Stochastic Mortality

1. StochasticMortality Andrew D Smith AndrewDSmith8@Deloitte.co.uk April 2005 Deloitte.

2. Abstract Rapid improvements in pensioner mortality have caught many in the insurance industry by surprise. Insurers have strengthened their reserves and increased the prices they charge for annuities. Classical mortality models are based on a binomial process, where any two individuals die independently of each other. On the other hand, medical breakthroughs and lifestyle changes such as diet and smoking, affect large parts of a population simultaneously, and therefore introduce dependencies between lives. The presentation quantifies the effect such dependencies can have the risk of large losses. The presentation goes on to develop the concept of a mortality term structure, which describes not only a pattern of deaths but also how insurers might revise their mortality estimates over the term of a contract. The presentation will show the impact of mortality revisions over a one year capital assessment, and on the value of annuity guarantees. The presentation concludes by extending mortality models to the individual level. Individual mortality models are relevant for assessing the cost of surrender risk on term assurance contracts, or the impact of guaranteed renewal terms. Deloitte.

3. How a Life Annuity Works cash flow Mr Adams and Mr Brown both buy annuities of £1000 Payable annually in arrears. Mr Adams dies aged 80 Mr Brown dies aged 90 Deloitte.

4. Reinsurance Contract Reinsurer Stop loss reinsurance contract Pays annuities once aggregate payout exceeds £2.5m Insurer 100 policyholders One-off premium aged 65 Annual pension until death Expected payments £2.12m customers Deloitte.

5. Distribution Tail:Alternative Models v1/2 = 5% thousands v1/2 = 2% v1/2 = 1% independent Deloitte.

6. Modelling Solvency • To model cash flows • we need to know only the number of deaths • But to model solvency tests • we need to know what mortality assumptions might be in use • at a future valuation date • need stochastic assumptions • Because its no use being solvent (with high probability) in the long term if you’re insolvent next year. Deloitte.

7. Regulator’s View “ we expect firms to consider … how estimates of longevity might change over time thereby affecting the future valuation of realistic liabilities” FSA insurance regulatory update - March 2005 In theory, also relevant for modelling mortality guarantees, such as guaranteed annuity options. Historically, mortality fluctuations have been as important as interest moves, dragging GAOs into the money. Deloitte.

8. Actuaries Debate the Right Assumptions for Today immediate annuity value @4.5% for 65 year old year of annuity price calculation Deloitte.

9. Modelling the Assumptions you Might Make Tomorrow • Underlying idea • proportional hazards • Distribution F of time of death • Hazard rate = F’(x)/[1-F(x)] • Actuaries call it “force of mortality” • Transformed Fnew(x) = 1- [1-F(x)]d • d = deterioration factor • IID Random deterioration factor Deloitte.

10. Mortality Term Structures initial valuation based on expected table assumptions gradually replaced by reality number of survivors valuation date cash flow date Deloitte.

11. The Olivier-Smith Model • Notation: • Take a homogeneous cohort • L(t) = number of survivors aged t • L(s,t) = Es{L(t)} • Conditional on information at time s Deloitte.

12. Olivier-Smith Model (cont) binomial model gamma deterioration factor gamma mean 1, variance v Proportional hazards produces biased deaths Bias correction so that L(s,t) is a martingale in s Deloitte.

13. Individual Mortality process Atp process At Atp is a finite variation previsible process Such that At-Atpis a martingale Slope of Atp, if it exists Is your own personal hazard rate Deloitte.

14. Stochastic Individual Mortality suppose my own hazard rate is a stochastic process … Consider my “state of health” as a Markov process dual previsible projection age Deloitte.

15. Properties of Individual Mortality • Consider A∞p– Atp • Conditional on information @ t • Either zero (if dead) • Or exponential(1) (if alive) • “A result which will be obvious to actuaries” (Rogers & Williams) Deloitte.

16. Individual Mortality Construction • Positive stochastic process μt • function of a Markov vector process • Integrated value Ct = ∫t μsds • Exponential RV L • T (time of death) defined by CT = L • Atp = min{CT, L} • Deep question: do all individual (totally inaccessible) models arise in this way? Deloitte.

17. Applications of Individual Mortality • Options on individual mortality • policy selection • options to lapse and re-enter • Guaranteed Annuity Options • Explaining select mortality table • common process μt • different starting populations Deloitte.

18. A Final Puzzle • You have written a portfolio of pensions business • policyholder annuity rate • = better of {9%, open market rate} • mixture of male and female policyholders • A well-intended regulator enforces unisex annuity pricing • does the GAO liability increase, decrease or stay the same? Deloitte.

19. Useful Links • SOA Presentation (Olivier & Jeffery) • Andrew Cairns Longevity Model • Affine Mortality (Schrager) • Annuity risks (Hibbert) • Mortality Projections (CMIB) Deloitte.

20. StochasticMortality Andrew D Smith AndrewDSmith8@Deloitte.co.uk April 2005 Deloitte.