Makeham's Mortality Law¶

Definition¶

Makeham's law specifies the force of mortality at age $y\ge0$ as $\mu_y=A+Bc^y$$B{c}^y$. In the standard increasing-age model, $A\ge0$$0$, $B>0$ and $c>1$$1$. These constants determine both an age-independent component and an exponentially increasing age component.

Interpretation¶

The constant term allows a background mortality rate, while the exponential term models increasing mortality at older ages. This is a model assumption, not a universal law of human mortality.

The figure plots the Examples parameters $A=0.001$$0.001$, $B=0.0001$ and $c=1.1$ on a logarithmic vertical scale. At young ages $\mu_y$ stays close to the constant $A$ (dashed orange); at older ages it follows the Gompertz term $Bc^y$, a straight line on this scale because $\log(Bc^y)$ is linear in $y$. At age 40 the two parts give $\mu_{40}\approx0.00553$$0.00553$.

Key Results¶

• For a life aged $x$,

  $${}_tp_x=\exp\left[-At-\frac{Bc^x(c^t-1)}{\log c}\right].$$

  Here $\log$ is the natural logarithm, because $\int_0^tBc^{x+s}\,\mathrm{d}s=Bc^x(c^t-1)/\log c$$1)/\log c$; a base-10 logarithm would give the wrong survival probability.

• Setting $A=0$ gives the Gompertz form.

• Mortality depends on attained age $x+t$$t$, so the starting age must be retained in calculations.

Examples¶

With $A=0.001$$0.001$, $B=0.0001$ and $c=1.1$$1.1$, the force at age 40 is $0.001+0.0001(1.1)^{40}\approx0.00553$$0.00553$. The survival formula integrates this changing force over the entire future interval.

Prerequisites¶

• Force of Mortality

Related Concepts¶

• Future Lifetime
• Life Table

Sources¶

• Life Insurance · Lecture 1

Aliases: Makeham law, Gompertz–Makeham mortality, Makeham's law, Gompertz–Makeham law Keywords: mu x = A + B c^x, Gompertz law (A = 0), exponentially increasing mortality, background mortality Symbols: integral, mu

Last updated: 17 October 2026 21:53