network which incorporates a payment system and a currencysystem known as XRP which is not based on proof-of-work like Monero and Dash A unique propertyof Ripple is that XRP is not compulsory for transactions on the network although it is encouragedas a bridge currency for more competitive cross border payments (Ripple 2017) The Ripple protocolis currently used by companies such as UBS Santander and Standard Chartered and increasinglybeing used by the financial services industry as technology in settlements Compared with Bitcoinit has advantages such as greater control over the system as it is not subject to the price volatility of theunderlying currencies and it has a more secure distributed authentication process

Summary statistics of the exchange rates and log returns of the exchange rates of the sevencryptocurrencies are given in Tables 1 and 2 respectively In Table 1 the summary statistics for theraw exchange rates of the cryptocurrencies we see a simple reflection of the ldquoworthrdquo or value of eachcurrency It can clearly be seen that the exchange rate of Dogecoin is the least significant and at thetime of writing (February 2017) the exchange rate is approximately $00002 USD to one DogecoinThis supports the evidence that Dogecoin is primarily used as a currency for online tipping ratherthan as a currency for standard payments It has the lowest minimum first quartile median mean

J Risk Financial Manag 2017 10 12 5 of 23

third quartile and maximum values In contrast being the most popular cryptocurrency Bitcoinhas the largest minimum first quartile median mean third quartile and maximum values whichshow its greater significance and higher ldquovaluerdquo to those with a vested interest in cryptocurrenciesThe exchange rates of all seven currencies are positively skewed with Litecoin Monero and Ripplebeing the most skewed In terms of kurtosis MaidSafeCoin shows less peakedness than that of thenormal distribution Bitcoin Dash and Dogecoin show levels similar to the normal distributionLitecoin Monero and Ripple have significantly greater peakedness than the normal distributionThe exchange rates of Dogecoin MaidSafeCoin and Ripple have the smallest variances and standarddeviations indicating that their low volatility can perhaps be explained by the low values of theexchange rates coupled with the fact that their range and interquartile ranges are very limitedOn the other hand Bitcoin Dash and Litecoinrsquos exchange rates show the greatest variance andstandard deviation

Table 1 Summary statistics of daily exchange rates of Bitcoin Dash Dogecoin Litecoin MaidSafeCoinMonero Ripple and Euro versus the US Dollar from 23 June 2014 until 28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum 192700 1178 0000 1269 0012 0235 0003 0626Q1 273600 2577 0000 3091 0020 0491 0006 0736

Median 415200 3623 0000 3662 0029 0811 0007 0779Mean 447400 5385 0000 3659 0046 2355 0008 0830

Q3 593000 7921 0000 4021 0074 1970 0008 0856Maximum 1140000 17560 0000 9793 0152 17590 0028 1207Skewness 0841 1006 0417 1363 0849 2108 2543 1127Kurtosis 3096 2992 3175 6621 2503 6526 10693 3067

SD 193241 3583 0000 1433 0032 3397 0004 0142Variance 37342159 12838 0000 2053 0001 11543 0000 0020

CV 0432 0665 0294 0392 0695 1443 0471 0171Range 946938 16385 0000 8524 0140 17358 0025 0581IQR 319400 5344 0000 0930 0054 1479 0002 0119

Table 2 Summary statistics of daily log returns of the exchange rates of Bitcoin Dash DogecoinLitecoin MaidSafeCoin Monero Ripple and the Euro versus the US Dollar from 23 June 2014 until28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum minus0159 minus0580 minus0385 minus0278 minus0404 minus0560 minus0299 minus0046Q1 minus0011 minus0019 minus0009 minus0010 minus0026 minus0026 minus0014 minus0004

Median minus0001 0003 0002 0000 minus0001 0002 0002 0Mean minus0001 minus0001 0000 0001 minus0002 minus0001 minus0000 minus000004

Q3 0008 0020 0015 0009 0023 0028 0017 0003Maximum 0205 0411 0188 0433 0241 0277 0288 0038Skewness 0758 minus1487 minus2506 0756 minus0478 minus1414 minus0401 minus0145Kurtosis 11568 26805 24434 22385 8520 13954 13818 2662

SD 0028 0051 0042 0042 0054 0062 0046 0006Variance 0001 0003 0002 0002 0003 0004 0002 000004

CV minus47976 minus84519 89782 45619 minus21499 minus54548 minus96585 minus143498Range 0364 0991 0573 0711 0645 0837 0587 0085IQR 0019 0039 0023 0019 0049 0054 0030 0007

Table 2 gives the summary statistics for the log returns of the exchange rates of the sevencryptocurrencies Here the log returns show some slightly different results Dash MaidSafeCoinand Monero have the lowest minimum values while Dash and Litecoin have the largest maximumsThe means and medians of the log returns of all seven currencies are similar and almost equal to zeroOnly the log returns of Bitcoin and Litecoin are positively skewed all others are negatively skewedwith Dogecoin being the most significant Log returns of all seven currencies have a peakednesssignificantly greater than that of the normal distribution with the most peaked being those of Dash

J Risk Financial Manag 2017 10 12 6 of 23

Dogecoin and Litecoin Noteworthy as much has been discussed about the volatility of Bitcoin returnsthe log returns of Bitcoin have the lowest variance and standard deviation of the seven cryptocurrenciesThose with the highest variation are Dash and perhaps unexpectedly MaidSafeCoin and Monero

Also shown in Tables 1 and 2 are the summary statistics of the exchange rates of the Euro and thesummary statistics of its log returns For the exchange rates the summary statistics for the Euro appearmuch smaller than those of Bitcoin but comparable to those of other cryptocurrencies For the logreturns of the exchange rates the summary statistics for the Euro generally appear smaller comparedto all the cryptocurrencies An exception is the coefficient of variation The magnitude of this statisticfor the Euro appears largest compared to all the cryptocurrencies

Fitting of a statistical distribution usually assumes that the data are independent andidentically distributed (ie randomness) have no serial correlation and have no heteroskedasticityWe tested for randomness using the difference sign and rank tests We tested for no serial correlationusing Durbin and Watson (1950 1951 1971)rsquos method We tested for no heteroskedasticity usingBreusch and Pagan (1979)rsquos test These tests showed that the log returns of the exchange rates of theseven cryptocurrencies can be assumed to be approximately independent and identically distributedhave no serial correlation and have no heteroskedasticity

3 Distributions Fitted

Having briefly examined the summary statistics for both the exchange rates and the log returns ofexchange rates of the seven cryptocurrencies we provide a visual representation of the distribution ofthe log returns Figure 1 shows the histograms of the daily log returns of the exchange rate (versus theUS Dollar) for all seven cryptocurrencies From the plots we find that the log returns in the cases ofall seven cryptocurrencies show significant deviation from the normal distribution Next we proceedto fit the parametric distributions to the data

Bitcoin

Log returns

Fre

quency

minus02 minus01 00 01 02

0300

Dash

Log returns

Fre

quency

minus06 minus02 00 02 04

0300

Dogecoin

Log returns

Fre

quency

minus04 minus02 00 01 02

0300

Litecoin

Log returns

Fre

quency

minus02 00 02 04

0300

MaidSafecoin

Log returns

Fre

quency

minus04 minus02 00 02

0300

Monero

Log returns

Fre

quency

minus06 minus04 minus02 00 02

0300

Ripples

Log returns

Fre

quency

minus03 minus01 01 02 03

0300

Figure 1 Histograms of daily log returns of the exchange rates of the seven cryptocurrencies versusthe US Dollar from 23 June 2014 until 28 February 2017

J Risk Financial Manag 2017 10 12 7 of 23

Let X denote a continuous random variable representing the log returns of the exchange rate ofthe cryptocurrency of interest Let f (x) denote the probability density function (pdf) of X Let F(x)denote the cumulative distribution function (cdf) of X We suppose X follows one of eight possibledistributions the most popular parametric distributions used in finance They are specified as follows

bull the Studentrsquos t distribution (Gosset 1908) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 and ν gt 0 where K(ν) =radic

νB (ν2 12) and B(middot middot)denotes the beta function defined by

B(a b) =int 1

0taminus1(1minus t)bminus1dt

bull the Laplace distribution (Laplace 1774) with

f (x) =1

2σexp

(minus| xminus micro |

σ

)for minusinfin lt x lt infin minusinfin lt micro lt infin and σ gt 0

bull the skew t distribution (Azzalini and Capitanio 2003) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

+2K2(ν)λ(xminus micro)

σ2 2F1

(12

1 + ν

2

32

minusλ2(xminus micro)2

σ2ν

)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin σ gt 0 and ν gt 0 where 2F1(a b c x) denotes theGauss hypergeometric function defined by

2F1 (a b c x) =infin

sumk=0

(a)k (b)k(c)k

xk

k

where (e)k = e(e + 1) middot middot middot (e + kminus 1) denotes the ascending factorialbull the generalized t distribution (McDonald and Newey 1988) with

f (x) =τ

2σν1νB (ν 1τ)

[1 +

1ν

∣∣∣∣ xminus micro

σ

∣∣∣∣τ]minus(ν+1τ)

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 ν gt 0 and τ gt 0bull the skewed Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =K (ν)

σ

1 +

1ν

[xminus micro

2σα

]2minus ν+1

2

if x le micro

1 +

1ν

[xminus micro

2σ (1minus α)

]2minus ν+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 and ν gt 0

J Risk Financial Manag 2017 10 12 8 of 23

bull the asymmetric Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =1σ

α

αlowastK (ν1)

1 +

1ν1

[xminus micro

2σαlowast

]2minus ν1+1

2

if x le micro

1minus α

1minus αlowastK (ν2)

1 +

1ν2

[xminus micro

2σ (1minus αlowast)

]2minus ν2+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 ν1 gt 0 and ν2 gt 0 where

αlowast =αK (ν1)

αK (ν1) + (1minus α)K (ν2)

bull the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ αradic

2πKminus12 (δγ)exp [β(xminus micro)]

[δ2 + (xminus micro)2

]minus1Kminus1

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2 and Kν(middot) denotesthe modified Bessel function of the second kind of order ν defined by

Kν(x) =

πcsc(πν)

2[Iminusν(x)minus Iν(x)] if ν 6isin Z

limmicrorarrν

Kmicro(x) if ν isin Z

where Iν(middot) denotes the modified Bessel function of the first kind of order ν defined by

Iν(x) =infin

sumk=0

1Γ(k + ν + 1)k

( x2

)2k+ν

where Γ(middot) denotes the gamma function defined by

Γ(a) =int infin

0taminus1 exp(minust)dt

bull the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ α12minusλ

radic2πKλ (δγ)

exp [β(xminus micro)][δ2 + (xminus micro)2

]λminus12Kλminus12

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2

Many of these distributions are nested the skew t distribution for λ = 0 is the Studentrsquos tdistribution the generalized t distribution for τ = 2 is the Studentrsquos t distribution the skewed Studentrsquost distribution for α = 12 is the Studentrsquos t distribution the asymmetric Studentrsquos t distribution forν1 = ν2 is the skewed Studentrsquos t distribution the generalized hyperbolic distribution for λ = minus12 isthe normal inverse Gaussian distribution and so on

All but one of the distributions (the Laplace distribution) are heavy tailed Heavy tails are commonin financial data The Studentrsquos t distribution is perhaps the simplest of the heavy tailed distributionsIt does not allow for asymmetry The skew t distribution due to Azzalini and Capitanio (2003) isan asymmetric generalization of the Studentrsquos t distribution The generalized t distribution due toMcDonald and Newey (1988) has two parameters controlling its heavy tails adding more flexibility

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 4 of 23

which distinguish it from Bitcoin its use of scrypt as a proof-of-work algorithm and a significantlyfaster confirmation time for transactions The former enables standard computational hardware toverify transactions and reduces the incentive to use specially designed hardware while the latterreduces transaction confirmation times to minutes rather than hours and is particularly attractive intime-critical situations (LiteCoin Project 2017)

MaidSafeCoin is a digital currency which powers the peer-to-peer Secure Access For Everyone(SAFE) network which combines the computing power of all its users and can be thought of asa ldquocrowd-sourced internetrdquo (MaidSafe 2017a) Each MadeSafe coin has a unique identity and thereexists a hard upper limit of 43 billion coins as opposed to Bitcoinrsquos 21 million As the currency isused to pay for services on the SAFE network the currency will be recycled meaning that in theorythe amount of MaidSafe coins will never be exhausted The process of generating new currencyis similar to other cryptocurrencies and in the case of the SAFE network it is known as ldquofarmingrdquo(MaidSafe 2017b) Users contribute their computing power and storage space to the network and arerewarded with coins when the network accesses data from their store (MaidSafe 2017b)

Monero (XMR) is a ldquosecure private untraceable currencyrdquo (Monero 2017) centred arounddecentralization and scalability that was launched in April 2014 The currency itself is completelydonation-based community driven and based entirely on proof-of-work Whilst transactions in thenetwork are private by default users can set their level of privacy allowing as much or as littleaccess to their transactions as they wish Although it employs a proof-of-work algorithm Monerois more similar to LiteCoin in that mining of the currency can be done by any modern computerand is not restricted to specially designed hardware It arguably holds some advantages over otherBitcoin-based cryptocurrencies such as having a dynamic block size (overcoming the problem ofscalability) and being a disinflationary currency meaning that there will always exist an incentive toproduce the Monero currency (Monero 2017)

Dogecoin (Dogecoin 2017) originated from a popular internet meme in December 2013 Createdby an Australian brand and marketing specialist and a programmer in Portland Oregon it initiallystarted off as a joke currency but quickly gained traction It is a variation on Litecoin running on thecryptographic scrypt enabling similar advantages over Bitcoin such as faster transaction processingtimes Part of the attraction of Dogecoin is its light-hearted culture and lower barriers to entry toinvesting in or acquiring cryptocurrencies One of the most popular uses for Dogecoin is the tipping ofothers on the internet who create or share interesting content and can be thought of as the next levelup from a ldquolikerdquo on social media or an ldquoupvoterdquo on internet forums This in part has arisen from thefact that it has now become too expensive to tip using Bitcoin

Ripple was originally developed in 2012 and is the first global real-time gross settlement network(RTGS) which ldquoenables banks to send real-time international payments across networksrdquo (Ripple 2017)The Ripple network is a blockchain network which incorporates a payment system and a currencysystem known as XRP which is not based on proof-of-work like Monero and Dash A unique propertyof Ripple is that XRP is not compulsory for transactions on the network although it is encouragedas a bridge currency for more competitive cross border payments (Ripple 2017) The Ripple protocolis currently used by companies such as UBS Santander and Standard Chartered and increasinglybeing used by the financial services industry as technology in settlements Compared with Bitcoinit has advantages such as greater control over the system as it is not subject to the price volatility of theunderlying currencies and it has a more secure distributed authentication process

Summary statistics of the exchange rates and log returns of the exchange rates of the sevencryptocurrencies are given in Tables 1 and 2 respectively In Table 1 the summary statistics for theraw exchange rates of the cryptocurrencies we see a simple reflection of the ldquoworthrdquo or value of eachcurrency It can clearly be seen that the exchange rate of Dogecoin is the least significant and at thetime of writing (February 2017) the exchange rate is approximately $00002 USD to one DogecoinThis supports the evidence that Dogecoin is primarily used as a currency for online tipping ratherthan as a currency for standard payments It has the lowest minimum first quartile median mean

J Risk Financial Manag 2017 10 12 5 of 23

third quartile and maximum values In contrast being the most popular cryptocurrency Bitcoinhas the largest minimum first quartile median mean third quartile and maximum values whichshow its greater significance and higher ldquovaluerdquo to those with a vested interest in cryptocurrenciesThe exchange rates of all seven currencies are positively skewed with Litecoin Monero and Ripplebeing the most skewed In terms of kurtosis MaidSafeCoin shows less peakedness than that of thenormal distribution Bitcoin Dash and Dogecoin show levels similar to the normal distributionLitecoin Monero and Ripple have significantly greater peakedness than the normal distributionThe exchange rates of Dogecoin MaidSafeCoin and Ripple have the smallest variances and standarddeviations indicating that their low volatility can perhaps be explained by the low values of theexchange rates coupled with the fact that their range and interquartile ranges are very limitedOn the other hand Bitcoin Dash and Litecoinrsquos exchange rates show the greatest variance andstandard deviation

Table 1 Summary statistics of daily exchange rates of Bitcoin Dash Dogecoin Litecoin MaidSafeCoinMonero Ripple and Euro versus the US Dollar from 23 June 2014 until 28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum 192700 1178 0000 1269 0012 0235 0003 0626Q1 273600 2577 0000 3091 0020 0491 0006 0736

Median 415200 3623 0000 3662 0029 0811 0007 0779Mean 447400 5385 0000 3659 0046 2355 0008 0830

Q3 593000 7921 0000 4021 0074 1970 0008 0856Maximum 1140000 17560 0000 9793 0152 17590 0028 1207Skewness 0841 1006 0417 1363 0849 2108 2543 1127Kurtosis 3096 2992 3175 6621 2503 6526 10693 3067

SD 193241 3583 0000 1433 0032 3397 0004 0142Variance 37342159 12838 0000 2053 0001 11543 0000 0020

CV 0432 0665 0294 0392 0695 1443 0471 0171Range 946938 16385 0000 8524 0140 17358 0025 0581IQR 319400 5344 0000 0930 0054 1479 0002 0119

Table 2 Summary statistics of daily log returns of the exchange rates of Bitcoin Dash DogecoinLitecoin MaidSafeCoin Monero Ripple and the Euro versus the US Dollar from 23 June 2014 until28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum minus0159 minus0580 minus0385 minus0278 minus0404 minus0560 minus0299 minus0046Q1 minus0011 minus0019 minus0009 minus0010 minus0026 minus0026 minus0014 minus0004

Median minus0001 0003 0002 0000 minus0001 0002 0002 0Mean minus0001 minus0001 0000 0001 minus0002 minus0001 minus0000 minus000004

Q3 0008 0020 0015 0009 0023 0028 0017 0003Maximum 0205 0411 0188 0433 0241 0277 0288 0038Skewness 0758 minus1487 minus2506 0756 minus0478 minus1414 minus0401 minus0145Kurtosis 11568 26805 24434 22385 8520 13954 13818 2662

SD 0028 0051 0042 0042 0054 0062 0046 0006Variance 0001 0003 0002 0002 0003 0004 0002 000004

CV minus47976 minus84519 89782 45619 minus21499 minus54548 minus96585 minus143498Range 0364 0991 0573 0711 0645 0837 0587 0085IQR 0019 0039 0023 0019 0049 0054 0030 0007

