The emphasis in this thesis is on the analysis of insurance contracts which combine traditional actuarial risk and financial risk. A simple example is a unit-linked pure endowment contract with guarantee. With this life insurance contract, the sum insured payable to the policy-holder at the term of the contract is contingent upon survival and not fixed a priori, but linked to the development of some stock index and guaranteed against falling below some amount. The actuarial risk in this contract stems from the uncertainty of not knowing whether or not the policy-holder will survive until the term of the contract, and the financial risk is related to the performance of the underlying index. Another example of an insurance contract with an inherent financial risk is a financial stop-loss contract. This reinsurance contract differs from traditional stop-loss contracts in that the insurer's total losses have both an insurance and a financial component.
In the first part of the thesis, we focus on the problem of hedging and pricing payment streams generated by unit-linked insurance contracts using the criterion of risk-minimization. A widely used approach is based on the assumption of risk-neutrality with respect to mortality, and we first demonstrate within simple discrete time models how this can be derived from the asymptotic behaviour of the insurer's loss from a portfolio of unit-linked contracts as the number of policies increases. Next, risk-minimizing hedging strategies are determined explicitly for a portfolio of independent identical unit-linked pure endowment contracts with guarantee in the special case where the financial market is described by the Cox-Ross-Rubinstein model. These results characterize the combined insurance and financial risk in the contracts and decompose this risk into a hedgeable part and a non-hedgeable part. In addition, we show how payment streams can be incorporated into the theory of risk-minimization in a continuous time set-up. This extension provides a framework for the analysis of insurance contracts that generate genuine payment streams. In this setting, risk-minimizing hedging strategies are worked out for general unit-linked life insurance contracts driven by a Markov jump process with a finite state space and for non-life insurance risk processes where claim amounts and premiums are affected by some traded price index, for example a claim inflation index.
The second part of the thesis deals with financial transformations of two classical actuarial premium calculation principles, the variance and standard deviation principles. The corresponding financial valuation principles were derived by Schweizer (1997) via indifference arguments which embedded their actuarial counterparts in a financial framework. Under the financial variance (or standard deviation) principle, the fair premium equals the expected value of the claim under the variance optimal martingale plus a safety-loading which is proportional to the variance (or standard deviation) of the non-hedgeable part of the claim. We complement existing results by deriving optimal hedging strategies for the two financial valuation principles when the discounted price process of the traded assets is a continuous semimartingale. For the variance principle, the optimal strategy differs from the mean-variance hedging strategy only by a correction term which is independent of the claim under consideration; for the standard deviation principle, the result is more complicated. Furthermore, we provide an alternative direct characterization of the financial standard deviation principle which does not involve an indifference argument.
A separate study is devoted to an investigation of the impact on the fair premiums of the amount of information available to the seller of the insurance contracts. This includes a comparison result of mean-variance hedging errors under two different filtrations, which is obtained via a projection argument for Hilbert spaces. In particular, this result allows the derivation of simple bounds on the fair premiums under the financial variance and standard deviation principles in the situation where the insurance claim involves two stochastically independent sources of randomness, purely financial risk and pure insurance risk. Explicit formulas are obtained for the fair premiums and the optimal trading strategies under different levels of information for unit-linked insurance contracts and for some reinsurance contracts with an inherent financial risk.
This page was last updated June 2000