The Gompertz-Makeham Law of Mortality

Abstract

In a number of earlier chapters I made reference to the Gompertz-Makeham (or just plain Gompertz) Law of Mortality. In particular, I used this law of mortality in Chapter 7 when I introduced and described Jared's tontine payout rate and in Chapter 2 when I discussed the probability density function of the “present value” of the tontine versus annuity cash-flow payout. Well, in this appendix I offer a brief explanation of this well-known law, as well as the analytic representation I used plus a bit of information about the person after whom it is named. For those interested the technical details, see Milevsky (2012), which is a popular book covering the most important equations in the field of retirement income planning. For starters, age-dependent mortality rates for adults – for example, those displayed in Table 5.3 but continued to older ages – seem rather arbitrary at first, but there is actually an underlying pattern to them. In particular, for people between the ages of twenty and ninety – mortality rates not only increase consistently every year with age, they actually increase by approximately 9% every year. In mathematical symbols, if the starting mortality or death rate per year was q percent at age y, (for example, q = 2% at age fifty) then it is q (1 + z ) percent in year ( y + 1) and then q (1 + z ) 2 percent in year ( y + 2), and then q(1 + z ) 3 percent in year ( y + 3), and so on, where z is approximately 9%. Human adult mortality rates – regardless of what particular group of humans or population you select and whether it is in the seventeenth or nineteenth or twenty-first century – are an exponentially increasing function of age with a (growth rate) parameter of 9%. What this also means (mathematically) is that if you take the logarithms of these annual mortality rates denoted by q , they can be approximated quite nicely by a straight line and determined by a slope and an intercept.