MULTI-POPULATION MORTALITY MODELLING: A Danish Case Study. Andrew Cairns Heriot-Watt University, and The Maxwell Institute, Edinburgh

Transcription

1 MULTI-POPULATION MORTALITY MODELLING: A Danish Case Study Andrew Cairns Heriot-Watt University, and The Maxwell Institute, Edinburgh Joint work with D. Blake, K. Dowd, M. Kallestrup-Lamb, C. Rosenskjold IAALS/PBSS, Oslo, June

2 Plan Introduction and motivation for multi-population modelling Modelling Danish sub-population mortality Economic capital 2

3 1. Motivation for multi-population modelling A: Risk assessment Multi-country (e.g. consistent demographic projections) Males/Females (e.g. consistent demographic projections) Socio-economic subgroups (e.g. blue or white collar) Smokers/Non-smokers Annuities/Life insurance Limited data learn from other populations reserving calculations; diversification benefits 3

4 Motivation for multi-population modelling B: Risk management for pension plans and insurers Retain systematic mortality risk; versus: Over-the-counter deals (e.g. longevity swap) Standardised mortality-linked securities linked to national mortality index < 100% risk reduction: basis risk 4

5 Multi-Population Challenges Data availability Data quality and depth Model complexity single population models can be complex 2-population versions are more complex multi-pop... Multi-population modelling requires (fairly) simple single-population models simple dependencies between populations 5

6 2. A New Case Study and a New Model Sub-populations differ from national population socio-economic factors other factors Denmark High quality data on ALL residents available (later data soon) Can subdivide population using covariates on the database 6

7 Danish Data What can we learn from Danish data that will inform us about other populations? Key covariates Wealth Income Affluence = Wealth+15 Income 7

8 Problem High income affluent and low mortality BUT Low income / not affluent, high mortality High wealth affluent and low mortality BUT Low wealth / not affluent, high mortality Empirical solution: use a combination Affluence, A = wealth +K income K = 15 seems to work well statistically as a predictor Low affluence, A, predicts poor mortality 8

9 Subdividing Data (after much experimentation!) Males resident in Denmark for the previous 12 months Divide population in year t into 10 equal sized Groups (approx) using affluence, A Individuals can change groups up to age 67 Group allocations are locked down at age 67 (better than not locking down at age 67) 9

10 Crude death rates 2005 Males Crude m(t,x); 2005 m(t,x) (log scale) Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group Age

11 Modelling the death rates, m k (t, x) m (k) (t, x) = pop. k death rate in year t at age x Population k, year t, age x log m (k) (t, x) = β (k) (x) + κ (k) 1 (t) + κ(k) (t)(x x) 2 (Extended CBD with a non-parametric base table, β (k) (x)) 10 groups, k = 1,..., 10 (low to high affluence) 21 years, t = 1985,..., ages, x = 55,..., 94 11

12 Model-Inferred Underlying Death Rates 2005 Males Crude m(t,x); 2005 Males CBD X Fitted m(t,x); 2005 m(t,x) (log scale) Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group m(t,x) (log scale) Age Age 12

13 Modelling the death rates, m k (t, x) log m (k) (t, x) = β (k) (x) + κ (k) 1 (t) + κ(k) (t)(x x) 2 Model fits the 10 groups well without a cohort effect Non-parametric β (k) (x) is essential to preserve group rankings Rankings are evident in crude data Bio-demographical reasonableness : more affluent healthier 13

14 Model-Inferred Underlying Death Rates 2005 Males Crude m(t,x); 2005 Males CBD X Fitted m(t,x); 2005 m(t,x) (log scale) Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group m(t,x) (log scale) Age Gap reduces from over 6 to 1.5 Age Or +17 years difference for Group 1, age 55; +11 at 67. Convergence way ahead for modelling very high ages??? 14

15 Life Expectancy for Groups 1 to 10 Males Period LE: Age 55 Males Period LE: Age Group 10 Group 9 Group 8 Group 7 Group 6 Group 5 Group 4 Group 3 Group 2 Group

16 Time series modelling t t + 1: Allow for correlation between κ (k) 1 (t + 1) and κ(k) (t + 1) 2 between groups k = 1,..., 10 Medium/long term: group specific period effects gravitate towards the national trend Bio-demographical reasonableness: groups should not diverge 16

17 Simulated Group versus Population Mortality, q(t, x) Total q(t,x) Group 2 T=2006 Corr = 0.58 Total q(t,x) Group 2 T=2010 Corr = 0.72 Total q(t,x) Group 2 T=2030 Corr = Group q(t,x) Group q(t,x) Group q(t,x) As T increases: +1 year; +5 years; +25years Scatterplots become more dispersed Shift down and to the left Correlation increasess 17

18 Forecast Correlations Deciles are quite narrow subgroups More diversified e.g. Blue collar pension plan equal proportions of groups 2, 3, 4 White collar pension plan equal proportions of groups 8, 9, 10 18

