Skewness in Financial Returns: Evidence from the Portuguese Stock Market

Abstract

Utility maximization is the principle behind investment choice. The conventional mean-variance (MV) equilibrium framework (two-parameter model) requires either the normality of the return distributions or quadratic utility functions. Researchers have proposed different statistical distributions for pricing financial assets. However, the pertinence of symmetry analysis exceeds the pure determination of the statistical distributions. The traditional CAPM assumes that investors care only about the mean and variance of returns, implying that upside and downside risks are viewed with equal distaste. Other authors have shown that CAPM-based valuation measures are problematic when market timing strategies and their subsequent non-normal returns are considered. Also, investors typically distinguish between upside and downside risk. Thus, the basic underpinnings of the CAPM are suspect, and its risk measure beta is equally dubious. Models that allow for some asymmetry of the returns (two or three-parameter models) and require logarithmic or cubic utility functions have been proposed. Rubinstein (1976) attempted to model the asymmetry in a portfolio context by deriving an equilibrium pricing formula “similar” to the traditional CAPM, although under the assumption that the market portfolio returns follow a lognormal distribution. However, the lognormal curve, although allowing for some asymmetry, “is a two-parameter family of distribution. In this sense, the lognormal is just as restrictive as the normal.“ (Sortino – Forsey, 1996, p. 38) Moreover, over relatively short time periods both approaches will yield identical estimates for systematic risk, which, in turn, will provide equal performance estimates. Markowitz (1959), realizing that investors frequently associate risk with the failure to achieve some minimum target return, offered an alternative risk proxy known as smivariance, which is calculated only in those periods where the returns are