# Actuarial mathematics for life contingent risks by D C M Dickson; Mary Hardy; H R Waters

By D C M Dickson; Mary Hardy; H R Waters

Balancing rigour and instinct, and emphasizing purposes, this contemporary textual content is perfect for college classes and actuarial examination preparation.

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Additional info for Actuarial mathematics for life contingent risks

Example text

In practice, it is very common for a life table to be presented, and in some cases even deﬁned, at integer ages only. In this form, the life table is a useful way of summarizing a lifetime distribution since, with a single column of numbers, it allows us to calculate probabilities of surviving or dying over integer numbers of years starting from an integer age. It is usual for a life table, tabulated at integer ages, to show the values of dx , where dx = lx − lx+1 , in addition to lx , as these are used to compute qx .

Show that ex = (eλ − 1)−1 . What conclusions do you draw about using this lifetime distribution to model human mortality? 02, calculate (a) (b) (c) (d) (e) px+3 , , p 2 x+1 , 3 px , 1 |2 qx . 7 Given that F0 (x) = 1 − 1 1+x for x ≥ 0, ﬁnd expressions for, simplifying as far as possible, (a) (b) (c) (d) (e) S0 (x), f0 (x), Sx (t), and calculate: p20 , and 10 |5 q30 . 001 x 2 for x ≥ 0, 38 Survival models ﬁnd expressions for, simplifying as far as possible, (a) f0 (x), and (b) µx . 9 Show that d t px = t px (µx − µx+t ) .

The extra term tends to improve the ﬁt of the model to mortality data at younger ages. In recent times, the Gompertz–Makeham approach has been generalized further to give the GM(r, s) (Gompertz–Makeham) formula, µx = h1r (x) + exp{h2s (x)}, where h1r and h2s are polynomials in x of degree r and s respectively. Adiscussion of this formula can be found in Forfar et al. (1988). Both Gompertz’ law and Makeham’s law are special cases of the GM formula. 3, we noted the importance of the force of mortality.