By D C M Dickson; Mary Hardy; H R Waters
Balancing rigour and instinct, and emphasizing functions, this contemporary textual content is perfect for college classes and actuarial examination preparation.
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Extra resources for Actuarial mathematics for life contingent risks
Calculate (a) (b) (c) (d) (e) (f) (g) the probability that a newborn life dies before age 60, the probability that a life aged 30 survives to at least age 70, the probability that a life aged 20 dies between ages 90 and 100, the force of mortality at age 50, the median future lifetime at age 50, the complete expectation of life at age 50, the curtate expectation of life at age 50. 2 The function G(x) = 18 000 − 110x − x2 18 000 has been proposed as the survival function S0 (x) for a mortality model.
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.
21) to evaluate this integral using integration by parts as ∞ ◦ ex = − d t px dt dt t 0 = − t t px ∞ 0 ∞ − t px dt . 2 we stated the assumption that limt→∞ t t px = 0, which gives ∞ ◦ ex = t px dt. 22) 30 Survival models Similarly, for E[Tx2 ], we have ∞ E[Tx2 ] = t 2 t px µx+t dt 0 ∞ =− d t px dt dt t2 0 = − t 2 t px ∞ 0 ∞ − t px 2t dt 0 ∞ =2 t t px dt. 23) 0 So we have integral expressions for E[Tx ] and E[Tx2 ]. For some lifetime distributions we are able to integrate directly. 23). The variance of Tx can then be calculated as ◦ V [Tx ] = E Tx2 − ex 2 .