analytic mortality laws (De Moivre, Gompertz, Makeham)
Before fast computers, carrying a full life table of a hundred numbers was awkward, and even today a tidy formula can capture the shape of human mortality with just two or three parameters. An analytic (or parametric) mortality law is exactly that: a mathematical expression for the force of mortality mu-x, or the survival function, that smoothly describes how death risk changes with age. Each named law is a different guess about that shape.
A few are classics. De Moivre's law assumes deaths are spread uniformly up to a maximum age omega, so mu-x = 1/(omega - x) — simple, but unrealistically flat. Constant force assumes mu-x equals a fixed c at every age, which gives an exponential lifetime with no ageing at all. Gompertz's law, mu-x = B times c-to-the-x, captures the real observation that adult mortality rises roughly exponentially — risk doubling roughly every eight years. Makeham's law adds a constant term, mu-x = A + B times c-to-the-x, where the A absorbs age-independent dangers like accidents. The Weibull form, mu-x proportional to a power of x, is popular for failure times in engineering.
These laws are valued for being compact, smooth, and easy to integrate into survival and annuity formulas, and for letting actuaries extrapolate sensibly into very old ages where data are thin. But they are deliberate simplifications: real human mortality has an infant hump and an accident hump in the late teens that Gompertz misses, and at the very oldest ages the exponential rise appears to slow. A law is a useful caricature, to be fitted and checked against data, never mistaken for the truth.
Under Gompertz with mu-x = B c-to-the-x and roughly B=0.0001, c=1.09, the force at age 40 is about 0.0033 and at 80 about 0.105 — a 30-fold rise across 40 years, mirroring real adult mortality.
Gompertz turns the empirical fact that adult death risk roughly doubles every 8 years into two tidy parameters.
No analytic law fits human mortality at all ages: Gompertz and Makeham work well in mid-to-late adult life but miss the infant and young-adult humps, and may overstate risk at extreme old age. Choose and fit the law to the age range you actually need.