Semivariance and semivolatility in portfolio selection: a unified framework for downside-risk portfolio models

This paper studies semivariance and semivolatility as downside-risk measures for portfolio selection. The analysis is motivated by the observation that downside-risk models are often presented as direct counterparts of mean-variance formulations, although different portfolio paradigms require risk measures with different structural properties. In particular, semivariance is a positively homogeneous functional of degree two and is therefore naturally suited to minimum-risk formulations, whereas semivolatility is positively homogeneous of degree one and is more appropriate for models based on risk budgeting, diversification ratios, and performance ratios. On this basis, we present a unified framework including minimum-risk portfolios, risk parity portfolios, most-diversified portfolios, and Sharpe/modified-Sortino-type ratio portfolios. For each class of models, we provide the mathematical formulation and discuss the role of convexity and homogeneity in the tractability of the associated optimization problems. The proposed framework offers a coherent interpretation of downside-risk portfolio design and provides a natural basis for empirical comparisons with classical variance-based benchmarks.