Force of Mortality¶

Definition¶

Under the course's absolutely continuous survival model, the force of mortality at attained age $x+t$ (the age reached $t$ years after age $x$) is

where $S_x(t)>0$ and the derivative exists. Equivalently, it is the limit of $\mathbb{P}(t<T_x\le t+h\mid T_x>t)/h$ as $h\downarrow0$$0$.

Interpretation¶

Mortality is measured per unit of time among those still alive. The force can exceed one; it is a rate and must be multiplied by a short duration to approximate a probability.

The figure uses the Examples force $0.02$, so $S_x(t)=\mathrm{e}^{-0.02t}$${\mathrm{e}}^{-0.02t}$. At $t=20$ the tangent to $S_x$ has slope $-f_x(20)$. Dividing the height $S_x(20)$ by this rate of decline gives the time the tangent needs to reach zero, $S_x(20)/f_x(20)=1/\mu_{x+20}=50$ years, so a high force means a tangent that is steep relative to the height of the curve. Because this force is constant, the tangent at any $t$ reaches zero 50 years later.

Here $T_x$ is future lifetime, $S_x(t)$ is its survivor function, and $f_x(t)$ is its probability density function. Dividing the density by the probability of still being alive expresses a death rate among survivors rather than among all lives originally insured.

Key Results¶

• ${}_tp_x=\exp(-\int_0^t\mu_{x+s}\,\mathrm{d}s)$ and $f_x(t)={}_tp_x\mu_{x+t}$${}_t{p}_x{\mu }_{x+t}$.
• A constant force $\lambda>0$ gives ${}_tp_x=\mathrm{e}^{-\lambda t}$${\mathrm{e}}^{-\lambda t}$, an exponential future lifetime.
• An increasing force means that the death rate among survivors rises with time survived (positive duration dependence); a decreasing force means that it falls (negative duration dependence).
• For a finite force at $x$, ${}_hq_x=\mu_xh+o(h)$$o(h)$, with little-$o$ notation.

Examples¶

At constant force $0.02$ per year, the one-year death probability is $1-\mathrm{e}^{-0.02}\approx0.01980$$0.01980$, rather than exactly $0.02$.

Prerequisites¶

• Future Lifetime
• Survivor Function
• Probability Density Function
• Derivative and Differentiability

Related Concepts¶

• Makeham's Mortality Law
• Constant Force Within a Year of Age

Sources¶

• Life Insurance · Lecture 1

Aliases: hazard rate, instantaneous mortality rate, hazard function, mortality intensity, intensity of mortality Keywords: mu x+t = f(t)/S(t), t p x = exp(-integral of mu), duration dependence, constant force gives an exponential lifetime, death rate among survivors Symbols: integral, lambda, mu

Last updated: 17 October 2026 21:53