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

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

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720 − 6x 24 Survival models As an alternative, we could use the relationship µx = − = d d log S0 (x) = − dx dx 1 1 log(1 − x/120) = 6 720(1 − x/120) 1 . 3 Let µx = Bcx , x > 0, where B and c are constants such that 0 < B < 1 and c > 1. This model is called Gompertz’ law of mortality. Derive an expression for Sx (t). 11), x+t Sx (t) = exp − Bcr dr . x Writing cr as exp{r log c}, x+t x+t Bcr dr = B x exp{r log c}dr x = = B exp{r log c} log c x+t x B cx+t − cx , log c giving Sx (t) = exp −B x t c (c − 1) .

First, we see an effective limiting age, even though, in principle there is no age to which the survival probability is exactly zero. 1, we see that although Sx (t) > 0 for all combinations of x and t, survival beyond age 120 is very unlikely. Second, we note that the survival functions are ordered according to age, with the probability of survival for any given value of t being highest for age 20 and lowest for age 80. For survival functions that give a more realistic representation of human mortality, this ordering can be violated, but it usually holds at ages of interest to insurers.

4), we also know survival probabilities for our individual from any future age x. 5) takes this a stage further. It shows that if we know survival probabilities from any age x (≥ 0), then we also know survival probabilities from any future age x + t (≥ x). Any survival function for a lifetime distribution must satisfy the following conditions to be valid. Condition 1. Sx (0) = 1; that is, the probability that a life currently aged x survives 0 years is 1. Condition 2. limt→∞ Sx (t) = 0; that is, all lives eventually die.

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