Survival Model

Study guide: Exam MAS-I

A Survival Model describes the distribution of a lifetime random variable $T \geq 0$ , representing the time until death or failure. The model is characterized by the survival function $S(t)$ , which gives the probability of surviving beyond time $t$ .

$S(t) = P(T > t)$

$= 1 - F(t)$

$f(t) = -S'(t)$

$= F'(t)$

$S(0) = 1$ , $S(\infty) = 0$ , and $S(t)$ is non-increasing
The conditional probability of dying between ages $x$ and $x+t$ , given alive at $x$ , uses actuarial notation: $_tq_x = P(T_x \leq t)$ and $_tp_x = P(T_x > t) = S(x+t)/S(x)$
$_{t}p_x = \dfrac{S(x+t)}{S(x)}$ and $_{t}q_x = 1 - {_t}p_x$
The Hazard Rate $\mu(t)$ (force of mortality) links $f$ , $S$ : $f(t) = \mu(t)\,S(t)$

Key relationship:

$S(t) = \exp\!\left(-\int_0^t \mu(s)\,ds\right)$

Probability of Surviving to Age 75 Given Alive at 70 Suppose $S(t) = e^{-0.02t}$ for $t \geq 0$ (constant force of mortality). Find the probability that a life aged 70 survives to age 75.

Answer ${_5}p_{70} = \frac{S(75)}{S(70)} = \frac{e^{-0.02(75)}}{e^{-0.02(70)}} = e^{-0.02(5)} = e^{-0.1} \approx 0.9048$

Finding the PDF from a Survival Function A survival function is given by $S(t) = \left(1 - \frac{t}{100}\right)^2$ for $0 \leq t \leq 100$ . Find $f(t)$ .

Answer $f(t) = -S'(t) = -2\left(1 - \frac{t}{100}\right)\cdot\left(-\frac{1}{100}\right) = \frac{2}{100}\left(1 - \frac{t}{100}\right)$