Best Position Sizing in Trading

8/11/2019 Best Position Sizing in Trading

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1 INTRODUCTION

to obtain an optimal gambling strategy, the Kelly criterion: maximise the expected value of the logarithm

of the gamblers wealth at each bet to achieve an asymptotically optimal growth rate; such a strategy alsominimises the expected time to reach a given wealth as demonstrated in B REIMAN(1961) in the case where

stock returns are assumed to be independent, identically distributed (i.i.d.). These results were applied to

investment management in LATANE(1959), who advised investors to maximise the geometric mean of their

portfolios. Later, the optimality of maximising expected log return was extended with no restrictions on the

distribution of the market process in ALGOET and COVER(1988) and, in B ROWNEand WHITT(1996), the

Kelly criterion was generalised to the case in which the underlying stochastic process is a simple random

walk in a random environment; OSORIO (2009) devised an analog to the Kelly criterion for fat tail returns

modelled by a Studentt-distribution and a log prospect rather than utility function.

In HAKANSSON (1970), closed form optimal consumption, investment and borrowing strategies were

obtained for a class of utility functions corresponding to an investor looking to maximise the expected utility

from consumption over time given an initial capital position and a known deterministic non-capital incomestream. ROL L (1973) studied the implications of growth optimum portfolios in terms of observed stock

returns for investors selecting such a portfolio. THORP (1971) applied the Kelly criterion of maximising

logarithmic utility to portfolio choice and compared it to mean variance portfolio theory, concluding that the

Kelly criterion does not always yield mean variance efficient portfolios.

SAMUELSON(1971, 1979) showed that while maximising the geometric mean utility at each stage may

be asymptotically optimal, this does not imply that such a strategy is optimal in finite time; he also high-

lighted the risk involved in using the Kelly criterion, namely that of excessive leverage leading to significant

drawdowns. Later on, more work on the properties of the Kelly criterion and its application to finance was

published in ROTANDO and THORP (1992); MACLEA N et al. (2004, 2010, 2011a,b); THORP (2006). The

concept of optimal f, which is an extension of the Kelly criterion was developed in VINCE (2007, 2009)

and the differences between the two approaches were detailed in VINCE (2011). Additionally, a number ofbooks on money management aimed at practitioners have analysed and backtested the Kelly criterion, the

optimal fand other approaches; see for example G EH M (1983); BALSARA (1992); GEH M (1995); JONES

(1999); STRIDSMAN (2003); MCDOWELL (2008). Finally, the optimalftechnique was applied to futures

trading and compared to more naive approaches in A NDERSON and FAFF (2004); LAJBCYGIER and LIM

(2007), each time with the conclusion that it resulted in leverage levels that would be unacceptable to most

investors; this leads to heuristic approaches such as using a fraction of the Kelly ratio (such as half Kelly

ratio).

The common feature of traditional money management techniques such as the Kelly criterion or the

optimalfis their focus on maximising wealth growth, whereas in practice, both individual traders and fund

managers are mostly concerned with maintaining a stable risk level through time and keeping their maxi-

mum drawdown below a chosen threshold; otherwise, they will most likely suffer significant redemptionsfrom investors or discontinue trading their strategy altogether. As such, maximising the rate of return is

usually secondary to controlling risk. However, there is very little available on this topic in the existing lit-

erature whether originating from academics or practitioners. Note that the practical shortcomings of utility

maximisation had already been noted in ROY (1952) before the introduction of the Kelly criterion as the

author explained that for the average investor, the first objective is to limit the risk of a disaster occurring:

In calling in a utility function to our aid, an appearance of generality is achieved at the cost of a loss of

practical significance and applicability in our results. A man who seeks advice about his actions will not be

grateful for the suggestion that he maximises expected utility.

The techniques presented in this article were born out of the need for position sizing rules that could be

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2 TAIL RISK CONTROL

computed and applied in practice and would result in consistent risk levels when implemented through varied

market conditions and changing trading strategy performance. Algorithms designed to control return tail riskare presented first, while drawdown control is tackled at a later stage. These techniques are then applied to

daily returns resulting from implementing a simple technical trading rule on the EURUSD and NZDMXN

currency pairs; a detailed analysis and comparison of the performance of each money management technique

concludes the article.

2 Tail Risk Control

When a trading strategy is applied to a given asset, the fluctuations in the volatility of the asset returns will

typically lead to changes in the volatility of the strategy returns. In practice, portfolio managers aim to limit

these variations and keep the tail risk of the strategy below a predetermined level by dynamically adjusting

trade size. This section presents techniques to achieve this objective.

