Academic journal article International Journal of Business

Dynamic versus Static Optimization of Hedge Fund Portfolios: The Relevance of Performance Measures

Academic journal article International Journal of Business

Dynamic versus Static Optimization of Hedge Fund Portfolios: The Relevance of Performance Measures

Article excerpt

I. INTRODUCTION

The seminal Markowitz's analysis is based on the first two moments of portfolio return and ignores the higher ones. By using an improved estimator ofthe covariance structure of hedge fund index returns, Amenc and Martellini (2002) prove that a portfolio including hedge funds can have a significantly smaller volatility (on an out-of-sample basis), while having almost the same mean return. This result suggests to incorporate hedge funds to get a mean-variance efficient portfolio. As illustrated by Cremers et al. (2005), hedge funds returns usually have higher means and lower standard deviations than standard assets, but they also have undesirable higher moment properties. For example, hedge funds returns can be negatively skewed. This is due to for instance to the non-linearity of their payoffs when they are generated by option-like strategies (see Goetzmann et al., 2002, 2003).

Many empirical studies have also proved that the mean-variance approach is no more valid when investors include hedge funds in their portfolios (Amenc and Martellini, 2002; Terhaar et al., 2003; McFall Lamm, 2003; Agarwal and Naik, 2004; Morton et al., 2006).

Empirical studies show that the assumption of normality in return distribution is not justified, in particular when dealing with hedge funds which have significant positive or negative skewness and high kurtosis (Fung and Hsieh, 1997; Ackerman et al., 1999; Brown et al., 1999; Caglayan and Edwards, 2001; Bacmann and Scholz, 2003; Agarwal and Naik, 2004).

One possible extension of the mean-variance analysis is the use of the expected utility theory. For a truncated utility at the order 4, we have to maximize a linear combination of the first four moments of the return. In this more general framework, the hedge fund portfolio is built according also to the skewness and kurtosis. This problem has been studied by Jurczenko and Maillet (2004), Malevergne and Sornette (2005), Anson et al. (2007) ... Martellini and Ziemann (2007) extend the Black-Litterman Bayesian approach when building a portfolio of hedge funds with non-trivial preferences about higher moments. Their results show that active style allocation provides a significant value in a hedge fund portfolio if we take account of non-normality and parameter uncertainty in hedge fund return distributions.

Since the demise of Long Term Capital Management (LTCM), downside risk measures have been introduced to take more account of the tails of the distributions. In most cases, they are based on the Value-at-Risk (VaR) approach. As proved by Goetzman et al. (2003), Agarwal and Naik (2004), the VaR can be high, due to important leverage effect. VaR also is more appropriate than the first four moments to measure extreme risks. VaR optimization has been previously studied by Litterman (1997a, 1997b), Lucas and Klaasen, (1998). Additionally, Favre and Galeano (2002) introduce a new Value-at-Risk method to adjust the volatility risk with the skewness and the kurtosis of the distribution of returns. They minimize the modified VaR at a given confidence level. However, as pointed by Lo (2001), VaR has several shortcomings. For instance, it does not indicate the magnitude of the potential losses below itself. Additionally, it is not a convex risk measure (see Artzner et al., 1999) and its minimization is rather involved.

For these reasons, Acerbi and Tasche (2002) introduce the expected shortfall (ES). This measure, also called CVaR, measures the amount of losses below the VaR, is coherent and leads to portfolio optimization problems that can be solved more easily. Indeed, Rockafellar and Uryasev (2000) prove that the CVaR minimization is equivalent to a convex optimization problem. Agarwal and Naik (2004) estimate ES from the HFR hedge fund indices. They show that downside risk is significantly underestimated by the mean-variance analysis, which suggests that mean-ES optimization must be introduced. …

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