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!
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}.