Constant Force Within a Year of Age¶

Definition¶

For integer age $x$ , constant-force interpolation, also called the constant force of mortality (CFM) assumption, takes $\mu_{x+s}=\mu_x^*$ for $0\le s<1$ $1$ . If the annual survival probability is $p_x>0$ $0$ , the required force is $\mu_x^*=-\log p_x$ $- lo g p_{x}$ . Different years may have different forces.

Interpretation¶

Survival declines exponentially within each annual interval. A piecewise constant force can match all integer-age entries without assuming one exponential lifetime distribution over every age.

Force of mortality from age 60 to 61 when one-year survival is 0.9: the constant force minus log 0.9 with its shaded area, and the dashed UDD force rising from 0.100 to 0.111.

The figure takes $p_{60}=0.9$ from the Examples. Under CFM the force is flat at $\mu_{60}^*=-\log0.9\approx0.105$ throughout the year; the shaded area up to $s$ is $s\mu_{60}^*$ , so ${}_sp_{60}=\mathrm{e}^{-s\mu_{60}^*}=0.9^s$ $0. 9^{s}$ . The dashed curve is the uniform distribution of deaths (UDD) force $q_{60}/(1-sq_{60})$ $s q_{60})$ , which rises from $0.100$ to about $0.111$ . Both forces have area $-\log0.9$ over the whole year, so they agree at $s=1$ $1$ , but at $s=\tfrac{1}{2}$ CFM gives $\sqrt{0.9}\approx0.94868$ and UDD gives $0.95$ .

Change $p_{60}$ and $s$ : the flat force $\mu^*_{60}=-\log p_{60}$ gives ${}_sp_{60}=p_{60}^s$ $p_{60}^{s}$ , while the dashed UDD force rises. Both have the same area over the year, so the two curves meet only at $s=0$ and $s=1$ $1$ .

A life table supplies $p_x=l_{x+1}/l_x$ $l_{x + 1} / l_{x}$ . Integrating a constant force of mortality gives exponential survival within the year; matching its one-year value to $p_x$ determines $\mu_x^*=-\log p_x$ $- lo g p_{x}$ .

Key Results¶

${}_sp_x=p_x^s$ and $l_{x+s}=l_xp_x^s$ for $0\le s\le1$ $1$ .
Within one interval, ${}_tp_{x+s}=p_x^t$ for $s+t\le1$ $1$ .
For an interval crossing an integer age, multiply survival probabilities for its successive pieces.
CFM and UDD agree at integer ages but generally differ at fractional ages.

Examples¶

With $p_{60}=0.9$ $0.9$ , half-year survival is $\sqrt{0.9}\approx0.94868$ under CFM, compared with $0.95$ under UDD.

Prerequisites¶

Uniform Distribution of Deaths

Sources¶

Life Insurance · Lecture 2

Aliases: CFM interpolation, exponential life-table interpolation, CFM, constant force of mortality assumption, constant force assumption Keywords: fractional age assumption, s p x = p x to the power s, mu* = -log p x, piecewise constant force of mortality, exponential survival within the year Symbols: mu, square root

Last updated: 17 October 2026 21:53