How to Price Hedge Funds:
From Two- to Four-Moment CAPM
Angelo Ranaldo α,ρ Laurent Favre βUBS Global Ast Management∗UBS Wealth Management*
Switzerland Switzerland
αAngelo Ranaldo, UBS Global Ast Management, Ast Allocation & Risk Management, Gessnerallee 3-5, P.O. Box, 8098 Zurich, Switzerland, Phone: ++4112353443, email: angelo.
ρContact author: Angelo Ranaldo.
β Laurent Favre, UBS Wealth Management, Ast Allocation Services, 8098 Zurich, Switzerland, e-mail: laurent-za.
∗The views expresd herein are tho of the authors and not necessarily tho of the UBS Bank which does not take on any responsibility about the contents and the opinions expresd in this paper. We are grateful to N. Amenc, J.-F. Bacmann, G. Ballocchi, G. Barone-Adesi, J. Benetti, F-S Lhabitant, R. Häberle, M. Ruffa, and R. Süttinger for their comments.
How to Price Hedge Funds:
From Two- to Four-Moment CAPM
Abstract
The CAPM model has rious difficulties to explain the past superior performance of most hedge funds. The purpo of this rearch is to analyze how to price hedge funds. We compare the traditional CAPM bad on the Markowitz mean-variance criterion with extensions of the CAPM that account for coskewness and cokurtosis. The key result is that the risk-return characteristics of hedge funds can differ widely. The u of a unique pricing model may be misleading. The beta is an exhaustive risk measure only for some hedge funds. Other hedge funds have significant coskewness and cokurtosis. The lack of consideration of higher moments may lead to an insufficient compensation for the investment risk.
Classification system G12: keywords: hedge funds, hedge fund, capm, skewness, kurtosis.
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First Version: July 23, 2002
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"Betters love skewness not risks, at the hor track !"
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Golec, Tamarkin (1998)
1.Introduction
In the last decade, hedge fund industry grew impressively. The ast under management1 grew from $170 billions in 1995 to more than 600 billions in December 2002. Prently, hedge funds are treated as an ast class per . Many studies show that hedge funds have a superior performance and that the introduction of hedge funds in a classical portfolio enhances the portfolio performance. The main reason advocated for the hedge fund superior performance relies on the skills of the hedge fund managers. On the other hand, some skepticism remains. First, hedge fund indices are broadly affected by the survivorship and performance measurement bias. Second, the nature of the return-generating process in hedge funds remains an unresolved issue. The attractive performance of hedge funds may be due to inadequate measurement techniques of the risk-return profile of hedge funds. The main aim of this rearch is indeed to investigate how to price hedge funds and, in particular, the validity of the traditional ast pricing models in measuring the risk-return trade-off in the hedge fund investment.
The Sharpe-Lintner-Mossin equilibrium model, usually called the Capital Ast Pricing Model (thereafter CAPM), is the commonly ud ast pricing model. This particular theoretical framework restricts the risk-return trade-off to a simple mean-variance relationship and / or to a quadratic utility function. However, the empirical evidence shows that the normality hypothesis has to be rejected for many hedge fund returns. Furthermore, a quadratic utility function for an investor implies an increasing risk aversion2. Instead, it is more reasonable to assume that risk aversion decreas with a wealth increa. In this paper, we consider some extensions of the traditional CAPM model that account for higher moment conditions and a more variegated structure of the risk premium concept. In particular, we examine the role of coskewness and cokurtosis in pricing hedge fund investments.
At this stage, the rearch in finance investigates if skewness and kurtosis have any relevance in explaining ast price returns. Skewness characterizes the degree of asymmetry of a distribution around its mean. Positive (negative) skewness indicates a distribution with an asymmetric tail extending toward more positive (negative) values. Kurtosis characterizes the relative peakness or flatness of a distribution compared with the normal distribution. Kurtosis higher (lower) than three indicates a distribution more peaked (flatter) than a normal distribution. Skewness and kurtosis of a given ast are also jointly analyzed with the skewness and kurtosis of the reference market. Similarl
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y to the so-called systematic risk or beta, some authors examine if there exists a systematic skewness and kurtosis and, if any, whether systematic skewness and kurtosis are priced in ast prices. Systematic skewness and kurtosis are also called coskewness
and cokurtosis (Christie-David and Chaudry [2001]). Provided that the market has a positive skewness of returns, investors will prefer an ast with positive coskewness. Cokurtosis measures the likelihood that extreme returns jointly occur in a given ast and in the market. The common characteristic of the models accounting for coskewness and cokurtosis is to incorporate higher moments in the ast pricing framework.wps云文档
In the literature, two main approaches have been investigated, namely the three-moment and four-moment CAPM. The theoretical specification of the three-moment CAPM is developed in Kraus and Litzenberger [1976], Ingersoll [1975], Jurcenzko and Maillet [2002], Gamba and Rossi [1998]. Other authors empirically study the three-moment CAPM. Barone-Adesi [1985] propos a Quadratic Model to test the three-moment CAPM. Harvey and Siddique [2000] find that the systematic skewness requires an average annual risk premium of 3.6% for US stocks. They also find that portfolios with high systematic skewness are compod of winner stocks (momentum effect). Harvey [2000] shows that skewness, coskewness and kurtosis are priced in the individual emerging markets
but not in developed markets. He obrves that volatility and returns in emerging markets are significantly positively related. But the significance of the volatility coefficient disappears when coskewness, skewness, and kurtosis are considered. Harvey’s explanation for this phenomenon is the low degree of integration of the emerging markets3.
