A Unifying Approach to Pricing Insurance and Financial Risk

The actuarial and financial approach to the pricing of risk remain different despite the increasingly direct interconnection of financial and insurance markets. The difference can be summarized as pricing based on classical risk theory vs. non-arbitrage pricing. However, comparable pricing principles are of importance when it comes to transferring insurance risk to financial markets and vice versa as it is done e.g. by alternative risk-transfer instruments for derivative products. Incompatibilities blur business opportunities or may open up the possibility to arbitrage. For these situations, the paper aims to bridge the gap between insurance and finance by extending the non-arbitrage pricing principle to insurance. The main obstacle that has be tackled is related to the incompleteness of the insurance market. It implies that equivalent martingale probabilities are not uniquely defined. By the information theoretical maximum entropy principle a sensible way to choose a particular equivalent martingale measure is found. This approach parallels the successful application of the maximum entropy principle in finance. The paper pays special attention to the role that investment opportunities beyond risk-fee investments play for insurance operations. Equivalent martingale probabilities for the combination of insurance operation risk-free investment and a risky investment are determined. They turn out to be connected to the Esscher measure. This recovers a generalized form of a well-known actuarial premium calculation principle. The sketched approach is further investigated for typical reinsurance structures like stop-loss and excess-of-loss reinsurance. Arbitrage-free reinsurance premiums are calculated. A numerical example stresses the influence that characteristics of risky investment opportunities have on arbitrage-free premiums.