Chapter 03

Chapter 03 - Introduction In this chapter, we study...

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FIN 3220A FIN 3220A Actuarial Models I Actuarial Models I Chapter 3 Life Annuities FIN 3220A (2005-2006) Page 2 Chapter 3 Life Annuities Introduction Introduction ± In this chapter, we study payments contingent on survival, as provided by various forms of life annuities ± A life annuity is a series of payments made continuously or at equal intervals while a given life survives ± It may be temporary (limited to a given years) or for whole life ± The payment intervals may commence immediately or may be deferred ± Payments may be due at the beginnings of the payment intervals ( annuities-due ) or at the ends of such intervals ( annuities- immediate ) ± Life annuities play major role in life insurance operations, pension systems, and disability or worker’s compensation insurance FIN 3220A (2005-2006) Page 3 Chapter 3 Life Annuities Continuous Life Annuities Continuous Life Annuities ± Consider annuities payable continuously at the rate of 1 per year ± This is of course an abstraction ± A whole life annuity provides for payments until death ± The PV of annuity payments to be made is ± Here, T is the future lifetime of ( x ) for all 0 T aT =≥ Y FIN 3220A (2005-2006) Page 4 Chapter 3 Life Annuities Continuous Life Annuities Continuous Life Annuities ± The distribution function of Y can be obtained from that for T as follows: () ( ) () Pr Pr Pr 1 Pr 1 ln 1 Pr ln 1 1 for 0 T TT T Fy y a y vy v y y T y =≤ = =− δ = δ ⎧⎫ −− δ ⎨⎬ δ ⎩⎭ δ =< < δδ Y Y
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FIN 3220A (2005-2006) Page 5 Chapter 3 Life Annuities Continuous Life Annuities Continuous Life Annuities ± Differentiating the d.f., we obtain the p.d.f. for Y () ln 1 ln 1 11 for 0 1 T T d fy Fy dy y d F dy y y = ⎧⎫ −− δ = ⎨⎬ δ ⎩⎭ δ =< < −δ δ δ YY FIN 3220A (2005-2006) Page 6 Chapter 3 Life Annuities Continuous Life Annuities Continuous Life Annuities ± A typical distribution function for the PV random variable for the whole life annuity FIN 3220A (2005-2006) Page 7 Chapter 3 Life Annuities Continuous Life Annuities Continuous Life Annuities ± The APV for a continuous whole life annuity is denoted by ± The post fixed subscript, x , indicates that ± The annuity ceases at the death of ( x ) ± The distribution of T ( x ) may depend on information available at age x ± Under aggregate mortality, we have x a 0 xt x t aE a p x t d t == µ + Y FIN 3220A (2005-2006) Page 8 Chapter 3 Life Annuities Continuous Life Annuities Continuous Life Annuities ± Using integration-by-parts with ± We obtain () () ( ) 1 ,, , t tt t tx df t de ft a e d ft vd t dt dt gt p dgt p x td t = = δ =− = µ + 0 0 0 0 0 x t t t t aa p x t d t ap vp d t vpd t Ed t + + = =
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FIN 3220A (2005-2006) Page 9 Chapter 3 Life Annuities Continuous Life Annuities Continuous Life Annuities ± The interpretation: ± A momentary payment of 1 dt made at time t ± Discounted at interest back to time zero by multiplying by v t ± Further multiplied by t p x to reflect the probability that a payment is made at time t ± This is the current payment form of APV for whole life annuity ±
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This note was uploaded on 05/21/2011 for the course FIN 3220 taught by Professor Cswong during the Spring '05 term at CUHK.

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Chapter 03 - Introduction In this chapter, we study...

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