Quantitative Finance

Chapter 8

Difference Equations for Life Annuities

8.1 CHAPTER SUMMARY

A life annuity is a product that pays its holder a constant monthly payment until his/her death. So it looks a little like a risky bond, but with “opposite” default risk. In this chapter, we analyze this product using already existing tools. No subsequent chapters of this book depend on this chapter, so it may be entirely omitted in the interest of time, or omitted on first reading.

8.2 Introduction

As we have already seen, a mortgage with initial amount X repaid in N equal payments each of A solves the difference equation boundary value problem

$V_{k+1}=(1+r)V_k-A$

$V_0=X,V_N=0$

With solution

$A=\frac{rX{(1+r)}^N}{{(1+r)}^N-1},$ (8.1)

If we turn this around, we can also consider it to be the equation ...