Examining Changes in Reserves Using Stochastic Interest Models

A fundamental problem in actuarial science is the determination of the reserves necessary to meet future obligations. Reserves are useful quantities because they summarize a vector of discounted cash flows. However, through this summarization, they mask the dynamic nature of interest rates. To study the effects on reserves of the dynamic nature of a stochastic interest environment, we look at a change in discounted reserves and study the potential short-term consequences of changes in the interest environment. Both the traditional linear ARIMA models and the newer nonlinear autoregressive conditionally heteroskedastic (ARCH) processes are used to model the force of interest stochastically. We find that, in general, the next period reserve is a function of the previous interest rate. However, this is not true when the force of interest can be modeled as a white noise process. Explicit formulas compute changes in discounted reserves for linear interest rate processes. For nonlinear processes, we describe some approximations and exact simulation algorithms for these computations. An empirical example using a U.S. Treasury security series is presented and evidence of ARCH behavior is found. KEY WORDS: ARIMA, reserve discounting, interest rates, simulation models, stochastic process.