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Unformatted text preview: Chapter 5 Life annuities Chapter 5 Life annuities ACTSC 232 Introduction to Actuarial Mathematics Tianxiang Shi Department of Statistics and Actuarial Science University of Waterloo Winter 2012 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Outline 1 Introduction 2 Annuities payable continuously 3 Annual life annuities 4 Annuities payable m times per year 5 Guaranteed annuities 6 Variable life annuities Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Introduction Overview Annuity : a regular series of payments. e.g. mortgage, pensions . Annuity certain : the amount, the number and the timing of the payments are fixed, not contingent on any other factors. Life annuity : the number, the amount and the timing of the payments, at least one of them , depend on the survival of a life Annuity type: annuitydue : payments made at the beginning of each time unit. annuityimmediate : payments made at the end of each time unit. Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Introduction Annuity certain review Annuity due and immediate Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Introduction Annuity certain review Annuity due and immediate a n ( a ( m ) n ) = 1 v n d ( d ( m ) ) , a n ( a ( m ) n ) = 1 v n i ( i ( m ) ) Continuous annuity a n = 1 v n Increasing annuity Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Introduction Annuity certain review Annuity due and immediate a n ( a ( m ) n ) = 1 v n d ( d ( m ) ) , a n ( a ( m ) n ) = 1 v n i ( i ( m ) ) Continuous annuity a n = 1 v n Increasing annuity ( I a ) n (( Ia ) n , ( I a ) n ) = a n n v n d ( i, ) , ( I a ) n = a n n v n Decreasing annuity Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Introduction Annuity certain review Annuity due and immediate a n ( a ( m ) n ) = 1 v n d ( d ( m ) ) , a n ( a ( m ) n ) = 1 v n i ( i ( m ) ) Continuous annuity a n = 1 v n Increasing annuity ( I a ) n (( Ia ) n , ( I a ) n ) = a n n v n d ( i, ) , ( I a ) n = a n n v n Decreasing annuity ( D a ) n (( Da ) n , ( D a ) n ) = n a n d ( i, ) , ( D a ) n = n a n Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Annuities payable continuously Whole life continuous annuity Whole life continuous annuity of $ 1 per year A benefit of $ 1 per year is payable continuously, conditional on the survival of ( x ) to the payment dates. Y = Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 5 Life annuities Annuities payable continuously Whole life continuous annuity Whole life continuous annuity of $ 1 per year A benefit of $ 1 per year is payable continuously, conditional on the survival of ( x ) to the payment dates....
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 Winter '08
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