MORTALITY IS ALIVE AND KICKING. Stochastic Mortality Modelling

Transcription

1 1 MORTALITY IS ALIVE AND KICKING Stochastic Mortality Modelling Andrew Cairns Heriot-Watt University, Edinburgh Joint work with David Blake & Kevin Dowd

2 2 PLAN FOR TALK Motivating examples Systematic and non-systematic mortality risk Mathematical concepts forward survival probabilities zero-coupon survivor bonds Short-rate versus Forward-rate models The Olivier-Smith model Generalisations and work in progress

3 3 November 2004: EIB/BNP Paribas longevity bond announced Form of index-linked bond Bond pays 50M S(t) at time t S(t) = proportion of cohort age 65 at time 0 surviving to time t. Reference population: England and Wales, males

4 4 How to price the BNP/EIB longevity bond? S(t) E[S(t)] with parameter uncertainty E[S(t)] without parameter uncertainty 5/95 percentile without parameter uncertainty 5/95 percentile with parameter uncertainty Data from Time, t

5 5 Role of stochastic mortality modelling How do we price this bond? In an arbitrage-free market how might the price of this bond evolve through time? What can we learn about pricing new longevity bonds?

6 6 OTHER MOTIVATING EXAMPLES Pension Plans: Recent poor equity returns, low interest focus on wider range of risks including longevity Life insurers: Annuity portfolios longevity risk Life insurers: deferred annuities with guarantees

7 7 STOCHASTIC MORTALITY n lives, probability p of survival, N survivors Unsystematic mortality risk: N p Binomial(n, p) risk is diversifiable, N/n p as n Systematic mortality risk: p is uncertain risk associated with p is not diversifiable

8 8 THE RUN-OFF PROBLEM Initial portfolio of N(0) identical annuitants N(0) large N(t) still alive at time t β.n(t) payable at t Interest-rate hedge hold βe[n(t)] units of zero-coupon bond maturing at time t. Systematic longevity risk cannot be hedged (yet) need to set up quantile (or CTE) reserves

9 9 Frequency Deterministic reserving ~12.6 Stochastic reserving 95% VaR ~

10 10 THE ANNUITY GUARANTEE PROBLEM Policyholder will retire at time T Lump sum F (T ) available for annuity purchase Market annuity rate: $ a(t ) per $ 1 of annuity Guaranteed purchase price of $ g per $ 1 of annuity Value of option at T is thus V (T ) = F (T ) max{a(t ) g, 0} g

11 11 Risk to individuals and pension plans Male Now age 35 Annuity purchase in 30 years Density High interest High mortality Low interest Low mortality Mortality Risk only Interest Risk only Annuity Price Mortality accounts for 25% of total risk

12 12 Risk to individuals and pension plans Male Now age 35 Annuity purchase in 30 years Density Interest Risk only GAO in the money Mortality + Interest Risk Annuity Price

13 13 MORTALITY MODEL Initial age x µ(t, x + t) ALIVE DEAD µ(t, y) = transition intensity at t, age y at t [ ] t S(t, x) = exp 0 µ(u, x + u)du = survivor index

14 14 Filtration: M t history of µ(u, y) up to time t, for all ages y M t individual histories Single individual aged x at time 0: I(t) = 1 if alive at t 0 otherwise

15 15 FORWARD SURVIVAL PROBABILITIES For T 1 > T 0, and any t p(t, T 0, T 1, x) = P r ( I(T 1 ) = 1 I(T 0 ) = 1, M t ) = E[S(T 1) M t ] E[S(T 0 ) M t ] p Q (t, T 0, T 1, x) = P r Q ( I(T1 ) = 1 I(T 0 ) = 1, M t ) For some Q P, the real-world measure

16 16 Zero-coupon survivor bonds B(t, T, x) = price at t for S(T, x) payable at T for simplicity: assume interest rates are zero The market is abritrage-free if there exists Q P under which the B(t, T, x) are martingales, for all T, x

17 17 B(t, T, x) = E Q [S(T, x) M t ] = p Q (t, 0, T, x) = p Q (1, 0, 1, x)... p Q (t, t 1, t, x) p Q (t, t, t + 1, x)... p Q (t, T 1, T, x) At t + 1: p Q (1, 0, 1, x)... p Q (t, t 1, t, x) p Q (t + 1, t, t + 1, x) p Q (t + 1, t + 1, t + 2, x)... p Q (t, T 1, T, x)

18 18 TWO TYPES OF MODEL Interest-rate modelling terminology: Short-rate models: model for dynamics of p Q (t, t 1, t, x) for all x Forward survival probabilities are output Forward-rate models: model for dynamics of p Q (t, T 1, T, x) T, x Forward survival probabilities are input

19 19 SHORT-RATE MODELS Good for pricing zero-coupon survivor bonds and longevity bonds e.g. by simulation Very few closed-form models that are biologically reasonable Pricing annuity guarantees difficult (i.e. evaluating V (T ) = F (T ) g computationally expensive) max{a(t ) g, 0} is

20 20 FORWARD-RATE MODELS NOT good for pricing zero-coupon survivor bonds and longevity bonds prices are input Prices are output Calculating a(t ) is easy Pricing annuity guarantees (more) straightforward

