trade sizing techniques for drawdown and tail risk control

Electronic copy available at: http://ssrn.com/abstract=2063848

Trade sizing techniques for drawdown and tail risk control

Issam S. STRUB ∗

The Cambridge Strategy (Asset Management) Ltd

Abstract

This article introduces three algorithms for trade sizing with the objective of controlling tail risk ormaximum drawdown when applied to a trading strategy. The first algorithm relies on historical volatilityestimates while the second uses tail risk estimates obtained by applying Extreme Value Theory (EVT) toestimate Conditional Value at Risk (CVaR); the third algorithm also uses Extreme Value Theory appliedto the drawdown distribution to compute the Conditional Drawdown at Risk (CDaR). These algorithmsare applied to 10 years of daily returns from a trend following strategy trading the EURUSD and NZD-MXN currency pairs. In each case, the performance of the algorithms is analysed in detail and comparedto the original strategy. The ability of these algorithms in terms of tail risk and drawdown control is eval-uated. The techniques presented in the article are readily applicable by investment managers to computeadequate trade size while maintaining a constant level of tail risk or limiting maximum drawdown to achosen value.

1 Introduction

Money management, also called position or trade sizing, consists in selecting an appropriate leverage levelto be applied to a given strategy, where the leverage level is defined as the ratio of the position market valuewith respect to the total assets under management (AUM) or account size. At a glance, it seems quite obvi-ous that the ability to dynamically adjust position size can result in increased profits; indeed, an investmentmanager able to reduce (respectively increase) leverage before periods of underperformance (respectivelyoutperformance) would obtain higher returns when compared to using a constant leverage. Practical evi-dence confirms that most traders tend to dynamically adjust their leverage level , usually relying on heuristicrules and being subject to behavioural biases, as demonstrated in LOCKE and MANN (2003), THALER andJOHNSON (1990). However, while a significant literature exists dealing with trading strategies profitabil-ity, there are fewer articles on money management and the limited literature on this topic often presentstechniques which are not always applicable in practice by traders. Additionally, these money managementtechniques rarely make use of modern econometrics and risk management tools such as Extreme ValueTheory or time series analysis.

Historically, money management has been applied to gambling as well as trading, typically with theaim of maximising growth rate. Indeed, in KELLY (1956), information theory and expected utility functiontheory (introduced in BERNOULLI (1738, 1954) in relation to the St. Petersburg paradox) were combined

∗Research Scientist, [email protected]

1


1 INTRODUCTION

to obtain an optimal gambling strategy, the Kelly criterion: maximise the expected value of the logarithmof the gambler’s 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 BREIMAN (1961) in the case wherestock returns are assumed to be independent, identically distributed (i.i.d.). These results were applied toinvestment management in LATANE (1959), who advised investors to maximise the geometric mean of theirportfolios. Later, the optimality of maximising expected log return was extended with no restrictions on thedistribution of the market process in ALGOET and COVER (1988) and, in BROWNE and WHITT (1996), theKelly criterion was generalised to the case in which the underlying stochastic process is a simple randomwalk in a random environment; OSORIO (2009) devised an analog to the Kelly criterion for fat tail returnsmodelled by a Student t-distribution and a log prospect rather than utility function.

In HAKANSSON (1970), closed form optimal consumption, investment and borrowing strategies wereobtained for a class of utility functions corresponding to an investor looking to maximise the expected utilityfrom consumption over time given an initial capital position and a known deterministic non-capital incomestream. ROLL (1973) studied the implications of growth optimum portfolios in terms of observed stockreturns for investors selecting such a portfolio. THORP (1971) applied the Kelly criterion of maximisinglogarithmic utility to portfolio choice and compared it to mean variance portfolio theory, concluding that theKelly criterion does not always yield mean variance efficient portfolios.

