A GENERALISATION OF THE SMITH-OLIVIER MODEL FOR STOCHASTIC MORTALITY

Transcription

1 1 A GENERALISATION OF THE SMITH-OLIVIER MODEL FOR STOCHASTIC MORTALITY Andrew Cairns Heriot-Watt University, Edinburgh

2 2 PLAN FOR TALK Two 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 THE ANNUITY GUARANTEE PROBLEM simplified Policyholder will retire at time T Lump sum k 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 ) = k g max{a(t ) g, 0} What is the price of the option at time t < T?

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 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

9 9 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 P r( I(t) = 1 I(0) = 1, M t ) = S(t, x) P r( I(t) = 1 I(0) = 1, M 0 ) = E[S(t, x) M 0 ]

10 10 FORWARD SURVIVAL PROBABILITIES Real-world probabilities, P For T 1 > T 0, and any t p P (t, T 0, T 1, x)=p r P ( I(T1 ) = 1 I(T 0 ) = 1, M t ) Pricing measure Q P = E P[S(T 1 ) M t ] E P [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 )

11 11 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

12 12 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 + 1, T 1, T, x)

13 13 TWO TYPES OF MODEL (Interest-rate terminology:) Short-rate models: (state variable X(t)) model for dynamics of p Q (t, t 1, t, x) for all x as a function of X(t) 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

14 14 SHORT-RATE MODELS: state variable X(t) Good for pricing zero-coupon survivor bonds and longevity bonds e.g. by simulation up to T B(t, T, x) = E Q [p Q (t + 1, t, t + 1, x) p Q (T, T 1, T, x) X(t) ] Very few biologically reasonable models have an analytical form for B(t, T, x) as a function of X(t)

15 15 SHORT-RATE MODELS: state variable X(t) Pricing annuity guarantees is difficult: Recall V (T ) = k g max {a(t ) g, 0} a(t ) = price at T for annuity of $1 per annum from time T. a(t ) a(t, X(T )) = u=t vt B(t, u, x, X(T ))

16 16 No analytical form for B(T, u, x, X(T )) evaluating V (T ) is computationally expensive Hence pricing annuity guarantees is difficult using short-rate models

17 17 FORWARD-RATE MODELS NOT good for pricing zero-coupon survivor bonds and longevity bonds prices are input Prices are output for all u T, B(T, u, x) is automatically available at T Calculating a(t ) is easy Pricing annuity guarantees (more) straightforward

18 18 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)

19 19 WHY GAMMA? 0 < p Q (t, T 1, T, x) < 1 0 < p Q (t + 1, T 1, T, x) < 1 Gamma + martingale property of p Q (u, t, T, x) for u = t, t + 1 implies 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 if α is large and p close to 1.) exact simulation in discrete time possible

20 20 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

21 21 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

22 22 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

23 23 logit(q_x) in 1961 logit(q_x) in 2002 logit(q_x) Linear Cubic Crude data logit(q_x) Age, x Age, x

24 24 Residual Sum of Squares Quality of polynomial fit Ages 40 to 90 Degree 2: Quadratic Degree 1: Linear Degree 3: Cubic Year

25 25 Estimated G(t,x): Age x=45 Estimated G(t,x): Age x=85 G(t,x) G(t,x) Year Year

26 26 Estimated G(t,x): Year t=1967 Estimated G(t,x): Year t=2000 G(t,x) G(t,x) Age, x Age, x

27 27 Contour plot: correlation between mortality improvements at different ages: G(t,x) and G(t,y) Age 2: y Age 1: x

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 alpha(x) alpha(x) Age, x

30 30 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

31 31 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

32 32 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

33 33 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

34 34 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

35 35 MVN Copula, rho=0.3 MVN Copula, rho=0.7 U_ U_ U_ U_1

36 36 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

37 37 FEASIBLE VALUES OF α(t ) Observation: The zero-density boundary must lie completely below the diagonal u 1 = u 2. Otherwise we don t have a copula. Lemma We require 1 α(t + 1) α(t ) g 2M 2 g 1 M 1

38 38 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.

39 39

40 40 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

41 41 Frequency Deterministic reserving ~12.6 Stochastic reserving 95% VaR ~

42 42 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 rate risk only Open Market Annuity Price Mortality accounts for 25% of total risk

43 43 Risk to individuals and pension plans Male Now age 35 Annuity purchase in 30 years Probability Density Interest rate risk only GAO in the money Mortality + Interest Risk Open Market Annuity Price