Technical University of Vienna

Risk PremiaAsymetric Tail Risk and Excess Returns

Student: Minja Petrovic

Lecturer: Stefan Gerhold

March 31, 2018

Abstract

The following seminar paper is based on the article ”Risk Premia: assymetric tailrisk and excess returns”, written by Y. Lemperiere, C. Deremble, T. T. Nyguen,P. Seager, M. Potters and J. P. Bouchaud. Its main focus is the explanation ofthe common belief about the risk premium together with some real life examples,which clearly show the weakness of such belief, and the new approach to the riskpremium, which is not dependent of the volatility of the market, but rather of theskewness of the distribution of returns.

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Contents

1 Introduction 31.1 What is Risk Premium? . . . . . . . . . . . . . . . . . . . . . . . . 31.2 CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Normal (Gaussian) distribution . . . . . . . . . . . . . . . . . . . . 5

2 The common belief about the risk premium 6

3 Risk Premium in global equity markets 83.1 No “volatility premium” in stock indices . . . . . . . . . . . . . . . 83.2 Downside tail risk and skewness premium . . . . . . . . . . . . . . . 9

4 The Fama-French factors 12

5 Risk premia in the FX world: the “Carry Trade” 14

6 A paradigmatic example: risk premia in option markets 15

7 Risk Premium is Skewness Premium 17

8 Discussion and conclusion 208.1 Skewness vs. Co-skewness . . . . . . . . . . . . . . . . . . . . . . . 208.2 Skewness, premia and crowded trades . . . . . . . . . . . . . . . . . 208.3 Diversifying skewness? . . . . . . . . . . . . . . . . . . . . . . . . . 21

Appendix 22

References 25

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1 Introduction

1.1 What is Risk Premium?

A risk premium is the return in excess of the risk-free rate of return an investmentis expected to yield. It is a form of compensation for investors who tolerate theextra risk, compared to that of a risk-free asset, in a given investment. [1]Example 1 : Let’s consider a game show, in which participants are asked to chooseone of two doors-one that hides 1000e and one that hides 0e. Suppose that thehost also offers the contestants to take 500e instead of choosing a door. The twooptions (choosing the door 1 or 2 or taking certain amount of money) have thesame expected value of 500e. That means that the no risk premium is offered forchoosing a door instead of taking the guaranteed 500e.A contestant unconcerned about risk is indifferent between the choices. A con-testant, who would accept the guaranteed amount, is called a risk-averse contes-tant. A risk-loving contestant will derive utility from the uncertainty and thereforechoose a door.If too many contestants choose to play safely (that is, they are risk-averse), thegame show may encourage selection of the riskier choice by offering a positive riskpremium. If the host offers 1600e behind the good door, increasing the expectedvalue of choosing between the doors to 800e, the risk premium becomes 300e (i.e.800e expected value - 500e guaranteed amount). [2]In other words, investors risk losing their money because of the uncertainty of apotential investment failure on the part of borrower in exchange for receiving extrareturns as a reward (“premium”) if their investments turns out to be profitable.Therefore, receiving a compensation is not a must, since the absence of a successfuloutcome is possible.

1.2 CAPM

The Capital Asset Pricing Model (CAMP) is a model that describes the relation-ship between the systematic risk β and the expected return of an asset. [3] Wewill consider the following formula:

E[Ri] = Rf + β(E[Rm]−Rf ),

where:

• E[Ri] is the expected return on the capital asset

• Rf is the risk-free rate of interest (such as interest arising from governmentbonds)

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• β is the sensitivity of the expected excess returns to the expected excessmarket returns (so known volatility of returns)

• E[Rm] is the expected market return.

The difference between the expected market return and the risk-free rate is some-times known as market premium, whereas the difference between the expectedreturn of the capital asset and the risk-free rate is the so called risk premium.Rewriting the formula in terms of risk premium yields:

E[Ri]−Rf = β(E[Rm]−Rf ),

which indicates that the individual risk premium equals to β times the marketpremium.In order to visualize the formula, we may observe the so called security market line(SML) that enables us to calculate the reward-to-risk ratio for a security in relationto that of the overall market. [4] The x-axis represents the risk in terms of β, andthe y-axis represents expected return. The market risk premium is determinedfrom the slope of SML (figure 1).The concept of β is central to the CAPM andthe SML. β, as a measure of systematic risk of a security, cannot be eliminated bydiversification. The beta value of 1 is considered as the overall market average. Ifβ has the value higher than 1, the risk level is above the market average and if thevalue is lower than 1, the level of risk is below the average.The SML is widely used by investors in evaluating a security for inclusion in aninvestment portfolio in terms of whether the security offers a favorable expectedreturn against its level of risk or not. If the security is plotted above the SML, itis considered under-evaluated because the position on the chart indicates that thesecurity offers a greater return against its inherent risk. On the other hand, if thesecurity is plotted below the SML, it is considered overvalued in price because theexpected return does not overcome the inherent risk.

