Spectral Measures of Risk: a Coherent Representation of Subjective Risk Aversion

We study a space of coherent risk measuresMφ 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 Mφ 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 Mφ for large N .