We can then modify the initial exposed to risk to a

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Unformatted text preview: state models 15/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Central exposed to risk Central exposed to risk Sometimes we actually know when deaths occur, say x + ti . We can then modify the initial exposed to risk to a central exposed to risk N c Ex (bi − ai ) (1 − di ) + = i =1 (ti − ai ) di i =1 where deaths contribute the period of length (ti − ai ) from x + ai to x + ti survivors contribute the period of length (bi − ai ) from x + ai to x + bi . c Note Ex = v is also called the observed total waiting at age x . 15/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Central exposed to risk c When the exact times of deaths are not available, but Ex is, the usual approch is to assume that deaths occur on average at age x + 1 so that the actuarial estimate becomes 2 qx = c Ex d + d 2 because under that assumption c Ex = Ex + d . 2 Proof: N Ex (1 − ai ) + = i :Di =1 (bi − ai ) i ;Di =0 N (1 − ti + ti − ai ) + = i :Di =1 16/30 (bi − ai ) i ;Di =0 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Central exposed to risk N (1 − ti + ti − ai ) di + = i =1 N N (1 − ti ) di + = i =1 N = = 17/30 (bi − ai ) (1 − di ) i (ti − ai ) di + i =1 i N c (1 − ti ) di + Ex = i =1 c Ex + 1− i =1 d /2 (bi − ai ) (1 − di ) 1 2 c di + Ex Actuarial Statistics – Module 6: Parametric models: Binomial and Po...
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This document was uploaded on 04/03/2014.

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