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Figure 1: Representation of the CAPM model and SML. [3]

When it comes to comparison of two similar securities offering approximately samereturn, the SML is used to determine which of the two securities has the leastamount of inherent market risk in relation to the expected return. It can alsobe used in the opposite way, i.e. seeing which security has the highest expectedreturn if their risks are approximately same.

1.3 Normal (Gaussian) distribution

The normal distribution is the most common type of distribution and it is oftenfound in stock market analysis. Given enough observations within a sample size, itis reasonable to assume that returns follow a normally distributed pattern, wheredata is equally distributed on the both sides of the peak, or the average value.Under normal circumstances, independently and identically distributed randomvariables, or outcomes, are said to converge to a standard normal distributionunder the central limit theorem. The standard normal distribution has two pa-rameters: the mean µ and the standard deviation σ, where µ=0 and σ=1.The skewness measures the symmetry of a distribution. The standard normal dis-tribution has skewness of zero, and is therefore symmetric. If the distribution hasa positive skewness, data piles up on the peak’s left side and curve’s tail pointsright, which means there were frequent small losses and a few large gains. In thecurve with negative skewness, data piles up on the peak’s right side and the tailpoints left, leading to the frequent small gains and a few large losses.

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Figure 2: Positive and negative skewness.[5]

Example 2: Let us consider the distribution of home costs in a community. Valuesrange between 100k and 1 mil e, with the average being 500k e. If the peak ison the left side of the average value, then many homes are sold for less than theaverage. The curve’s skewness is in this case positive and its tail points right.The kurtosis measures the tail end of a distribution and whether the distributionof a data set has skinny tails or fat tails in relation to the normal distribution.The standard normal distribution has a kurtosis of three, which indicates datathat follow a Gaussian distribution have neither fat nor thin tails. If a kurtosis isgreater than three, it is said that observed data has fat tails and if it is less thanthree, thin tails.Traditional portfolio strategies often rely on normal bell curves to make marketassumptions, but in reality markets tend not to behave “normally”. They actuallytend to have fat tails. Observed asset returns have typically had moves greaterthan three standard deviations beyond the mean. The risk rising from such asituation is known as “tail risk”.

2 The common belief about the risk premium

According to the modern finance theory the concept of “risk premium”, i.e. thatriskier investment should be more profitable on the long term, is based on the in-tuitively understandable argument. Given the choice between two assets with thesame expected return, any rational investor should prefer the less risky one. Thesmaller demand for the riskier asset should make its price drop, making its yieldgo up, until it becomes attractive enough to lure the investors despite its higherrisk. This idea is the heart of the CAPM, which states that the excess return of agiven stock (over the risk free rate) is proportional to β, i.e. to its covariance withthe market risk. 1

1Compare the section 1.2.

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Although the argument underlying the existence of risk premia is contigent, esti-mating and rationalizing their order of magnitude in financial markets is still verymuch a matter of debate. This is quite surprising in view of the importance of theissue of the asset management industry as a whole, and reveals how primitive ourunderstanding of financial markets and returns is. [6] The equity risk premium(ERP) is, according to the table number 1 found in section 3.2, to be found inthe range of 3-9% depending on the period and the country. According to Mehraand Prescott, however, these values are too high to be explained within a generalequilibrium model of the economy (also known as “equity premium puzzle”). 2

In addition, several authors have reported inverted relation between volatility ofa stock (β in CAPM) and its excess return. This leads us to the conclusion thatthe initial belief is not as rock solid as believed and that less volatile stocks mayactually be more profitable.The reason for this controversy may lay in the very definition of risk. Classicaltheories identify risk with volatility σ. This is partly resulting from the standardassumption of a Gaussian distribution for asset returns. But, in fact, fluctuationsare known to be strongly non Gaussian, and investors are concerned more aboutrare but plausible crashes that cause large negative drops in their wealth thanabout small fluctuations around the mean µ. These are, however, not captured byσ, but rather contribute to the negative skewness of the distribution. As a result,a new idea has emerged in the literature- a large contribution to the risk premiumis in fact a compensation for holding an asset that provides average returns, butmay occasionally erase a large fraction of the accumulated gains- which impliesthat risk premium might be more a question of skewness and negative tail eventsthan related to the volatility per se.We may formulate the story in the terms of utility function, which is supposedto describe investors’ preferences. Assuming that the “happiness” of the investoron his/her wealth W as a certain function U(W ), the preference for higher prof-its imposes that the function is monotonically increasing, whereas the conditionU ′′(W ) ≤ 0 for every W insures that a less volatile investment with the sameexpected gain is always preferred. Assuming that the proposed investment yieldsan uncertain final wealth W distributed around a mean E[W ] with the small fluc-tuations of variance σ2, and expending in powers of the difference W −E[W ], onefinds that the utility is given by :

E[U(W )] = U(E[W ])− 1

2λ2σ2 + · · · (1)

λ2 := −U ′′(E[W ]) ≥ 0

2Mehra, R. and Prescott, E.C., The equity risk premium: A puzzle. J. Monetary Economics,1985, 15, 145-161

