Semivariance, semicovariance matrix and lower partial moments for portfolio selection: Analytical and numerical results

The traditional quantitative risk measure in portfolio selection is the variance of the portfolio returns, a concept introduced by Harry Markowitz in his seminal work on Modern Portfolio Theory. However, variance penalizes both upside and downside deviations, which may not align with investor preferences, as the upside risk is generally desirable. Markowitz himself suggested focusing on downside risk, leading to the concept of semivariance, which only measures deviations below a certain threshold, typically the mean or a target return. Unlike variance, semivariance better captures investors’ aversion to losses. Despite its intuitive appeal, semivariance has been considered analytically intractable and numerically challenging, prompting scholars to explore various approximations over the years. In this paper, we show that optimization problems involving the semivariance or the more general Lower Partial Moments can be addressed as standard convex programs, with computational complexity comparable to the Mean-Variance framework. This breakthrough simplifies the use of downside risk measures, making them viable for both academic research and real-world applications. • An exact analytical expression for the symmetric semicovariance matrix is provided. • An alternative LPM representation is proposed without indicator functions. • Gradient and Hessian of LPMs are computed, where they exist. • Semivariance/LPM optimization is shown to be a convex, tractable problem. • The model supports real-world constraints while preserving convexity.