Optimal hedge fund allocation present challenges for hedge funds and FoHF managers

Research on optimal hedge fund allocation by Edhec Risk Institute and supported Newedge has potential implications for hedge fund and funds of hedge funds managers using portfolio optimisation.

Given that hedge fund returns are not distributed in a Gaussian manner, in the classic bell curve distribution around the mean, mean-variance optimisation techniques, which would be sub-optimal and impact negatively on the investor’s welfare, they need to be replaced by optimisation procedures that incorporate higher-order moments and co-moments.

Optimal portfolio decisions relating to hedge fund style allocation require estimates not only for co-variance parameters but also for co-skewness and co-kurtosis parameters. This is a considerable challenge that significantly augments the dimensionality issue that already exists with mean-variance analysis.

In a recent research paper (1) Edhec presented an application of enhanced estimators for higher-order co-moment parameters, introduced by Martellini and Ziemann (2010) in the context of hedge fund portfolio optimisation.

Edhec found using these improved estimates leads to a considerable improvement for investors in hedge funds. It also found it is only when enhanced estimators are used that portfolio selection with higher-order moments is consistently superior to mean-variance analysis from an out-of-sample perspective.

Diversification benefits
One of the principal reasons why asset owners generally are willing to include hedge funds in their portfolios is that they expect to achieve diversification benefits with respect to other existing investment possibilities.

The diversification argument states that investors can take advantage of the linear and non-linear exposure of hedge funds to a large variety of risk factors, including volatility, credit and liquidity risk, etc., to reduce the risk of their overall portfolio.

Many academics (see, for example, Terhaar et al (2002)) have stressed that mixing hedge funds with traditional assets leads to a reduction in the volatility of the traditional portfolio with a constant return level. This reduction in volatility originates from the fact that hedge funds (or at least some hedge fund strategies) present both low volatility and low correlation with traditional asset classes.

If they wish to capitalise fully on the benefits of diversification in a top-down approach, investors or funds of hedge funds managers must be able to rely on robust techniques for optimising portfolios that include hedge funds. Standard mean-variance portfolio selection techniques are known to suffer from a number of shortcomings, and the problems are exacerbated in the presence of hedge funds.

First, because hedge fund returns are not normally distributed (see, for example, Brooks and Kat (2002)), a mean-variance optimisation would be severely ill-adapted, except in the case of an investor who possesses quadratic preferences. For example, it can be shown through a statistical model integrating fatter tails than those of the normal distribution that minimising the second-order moment (volatility) can be accompanied by a significant increase in extreme risks (Sornette et al (2000)).

This finding is confirmed in Amin and Kat (2003), where the authors present empirical evidence that low volatility is generally obtained at the cost of lower skewness and higher kurtosis. As a result as stressed in Cremers et al (2005), in the presence of asymmetric and/or fat-tailed return distribution functions, the use of mean-variance analysis can potentially lead to a significant loss of utility for investors.

Extending techniques
As a consequence of the shortcomings of mean-variance optimisation, many attempts have been made to account for the specific risk features of hedge funds in a better way and to extend portfolio optimisation techniques in order to account for the presence of fat-tailed distributions, mostly by introducing some risk objective (for example, value-at-risk as in Favre and Galeano (2002), or conditional value-at-risk as in De Souza and Gokcan (2004) and Agarwal and Naik (2004)), that is more general than volatility, integrating the presence of non-trivial higher moments in asset returns. In the presence of non-normally distributed asset returns, optimal portfolio selection techniques require not only estimates for variance/co-variance parameters, but also estimates for higher-order moments and co-moments of the return distribution.

However, the need to estimate co-skewness and co-kurtosis parameters considerably exacerbates the dimensionality problem, which is already a serious concern in the context of co-variance matrix estimation. This concern is particularly acute in the hedge fund universe, where data is scarce, with a short history and low frequency, and where a number of performance biases are present (see for example Fung and Hsieh (1997, 2000, 2002)).

In this context, given the dramatic increase in dimensionality involved, one might wonder whether portfolio selection techniques that rely on higher-order moments can efficiently be implemented at all in realistic situations.

In a recent paper Martellini and Ziemann (2010) shed some light on this question by introducing improved estimators for the co-skewness and co-kurtosis parameters. They extend to the skewness and kurtosis dimensions several improved estimates that had been proposed for the co-variance matrix, including most notably the factor-based approach (Sharpe (1963)), the constant correlation approach (Elton and Gruber (1973)) and the statistical shrinkage approach (Ledoit and Wolf (2004)). In an empirical analysis based on US large-cap stock returns, Martellini and Ziemann (2010) subsequently find that the use of these enhanced estimates generates a significant improvement in investors’ welfare.

The Edhec research complemented these results by providing the first application of improved estimators for higher-order co-moment parameters in the context of hedge fund portfolio optimisation. It found that the use of these enhanced estimates generates a significant improvement for investors in hedge funds.

It also found that it is only when improved estimators are used that portfolio selection with higher-order moments dominates mean-variance analysis from an out-of-sample perspective. More specifically, we construct portfolios based on various hedge fund style indices using a fourth order approximation of expected CARA utility, using shrinkage estimators so as to alleviate the concern over robustness of purely sample-based estimates.

We find that the use of improved estimators leads to substantial increases in the investor’s utility as compared with using sample estimators.

We find extending the objective function to encompass higher-order moments of hedge fund return distribution can lead to value being destroyed, as opposed to added, when sample-based estimators are used. In our research paper, we introduce the improved estimators for hedge fund return co-variance, co-skewness and co-kurtosis parameters and then present our empirical analysis.

This article discusses an application of the improved estimators for higher-order co-moment parameters, introduced by Martellini and Ziemann (2010), in the context of hedge fund portfolio optimisation. We have found in recent research that the use of these enhanced estimates generates considerable improvements for investors in hedge funds.

We also find that it is only when improved estimators are used that portfolio selection with higher-order moments is consistently superior to mean-variance analysis from an out-of-sample perspective. Our results have important potential implications for hedge fund investors and hedge fund of funds managers who routinely use portfolio optimisation that incorporates higher moments without a formal analysis of the induced increase in parameter uncertainty and related lack of robustness of the results.

This article was written by Lionel Martellini, professor of finance at Edhec Business School and scientific director of EdhecRisk Institute.

The research from which this article was drawn was produced as part of the research chair on “Advanced Modelling for Alternative Investments” at EDHEC-Risk Institute sponsored by the Prime Brokerage Group at Newedge.

Footnote
1 Hitaj A, . Martellini and G Zambruno, June 2010, "Optimal Hedge Fund Allocation with Improved Estimates for Coskewness and Cokurtosis Parameters", Working Paper.

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