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Hence, with the small risk expansion, the reduction of utility is entirely describedby the variance σ2. But, as the intuition suggests, skewness does matter. Somelottery expreriments, for example, demonstrate that agents prefer lotteries withlarge potential gains, even when the expected gain is negative, to lotteries withsmall positive expected gain, but no possibilities of large payoffs. Analogously,we could expect agents to shy away from the investment with a high negativeskewness. The formula number (1), however, does not include the skewness; whichleads to the thought that the whole concept of the utility is insufficient to accountfor behavioural biases.In the following sections we will consider the correlation between the risk premiumand skewness of the strategy, which, as data shows, plays a greater role thanthe correlation with its volatility. We will investigate so called “risk premium”strategies in detail (not only in stocks, but also in equities, bonds, currencies,options and credit) and elicit an approximately linear relation between the Sharperatio of these strategies and their negative skewness. We will find, however, thatsome well-known strategies, such as trend following and the Fama-French “Highminus Low” factor, are not following this rule, suggesting that these strategies arenot risk premia, but genuine market anomalies.

3 Risk Premium in global equity markets

3.1 No “volatility premium” in stock indices

According to the standard belief, investing in the stock market rather than in risk-free instruments only makes sense if stocks provide better returns than the risk-freerate on the long term. Lately, the existence and the strength of the equity riskpremium (ERP) have been a matter of debate between the academic circles andthe asset management industry. The general consensus is that this premium existsand is in the range of 3-9%, with large fluctuations, both across time period andcountries. However, with such level of ERP, rational investors should put moneymassively in equities, and the demand for bonds, on the other hand, should bemuch weaker than it actually is, which should lead to the smaller risk premium.3

In order to estimate the ERP, it is common to compute the total returns of indices(dividends included), and subtract the risk-free rate, which is virtually equivalentto buying a future contract on indices. We will observe this cumulated differencefor various countries, using the 10Y government rate as a risk-free asset. Thisresult is, as expected, in favour of the existence of a risk premium, as it can beseen from the table 1 and figure 3.

3The so called ”Equity Premium Puzzle”. Compare with the section 2.

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Figure 3: Cumulated ERP since 1870, corresponding to a strategy where oneholds a long portfolio of indices with equal weight on all available contracts at anyinstant of time, i.e. 1 in 1870 and 27 in 2014. The t-stat is 4.2.

In the fourth column of the table 1, we give the average annual volatility of thecorresponding index, which allows us to compute the Sharpe ratio S = µ/σ of eachrisk premium, i.e. the average return earned in excess of the risk-free asset rateper unit of volatility. However, the value of the risk premium itself is indeed notmeaningful since it can be arbitrarily leveraged. Therefore, throughout this paper,we will use the a-dimensional Sharpe ratio to compare different risk premia. Thegreater it is, the better the portfolio’s risk-adjusted performance is. The figure 4shows a scatter plot of S as a function of the volatility of the index together with aregression line. We can see that the two quantities are in fact negatively correlated.This, once again, proves the “low-volatility” anomaly, i.e. that the risk-adjustedreturns of the low-volatility stocks is found to be higher than for high-volatilitystocks, not only within a single market, but across the international indices.

3.2 Downside tail risk and skewness premium

In this section we replot the cumulated ERP in the following way: instead of con-sidering the returns of the strategy in chronological order, we sort these returnsin terms of their absolute value and plot the cumulated ERP F (p) as a functionof a normalized rank p = k/N , starting from the smallest amplitude for k = 1and ending with the largest k = N . The figure 5 shows the result of such a repre-sentation in the case of the US since 1928. We can clearly see that small returnscontribute positively to the average, whereas the largest returns lead to a violentdrop of the P&L. In this figure, we also see the P&L Fs(p) that would have beenif the distribution of returns was exactly symmetrical around the mean, i.e. such

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Table 1: Average annualized risk premium in %, annualized volatility in %, Sharperatio, standard skewness ζ3 and ranked P&L skewness ζ∗ for various indices overthe 10Y bond rates, all based on montly returns.

Country Start date Premium Vol. Sharpe ζ3 ζ∗ Co-skew/USAustralia 1983 4.77 16.7 0.29 -2.0 -2.35 -0.29Canada 1934 4.44 14.8 0.30 -0.6 -1.68 -0.12

Germany 1870 4.23 19.6 0.21 1.7 -0.31 -0.15France 1898 6.65 19.7 0.34 1.7 1.05 -0.11

Hong Kong 1996 5.24 26.1 0.20 -0.20 -2.87 -0.11Mexico 2001 10.1 18.2 0.56 -0.5 -2.26 -0.32

The Netherlands 1957 4.47 16.9 0.27 -0.1 -1.7 -0.17Philippines 1996 -1.96 26.6 -0.07 0.1 -0.42 -0.11

UK 1958 5.71 18.8 0.30 1.0 -2.31 -0.13USA 1871 5.33 15.2 0.35 0.3 -0.32 -0.02

that the final point Fs(p = 1) coincides with F (p = 1). 4 In this case, the P&L isa monotonously increasing function of the normalized rank for large N. One alsofinds out that F (p) behaves as p3 for small p, which explains the strong curvaturein the figure below. The comparison between real and symmetries returns reveals,therefore, the strongly skewed nature of the ERP.This ranked amplitude P&L representation suggests a new general definition ofskewness as following:

ζ∗... = −100

∫ 1

0

dpF0(p),

where the arbitrary factor 100 is introduced such that the skewness is of order unity.

