Spectral risk measure

A spectral risk measure weights the quantiles of an outcome distribution by a risk aversion function chosen by whoever does the measuring. Carlo Acerbi formalized it in "Spectral measures of risk: a coherent representation of subjective risk aversion," Journal of Banking & Finance, vol. 26, no. 7, 2002, pp. 1505-1518. For an outcome X, with quantile q(p) at level p, it reads M(X) = - ∫ φ(p) q(p) dp over [0, 1], where φ is the spectrum. Acerbi shows that M is coherent in the sense of Artzner, Delbaen, Eber and Heath (1999) if and only if φ is positive, decreasing and integrates to 1. The decreasing condition says that a worse outcome never weighs less than a better one. Expected shortfall at level α is the case where φ equals 1/α on [0, α] and zero elsewhere; only the tail counts, and within it the user is neutral. VaR corresponds to a point mass at α, which is nothing like a decreasing function, hence its failure of subadditivity. Every spectral measure is a mixture of expected shortfalls at different levels. The exponential spectrum, φ(p) = k e^(-kp) / (1 - e^(-k)), comes up often, because a single parameter k sets the aversion. Acerbi also gives a discrete estimator, a weighted sum of order statistics, that stays coherent at any fixed sample size. The usual criticism is that the choice of φ is arbitrary: two teams starting from the same simulations can come out with very different capital figures depending on their spectrum. In insurance, distortion measures with a concave function, the Wang transform among them, are the same family written on the survival function rather than on the quantiles.