Mean-Semivariance Optimization: A Heuristic Approach

Academics and practitioners optimize portfolios using the mean-variance approach far more often than the meansemivariance approach, despite the fact that semivariance is often considered a more plausible measure of risk than variance. The popularity of the mean-variance approach follows in part from the fact that mean-variance problems have well-known closed-form solutions, whereas meansemivariance optimal portfolios cannot be determined without resorting to obscure numerical algorithms. This follows from the fact that, unlike the exogenous covariance matrix, the semicovariance matrix is endogenous. This article proposes a heuristic approach that yields a symmetric and exogenous semicovariance matrix, which enables the determination of mean-semivariance optimal portfolios by using the well-known closed-form solutions of mean-variance problems. The heuristic proposed is shown to be both simple and accurate. As is well known, Markowitz (1952) pioneered the issue of portfolio optimization with a seminal article, later expanded into a seminal book (Markowitz, 1959). Also well known is that at the heart of the portfolio-optimization problem, there is an investor whose utility depends on the expected return and risk of his portfolio, the latter quantified by the variance of returns. What may be less well known is that, from the very beginning, Markowitz favored another measure of risk: the semivariance of returns. In fact, Markowitz (1959) allocates the entire chapter IX to discuss semivariance, where he argues that “analyses based on S [semivariance] tend to produce better portfolios than those based on V [variance]” (see Markowitz, 1991, page 194). In the revised edition of his book (Markowitz, 1991), he goes further and claims that “semivariance is the more plausible measure of risk” (page 374). Later he claims that because “an investor worries about underperformance rather than overperformance, semideviation is a more appropriate measure of investor’s risk than variance” (Markowitz, Todd, Xu, and Yamane, 1993, page 307). Why, then, have practitioners and academics been optimizing portfolios for more than 50 years using variance as a measure of risk? Simply because, as Markowitz (1959) himself suggests, variance has an edge over semivariance “with respect to cost, convenience, and familiarity” (see Markowitz, 1991, page 193). He therefore focused his analysis on variance, practitioners, and academics followed his lead, and the rest is history. Familiarity, however, has become less of an issue over time. In fact, in both practice and academia, downside risk has been gaining increasing attention, and the many magnitudes that Javier Estrada is a Professor of Finance at the IESE Business School in Barcelona, Spain.