4Technically, this amounts to transforming the returns rt as m + εt(rt −m), where m is theaverage return and εt = ±1 an independent random sign for each t.

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Figure 4: Sharpe ratio of the excess return (over the 10Y bond) for each indexlisted in table 1 (with 10 other countries) as a function of the volatility of thatindex. We also plot the regression line, with a negative slope and a correlationcoefficient ρ ≈ −0.27. Note that the (flat) average Sharpe ratio over all contractsis S ≈ 0.3.

Figure 5: The ranked amplitude P&L representation: plot of the cumulated dailyP&L F (p) for the SPX (at constant risk) since 1928, as a function of the normalizedrank of the amplitude of the returns, p = k/N (in red). The standard chronologicalP&L corresponding to holding the US equity market index at constant risk is shownin black, that by construction ends at the same point. We also show for comparisonFs(p) (in green), corresponding to returns symmetries around the global mean (i.e.Fs(p = 1) = F (1), again by construction.) The minus of yellow area, enclosedbetween F (p) and Fs(p), corresponds to our definition of skewness.

The so far known definition of skewness can indeed be “awkward” for heavy-tailed

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distributions (such as those describing financial returns for example), since a fewextreme events can totally dominate the empirical determination of the skewnessζ3. Our new definition is not only intuitively clearer, but it can actually be definedeven if the mean of the distribution diverges. We also show in the appendix thatwhenever F0(p) has a single hump, as it is usually the case in financial applications,then the cumulative distribution of returns crosses its symmetrized version twiceand only twice, which means that investors with a utility U such that U ′′′ > 0 willshy away from the investments with negative ζ∗-i.e. they will demand higher riskpremium.Analysing the data given in the table 1 and values of ζ∗ (-1.47 in US market fordaily returns), we obtain a positive correlation between –ζ∗ and the ERP acrossdifferent countries. This is, however, not sufficient, since the standard skewnesson a monthly basis also gives us a positive correlation with ERP. Therefore, weextend our analysis to different contracts and different risk premia, and indeedfind that in most cases, excess returns lead to a humped shape function F0(p),meaning that returns of large amplitude have a negative mean. This will lead usto propose a possibly universal definition of “risk premium” in terms of skewnessof the returns- in a precise sense, the premium compensates for a tail risk that issystematically biased downwards.

4 The Fama-French factors

Here we consider the extended Fama-French model, which attributes excess stockreturns to 4 components, corresponding to 4 orthogonal portfolios: the alreadydiscussed ERP (i.e. market rate minus risk-free rate, MKT), SBM (”Small” capsminus ”Big” caps), UMD (”Up”-previous winers minus ”down”-previous losers)and HML (High book to price-”value” stocks minus Low book to price-”growth”stocks). Each of these is interpreted as a reward for a specific risk, even if the originof said risk is unclear. The first, long market portfolio is not market neutral, whilethe other three are constructed in such a way, in order to be orthogonal to themarket risk factor MKT. SMB consists of being long the small cap stocks andshort the large caps, which sounds like an intuitive risk premium strategy, sinceit seems probable that small caps have a greater expected future grow, but are”riskier” than large caps. However, table 2 shows us that the volatility of large capand small cap indices is practically the same in the US and that the volatility isactually smaller for small caps than for large caps in all other countries. Therefore,volatility alone is incapable of explaining the existence of possible SMB premium.If we now observe the above defined skewness of large, medium and small capindices in table 2, we clearly conclude that skewness does indeed become morenegative as we move form larger to smaller caps indices. In order to observe this

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issue a bit better, we use the ”decile” portfolios for the three Fama-French factorsin the US equity market. In table 3, we represent the volatility, skewness andSharpe ratio for being long each decile portfolio from 1950, for SMB, HML andUMD. The larger caps have much weaker skewness than small caps, confirmingthe conclusion above.

Table 2: Volatility in %/day and skewness of the daily returns oflarge(B)/medium(M)/small(S) cap indices in various countries. In the last col-umn we give Sharpe ratio of portfolios that are long on small caps, short on bigcaps and neutral on medium caps. Note that the volatility of small cap indicesis not larger than that of large cap indices, while their skewness is substantiallymore negative.

Index Start date Cap σ (%/day) ζ3 SR of SMBB 1.1 -0.66

S&P 1990 M 1.2 -1.12 0.32S 1.2 -1.16B 1.4 -0.78

France 1999 M 1.1 -1.66 0.40S 0.9 -2.40B 0.8 -0.90

New Zealand 1998 M 0.7 -1.23 0.39S 0.5 -1.06

Table 3: Volatility, skewness and Sharpe ratio of the daily returns of the decileportfolios corresponding to SMB, UMD and HML in the US equity market since1950.

