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

ISBN-10: 0511632436

ISBN-13: 9780511632433

ISBN-10: 0511633637

ISBN-13: 9780511633638

ISBN-10: 0511651694

ISBN-13: 9780511651694

ISBN-10: 0521118255

ISBN-13: 9780521118255

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 resources for Actuarial mathematics for life contingent risks**

**Sample text**

Under whole life insurance, the payment of the sum insured is certain, the uncertainty is in the timing. Under, say, 10-year term insurance, it is assumed that the majority of contracts will expire with no death beneﬁt paid. If the underwriting is not strict there is a risk of adverse selection by policyholders – that is, that very high-risk individuals will buy insurance in disproportionate numbers, leading to excessive losses. Since high sum insured contracts carry more risk than low sum insured, high sums insured would generally trigger more rigorous underwriting.

We deﬁne a select survival model and we derive some formulae for such a model. 2 Life tables Given a survival model, with survival probabilities t px , we can construct the life table for the model from some initial age x0 to a maximum age ω. We deﬁne a function {lx } for x0 ≤ x ≤ ω as follows. Let lx0 be an arbitrary positive number (called the radix of the table) and, for 0 ≤ t ≤ ω − x0 , deﬁne lx0 +t = lx0 t px0 . From this deﬁnition we see that for x0 ≤ x ≤ x + t ≤ ω, lx+t = lx0 x+t−x0 px0 = lx0 x−x0 px0 t px = lx t px , 41 42 Life tables and selection so that t px = lx+t /lx .

24) k=1 Note that the lower limit of summation is k = 1. Similarly, ∞ E[Kx2 ] = k 2 ( k px − k+1 px ) k=0 = (1 px − 2 px ) + 4(2 px − 3 px ) + 9(3 px − 4 px ) + 16(4 px − 5 px ) + · · · ∞ =2 ∞ k k px − k=1 k px k=1 ∞ =2 k k px − e x . k=1 34 Survival models As with the complete expectation of life, there are a few lifetime distributions that allow E[Kx ] and E[Kx2 ] to be calculated analytically. For more realistic models, such as Gompertz’, we can calculate the values easily using Excel or other suitable software.

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

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