# Actuarial Mathematics for Life Contingent Risks by David C. M. Dickson

By David C. M. Dickson

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Extra info for Actuarial Mathematics for Life Contingent Risks (International Series on Actuarial Science)

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Dx x d d g(t)dt = g(x). What about dx a dx (b) Deduce that a Hint: g(t)dt ? x o x + ex is an increasing function of x, and explain this result intuitively. 1 Summary In this chapter we deﬁne a life table. For a life table tabulated at integer ages only, we show, using fractional age assumptions, how to calculate survival probabilities for all ages and durations. We discuss some features of national life tables from Australia, England & Wales and the United States. We then consider life tables appropriate to individuals who have purchased particular types of life insurance policy and discuss why the survival probabilities differ from those in the corresponding national life table.

L30 (e) This probability is 5 | q30 . 00552. 3 Fractional age assumptions A life table {lx }x≥x0 provides exactly the same information as the corresponding survival distribution, Sx0 . 5 . Given values of lx at integer ages only, we need an additional assumption or some further information to calculate probabilities for non-integer ages or durations. Speciﬁcally, we need to make some assumption about the probability distribution for the future lifetime random variable between integer ages. We use the term fractional age assumption to describe such an assumption.

Note that together with d Condition 3 above, this means that dt Sx (t) ≤ 0 for all t > 0. Assumption 2. limt→∞ t Sx (t) = 0. Assumption 3. limt→∞ t 2 Sx (t) = 0. These last two assumptions ensure that the mean and variance of the distribution of Tx exist. These are not particularly restrictive constraints – we do not need to worry about distributions with inﬁnite mean or variance in the context of individuals’ future lifetimes. These three extra assumptions are valid for all distributions that are feasible for human lifetime modelling.