1 2 3 4 5 6 7 8 9 10SMB σ (%/day) 0.83 1.00 1.00 0.98 0.97 0.92 0.93 0.94 0.93 0.96

SMB ζ∗ -1.83 -1.53 -1.49 -1.52 -1.42 -1.45 -1.26 -1.28 -0.93 -0.39SMB SR 0.56 0.48 0.53 0.50 0.53 0.53 0.53 0.50 0.48 0.39

UMD σ (%/day) 1.48 1.21 1.05 0.99 0.95 0.93 0.92 0.94 1.01 1.23UMD ζ∗ 0.00 -0.14 0.00 -0.24 -0.24 -0.30 -0.58 -0.71 -0.79 -0.90UMD SR -0.70 0.17 0.33 0.37 0.39 0.46 0.45 0.62 0.53 0.67

HML σ (%/day) 1.05 0.97 0.93 0.95 0.94 0.92 0.91 0.97 0.98 1.10HML ζ∗ -0.70 -0.69 -0.52 -0.68 -0.52 -0.66 -0.59 -0.65 -0.55 -0.36HML SR 0.33 0.41 0.43 0.44 0.50 0.52 0.52 0.59 0.61 0.60

As a visual confirmation, we plot in figure 6 the ranked amplitude P&L of thethree Fama-French factors in the US since 1950. Apart from the special HMLcase, the findings give further credence to the idea that risk premium is actually

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not compensating for taking a symmetric risk (as measured by the volatility), butfor carrying risks for large loses.

Figure 6: The ranked amplitude P&L F (p) of the three Fama–French factors, SMB,UMD and HML. Note the familiar humped shape for SMB and UMD, revealingstrong negative skewness, but an inverted behavior for HML.

5 Risk premia in the FX world: the “Carry Trade”

The idea of the “carry trade” in the Forex world is following: one buys short-term debt in a county whose short-term rate is high using borrowed money froma country whose short-term rate is low. This allows the investor to pocket thedifference between the high interest and the low interest rate, in the absence ofadverse exchange rate moves. This excess gain is usually thought as a risk premum-one invests in the economy of a “risky” country while getting financed in a safecountry. Our questions remain the same:

• is the excess return compensating for volatility?

• is the strategy negatively skewed?

In order to answer these questions, we simulate this currency pair strategy on apool of 20 developed countries using data from Global Financial Data 5, both forspot currencies and interest rates, from January 1974 to January 2014. We useintra-bank rates when possible and central bank discount rates otherwise. Foreach day we consider all 20*19/2 currency ordered pairs, using the difference ofinterest rate in the two countries to order the pairs in a systematic way. Then,following the Fama-French construction, we make 10 deciles corresponding to pairs

5www.globalfinancialdata.com

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with the smallest interest rate difference to the largest, i.e. from the smallest tothe largest “carry”. After that, we compute the volatility, skewness and Sharperatio of these ten portfolios, as above (Table 5). Again, we find that the skewnessand Sharpe ratio are strongly correlated with ρ0 = −0.76, with a correlation beingsignificantly larger in the period from 1994 to 2014 than in the previous period(1974-1993). However, in this case there is also a substantial positive correlationbetween volatility and Sharpe ratio (ρ0=0.78). The Sharpe ratio of the fully diver-sified Carry trade, i.e. with an equal weight over all ordered FX pairs, is S = 0.85for a skewness ζ∗ = −0.94.

Table 4: Volatility, skewness and Sharpe ratio of the daily returns of the decileportfolios corresponding to FX Carry on G20 currencies since 1974. Decile 1 cor-responds to pairs with small interest rate difference whereas decile 10 correspondsto pairs with large interest rate difference.

1 2 3 4 5 6 7 8 9 10FX Carry(%/day) 0.45 0.47 0.45 0.50 0.55 0.58 0.64 0.68 0.74 0.93

FX Carry ζ∗ -0.41 -0.35 -0.60 -0.50 -0.87 -0.97 -0.86 -1.04 -0.81 -1.05FX Carry SR -0.28 -0.05 0.56 0.44 0.51 0.40 0.78 0.62 0.91 1.04

These results contrast with the conclusion of Jurek6, who finds that the prof-itability of the FX carry trade is not substantially degraded when hedged without-of-the-money currency options. Considering the general fact that the optionsare overpriced, this is somewhat surprising, but it leads Jurek to the conclusionthat the carry trade P&L is not related to the (implied) tail risk. However, we can’tconfirm these findings. Instead of that, we find that when hedging the carry tradestrategy with at-the-money instead of out-of-money options, the carry P&L totallyvanishes.7 Since at-the-money volatilities are usually cheaper than out-of-moneyvolatilities, we conclude that the carry trade P&L becomes negative when hedgedwith out-of-the-money currency options, which aligns with our general skewnessstory.

6 A paradigmatic example: risk premia in op-

tion markets

One of the most common examples of a risk-premium strategy is selling options,i.e. making insurance contracts against large movements of financial assets, such

6Jurek, J.W., Crash-neutral currency carry trades. J. Financial Econ., 2014, 113, 325-347.7Our data are, however, less reliable in this case.