SAMUELSON (1971, 1979) showed that while maximising the geometric mean utility at each stage maybe 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 significantdrawdowns. Later on, more work on the properties of the Kelly criterion and its application to finance waspublished in ROTANDO and THORP (1992); MACLEAN et al. (2004, 2010, 2011a,b); THORP (2006). Theconcept 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, theoptimal f and other approaches; see for example GEHM (1983); BALSARA (1992); GEHM (1995); JONES

(1999); STRIDSMAN (2003); MCDOWELL (2008). Finally, the optimal f technique was applied to futurestrading and compared to more naive approaches in ANDERSON and FAFF (2004); LAJBCYGIER and LIM

(2007), each time with the conclusion that it resulted in leverage levels that would be unacceptable to mostinvestors; this leads to heuristic approaches such as using a fraction of the Kelly ratio (such as half Kellyratio).

The common feature of traditional money management techniques such as the Kelly criterion or theoptimal f is their focus on maximising wealth growth, whereas in practice, both individual traders and fundmanagers 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 isusually 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 utilitymaximisation had already been noted in ROY (1952) before the introduction of the Kelly criterion as theauthor 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 ofpractical significance and applicability in our results. A man who seeks advice about his actions will not begrateful 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 variedmarket 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 todaily returns resulting from implementing a simple technical trading rule on the EURUSD and NZDMXNcurrency pairs; a detailed analysis and comparison of the performance of each money management techniqueconcludes 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 willtypically lead to changes in the volatility of the strategy returns. In practice, portfolio managers aim to limitthese variations and keep the tail risk of the strategy below a predetermined level by dynamically adjustingtrade 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) (BEDER (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 andselecting the quantile corresponding to the confidence level chosen (for example 95%). Unfortunately, VaRis concerned only with the number of losses that exceed the VaR confidence level and not the magnitudeof these losses; to obtain a more complete measure of large losses, one needs to examine the entire shapeof the left tail of the return distribution beyond the VaR threshold, which leads to the Conditional Value atRisk (CVaR) also referred to as Expected Shortfall, Tail VaR or Mean Shortfall (ARTZNER et al. (1999);CHRISTOFFERSEN (2003); HARMANTZIS et al. (2006); MCNEIL et al. (2005)). CVaR can be defined asthe average expected loss at a given confidence level; for example, at the 95% confidence level, the CVaRrepresents the average expected loss on the worst 5 days out of 100 whereas the VaR is the minimum losson those days. In mathematical terms, the CVaR for a daily return distribution F at a confidence level α isgiven by:

CVaRα = −E{X|X 6 −VaRα} (1)

where the VaR is defined by:VaRα = −F−1(1− α) (2)

Computing CVaR requires an explicit expression of the portfolio return distribution function F whichis usually unknown in practice. However, if historical daily returns are assumed to follow a normal (orGaussian) distribution, VaR and CVaR can be easily obtained from the standard deviation σ and mean µ ofreturns; for example, at the 95% level, standard deviation, VaR and CVaR are related by:

VaR ≃ 1.65× σ − µ and CVaR ≃ 2.07× σ − µ (3)

3

2.1 Tail Risk Measures 2 TAIL RISK CONTROL

−50 −45 −40 −35 −30 −25 −20 −15 −7.8−5 00123456789

101112131415

Daily Return (%)

Num

ber

of O

bser

vatio

ns

95% VaR

Generalised Pareto Distribution

Normal Distribution

Figure 1: Top: Comparison of Generalised Pareto and normal distribution. Note that the Generalised ParetoDistribution 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 usinga 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 bythe strategy, converting this volatility number to a VaR number using the above formula (3) and adjustingleverage in order to match the target VaR level. The historical volatility of the strategy can be computedusing the RiskMetrics exponentially weighted moving average introduced in ZANGARI (1996):

σ =

√√√√(1− λ)T∑t=1

λt−1(rt − r)2 (4)

where T is the length of the estimation window, λ the decay factor and r the mean return over the estimationwindow.

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 onthe 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 measuredusing tools originating from Extreme Value Theory (EVT), a branch of statistics dedicated to modellingextreme events introduced in BALKEMA and DE HAAN (1974); PICKANDS (1975). The central result inExtreme Value Theory states that the extreme tail of a wide range of distributions can be approximatelydescribed by the Generalised Pareto Distribution (GPD) with shape and scale parameters ξ and β:

Gξ,β(y) =

{1− (1 + ξy

β )− 1

ξ , for ξ = 0;

1− exp− y

β , for ξ = 0.(6)

where β > 0, and the support of Gξ,β is y > 0 when ξ > 0 and 0 6 y 6 −βξ when ξ < 0.

The shape and scale parameters ξ and β can be estimated using Maximum Likelihood Estimation (MLE)by fitting a GPD distribution to the tail of the return distribution after a given threshold u. Once this is done,the CVaR can be computed:

CVaRα =VaRα + β − ξu

1− ξ(7)

where the VaR for a GPD can be estimated by:

VaRα = u+β

ξ

((αN

Nu

)−ξ

− 1

)(8)

5

2.3 Extreme Value Theory based Position Sizing 2 TAIL RISK CONTROL

with N the 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, westart by filtering the daily returns and then apply Extreme Value Theory to the standardised residuals (seeMCNEIL and FREY (2000); NYSTROM and SKOGLUND (2005)), with a Generalised Pareto Distributionbeing fitted to the tails through Maximum Likelihood Estimation. Once this is done, we obtain the shape andscale parameters and replace these values in Equation (7) to compute the CVaR at the required confidencelevel. Extreme Value Theory has been used during the previous decade for risk management in finance witha notable increase in the number of publications on the subject since the recent financial crisis. We referto BALI (2003); BEIRLANT et al. (2004); CASCON and SHADWICK (2009); COLES (2001); DE HAAN andFERREIRA (2006); EMBRECHTS (2011); GHORBEL and TRABELSI (2008, 2009); GOLDBERG et al. (2008,2009); GUMBEL (2004); HUANG et al. (2012); LONGIN (2000); MCNEIL and FREY (2000); NYSTROM andSKOGLUND (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 EVTapproaches is illustrated in Figure 1. We consider 1000 daily returns for a stock and fit both a normaldistribution and a Generalised Pareto Distribution to the left tail of the daily returns. We can see that whileboth 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 whenusing volatility and a normal distribution assumption. Note that this is a pathological case which was chosenon purpose as the difference between the two methods is readily apparent. Still, relying on volatility andnormal distribution assumptions can lead to significantly underestimating the tail risk generated by a givenstrategy, 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 tothe left tail of the strategy returns; in practice, if 250 days are considered and the 95% confidence levelis desired, this means that the GPD has to be fitted to about 12 daily returns, a number which is typicallytoo low to guarantee convergence of the Maximum Likelihood Estimation method and which will cause ahigh 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 Carlosimulations (METROPOLIS and ULAM (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 tailrisk. Therefore, some form of historical simulation is highly preferable as it makes no assumption on thereturn distribution, instead relying on the past returns. However, as noted in PRITSKER (2006), such amethod presents two potential issues.

First, the required sample size to obtain a statistically significant distribution is usually considered to beat least 250 days; this, in turn, raises the potential issue of not being sensitive enough to recent returns whichpresumably contain the most relevant information to predict future returns. To circumvent this problem,the weighted historical simulation (WHS) method was developed in BOUDOUKH et al. (1998); this methodassigns 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 howto 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 volatilityof returns evolves through time and that periods of high and low volatility do not occur at randomly spacedintervals but rather tend to be clustered together. The filtered historical simulation (FHS) method presentedin BARONE-ADESI et al. (1999) is an attempt to combine the advantages of the historical and parametricmethods; the variance-covariance method attempts to capture conditional heteroskedasticity but assumes anormal distribution while the historical method does not assume a specific distribution but does not captureconditional 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 σt.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 sequenceof i.i.d. observations, we fit an AR(1) first order autoregressive model to the daily returns:

Rt+1 = c+ aRt + εt where εt = σtzt (9)

where we choose the standardised returns {zt} as following a Student’s t-distribution rather than a normalone to account for increased tail risk as the t-distribution has fatter tails.

To model the variation of the returns standard deviation, we can use a GARCH type model (BOLLER-SLEV (1986); ENGLE (1982, 2001); TAYLOR (1986)) such as the GARCH(1,1):

σ2t+1 = ω + αε2t + βσ2

t , with α+ β < 1 (10)

Alternately, the GARCH model can be replaced by its extension, the exponential GARCH (EGARCH)model developed in NELSON (1991); NELSON and CAO (1992) to capture the asymmetry in volatility in-duced by large positive and negative returns. Indeed, volatility usually increases more after a large drop thanafter a large increase due to the leverage effect (BLACK (1976)). This model is defined by:

lnσ2t+1 = ω + α(ϕεt + γ(|εt| − E|εt|)) + β lnσ2

t (11)

Once an AR(1)/GARCH(1,1) model has been fitted to the daily returns, the autocorrelation of thesquared returns is usually noticeably lower and these observations can now be used in a historical simu-lation method. The i.i.d. property is important for bootstrapping, as it allows the sampling procedure tosafely avoid the pitfalls of sampling from a population in which successive observations are serially depen-dent. We simulate a number of independent random trials (10,000 in this article) over a time horizon of 252trading days; unlike Monte Carlo simulations we do not make a specific distributional assumption regardingthe standardised returns {zt} and instead use the past returns data. Given a sequence of past returns we cancompute past standardised returns from observed returns and estimated standard deviations as the quotient

7

3 DRAWDOWN CONTROL

of the residual of the AR(1) model over the standard deviation. Once the historical standardised returns areknown, 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 togenerate a distribution of 2,520,000 daily returns to which left tail a GPD is fitted, eventually yielding theCVaR. The high number of residuals ensures the stability of the method, as the left tail contains 126,000returns for a 95% confidence level, which almost guarantees the convergence of the Maximum LikelihoodEstimation algorithm used to fit the GPD to the left tail of the simulated return distribution. This CVaRnumber can be converted into a VaR number under normal distribution assumptions using Equation (3) andtrade size adjusted through Equation (5). One of the advantages of using Extreme Value Theory to computethe CVaR is that the tail risk of the return distribution is measured much more accurately and less likely tobe 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 orCVaR at a given confidence level, the most adverse event from an investor or investment manager standpointis probably a significant drawdown in which a number of negative daily returns are clustered together overa given period time. Indeed, most investors have strict drawdown limits (such as 20%) upon which they willredeem part or the entirety of their investment in a given fund. Therefore, for a money manager, experiencinga 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 preventthem from collecting performance fees during a drawdown. Also, a manager trading a systematic strategywith proprietary or investor capital is likely to unnecessarily modify or discontinue the strategy if faced withan unacceptable drawdown; this can result in the loss of future performance as the changes may have beenunwarranted. This leads us to develop a money management technique to control the maximum drawdownencountered by a given strategy. Earlier work on drawdown control through portfolio optimisation can befound in CVITANIC and KARATZAS (1995); GROSSMAN and ZHOU (1993).

3.1 Drawdown Measures

The maximum drawdown experienced over a given period of time is defined as the largest peak to trough lossin Net Asset Value of a portfolio. If W (t) represents the portfolio value at time t, the maximum drawdownover a time interval [0, T ] is defined by:

MDD(T ) = max06t6T

( max06τ6t

W (τ)−W (t)) (12)

The historical maximum drawdown is a number which varies widely even for strategies presenting thesame mean and volatility and is based on the entire track record making difficult any comparison betweenstrategies run over different time lengths. Therefore, as noted in HARDING et al. (2003), considering adrawdown distribution with reference to a confidence level would be more practical. The distribution ofdrawdowns over a given time period of N days can be computed by computing the maximum drawdownfor blocks of N consecutive days from the track record of a strategy. As VaR and CVaR were definedfor 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 overlappingblocks 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 wouldbe the average expected drawdown beyond the 95th percentile. The modelling of the drawdown distributionhas been considered in CHEKHLOV et al. (2003, 2005); JOHANSEN and SORNETTE (2001); MENDES andBRANDI (2004); MENDES and LEAL (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 oneof 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 190drawdowns for each one of the 10,000 return series. The drawdowns are aggregated to generate a distributionof 1,900,000 drawdowns and a GPD is fitted to the right tail of this distribution containing the 5% largestdrawdowns 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 previoussections, we implement them on the daily returns generated by a systematic strategy applied to the EURUSDand NZDMXN currency pairs.

