Detecting Premium Portfolios in Higher-order Moments CAPM

Tepmony Sim

E-mail: tepmonysim_rupp@yahoo.com

M.sc. “QEM ”, Ca ’ Thesis

Foscari University of Venice

June 2010,

Executive Summary

Since the first introduction by Sharpe (1964), the Capital Asset Pricing Model (CAPM) has then been one of the most popular models for asset pricing, and is a cornerstone of financial economics. However, the CAPM suffers from several restrictive hypotheses such as the normality of return distributions. It has been criticized by many empirical evidences and this has led for further considerations on the extensions within the model. An extension which we regard in this thesis is to insert higher-order moments other than variance into capital asset pricing relation. The resulting model is the Higher- order Moments CAPM. This approach is initially proposed by Rubinstein (1973) and sequentially developed by Kraus and Litzenberger (1976), Fang and Lai (1997), Athayde and Flôres (1997 and 2000), and Jurczenko and Maillet (2006b).

In this dissertation, the Higher-order Moment CAPM which takes into account up to the fourth moment is considered. In Four-moment CAPM, to arrive at the four- moment CAPM fundamental relation, two specific portfolios, besides the riskless asset and the market portfolio, are assumed to exist. Our main purpose is to confirm their existence and show that when they exist, they are not unique. From the validity of four- moment CAPM fundamental relation, the roles of the third moment (skewness) and the fourth moment (kurtosis) can be investigated. To detect the two specific portfolios above as well as to investigate the role of skewness and kurtosis in current financial data, the Seemingly Unrelated Regression (SUR) method by Zellner (1962) is carried out.  It is also our interest to confirm the validity of Four-moment CAPM in another way. Following Fang and Lai (1997), Hwang and Satchell (1999) and Galagedera et al (2004), we can use the Cubic Market Model as a proxy. Our purpose of doing so is to compare the performance of the two models with the in-hand data.

Before we can go through the empirical part, some theoretical foundations are provided. In this part, we basically follow the works by Jurczenko and Maillet (2006b). Several notations used to represent and to compute the higher-order moments are also given. Moreover, a generalization of the univariate higher-order C-(co)moments to multivariate higher-order C-(co)moments is introduced as well. The systematic risk, systematic skewness and systematic kurtosis can be calculated by several means. They can be obtained from the cubic market model, the four-moment CAPM fundamental relation, or by their own definitions. Besides the C-moments these risk factors are also calculated by L-moments. Finally, a comparison of these calculations can thus be made.  Detecting Premium Portfolios in Higher-order Moments CAPM*

Abstract

In Four-moment CAPM, the roles of skewness and kurtosis can be investigated under the validity of the so-called four-moment CAPM fundamental relation. This relation assumes that, besides the riskless asset and the market portfolio, another two specific portfolios exist. We are to show that these portfolios exist but they are not unique. We also confirm the validity of the four-moment CAPM in another direction. Following Fang and Lai (1997), Hwang and Satchell (1999) and Galagedera et al (2004), we use the Cubic Market Model as a proxy of Four-moment CAPM. In the theoretical framework, various notations to represent and to compute the higher-order C-moments of assets ’ returns are introduced. The generalization of the univariate C-(co)moments to the multivariate C-(co)moments is also provided. Besides C-moments, the systematic risk, the systematic skewness, and the systematic kurtosis are also calculated by using L-(co)moments.

Keywords: CAPM, Higher-order Moments, Kurtosis, L-moments, Premium Portfolios,  Skewness, SUR.

JEL Classification: C01, C10, G11, G12.

Introduction

Since it was first introduced by Sharpe (1964), Lintner (1965) and Mossin (1966), the Capital Asset Pricing Model (CAPM) has then been one of the most popular models for asset pricing, and is a keystone of financial economics. This particular theoretical framework relates 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, which it bases on, has to be rejected for many financial data. A quadratic utility function for an investor, furthermore, implies an increasing risk aversion. Instead, it is more reasonable to assume that risk aversion decreases with an increase in wealth. Due to several inadequacies revealed by empirical tests, CAPM has been considered for further extensions by taking into account more factors additional to mean and variance.  Amongst these extensions, multifactor CAPM is included. The most prominent one of multifactor framework is size effect of Banz (1981). He finds that the size of a firm and the return on its common stock are inversely related.

Later on, Fama and French (1992) suggest three-factor model, which includes the capital size and book-to-market value into classical CAPM. The findings of Fama-French in their three-factor model suggest that small cap and value portfolios have higher expected returns -- and arguably higher expected risk -- than those of large cap and growth portfolios. Carhart (1997), who aims to study the persistence of mutual fund returns, then proposes four-factor model, which is an extension of Fama-French three-factor model by adding a new factor, one-year momentum in stock returns. In short term at least, the results do not support the existence of skilled or informed mutual fund portfolio managers.  Another appropriate approach, which we regard as the center of our interests, is to insert higher-order moments than variance in a pricing relation. The main feature of these models is to obtain, for any risky asset, a linear equilibrium relation between the expected rate of return and higher-order moments systematic risk measures.

