Stochastic orders and non-Gaussian risk factor models

Abstract

The main results of this paper are monotonicity statements about the risk measures value-at-risk (VaR) and tail value-at-risk (TVaR) with respect to the parameters of single and multi risk factor models, which are standard models for the quantification of credit and insurance risk. In the context of single risk factor models, non-Gaussian distributed latent risk factors are allowed. It is shown that the TVaR increases with increasing claim amounts, probabilities of claims and correlations, whereas the VaR is in general not monotone in the correlation parameters. To compare the aggregated risks arising from single and multi risk factor models, the usual stochastic order and the increasing convex order are used in this paper, since these stochastic orders can be interpreted as being induced by the VaR-concept and the TVaR-concept, respectively. To derive monotonicity statements about these risk measures, properties of several further stochastic orders are used and their relation to the usual stochastic order and to the increasing convex order are applied.