19 Forecast Correlations: Mortality Rates Correlation Between Group q(t,x) and Total q(t,x) Correlation Age 75 Blue Collar Plan White Collar Plan Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group Forecasting Horizon (years ahead) 19

20 Forecast Correlations: Cohort Survivorship Correlation Between Group S(t,65) and Total Population S(t,65) Rank Correlation Blue Collar Plan Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group Year 20

21 3. Hedging and Economic Capital Choices No hedging Hedge using own experience Hedge using standardised instrument: national mortality Basis Risk Two sources of basis risk considered here Population basis risk Sub-optimal choice of hedging instrument tradeoff: price vs basis risk 21

22 Economic capital relief using longevity options Population 1: national population; reference for hedge notional portfolio of males aged 65: A 1 =P.V. pension payments Population 2: hedger s own population portfolio of males aged 65: A 2 =P.V. pension payments Three choices: No hedging of A 2 Hedge A 2 with population 1 longevity swap A 1 Â1 Hedge A 2 with out-of-the-money option on A 1 (T ) Payoff at T = 20; underlying A 1 (T ) includes estimated t > T cashflows 22

23 Index Based Hedge: Payoffs Payoff to Hedger at T Cat Bond Payoff at T Payoff to Hedger Cat Bond Payoff Underlying PV National Annuity Underlying PV National Annuity 23

24 Impact of Hedging with T = 20 Option Cumulative Probability Present Value of (Un)Hedged Position: Cumulative Distribution Unhedged Option Longevity Swap PV Hedged Liability 24

25 Impact of Hedging with T = 20 Option Cumulative Probability Present Value of (Un)Hedged Position: Economic Capital 99.5% Runoff Add Economic Capital Best Estimate PV Hedged Liability 25 Unhedged Unhedged Option Longevity Swap

26 Impact of Hedging with T = 20 Option Cumulative Probability Present Value of (Un)Hedged Position: Economic Capital 99.5% Runoff Add Economic Capital Best Estimate Reduction in EC PV Hedged Liability 26 Unhedged Swap Unhedged Option Longevity Swap

27 Impact of Hedging with T = 20 Option Cumulative Probability Present Value of (Un)Hedged Position: Economic Capital 99.5% Runoff Add Economic Capital Best Estimate Reduction in EC PV Hedged Liability 27 Unhedged Swap Reduction in EC Option Unhedged Option Longevity Swap

28 Challenges 1 Simulation example assumes options priced at actuarially fair value But swap and option premiums might be more expensive Compare premium versus value of reduction in Economic Capital over multiple time periods 28

29 Challenges 2 Bull spread option: Choice of attachment/detachment points K 1, K 2 Maximum cat bond loss Capital markets capacity annuity liabilities Risk premiums Sub-optimal instrument basis risk 29

30 4. Summary Danish data allows insight into relative mortality dynamics between socio-economic sub-populations Conclusions for other countries likely to be similar Economic capital example is one of many potential risk management applications Working paper available soon. E: W: andrewc 30

31 Bonus Slides 31

32 Subdivided Data Exposures E (i) (t, x) for groups i = 1,..., 10 range from over 4000 down to 20 Deaths D (i) (t, x) range from 150 down to 6 Poisson risk is important Crude death rates ˆm (i) (t, x) = D (i) (t, x)/e (i) (t, x) 32

33 A specific model κ (i) (t) = 1 κ(i) 1 (t 1) + µ 1 + Z 1i (t) (random walk) ( ) ψ κ (i) 1 (t 1) κ 1(t 1) κ (i) (t) = 2 κ(i) 2 (t 1) + µ 2 + Z 2i (t) ( ) ψ κ (i) 2 (t 1) κ 2(t 1) (gravity between groups) where κ 1 (t) = 1 n n i=1 κ (i) 1 (t) and κ 2(t) = 1 n n i=1 κ (i) 2 (t) 33

34 A specific model κ (i) 1 (t) = κ (i) 1 (t 1) + µ 1 + Z 1i (t) ψ κ (i) 2 (t) = κ (i) 2 (t 1) + µ 2 + Z 2i (t) ψ Model structure ( κ 1 (t), κ 2 (t)) bivariate random walk ( ) κ (i) 1 (t 1) κ 1 (t 1) ( ) κ (i) 2 (t 1) κ 2 (t 1) Each κ (i) 1 (t) κ 1(t) AR(1) reverting to 0 Each κ (i) 2 (t) κ 2(t) AR(1) reverting to 0 β (i) (x) vs β (j) (x) intrinsic group differences 34

35 Non-trivial correlation structure: between different ages and groups κ (i) 1 (t) = κ (i) 1 (t 1) + µ 1 +Z 1i (t) ψ κ (i) 2 (t) = κ (i) 2 (t 1) + µ 2 +Z 2i (t) ψ ( ) κ (i) 1 (t 1) κ 1 (t 1) ( ) κ (i) 2 (t 1) κ 2 (t 1) The Z ki are multivariate normal, mean 0 and Cov(Z ki, Z lj ) = v kl ρv kl ρ = cond. correlation between κ (i) 1 for i = j for i j (t) and κ(j) (t) 1 etc. 35

