Tail risk of hedge funds: an extreme value application

1.1 Problem description Investors seek to maximise returns and to minimise risk. As risk is man- ageable but returns are not, these objectives can best be achieved through risk measurement/management techniques. In this regard, the concept of diversification plays a central role in modern portfolio theory. It follows that investors' welfare can be improved by allocating wealth among a large number of different assets. Ideally, any poorly performing asset can even- tually be compensated by for positive performance from other assets in the portfolio. To put it differently, the idiosyncratic risk of a single asset can be diversified away leading to lower portfolio risk and thus a higher risk adjusted portfolio return. Obviously, a necessary condition for risk diversi- fication to work is that asset returns do not depend on each other. Under the assumption of normally distributed returns, a standard assumption in finance, risk and dependence can be expressed by volatility and correlation respectively. Low volatility and low correlation with other assets offers diversification benefits to investors. These two features, together with historically good performance may explain the increasing attractiveness of hedge funds among institutional and retail investors in recent years. In the last decade the hedge fund industry has been the fastest growing asset class in the financial sector. Despite the decade-long bull market in the 1990s and liquidity/credit crises in the late 1990s, hedge fund investing has been gaining popularity among various types of investors. HFR (2007) estimates that the total net assets in hedge funds are approximately USD 1.4 trillion as of the fourth quarter 2006. As a result of this growth, an increasing number of studies describing the various hedge fund characteristics, performance comparison with other asset classes, and their overall contribution in institutional portfolios has been produced. Some of the early works are the monographs of Lederman and Klein (1995), Crerend (1998), Jaffer (1998), Lake (1999) as well as the studies of Ackermann, McEnally, and Ravenscraft (1999) and Fung and Hsieh (1997). Other monographs such as Jaffer (2003) focus entirely on the properties of fund of hedge funds. The risk and diversification benefits of hedge funds have been studied in many different ways. Two major events at the end of 1990s; the near collapse of Long-Term Capital Management and the Asian crisis, have led regulatory authorities to focus more on studying the risk inherent in hedge fund strategies. Brown, Goetzmann, and Park (1998) examine the involve- ment of hedge funds in the Asian crisis of 1997-1998, and the Report of the President's Working Group on Financial Markets (1999) deals extensively with the Long-Term Capital Management incident in 1998 and highlights the potential risks of excessive use of leverage. The general role played by hedge funds in financial market dynamics has been studied in Eichengreen, Mathieson, Sharma, Chadha, Kodres, and Jansen (1998). The investment risk of hedge funds, their unique risk properties stand alone as well as in portfolio context have been analysed with standard risk management tools typically assuming implicitly or explicitly normally dis- tributed returns. For example, Edwards and Liew (1999) show that adding hedge funds to traditional portfolios increases the Sharpe ratio of those port- folios. Purcell and Crowley (1999) show that hedge funds outperform tra- ditional assets in times of down markets. Diversification benefits of adding hedge funds are also found in Crerend (1998) and Agarwal and Naik (2000) as well as in Géhin and Vaissié (2005). In these studies a significant upward shift of efficient frontier and reduction in risk measures is observed. However, hedge funds pose a challenge to standard risk measures based on normally distributed returns. Recent evidence (see e.g. Schmidhuber and Moix 2001, Brooks and Kat 2002) casts doubt on the validity of volatility and correlation as appropriate risk measures for hedge funds. Indeed, the returns of hedge fund indices are not normally distributed and have exhib- ited unusual levels of skewness and kurtosis. The asymmetric properties of hedge fund returns are investigated in Anson (2002a), Ineichen and Johansen (2002), and Ineichen (2002). These characteristics are consistent with the complex trading strategies used by hedge funds which present option-like payoffs (see e.g. Fung and Hsieh 1997, Fung and Hsieh 2001, Mitchel and Pulvino 2001, Fung and Hsieh 2002c, Agarwal, Fung, Loon, and Naik 2004). Clearly, volatility and correlation do not provide su±cient information about risk and dependence when the normality assumption is violated. As a consequence, applying symmetric measures on hedge funds may lead to erroneous conclusions. One potential solution to overcome the problem of non-normality in hedge fund returns is to apply methods that take the asym- metry in return distribution into account. For instance, Bacmann and Pache (2004) apply downside deviation, Keating and Shadwick (2002) make use of the Omega function and Favre and Signer (2002) propose the use of a mod- ified Value-at-Risk based on Cornish-Fisher expansion. In this thesis, the use of Extreme Value Theory (EVT) is advocated. This area of statistics enables the estimation of tail probabilities regardless of the underlying distribution of hedge fund returns. The fact that it focuses on the tail returns rather than their means, makes modelling of the whole time series of returns unnecessary. Consequently, the estimation of Value- at-Risk and Expected Shortfall can be done under fairly general types of distributions. This thesis contributes to the growing literature on risk associated with hedge funds in two main directions. Firstly, it carefully examines the tail risk of individual hedge fund strategies and of portfolios built with stocks, bonds and hedge funds using EVT. Consequently, the first objective is to evaluate the size of return asymmetry in order to quantify a potential ten- dency for extreme losses among various hedge fund strategies. The second objective follows the first one as it attempts to quantify eventual benefits of the inclusion of hedge funds in a traditional portfolio (stocks and bonds) depending on the initial composition of the portfolio and on the type of hedge funds added. Several papers (Lhabitant 2001, Blum, Dacorogna, and Jaeger 2003, Gupta and Liang 2003) have already used Value-at-Risk derived from EVT in the context of single funds or hedge fund indices. Bacmann and Gawron (2005) evaluates portfolio risk by allocating fund of hedge funds only. Secondly, the thesis further measures the dependence between hedge funds and traditional investments in periods of distressed markets. In such periods, correlation breaks down and investors' ability to diversify dimin- ishes because the asset dependence is much higher than in periods of market quiescence. For this purpose the main objective is to test explicitly the ex- istence of asymptotic dependence among hedge funds as well as between hedge funds and traditional investments.