I've a question regarding the calculation of joint life mortality probabilities based on a "last survivor" principle, i.e. both borrowers need to be dead for the event to be triggered.
If I have Qx tables for 2 borrowers (representing the probability each borrower will die aged x exact), I figure the way of calculating the probability that both borrowers are dead will be as follows:
(1) Calculate the survival probabilities for each borrower. Performed using (1-Qx at that point in time) multiplied by the value preceding it, i.e. chaining these probabilities together
(2) Calculate the probability of borrower 1 already being dead (1 – survival probability at that point) multiplied by the probability that borrower 2 dies in that period (simply the Qx value)
(3) The reverse of (2), i.e. probability borrower 2 is already dead multiplied by probability borrower 1 dies in that period
(4) Calculate the probability both borrowers die in that period – calculated as the Qx values for each, multiplied together
Finally, sum together (2), (3) and (4).
The issue with this is, the probabilities will add to 3 (given that each individual probability eventually hits 1) which doesn't make sense, the overall probability must be 1.
Could someone help address this question?
Thanks!
Best Answer
There are two individuals: A and B. Let $T_A$ and $T_B$ denote the time of death of individual $A$ and $B$ respectively measure from say, today. $T^*=T_A\wedge T_B:=\max(T_A,T_B)$ denotes the time at group compose by $A$ and $B$ dies (both become death).
Let $S_A(t)=P[T_A>t]$ be the survival function of $A$; similar definition for $T_B$. Under the assumption that the "lives" $A$ and $B$ are independent one gets:
$$\begin{align} P[T^*>t]&=P[\max(T_A,T_B)>t]=P[T_A>t]+P[T_B>t]-P[\{T_A>t\}\cap\{T_B>t\}]\\ &=P[T_A>t]+P[T_B>t]-P[T_A>t]\,P[T_B>t]\\ &= S_A(t)+S_B(t)-S_A(t)S_B(t)\tag{1}\label{one} \end{align}$$
The next step now is to write expressions form $S_A$ (and $S_B$) in terms of the values one usually sees in actuarial tables: $l_x$, $d_x$, $p_x$, $q_x$. For simplicity, suppose $A$ and $B$ have ages $a$ and $b$ and that their survival can be obtained from a common actuarial table (they live in the same country, have similar jobs, etc) Using the number of lives $l_x$ we can se $\{l_x:a\leq x\leq \omega\}$ and $\{l_y: b\leq y\leq \omega\}$ to build $S_A$ and $S_B$ respectively, and from these, you can build $S^*(t)=P[T^*>t]$ using the expression \eqref{one}.