On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation

Recent studies in the empirical finance literature have reported evidence of two types of asymmetries in the joint distribution of stock returns. The first is skewness in the distribution of individual stock returns. The second is an asymmetry in the dependence between stocks: stock returns appear to be more highly correlated during market downturns than during market upturns. In this article we examine the economic and statistical significance of these asymmetries for asset allocation decisions in an out-of-sample setting. We consider the problem of a constant relative risk aversion (CRRA) investor allocating wealth between the riskfree asset, a small-cap portfolio, and a large-cap portfolio. We use models that can capture time-varying moments up to the fourth order, and we use copula theory to construct models of the time-varying dependence structure that allow for different dependence during bear markets than bull markets. The importance of these two asymmetries for asset allocation is assessed by comparing the performance of a portfolio based on a normal distribution model with a portfolio based on a more flexible distribution model. For investors with no short-sales constraints, we find that knowledge of higher moments and asymmetric dependence leads to gains that are economically significant and statistically significant in some cases. For short sales-constrained investors the gains are limited. keywords: asymmetry, copulas, density forecasting , forecasting , normality, stock returns This article is a revision of Chapter IV of my Ph.D. dissertation [Patton (2002)]. I would like to thank Sean Campbell, Rob Engle, Raffaella Giacomini, Clive Granger, Kris Jacobs, Bruce Lehmann, two anonymous referees, and seminar participants at the Econometric Society meetings, the 2002 Inquire UK annual seminar in Bournemouth, and the October 2002 Extremal Events in Finance conference in Montreal for helpful comments. Special thanks are due to Allan Timmermann for many useful discussions on this topic. All remaining deficiencies are my responsibility. Many thanks go to Vince Crawford and the UCSD Economics Experimental and Computational Laboratory for providing the computational resources required for this project. Financial support from the IAM Programme in Hedge Fund Research at LSE and the UCSD Project in Econometric Analysis fellowship is gratefully acknowledged. Address correspondence to Andrew J. Patton, Financial Markets Group, London School of Economics, Houghton Street, London WC2A 2AE, UK, or e-mail: [email protected]. Journal of Financial Econometrics, Vol. 2, No. 1, pp. 130--168 a 2004 Oxford University Press; all rights reserved. DOI: 10.1093/jjfinec/nbh006 at Penylvania State U niersity on Feruary 0, 2013 http://jfecrdjournals.org/ D ow nladed from Recent studies in the empirical finance literature have reported evidence of two types of asymmetries in the joint distribution of stock returns. The first is skewness or asymmetry in the distribution of individual stock returns, which has been reported by numerous authors over the last three decades. Evidence that stock returns exhibit some form of asymmetric dependence has been reported by several authors in recent years [see Erb, Harvey, and Viskanta (1994), Longin and Solnik (2001), Ang and Bekaert (2002), Ang and Chen (2002), Campbell, Koedijk, and Kofman (2002), and Bae, Karolyi, and Stulz (2003)]. The presence of either of these asymmetries violates the assumption of elliptically distributed asset returns, which underlies traditional mean-variance analysis [see Ingersoll (1987)]. In this article we examine the economic and statistical significance of these two asymmetries for asset allocation decisions in an out-of-sample setting. This article can thus be viewed as an attempt to address the suggestions of Harvey and Siddique (1999) and Longin and Solnik (2001), who propose investigating the impact of conditional skewness (Harvey and Siddique) and asymmetric dependence (Longin and Solnik) on portfolio choices. Theoretical justification for the importance of distributional asymmetries may be found in Arrow (1971), who suggests that a desirable property of a utility function is that it exhibits nonincreasing absolute risk aversion. Under nonincreasing absolute risk aversion investors can be shown to have a preference for positively skewed portfolios. The skewness of a