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Lecture%208%20Annotated - Life Insurance and Superannuation...

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Life Insurance and Superannuation Models Week 8: Insurance and Annuities for Multiple Lives April 16, 2011 1 / 32
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Week 8: Insurance and Annuities for Multiple Lives Summary of Lecture Joint distributions of future lifetimes Statuses: Joint life status Last-survivor status Insurances and annuities involving multiple lives Evaluation using special mortality laws Simple reversionary annuities Dependent lifetime models References Chapter 9 (Bowers, et al.) or Chapter 8 (Gerber). ACTL3002: Week 8 2
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Joint Distribution of Future Lifetimes Consider the case of two lives currently aged x and y with respective future lifetimes T ( x ) and T ( y ) . Joint distribution function: F T ( x ) T ( y ) ( s , t ) = P ( T ( x ) s , T ( y ) t ) . Joint p.d.f.: f T ( x ) T ( y ) ( s , t ) = 2 F T ( x ) T ( y ) ( s , t ) s t Joint survival function: S T ( x ) T ( y ) ( s , t ) = P ( T ( x ) > s , T ( y ) > t ) ACTL3002: Week 8 3
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The Case of Independence Joint distribution function: F T ( x ) T ( y ) ( s , t ) = F T ( x ) ( s ) F T ( y ) ( t ) . Joint p.d.f: f T ( x ) T ( y ) ( s , t ) = f T ( x ) ( s ) f T ( y ) ( t ) . Joint survival function: S T ( x ) T ( y ) ( s , t ) = S T ( x ) ( s ) S T ( y ) ( t ) . ACTL3002: Week 8 4
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The Joint Life Status This is a status that survives as long as all members are alive, and therefore fails upon the first death. Notation: ( x 1 x 2 ... x m ) For two lives: T ( xy ) = min [ T ( x ) , T ( y )] Distribution function: F T ( xy ) ( t ) = t q xy = P ( min [ T ( x ) , T ( y )] t ) = 1 P ( min [ T ( x ) , T ( y )] > t ) = 1 P ( T ( x ) > t , T ( y ) > t ) = 1 S T ( x ) T ( y ) ( t , t ) = 1 t p xy , where t p xy = P ( T ( x ) > t , T ( y ) > t ) = S T ( xy ) ( t ) is the probability that both lives ( x ) and ( y ) survive after t years. ACTL3002: Week 8 5
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The Case of Independence Alternative expression for distribution function: F T ( xy ) ( t ) = F T ( x ) ( t ) + F T ( y ) ( t ) F T ( x ) T ( y ) ( t , t ) In the case where T ( x ) and T ( y ) are independent: t p xy = P ( T ( x ) > t , T ( y ) > t ) = P ( T ( x ) > t ) P ( T ( y ) > t ) = t p x × t p y and t q xy = t q x + t q y t q x × t q y . Remember this (even in the case of independence): t q xy negationslash = t q x × t q y . ACTL3002: Week 8 6
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The density of T ( xy ) The density of T ( xy ) can be obtained by differentiating F T ( xy ) ( t ) = F T ( x ) ( t ) + F T ( y ) ( t ) F T ( x ) T ( y ) ( t , t ) in which case, we have: f T ( xy ) ( t ) = f T ( x ) ( t ) + f T ( y ) ( t ) dF T ( x ) T ( y ) ( t , t ) dt , where dF T ( x ) T ( y ) ( t , t ) dt = d dt integraldisplay t 0 integraldisplay t 0 f T ( x ) T ( y ) ( u , v ) dudv = integraldisplay t 0 f T ( x ) T ( y ) ( t , v ) dv + integraldisplay t 0 f T ( x ) T ( y ) ( u , t ) du . The latter equation can be obtained using Leibnitz’ rule of integration see Appendix 5 of Bowers, et al. One can then show in the case of independence: f T ( xy ) ( t ) = t p x × t p y ( μ x + t + μ y + t ) . ACTL3002: Week 8 7
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Force of mortality of T ( xy ) Define the force of mortality (similar manner to any r.v.): μ x + t : y + t = f T ( xy ) ( t ) 1 F T ( xy ) ( t ) = f T ( xy ) ( t ) S T ( xy ) ( t ) = f T ( xy ) ( t ) t p xy In the case of independence, we have: μ x + t : y + t = t p x × t p y ( μ x + t + μ y + t ) t p x × t p y = μ x + t + μ y + t The force of mortality of the joint life status is the sum of the individuals’ force of mortality when lives are independent.
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