Table 2 gives the summary statistics for the log returns of the exchange rates of the sevencryptocurrencies Here the log returns show some slightly different results Dash MaidSafeCoinand Monero have the lowest minimum values while Dash and Litecoin have the largest maximumsThe means and medians of the log returns of all seven currencies are similar and almost equal to zeroOnly the log returns of Bitcoin and Litecoin are positively skewed all others are negatively skewedwith Dogecoin being the most significant Log returns of all seven currencies have a peakednesssignificantly greater than that of the normal distribution with the most peaked being those of Dash

J Risk Financial Manag 2017 10 12 6 of 23

Dogecoin and Litecoin Noteworthy as much has been discussed about the volatility of Bitcoin returnsthe log returns of Bitcoin have the lowest variance and standard deviation of the seven cryptocurrenciesThose with the highest variation are Dash and perhaps unexpectedly MaidSafeCoin and Monero

Also shown in Tables 1 and 2 are the summary statistics of the exchange rates of the Euro and thesummary statistics of its log returns For the exchange rates the summary statistics for the Euro appearmuch smaller than those of Bitcoin but comparable to those of other cryptocurrencies For the logreturns of the exchange rates the summary statistics for the Euro generally appear smaller comparedto all the cryptocurrencies An exception is the coefficient of variation The magnitude of this statisticfor the Euro appears largest compared to all the cryptocurrencies

Fitting of a statistical distribution usually assumes that the data are independent andidentically distributed (ie randomness) have no serial correlation and have no heteroskedasticityWe tested for randomness using the difference sign and rank tests We tested for no serial correlationusing Durbin and Watson (1950 1951 1971)rsquos method We tested for no heteroskedasticity usingBreusch and Pagan (1979)rsquos test These tests showed that the log returns of the exchange rates of theseven cryptocurrencies can be assumed to be approximately independent and identically distributedhave no serial correlation and have no heteroskedasticity

3 Distributions Fitted

Having briefly examined the summary statistics for both the exchange rates and the log returns ofexchange rates of the seven cryptocurrencies we provide a visual representation of the distribution ofthe log returns Figure 1 shows the histograms of the daily log returns of the exchange rate (versus theUS Dollar) for all seven cryptocurrencies From the plots we find that the log returns in the cases ofall seven cryptocurrencies show significant deviation from the normal distribution Next we proceedto fit the parametric distributions to the data

Bitcoin

Log returns

Fre

quency

minus02 minus01 00 01 02

0300

Dash

Log returns

Fre

quency

minus06 minus02 00 02 04

0300

Dogecoin

Log returns

Fre

quency

minus04 minus02 00 01 02

0300

Litecoin

Log returns

Fre

quency

minus02 00 02 04

0300

MaidSafecoin

Log returns

Fre

quency

minus04 minus02 00 02

0300

Monero

Log returns

Fre

quency

minus06 minus04 minus02 00 02

0300

Ripples

Log returns

Fre

quency

minus03 minus01 01 02 03

0300

Figure 1 Histograms of daily log returns of the exchange rates of the seven cryptocurrencies versusthe US Dollar from 23 June 2014 until 28 February 2017

J Risk Financial Manag 2017 10 12 7 of 23

Let X denote a continuous random variable representing the log returns of the exchange rate ofthe cryptocurrency of interest Let f (x) denote the probability density function (pdf) of X Let F(x)denote the cumulative distribution function (cdf) of X We suppose X follows one of eight possibledistributions the most popular parametric distributions used in finance They are specified as follows

bull the Studentrsquos t distribution (Gosset 1908) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 and ν gt 0 where K(ν) =radic

νB (ν2 12) and B(middot middot)denotes the beta function defined by

B(a b) =int 1

0taminus1(1minus t)bminus1dt

bull the Laplace distribution (Laplace 1774) with

f (x) =1

2σexp

(minus| xminus micro |

σ

)for minusinfin lt x lt infin minusinfin lt micro lt infin and σ gt 0

bull the skew t distribution (Azzalini and Capitanio 2003) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

+2K2(ν)λ(xminus micro)

σ2 2F1

(12

1 + ν

2

32

minusλ2(xminus micro)2

σ2ν

)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin σ gt 0 and ν gt 0 where 2F1(a b c x) denotes theGauss hypergeometric function defined by

2F1 (a b c x) =infin

sumk=0

(a)k (b)k(c)k

xk

k

where (e)k = e(e + 1) middot middot middot (e + kminus 1) denotes the ascending factorialbull the generalized t distribution (McDonald and Newey 1988) with

f (x) =τ

2σν1νB (ν 1τ)

[1 +

1ν

∣∣∣∣ xminus micro

σ

∣∣∣∣τ]minus(ν+1τ)

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 ν gt 0 and τ gt 0bull the skewed Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =K (ν)

σ

1 +

1ν

[xminus micro

2σα

]2minus ν+1

2

if x le micro

1 +

1ν

[xminus micro

2σ (1minus α)

]2minus ν+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 and ν gt 0

J Risk Financial Manag 2017 10 12 8 of 23

bull the asymmetric Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =1σ

α

αlowastK (ν1)

1 +

1ν1

[xminus micro

2σαlowast

]2minus ν1+1

2

if x le micro

1minus α

1minus αlowastK (ν2)

1 +

1ν2

[xminus micro

2σ (1minus αlowast)

]2minus ν2+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 ν1 gt 0 and ν2 gt 0 where

αlowast =αK (ν1)

αK (ν1) + (1minus α)K (ν2)

bull the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ αradic

2πKminus12 (δγ)exp [β(xminus micro)]

[δ2 + (xminus micro)2

]minus1Kminus1

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2 and Kν(middot) denotesthe modified Bessel function of the second kind of order ν defined by

Kν(x) =

πcsc(πν)

2[Iminusν(x)minus Iν(x)] if ν 6isin Z

limmicrorarrν

Kmicro(x) if ν isin Z

where Iν(middot) denotes the modified Bessel function of the first kind of order ν defined by

Iν(x) =infin

sumk=0

1Γ(k + ν + 1)k

( x2

)2k+ν

where Γ(middot) denotes the gamma function defined by

Γ(a) =int infin

0taminus1 exp(minust)dt

bull the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ α12minusλ

radic2πKλ (δγ)

exp [β(xminus micro)][δ2 + (xminus micro)2

]λminus12Kλminus12

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2

Many of these distributions are nested the skew t distribution for λ = 0 is the Studentrsquos tdistribution the generalized t distribution for τ = 2 is the Studentrsquos t distribution the skewed Studentrsquost distribution for α = 12 is the Studentrsquos t distribution the asymmetric Studentrsquos t distribution forν1 = ν2 is the skewed Studentrsquos t distribution the generalized hyperbolic distribution for λ = minus12 isthe normal inverse Gaussian distribution and so on

All but one of the distributions (the Laplace distribution) are heavy tailed Heavy tails are commonin financial data The Studentrsquos t distribution is perhaps the simplest of the heavy tailed distributionsIt does not allow for asymmetry The skew t distribution due to Azzalini and Capitanio (2003) isan asymmetric generalization of the Studentrsquos t distribution The generalized t distribution due toMcDonald and Newey (1988) has two parameters controlling its heavy tails adding more flexibility

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 5 of 23

third quartile and maximum values In contrast being the most popular cryptocurrency Bitcoinhas the largest minimum first quartile median mean third quartile and maximum values whichshow its greater significance and higher ldquovaluerdquo to those with a vested interest in cryptocurrenciesThe exchange rates of all seven currencies are positively skewed with Litecoin Monero and Ripplebeing the most skewed In terms of kurtosis MaidSafeCoin shows less peakedness than that of thenormal distribution Bitcoin Dash and Dogecoin show levels similar to the normal distributionLitecoin Monero and Ripple have significantly greater peakedness than the normal distributionThe exchange rates of Dogecoin MaidSafeCoin and Ripple have the smallest variances and standarddeviations indicating that their low volatility can perhaps be explained by the low values of theexchange rates coupled with the fact that their range and interquartile ranges are very limitedOn the other hand Bitcoin Dash and Litecoinrsquos exchange rates show the greatest variance andstandard deviation

Table 1 Summary statistics of daily exchange rates of Bitcoin Dash Dogecoin Litecoin MaidSafeCoinMonero Ripple and Euro versus the US Dollar from 23 June 2014 until 28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum 192700 1178 0000 1269 0012 0235 0003 0626Q1 273600 2577 0000 3091 0020 0491 0006 0736

Median 415200 3623 0000 3662 0029 0811 0007 0779Mean 447400 5385 0000 3659 0046 2355 0008 0830

Q3 593000 7921 0000 4021 0074 1970 0008 0856Maximum 1140000 17560 0000 9793 0152 17590 0028 1207Skewness 0841 1006 0417 1363 0849 2108 2543 1127Kurtosis 3096 2992 3175 6621 2503 6526 10693 3067

SD 193241 3583 0000 1433 0032 3397 0004 0142Variance 37342159 12838 0000 2053 0001 11543 0000 0020

CV 0432 0665 0294 0392 0695 1443 0471 0171Range 946938 16385 0000 8524 0140 17358 0025 0581IQR 319400 5344 0000 0930 0054 1479 0002 0119

Table 2 Summary statistics of daily log returns of the exchange rates of Bitcoin Dash DogecoinLitecoin MaidSafeCoin Monero Ripple and the Euro versus the US Dollar from 23 June 2014 until28 February 2017

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum minus0159 minus0580 minus0385 minus0278 minus0404 minus0560 minus0299 minus0046Q1 minus0011 minus0019 minus0009 minus0010 minus0026 minus0026 minus0014 minus0004

Median minus0001 0003 0002 0000 minus0001 0002 0002 0Mean minus0001 minus0001 0000 0001 minus0002 minus0001 minus0000 minus000004

Q3 0008 0020 0015 0009 0023 0028 0017 0003Maximum 0205 0411 0188 0433 0241 0277 0288 0038Skewness 0758 minus1487 minus2506 0756 minus0478 minus1414 minus0401 minus0145Kurtosis 11568 26805 24434 22385 8520 13954 13818 2662

SD 0028 0051 0042 0042 0054 0062 0046 0006Variance 0001 0003 0002 0002 0003 0004 0002 000004

CV minus47976 minus84519 89782 45619 minus21499 minus54548 minus96585 minus143498Range 0364 0991 0573 0711 0645 0837 0587 0085IQR 0019 0039 0023 0019 0049 0054 0030 0007

Table 2 gives the summary statistics for the log returns of the exchange rates of the sevencryptocurrencies Here the log returns show some slightly different results Dash MaidSafeCoinand Monero have the lowest minimum values while Dash and Litecoin have the largest maximumsThe means and medians of the log returns of all seven currencies are similar and almost equal to zeroOnly the log returns of Bitcoin and Litecoin are positively skewed all others are negatively skewedwith Dogecoin being the most significant Log returns of all seven currencies have a peakednesssignificantly greater than that of the normal distribution with the most peaked being those of Dash

J Risk Financial Manag 2017 10 12 6 of 23

Dogecoin and Litecoin Noteworthy as much has been discussed about the volatility of Bitcoin returnsthe log returns of Bitcoin have the lowest variance and standard deviation of the seven cryptocurrenciesThose with the highest variation are Dash and perhaps unexpectedly MaidSafeCoin and Monero

Also shown in Tables 1 and 2 are the summary statistics of the exchange rates of the Euro and thesummary statistics of its log returns For the exchange rates the summary statistics for the Euro appearmuch smaller than those of Bitcoin but comparable to those of other cryptocurrencies For the logreturns of the exchange rates the summary statistics for the Euro generally appear smaller comparedto all the cryptocurrencies An exception is the coefficient of variation The magnitude of this statisticfor the Euro appears largest compared to all the cryptocurrencies

Fitting of a statistical distribution usually assumes that the data are independent andidentically distributed (ie randomness) have no serial correlation and have no heteroskedasticityWe tested for randomness using the difference sign and rank tests We tested for no serial correlationusing Durbin and Watson (1950 1951 1971)rsquos method We tested for no heteroskedasticity usingBreusch and Pagan (1979)rsquos test These tests showed that the log returns of the exchange rates of theseven cryptocurrencies can be assumed to be approximately independent and identically distributedhave no serial correlation and have no heteroskedasticity

3 Distributions Fitted

Having briefly examined the summary statistics for both the exchange rates and the log returns ofexchange rates of the seven cryptocurrencies we provide a visual representation of the distribution ofthe log returns Figure 1 shows the histograms of the daily log returns of the exchange rate (versus theUS Dollar) for all seven cryptocurrencies From the plots we find that the log returns in the cases ofall seven cryptocurrencies show significant deviation from the normal distribution Next we proceedto fit the parametric distributions to the data

Bitcoin

Log returns

Fre

quency

minus02 minus01 00 01 02

0300

Dash

Log returns

Fre

quency

minus06 minus02 00 02 04

0300

Dogecoin

Log returns

Fre

quency

minus04 minus02 00 01 02

0300

Litecoin

Log returns

Fre

quency

minus02 00 02 04

0300

MaidSafecoin

Log returns

Fre

quency

minus04 minus02 00 02

0300

Monero

Log returns

Fre

quency

minus06 minus04 minus02 00 02

0300

Ripples

Log returns

Fre

quency

minus03 minus01 01 02 03

0300

Figure 1 Histograms of daily log returns of the exchange rates of the seven cryptocurrencies versusthe US Dollar from 23 June 2014 until 28 February 2017

J Risk Financial Manag 2017 10 12 7 of 23

Let X denote a continuous random variable representing the log returns of the exchange rate ofthe cryptocurrency of interest Let f (x) denote the probability density function (pdf) of X Let F(x)denote the cumulative distribution function (cdf) of X We suppose X follows one of eight possibledistributions the most popular parametric distributions used in finance They are specified as follows

bull the Studentrsquos t distribution (Gosset 1908) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 and ν gt 0 where K(ν) =radic

νB (ν2 12) and B(middot middot)denotes the beta function defined by

B(a b) =int 1

0taminus1(1minus t)bminus1dt

bull the Laplace distribution (Laplace 1774) with

f (x) =1

2σexp

(minus| xminus micro |

σ

)for minusinfin lt x lt infin minusinfin lt micro lt infin and σ gt 0

bull the skew t distribution (Azzalini and Capitanio 2003) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

+2K2(ν)λ(xminus micro)

σ2 2F1

(12

1 + ν

2

32

minusλ2(xminus micro)2

σ2ν

)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin σ gt 0 and ν gt 0 where 2F1(a b c x) denotes theGauss hypergeometric function defined by

2F1 (a b c x) =infin

sumk=0

(a)k (b)k(c)k

xk

k

where (e)k = e(e + 1) middot middot middot (e + kminus 1) denotes the ascending factorialbull the generalized t distribution (McDonald and Newey 1988) with

f (x) =τ

2σν1νB (ν 1τ)

[1 +

1ν

∣∣∣∣ xminus micro

σ

∣∣∣∣τ]minus(ν+1τ)

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 ν gt 0 and τ gt 0bull the skewed Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =K (ν)

σ

1 +

1ν

[xminus micro

2σα

]2minus ν+1

2

if x le micro

1 +

1ν

[xminus micro

2σ (1minus α)

]2minus ν+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 and ν gt 0

J Risk Financial Manag 2017 10 12 8 of 23

bull the asymmetric Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =1σ

α

αlowastK (ν1)

1 +

1ν1

[xminus micro

2σαlowast

]2minus ν1+1

2

if x le micro

1minus α

1minus αlowastK (ν2)

1 +

1ν2

[xminus micro

2σ (1minus αlowast)

]2minus ν2+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 ν1 gt 0 and ν2 gt 0 where

αlowast =αK (ν1)

αK (ν1) + (1minus α)K (ν2)

bull the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ αradic

2πKminus12 (δγ)exp [β(xminus micro)]

[δ2 + (xminus micro)2

]minus1Kminus1

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2 and Kν(middot) denotesthe modified Bessel function of the second kind of order ν defined by

Kν(x) =

πcsc(πν)

2[Iminusν(x)minus Iν(x)] if ν 6isin Z

limmicrorarrν

Kmicro(x) if ν isin Z

where Iν(middot) denotes the modified Bessel function of the first kind of order ν defined by

Iν(x) =infin

sumk=0

1Γ(k + ν + 1)k

( x2

)2k+ν

where Γ(middot) denotes the gamma function defined by

Γ(a) =int infin

0taminus1 exp(minust)dt

bull the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ α12minusλ

radic2πKλ (δγ)

exp [β(xminus micro)][δ2 + (xminus micro)2

]λminus12Kλminus12

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2

Many of these distributions are nested the skew t distribution for λ = 0 is the Studentrsquos tdistribution the generalized t distribution for τ = 2 is the Studentrsquos t distribution the skewed Studentrsquost distribution for α = 12 is the Studentrsquos t distribution the asymmetric Studentrsquos t distribution forν1 = ν2 is the skewed Studentrsquos t distribution the generalized hyperbolic distribution for λ = minus12 isthe normal inverse Gaussian distribution and so on

All but one of the distributions (the Laplace distribution) are heavy tailed Heavy tails are commonin financial data The Studentrsquos t distribution is perhaps the simplest of the heavy tailed distributionsIt does not allow for asymmetry The skew t distribution due to Azzalini and Capitanio (2003) isan asymmetric generalization of the Studentrsquos t distribution The generalized t distribution due toMcDonald and Newey (1988) has two parameters controlling its heavy tails adding more flexibility

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 6 of 23

Dogecoin and Litecoin Noteworthy as much has been discussed about the volatility of Bitcoin returnsthe log returns of Bitcoin have the lowest variance and standard deviation of the seven cryptocurrenciesThose with the highest variation are Dash and perhaps unexpectedly MaidSafeCoin and Monero

Also shown in Tables 1 and 2 are the summary statistics of the exchange rates of the Euro and thesummary statistics of its log returns For the exchange rates the summary statistics for the Euro appearmuch smaller than those of Bitcoin but comparable to those of other cryptocurrencies For the logreturns of the exchange rates the summary statistics for the Euro generally appear smaller comparedto all the cryptocurrencies An exception is the coefficient of variation The magnitude of this statisticfor the Euro appears largest compared to all the cryptocurrencies

Fitting of a statistical distribution usually assumes that the data are independent andidentically distributed (ie randomness) have no serial correlation and have no heteroskedasticityWe tested for randomness using the difference sign and rank tests We tested for no serial correlationusing Durbin and Watson (1950 1951 1971)rsquos method We tested for no heteroskedasticity usingBreusch and Pagan (1979)rsquos test These tests showed that the log returns of the exchange rates of theseven cryptocurrencies can be assumed to be approximately independent and identically distributedhave no serial correlation and have no heteroskedasticity

3 Distributions Fitted

Having briefly examined the summary statistics for both the exchange rates and the log returns ofexchange rates of the seven cryptocurrencies we provide a visual representation of the distribution ofthe log returns Figure 1 shows the histograms of the daily log returns of the exchange rate (versus theUS Dollar) for all seven cryptocurrencies From the plots we find that the log returns in the cases ofall seven cryptocurrencies show significant deviation from the normal distribution Next we proceedto fit the parametric distributions to the data

Bitcoin

Log returns

Fre

quency

minus02 minus01 00 01 02

0300

Dash

Log returns

Fre

quency

minus06 minus02 00 02 04

0300

Dogecoin

Log returns

Fre

quency

minus04 minus02 00 01 02

0300

Litecoin

Log returns

Fre

quency

minus02 00 02 04

0300

MaidSafecoin

Log returns

Fre

quency

minus04 minus02 00 02

0300

Monero

Log returns

Fre

quency

minus06 minus04 minus02 00 02

0300

Ripples

Log returns

Fre

quency

minus03 minus01 01 02 03

0300

Figure 1 Histograms of daily log returns of the exchange rates of the seven cryptocurrencies versusthe US Dollar from 23 June 2014 until 28 February 2017

J Risk Financial Manag 2017 10 12 7 of 23

Let X denote a continuous random variable representing the log returns of the exchange rate ofthe cryptocurrency of interest Let f (x) denote the probability density function (pdf) of X Let F(x)denote the cumulative distribution function (cdf) of X We suppose X follows one of eight possibledistributions the most popular parametric distributions used in finance They are specified as follows

bull the Studentrsquos t distribution (Gosset 1908) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 and ν gt 0 where K(ν) =radic

νB (ν2 12) and B(middot middot)denotes the beta function defined by

B(a b) =int 1

0taminus1(1minus t)bminus1dt

bull the Laplace distribution (Laplace 1774) with

f (x) =1

2σexp

(minus| xminus micro |

σ

)for minusinfin lt x lt infin minusinfin lt micro lt infin and σ gt 0

bull the skew t distribution (Azzalini and Capitanio 2003) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

+2K2(ν)λ(xminus micro)

σ2 2F1

(12

1 + ν

2

32

minusλ2(xminus micro)2

σ2ν

)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin σ gt 0 and ν gt 0 where 2F1(a b c x) denotes theGauss hypergeometric function defined by

2F1 (a b c x) =infin

sumk=0

(a)k (b)k(c)k

xk

k

where (e)k = e(e + 1) middot middot middot (e + kminus 1) denotes the ascending factorialbull the generalized t distribution (McDonald and Newey 1988) with

f (x) =τ

2σν1νB (ν 1τ)

[1 +

1ν

∣∣∣∣ xminus micro

σ

∣∣∣∣τ]minus(ν+1τ)

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 ν gt 0 and τ gt 0bull the skewed Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =K (ν)

σ

1 +

1ν

[xminus micro

2σα

]2minus ν+1

2

if x le micro

1 +

1ν

[xminus micro

2σ (1minus α)

]2minus ν+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 and ν gt 0

J Risk Financial Manag 2017 10 12 8 of 23

bull the asymmetric Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =1σ

α

αlowastK (ν1)

1 +

1ν1

[xminus micro

2σαlowast

]2minus ν1+1

2

if x le micro

1minus α

1minus αlowastK (ν2)

1 +

1ν2

[xminus micro

2σ (1minus αlowast)

]2minus ν2+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 ν1 gt 0 and ν2 gt 0 where

αlowast =αK (ν1)

αK (ν1) + (1minus α)K (ν2)

bull the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ αradic

2πKminus12 (δγ)exp [β(xminus micro)]

[δ2 + (xminus micro)2

]minus1Kminus1

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2 and Kν(middot) denotesthe modified Bessel function of the second kind of order ν defined by

Kν(x) =

πcsc(πν)

2[Iminusν(x)minus Iν(x)] if ν 6isin Z

limmicrorarrν

Kmicro(x) if ν isin Z

where Iν(middot) denotes the modified Bessel function of the first kind of order ν defined by

Iν(x) =infin

sumk=0

1Γ(k + ν + 1)k

( x2

)2k+ν

where Γ(middot) denotes the gamma function defined by

Γ(a) =int infin

0taminus1 exp(minust)dt

bull the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ α12minusλ

radic2πKλ (δγ)

exp [β(xminus micro)][δ2 + (xminus micro)2

]λminus12Kλminus12

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2

Many of these distributions are nested the skew t distribution for λ = 0 is the Studentrsquos tdistribution the generalized t distribution for τ = 2 is the Studentrsquos t distribution the skewed Studentrsquost distribution for α = 12 is the Studentrsquos t distribution the asymmetric Studentrsquos t distribution forν1 = ν2 is the skewed Studentrsquos t distribution the generalized hyperbolic distribution for λ = minus12 isthe normal inverse Gaussian distribution and so on

All but one of the distributions (the Laplace distribution) are heavy tailed Heavy tails are commonin financial data The Studentrsquos t distribution is perhaps the simplest of the heavy tailed distributionsIt does not allow for asymmetry The skew t distribution due to Azzalini and Capitanio (2003) isan asymmetric generalization of the Studentrsquos t distribution The generalized t distribution due toMcDonald and Newey (1988) has two parameters controlling its heavy tails adding more flexibility

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 7 of 23

Let X denote a continuous random variable representing the log returns of the exchange rate ofthe cryptocurrency of interest Let f (x) denote the probability density function (pdf) of X Let F(x)denote the cumulative distribution function (cdf) of X We suppose X follows one of eight possibledistributions the most popular parametric distributions used in finance They are specified as follows

bull the Studentrsquos t distribution (Gosset 1908) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 and ν gt 0 where K(ν) =radic

νB (ν2 12) and B(middot middot)denotes the beta function defined by

B(a b) =int 1

0taminus1(1minus t)bminus1dt

bull the Laplace distribution (Laplace 1774) with

f (x) =1

2σexp

(minus| xminus micro |

σ

)for minusinfin lt x lt infin minusinfin lt micro lt infin and σ gt 0

bull the skew t distribution (Azzalini and Capitanio 2003) with

f (x) =K(ν)

σ

[1 +

(xminus micro)2

σ2ν

]minus(1+ν)2

+2K2(ν)λ(xminus micro)

σ2 2F1

(12

1 + ν

2

32

minusλ2(xminus micro)2

σ2ν

)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin σ gt 0 and ν gt 0 where 2F1(a b c x) denotes theGauss hypergeometric function defined by

2F1 (a b c x) =infin

sumk=0

(a)k (b)k(c)k

xk

k

where (e)k = e(e + 1) middot middot middot (e + kminus 1) denotes the ascending factorialbull the generalized t distribution (McDonald and Newey 1988) with

f (x) =τ

2σν1νB (ν 1τ)

[1 +

1ν

∣∣∣∣ xminus micro

σ

∣∣∣∣τ]minus(ν+1τ)

for minusinfin lt x lt infin minusinfin lt micro lt infin σ gt 0 ν gt 0 and τ gt 0bull the skewed Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =K (ν)

σ

1 +

1ν

[xminus micro

2σα

]2minus ν+1

2

if x le micro

1 +

1ν

[xminus micro

2σ (1minus α)

]2minus ν+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 and ν gt 0

J Risk Financial Manag 2017 10 12 8 of 23

bull the asymmetric Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =1σ

α

αlowastK (ν1)

1 +

1ν1

[xminus micro

2σαlowast

]2minus ν1+1

2

if x le micro

1minus α

1minus αlowastK (ν2)

1 +

1ν2

[xminus micro

2σ (1minus αlowast)

]2minus ν2+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 ν1 gt 0 and ν2 gt 0 where

αlowast =αK (ν1)

αK (ν1) + (1minus α)K (ν2)

bull the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ αradic

2πKminus12 (δγ)exp [β(xminus micro)]

[δ2 + (xminus micro)2

]minus1Kminus1

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2 and Kν(middot) denotesthe modified Bessel function of the second kind of order ν defined by

Kν(x) =

πcsc(πν)

2[Iminusν(x)minus Iν(x)] if ν 6isin Z

limmicrorarrν

Kmicro(x) if ν isin Z

where Iν(middot) denotes the modified Bessel function of the first kind of order ν defined by

Iν(x) =infin

sumk=0

1Γ(k + ν + 1)k

( x2

)2k+ν

where Γ(middot) denotes the gamma function defined by

Γ(a) =int infin

0taminus1 exp(minust)dt

bull the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ α12minusλ

radic2πKλ (δγ)

exp [β(xminus micro)][δ2 + (xminus micro)2

]λminus12Kλminus12

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2

Many of these distributions are nested the skew t distribution for λ = 0 is the Studentrsquos tdistribution the generalized t distribution for τ = 2 is the Studentrsquos t distribution the skewed Studentrsquost distribution for α = 12 is the Studentrsquos t distribution the asymmetric Studentrsquos t distribution forν1 = ν2 is the skewed Studentrsquos t distribution the generalized hyperbolic distribution for λ = minus12 isthe normal inverse Gaussian distribution and so on

All but one of the distributions (the Laplace distribution) are heavy tailed Heavy tails are commonin financial data The Studentrsquos t distribution is perhaps the simplest of the heavy tailed distributionsIt does not allow for asymmetry The skew t distribution due to Azzalini and Capitanio (2003) isan asymmetric generalization of the Studentrsquos t distribution The generalized t distribution due toMcDonald and Newey (1988) has two parameters controlling its heavy tails adding more flexibility

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 8 of 23

bull the asymmetric Studentrsquos t distribution (Zhu and Galbraith 2010) with

f (x) =1σ

α

αlowastK (ν1)

1 +

1ν1

[xminus micro

2σαlowast

]2minus ν1+1

2

if x le micro

1minus α

1minus αlowastK (ν2)

1 +

1ν2

[xminus micro

2σ (1minus αlowast)

]2minus ν2+1

2

if x gt micro

for minusinfin lt x lt infin minusinfin lt micro lt infin 0 lt α lt 1 ν1 gt 0 and ν2 gt 0 where

αlowast =αK (ν1)

αK (ν1) + (1minus α)K (ν2)

bull the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ αradic

2πKminus12 (δγ)exp [β(xminus micro)]

[δ2 + (xminus micro)2

]minus1Kminus1

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2 and Kν(middot) denotesthe modified Bessel function of the second kind of order ν defined by

Kν(x) =

πcsc(πν)

2[Iminusν(x)minus Iν(x)] if ν 6isin Z

limmicrorarrν

Kmicro(x) if ν isin Z

where Iν(middot) denotes the modified Bessel function of the first kind of order ν defined by

Iν(x) =infin

sumk=0

1Γ(k + ν + 1)k

( x2

)2k+ν

where Γ(middot) denotes the gamma function defined by

Γ(a) =int infin

0taminus1 exp(minust)dt

bull the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f (x) =(γδ)λ α12minusλ

radic2πKλ (δγ)

exp [β(xminus micro)][δ2 + (xminus micro)2

]λminus12Kλminus12

(αradic

δ2 + (xminus micro)2)

for minusinfin lt x lt infin minusinfin lt micro lt infin minusinfin lt λ lt infin δ gt 0 α gt 0 and β gt 0 where γ =radic

α2 minus β2

Many of these distributions are nested the skew t distribution for λ = 0 is the Studentrsquos tdistribution the generalized t distribution for τ = 2 is the Studentrsquos t distribution the skewed Studentrsquost distribution for α = 12 is the Studentrsquos t distribution the asymmetric Studentrsquos t distribution forν1 = ν2 is the skewed Studentrsquos t distribution the generalized hyperbolic distribution for λ = minus12 isthe normal inverse Gaussian distribution and so on

All but one of the distributions (the Laplace distribution) are heavy tailed Heavy tails are commonin financial data The Studentrsquos t distribution is perhaps the simplest of the heavy tailed distributionsIt does not allow for asymmetry The skew t distribution due to Azzalini and Capitanio (2003) isan asymmetric generalization of the Studentrsquos t distribution The generalized t distribution due toMcDonald and Newey (1988) has two parameters controlling its heavy tails adding more flexibility

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 9 of 23

The skewed Studentrsquos t distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquost distribution with the scale allowed to be different on the two sides of micro This distribution is useful ifpositive log returns have a different scale compared to negative log returns The asymmetric Studentrsquost distribution due to Zhu and Galbraith (2010) is a generalization of the Studentrsquos t distribution withthe scale as well as the tail parameter allowed to be different on the two sides of micro This distribution isuseful if positive log returns also have a different heavy tail behavior compared to negative log returnsThe generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavytails It is popular in finance because it contains several heavy tailed distributions as particular cases

The maximum likelihood method was used to fit each distribution If x1 x2 xn is a randomsample of observed values on X and if Θ = (θ1 θ2 θk)

primeare parameters specifying the distribution

of X then the maximum likelihood estimates of Θ are those maximizing the likelihood

L (Θ) =n

prodi=1

f (xi Θ)

or the log likelihood

ln L (Θ) =n

sumi=1

ln f (xi Θ)

where f (middot) denotes the pdf of X We shall let Θ =(

θ1 θ2 θk

)primedenote the maximum likelihood

estimate of Θ The maximization was performed using the routine optim in the R software package(R Development Core Team 2017) The standard errors of Θ were computed by approximating thecovariance matrix of Θ by the inverse of observed information matrix ie

cov(

Θ)asymp

part2 ln Lpartθ2

1

part2 ln Lpartθ1partθ2

middot middot middot part2 ln Lpartθ1partθk

part2 ln Lpartθ2partθ1

part2 ln Lpartθ2

2middot middot middot part2 ln L

partθ2partθk

part2 ln Lpartθkpartθ1

part2 ln Lpartθkpartθ2

middot middot middot part2 ln Lpartθ2

k

minus1∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Θ=Θ

Many of the fitted distributions are not nested Discrimination among them was performed usingvarious criteria

bull the Akaike information criterion (Akaike 1974) defined by

AIC = 2kminus 2 ln L(

Θ)

bull the Bayesian information criterion (Schwarz 1978) defined by

BIC = k ln nminus 2 ln L(

Θ)

bull the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by

CAIC = minus2 ln L(

Θ)+ k (ln n + 1)

bull the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by

AICc = AIC +2k(k + 1)nminus kminus 1

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 10 of 23

bull the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by

HQC = minus2 ln L(

Θ)+ 2k ln ln n

The five discrimination criteria above used to discriminate between and determine the bestfitting distribution all utilise the maximum likelihood estimate In all cases the smaller the valuesof the criteria the better the fit The Akaike information criterion comprises two parts the bias(log likelihood) and variance (parameters) (Hu 2007) and the larger the log likelihood the better thegoodness of fit However the criterion includes a penalty term which is dependent on the numberof parameters in the model This penalty term increases with the number of estimated parametersand discourages overfitting The Bayesian information criterion is very similar to that of the Akaikeinformation criterion the only difference being that the penalty term is not twice the number ofestimated parameters but instead is the number of parameters multiplied by the natural logarithm ofthe number of observed data points The two criteria possess different properties and thus their use isalso dependent on different factors In addition the Bayesian information criterion is asymptoticallyefficient while the Akaike information criterion is not (Vrieze 2012) The consistent Akaike informationcriterion and the corrected Akaike information criterion are also very similar to both the Akaike andBayesian information criteria The former acts as a direct extension of the Akaike information criterionin that it is asymptotically efficient and still includes a penalty term which penalises overfitting morestrictly (Bozdogan 1987) On the other hand the latter includes a correction for small sample biasand includes an additional penalty term which is a function of the sample size (Anderson et al 2010)Finally an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn informationcriterion The expression for the Hannan-Quinn criterion is the same as that for the Akiake informationcriterion however the parameter term is multiplied by the double logarithm of the sample sizeIn general the Hannan-Quinn criterion penalises models with a greater number of parameters morecompared to both the Akaike and Bayesian information criteria However it tends to show signs ofoverfitting when the sample size is small (McNelis 2005) For a more detailed discussion on thesecriteria see Burnham and Anderson (2004) and Fang (2011)

Apart from the five criteria various other measures could be used to discriminate betweennon-nested models These could include

bull the Kolmogorov-Smirnov statistic (Kolmogorov 1933 Smirnov 1948) defined by

KS = supx

∣∣∣∣∣ 1n n

sumi=1

I xi le x minus F(x)

∣∣∣∣∣

where I middot denotes the indicator function and F(middot) the maximum likelihood estimate of F(x)bull the Anderson-Darling statistic (Anderson and Darling 1954) defined by

AD = minusnminusn

sumi=1

ln F

(x(i))+ ln

[1minus F

(x(n+1minusi)

)]

where x(1) le x(2) le middot middot middot le x(n) are the observed data arranged in increasing orderbull the Cramer-von Mises statistic (Cramer 1928 Von Mises 1928) defined by

CM =1

12n+

n

sumi=1

[2iminus 1

2nminus F

(x(i))]

Once again the smaller the values of these statistics the better the fit The use of these statistics inSection 4 instead of the five criteria led to the same conclusions

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 11 of 23

4 Results

In this section we provide our analysis in terms of the best fitting distributions and the results forthe log returns of the different cryptocurrencies The results provided are in terms of log likelihoodvalues information criteria goodness of fit tests probability plots quantile plots plots of twoimportant financial risk measures back-testing using Kupiecrsquos test and dynamic volatility

41 Fitted Distributions and Results

The eight distributions in Section 3 were fitted to the data described in Section 2 The methodof maximum likelihood was used The parameter estimates and their standard errors for the bestfitting distributions in each case are given in Table 3 The log likelihood values and the values ofAIC AICc BIC HQC and CAIC for the fitted distributions (for each of the seven cryptocurrencies)are shown in Tables 4ndash10 We find that there is no one best fitting distribution jointly for all sevencryptocurrencies However we find that for Bitcoin (the most popular cryptocurrency) and its nearestrival LiteCoin the generalized hyperbolic distribution gives the best fit For three out of the sevenDash Monero and Ripple the normal inverse Gaussian distribution gives the best fit For Dogecoinand MaidSafeCoin the best fitting distributions are the generalized t and Laplace distributionsrespectively The adequacy of the best fitting distributions is assessed in terms of Q-Q plots P-P plotsthe one-sample Kolmogorov-Smirnov test the one-sample Anderson-Darling test and the one-sampleCramer-von Mises test

Table 3 Best fitting distributions and parameter estimates with standard errors given in brackets

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic

micro = minus0001 (0000)δ = 0003 (0001)α = 29644 (3707)β = 0530 (1305)λ = 0220 (0010)

Dash Normal inverse Gaussian

micro = 0004δ = 0025α = 10714β = minus2100

Dogecoin Generalized t

micro = 0002 (0000)σ = 0014 (0002)p = 0893 (0094)ν = 3768 (1269)

Litecoin Generalized hyperbolic

micro = 0000 (0000)δ = 0006 (0001)α = 10517 (2021)β = 0412 (0801)λ = minus0186 (0078)

MaidSafeCoin Laplace micro = minus0001 (0001)σ = 00368 (0001)

Monero Normal inverse Gaussian

micro = 0005δ = 0040α = 11164β = minus1705

Ripple Normal inverse Gaussian

micro = 0003δ = 0018α = 8729β = minus1670

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

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minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

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Expe

cted

00 02 04 06 08 10

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02

04

06

08

10

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cted

00 02 04 06 08 10

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02

04

06

08

10

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cted

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02

04

06

08

10

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cted

00 02 04 06 08 10

00

02

04

06

08

10

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Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

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00

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01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

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10

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05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

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00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

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10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

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minus0

10

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05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

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minus0

10

minus0

05

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05

01

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q

Va

R (

q)

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minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

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04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

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05

00

0

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Euro

00 02 04 06 08 10

minus0

12

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06

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ES

(q

)

00 02 04 06 08 10

minus0

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06

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02

00

0

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Euro

00 02 04 06 08 10

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02

00

0

q

ES

(q

)

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minus0

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06

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02

00

0

Litecoin

Euro

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0

q

ES

(q

)

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minus0

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10

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05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

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0

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ES

(q

)

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minus0

20

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15

minus0

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05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

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10

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05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

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10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 12 of 23

Table 4 Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus23039 minus46018 minus46018 minus45873 minus45963 minus45843Laplace minus22893 minus45747 minus45747 minus45650 minus45710 minus45630Skew t minus23040 minus46000 minus45999 minus45805 minus45926 minus45765

GT minus23257 minus46433 minus46433 minus46238 minus46359 minus46198SST minus23040 minus46000 minus45999 minus45805 minus45926 minus45765AST minus23043 minus45986 minus45986 minus45743 minus45894 minus45693NIG minus23160 minus46239 minus46239 minus46045 minus46165 minus46005GH minus23257 minus46415 minus46414 minus46171 minus46322 minus46121

Table 5 Fitted distributions and results for daily log returns of the exchange rates of Dash from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus17048 minus34036 minus34036 minus33890 minus33981 minus33860Laplace minus16894 minus33748 minus33747 minus33650 minus33711 minus33630Skew t minus17074 minus34068 minus34068 minus33873 minus33994 minus33833

GT minus17089 minus34097 minus34097 minus33903 minus34023 minus33863SST minus17075 minus34070 minus34069 minus33875 minus33996 minus33835AST minus17075 minus34051 minus34050 minus33808 minus33958 minus33758NIG minus17104 minus34128 minus34128 minus33934 minus34054 minus33894GH minus17105 minus34109 minus34108 minus33866 minus34016 minus33816

Table 6 Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus20377 minus40694 minus40694 minus40548 minus40638 minus40518Laplace minus19859 minus39678 minus39678 minus39581 minus39641 minus39561Skew t minus20377 minus40675 minus40674 minus40480 minus40601 minus40440

GT minus20519 minus40958 minus40958 minus40764 minus40884 minus40724SST minus20377 minus40674 minus40673 minus40479 minus40600 minus40439AST minus20396 minus40691 minus40691 minus40448 minus40599 minus40398NIG minus20481 minus40881 minus40881 minus40687 minus40807 minus40647GH minus20522 minus40943 minus40942 minus40700 minus40850 minus40650

Table 7 Fitted distributions and results for daily log returns of the exchange rates of Litecoin from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus21133 minus42207 minus42207 minus42061 minus42151 minus42031Laplace minus20200 minus40360 minus40360 minus40263 minus40323 minus40243Skew t minus21136 minus42193 minus42193 minus41998 minus42119 minus41958

GT minus21257 minus42434 minus42434 minus42239 minus42360 minus42199SST minus21135 minus42191 minus42191 minus41996 minus42117 minus41956AST minus21136 minus42172 minus42171 minus41928 minus42079 minus41878NIG minus21239 minus42399 minus42399 minus42204 minus42325 minus42164GH minus21306 minus42512 minus42511 minus42269 minus42419 minus42219

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

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minus03

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00

01

02

03

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Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

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minus02

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01

02

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04

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Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

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00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 13 of 23

Table 8 Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoinfrom 23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus15332 minus30603 minus30603 minus30457 minus30548 minus30427Laplace minus15404 minus30768 minus30768 minus30671 minus30731 minus30651Skew t minus15333 minus30586 minus30585 minus30391 minus30512 minus30351

GT minus15419 minus30759 minus30758 minus30564 minus30685 minus30524SST minus15333 minus30585 minus30585 minus30391 minus30511 minus30351AST minus15334 minus30567 minus30567 minus30324 minus30475 minus30274NIG minus15390 minus30700 minus30700 minus30506 minus30626 minus30466GH minus15424 minus30749 minus30748 minus30506 minus30656 minus30456

Table 9 Fitted distributions and results for daily log returns of the exchange rates of Monero from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus14386 minus28712 minus28712 minus28566 minus28656 minus28536Laplace minus14322 minus28605 minus28605 minus28507 minus28568 minus28487Skew t minus14400 minus28719 minus28719 minus28525 minus28645 minus28485

GT minus14405 minus28729 minus28729 minus28535 minus28655 minus28495SST minus14397 minus28714 minus28714 minus28520 minus28640 minus28480AST minus14406 minus28712 minus28711 minus28469 minus28619 minus28419NIG minus14419 minus28758 minus28757 minus28563 minus28684 minus28523GH minus14420 minus28740 minus28740 minus28497 minus28648 minus28447

Table 10 Fitted distributions and results for daily log returns of the exchange rates of Ripple from23 June 2014 until 28 February 2017

Distribution minus ln L AIC AICC BIC HQC CAIC

Student t minus18676 minus37291 minus37291 minus37145 minus37236 minus37115Laplace minus18319 minus36599 minus36599 minus36502 minus36562 minus36482Skew t minus18695 minus37311 minus37310 minus37116 minus37237 minus37076

GT minus18706 minus37331 minus37331 minus37137 minus37257 minus37097SST minus18693 minus37305 minus37305 minus37111 minus37231 minus37071AST minus18697 minus37294 minus37294 minus37051 minus37202 minus37001NIG minus18756 minus37433 minus37432 minus37238 minus37358 minus37198GH minus18759 minus37418 minus37418 minus37175 minus37326 minus37125

The best fitting distributions show the following log returns of the exchange rates ofMaidSafeCoin have light tails log returns of the exchange rates of Dogecoin have heavy tails logreturns of the exchange rates of Bitcoin Dash Litecoin Monero and Ripple have semi heavy tailsAmong the last five the asymmetry of tails as measured by β is negative for Dash Monero and RippleOf these the largest asymmetry is for Dash followed by Monero and then Ripple The asymmetry oftails as measured by β is positive for Bitcoin and Litecoin Of these the larger asymmetry is for Bitcoin

It is surprising that a light tailed distribution (the Laplace distribution) gives the best fit forlog returns of the exchange rates of MaidSafeCoin However many authors have found that theLaplace distribution can provide adequate fits to financial data Linden (2001) models stock returnsby the Laplace distribution Linden (2005) models the realized stock return volatility by the Laplacedistribution Aquino (2006) establishes that the Laplace distribution characterizes the price movementsfor the Philippine stock market Podobnik et al (2009) show Laplace distribution fits for the stockgrowth rates for the Nasdaq Composite and the New York Stock Exchange Composite

All of the cryptocurrencies show substantially larger volatility than the exchange rate of theEuro Nevertheless we also observe substantial differences within the group of cryptocurrencies

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

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minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailedGiven that traditional financial instruments usually exhibit heavy tails this is a slightly surprisingbut also a very important result

42 Q-Q Plots

The Q-Q plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 2 For Bitcoin the best fitting distribution captures the middle and lower parts of the data wellbut not the upper tail For Dash the best fitting distribution captures the middle lower and upperparts of the data well For Dogecoin the best fitting distribution captures the middle part of the datawell but not the lower or upper tails For Litecoin the best fitting distribution captures the middlelower and upper parts of the data well For MaidSafeCoin the best fitting distribution captures themiddle and upper parts of the data well but not the lower tail For Monero the best fitting distributioncaptures the middle and upper parts of the data well but not the lower tail For Ripple the best fittingdistribution captures the middle part of the data well but not the lower or upper tails

minus01 00 01 02

minus01

00

01

02

Observed

Exp

ecte

d

minus06 minus04 minus02 00 02 04

minus06

minus04

minus02

00

02

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02 03

minus04

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

minus03 minus02 minus01 00 01 02 03 04

minus03

minus02

minus01

00

01

02

03

04

Observed

Exp

ecte

d

minus04 minus03 minus02 minus01 00 01 02

minus04

minus03

minus02

minus01

00

01

02

Observed

Exp

ecte

d

minus04 minus02 00 02

minus04

minus02

00

02

Observed

Exp

ecte

d

Figure 2 Cont

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 15 of 23

minus03 minus02 minus01 00 01 02 03

minus03

minus02

minus01

00

01

02

03

Observed

Exp

ecte

d

Figure 2 The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

43 P-P Plots

The P-P plots for the best fitting distribution for each of the seven cryptocurrencies are shown inFigure 3 For each cryptocurrency the best fitting distribution captures the middle lower and upperparts of the data well

