Thu, 23/02/2006 - 06:31
Using Monte Carlo analysis to explore the limitations of drawdown statistics in comparing active currency indices.
Downside risk measures have received renewed attention recently. The most frequently quoted statistic of that sort within the hedge funds world is probably maximum drawdown; that is, over a defined period, the largest decline of the investment having once reached a peak. In simple terms it quantifies the biggest cumulative loss an investor could have made by entering and exiting the investment at the worse possible time. Despite its practitioner's appeal, only a few academic articles have been written on the topic (Douady et al, 2000; Magdon-Ismail et al, 2004).
The analytical formula specifying the expected maximum drawdown is quite complex and as a consequence other authors have preferred using Monte-Carlo simulations (Burghardt et al, 2003; Harding et al, 2003). The expected maximum drawdown, as well as percentiles, can then be worked out as a function of a given investment mean standard deviation and period length. The goal of this article is to quantify the empirical maximum drawdown of a variety of active currency indices and then compare retrospectively to the values obtained from a normal distribution with corresponding mean and standard deviation. Interesting differences between currencies indices are then highlighted which in turn open up avenues for further research.
About active currency indices
Active currency indices can be very useful when trying to analyse the performance of foreign exchange managers. There are several such indices in existence including those provided by Stark, CISDM and Parker to mention just a few. Even though these indices possess relatively high correlation with one another over the period December 1989 to January 2005, Figure 1, differences are apparent especially with respect to the manager composition and the method by which the index returns are calculated. For example, the CISDM2 tracks over 40 currency programmes and compiles the returns on a median basis; whereas Stark3 uses a money-weighted approach and the Parker index4 includes 63 on an equallyweighted basis.
The differences between the indices are further highlighted when performance over the period in question is examined, Figure 2, this is true in terms of annualised mean returns and standard deviation. It is also apparent when examining the maximum drawdown statistics which range from -9.18% (Parker) through to -20.86% (Stark) and up to -28.69% (CISDM). Although one needs to question the extent to which these variations are due to the idiosyncrasies, in composition and constitution, of the individual indices. It could also be a function of survivorship or selection bias which would artificially inflate or deflate the returns of any of these indices, but this aspect falls outside of the scope of this article.
Yet some clarity could be provided by examining a totally transparent index, namely the AFX. Developed by Lequeux and Acar (1998) the AFX index is based on three moving averages of lengths 32, 61 and 117 days and applied to a Bank of International Settlements (BIS) weighted portfolio of currencies. The performance statistics of the AFX can also be seen in Figure 2. The AFX index provided here is net of transaction costs5 but gross of fees. By twice leveraging the AFX index, denoted as 'AFX_2', and adding the Usd risk free rate, 'AFX_2_RF', the index provides a comparable risk/return profile to that of the other currency indices. As the AFX is fully replicable and therefore completely transparent, it provides a better understanding of the market inefficiencies being exploited and should aid in the assessment of risk.
A recent study by Middleton (2005) has shown that the majority of individual currency programmes, managed by either Commodity Trading Advisors or Overlay, show statistically significant correlation with the AFX index. For completeness purposes the performance of a passive buy and hold strategy (B&H) applied to a BIS-weighted portfolio is also shown, where positions are rolled using CME Futures contracts.
Are maximum drawdowns consistent with normally distributed returns?
Our study investigates whether the maximum drawdowns seen empirically are compatible with normally distributed returns assuming ex-post knowledge of the mean returns and standard deviation over the period. To calculate the maximum drawdown percentiles a Monte-Carlo experiment was conducted based on the historical mean and volatility parameters observed over the period December 1989 to January 2005. The simulations were performed 5,000 times and the 181-month maximum drawdown statistics collected. The 5th and 95th percentiles, together with the median and empirical maximum drawdown statistics are plotted in the first thing one can observe from Figure 3
is that the confidence interval surrounding the maximum drawdown statistic is large, for example at the 5 and 95 percentile level the maximum drawdown values for the Stark index ranged from -38.07% to -14.54%. Interestingly the maximum drawdown of a passive currency investment falls below the 5th percentile.
