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Risk Measures for Hedge Funds: a Cross-Sectional Approach Abstract Forthcoming: European Financial Management Risk Measures for Hedge Funds: A Cross-Sectional Approach by Bing Liang and Hyuna Park * This version: September 2006 Abstract This paper analyzes the risk-return trade-off in the hedge fund industry. We compare semi-deviation, value-at-risk (VaR), Expected Shortfall (ES) and Tail Risk (TR) with standard deviation at the individual fund level as well as the portfolio level. Using the Fama and French (1992) methodology and the combined live and defunct hedge fund data from TASS, we find that the left-tail risk captured by Expected Shortfall (ES) and Tail Risk (TR) explains the cross-sectional variation in hedge fund returns very well, while the other risk measures provide statistically insignificant or marginally significant results. During the period between January 1995 and December 2004, hedge funds with high ES outperform those with low ES by an annual return difference of 7%. We provide empirical evidence on the theoretical argument by Artzner et al. (1999) that ES is superior to VaR as a downside risk measure. We also find the Cornish-Fisher (1937) expansion is superior to the nonparametric method in estimating ES and TR. Key word s: hedge funds, expected shortfall, tail risk, conditional VaR, Cornish-Fisher expansion JEL classification : G11, G12, C31 * We thank Daniel Giamouridis, Hossein Kazemi, Bernard J. Morzuch, Mila Getmansky Sherman, and an anonymous referee for helpful comments and suggestions. We are responsible for any error. * Bing Liang is an associate professor of finance at the Department of Finance & Operations Management, Isenberg School of Management, University of Massachusetts, 121 Presidents Drive, Amherst, MA 01003-9310, Phone: (413) 545-3180, Fax: (413) 545-3858, E-mail: [email protected] . Hyuna Park is a Ph.D. candidate in finance at the Isenberg School of Management, University of Massachusetts, 121 Presidents Drive, Amherst, MA 01003-9310, Phone: (413) 348-9116, E-mail: [email protected] . 1. Introduction One of the most important developments in modern finance theory is the ability to model risk in a quantifiable fashion. This is important because if we know how to measure and price financial risk correctly, we can properly value risky assets (Copeland and Weston, 1992). Since the seminal work of Markowitz (1952), standard deviation of returns has been one of the best-known measures for risk. The Markowitz model, devised mainly for the long-only portfolios in the U.S. equity market, is based on the assumption that investors’ utility curves are a function of the expected return and standard deviation of returns only. Therefore, higher moments, if ever exist in the return distribution, can be ignored. Is this assumption still valid when investors include hedge funds in their portfolios? Interest in alternative investments has grown rapidly since the recent downturn in the U.S. equity market. 1 There is increasing evidence that hedge funds offer higher mean returns and lower standard deviations than traditional assets, but they also give investors undesirable higher moment characteristics (Cremers, Kritzman, and Page (2005) and Alexiev (2005)). After the demise of Long Term Capital Management (LTCM) in August 1998, downside risk management and risk-adjusted performance measurement have been strongly emphasized in this industry. The source of this negatively skewed payoff is well documented in the literature. Hedge funds implement dynamic, option-like strategies, trade derivative securities, and have a fee structure that generates non-linear payoffs (Goetzmann, Ingersoll, Spiegel and Welch (2002), Spurgin (2001), Mitchell and Pulvino (2001), Goetzmann, Ingersoll 2 and Ross (2003), Taleb (2004) and Chan, Getmansky, Haas and Lo (2005)). All these facts make standard deviation an incomplete measure of a hedge fund’s risk. Although most researchers agree that traditional risk management tools cannot capture many of the risk exposures of hedge fund investments, an alternative framework is not yet well established. Traditional risk measures are still dominant among practitioners (see the survey result for the fund of hedge funds industry by Amenc, Giraud, Martellini and Vaissie (2004)). However, academic research is beginning to examine downside risk, asymmetric volatility, semi-deviation, extreme value analysis, regime-switching, jump processes, and so on. Semi-deviation considers standard deviation only over negative outcomes and is of interest because investors only dislike downside volatility. Estrada (2001) argues that semi-deviation combines information provided by two statistics: standard deviation and skewness. Empirically, the semi-deviation has been reported to explain the cross-section of returns of emerging markets and the cross-section of internet stocks returns (Harvey (2000) and Estrada (2000, 2001)). Another important alternative is Value-at-Risk (VaR). VaR is the worst loss that can happen over a specified horizon at a specified