2.1 Tail Risk Measures

A common measure of tail risk is Value at Risk (VaR) (B EDER (1995); DUFFIE and PAN (1997); JORION

(2006)), which is defined as the minimum loss experienced over a given time horizon with a given prob-

ability. When applied to historical daily returns, VaR can be computed by ordering the daily returns and

selecting the quantile corresponding to the confidence level chosen (for example 95%). Unfortunately, VaR

is concerned only with the number of losses that exceed the VaR confidence level and not the magnitude

of these losses; to obtain a more complete measure of large losses, one needs to examine the entire shape

of the left tail of the return distribution beyond the VaR threshold, which leads to the Conditional Value at

Risk (CVaR) also referred to as Expected Shortfall, Tail VaR or Mean Shortfall (A RTZNER et al. (1999);

CHRISTOFFERSEN (2003); HARMANTZIS et al. (2006); M CNEI L et al. (2005)). CVaR can be defined as

the average expected loss at a given confidence level; for example, at the 95% confidence level, the CVaR

represents theaverage expected loss on the worst 5 days out of 100 whereas the VaR is the minimum loss

on those days. In mathematical terms, the CVaR for a daily return distributionFat a confidence level is

given by:

CVaR= E{X|X VaR} (1)

where the VaR is defined by:

VaR= F1(1 ) (2)

Computing CVaR requires an explicit expression of the portfolio return distribution function F which

is usually unknown in practice. However, if historical daily returns are assumed to follow a normal (or

Gaussian) distribution, VaR and CVaR can be easily obtained from the standard deviation and mean of

returns; for example, at the 95% level, standard deviation, VaR and CVaR are related by:

VaR 1.65 and CVaR 2.07 (3)

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2.1 Tail Risk Measures 2 TAIL RISK CONTROL

50 45 40 35 30 25 20 15 7.85 00

1

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6

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Daily Return (%)

Numb

erofObservations

95% VaR

Generalised Pareto Distribution

Normal Distribution

Figure 1: Top: Comparison of Generalised Pareto and normal distribution. Note that the Generalised Pareto

Distribution models the left tail of the daily returns much more accurately than the normal distribution.

Bottom: 95 % CVaR for each distribution. The CVaR is represented by the shaded area under the green

(GPD) or red (normal distribution) curve. In the present case, it is apparent that the CVaR computed using

a normal distribution underestimates the downside risk when compared to a GPD.

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2.2 Volatility based Position Sizing 2 TAIL RISK CONTROL

2.2 Volatility based Position Sizing

The first position sizing method consists in computing the historical volatility of daily returns generated by

the strategy, converting this volatility number to a VaR number using the above formula (3) and adjusting

leverage in order to match the target VaR level. The historical volatility of the strategy can be computed

using the RiskMetrics exponentially weighted moving average introduced in Z ANGARI(1996):

=

(1 ) Tt=1

t1(rt r)2 (4)

where Tis the length of the estimation window, the decay factor and rthe mean return over the estimation

window.

Once the volatility has been computed it can be converted into a VaR number using Equation (3) and theleverage or position size is adjusted through the formula:

Leverage Adjustment= Target VaR

Current VaR (5)

This process is typically implemented with a chosen frequency (daily, weekly, monthly) depending on

the average holding period of the trading strategy.

2.3 Extreme Value Theory based Position Sizing

2.3.1 Extreme Value Theory

The previous money management method relies on the assumption that daily strategy returns are normallydistributed. However, in practice, this is unlikely to be the case and tail risk can be more accurately measured

using tools originating from Extreme Value Theory (EVT), a branch of statistics dedicated to modelling

extreme events introduced in BALKEMA and DE HAA N (1974); PICKANDS (1975). The central result in

Extreme Value Theory states that the extreme tail of a wide range of distributions can be approximately

described by theGeneralised Pareto Distribution (GPD) with shape and scale parameters and :

G,(y) =

1 (1 + y

)

1

, for= 0;

1 expy , for= 0.

(6)

where >0, and the support ofG, isy 0when 0and0 y

when


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2.3 Extreme Value Theory based Position Sizing 2 TAIL RISK CONTROL

withNthe total number of observations and Nu the number of observations exceeding the threshold u.

Note that the preceding results requires that observations be independent and identically distributed,which is often not the case for daily returns as they present some level of autocorrelation. Therefore, we

start by filtering the daily returns and then apply Extreme Value Theory to the standardised residuals (see

MCNEI L and FRE Y (2000); NYSTROM and SKOGLUND (2005)), with a Generalised Pareto Distribution

being fitted to the tails through Maximum Likelihood Estimation. Once this is done, we obtain the shape and

scale parameters and replace these values in Equation (7) to compute the CVaR at the required confidence

level. Extreme Value Theory has been used during the previous decade for risk management in finance with

a notable increase in the number of publications on the subject since the recent financial crisis. We refer

to BAL I(2003); BEIRLANT et al. (2004); CASCONand SHADWICK(2009); COLES(2001); DE HAA Nand

FERREIRA (2006); EMBRECHTS(2011); GHORBELand TRABELSI (2008, 2009); GOLDBERGet al. (2008,

2009); GUMBEL(2004); HUANGet al. (2012); L ONGIN(2000); MCNEI Land FRE Y(2000); NYSTROM and

SKOGLUND(2005) for a sample of publications dealing with Extreme Value Theory and its applications tofinancial risk modelling.