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as stock indices. In this case, there is a built-up asymmetry between the risk takenby the insurer (the option seller) and the option buyer. The option seller bearsthe risk of arbitrarily large losses, and hence the skewness of his/her P&L is (al-most by definition) negatively skewed, even when hedging in a world where perfectreplication is not possible. On the other hand, skewness of the buyer is positive.Therefore, it is logical to expect that option sellers demand a “skew premium” asa compensation for the asymmetric risk they agree to cover. In order to confirmthis intuitively driven concept, we compute the performance of selling varianceswaps- a basket of options whose return is directly proportional to the differencebetween realized and implied variance. The observed difference is exactly the riskpremium we seek to elicit, so these baskets are perfectly suited to our purpose.

Figure 7: Performance of a risk managed short-option strategy, equi-weighted onS&P 500, DAX, Crude oil, Gold, EURXUSD and GBPXUSD variance swaps fromJanuary 1996 until January 2014. The Sharpe ratio is 1.26. We also show theranked P&L which reveals the strongly skewed nature of selling volatility.

As the risk premium should be a universal feature valid across the board, wetest this risk premium strategy on a variety of products, from commodity (CrudeOil, Gold) and currency options (EURXUSD, GBPXUSD) to stock index varianceswaps (S&P 500, DAX) since January 1996. Selling each of these variance swapcontacts and risk-managing using the last 20-day’s volatility of the P&L itselfleads to the global performance shown in figure 7. As we can see, the return ispositive, with a t-stat over 4 and the strategy is strikingly skewed, as expectedfor this paradigmatic risk premium strategy. In table 5 we summarize the Sharperatio and the skewness ζ∗ of all six contracts. As shown in table, the skewness isindeed negative in all cases. It is, actually, substantially larger than in all examplesconsidered in this paper so far. The performance of the individual short variance

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swap strategies is also somewhat better than all the risk premia analysed above.Interestingly, shorting the VIX future instead leads to a P&L that is both lessskewed and with a lower Sharpe ratio, as it can be seen in figure 8. This onceagain suggests that there might be a systemic relation between risk premia andskewness. This is a point we want to discuss further.

Table 5: Sharpe ratio and skewness of short var-swap, risk managed strategies ondifferent underlyings since January 1996.

Underlying Sharpe ratio ζ∗

S&P 500 1.47 -4.64DAX 0.79 -5.53

Crude oil 0.74 -5.28Gold 0.71 -4.79

EURXUSD 0.88 -5.32GBPXUSD 0.38 -5.01

Total 1.26 -4.61

7 Risk Premium is Skewness Premium

The consistent picture behind all the empirical results in this paper is that “riskpremium” is in fact very weakly related to the volatility of an investment, if atall. It is rather related to its skewness, or more precisely, to the fact that largestreturns of that investment are strongly biased downwards. For example, the “LowVolatility” puzzle goes against the intuitive argument that more (volatility) riskshould be accompanied by higher average returns. Similarly, we find no evidenceof a higher volatility in the small caps, large momentum or high value stocks thatwould explain the excess returns associated to this factor. On the other hand,in all these situations, skewness seems to play an important role. In table 6 wesummarize the correlation between Sharpe ratios, volatility and skewness for allthe strategies discussed above.In order to reveal possibly universal relation between excess returns and skewness,we summarize all our results above in a single scatter plot in figure 8, where weshow the Sharpe ratios of different portfolios/strategies as a function of their neg-ative skewness. Quite remarkably, all but one scatter around the regression line,representing the final result of this paper. The glaring exception is the 50-daytrend following strategy on a diversed set of futures contracts since 1960. Theother outliners, although less clear-cut, are the Fama-French HML factor and theLow-Volatility (LoV) strategy, which also have positive skewness.

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Table 6: Correlation coefficient ρ between volatility and Sharpe ratio, and betweenskewness and Sharpe ratio for all strategies investigated in this paper together withLoV and a 50-day trend following strategy.

Underlying Vol./SR corr. Skewness/SR corr.Bonds -0.69 -0.36

Int. IDX -0.45 -0.38SMB -0.42 -0.89UMD -0.63 -0.85

FX Carry +0.78 -0.76HML +0.03 +0.64LoV -0.98 +0.23

TREND +0.23 +0.58

Figure 8: Sharpe ratio vs Skewness of all assets and/or strategies considered in thispaper. The plain line S = 1/3 − ζ∗/4 corresponds to the regression line throughall positively skewed risk premia marked with filled symbols. The two dotted linescorrespond to a 2-σ channel, computed with the errors on the SR and on theskewness of the Fama–French strategies.

We want to point out, however, that the figure 8 is only a suggestive summaryof our empirical results and it does not follow from the prediction of a valuationtheory. 8 It is also not intended to establish a convincing linear relation betweenskewness and Sharpe ratios, but to suggest a relevant correlation. Figure 8 notonly efficiently captures our results, but also suggests a rule to classify and rankdifferent investment strategies. It is indeed tempting to define a risk premium

8Compare the discussion in section 8.2.