Berenyi [2002], Christie-David and Chaudry [2001], Chung and Johnson [2001], Hwang and Satchell [1999], Jurczenko and Maillet [2002], Galagedera, Henry and Silvapulle [2002] propo the u of the Cubic Model as a test for coskewness and cokurtosis. Berenyi [2002] applies the four-moment CAPM to mutual fund and hedge fund data. He shows that volatility is an insufficient measure for the risk for hedge funds and for medium risk aver agents. Christie-David and Chaudry [2001] employ the four-moment CAPM on the future markets. They show that systematic skewness and systematic kurtosis increa the explanation power of the return generating process of future markets. Hwang and Satchell [1999] investigate coskewness and cokurtosis in emerging markets. Using a GMM approach, they show that the systematic kurtosis explains better the emerging market returns than the systematic skewness. Chung and Johnson [2001] compare the four-moment CAPM with the Fama-French two factors model. Dittmar [2002] analyzes skewness and kurtosis across industry indices.
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The question how to price hedge funds is the main motivation of this study. The two-moment assumption underpinning the standard CAPM strikes with the empirical characteristics of hedge fund returns. Consistently, we investigate if the first two moments are enough to fully explain the risk-return characteristics of the hedge funds. To do this, we extend the two-moment CAPM in the three-moment and four-moment CAPM, i.e. less restrictive forms of the traditional CAPM that accommodate systematic volatility (i.e. beta), systematic skewness, and systematic
竞相开放kurtosis. Finally, we examine how the required rate of return for hedge funds changes according to the different pricing models.
Our paper is organized as follows. In Chapter 2, we explain the economic arguments behind coskewness and cokurtosis. In Chapter 3, we derive the three-moment and four-moment CAPM models from the expected utility function. In Chapter 4, we empirically analyze whether beta, coskewness, and cokurtosis are priced. Concluding remarks follow thereafter.
2.Arguments for the existence of coskewness and cokurtosis
The existence of skewness and kurtosis in ast return distributions is well known. Here, the rearch focus is instead on the existence of coskewness and cokurtosis and, if any, their relevance
衣原体治疗in modeling ast pricing. The source of coskewness and cokurtosis in ast return distributions is esntially twofold. On the one hand, peculiar return distribution patterns may be originated from the u of specific trading strategies. Hedge fund managers pursue varied hedging and arbitrage strategies that engender pay-off profiles extremely different from traditional asts. On the other hand, skewed and / or kurtotic return distributions may be en as the statistical expression of market inefficiency and market frictions. Specifically, non-normal return distributions may be due to illiquidity, lack of divisibility, and low information transparency. All the factors contrast with the assumptions underpinning the standard CAPM model4. Throughout this ction, we discuss how and why the factors reprent eligible sources of coskewness and cokurtosis between hedge funds and the market portfolio.
The u of specific investing strategies. It is worth emphasizing that trading strategies applied by hedge fund managers engender return distribution typically different from equity market or mutual fund returns. Here, we mention only three factors that can generate coskewness and cokurtosis. First, hedge funds are often able to protect investors against declining markets. Hedge fund managers pursue downside protection by utilizing a variety of hedging strategy and investment styles. As a result, some hedge funds generate non-negative returns even in declining markets. Sec
ond, the u of leverage and derivatives contributes to the realization of particular risk-performance profiles characterized by low correlation with traditional ast markets. The hedge fund trading strategies widely benefit from options, option-like trading strategies, and, in general, financial engineering. Agarwal and Naik [2000, 2002] shows that writing and buying at and out-of-the money options increas significantly the explanation power of hedge fund returns. Fung and Hsieh [1997] shows that CTA's payoffs can be reprented like a stradlle payoff of lookback options. Third, hedge funds are less regulated than mutual funds. The weaker restrictions allow short lling to boost performance and reduce volatility.
Illiquidity. Typically, illiquid asts do not allow trading any volume size with an immediate execution and / or without price impact. Hedge funds are generally considered illiquid