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Expe

cted

Figure 3 Cont

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 16 of 23

00 02 04 06 08 10

00

02

04

06

08

10

Observed

Exp

ecte

d

Figure 3 The P-P plots of the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017

44 Goodness of Fit Tests

The p-values of the one-sample Kolmogorov-Smirnov test for the best fitting distributionslisted in Table 3 are 0095 0124 0164 0094 0119 0162 and 0056 The corresponding p-valuesof the one-sample Anderson-Darling test are 0103 0144 0154 0120 0051 0157 and 0176The corresponding p-values of the one-sample Cramer-von Mises test are 0082 0116 0196 00850168 0088 and 0122 Hence all of the best fitting distributions are adequate at the five percentsignificance level

45 VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular financial risk measures(Kinateder 2015 2016) If F(middot) denotes the cdf of the best fitting distribution then VaR and EScorresponding to probability q can be defined by

VaR(q) = Fminus1 (q)

and

ES(q) =1q

int q

0VaR(u)du

respectively for 0 lt q lt 1 Plots of VaR (q) and ES (q) for the best fitting distributions for the sevencryptocurrencies are shown in Figures 4 and 5 Also shown in the figures are estimates of VaR (q)and ES (q) for daily log returns of the Euro computed using the same best fitting distributions It isclear that each cryptocurrency is riskier than the Euro With respect to the upper tail of VaR LitecoinMainSafecoin and Monero appear to have the largest risks With respect to the lower tail of VaRMonero appears to have the largest risk Monero also has the largest risk with respect to the lowertail of ES

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 17 of 23

00 02 04 06 08 10

minus0

05

00

00

05

q

Va

R (

q)

00 02 04 06 08 10

minus0

05

00

00

05

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Dash

Euro

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

10

minus0

05

00

00

05

01

0

Dogecoin

Euro

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

0minus

00

500

000

501

001

5

Litecoin

Euro

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

q

VaR

(q)

00 02 04 06 08 10

minus01

5minus

01

0minus

00

500

000

501

001

5

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

q

Va

R (

q)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

00

05

01

00

15

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

q

Va

R (

q)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

00

05

01

0

Ripple

Euro

Figure 4 Value at risk for the best fitting distributions for daily log returns of the exchange rates ofBitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second row right)MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014 until28 February 2017 Also shown are the values at risk for daily log returns of the exchange rates of theEuro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 18 of 23

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Bitcoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Dash

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Dogecoin

Euro

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

12

minus0

10

minus0

08

minus0

06

minus0

04

minus0

02

00

0

Litecoin

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

MaidSafeCoin

Euro

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

20

minus0

15

minus0

10

minus0

05

00

0

Monero

Euro

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

q

ES

(q

)

00 02 04 06 08 10

minus0

15

minus0

10

minus0

05

00

0

Ripple

Euro

Figure 5 Expected shortfall for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017 Also shown are the expected shortfall values for daily log returns of theexchange rates of the Euro computed using the same best fitting distributions

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 19 of 23

46 Kupiecrsquos test

The p-values of Kupiecrsquos test for the best fitting distribution for each of the seven cryptocurrenciesare shown in Figure 6 The p-values (points above 005 in Figure 6) suggest that the out of sampleperformance of VaR can be considered accurate at the corresponding values of q for any of the bestfitting distributions The motivation for using the Kupiecrsquos test is that we can give predictions ofthe exchange rate of the different cryptocurrencies including predictions for the extreme worst casescenario and the extreme best case scenario

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

00

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

090 092 094 096 098 100

02

04

06

08

10

q

pminusva

lue

Figure 6 Cont

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 20 of 23

090 092 094 096 098 1000

20

40

60

81

0q

pminusva

lue

Figure 6 Kupiecrsquos p-values for the best fitting distributions for daily log returns of the exchange ratesof Bitcoin (first row left) Dash (first row right) Dogecoin (second row left) Litecoin (second rowright) MaidSafeCoin (third row left) Monero (third row right) and Ripple (last row) from 23 June 2014until 28 February 2017

47 Dynamic Volatility

So far in this section we have supposed that volatility is represented by a fixed parameter ofa distribution Often in financial series volatility varies throughout time This could be accommodatedin various ways One is to treat volatility itself as a random variable Another is to let volatility dependon some covariates including time as in GARCH models for example In the first case the interest willbe on the distribution of volatility

For the seven cryptocurrencies and the Euro we computed standard deviations of daily logreturns of the exchange rates over windows of width 20 days Histograms of these standard deviationsare shown in Figure 7 We see that the distribution is skewed for the seven cryptocurrencies and theEuro The range of volatility appears smallest for the Euro and second smallest for Bitcoin The rangeappears largest for Dogecoin Litecoin and Monero

Bitcoin

SD

Fre

quency

001 003 005 007

06

Dash

SD

Fre

quency

002 006 010 014

06

Dogecoin

SD

Fre

quency

000 005 010 015

06

14

Litecoin

SD

Fre

quency

000 005 010 015

06

MaidSafecoin

SD

Fre

quency

002 006 010

06

Monero

SD

Fre

quency

005 010 015

06

Ripples

SD

Fre

quency

000 004 008 012

010

Euro

SD

Fre

quency

0000 0005 0010 0015

040

Figure 7 Histograms of standard deviations of daily log returns of the exchange rates of the sevencryptocurrencies over windows of width 20 days Also shown is the histogram of standard deviationsof daily log returns of the exchange rates of the Euro over windows of width 20 days

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

References

Akaike Hirotugu 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control19 716ndash23

Anderson D R K P Burnham and G C White 2010 Comparison of Akaike information criterion and consistentAkaike information criterion for model selection and statistical inference from capture-recapture studiesJournal of Applied Statistics 25 263ndash82

Anderson Theodore W and Donald A Darling 1954 A test of goodness of fit Journal of the American StatisticalAssociation 49 765ndash69

Aquino Rodolfo Q 2006 Efficiency of the Philippine stock market Applied Economics Letters 13 463ndash70Azzalini Adelchi and Antonella Capitanio 2003 Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew t distribution Journal of the Royal Statistical Society B 65 367ndash89Barndorff-Nielsen Ole 1977 Exponentially decreasing distributions for the logarithm of particle size Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353 401ndash9Bitcoin Project 2017 Frequently Asked Questions Available online httpsbitcoinorgenfaqwhat-is-bitcoin

(accessed on 2 February 2017)Bozdogan Hamparsum 1987 Model selection and Akaikersquos Information Criterion (AIC) The general theory and

its analytical extensions Psychometrika 52 345ndash70Breusch Trevor S and Adrian R Pagan 1979 A simple test for heteroscedasticity and random coefficient

variation Econometrica 47 1287ndash94Briere Marie Kim Oosterlinck and Ariane Szafarz 2015 Virtual currency tangible return Portfolio diversification

with Bitcoins Journal of Asset Management 16 365ndash73Bruneau Gabriel and Kevin Moran 2017 Exchange rate fluctuations and labour market adjustments in Canadian

manufacturing industries Canadian Journal of Economics 50 72ndash93Burnham Kenneth P and David R Anderson 2004 Multimodel inference Understanding AIC and BIC in model

selection Sociological Methods and Research 33 261ndash304

J Risk Financial Manag 2017 10 12 21 of 23

5 Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the US Dollar usingeight of the most popular parametric distributions in finance From our analysis of over two and a halfyears of data it is clear that most cryptocurrencies exhibit heavy tails Using the discrimination criteriaof the log likelihood AIC AICc BIC HQC and CAIC the results obtained show that none of thedistributions used give the best fit jointly across the data for all of the cryptocurrencies Instead we findthat the generalized hyperbolic distribution gives the best fit for the Bitcoin and LiteCoin the normalinverse Gaussian distribution gives the best fit for Dash Monero and Ripple the generalized tdistribution gives the best fit to Dogecoin the Laplace distribution gives the best fit to MaidSafeCoin

Implications of these results are in the area of risk management where one may need to computethe VaR and ES for risk but also for investment purposes To our knowledge this is the first studyinvestigating the statistical properties of cryptocurrencies going beyond Bitcoin and the traditionalcurrencies Indeed there is much scope for future work and possible extensions could include (i) usingGARCH type processes to model the log returns of cryptocurrencies for example the distributionsmentioned in Section 3 can be used for modeling the innovation processes (ii) using multivariateprocesses to model the joint distribution of the log returns of cryptocurrencies (iii) using nonparametricor semiparametric distributions to analyze the exchange rates of cryptocurrencies

Acknowledgments Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat forEducation Research and Innovation within the context of the European COST network ldquoMathematics for IndustryrdquoAll authors would like to thank the two referees and the Editor for careful reading and comments which greatlyimproved the paper

Author Contributions All authors contributed equally to the work

Conflicts of Interest The authors declare no conflict of interest

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ccopy 2017 by the authors Licensee MDPI Basel Switzerland This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (httpcreativecommonsorglicensesby40)