The significant improvement in tail risk modelling between the volatility/normal distribution and EVT

approaches is illustrated in Figure 1. We consider 1000 daily returns for a stock and fit both a normal

distribution and a Generalised Pareto Distribution to the left tail of the daily returns. We can see that while

both techniques yield similar VaR numbers at the 95% confidence level (in this case 7.8%), the 95% CVaR,

which can be visually identified as the area under a given distribution curve left of the 95% VaR threshold,

is significantly higher (by a factor 2.4) when computed using the Generalised Pareto Distribution than when

using volatility and a normal distribution assumption. Note that this is a pathological case which was chosen

on purpose as the difference between the two methods is readily apparent. Still, relying on volatility and

normal distribution assumptions can lead to significantly underestimating the tail risk generated by a given

strategy, a dangerous situation to be in for any investment manager.

2.3.2 Filtered Historical Simulation

Applying Extreme Value Theory to tail risk estimation requires fitting a Generalised Pareto Distribution to

the left tail of the strategy returns; in practice, if 250 days are considered and the 95% confidence level

is desired, this means that the GPD has to be fitted to about 12 daily returns, a number which is typically

too low to guarantee convergence of the Maximum Likelihood Estimation method and which will cause a

high sensitivity to changes in historical returns. To circumvent these issues, simulations can be employed,

generating a much larger number of daily returns to which left tail a GPD can be fitted more easily.

Choosing the appropriate simulation method is not necessarily straightforward. Indeed, if Monte Carlo

simulations (METROPOLIS and ULA M (1949)) are selected, a distribution of returns has to be specified,usually a normal distribution, which negates the advantage of using Extreme Value Theory to estimate tail

risk. Therefore, some form of historical simulation is highly preferable as it makes no assumption on the

return distribution, instead relying on the past returns. However, as noted in PRITSKER (2006), such a

method presents two potential issues.

First, the required sample size to obtain a statistically significant distribution is usually considered to be

at least 250 days; this, in turn, raises the potential issue of not being sensitive enough to recent returns which

presumably contain the most relevant information to predict future returns. To circumvent this problem,

theweighted historical simulation (WHS) method was developed in BOUDOUKHet al. (1998); this method

assigns probabilistic weights to the daily returns which decay exponentially with a chosen decay factor over

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2.3 Extreme Value Theory based Position Sizing 2 TAIL RISK CONTROL

time; thus recent returns have more influence than the more distant ones. Unfortunately, it is not clear how

to select the correct time constant; also, an unintended consequence is that extreme events, which by natureoccur rarely, might end up being discounted.

Second, the historical simulation method assumes that daily returns are independent and identically dis-

tributed through time, which is not particularly realistic. Indeed, it is commonly observed that the volatility

of returns evolves through time and that periods of high and low volatility do not occur at randomly spaced

intervals but rather tend to be clustered together. The filtered historical simulation (FHS) method presented

in BARONE-A DESI et al. (1999) is an attempt to combine the advantages of the historical and parametric

methods; the variance-covariance method attempts to capture conditional heteroskedasticity but assumes a

normal distribution while the historical method does not assume a specific distribution but does not capture

conditional heteroskedasticity. The FHS method relies on a model based approach for the volatility, typi-

cally using a GARCH type model, while remaining model free in terms of the distribution. In particular,

this method has the notable advantage of being able to simulate extreme losses even if they are not presentin the historical returns used for the simulation.

2.3.3 Practical Implementation

We begin by implementing the FHS method on a series of daily returns Rt with standard deviation t.

As mentioned above, the historical simulation method assumes that daily returns are i.i.d. through time;

however, significant autocorrelation can often be found in the daily squared returns. To produce a sequence

of i.i.d. observations, we fit an AR(1) first order autoregressive model to the daily returns:

Rt+1= c + aRt+ t wheret= tzt (9)

where we choose the standardised returns{zt}as following a Students t-distribution rather than a normalone to account for increased tail risk as thet-distribution has fatter tails.

To model the variation of the returns standard deviation, we can use a GARCH type model (B OLLER-

SLEV(1986); ENGLE(1982, 2001); TAYLOR(1986)) such as the GARCH(1,1):

2t+1= + R2t +

2t , with +


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3 DRAWDOWN CONTROL

of the residual of the AR(1) model over the standard deviation. Once the historical standardised returns are

known, we generate future returns by drawing standardised returns with replacement. Eventually, we endup with 10,000 daily return series, each covering 252 trading days. These daily returns are aggregated to

generate a distribution of 2,520,000 daily returns to which left tail a GPD is fitted, eventually yielding the

CVaR. The high number of residuals ensures the stability of the method, as the left tail contains 126,000

returns for a 95% confidence level, which almost guarantees the convergence of the Maximum Likelihood

Estimation algorithm used to fit the GPD to the left tail of the simulated return distribution. This CVaR

number can be converted into a VaR number under normal distribution assumptions using Equation (3) and

trade size adjusted through Equation (5). One of the advantages of using Extreme Value Theory to compute

the CVaR is that the tail risk of the return distribution is measured much more accurately and less likely to

be underestimated than when relying only on the volatility based method described earlier on.

