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Dying on Birthdays
By Eugene McGovern
A fixture of life contingencies is the following definition of the notation :
Where T(x) is the complete-future-life random variable for (x) - a life aged exactly x. is the cumalative distribution function of T(x). X is the age-at-death random variable, so . (This is the notation of Bowers et al.)
In practice, however, this is not the definition we use for . The definition we use is
Note the strict inequality. That this is the definiation we use becomes clear when we verbalize as
The discrepancy between and our verbalizations for is most pointed when both x and t are integers. In that case, "now" is the birthday of the life, and we are interested in the birthday. Then we verbalize (paraphrase) as follows:
If we were using the definition
these verbalizations would be that is
This discrepancy between our definition and our practice is worrisome in view of how we calculate the EPV of term assurances and endowments.
Suppose on his birthday the life purchased both a t-year term assurance and a t-year pure endowment. In our calculations, we note that exactly one of these contracts will pay, and we take to be the probability that the term assurance will pay and to be the probability the pure endowment pays. In the case that the life dies on his birthday,
This is not a problem of semantics. Our practice and our verbalizations are in agreement. They agree on the fact that if death takes place on a birthday, then the life survived to that birthday, and death did not occur before the birthday.
In principle, the range of and the domain of is the continuum (0, x). In principle, however, birthdates and death dates are known, but birth times and death times are not known, so the range of , and the domain of is made discrete by letting t increase each 24 hours, taking on 364 values between consecutive integers. Giving t in years, with three decimal places, will distinguish these 364 values from each other.
This discretization does not, however, resolve the conflict with our verbalizations. If (41) dies on his 64th birthday, we will say he survived to his 64th birthday, and T(41) = 23.000.
The remedy is to introduce a new function, and to define as
is not a cumulative distribution function, but it is differentiable to the same extent as is differentiable. Its derivative, is not a density function, but has all the properties we need.
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