Future Lifetime¶

Definition¶

For a life currently aged $x\ge0$ $0$ , the future lifetime $T_x$ is the remaining time until death, conditional on being alive at age $x$ . If $T_0$ denotes age at death and $S_0(x)=\mathbb{P}(T_0>x)>0$ $0$ , then

\mathbb{P}(T_x>t)=\frac{S_0(x+t)}{S_0(x)},\qquad t\ge0.

The course assumes an absolutely continuous lifetime distribution. Thus $T_x$ is a continuous random variable, not a single number; its mean $\mathbb{E}[T_x]$ is the expected future lifetime.

Interpretation¶

The model updates the lifetime distribution when a person survives to a specified age. Two people at different ages generally have different remaining-lifetime distributions.

Survivor function S_0(y) = 1 - y/100 drawn as a falling blue line, with the rescaled curve P(T_60 > t) dashed in red from age 60 and reaching 0.75 at age 70.

The figure uses the uniform example: the blue line is $S_0(y)=1-y/100$ $y /100$ , and the dashed red line is $\mathbb{P}(T_{60}>t)$ drawn at attained age $60+t$ $t$ . Conditioning on survival to 60 rescales the remaining part of $S_0$ by $1/S_0(60)$ , so the new curve starts again at 1 and still reaches 0 at age 100; at age 70 it equals $S_0(70)/S_0(60)=0.3/0.4=0.75$ $0.75$ .

Move $x$ and $t$ : $\mathbb P(T_x>t)=S_0(x+t)/S_0(x)$ rescales the rest of $S_0$ so that it starts again at 1. Under a constant force the rescaled curve is the newborn's curve once more.

The survivor function $S_0$ gives survival from birth. By conditional probability, survival another $t$ years given survival to age $x$ is $S_0(x+t)/S_0(x)$ $t) / S_{0} (x)$ , which is the remaining-lifetime survivor function.

Key Results¶

Writing $S_x(t)=\mathbb{P}(T_x>t)$ $t)$ , conditioning gives $S_x(t+u)=S_x(t)S_{x+t}(u)$ whenever the conditioning probability is positive.
The density is $f_x(t)=-S_x'(t)$ almost everywhere.
A distribution of $T_x$ that does not depend on $x$ is a special property of a constant force of mortality (the memoryless property), not a general assumption.

Examples¶

If $T_0$ is uniform on $(0,100)$ , a survivor aged 60 has $T_{60}$ uniform on $(0,40)$ . Their probability of surviving another 10 years is $30/40=0.75$ $0.75$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 1

Aliases: remaining lifetime, residual lifetime of a life, future lifetime random variable, time until death, residual lifetime, time to death Keywords: S0(x+t)/S0(x), conditioning on survival to age x, age at death T0, memoryless property at constant force

Last updated: 17 October 2026 21:53