3 Drawdown Control

While the previous section outlined money management tools to control tail risk, defined as daily VaR or

CVaR at a given confidence level, the most adverse event from an investor or investment manager standpoint

is probably a significant drawdown in which a number of negative daily returns are clustered together over

a given period time. Indeed, most investors have strict drawdown limits (such as 20%) upon which they will

redeem part or the entirety of their investment in a given fund. Therefore, for a money manager, experiencing

a significant drawdown can lead to a drop in AUM which itself results in a loss of management fees; addi-

tionally, most fund managers who charge performance fees have high watermarks in place which prevent

them from collecting performance fees during a drawdown. Also, a manager trading a systematic strategy

with proprietary or investor capital is likely to unnecessarily modify or discontinue the strategy if faced with

an unacceptable drawdown; this can result in the loss of future performance as the changes may have been

unwarranted. This leads us to develop a money management technique to control the maximum drawdown

encountered by a given strategy. Earlier work on drawdown control through portfolio optimisation can be

found in CVITANICand KARATZAS(1995); GROSSMANand ZHO U(1993).

3.1 Drawdown Measures

The maximum drawdown experienced over a given period of time is defined as the largest peak to trough loss

in Net Asset Value of a portfolio. IfW(t)represents the portfolio value at time t, the maximum drawdownover a time interval[0, T]is defined by:

M DD(T) = max0tT

(max0t

W() W(t)) (12)

The historical maximum drawdown is a number which varies widely even for strategies presenting the

same mean and volatility and is based on the entire track record making difficult any comparison between

strategies run over different time lengths. Therefore, as noted in HARDING et al. (2003), considering a

drawdown distribution with reference to a confidence level would be more practical. The distribution of

drawdowns over a given time period ofNdays can be computed by computing the maximum drawdown

for blocks ofN consecutive days from the track record of a strategy. As VaR and CVaR were defined

for a daily return drawdown, the Drawdown at Risk (DaR) and Conditional Drawdown at Risk (CDaR)

at a given confidence level can be obtained from the drawdown distribution. For example, the 63 days

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3.2 Practical Implementation 4 APPLICATIONS

DaR at the 95% confidence level will be obtained by subdividing the historical daily returns in overlapping

blocks of 63 consecutive daily returns, computing the maximum drawdown for each block thus forming thedrawdown distribution and taking the 95th percentile of this distribution. Similarly the 63 day CDaR would

be the average expected drawdown beyond the 95th percentile. The modelling of the drawdown distribution

has been considered in CHEKHLOVet al. (2003, 2005); JOHANSENand SORNETTE (2001); MENDES and

BRANDI(2004); MENDESand LEA L (2005).

3.2 Practical Implementation

We construct a position sizing algorithm for drawdown control as was done earlier for tail risk control.

Starting with a given number of daily historical returns such as 252 days, we apply an AR(1)/GARCH(1,1)

filtering process and using FHS to simulate 10,000 daily return series of 252 days each. For each one

of these return series, we generate a drawdown distribution by computing the maximum drawdown foroverlapping blocks of consecutive daily returns of a given length (such as 63 days) thereby resulting in 190

drawdowns for each one of the 10,000 return series. The drawdowns are aggregated to generate a distribution

of 1,900,000 drawdowns and a GPD is fitted to the right tail of this distribution containing the 5% largest

drawdowns which yields the CDaR at the 95% confidence level. This number is compared to a set 95%

CDaR target and the leverage is adjusted in consequence using a similar formula as for tail risk control:

Leverage Adjustment= Target CDaR

Current CDaR (13)

4 Applications

In order to analyse the effectiveness and performance of the trade sizing algorithms defined in the previous

sections, we implement them on the daily returns generated by a systematic strategy applied to the EURUSD

and NZDMXN currency pairs.

4.1 Trading Strategy

The trading strategy used in this article is a typical breakout trend following strategy, similar to strategies

commonly used in futures and currency trading; it is based on a moving average with a 2 standard deviationband; on any given day, if the price is above (resp. below) the upper (resp. lower) band, a long (resp. short)

position is initiated, whereas if the price is between the two bands, no action is taken and the previous

day trade direction is maintained. The strategy is traded over a 10 year period going from January 2001to December 2010 which will be referred to as Year 1 to Year 10 in the following. The EURUSD and

NZDMXN currency pairs were selected as they demonstrate different return profiles with NZDMXN being

typically more volatile than EURUSD; also, the strategy performance is significantly higher for EURUSD

than for NZDMXN , which gives us the opportunity to apply the money management algorithms in different

settings. Indeed, looking at Tables 1 to 4 which summarise the performance for the original strategy as well

as the money management techniques, we can see that the Sharpe ratio is 0.79 for the EURUSD strategy

and 0.25 for the NZDMXN strategy. The maximum drawdown is also higher for the NZDMXN strategy at

23.51% compared to 15.25% for the EURUSD strategy.