A further observation can be made about the impact that leverage has on the maximum drawdown statistic. Whereas the return to risk ratio of the AFX index is identical regardless of leverage, see AFX and AFX_2 in Figure 2, the leverage does affect the maximum drawdown. For example, the empirical maximum drawdown of the original AFX was -11.19%, however twice leveraging the index results in a maximum drawdown of -21.13% i.e. the relationship is not strictly linear. In addition Figure 3 also demonstrates the impact of the risk free rate on the maximum drawdown statistic. Using the 5th percentile values of AFX_2 and AFX_2_RF, - 53.80% and -41.20% respectively, we see quite clearly how the deposit rate effectively 'cushions' substantially the maximum drawdown statistic over long period of times.
The results of the Monte Carlo experiment lso show the significance of the maximum drawdown observed empirically for each of the three indices. For example the -28.69% maximum drawdown of the CISDM index is larger [in magnitude] than that expected, as indicated by the median value, but fails to be significantly large at the 5% level. Whereas the -20.26% maximum drawdown experienced by the Stark index over the period in question is lower than that expected but again is not significantly low at the confidence level set. However, the maximum drawdown observed empirically for the Parker Index was much smaller than that expected, -9.18% versus -13.65%, and beyond the 95th percentile.
Possible reasons for the 'abnormal' drawdown
There could be a few explanations for these 'abnormal' maximum drawdowns. Firstly, the underlying distribution of returns may exhibit fat tails and that is apparent from the large excess kurtosis displayed for the three 'managers' indices. Secondly, the volatility of returns may not have been constant though time because of at least three factors: stochastic market volatility; varying degree of leverage from single managers; and changing composition of the index which has on the most part increased the number of programmes. Figure 4 highlights for instance that the Stark two years rolling volatility dropped from almost 25% in December 1991 to less than 5% in January 2005 whereas the market volatility, as measured by the passive basket of currency pairs, has only gone down from 9.2% to 6.3%.
Having exhibited these stylised facts, it is legitimate to ask if the managers' indices returns are comparable through time.Keeping this in mind, the losses exhibited in 2004 may actually exceed those generated in 1994 at constant risk exposure. On the other hand, the AFX index has tended to capture a more stable fraction (at around 85% over the total period) of the market volatility. For both of these reasons, smaller kurtosis and more stable volatility, the AFX index may represent a more accurate measure of the risk-adjusted performance of active currency managers.
Conclusion: drawdown statistics may not be comparable
This article has highlighted some basic properties of the maximum drawdown statistic. When applied to currency hedge funds indices, it becomes clear that results are not comparable for a number of reasons. As a consequence, transparent proxies may be useful for analysing the performance of foreign exchange managers, including the quantification of maximum drawdown.
Footnotes:
1. The opinions expressed in the article are those of the authors, and not of the authors' employer.
2. ww.marhedge.com/benchmarks/benchmarks.asp ?f=ta_currency
3. www.iasg.com/SnapshotPT.asp?ID=282
4. www.parkerglobal.com/fxindex.htm
5. The following bid/ask spreads have been applied:
Eur/Usd: 0.023%, Usd/Jpy: 0.029%, Usd/Chf:0.042%, Gbp/Usd: 0.027%, Eur/Jpy: 0.030%, Eur/Gbp: 0.043%, Eur/Chf:0.019%.
Author: Emmanuel Acar has researched the foreign exchange markets for the past 15 years mostly on the trading side. Prior to joining Bank of America, Emmanuel worked at Citibank within the FX Engineering Group. He was a proprietary trader and portfolio manager for almost ten years at Dresdner Kleinwort Benson, BZW and BNP Amy Middleton works at Bank of America assisting clients in the analysis of currency exposures. Her main areas of interest include hedging, forecasting techniques and trading style analysis. She has published articles in several periodicals, frequently speaks at conferences and has a Masters in Finance from Birkbeck University.
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