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

We study a space of coherent risk measures Mphi obtained as certain expansions of coherent elementary basis measures. In this space, the concept of "risk aversion function" phi 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 phi for Mphi 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 Mphi((N)) 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 Mphi for large N. (C) 2002 Elsevier Science B.V. All rights reserved.