4.1 Trading Strategy

The trading strategy used in this article is a typical breakout trend following strategy, similar to strategiescommonly 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 previousday 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 andNZDMXN currency pairs were selected as they demonstrate different return profiles with NZDMXN beingtypically more volatile than EURUSD; also, the strategy performance is significantly higher for EURUSDthan for NZDMXN , which gives us the opportunity to apply the money management algorithms in differentsettings. Indeed, looking at Tables 1 to 4 which summarise the performance for the original strategy as wellas the money management techniques, we can see that the Sharpe ratio is 0.79 for the EURUSD strategyand 0.25 for the NZDMXN strategy. The maximum drawdown is also higher for the NZDMXN strategy at23.51% compared to 15.25% for the EURUSD strategy.

9

4.1 Trading Strategy 4 APPLICATIONS

0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

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


0 1 2 3 4 5 6 7 8 9 10100

150

200

250

300

350

400

Time (Years)

Net

Ass

et V

alue

(B

ase

100)

Original ReturnsVolatility based trade sizingEVT 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

3

3.5

Time (Years)

Leve

rage

Adj

ustm

ent F

acto

r

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 andEVT based position sizing methods. Bottom: Evolution of the leverage adjustment factor for the volatilityand EVT based position sizing methods applied to the EURUSD strategy.

11


0 1 2 3 4 5 6 7 8 9 1090

100

110

120

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190

Time (Years)

Net

Ass

et V

alue

(B

ase

100)

Original ReturnsVolatility based trade sizingEVT 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)

Leve

rage

Adj

ustm

ent F

acto

r

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 andEVT based position sizing methods. Bottom: Evolution of the leverage adjustment factor for the volatilityand EVT based position sizing methods applied to the NZDMXN strategy.

12


0 5 10 150

20

40

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200

63 Day Drawdowns (%)

Fre

quen

cy

0 5 10 15 200

20

40

60

80

100

120

63 Day Drawdowns (%)

Fre

quen

cy

Figure 5: Top: Distribution of 63 day drawdowns for the EURUSD strategy. Bottom: Distribution of 63 daydrawdowns 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 andNZDMXN strategy with the objective of maintaining a constant tail risk level over time set at a 95% VaR of1.5%. For the volatility based technique, the historical volatility is computed at the end of each week usingthe RiskMetrics exponentially weighted moving average presented in Equation (4) and transformed into aVaR level resulting in a leverage adjustment coefficient which is applied to the strategy over the followingweek. The typical parameters, recommended in ZANGARI (1996) are used: λ = 0.94, T = 74 days;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 dailyreturns 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 standardisedresiduals 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 andgenerates after aggregation of the 10,000 series of 252 daily returns, one series of 2,520,000 daily returns towhich left tail beyond the 5% threshold a GDP distribution is fitted. From the shape and scale parametersof 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 ourtarget VaR level of 1.5% if the distribution was following a normal distribution. This means that if in factthe return distribution has a thicker left tail than a normal distribution, this will be taken into account as the95% 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 basedalgorithm:

1. At the end of week N , 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 week N + 1.

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

For the EVT based algorithm:

1. At the end of week N , 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 oneseries 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 returnseries, yielding the shape and scale parameters.

4. Compute the 95% CVaR corresponding to the GPD parameters values and convert the CVaR into aVaR 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 oflimiting the maximum drawdown over each year to a set value, in this case chosen as 10%. Similarly to theEVT based technique for tail risk control, we start by filtering the previous 252 days and generating 10,000series of 252 daily returns each using FHS. These daily returns are decomposed in blocks of 63 (whichrepresents about 3 months) consecutive days from day 1 to day 63, day 2 to day 64, etc; a block length of 3month 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 iscomputed for each block yielding a distribution of 190 drawdowns for each of the 10,000 series; thesedrawdowns are aggregated to yield one series of 1,900,000 drawdowns to which right tail a GPD distributionis fitted, yielding the 95% CDaR from which the leverage factor is computed. The interest of using EVTto estimate the CDaR is apparent from Figure 5, which shows the 63 day drawdown distribution for eachstrategy; both drawdown distributions present a right tail which is significantly longer than the left tail andwhich would not be measured accurately with a normal distribution; therefore, it is crucial to fit a GPD tothe 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 of week N , 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 seriesof 1,900,000 drawdowns.

5. Using MLE, fit a GPD distribution to the right tail (5% largest drawdowns) of the simulated returnseries, yielding the shape and scale parameters.

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

15

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 andthe drawdown control technique are summarised in Tables 1 and 2 for the EURUSD strategy and Tables 3and 4 for the NZDMXN strategy.

The effectiveness of the tail risk control techniques can be evaluated by looking at the fluctuations ofthe 95% Var when the techniques are applied. For the original strategy, the 95% VaR varies widely goingfrom 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 arange of 1.32% to 1.66% for the EURUSD strategy and 1.32% to 1.68% for the NZDMXN strategy, therebydemonstrating the ability of this technique to stabilise the 95% VaR around its target value of 1.5%. For theEVT based technique, the 95% VaR fluctuates from 0.93% to 1.32% for the EURUSD strategy and from1.08% to 1.66% for the NZDMXN strategy, which can be explained since the method does not target aconstant 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 volatilitybased technique compared to the EVT based technique. This means that the first method is more responsiveto changes in VaR levels but would also incur higher transaction costs due to the frequent rebalancing. Theleverage factor is usually lower for the EVT based technique, due to the use of EVT for tail risk computationwhich 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 almostexactly at the targeted level, being 1.50% and 1.52% for the EURUSD and NZDMXN strategy. For the EVTbased strategy, the VaR is lower at 1.33% and 1.38% respectively. However, the 95% CVaR levels whenusing the volatility based strategy are 2.10% and 2.07% which is higher than the CVaR corresponding tothe 1.5% VaR target for a normal distribution; indeed, from Equation (3), the 95% CVaR corresponding toa 95% VaR of 1.5% for a normal distribution is 1.89%, which serves as target CVaR for the EVT basedalgorithm. This target CVaR level is approximately equal to the overall CVaR over the 10 year period forthe EVT based technique which yields a CVaR of 1.94% for both strategies. Thus, we have the confirmationthat the EVT based algorithm adjusts the leverage factor to reach a CVaR target whereas the volatility basedalgorithm simply focuses on maintaining the VaR at its chosen value without accounting for the changes intail 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 riskcontrol do not come at the expense of performance as the Sharpe ratios for the tail control techniques areslightly 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 onthe maximum drawdown sustained by the strategy each year. This is the objective of the drawdown controltechnique which adjust the leverage factor to target a 10% CDaR at a 95% confidence level computed overa 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% rangefor the EURUSD strategy and a 5.55% to 10.86% range for the NZDMXN strategy whereas the maximum

16

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 tosuffer from high transaction costs when implemented in practice. Once again, the Sharpe ratio for the CDaRbased 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 ordrawdown rather than attempting to maximise return or expected utility at any cost as is the case for mostmoney management techniques available in the existing literature. Indeed, the main concern of investmentprofessionals 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 wouldusually trigger significant redemptions or would lead the investment manager to stop trading the strategyaltogether.

The first two methods aim to maintain a stable level of tail risk through time, using either historicalvolatility or Extreme Value Theory to measure tail risk. Both methods were applied to two sets of dailyreturns generated by applying a typical trend following strategy to the EURUSD and NZDMXN currencypairs over a 10 year period, and demonstrated the ability to target a given VaR level for the volatility basedtechnique or a given CVaR level for the EVT based technique. The EVT based technique, which considersthe entire left tail of the return distribution at a given confidence level, is superior to the volatility basedtechnique which is oblivious to the size of losses beyond the VaR threshold and therefore can result in ahigher 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 overlappingblocks of consecutive returns and calculating the maximum drawdown for each block, yielding a drawdowndistribution from which the average expected drawdown beyond a certain confidence level (CDaR) can beobtained. 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 techniqueachieves its objective when applied to the two strategies as the maximum drawdown for each year remainsaround or below the targeted level.

