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Deloitte. Stochastic Mortality Andrew D Smith April 2005.

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1 Deloitte. Stochastic Mortality Andrew D Smith April 2005

2 Deloitte. 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.

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

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

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

6 Deloitte. 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 youre insolvent next year.

7 Deloitte. Regulators 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.

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

9 Deloitte. 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 F new (x) = 1- [1-F(x)] d –d = deterioration factor IID Random deterioration factor

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

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

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

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

14 Deloitte. 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

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

16 Deloitte. Individual Mortality Construction Positive stochastic process μ t –function of a Markov vector process Integrated value C t = t μ s ds Exponential RV L T (time of death) defined by C T = L A t p = min{C T, L} Deep question: do all individual (totally inaccessible) models arise in this way?

17 Deloitte. 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 μ t –different starting populations

18 Deloitte. 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?

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

20 Deloitte. Stochastic Mortality Andrew D Smith April 2005

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