Summary

In this chapter we consider the effect on annuity and insurance valuation of interest rates varying with the duration of investment, as summarized by a yield curve, and of uncertainty over future interest rates, which we will model using stochastic interest rates. We introduce the concepts of diversifiable and non-diversifiable risk and give conditions under which mortality risk can be considered to be diversifiable. In the final section we demonstrate the use of Monte Carlo methods to explore distributions of uncertain cash flows and loss random variables through simulation of both future lifetimes and future interest rates.

The yield curve

In practice, at any given time interest rates vary with the duration of the investment; that is, a sum invested for a period of, say, five years, would typically earn a different rate of interest than a sum invested for a period of 15 years or a sum invested for a period of six months.

Let v(t) denote the current market price of a t year zero-coupon bond; that is, the current market price of an investment which pays a unit amount with certainty t years from now. Note that, at least in principle, there is no uncertainty over the value of v(t) although this value can change at any time as a result of trading in the market.