By D C M Dickson; Mary Hardy; H R Waters
Balancing rigour and instinct, and emphasizing purposes, this contemporary textual content is perfect for collage classes and actuarial examination preparation.
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Again, we substitute y = 1 − t/(120 − x) giving E Tx2 = 2(120 − x)2 = 2(120 − x)2 1 (y1/6 − y7/6 ) dy 0 6 6 − . 23773))2 . 5092 . Since we know under this model that all lives will die before age 120, it makes sense that the uncertainty in the future lifetime should be greater for younger lives than for older lives. 6 is that we can obtain formulae for ◦ quantities of interest such as ex , but for many models this is not possible. For example, when we model mortality using Gompertz’ law, there is no explicit ◦ formula for ex and we must use numerical integration to calculate moments of Tx .
1. 1 Kx and ex In many insurance applications we are interested not only in the future lifetime of an individual, but also in what is known as the individual’s curtate future lifetime. The curtate future lifetime random variable is deﬁned as the integer part of future lifetime, and is denoted by Kx for a life aged x. If we let denote the ﬂoor function, we have Kx = Tx . We can think of the curtate future lifetime as the number of whole years lived in the future by an individual. As an illustration of the importance of curtate future lifetime, consider the situation where a life aged x at time 0 is entitled to payments of 1 at times 1, 2, 3, .
Then, when all of these factors have been modelled, they must be combined to set a premium. Each year or so, the actuary must determine how much money the insurer or pension plan should hold to ensure that future liabilities will be covered with adequately high probability. This is called the valuation process. For with-proﬁt insurance, the actuary must determine a suitable level of bonus. The problems are rather more complex if the insurance also covers morbidity risk, or involves several lives.
Actuarial mathematics for life contingent risks by D C M Dickson; Mary Hardy; H R Waters