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strategy as one that compensates for the skewness of the returns, in the sense thatits Sharpe ratio lies close to the “skew-rewarding” line. Engagingly, the aggre-gate returns of hedge fund strategies fall on this line, suggesting that a signififcantfraction of hedge fund strategies is indeed risk premia. Strategies that lie signif-icantly below this line take too much tail risk for the amount of excess return.Contrarily, strategies that lie significantly above this line, in particular those withpositive skewness (such as trend following for example), seem to get the best ofboth worlds. By the same token, these strategies cannot be meaningfully classifiedas risk premia 9, but rather as general market anomalies.

Figure 9: The skew-reward trade-off line and different risk premium dynamicalpaths.

The figure 9 shows the skew-reward trade-off line and different risk premium dy-namical paths:

• Path 1: starts with a symmetric asset that pays dividends and thereforeattracts investors. The trade becomes crowded and this creates potentialdeleveraging spirals and downside risk trade.

• Path 2: corresponds to an initially skewed payoff (like options) that movesup in Sharpe ratio as a result of skewness aversion and a dearth of optionsellers.

• Path with a question mark (?): might describe the fate of pure alpha strate-gies as they become more crowded and prone to deleveraging spirals of type

9Barring, of course, the existence of an unidentified risk factor that has never materialized,but that would contribute to a negative skewness if it did.

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discussed in Brunnermeier and Pedersen. 10

8 Discussion and conclusion

8.1 Skewness vs. Co-skewness

Our central result is that skewness is indeed the main determinant of risk premia(see figure 8), with approximately linear relation between the Sharpe ratio S ofa risk strategy and its skewness ζ∗. From a formal point of view, this can beinterpreted within the classical framework of utility theory: provided that thethird derivative of the utility function is positive, skewness-comparable P&Ls canbe ordered, and negatively skewed strategies should be compensated by higherreturns to remain attractive. Previous attempts to formally include the effect ofskewness in valuation theories have led to an extended formulation of the CAPM,where the co-skewness of a given stock with the market should determine the excessreturn instead of skewness itself. This is, however, not the route we took in thispaper for several reasons. Firstly, market anomalies are numerous and strong (asshown above through the sections) and theoretical predictions based on arbitrageand rational behavior are not very compelling. Secondly, our result do not concernthe behavior of individual stocks but rather focus on the profitability of ”factors”in wide sense (Fama-French, bonds, FX and carry trades). Transposing the co-skewness idea would require the definition of the global risk factor that drives allrisk premia, much as the market factor drives individual stocks.

8.2 Skewness, premia and crowded trades

Instead of trying to define a formal argument supporting our findings, we discussthe mechanism enforcing the trade-off between skewness and excess returns in arather intuitive way. Our work actually suggests that there may in fact be twofundamentally different ways in which risk premia reach the skew-reward trade-offline in figure 8.

• Positive returns generate skewness: In many situations, one buys anasset because of an expected steam of payments, such as dividends for stocks,coupons for bonds, etc. This intrinsic source of returns attracts a crowd ofinvestors that both generates a price increase and creates the risk of crash,induced by a self-fulfilling panic or ”bank run” mechanism, due to the crowd-edness of the trade.

10Brunnermeier, M.K. and Pedersen, L.H., Market liquidity and funding liquidity. Rev. Fi-nanc. Stud., 2009, 22, 2201-2238.

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One of the obvious examples are the stock markets themselves-the prospectof dividends attracts investors, but the ”madness of the crowd” may leadto a crash. As illustrated in figure 9 crowdedness decreases the returns andincreases downside tail risk. Both of these effects stabilize the market aroundan acceptable skewness/excess return trade-off.

• Skewness demands risk premium: In other situations, downside tailrisks preexist and lead to excess returns. A perfect example is provided byoption markets. Since options are insurance contracts, their payoff profile isskewed by construction-negatively for options sellers and positively for optionbuyers. In an efficient market, the fair price of options is such that theiraverage payoff is 0, there is no risk premium and the Sharpe ratio of long orshort options is 0. However, an investor’s desire for positive skewness and thepresence of fat tails lead to a rise in market price of options as investors seekinsurance against sudden adverse price movements. The negative skewness,arising from the rare large payoffs, remains constant and ever present. TheSharpe ratio of the short option position rises to reflect the extra premiumrequired to act as insurer in an inefficient market.

8.3 Diversifying skewness?

Our results provide an objective definition of risk premia and a criterion to accesstheir quality based on the comparison between Sharpe ratio and skewness, as sug-gested by figure 8. However, not all excess returns can be classified as risk premia(for example trend following). On the other hand, if the different risk premia aredecorrelated, then skewness can be diversified away, allowing risk premia portfoliosto move above the regression line in figure 8. This might be something that good”alterantive beta” managers should strive to achieve.

