A Note on the Link between Asymmetric Risk and Shortfall Risk

Abstract

Practitioners often choose intuitive measures of risk which are possibly neither coherent nor consistent with the second order stochastic dominance (SSD) in order to reflect their risk characteristic (i.e., aversion to losses and preference to gains). An example of such risk measures which have such characteristic is the exponential weighted mean square risk EWMSR). On the other hand, academics have proposed to use SSD preserving risk measures such as the shortfall probability which is a special case of mean-lower partial moments or the expected shortfall. The purpose of this note is threefold. First, we propose the linear exponential (LINEX) weighed variance as a measure of asymmetric risk. Like the EWMSR, the LINEX risk measure gives heavier penalty to losses than gains. However, the penalty to losses is higher for the LINEX risk measure than the EWMSR. Second, using the large deviations theory, we show that given a sufficiently large portfolio, investors who are inverse to shortfall risk will reasonably choose to maximize the Lagrangian utility function of the one-period portfolio return associated with an symmetric risk measure. Third, we provide a measure of the deviation between the minimum shortfall risk and a minimum asymmetric risk; and whence propose conditions under which the optimal asymmetric risk and the optimal shortfall risk can be reconciled. This suggests a possibility to approximate a SSD inconsistent risk measure of a large portfolio with a SSD consistent one.

Keywords: Linear exponential weighted mean square risk, Exponential weighted mean square risk, second order stochastic dominance, Large deviations, asymmetric risk, shortfall risk, utility function, optimal portfolio

JEL Classification: C190, C590, C130