1. Andrew D Smith AndrewDSmith8@Deloitte.co.uk April 2005
Stochastic Mortality
Andrew D Smith
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.
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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
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4. Reinsurance Contract Deloitte. 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
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5. Distribution Tail: Alternative Models
thousands
v1/2 = 2%
v1/2 = 1%
independent
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6. Modelling Solvency To model cash flows But to model solvency tests
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.
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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 for 65 year old
year of annuity price calculation
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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
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10. Mortality Term Structures
initial valuation based on
expected table
assumptions gradually replaced by reality
number of survivors
valuation
date
cash flow date
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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
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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
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13. Individual Mortality Deloitte. process Atp process At
Atp is a finite variation previsible process
Such that At-Atp is a martingale
Slope of Atp, if it exists
Is your own personal hazard rate
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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
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15. Properties of Individual Mortality
Consider A∞p – Atp
Conditional on t
Either zero (if dead)
Or exponential(1) (if alive)
“A result which will be obvious to actuaries” (Rogers & Williams)
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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?
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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
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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. Andrew D Smith AndrewDSmith8@Deloitte.co.uk April 2005
Stochastic Mortality
Andrew D Smith
April 2005
Deloitte.