How Superadditive Can a Risk Measure Be?

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract 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. Permanent repository link

Reducing model risk via positive and negative dependence assumptions

“…The analytical computation of worst-possible bounds on Expected Shortfall (ES) is in general straightforward, while for the best-case ES partial analytical results can be found in Wang and Wang (2011) and Bernard et al (2014). For several classes of risk measures (including convex and distortion risk measures) Wang et al (2014) provide a systematic way to compute the worst (and best) possible bounds across any homogeneous portfolio.…”

A Review and Some Complements on Quantile Risk Measures and Their Domain

“…Assertion (1) of Corollary 4 extends a result of Wang et al (2015), who considered risk measures that are defined on a common convex cone containing L ∞ .…”

On aggregation sets and lower-convex sets

“…, n} = sup Y ∈B n (F ) VaR p (Y ). It can be easily verified that as n → ∞, the limit of sup Y ∈B n (F ) VaR p (Y ) exists; see ( [22] Proposition 2.1). Since ES p preserves the convex order and…”