Mean-Coherent Risk and Mean-Variance Approaches in Portfolio Selection: An Empirical Comparison

Abstract

We empirically analyze the implementation of coherent risk measures in portfolio selection. First, we compare optimal portfolios obtained through mean-coherent risk optimization with corresponding mean-variance portfolios. We find that, even for a typical portfolio of equities, the outcomes can be statistically and economically different. Furthermore, we apply spanning tests for the mean-coherent risk efficient frontiers, which we compare to their equivalents in the meanvariance framework. For portfolios of common stocks the outcomes of the spanning tests seem to be statistically the same.Keywords: portfolio choice, mean variance, mean coherent risk, comparison. JEL Classification: G11. I IntroductionThere is an ongoing debate in the financial literature on which risk measure to use in risk management and portfolio choice. As some risk measures are more theoretically appealing, others are easier to implement practically. For a long time, the standard deviation has been the predominant measure of risk in asset management. Mean-variance portfolio selection via quadratic optimization, introduced by Markowitz (1952), used to be the industry standard (see, for instance, Tucker et al. (1994)). Two justifications for using the standard deviation in portfolio choice can be given. First, an institution can view the standard deviation as a measure of risk, which needs to be minimized to limit the risk exposure. Second, a mean-variance portfolio maximizes expected utility of an investor if the utility index is quadratic or asset returns jointly follow an elliptically symmetric distribution. 1 Despite the computational advantages, the variance is not a satisfactory risk measure from the risk measurement perspective. First, mean-variance portfolios are not consistent with second-order stochastic dominance (SDD) and, thus, with the benchmark expected utility approach for portfolio selection. Second, but not independently, as a symmetric risk measure, the variance penalizes gains and losses in the same way. Artzner et al. (1999) give an axiomatic foundation for so-called coherent risk measures. They propose that a "rational" risk measure related to capital requirements 2 should be monotonic, subadditive, linearly homogeneous, and translation invariant. Tasche (2002) and Kusuoka (2001) demonstrate that a Choquet expectation with a concave distortion function represents a general class of coherent risk measures. Moreover, with some additional regularity restrictions, as imposed by Kusuoka (2001), the class of coherent risk measures becomes consistent with the second order stochastic dominance principle and thus generates portfolios consistent with the expected utility paradigm, see, for example, Ogryczak andRuszczyński (2002) andDe Giorgi (2005).The class of coherent risk measures generalizes expected shortfall, a co- Ingersoll (1987). 2 The capital requirements are relevant for asset management since they are directly applied to financial institutions, see the Basel Accord (1999).3 herent risk measure which received a...

Testing for Mean-Coherent Regular Risk Spanning

“…Moreover, for a moderate levels of skewness and kurtosis in the index return distributions the different risk measures are statistically equivalent and can be used interchangeably. This is in line with findings in Polbennikov and Melenberg (2005) who perform a systematic comparison of the mean-variance and the mean-CRR approaches in portfolio management. De Giorgi and Post (2004), who test mean-variance and mean-CRR efficiency of the US market index for the beta and momentum portfolios, obtain similar results: the mean-variance and mean-CRR performance measures as well as the test statistics are comparable.…”

“…Moreover, as explained in Bassett et al (2004), a CRR measure can be approximated by a finite sum of expected shortfalls. A sample analog of a mean-CRR problem with this finite sum approximation can be reformulated as a linear program and efficiently solved, see Portnoy and Koenker (1997), Rockafellar and Uryasev (2000), and Polbennikov and Melenberg (2005), making mean-CRR optimal portfolio choice also practically feasible. In summary, a CRR measure is a natural choice for a risk measure in case of a portfolio choice based on a mean-risk trade-off.…”

“…In summary, a CRR measure is a natural choice for a risk measure in case of a portfolio choice based on a mean-risk trade-off. In the companion paper, Polbennikov and Melenberg (2005), we also derive the asymptotic distribution of the mean-CRR portfolio weights θ and consider special cases of a point mass approximation of a CRR measure and expected shortfall. In the remainder of this section, we consider the risk contribution results obtained by Bertsimas et al (2004) and Tasche (1999) for the case of expected shortfall and generalize them to a general CRR measure.…”

“…As shown in Polbennikov and Melenberg (2005), a sample analog of a the mean-PMA CRR portfolio problem can be reformulated as a linear program and efficiently solved with existing numerical algorithms. The spanning test results of this section are applicable for the mean-PMA CRR spanning as well.…”

“…We report the relevant formulas for a mean-CRR market portfolio in the Appendix D, referring for the derivation details to the paper by Polbennikov and Melenberg (2005). The considered cases are mean-CRR, with as special cases mean-expected shortfall, and mean-PMA CRR.…”