Joint Life

Study guides: Exam ALTAM, Exam MAS-I

Joint Life models describe the survival of two (or more) lives simultaneously, focusing on the time until the first death. Given two independent lives aged $x$ and $y$ with future lifetimes $T_x$ and $T_y$ , the joint-life future lifetime is $T_{xy} = \min(T_x, T_y)$ .

$_{t}p_{xy} = P(T_{xy} > t)$

$= {_t}p_x \cdot {_t}p_y \quad \text{(independence)}$

$_{t}q_{xy} = 1 - {_t}p_{xy}$

$= 1 - {_t}p_x \cdot {_t}p_y$

$_{t}p_{\overline{xy}} = {_t}p_x + {_t}p_y - {_t}p_{xy}$

$\ddot{a}_x + \ddot{a}_y = \ddot{a}_{xy} + \ddot{a}_{\overline{xy}}$

The last survivor status $T_{\overline{xy}} = \max(T_x, T_y)$ lasts until the second death. It is not the complement of the joint-life status: its survival probability is ${_t}p_x + {_t}p_y - {_t}p_{xy}$ , which holds with or without independence and becomes ${_t}p_x + {_t}p_y - {_t}p_x\,{_t}p_y$ for independent lives
Relationship: $T_{xy} + T_{\overline{xy}} = T_x + T_y$ , so $\mathring{e}_{xy} + \mathring{e}_{\overline{xy}} = \mathring{e}_x + \mathring{e}_y$
The complete joint-life expectancy: $\mathring{e}_{xy} = \int_0^\infty {_t}p_{xy}\,dt$
Under independence, the joint survival function factors into the product of the individual survival functions
Annuities on two lives: the joint-life annuity $\ddot{a}_{xy}$ pays 1 at the start of each year until the first of the two dies, the last-survivor annuity $\ddot{a}_{\overline{xy}}$ until both have died (see Life Annuity). Paying 1 to each life while it is alive is the same as paying 1 while either is alive plus a second 1 while both are — so if annuities on $(40)$ and $(50)$ cost 15 and 13 and the last-survivor annuity costs 20, the joint-life annuity costs $15 + 13 - 20 = 8$

Probability Both, and At Least One, Survive 10 Years Two independent lives, aged 40 and 50, have $_{10}p_{40} = 0.95$ and $_{10}p_{50} = 0.88$ . Find the probability that both survive 10 years, and the probability that at least one of them does.

Answer $_{10}p_{40:50} = {_{10}p_{40}} \cdot {_{10}p_{50}} = 0.95 \times 0.88 = 0.836$ $_{10}p_{\overline{40:50}} = 0.95 + 0.88 - 0.836 = 0.994$ The last-survivor probability is not $1 - 0.836 = 0.164$ : that is the probability that at least one of the two dies.

Probability That the Joint Life Status Fails Within 5 Years Under a constant force of mortality, $\mu_x = 0.04$ and $\mu_y = 0.03$ . Find $_{5}q_{xy}$ , the probability that the joint-life fails within 5 years.

Answer With constant forces, ${_t}p_x = e^{-0.04t}$ and ${_t}p_y = e^{-0.03t}$ . $_{5}p_{xy} = e^{-0.04(5)} \cdot e^{-0.03(5)} = e^{-0.20} \cdot e^{-0.15} = e^{-0.35} \approx 0.7047$ $_{5}q_{xy} = 1 - 0.7047 = 0.2953$