Commodity Trading Advisors: Risk, Performance Analysis, and Selection Chapter 22 pdf

377 CHAPTER 22 Risk-Adjusted Returns of CTAs: Using the Modified Sharpe Ratio Robert Christopherson and Greg N. Gregoriou M any institutional investors use the traditional Sharpe ratio to examine the risk-adjusted performance of CTAs. However, this could pose prob- lems due to the nonnormal returns of this alternative asset class. A modi- fied VaR and modified Sharpe ratio solves the problem and can provide a superior tool for correctly measuring risk-adjusted performance. Here we rank 30 CTAs according to the Sharpe and modified Sharpe ratio and find that larger CTAs possess high modified Sharpe ratios. INTRODUCTION The assessment of portfolio performance is fundamental for both in- vestors and funds managers, as well as commodity trading advisors (CTAs). Traditional portfolio measures are of limited value when applied to CTAs. For instance, applying the traditional Sharpe ratio will overstate the excess reward per unit of risk as measure of performance, with risk represented by the variance (standard deviation) because of the non- normal returns of CTAs. The mean-variance approach to the portfolio selection problem devel- oped by Markowitz (1952) has been criticized often due to its utilization of variance as a measure of risk exposure when examining the nonnormal returns of CTAs. The value at risk (VaR) measure for financial risk has become accepted as a better measure for investment firms, large banks, and pension funds. As a result of the recurring frequency of down mar- kets since the collapse of Long-Term Capital Management (LTCM) in August 1998, VaR has played a paramount role as a risk management tool and is considered a mainstream technique to estimate a CTA’s expo- sure to market risk. c22_gregoriou.qxd 7/27/04 12:06 PM Page 377 378 PROGRAM EVALUATION, SELECTION, AND RETURNS With the large acceptance of VaR and, specifically, the modified VaR as a relevant risk management tool, a more suitable portfolio performance measure for CTAs can be formulated in term of the modified Sharpe ratio. 1 Using the traditional Sharpe ratio to rank CTAs will under- estimate the tail risk and overestimate performance. Distributions that are highly skewed will experience greater-than-average risk underestimation. The greater the distribution is from normal, the greater is the risk under- estimation. In this chapter we rank 30 CTAs according to the Sharpe ratio and modified Sharpe ratio. Our results indicate that the modified Sharpe ratio is more accurate when examining nonnormal returns. Nonnormality of returns is present in the majority of CTA subtype classifications. LITERATURE REVIEW Many CTAs produce statistical reports that include the traditional Sharpe ratio, which can be misleading because funds will look better in terms of risk-adjusted returns. The drawback of using a traditional Sharpe ratio is that it does not distinguish between upside and downside risk. VaR has emerged in the finance literature as a ubiquitous measure of risk. However, its simple version presents some limitations. Methods to measure VaR such as, the Delta-Normal method described in Jorion (2000), are simple and easy to apply. However, the formula has a drawback since the assumption of normality of the distributions is violated due to the use of short-selling and derivatives strategies such as futures contracts fre- quently used by CTAs. Several methods have been proposed recently to correctly assess the VaR for nonnormal returns (Rockafellar and Uryasev 2001). Using a condi- tional VaR for general loss distributions, Agarwal and Naik (2004) con- 1 The standard VaR, which assumes normality and uses the traditional standard deviation measure, looks only at the tails of the distribution of the extreme events. This is common when examining mutual funds, but when applying this technique to funds of hedge funds, difficulty arises because of the nonnormality of returns (Favre and Galeano 2002a, b). The modified VaR takes into consideration the mean, standard deviation, skewness, and kurtosis to correctly evaluate the risk- adjusted returns of funds of hedge funds. Computing the risk of a traditional invest- ment portfolio consisting of 50 percent stocks and 50 percent bonds with the traditional standard deviation measure could underestimate the risk in excess of 35 percent (Favre and Singer 2002). c22_gregoriou.qxd 7/27/04 12:06 PM Page 378 Risk-Adjusted Returns of CTAs 379 struct a mean conditional VaR demonstrating that mean-variance analysis underestimates tail risk. Favre and Galeano (2002b) also have developed a technique to properly assess funds with nonnormal distributions. They demonstrate that the modified VaR (MVaR) does considerably improve the accuracy of the traditional VaR. The difference between the modified VaR and the traditional VaR is that the latter only considers the mean and stan- dard deviation, while the former takes into account higher moments such as skewness and kurtosis. The modified VaR allows one to calculate a modified Sharpe ratio, which is more suitable for CTAs. For example, when two portfolios have the same mean and standard deviation, they still may be quite different due to their extreme loss potential. If a traditional portfolio of stocks and bonds was equally split, using the standard deviation as opposed to modified VaR to calculate risk-adjusted performance could underestimate the risk by more than 35 percent (Favre and Galeano 2002b). DATA AND METHODOLOGY The data set consists of 164 CTAs who reported monthly performance fig- ures, net of all fees, to the Barclay Trading Group database. The data spans the period January 1997 to November 31, 2003, for a total of 83 months. We selected this period because of the extreme market event of August 1998 (Long-Term Captial Management collapse) as well as the September 11, 2001, attacks. From this we extracted and ranked the top 10, middle 10, and bottom 10 funds according to ending assets under management. We use this comparison to see if there exist any differences between groups in terms of the Sharpe and modified Sharpe ratio. We use the Extreme metrics soft- ware available on the www.alternativesoft.com web site to compute the results using a 99 percent VaR probability, and we assume that we are able to borrow at a risk-free rate of 0 percent. The difference between the traditional and modified Sharpe ratio is that, in the latter, the standard deviation is replaced by the modified VaR in the denominator. The traditional Sharpe ratio, generally defined as the excess return per unit of standard deviation, is represented by this equation: (22.1) where R P = return of the portfolio R F = risk-free rate and s = standard deviation of the portfolio Sharpe Ratio = −RR pF σ c22_gregoriou.qxd 7/27/04 12:06 PM Page 379 380 PROGRAM EVALUATION, SELECTION, AND RETURNS A modified Sharpe ratio can be defined in terms of modified VaR: (22.2) The derivation of the formula for the modified VaR is beyond the scope of this chapter. Readers are guided to Favre and Galeano (2002b) and Christoffersen (2003) for a more detailed explanation. EMPIRICAL RESULTS Descriptive Statistics Table 22.1 displays monthly statistics on CTAs during the examination period, including mean return, standard deviation, skewness, excess kurtosis, and compounded returns. The average of the compounded returns and mean monthly returns is greatest in the top group (Panel A) and the lowest in the bottom group, as expected. In addition, we find that negative skewness is more pronounced in the bottom group, yielding more negative extreme returns, whereas the middle group (Panel B) has the greatest positive skewness. A likely explanation is that the middle-size CTA may better control skewness dur- ing down markets and will have on average fewer negative monthly returns. Large CTAs may have a harder time getting in and out of invest- ment positions. The