Stochastic Life Contingencies with Solvency Considerations

The extension of the theory of life contingencies to a stochastic interest environment and its application to solvency valuation are discussed. Although life contingencies are widely used in traditional actuarial valuations of life insurance contracts, certain complications arise in a stochastic interest environment that are not evident when using traditional deterministic interest assumptions. In particular, many insurance functions can no longer be expressed in a simple form, resulting in a loss of the intuitive appeal of these functions. In this paper, a stochastic interest environment is introduced and analyzed in terms of its effects on insurance functions. Although the model is less general than others introduced in the literature, it is sufficiently flexible to handle the volatility and certain autocorrelation aspects of interest series. Its main advantage is the simple form of the resulting insurance functions and, hence, its intuitive appeal. To examine the performance of a block of business, the assets as well as the liabilities are considered. For liabilities of a block of business under a common stochastic interest environment, limit theorems for approximating the behavior of sums of policies are no longer readily available. Even if the mortality experiences of the policies are independent, the liabilities are not independent because of the common interest environment. By considering assets as well as liabilities, matching of cash flows reduces the volatility of surplus, defined to be assets in excess of liabilities. In fact, under an extreme type of matching, limit laws for sums of homogeneous policies can be described under more general interest environments than those described above. OVERVIEW This paper addresses the stochastic theory of the valuation of a risk-taking enterprise from a solvency perspective. For concreteness, the enterprise is assumed to be a life insurance company. In the development, the incorporation of elements of financial economics is emphasized where possible. However, it is not the intent of this paper to show how weU-developed