portfolio of two assets is a function of the skewness of the individual assets, and two ‘‘coskewness’’ terms. Asymmetry in the dependence structure can be shown [see Patton (2002)] to lead to nonzero coskewness and thus impact the skewness of the portfolio return. This suggests that risk-averse investors will have preferences over alternative dependence structures. Ang, Chen, and Xing (2002) report empirical evidence in support of this. We examine the problem of an investor with constant relative risk aversion (CRRA) allocating wealth between the risk-free asset, the Center for Research in Security Prices (CRSP) small cap and large cap indices, comprised of the 1st and 10th decile of U.S. stocks sorted by market capitalization. We use monthly data from January 1954 to December 1989 to develop the models, and data from January 1990 to December 1999 for out-of-sample forecast evaluation. This problem is representative of that of choosing between a high risk--high return asset and a lower risk--lower return asset, as the annualized mean and standard deviation on these indices over the sample were 9.95% and 21.29% for the small caps, and 7.97% and 14.29% for the large caps. Our motivation for studying a problem involving two stocks rather than a stock and a bond, as in numerous previous studies, is that evidence of asymmetric dependence has so far been reported only 1 See Kraus and Litzenberger (1976), Friend and Westerfield (1980), Singleton and Wingender (1986), Lim (1989), Richardson and Smith (1993), Harvey and Siddique (1999, 2000), and Aı̈t-Sahalia and Brandt (2001), among others. Peri o (1999) finds no such evidence. 2 Utility functions that exhibit nonincreasing absolute risk aversion include the constant absolute risk aversion utility function, and the constant relative risk aversion utility function, see Huang and Litzenberger (1988). PATTON | Out-of-Sample Importance of Skewness 131 at Penylvania State U niersity on Feruary 0, 2013 http://jfecrdjournals.org/ D ow nladed from for equity returns. The presence or absence of asymmetric dependence between equity and bond returns is yet to be established. We use models of the asset returns that can capture the empirically observed time-varying means and variances of stock returns, and also the presence of (possibly time-varying) skewness and kurtosis, as in Hansen (1994) and Jondeau and Rockinger (2003). Further, we employ models of the dependence structure (or copula) that allow for, but do not impose different dependence during bear markets than bull markets, and allow for changes in this dependence structure through time. A thorough introduction to copula theory is presented in Schweizer and Sklar (1983), Joe (1997), and Nelsen (1999). The importance of skewness and asymmetric dependence for asset allocation is measured by comparing the performance of a portfolio based on a bivariate normal distribution model with a portfolio based on a model developed using copula theory. We compute the amount that an investor could be charged to make him/her indifferent between two competing portfolios, as in West, Edison, and Cho (1993), Ang and Bekaert (2002), and others. The significance of the differences in portfolio performance are tested using bootstrap methods. We find evidence in most cases that nonnormalities in the marginal distributions and copula do have important economic implications for asset allocation, however, the statistical significance of the improvement is only moderate. Gains are generally only present for investors that are not short-sales constrained, such as hedge funds. This article is essentially trying to test three hypotheses simultaneously: (1) Are these asymmetries present in this dataset? (2) Are these asymmetries predictable out-of-sample? (3) Can we make better portfolio decisions by using forecasts of these asymmetries than we can by ignoring them? If the answer to any of these questions is ‘‘no,’’ then we would conclude that the out-of-sample importance of these asymmetries for asset allocation is zero. The distinction between insample and out-of-sample significance is an important one. Finding that a more flexible distribution model fits the data better in-sample does not imply that it will lead to better out-of-sample portfolio decisions than those based on a simpler model. In fact, a common finding in the point forecasting literature is that more complicated models often provide poorer forecasts than