13_AS_6_lec_a

# 1 bi qx i 1 1030 actuarial statistics module 6

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Unformatted text preview: (bi −ai qx +ai ) E [D ] = i =1 N [1−ai qx +ai − (bi −ai px +ai ) (1−bi qx +bi )] = i =1 N [(1 − ai ) qx − (bi −ai px +ai ) (1 − bi ) qx ] = i =1 N [(1 − ai ) qx − [1 − E [Di ]] (1 − bi ) qx ] = i =1 10/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model The actuarial estimate Now we substitute E [Di ] = di and E [D ] = d to get N (1 − ai ) qx − d= i =1 (1 − bi ) qx i ;Di =0 which gives a moment estimate qx of qx = d N i =1 (1 − ai ) − i ;Di =0 (1 This is what is called the “actuarial estimate” 11/30 − bi ) Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model The actuarial estimate Initial exposed to risk We can rewrite the actuarial estimate as qx = dx Ex where the Initial exposed to risk Ex is deﬁned as N N (1 − ai ) − Ex = i =1 (1 − bi ) = i ;Di =0 (1 − ai ) + i :Di =1 (bi − ai ) i ;Di =0 Note: deaths contribute the period of length (1 − ai ) from x + ai to x + 1 survivors contribute the period of length (bi − ai...
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