Economic Capital and the Aggregation of Risks Using Copulas

Abstract

Insurance companies measure and manage capital across a broad range of diverse business products. Thus there is a need for the aggregation of the losses from the various business lines whose risk distributions vary. Risk dependencies between losses from different business lines have long been recognised in the insurance industry as integral factors driving the insurer's aggregate loss process. However, in the past, there has been limited attempt at adequately modelling the dependence structure to be factored in the aggregation process for capital determination purposes. The current industry standard is to solely use linear correlations to describe the dependence structure. While being computationally convenient and straightforward to understand, linear correlations fail to capture all the dependence structure that exist between losses from multiple business lines. Other more general dependence modelling techniques such as copulas have become popular recently. In this paper, we address the issue of the aggregation of risks using copula models. Copulas can be used to construct joint multivariate distributions of the losses and provide a rather flexible and realistic model of allowing for the dependence structure, while separating the effects of peculiar characteristics of the marginal distributions such as thickness of tails. This modelling structure allows us to explore the impact of dependencies of risks on the total required economic capital. Using numerical illustrations based on Australian general insurance data, the sensitivities of the capital requirement to the choice of the copula and other modelling assumptions are investigated. The related issue of the diversification benefit from operating multiple business lines in the context of aggregation of risks by copulas is also explored. The key conclusion is that there is a large variation in the capital requirement as well as diversification benefit under different copula assumptions. The results of this paper serve as a reminder to actuaries and other industry practitioners of the significance of choosing an appropriate aggregation model for capital purposes.

Keywords: capital at risk, risk measures, risk aggregation, dependence, and copulas

JEL Classification: G22