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4.1 Trading Strategy 4 APPLICATIONS

0 5 10 15 200.2

0

0.2

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0.6

0.8

Lag

SampleAutocorrelation

0 5 10 15 200.2

0

0.2

0.4

0.6

0.8

Lag

SampleAutocorrelation

Figure 2: Top: The autocorrelation function of the squared daily returns for the EURUSD strategy reaches

significant values through time, thus preventing the use of unfiltered data for historical simulation. Bottom:

Autocorrelation function of the standardised residuals after filtering with an AR(1)/GARCH(1,1) model; theautocorrelation has been almost entirely removed. 10


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100

150

200

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Time (Years)

NetAssetValue

(Base

100)

Original ReturnsVolatility based trade sizing

EVT based trade sizingCDaR based trade sizing

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

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3.5

Time (Years)

LeverageAdjustmentFactor

Volatility based trade sizingEVT based trade sizingCDaR based trade sizing

Figure 3: Top: Evolution of the Net Asset Value for the original EURUSD strategy and the volatility and

EVT based position sizing methods. Bottom: Evolution of the leverage adjustment factor for the volatility

and EVT based position sizing methods applied to the EURUSD strategy.11


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0 1 2 3 4 5 6 7 8 9 1090

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190

Time (Years)

NetAssetValue

(Base

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Original ReturnsVolatility based trade sizing

EVT based trade sizingCDaR based trade sizing

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

Time (Years)

LeverageAdjustmentFactor

Volatility based trade sizingEVT based trade sizingCDaR based trade sizing

Figure 4: Top: Evolution of the Net Asset Value for the original NZDMXN strategy and the volatility and

EVT based position sizing methods. Bottom: Evolution of the leverage adjustment factor for the volatility

and EVT based position sizing methods applied to the NZDMXN strategy.12


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0 5 10 150

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63 Day Drawdowns (%)

Frequency

0 5 10 15 200

20

40

60

80

100

120

63 Day Drawdowns (%)

Frequency

Figure 5: Top: Distribution of 63 day drawdowns for the EURUSD strategy. Bottom: Distribution of 63 day

drawdowns for the NZDMXN strategy.

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4.2 Tail Risk Control Techniques 4 APPLICATIONS

4.2 Tail Risk Control Techniques

We apply the the volatility and EVT based tail risk control techniques presented earlier to the EURUSD and

NZDMXN strategy with the objective of maintaining a constant tail risk level over time set at a 95% VaR of

1.5%. For the volatility based technique, the historical volatility is computed at the end of each week using

the RiskMetrics exponentially weighted moving average presented in Equation (4) and transformed into a

VaR level resulting in a leverage adjustment coefficient which is applied to the strategy over the following

week. The typical parameters, recommended in ZANGARI (1996) are used: = 0.94, T = 74days;however,r is taken to be the mean of daily returns over the previous 74 days rather than zero.

For the EVT based algorithm, the first step consists in removing the autocorrelation from the daily

returns by applying an AR(1)/GARCH(1,1) filtering process. Figure 2 illustrates the high level of autocor-

relation in the squared returns and its almost complete removal after filtering; as a result, the standardised

residuals can be considered approximately i.i.d. and used as input in the FHS algorithm to generate simu-lated return series. This process is applied weekly to the previous 252 daily returns from the strategy and

generates after aggregation of the 10,000 series of 252 daily returns, one series of 2,520,000 daily returns to

which left tail beyond the 5% threshold a GDP distribution is fitted. From the shape and scale parameters

of the fitted GPD distribution a 95% CVaR is obtained and then converted into a 95% VaR using Equation

(3). This ensures that the actual CVaR of the strategy is adjusted to match the CVaR corresponding to our

target VaR level of 1.5%if the distribution was following a normal distribution . This means that if in fact

the return distribution has a thicker left tail than a normal distribution, this will be taken into account as the

95% CVaR measured by EVT will be higher and leverage will be reduced in consequence.

4.2.1 Algorithmic presentation

The previous tail risk control techniques can be described in algorithmic form. For the volatility based

algorithm:

1. At the end of weekN, select the previous 74 daily returns and generate the volatility using Equa-

tion (4).

2. Using Equation (3), convert the volatility into a 95% VaR.

3. Compute the Leverage Adjustment corresponding to a target 95% VaR of 1.5% using Equation (5).

4. Apply the Leverage Adjustment to the strategy during weekN+ 1.

5. At the end of weekN+ 1, repeat the algorithm starting from step 1.

For the EVT based algorithm:

1. At the end ofweekN, select the previous 252 daily returns and filter them using an AR(1)/GARCH(1,1)

model; check that the autocorrelation has been brought to a sufficiently low level for the i.i.d. assump-

tion to be valid.

2. Using the AR(1)/GARCH(1,1) model, simulate 10,000 daily returns of 252 days each, generating one

series of 2,520,000 returns after aggregation.

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4.3 Drawdown Control Technique 4 APPLICATIONS

3. Using MLE, fit a GPD distribution to the left tail (5% worst daily returns) of the simulated return

series, yielding the shape and scale parameters.

4. Compute the 95% CVaR corresponding to the GPD parameters values and convert the CVaR into a

VaR number using Equation (3).

