MULTI-DIMENSIONAL RISK AND MEAN-KURTOSIS PORTFOLIO OPTIMIZATION.

Journal of Financial Management and Analysis, 21(2):2008:47-56 (c) Om Sai Ram Centre for Financiat Management Research

MULTIDIMENSIONAL RISK AND MEAN-KURTOSIS PORTFOLIO OPTIMIZATION
JAMES STACEY, Ph.D. (Physics), M.B.A. Faculty Member Faculty of Business Administration Memorial University of Newfoundland SI. John's, CANADA A B3X5 Abstract
The mciin-variance portfolio optimization theory of Markowitz assumes that stock returns are distributed according to normal prohabiiiiy density functions (pdfs). In reality, slock returns are more accurately described by lepiokurtic pdts which have kurtosis greater than zero. Stocks with Icptokurtic distributions of returns are conventionally considered to be inherently more risky lliLin stocks with normal pdfs. This paper examines portfolio optimization using the kurtosis as a risk measure. Maximizing the kurlosis as a function of portfolio weights is equivalent to maximizing the probability of large fluctuations from the mean which is counicr-intuitive and contrary to normal practice. It is argued that risk is multidimensional and that ihe kurtosis is an ambiguous multi-dimensional risk measure. Key Wonls: Portfolio; Optimization: Kurtosis; Riskmeasure JEL Classification; C12. C13. C6I. C87. G32, N20

Introduction The mean-variance portfolio optimization theory of Markowitz' follows from the assumption that investors arc altcmpting to maximize their expected utility of returns Irom an investment portfolio, as described by Elton and Gruber^. The expected utility is described in terms of returns and variances, and the mean-variance approach lo portfolio management holds exactly when investors are expected utility maximizers, prefer more wealth to less, are risk averse, and either security returns are normally distributed or utility functions arc quadratic. Elton and Cirubcr review other criteria for portfolio selection; in particular, the geometric mean return, safety first, stochastic dominance, and analysis in terms of characteristics of the return distribution. In this introduction Elton and Gruber's discussion of these alternatives are briefly summarized. Prelude The geometric mean return criterion is viewed by Elton and Gruber as maximizing the expected value of

terminal wealth. Under additional alternative assumptions, this criterion leads to the selection of a portfolio that is on the efficient set. These results are consistent with mean-variance theory. Investors employing the safety first criterion use simple decision rules that concentrate on avoiding bad outcomes. Portfolios that optimize a safety first criterion also often lie on the efficient set, also consistent with mean-variance theory. Another alternative to mean variance theory is stochastic dominance (see, for example, Whitmore and Findlay'). The most general form of stochastic dominance makes no assumptions about the underlying distribution of returns. In addition, it is not necessary to assume any specific form of the investors' utility functions. There are three progressively stronger assumptions regarding investor behaviour employed by stochastic dominance: first-order stochastic dominance assumes investors prefer more to less; second order stochastic dominance assumes, in addition to preferring more to less, investors

An earlier version of this paper won the Best Paper Award in Ihe Finance Division of the 2007 Adminislralive Sciences As.socialIDn r Canada Conference in Ottawa for which Professor Ted Neave of Queen's University wa.s Ihc discussant. The atithor would like to Ihank Professor Alex Faseruk for helpful comments and to Dr. Larry Bauer for providing the slock return data cxtfiiclcd from the CRSP database, as well as the journal referees for comments that significantly enriched the presentation of Ihis work. Any remaining errors arc the sole responsibility of the author. The author owns full responsibility for the contents of the paper.

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JOURNAL OF RNANCIAL MANAGEMENT AND ANALYSIS

are risk averse; and third-order stochastic dominance adds to first- and second-order stochastic dominance that investors have decreasing absolute risk aversion. When returns are normally distributed, second-order stochastic dominance defines an optimal set of portfolios that is consistent with the efficient set produced hy meanvariance analysis. Finally, Elton and Gruber briefly discuss selecting portfolios on the basis of the first three moments of retum distributions, the mean (first moment), variance (second moment), and skewness (third moment). It is suggested the portfolio problem is best represented in threedimensional space with mean on one axis, variance on the second, and skewness on the third. The efficient set would then be the outer shell of the feasible set with maximum mean retum, minimum variance, and maximum skewness. Early work in this approach is due to Samuelson''. When the underlying distribution of returns is symmetric, the skewness is zero. This work substitutes the fourth moment of the probability distribution for the skewness and explores some of the issues that arise relating to the portfolio optimization problem in three dimensions. The kurtosis is defined as a shape measure and used as a proxy for the leptokurtosity of the underlying probability density function. The problem of calculating the joint movements of retums is significant for the kurtosis, but a brute force optimization approach is currently feasible (barely) with desktop computing systems, these systems having become more than an order of magnitude more powerful over the past decade. The motivation for the present work is that alternative approaches outlined above appear to be consistent with the mean-variance approach. Stochastic dominance has great power theoretically and is applicable regardless of tbe underlying distribution of stock returns, but is difficult to apply. Tbe present study seeks to examine how introducing tbe shape of the underlying distribution of returns might impact on portfolio analysis and risk. The novelty of this work is not only in the results obtained by the portfolio optimization but in the identification of additional dimensionCs) to the quantification of risk. This provides another avenue of potentially fruitful insight into the nature of idiosyncratic risk and the important distinction that must be drawn between risk and uncertainty. Lusk, Halperin and Bem' give a detailed survey of recent research into reformulation of the capital

