Discrete annuities using truncate stochastic interest rates: The case of a Vasicek and Ho-Lee model

A subject often recurring in the recent financial and actuarial research, is the investigation of present value functions with stochastic interest rates. Only in the case of uncomplicated payment streams and rather basic interest rate models, an exact analytical result for the distribution function is available. In the present contribution, we introduce the concept of truncate stochastic interest rates, useful to adapt general stochastic models to specific financial requirements, and we show how to obtain in that case analytical results for bounds for the present value. We employ our method in extension for the Ho-Lee model and the Vasicek model. We illustrate the accuracy of the approximations graphically, and we use the bounds to estimate the Value-at-Risk.