Unhappy returns

The man in George Outram's poem sells an old lady an annuity, relying on an insurance company's life-table to set the price. Unfortunately she beats the mean and years later he is left bemoaning the fact that he still has to pay her her annuity, the premium being long gone.

In setting premiums for life insurance, annuities and so forth, one has to be careful about two matters. The first is that the premiums should be set at a level that will not leave the insurer out of pocket in the long run. The second is that the fund is secure against a run of possible bad luck. This latter issue is known as the problem of gambler's ruin and is a very old one in statistics, having formed one of the subjects of Pascal's famous exchange of letters with Fermat. It was also treated by Christiaan Huygens in the book John Arbuthnot translated into English, De Ratiociniis in Ludo Aleae (1657). Given a very large number of persons insured, provided that life expectancies and other quantities related to them have been calculated correctly, the contingency required in percentage terms to cover a run against the fund is small.