5. Compute the Leverage Adjustment corresponding to a target 95% VaR of 1.5% using Equation (5).



4.3 Drawdown Control Technique

The drawdown control technique is applied to the EURUSD and NZDMXN strategy with the objective of

limiting the maximum drawdown over each year to a set value, in this case chosen as 10%. Similarly to the

EVT based technique for tail risk control, we start by filtering the previous 252 days and generating 10,000

series of 252 daily returns each using FHS. These daily returns are decomposed in blocks of 63 (which

represents about 3 months) consecutive days from day 1 to day 63, day 2 to day 64, etc; a block length of 3

month was selected as it is a good estimate of the length of the worst drawdowns generated by the strategy;

using higher block lengths such as 1 year would result in underleveraging. The maximum drawdown is

computed for each block yielding a distribution of 190 drawdowns for each of the 10,000 series; these

drawdowns are aggregated to yield one series of 1,900,000 drawdowns to which right tail a GPD distribution

is fitted, yielding the 95% CDaR from which the leverage factor is computed. The interest of using EVT

to estimate the CDaR is apparent from Figure 5, which shows the 63 day drawdown distribution for each

strategy; both drawdown distributions present a right tail which is significantly longer than the left tail and

which would not be measured accurately with a normal distribution; therefore, it is crucial to fit a GPD to

the right tail in order to correctly estimate the CDaR.

4.3.1 Algorithmic presentation

The drawdown control technique can be described in algorithmic form:

1. At the end ofweekN, select the previous 252 daily returns and filter them using an AR(1)/GARCH(1,1)

model; check that the autocorrelation has been brought to a sufficiently low level for the i.i.d. assump-

tion to be valid.

2. Using the AR(1)/GARCH(1,1) model, simulate 10,000 daily returns of 252 days each.

3. Decompose each series of 252 daily returns into 190 overlapping blocks of 63 consecutive days.

4. Compute the maximum drawdown for each block, and aggregated all the drawdowns into one series

of 1,900,000 drawdowns.

5. Using MLE, fit a GPD distribution to the right tail (5% largest drawdowns) of the simulated return

series, yielding the shape and scale parameters.

6. Compute the 95% CDaR corresponding to the GPD parameters values.

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4.4 Results analysis 4 APPLICATIONS

7. Compute the Leverage Adjustment corresponding to a target 95% CDaR of 10% using Equation (13).



4.4 Results analysis

The performance data for the original strategy, the volatility and EVT based tail risk control techniques and

the drawdown control technique are summarised in Tables 1 and 2 for the EURUSD strategy and Tables 3

and 4 for the NZDMXN strategy.

The effectiveness of the tail risk control techniques can be evaluated by looking at the fluctuations of

the 95% Var when the techniques are applied. For the original strategy, the 95% VaR varies widely going

from 0.69% in Year 6 to 1.39% in Year 8 for the EURUSD strategy and from 1.13% in Year 10 to 1.78%in Year 9 for the NZDMXN strategy. These variations are reduced for the volatility based technique with a

range of 1.32% to 1.66% for the EURUSD strategy and 1.32% to 1.68% for the NZDMXN strategy, thereby

demonstrating the ability of this technique to stabilise the 95% VaR around its target value of 1.5%. For the

EVT based technique, the 95% VaR fluctuates from 0.93% to 1.32% for the EURUSD strategy and from

1.08% to 1.66% for the NZDMXN strategy, which can be explained since the method does not target a

constant VaR but a constant CVaR and accounts for the entire tail risk rather than simply the 5% quantile.

Also, Figures 3 and 4 show that the leverage adjustment factors vary much more abruptly for the volatility

based technique compared to the EVT based technique. This means that the first method is more responsive

to changes in VaR levels but would also incur higher transaction costs due to the frequent rebalancing. The

leverage factor is usually lower for the EVT based technique, due to the use of EVT for tail risk computation

which typically results in higher tail risk estimates than when relying on volatility.Over the 10 year period, the realised 95% VaR when using the volatility based technique is almost

exactly at the targeted level, being 1.50% and 1.52% for the EURUSD and NZDMXN strategy. For the EVT

based strategy, the VaR is lower at 1.33% and 1.38% respectively. However, the 95% CVaR levels when

using the volatility based strategy are 2.10% and 2.07% which is higher than the CVaR corresponding to

the 1.5% VaR target for a normal distribution; indeed, from Equation (3), the 95% CVaR corresponding to

a 95% VaR of 1.5% for a normal distribution is 1.89%, which serves as target CVaR for the EVT based

algorithm. This target CVaR level is approximately equal to the overall CVaR over the 10 year period for

the EVT based technique which yields a CVaR of 1.94% for both strategies. Thus, we have the confirmation

that the EVT based algorithm adjusts the leverage factor to reach a CVaR target whereas the volatility based

algorithm simply focuses on maintaining the VaR at its chosen value without accounting for the changes in

tail risk beyond the VaR threshold. In practice, controlling the entire left tail is preferable and the EVT basedmethod would be considered superior to its volatility based counterpart. Additionally, these gains in tail risk

control do not come at the expense of performance as the Sharpe ratios for the tail control techniques are

slightly higher than for the original strategy.

While the previous methods allow to stabilise tail risk at a set level, they do not have a direct effect on

the maximum drawdown sustained by the strategy each year. This is the objective of the drawdown control

technique which adjust the leverage factor to target a 10% CDaR at a 95% confidence level computed over

a 3 months period, the aim being to keep the maximum drawdown for each year around or below 10%.

