Symmetry vs. Asymmetry, Stochastic Dominance Optimization, and The Impact on Capital Asset Pricing

Abstract

This paper models the risk tradeoff between symmetry and asymmetry in the distribution of asset return as a core consideration for rational investor decision making under uncertainty and presents a closed-form solution for determining the optimal market portfolio using a classic utility maximization framework and stochastic dominance optimization. The optimization yields a portfolio separation for both risk-averse investors as well as those who also possess a preference for upside gains and an aversion to downside losses. In equilibrium, in addition to requiring compensation for bearing volatility risk – which is symmetrical in gains and losses – investors also require additional compensation for bearing the systematic risk associated with distributional asymmetries. The model serves as an expansion of CAPM and, as is demonstrated both theoretically and empirically, solves the long-lasting puzzle of the beta anomaly.

Keywords: Swap-Variance, Asymmetry, Expected Utility, Stochastic Dominance, Gain-Preference, Loss Aversion, CAPM

JEL Classification: D81, G02, G11, G12.