21 21 THE OLIVIER & SMITH MODEL p Q (t + 1, T 1, T, x) = Discrete time G(t + 1) Gamma(α, α) under Q p Q (t, T 1, T, x) b(t,t 1,T,x)G(t+1) b(t + 1, T 1, T, x) = bias correction factors E Q [ pq (t + 1, t, T, x) M t ] = pq (t, t, T, x)

22 22 WHY GAMMA? 0 < p Q (t, T 1, T, x) < 1 0 < p Q (t + 1, T 1, T, x) < 1 Analytical properties of the Gamma distribution b(t, T, T + 1, x) = αp Q(t, t, T, x) ( 1/α p Q (t, T, T + 1, x) 1/α 1 ) log p Q (t, T, T + 1, x) (Note b 1 is α is large and p close to 1.) exact simulation in discrete time possible

23 23 STATISTICALLY IS IT A GOOD MODEL? Same G(t + 1) applies to all p Q (t + 1, T 1, T, x) Single-factor model No flexibility over the volatility term structure (except through the choice of α) Model testable hypothesis

24 24 STATISTICALLY IS IT A GOOD MODEL? Problem: there is no market no forward survival probabilities Compromise: Concentrate on observed 1-year survival probabilities Assumption: p Q (t, t, t + 1, x t) = θ(x)p Q (t, t 1, t, x (t 1)) θ(x) = age x predicted improvement

25 25 Data: England and Wales mortality, males, Individual calendar years smoothed first G(t, x) calculated for each year t and age x Results: First factor explains 80% of variability For a single x: Estimate α(x) = 1/V ar[g(t, x)] α(x) is clearly dependent on x

26 26 logit(q_x) in 1961 logit(q_x) in 2002 logit(q_x) Linear Cubic Crude data logit(q_x) Age, x Age, x

27 27 Residual Sum of Squares Quality of polynomial fit Ages 40 to 90 Degree 2: Quadratic Degree 1: Linear Degree 3: Cubic Year

28 28 Estimated G(t,x): Age x=45 Estimated G(t,x): Age x=85 G(t,x) G(t,x) Year Year

29 29 Estimated G(t,x): Year t=1967 Estimated G(t,x): Year t=2000 G(t,x) G(t,x) Age, x Age, x

30 30 Contour plot: correlation between mortality improvements at different ages Age Age 1

31 31 Estimated G(t,x): Age x=45 Estimated G(t,x): Age x=85 G(t,x) G(t,x) Year Year

32 32 Estimated alpha(x) alpha(x) Age, x

33 33 A GENERALISED OLIVIER-SMITH MODEL Solution: use copulas Stage 1: one-year spot survival probabilities p Q (t + 1, t, t + 1, x) = p Q (t, t, t + 1, x) b(t,t,t+1,x)g(t+1,x) G(t + 1, x) Gamma ( ) ( α(x), α(x) ) under Q cor G(t + 1, x 1 ), G(t + 1, x 2 ) = ρ(x 1, x 2 ) {G(t + 1, x) : x l x x u } generated e.g. using the multivariate Gaussian copula

34 34 Stage 2A: all spot survival probabilities p Q (t + 1, T 1, T, x) = p Q (t, T 1, T, x) b(t,t 1,T,x)G(t+1,x) G(t + 1, x) Gamma ( α(x), α(x) ) under Q Same G(t + 1, x) for each T cor ( G(t + 1, x 1 ), G(t + 1, x 2 ) ) = ρ(x 1, x 2 ) {G(t + 1, x) : x l x x u } generated e.g. using the multivariate Gaussian copula

35 35 Stage 2B: all forward survival probabilities p Q (t + 1, t, T, x) = p Q (t, t, T, x) g(t,t,x)g(t+1,t,x) G(t+1, T, x) Gamma ( ) α(t, x), α(t, x) underq Different G(t + 1, T, x) for each (T, x) Specified correlation structure {G(t + 1, T, x) : x l x x u ; T > t} generated e.g. using the multivariate Gaussian copula

36 36 ONGOING ISSUES Problem: all 0 < p Q (t + 1, t, T, x) < 1 BUT with small probability Gaussian copula p Q (t + 1, t, T, x) not decreasing with T

37 37 Some thoughts on how to resolve this: Let p Q (t, t, T, x) g(t,t,x)g(t+1,t,x) e M 1g 1 G 1 p Q (t, t, T + 1, x) g(t,t +1,x)G(t+1,T +1,x) e M 2g 2 G 2 M 2 > M 1 Note g 1, g 2 1 Require M 1 g 1 G 1 < M 2 g 2 G 2

38 38 MVN Copula, rho=0.3 MVN Copula, rho=0.7 U_ U_ U_ U_1

39 39 M 1 g 1 G 1 < M 2 g 2 G 2 constraints on copula: alpha = 50 alpha = 500 U_ DENSITY = 0 U_ U_ U_1 e.g. M 1 g 1 /M 2 g 2 = 0.9, α 1 = α 2

40 40 CONCLUSIONS Provided we can find a suitable copula... ( simulation of U(T, x) for all (T, x) easy) generalised Olivier-Smith model could prove a useful tool for modelling stochastic mortality.