Coherent Distortion Risk Measures—A Pitfall

Concave distortion risk measures were introduced in the actuarial literature by Wang (1996). Loosely speaking, such a risk measure assigns a ”distorted expectation” to any distribution function. The expectation is distorted by a so-called ”distortion function”. Concavity of the distortion function ensures that the risk measure preserves stop-loss order. Consider two random couples with identical marginal distributions but of which the dependency structure differs. Assume that the covariance of the second couple exceeds the covariance of the first one. Let us now consider a risk measure for the sum of the components of each couple. One would expect that any reasonable risk measure will lead to a smaller real number for the sum of the components of the first couple. However, we will demonstrate that this property does not hold in general for concave distortion risk measures. Moreover, for any such risk measure, it is possible to construct an example where the correlation order is not preserved. Despite this theoretical result, some simulation-based testing indicates that most well-known concave distortion risk measures for sums of random couples with given marginals frequently do preserve the order of the correlations.