36 Mortality Fan Charts Including Parameter Uncertainty Mortality Rates: Age 75 q(t,x) Group 1 Group Year 36

37 Comments Model is very simple One gravity parameter, 0 < ψ < 1 One between-group correlation parameter, 0 < ρ < 1 Many generalisations are possible But more parameters + more complex computing This simple model seems to fit quite well. Nevertheless work in progress 37

38 Prior distributions As uninformative as possible µ 1, µ 2 improper uniform prior {v ij } Inverse Wishart ρ Beta(2, 2) ψ Beta(2, 2) State variables and process parameters estimated using MCMC (Gibbs + Metropolis-Hastings) 38

39 Posterior Distributions and 95% Credibility Intervals Kappa_1 Drift, mu_1 Kappa_2 Drift, mu_2 Cumulative Posterior Probability Cumulative Posterior Probability mu_ mu_2 Note: global improvement rate = µ 1 39

40 Posterior Distributions and 95% Credibility Intervals Between Group Correlation, rho Gravity Parameter, psi Cumulative Posterior Probability Cumulative Posterior Probability rho psi 40

41 Economic capital relief using longevity options Population 1: national population; reference for hedge Population 2: hedger s own population Option payoff at T based on Pop 1 cashflows up to T Estimated Pop 1 cashflows after T (commutation) Option Special purpose vehicle Longevity CAT bond BE = best estimate liability at time 0 EC = additional Economic Capital to cover e.g. 99.5% runoff EC 0 = EC without hedge EC 1 = EC with index-based option hedge 41

42 Practical issues Structure of the hedging instrument Price / risk premium payable by hedger Tradeoff: Hedger Customised Full term Uncapped payoff Swap Counterparty Index Medium term Limited loss Cat Bond format Tradeoff basis risk: sub-optimal hedge instrument + pop. 42

43 Longevity Option S 1 (t, x) = Population 1 Survivor Index A 1 = t=1 vt S 1 (t, x) (PV Population 1 Annuity) Underlying index for option: A 1 (T ) = E C [A 1 F T ] E C expected cashflows after T calculated on a basis prescribed at time 0 Best estimate of the PV given information up to time T? S 2 (t, x) = Population 2 Survivor Index A 2 = t=1 vt S 2 (t, x) (PV Population 2 Annuity) 43

44 Longevity Swaps and Option Hedging instrument payoffs (discounted to time 0) Longevity swap A: A 1 E Q [A 1 ] (cashflow hedge) Longevity swap B: A 1 (T ) E Q [A 1 ] (payable at T ) Bull spread, attachment points K 1 < K 2 B(T ) = (A 1 (T ) K 1 ) + (A 1 (T ) K 2 ) + (payable at T ) PV Bull spread hedge Y (T, K 1, K 2, h) = A 2 h (B(T ) E Q [B(T )]) E Q pricing measure at time 0 44

45 Hedging Example Hedgers population: Blue Collar annuity portfolio Equal proportions of groups 2, 3, 4 Age 65 cohort, males Index hedge linked to national mortality Age 65 cohort, males Attachment points: 13.5, 14 45

46 Index Based Hedge: the Underlying Cumulative Probability Attachment Points Underlying PV National Annuity 46

47 Index Based Hedge:Payoffs Payoff to Hedger at T Cat Bond Payoff at T Payoff to Hedger Cat Bond Payoff Underlying PV National Annuity Underlying PV National Annuity Cat Bond maximum loss =$0.5 Billion, $12 Billion liabilities (E[A 2 ]) Notional $1Bn 47

48 Recap: Hedging choices No hedge Hedge Population 2 A 2 using Population 1 longevity swap bull spread with T = bull spread with T = 20 48

49 Unhedged Longevity Swap Option oo Option 20 yr Cumulative Probability Unhedged Longevity Swap Option oo Option 20 yr 10.5 PV Hedged Liability Impact of Hedging with Longevity Swap PV Unhedged Liability PV Hedged Liability

50 = Bull Spread payable on last death Unhedged Longevity Swap Option oo Option 20 yr Cumulative Probability Unhedged Longevity Swap Option oo Option 20 yr 10.5 PV Hedged Liability Impact of Hedging with T PV Unhedged Liability PV Hedged Liability

51 = 20 Bull Spread Unhedged Longevity Swap Option oo Option 20 yr Cumulative Probability Unhedged Longevity Swap Option oo Option 20 yr 10.5 PV Hedged Liability Impact of Hedging with T PV Unhedged Liability PV Hedged Liability

52 13.5 Impact of Hedging on Economic Capital Reduction in Economic Capital Economic Capital Unhedged Longevity Swap Option oo Option 20 yr Option 10 yr PV Hedged Liability % quantiles PV Unhedged Liability /