ACTSC232-Ch5

# ACTSC232-Ch5 - Chapter 5 Life annuities Chapter 5 Life...

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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([email protected]) 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([email protected]) 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: annuity-due : payments made at the beginning of each time unit. annuity-immediate : payments made at the end of each time unit. Tianxiang Shi([email protected]) Chapter 5 Life annuities Introduction Annuity certain review Annuity due and immediate Tianxiang Shi([email protected]) 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 δ Increasing annuity Tianxiang Shi([email protected]) 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 δ Increasing annuity ( I ¨ a ) n (( Ia ) n , ( I ¯ a ) n ) = ¨ a n- n · v n d ( i,δ ) , ( ¯ I ¯ a ) n = ¯ a n- n · v n δ Decreasing annuity Tianxiang Shi([email protected]) 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 δ Increasing annuity ( I ¨ a ) n (( Ia ) n , ( I ¯ a ) n ) = ¨ a n- n · v n d ( i,δ ) , ( ¯ I ¯ a ) n = ¯ a n- n · v n δ Decreasing annuity ( D ¨ a ) n (( Da ) n , ( D ¯ a ) n ) = n- a n d ( i,δ ) , ( ¯ D ¯ a ) n = n- ¯ a n δ Tianxiang Shi([email protected]) 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([email protected]) 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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## This note was uploaded on 03/30/2012 for the course ACTSC 232 taught by Professor Matthewtill during the Winter '08 term at Waterloo.

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ACTSC232-Ch5 - Chapter 5 Life annuities Chapter 5 Life...

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