This paper determines the optimal asset allocation of pure endowments insurance contracts, maximizing the expected utility of terminal surplus under a budget constraint. The market resulting from the combination of insurance and financial products is incomplete owing to the unhedgeable mortality of the insured population, modelled by a Poisson process. For a given equivalent measure, the optimal wealth process is obtained by the martingale approach, and the investment strategy replicating at best this process is obtained either by martingale decomposition or by dynamic programming. The paper also illustrates this method for CARA and CRRA utility functions.

Due to the presence of mortality risk, which is not yet hedgeable by traditional financial tools, the combination of life insurance and financial markets is incomplete and standard utility optimization methods have to be used with care. The contribution of this paper is precisely to show how the celebrated martingale approach, developed by Cox and Huang (1989) may be used to handle asset allocation problems in life insurance business. Two classical ways are usually exploited to study the optimal asset allocation of insurance contracts. The first one is the martingale approach, as already mentioned above. However, the existing literature based on this orientation, neglects the mortality risk. Interested readers may refer to Boulier et al. (2001) and Deelstra et al. (2003 and 2004) for examples of management and design of a pension fund. The second method relies on stochastic control and the resolution of the Hamilton Jacobi Bellman equation. This approach was successfully applied to the management and pricing of a wide variety of insurance contracts with exponential CARA (Constant Absolute Risk Aversion) utility. Some results of this paper will be compared with those obtained by Young and Zariphopoulou (2002). Another application of stochastic control is the management of one-life annuity studied by Menoncin et al. (2004)..

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