The CDaR based technique reaches this objective as maximum drawdowns are in a 6.40% to 10.95% range

for the EURUSD strategy and a 5.55% to 10.86% range for the NZDMXN strategy whereas the maximum

16


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5 CONCLUSION

drawdowns for the original strategies fluctuated from 5.52% to 15.25% and from 7.21% to 17.40% respec-

tively. This demonstrates the ability to control maximum drawdown by using the CDaR based algorithm.The evolution of the leverage factor for the CDaR based algorithm is quite smooth, making it less likely to

suffer from high transaction costs when implemented in practice. Once again, the Sharpe ratio for the CDaR

based technique is slightly higher than for the original strategies.

5 Conclusion

A number of money management techniques were presented, with the aim of controlling either tail risk or

drawdown rather than attempting to maximise return or expected utility at any cost as is the case for most

money management techniques available in the existing literature. Indeed, the main concern of investment

professionals is to remain at or below certain risk constraints set either internally or by investors; as such,

maximising expected utility is only secondary to controlling risk as a breach of these risk limits would

usually trigger significant redemptions or would lead the investment manager to stop trading the strategy

altogether.

The first two methods aim to maintain a stable level of tail risk through time, using either historical

volatility or Extreme Value Theory to measure tail risk. Both methods were applied to two sets of daily

returns generated by applying a typical trend following strategy to the EURUSD and NZDMXN currency

pairs over a 10 year period, and demonstrated the ability to target a given VaR level for the volatility based

technique or a given CVaR level for the EVT based technique. The EVT based technique, which considers

the entire left tail of the return distribution at a given confidence level, is superior to the volatility based

technique which is oblivious to the size of losses beyond the VaR threshold and therefore can result in a

higher overall tail risk than intended.

The third method focuses on drawdown control, by adjusting the leverage factor based on the Condi-

tional Drawdown at Risk level generated by the strategy. The CDaR is computed by considering overlapping

blocks of consecutive returns and calculating the maximum drawdown for each block, yielding a drawdown

distribution from which the average expected drawdown beyond a certain confidence level (CDaR) can be

obtained. Considering the drawdown distribution rather than the maximum drawdown over the entire pe-

riod results in a more stable and robust estimate of potential drawdown. The drawdown control technique

achieves its objective when applied to the two strategies as the maximum drawdown for each year remains

around or below the targeted level.

17


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5 CONCLUSION

Return(%)

Volatility(%)

Max.

Drawdown(%)

Strategy

Orig.

Vol.

EVT

CDaR

Orig.

Vol.

EVT

CDaR

Orig

.

Vol.

EVT

CDaR

Year1

2.28

3.1

9

0.5

6

1.9

4

10.96

15.2

6

12.0

2

9.7

5

8.08

11.4

9

9.4

7

6.8

8

Year2

25.18

43.9

0

32.2

8

32.4

2

10.83

15.7

6

13.2

7

12.7

6

6.00

8.3

4

7.7

4

6.7

4

Year3

1.48

1.0

0

-0.3

8

5.6

1

11.99

15.4

0

15.1

6

11.0

6

15.25

18.9

7

18.4

4

10.8

2

Year4

10.19

16.7

6

9.1

2

8.4

5

10.06

14.3

9

14.2

1

11.1

7

6.46

10.2

7

12.5

9

10.0

5

Year5

3.24

7.2

7

6.0

9

3.3

6

9.75

15.2

2

15.1

0

12.7

9

5.52

8.6

3

7.9

1

6.9

0

Year6

2.80

5.6

6

2.1

7

1.7

3

8.16

15.3

7

11.0

2

9.7

6

8.59

16.9

4

12.0

2

10.5

0

Year7

6.87

15.7

1

12.5

5

10.1

8

8.00

15.8

7

13.9

6

11.3

4

3.65

10.1

0

8.3

1

6.4

0

Year8

26.45

35.7

4

30.9

1

27.6

8

14.67

15.8

9

12.2

8

11.7

2

13.19

9.9

6

8.1

5

7.7

4

Year9

-0.51

0.0

4

2.3

2

1.0

8

12.64

14.3

4

13.4

8

9.8

9

11.60

12.1

9

11.8

4

9.0

9

Year10

12.43

19.7

1

16.9

4

15.9

2

12.73

15.1

4

14.2

1

12.5

3

12.35

12.4

5

13.2

6

10.9

5

Year110

8.84

14.3

6

10.9

9

10.6

5

11.14

15.2

8

13.5

4

11.3

5

15.25

18.9

7

18.4

4

10.9

5

Table1:PerformancedatafortheEURUSDorig

inalstrategyandthevolatilityand

EVTbasedstrategies.

18


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5 CONCLUSION

95%VaR(%)

95%CVaR(%)

SharpeRatio

Strategy

O

rig.

Vol.

EVT

CDaR

Orig

.

Vol.

EVT

CDaR

Orig.

Vol.

