We study a space of coherent risk measures M_? obtained as certain expansions of coherent elementary basis measures. In this space, the concept of “Risk Aversion Function” ? naturally arises as the spectral representation of each risk measure in a space of functions of confidence level probabilities. We give necessary and su .cient conditions on ? for M ? 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) 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_? for large N.Finally,wefind in our results some interesting and not yet fully investigated relationships with certain results known in insurance mathematical literature.

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Expected　Shortfall　Risk　measure　value-at-risk　VaR　conditional　CVaR　coherence　quantile　sub-additivity　
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