simple misspecified models [see Weigend and Gershenfeld (1994), Swanson and White (1995, 1997), and Stock and Watson (1999)]. In this article we consider both unconstrained and short sales-constrained estimates of the optimal portfolio weight. The first reason for doing so is economically motivated: many market participants face the constraint that they are unable to short sell stocks or to borrow and invest the proceeds in stocks, while others, such as hedge funds, actively take both long and short positions. The second reason is statistically motivated: the optimal portfolio weight given a density forecast is itself only an estimate of the true optimal portfolio weight. By ensuring that our estimate always lies in the interval [0, 1], we employ a type of ‘‘insanity filter’’ that prevents the investor from taking an extreme position in the market. Such constraints have been found to improve the out-of-sample performance of optimal portfolios based on parameter estimates [see Frost and Savarino (1988) 132 Journal of Financial Econometrics at Penylvania State U niersity on Feruary 0, 2013 http://jfecrdjournals.org/ D ow nladed from and Jagannathan and Ma (2002)]. One could also consider an intermediate filter that allows for some limited amount of short selling, but we do not explore such a possibility here. Much of the existing work on asset allocation focused on special cases where the combination of utility function and distribution model were such that an analytical solution for the optimal portfolio decision exists [see Kandel and Stambaugh (1996) or Campbell and Viceira (1999), among others]. Brandt (1999) and Aı̈t-Sahalia and Brandt (2001) overcome the problem of the appropriate distributional assumption to combine with a given utility function by using the method of moments and the first-order conditions of the investor’s optimization problem to obtain an optimal portfolio decision. Detemple, Garcia and Rindisbacher (2003) present a sophisticated new method for finding optimal portfolio weights from empirically relevant models. In this article we combine density models that are shown to adequately describe the statistical properties of the asset returns with the CRRA utility function. One of the costs of using flexible parametric models for the joint distribution of stock returns is that we are forced by computational constraints to be relatively unsophisticated in other aspects of the project. First, we ignore the effects of parameter estimation uncertainty on the investor’s decision problem, though this was found to be important by Kandel and Stambaugh (1996). Also, we only consider the investor’s problem for the one-period-ahead investment horizon, thus ignoring the hedging component of the optimal portfolio weight [see Merton (1971)]. Empirical evidence on the importance of the hedging component is mixed: Brandt(1999),CampbellandViceira(1999),andDetemple,Garcia,andRindisbacher (2003) find it to be important, whereas Aı̈t-Sahalia and Brandt (2001) and Ang and Bekaert (2002) find only weak evidence. The remainder of the article is structured as follows. In Section 1 we provide a brief introduction to copula theory and its use in the density forecasting of stock returns. In Section 2 we present the investor’s decision problem in detail. Section 3 presents the empirical results on the asset allocation problem for a portfolio of a small-cap index and a large-cap index: the models employed, comparisons of portfolio weights, and tests for improvements in portfolio performance. We conclude in Section 4. In Appendix A we present some details of the optimization procedure and in Appendix B we provide the functional forms of the copula models considered. 1 FLEXIBLE MULTIVARIATE DISTRIBUTION MODELS USING COPULAS In this article we use copula theory to develop flexible parametric models of the joint distribution of returns. Suppose we have two (scalar) random variables of interest, Xt and Yt, and some exogenous variables Wt. The variables’ joint conditional distribution is (Xt, Yt)jF tÿ1 Ht1⁄4Ct (Ft, Gt), where Ht is some conditional bivariate distribution function, with conditional univariate distributions of Xt and Yt being Ft and Gt, the conditional copula being Ct, and F tÿ1 is the information set PATTON | Out-of-Sample Importance of Skewness 133 at Penylvania State U niersity on Feruary 0, 2013 http://jfecrdjournals.org/ D ow nladed from defined as F t sðZtÞ; for Zt 