EVT

CDaR

Year1

1

.04

1.5

0

1.1

4

0.9

6

1.47

2.0

9

1.6

9

1.3

0

0.2

1

0.2

1

0.0

5

0.2

0

Year2

1

.03

1.3

9

1.2

8

1.2

1

1.48

2.1

0

1.8

5

1.7

7

2.3

3

2.7

9

2.4

3

2.5

4

Year3

1

.23

1.6

4

1.5

7

1.1

3

1.52

1.9

5

1.9

7

1.4

6

0.1

2

0.0

7

-0.0

2

0.5

1

Year4

1

.01

1.3

2

1.4

6

1.1

7

1.21

1.6

7

1.8

2

1.4

2

1.0

1

1.1

7

0.6

4

0.7

6

Year5

1

.02

1.5

4

1.5

4

1.2

5

1.39

2.0

7

2.1

4

1.8

6

0.3

3

0.4

8

0.4

0

0.2

6

Year6

0

.69

1.4

0

0.9

3

0.8

1

1.14

2.1

1

1.5

7

1.3

9

0.3

4

0.3

7

0.2

0

0.1

8

Year7

0

.80

1.6

2

1.3

2

1.1

6

1.29

2.4

4

2.2

7

1.8

4

0.8

6

0.9

9

0.9

0

0.9

0

Year8

1

.39

1.6

6

1.1

6

1.1

7

2.28

2.1

7

1.7

1

1.6

0

1.8

0

2.2

5

2.5

2

2.3

6

Year9

1

.32

1.4

9

1.2

8

0.9

6

1.70

1.9

7

1.8

9

1.3

8

-0.0

4

0.0

0

0.1

7

0.1

1

Year10

1

.09

1.4

2

1.3

1

1.2

0

1.77

2.1

2

2.0

1

1.7

5

0.9

8

1.3

0

1.1

9

1.2

7

Year110

1

.08

1.5

0

1.3

3

1.1

1

1.58

2.1

0

1.9

4

1.6

1

0.7

9

0.9

4

0.8

1

0.9

4

Table2:PerformancedatafortheEURUSDorig

inalstrategyandthevolatilityand

EVTbasedstrategies.

19


20/27

5 CONCLUSION

Return(%)

Volatility(%)

Max.

Drawdown(%)

Strategy

Orig.

Vol.

EVT

CDaR

Orig.

Vol.

EVT

CDaR

Orig.

Vol.

EVT

CDaR

Year1

-5.40

-3.4

2

-3.0

8

-3.1

3

13.3

5

14.8

2

11.8

9

9.3

9

14.0

4

13.9

5

12.0

1

9.8

0

Year2

43.16

49.2

1

48.8

1

32.6

8

15.2

3

15.9

0

16.6

7

12.8

2

8.0

7

8.0

3

8.6

0

8.1

3

Year3

2.72

-0.6

8

0.1

2

-0.4

1

14.8

5

15.6

3

15.2

4

12.4

2

10.4

5

13.1

7

12.1

9

9.7

3

Year4

-10.06

-9.1

6

-10.8

7

-7.5

9

12.1

7

14.1

2

13.1

9

7.6

6

14.3

8

14.7

4

16.5

1

10.2

9

Year5

-0.93

-1.7

1

-2.6

0

-1.4

0

11.2

5

15.7

8

15.3

7

9.0

1

13.3

5

17.1

7

17.2

0

10.8

6

Year6

4.62

6.7

2

4.8

4

2.8

2

12.6

9

15.2

2

14.6

0

9.9

6

7.2

1

8.2

6

7.9

6

5.5

5

Year7

5.57

6.3

0

7.9

0

4.8

3

12.7

6

15.7

9

14.5

4

10.4

9

13.5

9

15.5

2

14.6

1

10.5

6

Year8

11.08

16.5

2

16.5

8

9.0

2

18.1

1

16.3

2

17.3

1

10.2

4

14.8

6

8.4

5

10.3

1

6.7

1

Year9

-3.86

1.5

4

-5.6

4

-2.9

4

17.3

9

14.6

2

9.7

3

5.7

9

17.4

0

12.9

6

10.9

5

6.8

3

Year10

-1.67

-3.0

4

-3.1

8

-1.4

2

10.5

6

15.4

7

12.2

8

7.2

9

8.8

2

13.1

5

11.4

1

6.6

7

Year110

3.44

5.0

4

4.0

7

2.6

0

14.0

3

15.3

7

14.2

6

9.7

3

23.5

1

24.2

9

24.7

9

18.2

5

Table3:Perfor

mancedatafortheNZDMXNoriginalstrategyandthevolatilityandEVTbasedstrategies.

20


21/27


22/27

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About the author:

DR ISSAM STRUB: Dr Strub is a senior member of the Cambridge Strategy research group where he

works on quantitative strategies as well as asset allocation and risk management tools; he has authored a

number of research articles in financial and scientific journals and has been an invited speaker at financial

conferences and roundtables. Prior to joining the Cambridge Strategy, Dr Strub was a graduate student at the

University of California, Berkeley, where he conducted research in an array of fields ranging from Partial

Differential Equations and Fluid Mechanics to Scientific Computing and Optimisation; he obtained a Ph.D.

in Engineering from the University of California in 2009.

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