I've been able to figure out how a basic life table works.
But I would like to know how to extend it to calculate the life expectancy of the last survivor of a married couple.
(I understand there are some complexities from the fact that deaths of married couples may not be independent, but I can ignore that here...)
With a simple life table, you have a column of values named $m_x$, which represent the mortality rate at age x. (I'm using unisex mortality rates, so I don't need to worry about the sex of the person.)
You also have a column of values named $l_x$, which is the number of persons alive at age x (where $l_0$ starts at 100,000). So, $$l_x = l_{x-1}(1 - m_{x-1})$$
$d_x$ contains the number of persons who died in the interval (x, x+1), which is $l_xm_x$
$L_x$ contains the total number of person-years lived by the cohort in the interval (x, x+1), which is the equal to $l_{x+1} + 0.5d_x$. (Each person who lives to x+1 contributes one year, and each person who dies in x contributes half a year, on average.)
$T_x$ contains the total number of person-years lived by the cohort from age x until all members of the cohort have died. I.e., the sum of $L_x$ through $L_n$ (where n is the total number of rows).
Finally, $e_x$ contains the life expectancy of a person at age x, computed as $e_x = \frac{T_x}{l_x}$.
This all makes sense to me.
But, how would I use this approach to calculate the "joint and survivor" life expectancy of two people (e.g., a married couple) assuming their deaths are independent? I.e., how long would you expect at least one of those two people to still be alive?
A few simplifications for my specfic question...
- I'm using a single unisex mortality rate, so I don't need to worry about the sex of each person.
- I can assume the deaths are independent
- I can assume that both people are the same age