confidence level. However, VaR is only as accurate as its inputs. VaR based risk management has been criticized regarding the failure of Long Term Capital Management (LTCM). Jorion (2000b) applies a VaR approach to analyze the failure of LTCM and concludes that LCTM relied too much on short-term history and significantly underestimated its risk profile. Gupta and Liang (2005) adopt the extreme value theory (EVT) to estimate VaR and address capital adequacy and risk estimation issues in the hedge fund industry. Using 3 three times the 99% 1-month VaR as the required equity capital for hedge funds, they find that the majority of hedge funds are adequately capitalized. They also compare VaR with traditional risk measures in evaluating hedge fund risk. They conclude that VaR is better than standard deviation due to negative skewness and high kurtosis in hedge fund returns. Bali and Gokcan (2004) estimate VaR for hedge fund portfolios using a normal distribution, a fat-tailed generalized error distribution (GED), the Cornish-Fisher (CF) expansion, and the extreme value theory (EVT). They use the HFR (Hedge Fund Research) indexes and find that the EVT approach and the CF expansion capture tail risk better than the other approaches. Recently, Bali, Gokcan and Liang (2006) examine the cross-sectional relation between hedge fund return and risk measured by VaR in an asset pricing framework. They estimate VaR using both an empirical distribution and the Cornish-Fisher (CF) expansion to incorporate the higher-order moments in fund returns. They find a significant positive relation between VaR and the expected returns on live funds. VaR, however, is subject to severe criticism. Lo (2001) points out that only several years of historical data may not show the distribution of returns and questions the usefulness of VaR based risk management. In addition to this empirical problem, VaR has theoretical shortcomings. It does not provide the magnitude of the possible losses below the threshold it identifies. A portfolio’s VaR is the maximum loss that the investors might suffer during a time horizon at a specified confidence level. It should be noted that there is still a small but nonzero probability that investors can experience a loss more than VaR. In other words, VaR does not give any information on how big the 4 loss can be when that level is breached. Expected Shortfall(ES) measures this amount. It is the expected amount of loss conditional on the fact that VaR is exceeded. ES is also known as tail conditional expectation, conditional loss or tail loss (see Jorion (2000a)). ES is sometimes called conditional VaR (CVaR) (see Agarwal and Naik (2004) and Alexander and Baptista (2004)). 2 There is another theoretical reason why we prefer ES to VaR as an alternative risk measure. VaR has some mathematical irregularity, such as lack of convexity and monotonicity, as well as reasonable continuity (see Artzner, Delbaen, Eber and Heath (1999), Uryasev (2000), and Alexander and Baptista (2004) for details). 3 Recognizing these shortcomings of VaR, Agarwal and Naik (2004) adopt ES in their empirical tests. They use the empirical distributions of returns on the HFR hedge fund indexes to estimate ES. They find that downside risk is significantly underestimated in the mean- variance framework and suggest mean-ES optimization as an alternative. 4 We use another downside risk measure called tail risk (TR) in this paper. It is introduced by Bali, Demirtas and Levy (2005) to explain the time-series variation in market returns. TR measures deviation from mean only when the return is lower than VaR. 5 Despite the debate on downside risk in hedge funds, empirical evidence is very scarce. Academic literature on hedge funds has been mainly focused on performance measurement. To the best of our knowledge, this is the first paper that empirically compares alternative risk measures to explain the cross-section of hedge fund returns using individual hedge fund data under the asset pricing framework. 6 We recognize that this is mainly due to the short history of the hedge fund industry and that the historical 5 return distribution may not reveal the true risk in hedge funds. In addition, reporting to a database is voluntary, which can cause biases in empirical research. 7 This, however, should not be a reason why researchers disregard available hedge fund data to measure downside risk and stay away from analyzing cross-sectional variation in hedge fund returns. 8 In fact, the goal of this paper is to examine whether available data on hedge funds can reveal the risk-return trade-off and, if so, which risk measure best captures the cross-sectional variation in hedge fund returns. We adopt research methodology from the traditional asset pricing literature. Similar to Fama and French (1992), we first sort individual hedge funds by a risk measure at the end of each period and form decile portfolios. Then we compare the rate of return on the most risky portfolio with that from the least risky portfolio during the following period.
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