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How Superadditive Can a Risk Measure Be?

Wang, R., Bignozzi, V. & Tsanakas, A. (2015). How Superadditive Can a Risk Measure Be?. SIAM Journal on Financial Mathematics, 6(1), pp. 776-803. doi: 10.1137/140981046


In this paper, we study the extent to which any risk measure can lead to superadditive risk assessments, implying the potential for penalizing portfolio diversification. For this purpose we introduce the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure, by yielding the worst-possible diversification ratio across dependence structures. One of the main contributions is demonstrating that, for a wide range of risk measures, the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extremeaggregation measure induced by a distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. In particular, the extreme-aggregation measure induced by a convex shortfall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management.

Publication Type: Article
Additional Information: Copyright Society for Industrial and Applied Mathematics 2015
Publisher Keywords: distortion risk measures; shortfall risk measures; expectiles; dependence uncertainty; risk aggregation; diversification
Subjects: H Social Sciences > HD Industries. Land use. Labor > HD61 Risk Management
Departments: Bayes Business School > Actuarial Science & Insurance
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