In this dissertation, we consider some extensions of the traditional mean- variance framework that account for higher-order moment conditions and a more variegated structure of the risk premium concept. In particular, we examine the roles of skewness and kurtosis in pricing the recent financial data. Skewness characterizes the degree of asymmetry of a distribution around its mean. Negative (positive) skewness indicates a distribution with an asymmetric tail extending towards more negative (positive) values. Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution. In standard definition, kurtosis higher (lower) than three indicates a distribution which is more peaked (flatter) than a normal one. Similarly to the so-called systematic risk or beta, it is possible to test for a systematic skewness and systematic kurtosis. Systematic skewness and kurtosis are also known as co-skewness and co-kurtosis (Christie-David and Chaudry, 2001). Provided that the market has a positive skewness of returns, investors will prefer an asset with positive coskewness. Cokurtosis measures the likelihood that extreme returns jointly occur in a given asset and in the market; and thus investors prefer small co-kurtosis. The common characteristic of the models accounting for co-skewness and co-kurtosis is that they incorporate higher moments in the asset pricing framework. In the literature, two main approaches have been investigated: three- moment and four-moment CAPM. The theoretical higher-order moments CAPM is initially proposed by Rubinstein (1973), and, subsequently, developed by Ingersoll (1975), Kraus and Litzenberger (1976), Athayde and Flôres (1997 and 2000), and Jurcenzko and Maillet (2001 and 2006b). Other authors empirically study the validity of the higher-order moments CAPM such as, often, three- moment and four-moment CAPM. For three-moment CAPM, Barone-Adesi (1985) proposes a quadratic model to test the three-moment CAPM, while 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 composed 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 observes that volatility and returns in emerging markets are significantly positively related. But the significance of the volatility coefficient disappears when co-skewness, skewness, and kurtosis are considered. Harvey ’s explanation for this phenomenon is the low degree of integration of the emerging markets. When accounting up to the fourth moment, Berényi (2002), Christie-David and Chaudry (2001), Chung, Johnson and Schill (2006), Fang and Lai (1997), Hwang and Satchell (1999), Galagedera, Henry and Silvapulle (2002) propose the use of the Cubic Market Model as a test for coskewness and cokurtosis. Berényi (2002) applies the four-moment CAPM to mutual fund and hedge fund data, and he then shows that volatility is an insufficient measure of risk for hedge funds and for medium risk averse agents.

Christie-David and Chaudry (2001) employ the four-moment CAPM on the future markets, where they find that systematic skewness and systematic kurtosis increase the explanatory power of the return generating process of future markets. Fang and Lai (1997), in purpose to corporate the effect of kurtosis, apply the four-moment CAPM on New York Stock Exchange (NYSE). They find that the expected rate of return is not only related to the systematic variance but also to the systematic skewness and systematic kurtosis. Hwang and Satchell (1999) investigate co-skewness and co-kurtosis in emerging markets. They show that systematic kurtosis is better than systematic skewness in explaining emerging market returns.

Following Jurzenko and Maillet (2006b), in Four-moment CAPM, we attempt to detect the two premium portfolios Z1m and Z2m introduced in the so- called Four-moment CAPM Fundamental Relation. These two portfolios are defined as such that: Z1m possesses zero-covariance and zero cokurtosis and has unitary coskewness with market portfolio, and Z2m  possesses zero-covariance and zero coskewness and has unitary cokurtosis with market portfolio. We are to show that these portfolios always exist, however, they are not unique. We propose some methods to elicit the appropriate ones. It is straightforward that when we can find these two premium portfolios, the effect of skewness and kurtosis can be examined. Besides, we also wish to test the validity of the four-moment CAPM in another way. We depart from testing the mean-variance CAPM, then the three-moment CAPM and finally the four-moment CAPM by  using, respectively, the linear market model, the quadratic market model and the cubic market model as the proxies.

In the theoretical framework, various notations for presenting and computing the portfolio returns are introduced. Moreover, a generalization of scalar C-moments of returns to multivariate case is also provided. In addition to conventional moments (C-moments) used in traditional way in higher-order moments CAPM, we also introduce robust moments -- called linear moments (L-moments). The main advantage of L-moments over C-moments is that L-moments, being linear functions of the data, suffer less from the effects of sampling variability: L-moments are more robust than C-moments to outliers in the data and enable more secure inferences to be made from small sample about an underlying probability distribution (see, for instance, Hosking, 1990; Hosking and Wallis, 1987; Ulrych et al, 2000). L-moments sometimes yield more efficient parameter estimates than the maximum likelihood estimates (see Hosking, 1990). Furthermore, L-moments exhibit some specifically fascinating features for financial applications. For example, they have abilities to reduce the so-called Hamburger moment problem (see Jondeau and Rockinger, 2003a; and Jurczenko and Maillet, 2006a); and they are also coherent shape measures of risk (see Artzner, Delbaen, Eber and Heath, 1999).

To obtain the estimations of all wanted parameters, we carry out throughout the thesis the so-called Seemingly Unrelated Regression (SUR) method by Zellner (1962). As pointed out by the author, it is only under special conditions that classical least-squares applied equation-by-equation yields efficient coefficient estimators. For conditions generally encountered, SURE are at least asymptotically more efficient than single-equation least-squares estimators.

The organization of the thesis is as follows. In Chapter 2, we discuss the literature and selected research papers of interest as well as the central theory behind CAPM. In Chapter 3, we present the theoretical framework used for the analysis. Chapter 4 presents the econometric methods used. We devote Chapter 5 for discussing the properties of the data material and for presenting the results obtaining by using econometric methods in previous part. The last part, Chapter 6, the conclusions are made; and the rests are for references and appendices.

Download