{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture%208%20Annotated

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

This preview shows pages 1–10. Sign up to view the full content.

Life Insurance and Superannuation Models Week 8: Insurance and Annuities for Multiple Lives April 16, 2011 1 / 32

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern