A Portfolio Performance Index and Its Implications

Abstract

Cramer's Large Deviation Theorem is used to formalize a modern time series variant of the Safety-First, loss aversion criterion, providing a behavioral foundation for a new portfolio performance index. When returns are normally distributed, the performance index is proportional to the squared Sharpe Index, so the mean-variance efficient tangency portfolio is optimal, and the CAPM results when all investors behave this way. When returns are non-normally distributed, the index generalizes to a different, preference parameter-free formula, which is easily estimable without prior knowledge of the distribution. The tangency portfolio is no longer optimal; what results instead is a non-CAPM relationship between an asset's expected excess return and the return's covariance with an exponential function of the performance index maximizing portfolio. The endogenous loss aversion induces a preference not only for higher mean and lower standard deviation, like the Sharpe Index, but also for higher skewness and lower kurtosis in non-normal cases, with the potential to explain the demand for portfolio insurance and/or other skewness enhancing strategies.