Examining Changes in Reserves Using Stochastic Interest Models

Introduction On balance sheets, assets must equal liabilities plus net worth of the firm. An important component of these liabilities for insurers is the reserve, the portion of a firms' assets set aside to meet future uncertain obligations arising from insurance contracts. Although the obligations of each contract are contingent upon uncertain future events, and thus may be modeled stochastically, the reserve set aside is a single number. There are limitations when using a single number to summarize a stochastic quantity. However, reserves play a prominent role in financial statements and thus these quantities are important to managers of insurance organizations. Several important problems in actuarial science rely heavily on the determination of a reserve. To illustrate, if a company or a block of business is to be traded on the open market, a value must be determined for the associated set of obligations. Thus, it is useful to think of a reserve as the "value" associated with a set of stochastic obligations. As another application, reserves traditionally have been used as a measure of an organization's financial strength. In this context, the reserve should be larger than the "value" of obligations, because a conservative approach should be taken for assessing potential future obligations. Life insurance and annuity reserves are calculated by summarizing discounted cash flows, where the discounting is done with respect to investment earnings, as well as decrements due to mortality, disability, policy lapse, and so on, that may be applicable to a particular policy. For brevity, this articles considers only investment earnings and the mortality decrement. Extensions to the multidecrement case are straightforward. In the traditional insurance literature, as in Jordan (1967), the deterministic assumption dominates the development of the theory of life contingencies. Namely, mortality occurs according to a known mortality table and the interest rate is assumed to have a deterministic value. One step further is to allow the age at death to be a random variable, although the interest rate is assumed to be deterministic--a "semi-stochastic" approach followed in Bowers et al. (1986). Insurance literature has generalized the traditional theory of life contingencies by introducing stochastic variation in interest rates (Boyle, 1976; Waters, 1978; Panjer and Bellhouse, 1980; Bellhouse and Panjer, 1981; Giaccotto, 1986; Dhaene, 1989; Frees, 1990; and Beekman and Fuelling, 1990, 1991). Additional stochastic interest models from financial economics are discussed below. This article computes reserves as (conditional) expectations of sums of future cash flows. Motivation for this approach can be found in, for example, Bowers et al. (1986) for the semi-stochastic approach and Buhlmann (1992) for models using stochastic interest. Here, we are primarily concerned with quantifying changes in reserves from one financial period to the next. Changes in reserves could be used to quantify the amount of profit released, as in Ramlau-Hansen (1988), which studied gains and losses emerging from margins built into mortality and other decrements and ignored those arising from investments. To complement that work, we focus on changes arising from stochastic interest rates and do not explicitly consider margins built into other decrement rates. Changes in value of future obligations due to dynamic models of interest have been considered extensively in the financial economics literature, in particular as part of immunization theory, which deals with instantaneous changes in value. Here, we examine changes in value from one financial period to the next. The new idea of examining changes in reserves can be illustrated by considering the following simple scenario. Let {[y.sub. s]} represent the random force of interest in the sth period. As argued in Frees (1990), [y.sub. …