bottom group (Panel C) has the highest volatility (standard devia- tion 32.56 percent) and lowest compounded returns (18.29 percent), likely attributable to CTAs taking on more risk to achieve greater returns. Performance Discussion Table 22.2 presents market risk and performance results. First, observe that the top group (Panel A) has, in absolute value, the lowest normal and mod- ified VaR (i.e., is less exposed to extreme market losses). Furthermore, the bottom group (Panel C) has in absolute value the highest normal and mod- ified VaR, implying that CTAs with small assets under management are more susceptible to extreme losses. This is not surprising, because they have the lowest monthly average returns, as seen in Table 22.1. Concerning performance, the bottom group has the lowest traditional modified and modified Sharpe ratios. It appears that large CTAs do a bet- ter job of controlling risk-adjusted performance than can small CTAs. Com- paring the results of the traditional and the modified Sharpe ratios, we find that the traditional Sharpe ratio is higher, confirming that tail risk is under- estimated when using the traditional Sharpe ratio. Modified Sharpe Ratio MVaR = −RR pF c22_gregoriou.qxd 7/27/04 12:06 PM Page 380 Risk-Adjusted Returns of CTAs 381 TABLE 22.1 Descriptive Statistics Average Average Assets Annualized Annualized Compounded Fund (Ending Return Std. Dev. Excess Return Name Millions $) (%) (%) Skewness Kurtosis (%) Panel A: Subsample 1: Top 10 CTAs Bridgewater Associates 6,831.00 11.88 9.75 −0.10 −0.60 119.38 Campbell & Co., Inc. 5,026.00 14.16 13.70 −0.40 0.10 148.53 Vega Asset Management (USA) LLC 2,054.68 9.21 4.60 −1.50 5.00 87.28 Grossman Asset Management 1,866.00 15.64 15.28 −0.10 −0.30 170.81 UBS O’Connor 1,558.00 8.31 8.54 0.30 0.70 73.02 Crabel Capital Management, LLC 1,511.00 7.74 6.31 1.10 3.70 68.29 FX Concepts, Inc. 1,480.00 10.79 15.26 0.30 −0.10 94.63 Grinham Managed Funds Pty., Ltd. 1,280.00 11.69 10.01 0.50 −0.10 116.34 Rotella Capital Management Inc. 1,227.95 11.63 12.19 0.30 0.30 112.10 Sunrise Capital Partners 1,080.96 13.77 13.75 0.90 0.50 142.03 Average 2,391.62 11.48 10.94 0.13 0.92 113.24 Panel B: Subsample 2: Middle 10 CTAs Compucom Finance, Inc. 53.00 9.90 22.18 0.50 0.50 68.12 Marathon Capital Growth Ptnrs., LLC 50.10 13.73 14.78 0.00 1.30 139.11 DynexCorp Ltd. 50.00 7.47 12.17 0.10 −0.70 59.25 ARA Portfolio Management Company 47.70 7.05 17.24 −0.10 0.90 47.08 c22_gregoriou.qxd 7/27/04 12:06 PM Page 381 382 PROGRAM EVALUATION, SELECTION, AND RETURNS TABLE 22.1 (continued) Average Average Assets Annualized Annualized Compounded Fund (Ending Return Std. Dev. Excess Return Name Millions $) (%) (%) Skewness Kurtosis (%) Panel B: Subsample 2: Middle 10 CTAs (continued) Blenheim Capital Mgmt., LLC 46.50 21.66 37.22 −0.10 −0.20 181.17 Quality Capital Management, Ltd. 46.00 13.06 16.34 0.20 −0.40 124.74 Sangamon Trading, Inc. 46.00 9.06 7.30 1.80 6.70 83.40 Willowbridge Associates, Inc. 45.80 14.38 42.44 0.90 4.80 48.89 Clarke Capital Management, Inc. 43.20 16.19 17.41 0.60 0.90 175.78 Millburn Ridgefield Corporation 42.94 5.91 17.47 1.00 0.70 36.04 Average 47.12 11.84 20.46 0.49 1.45 96.36 Panel C: Subsample 3: Bottom 10 CTAs Muirlands Capital Management LLC 0.40 16.10 24.11 0.20 -0.70 149.13 Minogue Investment Co. 0.40 9.27 41.88 1.70 8.30 8.10 Shawbridge Asset Mgmt. Corp. 0.22 15.66 33.88 1.00 3.00 102.94 International Trading Advisors, B.V.B.A. 0.20 −6.33 12.22 −1.10 8.10 −38.83 Be Free Investments, Inc. 0.20 14.95 20.49 −1.50 5.70 140.79 Lawless Commodities, Inc. 0.10 −11.10 43.02 −1.70 7.80 −77.22 District Capital Management 0.10 13.80 34.68 −0.50 1.20 67.73 Venture I 0.10 −1.42 21.19 −2.50 11.80 −22.91 Marek D. Chelkowski 0.10 −15.91 78.29 −0.30 0.50 −95.98 Robert C. Franzen 0.10 −8.94 15.79 −2.00 4.70 −50.81 Average 0.19 2.61 32.56 −0.67 5.04 18.29 c22_gregoriou.qxd 7/27/04 12:06 PM Page 382 Risk-Adjusted Returns of CTAs 383 TABLE 22.2 Performance Results Fund Normal