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Appendix

Ranked amplitude P&Ls and an alternative definition ofskewnessLet us consider a random variable r (the returns) with a certain probability densityP (r). We assume that r has been standardized (i.e. r has zero mean and unitvariance). We will denote x = |r| ≥ 0 as the amplitude of r. From the definitionof the ranked P&L function F0(p), one has:

F0(p) =

∫ x(p)

0

dy y[P (y)− P (−y)], (A1)

where x(p) is the p-quantile of |r|, defined as p =∫ x(p)0

dr(P (r) +P (−r)). We willintroduce the symmetric and antisymmetric contributions to P as:

Ps(r) = P (r) + P (−r) , Pa(r) = P (r)− P (−r). (A2)

Note that for generic distributions, P (r) ≈r→0 P (0) − P ′(0) + · · · , leading tolowest order (when p→0) to x(p) ≈ p/2P (0) and therefore

F0(p) ∼ −P ′(0)/12× (p/P (0))3, (A3)

i.e. a generic αp3 behavior for small p.Now our new definition of skewness is:

ζ∗... = −100

∫ 1

0

dpF0(p) = −100

∫ ∞0

dx Ps(x)

∫ x

0

dy Pa(y). (A4)

It is interesting to give an alternative, intuitive interpretation of this definition.After simple manipulations, one finds:

F0(p) = E[|r| | r < −x(p)]− E[|r| | r > −x(p)]. (A5)

F0(p) therefore compares the average amplitude of large negative and large positivereturns. ζ∗ is an average of this difference over all possible quantile choices. Itmight also be useful to relate ζ∗ to the standard definition of skewness ζ3 (definedthrough the third cumulant of a distribution) in the limit of weakly non-Gaussiandistributions:

P (r) = [1− ζ33!

d3

dr3+κ

4!

d4

dr4+ · · · ] 1√

2πe

−r22 , (A6)

where κ is the kurtosis; finally leading to

ζ∗ =25

6πζ3(1−

κ

24+ · · · ) ≈ 1.273ζ3(1−

κ

24+ · · · ). (A7)

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Note that ζ∗ does not require the existence of the third moment of the distribution,and remains well defined for all distributions with a finite first moment. In fact,an even weaker condition is sufficient: the distribution should fall out faster than|r|−3/2.It is interesting to compare ζ∗ to the classical definition of skewness is a concretecase. We choose an asymmetric Student-t distribution11:

P (r) = N (1 +r√

ν++ν−2

+ r2)ν−+1

2 (1− r√ν++ν−

2+ r2

)ν++1

2 (A8)

which behaves asymptotically as:

P (r) ∼r→±∞ const.|r|−1−ν± . (A9)

The classical skewness ζ3 is finite only when ν± > 3, whereas ζ∗ remains finite aslong as ν± > 1/2. While the latter condition is always satisfied by financial data,many authors have reported that ν± is actually close to 3 for most markets. As anumerical exercise, we compute both ζ3 and ζ∗ as a function of ν+ > 3, for a fixedvalue of ν− = 3.5. The results are shown in figure 10. Clearly, the skewness of thedistribution is positive when ν+ < ν− and negative otherwise. What we see is that,as expected from the above general formula, ζ∗ and ζ3 behave similarly when theyare both small. However, as ν+ decreases towards 3, ζ3 diverges whereas remainswell behaved. In the opposite direction, we see that ζ3 quickly saturates as ν+increases, while ζ∗ continues to decrease. Therefore, ζ∗ is a better discriminant ofthe asymmetry of the distribution.Finally, let us assume that F0(p) is a humped function of p with a single maximum,corresponding to a negatively skewed distribution. This means that necessarilyPa(y > 0) is negative for large enough y > y∗ and positive for smaller y’s, and vice-versa for y < 0. Now the cumulative function of P compared to its symmetrizedversion is precisely the cumulative of Pa(y):

G(y) =

∫ y

−∞dr[P (r)− Ps(r)] =

∫ y

−∞drPa(r). (A10)

Since Pa(r) vanishes three times and is positive for large negative r, it is clearthat G(y) has two symmetric maxima located at ±y∗ and a minimum for y =0. Now using 0 = F0(p = 1) < y∗

∫∞0drPa(r) and G(0) = −

∫∞0drPa(r), one

immediately finds that G(0) < 0 for non degenerate distributions. This provesthat G(y) crosses zero twice and only twice, and therefore that P (r) and Ps(r) are

11Defined as in ones, M.C. and Faddy, M.J., A skew extension of the t distribution, withapplications.J. Roy. Stat. Soc., Ser. B, 2003, 65, 159–174.

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skewness-comparable in the sense of Oja12 whenever F0p has a unique maximum(or minimum).

Figure 10: ζ3 and ζ∗ as a function of ν+ for a fixed value of ν− = 3.5 for theasymmetric Student-t distribution defined in the text.

12Oja, H., On location, scale, skewness and kurtosis of univariate distributions.Scand. J. Stat.,1981, 8, 154–168.

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References

[1] www.investopedia.com/terms/r/riskpremium.asp.

[2] en.wikipedia.org/wiki/Risk_premium.

[3] corporatefinanceinstitute.com/resources/knowledge/finance/

what-is-capm-formula.

[4] en.wikipedia.org/wiki/Capital_asset_pricing_model.

[5] www.quora.com/What-does-SKEWED-DISTRIBUTION-mean.

[6] www.tandfonline.com/doi/full/10.1080/14697688.2016.1183035.

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