Deloitte. Stochastic Mortality Andrew D Smith [email protected] April 2005.
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Transcript of Deloitte. Stochastic Mortality Andrew D Smith [email protected] April 2005.
Deloitte.
AbstractRapid 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.
How a Life Annuity Works
0
500
1000
65 70 75 80 85 90 95 100
ExpectedMr AdamsMr Barber
Mr Adams and Mr Brown both buy annuities of £1000Payable annually in arrears.Mr Adams dies aged 80Mr Brown dies aged 90
cash
flow
Deloitte.
Reinsurance Contract
Insurer
customers
100 policyholdersOne-off premium aged 65Annual pension until deathExpected payments £2.12m
Reinsurer
Stop loss reinsurance contractPays annuities once aggregate payout exceeds £2.5m
Deloitte.
Distribution Tail:Alternative Models
2000
2500
3000
3500
4000
4500
5000
5500
6000
95% 96% 97% 98% 99% 100%
independent
v1/2 = 1%
v1/2 = 2%
v1/2 = 5%
thou
sand
s
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 you’re insolvent next year.
Deloitte.
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 asguaranteed annuity options. Historically, mortality fluctuations have been as important as interest moves, dragging GAOs into the money.
Deloitte.
Actuaries Debate the Right Assumptions for Today
12
12.5
13
13.5
14
2000 2005 2010 2015 2020 2025 2030
Short CohortMedium CohortLong Cohort
year of annuity price calculationimm
ed
iate
annuit
y v
alu
e @
4.5
% f
or
65
year
old
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 Fnew(x) = 1- [1-F(x)]d
– d = deterioration factor
• IID Random deterioration factor
Deloitte.
Mortality Term Structuresnum
ber
of
surv
ivors
0 1 2 3 4 5
01
23
45
01234567
8
9
10
cash flow date
valuationdate
initial valuation based onexpected table
assumptions gradually replaced by reality
Deloitte.
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.
Olivier-Smith Model (cont)
vv
sGts
tsL
ssL
tsL
ssL
tsL
tsLvts
vvsG
tsL
tsL
tsL
tsL
ssL
ssLssLBinssL
)1,1(
),1(
),1(
),1(
),1(
)1,1(ln),(
),(~)(
)1,1(
),1(
)1,(
),(
)1,1(
),1(),1,1(~),(
1
1
11
)()*,(
binomial model
gamma deterioration factor
Proportional hazards produces biased deathsBias correction so that L(s,t) is a martingale in s
gamma mean 1, variance v
Deloitte.
Individual Mortality
0
1
2
65 70 75 80 85 90 95 100 105 110
Death Indicator
Dual PrevisibleProjection
process At
process Atp
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
Deloitte.
Stochastic Individual Mortality
0
0.5
1
1.5
2
2.5
3
65 70 75 80 85 90 95 100 105 110
suppose my own hazard rateis a stochastic process …
Consider my “state of health”as a Markov process
dual pre
vis
ible
pro
ject
ion
age
Deloitte.
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.
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.
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.
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.
Useful Links
• SOA Presentation (Olivier & Jeffery)• Andrew Cairns Longevity Model• Affine Mortality (Schrager)• Annuity risks (Hibbert)• Mortality Projections (CMIB)