1⁄2Xt, Yt, Wt, Xtÿ1, Ytÿ1, Wtÿ1, . . . , Xtÿj, Ytÿj, WtÿjŠ 0. We will denote the distribution (cdf ) of a random variable using an uppercase letter and the corresponding density (pdf ) using a lowercase letter. A copula is any multivariate distribution function that has Uniform (0,1) marginal distributions. It links together two (or more) marginal distributions to form a joint distribution. The marginal distributions that it couples can be of any type: a normal and an exponential, or a Student’s t and a Uniform, for example. The theory of copulas dates back to Sklar (1959) and since then numerous applications have appeared in the statistics literature and more recently also in the analysis of economic data. The main theorem in copula theory is that of Sklar (1959), presented below for the conditional case. For an introduction to copula theory see Joe (1997) and Nelsen (1999). Theorem 1 (Sklar’s theorem for continuous conditional distributions). Let F be the conditional distribution of XjZ, G be the conditional distribution of YjZ, and H be the joint conditional distribution of (X, Y)jZ. Assume that F and G are continuous in x and y, and let Z be the support of Z. Then there exists a unique conditional copula C such that Hðx, yjzÞ 1⁄4 CðFðxjzÞ, GðyjzÞjzÞ, 8ðx, yÞ 2 R R and each z 2 Z: ð1Þ Conversely, if we let F be the conditional distribution of XjZ, G be the conditional distribution of YjZ, and C be a conditional copula, then the function H defined by Equation (1) is a conditional bivariate distribution function with conditional marginal distributions F and G. Sklar’s theorem allows us to decompose a bivariate distribution, Ht, into three components: the two marginal distributions, Ft and Gt, and the copula, Ct. The density function equivalent of Equation (1) is obtained quite easily, provided that Ft and Gt are differentiable, and Ht and Ct are twice differentiable: htðx, yjzÞ 1⁄4 ftðxjzÞ gtðyjzÞ ctðu, vjzÞ, 8ðx, y, zÞ 2 R R Z, ð2Þ where u Ft(xjz), and v Gt(yjz). Taking logs of both sides we obtain log htðx, yjzÞ 1⁄4 log ftðxjzÞ þ log gtðyjzÞ þ log ctðu, vjzÞ ð3Þ and so the joint log-likelihood is equal to the sum of the marginal log-likelihoods and the copula log-likelihood. For the purposes of multivariate density modeling, the copula representation allows for great flexibility in the specification: we may model the individual variables using whichever marginal distributions provide the best fit and then work on modeling the dependence structure via a model for the copula. The estimation of multivariate time-series models constructed using 3 In statistics, see Clayton (1978), Cook and Johnson (1981), Oakes (1989) and Genest and Rivest (1993). In economics and finance, see Li (2000), Embrechts, Höing, and Juri (2001), Patton (2001a, 2001b), Rockinger and Jondeau (2001), Chen and Fan (2002a, 2002b), Mashal and Zeevi (2002), Miller and Liu (2002), Junker and May (2002), Fermanian and Scaillet (2003), and Rosenberg (2003). 134 Journal of Financial Econometrics at Penylvania State U niersity on Feruary 0, 2013 http://jfecrdjournals.org/ D ow nladed from copulas is discussed in Patton (2001a) for the parametric case and Fermanian and Scaillet (2003) for the nonparametric case. 2 THE INVESTOR’S OPTIMIZATION PROBLEM The utility functions we assume for our hypothetical investors are from the class of CRRA utility functions: UðgÞ 1⁄4 ð1ÿ gÞ ÿ1 ðP0 ð1þ vxXt þ vyYtÞÞ if g 61⁄4 1 logðP0 ð1þ vxXt þ vyYtÞÞ if g 1⁄4 1, ( ð4Þ where P0 is the initial wealth, Xt and Yt represent the continuously compounded excess return (over the risk-free rate) on the small cap and large cap indices, respectively, and vi is the proportion of wealth in asset i. The degree of relative risk aversion (RRA) is denoted by g. For this utility function, the initial wealth does not affect the choice of optimal weight and so we set P0 1⁄4 1. We consider five different levels of relative risk aversion: g 1⁄4 1, 3, 7, 10, and 20. A similar range of risk aversion levels was also considered in Campbell and Viceira (1999) and Aı̈t-Sahalia and Brandt (2001). While there exist other utility functions that place higher weight on tail events or asymmetries in the distribution of payoffs, we focus on the CRRA utility because of its prominence in the finance literature. If gains are found using the CRRA utility function then they may be thought of as a conservative estimate of the possible gains using other, more sensitive, utility functions. The setup of the investor’s problem is as follows. Let the excess returns on the two risky assets under consideration be denoted Xt and Yt, with some joint distribution, Ht, with associated marginal distributions, Ft and Gt, and copula, Ct. We will develop density forecasts of this joint distribution ---F̂tþ1, Ĝtþ1, and the conditional copula, Ĉtþ1 ---and use them to compute the optimal weights, v tþ1 1⁄2v x;tþ1, v y;tþ1Š, for the portfolio. The optimal weights are found by maximizing the expected utility of the end-of-period wealth under the estimated probability density: v tþ1 arg max v2W EĤtþ1 1⁄2Uð1þ vxXtþ1 þ vyYtþ1ފ 1⁄4 arg max v2W Z Z Uð1þ vxxþ vyyÞ f̂tþ1ðxÞ ĝtþ1ðyÞ ĉtþ1ðF̂tþ1ðxÞ, Ĝtþ1ðyÞÞ dx dy, ð5Þ where W is some compact subset of R for the unconstrained investor and W1⁄4 {(vx, vy)2 [0, 1] : vx þ vy 1} for the short sales-constrained investor. The investor is assumed to estimate the model of the conditional distribution of excess returns using maximum likelihood and then optimize the portfolio’s weight using the predicted conditional distribution of returns. Work from the forecasting and estimation literature suggests that the parameter estimation stage and the forecast evaluation stage should both use the same objective function PATTON | Out-of-Sample Importance of Skewness 135 at Penylvania State U niersity on Feruary 0, 2013 http://jfecrdjournals.org/ D ow nladed from [see Granger (1969), Weiss (1996), and Skouras (2001)]. We use maximum-likelihood estimation for computational tractability. The double-integral defining the expected utility of wealth does not have a closed-form solution for our case. We use 10,000 Monte Carlo replications to estimate the value of this integral, which must be done for each point in the outof-sample period. The objective function was found to be well behaved (smooth and having a unique global optimum) for our choices of utility functions and density models and so we employed the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to locate the optimum, v tþ1, at each point in time. Further details on this procedure may be found in Appendix A. One concern that may arise in this design is the existence of EĤtþ1 1⁄2Uð1þ vxXtþ1 þ vyYtþ1ފ for certain density models. Given the CRRA utility, any density model that assigns positive probability to the case of bankruptcy would preclude the existence of EĤtþ1 1⁄2UŠ. All of the above specifications will assign some (extremely small) positive probability to bankruptcy. We deal with this by modifying the left tail of the distribution: we apply a logistic transformation to the lower tail of the portfolio return distribution so that all probability mass assigned to the region (ÿ1, «) is relocated to the region (0, «), where « is some extremely small positive number. 3 A PORTFOLIO OF SMALL CAP AND LARGE CAP STOCKS In this section we consider an investor with constant relative risk aversion facing the problem of allocating wealth between two assets: a portfolio of low market capitalization stocks (‘‘small caps’’) and a portfolio of high market capitalization stocks (‘‘large caps’’). These two assets were chosen as being representative of the general problem of balancing a portfolio comprised of a high risk----high return asset and a lower risk----lower return asset. 3.1 Description of the Data We use monthly data from the CRSP on the top 10% and bottom 10% of stocks sorted by market capitalization to form indices ---the ‘‘large cap’’ and ‘‘small cap’’ indices, from January 1954 to December 1999, yielding 552 observations. These data were also analyzed in a different context by Perez-Quiros and Timmermann (2001). We reserve the last 120 observations, from January 1990 to December 1999, for the out-of-sample evaluation of the models. Descriptive statistics on the two portfolios are presented in Table 1. The small cap index generally exhibited slightly positive skewness, while the large cap index exhibited negative skewness. Both indices exhibited excess kurtosis. The Jarque-Bera statistic indicates that neither series is unconditionally normal, and the unconditional correlation coefficient indicates a high degree of linear dependence, as expected. Table 1 also reveals that the small cap index had a higher mean and higher volatility than the large cap index over the total sample 136 Journal of Financial Econometrics at Penylvania State U niersity on Feruary 0, 2013 http://jfecrdjournals.org/ D ow nladed from