Understanding the Mathematics of Personal Finance
PAYMENTS FOR COUPLES
In the case of a married couple, there are several options for structuring the payouts of a life annuity. The first option is what has already been discussed—a defined amount for the life of the annuity holder with or without a guaranteed minimum number of payments. The second option is a defined amount, with or without a guaranteed minimum number of payments, until both people have died. The third option is for a defined amount, with or without a guaranteed minimum number of payments, until one of the couple has died and then a second, usually lower, defined amount until the second of the couple has died.
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Figure 11.2 Expected life spans for men, women, and (same age) couples. |
Figure 11.2 shows the expected life spans for men and women (taken directly from the Life Insurance chapter) and also two new curves: first to die and last to die. In this graph, the first to die and last to die curves were generated assuming that both the man and the woman are the same age. For same sex couples, there would be either a single men or a single women curve, surrounded by first to die and last to die curves.
As you can see from the figure, men statistically don’t live as long as women. However, if we are just interested in when one of the couple will die—regardless of which one—then we can expect a younger demise than we would for either party individually. This result just arises from the probabilities: It’s analogous to the probability of picking one green ball out of a bowl with one green ball and 1,000 red balls as compared with the probability of picking one green ball out of a bowl with two green balls and 1,000 red balls. When we raise the number of opportunities for our event of interest happening while keeping constant the number of opportunities for our event of interest not happening, the probability of our event of interest happening goes up. In this case, the probability of an early death increases because we are watching two people rather than just one person.
On the other hand, the expected age of the last to die goes up. The reasoning here is the same as the reasoning in the last paragraph. If I were to look at three people rather than two, I would expect to see an expected still younger first to die and a still older last to die. If I looked at, say, 10,000 60-year-olds, I would expect to see at least one of them dying this year and at least one of them lasting to 100.
Figure 11.2 is a snapshot of one out of all of the possible sets of curves; all couples are not of the same age. Table 11.4 is an abbreviated part of the table of tax multiples (or equivalently e values in the Life Tables) for man-woman couples of differing ages. What these tables tell us is the expected number of years from the
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Table 11.4 First and Last to Die Multipliers for Couples
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start of a payout of an annuity when we can expect to see the first of the couple die and the expected number of years when we can see the second of the couple to die.
These tables are used in the following way. Suppose a 65-year-old man and his 60-year-old wife were to buy an immediate fixed annuity today. The annuity pays a certain amount monthly until the first of the couple dies, then pays, say, 75% of that amount until the second of the couple dies. We can think of what’s needed as two separate annuities. First, we need an immediate life annuity starting today for the initial amount with an expected payout period of 13.9 years. Second, we need a deferred life annuity, starting 13.9 years from now, with an expected payout period of 26.1 - 13.9 = 12.2 years for 75% of the initial amount.
A defined benefit pension that some workers receive upon retirement looks a bit like the annuities above but has some differences. On retirement, the worker is usually given the options of either receiving a full amount for his or her lifetime or receiving a reduced amount until the last to die of the couple passes away. In the former case, the retiree is receiving a conventional immediate fixed annuity based on his or her expected date of death and the pension fund accrued for this worker. In the latter case, the reduced amount is based on the employer taking the same premium and figuring that it will be paid until the expected date of death of the sex of the second to die.