Modified Normal Modified Name VaR (%) VaR (%) Sharpe Ratio Sharpe Ratio Panel A: Subsample 1: Top 10 CTAs Bridgewater Associates −6.42 −6.28 0.09 0.10 Campbell & Co., Inc. −8.17 −9.13 0.13 0.12 Vega Asset Management (USA) LLC −1.33 −2.64 0.60 0.30 Grossman Asset Management −8.99 −8.94 0.11 0.11 UBS O’Connor −3.91 −3.75 0.25 0.26 Crabel Capital Management, LLC −2.85 −2.33 0.24 0.29 FX Concepts, Inc. −9.22 −8.09 0.10 0.11 Grinham Managed Funds Pty., Ltd. −5.66 −4.23 0.16 0.22 Rotella Capital Management Inc. −7.33 −6.54 0.12 0.14 Sunrise Capital Partners −8.08 −4.89 0.11 0.18 Average −6.20 −5.68 0.19 0.18 Panel B: Subsample 2: Middle 10 CTAs Compucom Finance, Inc. −11.07 −12.66 −0.03 −0.03 Marathon Capital Growth Ptnrs., LLC −10.69 −9.36 0.11 0.10 DynexCorp Ltd. −6.83 −7.60 0.01 0.02 ARA Portfolio Management Company −12.24 −10.98 0.06 0.05 Blenheim Capital Mgmt, LLC −21.76 −21.49 0.08 0.08 Quality Capital Management, Ltd. −8.81 −9.85 0.11 0.13 Sangamon Trading, Inc. −2.19 −4.01 0.23 0.12 Willowbridge Associates, Inc. −3.54 −32.94 0.03 0.02 Clarke Capital Management, Inc. −8.32 −9.94 0.12 0.10 Millburn Ridgefield Corporation −7.21 −12.67 0.07 0.04 Average −9.27 −13.15 0.08 0.06 c22_gregoriou.qxd 7/27/04 12:06 PM Page 383 384 PROGRAM EVALUATION, SELECTION, AND RETURNS CONCLUSION It is of critical importance to understand that complications will arise when a traditional measure of risk-adjusted performance, such as the Sharpe ratio, is used on the nonnormal returns of CTAs. Institutional investors must use the modified Sharpe ratio to measure the risk-adjusted returns cor- rectly. The modified VaR is better in the presence of extreme returns because the normal VaR considers only the first two moments of a distri- bution, namely mean and standard deviation. The modified VaR, however, takes into consideration the third and fourth moments of a distribution, skewness and kurtosis. Using both the modified Sharpe and modified VaR will enable investors to more accurately assess CTA performance. In many cases, if the modified Sharpe ratio is used to examine normally distributed assets, they will be ranked in the same exact order as if the traditional Sharpe ratio was used. This occurs because the modified VaR converges to the classical VaR if skewness equals zero and excess kurtosis equals zero. The statistics presented can be applied to all CTA classifications dis- playing nonnormal returns. We believe many institutional investors want- ing to add CTAs to traditional stock and bond portfolios must request additional and more appropriate statistics, such as the modified Sharpe ratio, to analyze the returns of CTAs. TABLE 22.2 (continued) Fund Normal Modified Normal Modified Name Var (%) Var (%) Sharpe Ratio Sharpe Ratio Panel C: Subsample 3: Bottom 10 CTAs Muirlands Capital Management LLC −13.90 −15.98 0.03 0.03 Minogue Investment Co. −24.62 −29.99 −0.01 −0.01 Shawbridge Asset Mgmt. Corp. −18.66 −22.18 0.03 0.04 International Trading Advisors, B.V.B.A. −21.31 −10.86 −0.01 −0.00 Be Free Investments, Inc. −24.37 −14.15 0.06 0.03 Lawless Commodities, Inc. −52.03 −29.80 −0.11 −0.06 District Capital Management −29.99 −24.05 0.02 0.02 Venture I −26.46 −13.79 −0.06 −0.03 Marek D. Chelkowski −44.79 −40.25 −0.10 −0.09 Robert C. Franzen −11.90 −8.34 −0.09 −0.06 Average −26.80 −20.94 −0.02 -0.01 c22_gregoriou.qxd 7/27/04 12:06 PM Page 384 . (Rockafellar and Uryasev 2001). Using a condi- tional VaR for general loss distributions, Agarwal and Naik (2004) con- 1 The standard VaR, which assumes normality and uses the traditional standard deviation. standard deviation, is represented by this equation: (22. 1) where R P = return of the portfolio R F = risk-free rate and s = standard deviation of the portfolio Sharpe Ratio = −RR pF σ c22_gregoriou.qxd. namely mean and standard deviation. The modified VaR, however, takes into consideration the third and fourth moments of a distribution, skewness and kurtosis. Using both the modified Sharpe and modified