Pricing insurance contracts with an existing portfolio as background risk

Abstract

We develop and investigate a premium principle that explicitly takes into account the impact of a new risk on some insurer's existing portfolio. Specifically, we propose the notion of an indifference premium for a new risk conditioned on an existing portfolio acting as background risk. The resulting premium rule, which in our case depends on the joint distribution of the new risk and the existing portfolio, is analyzed in detail with respect to its mathematical properties. In order to underline the differences between our approach and the literature on law-invariant premium rules, special attention is given to the indifference premium behaviour with respect to some well-known dependence concepts. Axiomatic and continuity properties of the proposed indifference premium rule are also investigated. To demonstrate the practical relevance of our approach, we consider a portfolio of exchangeable risks and investigate the role of the portfolio's dimension on the price of a risk to be added. This illustrates the (limits of) diversification benefits under the flexible exchangeability assumption on the joint distribution of a sequence of risks.