Spectral measures of risk: A coherent representation of subjective risk aversion

Abstract

We study a space of coherent risk measures M φ obtained as certain expansions of coherent elementary basis measures. In this space, the concept of “risk aversion function” φ naturally arises as the spectral representation of each risk measure in a space of functions of confidence level probabilities. We give necessary and sufficient conditions on φ for M φ to be a coherent measure. We find in this way a simple interpretation of the concept of coherence and a way to map any rational investor's subjective risk aversion onto a coherent measure and vice-versa. We also provide for these measures their discrete versions M ( N) φ acting on finite sets of N independent realizations of a r.v. which are not only shown to be coherent measures for any fixed N, but also consistent estimators of M φ for large N.