Computing and Estimating Distortion Risk Measures: How to Handle Analytically Intractable Cases?

Abstract

In insurance data analytics and actuarial practice, distortion risk measures are used to capture the riskiness of the distribution tail. Point and interval estimates of the risk measures are then employed to price extreme events, to develop reserves, to design risk transfer strategies, and to allocate capital. Often the computation of those estimates relies on Monte Carlo simulations, which, depending upon the complexity of the problem, can be very costly in terms of required expertise and computational time. In this article, we study analytic and numerical evaluation of distortion risk measures, with the expectation that the proposed formulas or inequalities will reduce the computational burden. Specifically, we consider several distortion risk measures––value-at-risk (VaR), conditional tail expectation (cte), proportional hazards transform (pht), Wang transform (wt), and Gini shortfall (gs)––and evaluate them when the loss severity variable follows shifted exponential, Pareto I, and shifted lognormal distributions (all chosen to have the same support), which exhibit common distributional shapes of insurance losses. For these choices of risk measures and loss models, only the VaR and cte measures always possess explicit formulas. For pht, wt, and gs, there are cases when the analytic treatment of the measure is not feasible. In the latter situations, conditions under which the measure is finite are studied rigorously. In particular, we prove several theorems that specify two-sided bounds for the analytically intractable cases. The quality of the bounds is further investigated by comparing them with numerically evaluated risk measures. Finally, a simulation study involving application of those bounds in statistical estimation of the risk measures is also provided.

Acknowledgments

The authors are very appreciative of valuable insights and useful comments provided by two anonymous referees, which helped to substantially improve the article. Also, much of this work was completed while the first author was a Ph.D. student in the Department of Mathematical Sciences at the University of Wisconsin–Milwaukee.