Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

Abstract

In this paper, we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures is asymptotically equivalent if the ratio of the worst‐case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a noncoherent risk measure and a coherent risk measure, as the worst‐case value of a noncoherent risk measure under dependence uncertainty is typically difficult to obtain. The main contribution of this paper is to establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.

Dual Utilities Under Dependence Uncertainty

“…Remark 4.4. The asymptotics as n → ∞ of the worst-case aggregate values of dual utilities (under the term distortion risk measures) has been studied in Wang et al (2015) and Cai et al (2018) under different settings. In particular, Cai et al (2018) showed that the ratio…”

Worst-Case Range Value-at-Risk with Partial Information

“…For aggregate risk models, that is, n > 1 in (1), finding bounds for quantities related to the sum of random variables with the knowledge of marginal distributions is typically called the Fréchet problem, where the complete uncertainty of the dependence is typically assumed. For recent research on Fréchet problems for VaR and convex risk measures, see , Wang et al (2013), Bernard et al (2014) and Cai et al (2017). For VaR bounds with partial dependence information in addition to the marginal information, see Bernard et al (2016bBernard et al ( , 2017a, Bernard and Vanduffel (2015) and Puccetti et al (2016Puccetti et al ( , 2017, amongst others.…”

Collective Risk Models With Dependence Uncertainty

“…The equivalence (2.2) was shown in Puccetti and Rüschendorf (2014) and Puccetti et al (2013) under different conditions, and with generality in . For equivalence of type (2.2) under the setting of inhomogeneous marginal distributions and for risk measures other than VaR and ES, see Embrechts et al (2015), and Cai et al (2017). The convergence rate of (2.2) is given in the following lemma:…”