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ACTSC232-Ch3 - Chapter 3 Life tables and selection Chapter...

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Chapter 3 Life tables and selection Chapter 3 Life tables and selection ACTSC 232 Introduction to Actuarial Mathematics Tianxiang Shi Department of Statistics and Actuarial Science University of Waterloo Winter 2012 Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Outline 1 Life tables 2 Fractional age assumptions UDD assumption CFM assumption 3 Select and ultimate survival models Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Life tables Heuristic analysis on a life table What a life table looks like? Age x l x d x q x e x 0 100,000 671 0.006713 77.7 1 99,329 44 0.000444 77.2 2 99,285 30 0.000300 76.3 . . . . . . . . . . . . . . . l 0 : initial number (say 100,000 newborns). TEMPORARILY consider l x : # of survivors up to age x d x : # of death between age x and age x + 1 Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Life tables US/Canada life tables 2006 US life tables http: //www.cdc.gov/nchs/data/nvsr/nvsr58/nvsr58_21.pdf 2002 Canada life tables http://www.statcan.gc.ca/pub/84-537-x/ 84-537-x2006001-eng.htm Some comments tabulated at integer ages only , and containing l x , d x , q x and other functions. l 0 : radix of the table. Arbitrary positive number, usually large. the maximum age is NOT necessarily the limiting age. Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Life tables Survivors to age x ( L x ) Expected number of survivors Denote L x as the number of survivors up to age x among l 0 newborns. Let I k = 1 , if the k th life survives to age x (with prob. x p 0 ) 0 , otherwise (with prob. x q 0 ) Then, L x = l 0 X k =1 I k . Define l x := E [ L x ] If I k are independent, L x Bin ( l 0 , x p 0 ) Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Life tables t p x and l x The mean of L x l x = E [ L x ] = l 0 X k =1 E [ I k ] = l 0 · x p 0 In general t p x = x + t p 0 x p 0 = l x + t /l 0 l x /l 0 = l x + t l x or l x + t = l x · t p x In particular l x +1 = l x · p x , and p x = l x +1 l x Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Life tables Death between ages x and x + n ( n D x ) Expected number of death between ages x and x + n Denote n D x as the number of death between ages x and x + n among l 0 newborns. Let u k = 1 , k th life dies b/t ages x & x + n (w.p. x | n q 0 ) 0 , otherwise (w.p. 1 - x | n q 0 ) Then, n D x = L x - L x + n = l 0 X k =1 u k . Define n d x := E [ n D x ] and denote: d x = 1 d x If u k are independent, L x Bin ( l 0 , x | n q 0 ) Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Life tables n q x , n d x and l x The mean of n D x n d x = E [ n D x ] = E [ L x - L x + n ] = l x - l x + n or n d x = l 0 X k =1 E [ u k ] = l 0 · x | n q 0 = l x · n q x In particular l x = l x +1 + d x , and q x = d x l x Other property n d x = d x + d x +1 + · · · + d x + n - 1 Tianxiang Shi([email protected])
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Chapter 3 Life tables and selection Life tables Other functions k | n q x in terms of l x k | n q x = k p x - k + n p x = l x + k - l x + k + n l x Total expected # of years lived beyond age x by those survivors of l 0 newborns T * x = Z 0 l x + t dt = l x · ˚ e x Note: T * x corresponds to T x in the US life table.
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