This article introduces three algorithms for trade sizing with the objective of controlling tail risk or maximum drawdown when applied to a trading strategy. (Page 1)
In each case, the performance of the algorithms is analysed in detail and compared to the original strategy. The ability of these algorithms in terms of tail risk and drawdown control is evaluated. The techniques presented in the article are readily applicable by investment managers to compute adequate trade size while maintaining a constant level of tail risk or limiting maximum drawdown to a chosen value. (Page 1)
Money management, also called position or trade sizing, consists in selecting an appropriate leverage level to be applied to a given strategy, where the leverage level is defined as the ratio of the position market value with respect to the total assets under management (AUM) or account size. (Page 1)
Indeed, in KELLY (1956), information theory and expected utility function theory (introduced in BERNOULLI (1738, 1954) in relation to the St. Petersburg paradox) were combined (Page 1)
to obtain an optimal gambling strategy, the Kelly criterion: maximise the expected value of the logarithm of the gambler’s wealth at each bet to achieve an asymptotically optimal growth rate; such a strategy also minimises the expected time to reach a given wealth as demonstrated in BREIMAN (1961) in the case where stock returns are assumed to be independent, identically distributed (i.i.d.). (Page 2)
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. (Page 2)
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 highlighted the risk involved in using the Kelly criterion, namely that of excessive leverage leading to significant drawdowns. (Page 2)
The common feature of traditional money management techniques such as the Kelly criterion or the optimal f is 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 maximum drawdown below a chosen threshold; otherwise, they will most likely suffer significant redemptions from investors or discontinue trading their strategy altogether. (Page 2)
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.” (Page 2)
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. (Page 3)
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 probability. (Page 3)
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 (ARTZNER et al. (1999); CHRISTOFFERSEN (2003); (Page 3)
CVaR can be defined as the average expected loss at a given confidence level; for example, at the 95% confidence level, the CVaR represents the average expected loss on the worst 5 days out of 100 whereas the VaR is the minimum loss on those days. (Page 3)
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 σ and mean µ of returns; for example, at the 95% level, standard deviation, VaR and CVaR are related by (Page 3)
The previous money management method relies on the assumption that daily strategy returns are normally distributed. 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 HAAN (1974); PICKANDS (1975). (Page 5)
Note that the preceding result requires that observations be independent and identically distributed, which is often not the case for daily returns as they present some level of autocorrelation. (Page 6)
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. (Page 6)
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; additionally, 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. (Page 8)
This leads us to develop a money management technique to control the maximum drawdown encountered by a given strategy. (Page 8)
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. (Page 8)
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. (Page 8)
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. (Page 9)
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: (Page 9)
We apply the the volatility and EVT based tail risk control techniques presented earlier to the different strategies with the objective of maintaining a constant tail risk level over time set at a 95% VaR of 1.5%. (Page 16)
The drawdown control technique is applied to the strategies 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. (Page 17)
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%, 1.52%, 1.48% and 1.56% for the different strategies. (Page 18)
For the EVT based strategy, the VaR is lower at 1.33%, 1.38%, 1.51% and 1.39% respectively. However, the 95% CVaR levels when using the volatility based strategy are 2.10%, 2.07%, 2.03% and 2.36% 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. (Page 18)
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 based method would be considered superior to its volatility based counterpart. (Page 18)
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 except for the 10 year future strategy for which they are slightly lower. (Page 18)
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%. (Page 19)
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. (Page 19)
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. (Page 19)
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 some typical systematic strategies applied to currencies and futures 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. (Page 19)
The third method focuses on drawdown control, by adjusting the leverage factor based on the Conditional 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. (Page 19)
Considering the drawdown distribution rather than the maximum drawdown over the entire period 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. (Page 19)
Additionally, these gains in tail risk and drawdown control do not come at the expense of performance as the Sharpe ratios for these trade sizing methods are slightly higher than for the original strategy except for the 10 year future strategy for which they are marginally (Page 19)