17

5 CONCLUSION

Ret

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data

fort

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orig

inal

stra

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18

5 CONCLUSION

95%

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19

5 CONCLUSION

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20

5 CONCLUSION

95%

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(%)

95%

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aR(%

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27

Tabl

e4:

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data

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.

21

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About the author:

DR ISSAM STRUB: Dr Strub is a senior member of the Cambridge Strategy research group where heworks on quantitative strategies as well as asset allocation and risk management tools; he has authored anumber of research articles in financial and scientific journals and has been an invited speaker at financialconferences and roundtables. Prior to joining the Cambridge Strategy, Dr Strub was a graduate student at theUniversity of California, Berkeley, where he conducted research in an array of fields ranging from PartialDifferential Equations and Fluid Mechanics to Scientific Computing and Optimisation; he obtained a Ph.D.in Engineering from the University of California in 2009.

Disclaimer: Some services are not available to private inexperienced investors. Services may also not be available to certain investors due to regulatory or other constraints either in the UK orelsewhere. The information contained herein is not targeted at the residents of any particular country and is not intended for distribution to, or use by, any person in any country or jurisdiction wheresuch distribution or use would be contrary to local law or regulatory requirements. Information contained herein may be subject to change without notice. Investments in securities for financialinstruments (which include contracts for differences, futures, options spot and forward foreign exchange and off-exchange contracts) can fluctuate in value and you should be aware that you maynot realise the initial amount invested and may incur additional liabilities. As investments in securities or financial instruments may entail above average risks you should carefully consider whetheryour financial circumstances permit you to invest and if necessary seek the advice of an independent financial adviser. Foreign exchange denominated securities and financial instruments are subjectto fluctuations in exchange rates that may have a positive or negative effect on the value, price or income derived from the securities or financial instrument concerned. Past performance is not areliable indicator of future performance. You are advised that The Cambridge Strategy (Asset Management) Limited is unable to provide advice as to tax consequences of a particular investmentor investment strategy and you are advised to seek professional advice in this respect. This document is issued by the Cambridge Strategy (Asset Management) Limited and is not a solicitationor instruction to invest. It is provided for information purposes only. The Cambridge Strategy (Asset Management) Limited is authorised and regulated by the Financial Services Authority andregistered with the SEC (US) and the SFA (HK). The Cambridge Strategy (Asset Management) Limited is exempt from the requirement to hold an Australian financial services licence under theCorporations Act 2001 (Cth) (Class Order 03/1099) in respect of the provision of financial services. It is regulated by the Financial Services Authority (FSA) under UK laws, which differ fromAustralian laws. Australian Investors: These materials are provided by a representative of The Cambridge Strategy (Asset Management) Limited (’Cambridge’) and is intended for wholesaleclients as defined in the Corporations Act 2001 (Cth). The information in this presentation is current unless stated otherwise and Cambridge is not under any obligation to update the informationto the extent that it is or becomes out of date or incorrect. It is confidential and has been prepared by Cambridge solely for use in connection with its Programmes. This information must not bemade available, published or distributed to any third party without the prior consent of Cambridge. This information has been prepared without taking into account anyone’s objectives, financialsituation or needs so before acting on it each person should consider its appropriateness to their circumstances before making any investment decision. In particular, a person should consider theProgramme’s investment objectives, risks, fees and other charges. Each person should carefully read and consider any offer documentation before making an investment decision. The information inthis presentation is indicative and may change with market fluctuations. It does not purport to be a comprehensive statement or description of any markets or securities referred to within. Cambridgeassumes no fiduciary responsibility or liability for any consequences financial or otherwise arising from any reliance on this information. Each person should make their own appraisal of the risksand should consult to the extent necessary, their own legal, financial, tax, accounting and other professional advisors in this respect to any investment in the Programmes. This document is not asolicitation or instruction to invest. It is provided for information purposes only.

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trade sizing techniques for drawdown and tail risk control

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