asset pricing model (CAPM), citing how Markowitz' conceptualization of risk as the variability of retums quickly led to the CAPM; the importance of Knight's treatise on Risk and Uncertainty defining risk as a quantity susceptible to measurement and associating uncertainty with the non-measurable or nonquantitative; Sharpe's identification of this "uncertainty dimension" (non-systematic, unique, asynchronous or idiosyncratic risk) with the residuals of a CAPM leastsquares regression analysis; and the divergence of opinion regarding the calculus of risk according to a standard deviation basis (direct summation of risk variables) or a variance basis (summation in quadrature of risk variables). Lusk et al. cite the analysis of RolT (in his landmark article simply entitled R-) that tiie correlation coefficient is "too low for the CAPM to be taken seriously as a predictive tool" and Roll's suggestion that the separation between what is expected from CAPM and its residuals (the "distributional mixture") is "perhaps due to traders acting on private information." This is consistent with the findings of Piotroski and Roulstonc that idiosyncratic risk is positively associated with insider trading. Lusk et al. then go on to present a detailed analysis of tbe corporate social responsibility dimension of a firm's market profile and its contribution to understanding idiosyncratic risk. What is most interesting for the present paper is Roll's suggestion that other moments of the underlying distributional mixture, in particular the kurtosis, seem to he structurally aligned with possible private information fiows. Multi-Dimensional Risk The standard deviation o of stock return data, a common measure of volatility, is a popular measure of risk for the given stock. The standard deviation o of a portfolio is the square root of the variance of the portfolio defined by the equation

0)
where a^^^ is tbe variance-covariance matrix element between stocks rand s (often notated as a^ ) and w^ and w^ are the portfolio weights for stocks r and .v respectively. The variance is also called the second central moment \i^_ of the probability distribution. The third moment provides a measure of the skewness of the probability

MULTI-DIMENSIONAL RISK AND MEAN-KURTOSIS PORTFOLIO OPTIMIZATION

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distribution. With the symmetric pdfs that are assumed to underlie stock returns in this paper, the skewness is zero. The portfolio optimization questions of interest to this paper arise only from the introduction of the fourth moment. The fourth central moment of the portfolio is defined as (2) wliere E (3) is the expectation value of the fourth moment of the portfolio of returns r, the weights w for each stock in the portfolio, and the variance of the stock portfolio given by Equation 1. X will be considered shortly as a risk variable analogous to a. The kurtosis of the portfolio is given by
(4)

Mean-Kurtosis Portfolio Optimization Portfolio optimization with kurtosis as the risk measure becomes a matter of maximizing the kurtosis. This is equivalent to maximizing the probability of large fluctuations from ihe mean (i.e. maximizing the size of the tails of the distribution). This approach is counterintuitive and contrary to normal practice. The alternative, minimizing the tails of the distribution, will simply yield the mean-variance theory of Markowitz and provides no further insights. Tracking the joint movement of stock returns with the variance involves the calculation of the variance-covariance matrix a- of dimension N^ where N is the number of stocks. Tracking the joint movement of stock returns with the kurtosis involves ihe calculation of the fourlh-rank tensor E^,^ with N* different entries. One objective function that maximizes kurtosis is to minimize

(6)
subject to the constraints that the return of the portfolio is equal to the required rate of return /i*:

r.>M> which can be simplified as

[a'J

the sum of all weights in the portfolio is unity:
-3

(5)
and all the weights are positive (no short selling):

(8) (9)

In this paper, both o and X. are employed as measures of risk. The first question addressed in this paper is whether X is proportional to risk. As X goes to zero, the stock's distribution of returns becomes increasingly lcptokurtic. If the underlying distribution of stock returns is a Levy pdf tbe higher moments are not defined. However, Stacey and Faseruk' have constructed univariate leptokurtic pdfs called fractal normal (or Formal) …