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“Self-Annuitization and Ruin in Retirement”, Moshe Arye Milevsky and Chris Robinson, October 2000

JAN DHAENE,* MARC J. GOOVAERTS, AND ROB KAAS The present paper is a very interesting one because it introduces financial modeling into classical actuarial problems in a way that is both very nice and relatively easy to grasp. The paper derives an approximation to the distribution of Asian options and formulates very nicely the connection with the idea behind ruin probabilities in case interest is described by a Wiener process. The results obtained are attractive because, to some extent, they demystify the stochastic discount process. While Wiener measures should be familiar tools to actuaries certifying the accounts of financial conglomerates, one such actuary wasn’t ashamed to state that he only knew about the Wiener Sängerknaben. One of the merits of this paper is that it introduces a Wiener measure into a mortality model with stochastic interest returns. In addition, the mathematically equivalent problem of pricing Asian options is tackled. Deriving approximations for net present values under stochastic return models is necessary because the known analytical distributions for problems of this kind are scarce and difficult to apply in practice. To determine the distribution of a stochastic annuity (nonstochastic mortality but random return) is an old mathematical problem, totally equivalent to the so-called Callogero model of mathematical physics, which was solved in the early 1970s. Attempts to generalize the results have been unsuccessful in mathematical physics for about 30 years. To indicate the importance of having approximations to this very important actuarial problem, the interested reader is referred to De Schepper et al. (1994), formula (3.1) which provides us with the distribution density of a stochastic annuity: