the omega glory

At work, some colleagues and I got into an email exchange about mortality tables. It was prompted by a passage in an old probability textbook. Feller, I think. The relevant passage:

It is impossible to measure the life span of an atom or a person without some error, but for theoretical purposes it is expedient to imagine that these quantities are exact numbers. The question then arises as to which numbers can represent the lifespan of a person. Is there a maximal age beyond which life is impossible, or is any age conceivable? We hesitate to admit that man can grow 1000 years old, and yet current actuarial practice admits no bounds to the possible duration of life. According to formulas on which modern mortality tables are based, the proportion of men surviving 1000 years is of the order of magnitude of one in 10^10^36 — a number with 10^27 billions of zeros. This statement does not make sense from a biological or sociological point of view, but considered exclusively from a statistical standpoint it certainly does not contradict any experience. There are fewer than 10^10 people born in a century. To test the contention statistically, more than 10^10^35 centuries would be required, which is considerably more than 10^10^34 lifetimes of the earth. Obviously, such extremely small probabilities are compatible with our notion of impossibility. Their use may appear utterly absurd, but it does no harm and is convenient in simplifying many formulas. Moreover, if we were seriously to discard the possibility of living 1000 years, we should have to accept the existence of maximum age, and the assumption that it should be possible to live x years and impossible to live x years and two seconds is as unappealing as the idea of unlimited life.

 Before going on, I should note one mistake in the passage above. Current actuarial practice is (and, I believe, was as of the time that Feller was written) to use mortality tables that did have a maximal age. Modern mortality tables generally have an omega -- that age at which q (the probability of dying within a year) is 1.

At any rate, the question being alluded to is whether it makes more sense to have an omega or to assume that there is no upper bound on potential lifespan. For practical purposes, it doesn't matter. There is clearly a mathematical difference between having q=1 at age 120 and having q=.99999999999999 at age 120, growing monotonically, and converging to 1 as age approaches infinity. But in the world of insurance (and its place in finance), it doesn't matter.

Conceptually, I prefer the notion that there is no omega, but q's get arbitrarily close to 1. It just makes more sense to me. But, based on the conversation at work, I am in the minority.