MODELLING AND MANAGEMENT OF MORTALITY RISK

Transcription

1 1 MODELLING AND MANAGEMENT OF MORTALITY RISK Stochastic models for modelling mortality risk ANDREW CAIRNS Heriot-Watt University, Edinburgh and Director of the Actuarial Research Centre Institute and Faculty of Actuaries

2 2 Actuarial Research Centre (ARC) The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuaries network of actuarial researchers around the world. The ARC seeks to deliver research programmes that bridge academic rigour with practitioner needs by working collaboratively with academics, industry and other actuarial bodies. The ARC supports actuarial researchers around the world in the delivery of cutting-edge research programmes that aim to address some of the significant challenges in actuarial science.

3 3 Actuarial Research Centre (ARC) Current research programmes ( ) Modelling, Measurement and Management of Longevity and Morbidity Risk Use of Big Health and Actuarial Data for understanding Longevity and Morbidity Minimising Longevity and Investment Risk While Optimising Future Pension Plans

4 4 Stochastic models for modelling mortality risk: Plan Introduction, motivation, problems Modelling Criteria for a good model Comparison of 8 models Robustness Graphical diagnostics Applications

5 5 The Problem 2017: What we know as the facts: Life expectancy is increasing. Future development of life expectancy is uncertain. Longevity risk Systematic risk for pension plans and annuity providers

6 6 The Problem Example: UK Defined-Benefit Pension Plans Before 2000: High equity returns masked impact of longevity improvements After 2000: Poor equity returns, low interest rates Decades of longevity improvements now a problem

7 England and Wales males mortality (log scale) 7 Age = 25 Age = 45 Mortality rate Mortality rate Year Year Age = 65 Age = 85 Mortality rate Mortality rate Year Year

8 8 Graphical diagnostics Mortality is falling Different improvement rates at different ages Different improvement rates over different periods Improvements are random Short term fluctuations Long term trends All stylised facts

9 9 STOCHASTIC MORTALITY n lives, probability p of survival, N survivors Unsystematic mortality risk: N p Binomial(n, p) risk is diversifiable, Systematic mortality risk: N/n p as n p is uncertain risk associated with p is not diversifiable Longevity Risk: the risk that in aggregate people live longer than anticipated.

10 10 Why do we need stochastic mortality models? Data future mortality is uncertain Good risk management Setting risk reserves Regulatory capital requirements (e.g. Solvency II) Life insurance contracts with embedded options Pricing and hedging mortality-linked securities

11 11 Modelling Aims: to develop the best models for forecasting future uncertain mortality; general desirable criteria complexity of model complexity of problem; longevity versus brevity risk; measurement of risk; valuation of future risky cashflows.

12 12 Management Aims: active management of mortality and longevity risk; internal (e.g. product design; natural hedging) over-the-counter deals (OTC) securitisation part of overall package of good risk management.

13 13 Modelling Stochastic Mortality Many models to choose from Limited data model and parameter risk Important to take the time to analyse models thoroughly No single model is best for all datasets and applications

14 14 Model Selection Criteria Positive mortality rates Consistent with historical data Biologically reasonable and plausible forecasts Robust parameter estimates and forecasts Straightforward to implement Parsimonious Generates sample paths Can include parameter uncertainty Cohort effect if appropriate Non-trivial correlation structure Not used as a black box

15 15 Consistent with historical data Model fit consistent with i.i.d. Poisson assumption goodness of fit tests graphical diagnostics Compare models using likelihoods and the Bayes Information Criterion (BIC) Future versus past patterns of randomness Backtesting

16 16 Biologically reasonable and plausible forecasts Biologically reasonable e.g. inverted mortality curve?? strong mean reversion?? time horizon matters Plausible forecasts trend and degree of uncertainty

17 17 Robustness What happens if I change the age range? What happens if I add one extra calendar year? Revised parameter estimates and forecasts should be similar to old

18 18 Not a black box Understand the advantages and disadvantages of each model Understand the limitations and assumptions of each model Better understanding of the model Better understanding of the risks Good risk management practice

19 19 Measures of mortality q(t, x) = underlying mortality rate in year t at age x m(t, x) = underlying death rate Assume q(t, x) = 1 exp[ m(t, x)] Poisson model: Exposures: E(t, x) Actual deaths: D(t, x) Poisson (m(t, x)e(t, x)) in year t, age x last birthday

20 20 The Lee-Carter (1992) model Component 1: β (1) x Age effect log m(t, x) = β x (1) + β x (2) Baseline log-mortality curve (κ t = 0) κ t

21 21 Component 2: β x (2) κ t Age-period component κ t : period effect log m(t, x) = β x (1) + β x (2) changes with time, t mortality improvements β (2) x : age effect dictates relative rates of improvement at different ages κ t

22 22 The Lee-Carter (1992) model log m(t, x) = β x (1) + β x (2) Time series model for κ t (e.g. random walk) Single κ t for all ages Future T : St.Dev.[log m(t, x)] = β (2) x St.Dev.[κ T ] κ t

23 23 Comparison of Eight Models Cairns, et al (2009) North American Actuarial Journal 8 models Historical data Backtesting Plausibility of forecasts

24 24 General class of models OR β (k) x κ (k) t log m(t, x) = β (1) x κ (1) t γ (1) logit q(t, x) = β (1) x κ (1) t γ (1) = age effect for component k = period effect for component k γ (k) t x = cohort effect for component k t x β (N) x t x β (N) x κ (N) t γ (N) t x κ (N) t γ (N) t x

25 25 Lee-Carter family β (k) x log m(t, x) = β (1) x κ (1) t γ (1) = non-parametric age effects not smooth (can be smoothed) t x β (N) x κ (N) t γ (N) t x κ (k) t and γ (k) t x = random period and cohort effects

26 26 M1: Lee-Carter (1992) model (LC) N = 2 components β x (1), β(2) x κ (2) t κ (1) t 1 log m(t, x) = β x (1) + β x (2) κ (2) t age effects single random period effect # parameters = 2 n ages + n years

27 27 Cohort Effects (e.g. Willetts, 2004) Annual mortality improvement rates (Engl. & Wales, males) Age Annual improvement rate (%) 4% 3% 2% 1% 0% 1% 2% Year

28 28 M2: Renshaw-Haberman (2006) model (RH) log m(t, x) = β (1) x + β (2) x κ (2) t + β (3) x γ (3) t x N = 3 components β (1) x κ (2) t, β(2) x, β(3) x age effects single random period effect γ (3) t x single cohort effect

29 29 M3: Age-Period-Cohort model (APC) N = 3 components Special case of R-H model log m(t, x) = β (1) x + κ (2) t + γ (3) t x β (1) x age effect; β (2) x = β (3) x = 1 κ (2) t single random period effect γ (3) t x single random cohort effect

30 30 Background M1: Lee-Carter First (??) stochastic mortality model Simple and robust Reasonable fit over a wide range of ages M2: Renshaw-Haberman Incorporation of a cohort effect M3: APC Roots in medical statistics, pre Lee-Carter Simpler and more robust than R-H

31 31 M4: P-splines family Age-Period models log m(t, x) = k,l β (k) x κ (l) t γ (k,l) t x where β (k) x and κ (l) t are B-spline basis functions γ (k,l) t x are constant in t x for each (k, l)

32 32 Background M4: Age-Cohort P-splines model Data are noisy Underlying m(t, x) is smooth Model parsimonious, non-parametric fit Output: confidence intervals for underlying smooth surface (Non-parametric generalisation of linear regression)

33 33 CBD family logit q(t, x) = log q(t, x) 1 q(t, x) β (k) x = β (1) x κ (1) t γ (1) = parametric age effects t x β (N) pre-specified, e.g. constant, linear, quadratic in x x κ (N) t γ (N) t x κ (k) t and γ (k) t x = random period and cohort effects

34 34 M5: Cairns-Blake-Dowd (2006) model (CBD-1) logit q(t, x) = κ (1) t + κ (2) t (x x) = 2 N = 2 components i=1 β x (i) κ (i) t γ (i) t x β x (1) = 1, β x (2) κ (1) t, κ (2) t = (x x) age effects correlated random period effects γ (1) t x = γ (2) t x 1 (model has no cohort effect)

35 35 Background M5: CBD-1 Designed to take advantage of simple structure at higher ages focus on pension plan longevity risk Two random period effects allows different improvements at different ages at different times Simple and robust, good at bigger picture

36 36 Case study: England and Wales males log q_y/(1 q_y) Age of cohort at the start of 2002 q y = mortality rate at age y in 2002 Data suggests logit q y = log q y /(1 q y ) is linear

37 37 M6-M8: Cohort-effect extensions to CBD-1 M6: M7: M8: logit q(t, x) = κ (1) t + κ (2) t (x x) + γ (3) t x logit q(t, x) = κ (1) t + κ (2) t (x x) +κ (3) { t (x x) 2 σx 2 } (4) + γ t x logit q(t, x) = κ (1) t + κ (2) t (x x) + γ (3) t x(x c x)

38 38 Background M6-M8: CBD-2/3/4 Developed during the course of the bigger study Build on the advantages of M1-M5 Avoid the disadvantages of M1-M5 Models focus on the higher ages

39 39 Past and Present: Modelling Genealogy APC model (M3) Eilers/Marx P-splines Lee-Carter (M1) Currie/Richards (M4) 2-D P-splines Hyndman et al. Booth et al. DDE APC model (M3) Renshaw-Haberman (M2) CBD-1 (M5) CBD-2 (M6) CBD-3 (M7) Multipopulation Multipopulation Plat CBD-5 (M9) Time CBD-4 (M8) CBD-R Mavros et al.

40 40 Quantitative Criteria Bayes Information Criterion (BIC) Model k: ˆl k = model maximum likelihood BIC penalises over-parametrised models Model k: BIC k = ˆl k 1 2 n k log N n k = number of parameters (effective) N = number of observations

41 41 Maximum Likelihood Estimation Usual approach: Stage 1: estimate the β (k) x, κ(k) t, γ (k) t x without reference to the stochastic models governing the period and cohort effects. Stage 2: fit a stochastic model to the ˆκ (k) t Okay for large populations Smaller populations: exercise caution, κ(k) t, γ (k) t x subject to estimation error β (k) x and ˆγ (k) t x

42 42 Alternatives to 2-stage MLE 1-stage MLE Models for κ (k) t, γ (k) t x specified in advance Full Bayesian model (e.g. Czado et al.) Models for κ (k) t, γ (k) t x specified in advance Output includes posterior distributions for model parameters plus latent β x (k), κ(k) t, γ (k) t x

43 43 2-Stage MLE: Application to 8 Models England and Wales males Ages Exclusions : ages (not available) 1886 cohort (unreliable exposures) Cohorts with 4 or fewer data points (overfitting)

44 44 Typical parameter estimation results: M3-APC Age Effect, beta Age Period Effect, Kappa Year Cohort Effect, Gamma Year of Birth

45 45 Model Max log-lik. # parameters BIC (rank) M1: LC M2: RH M3: APC M4: P-Splines M5: CBD M6: CBD M7: CBD M8: CBD

46 46 The BIC doesn t tell us the whole story... Qualitative Criteria Graphical diagnostics Poisson model (t, x) cells are all independent. Standardised residuals: Z(t, x) = D(t, x) ˆm(t, x)e(t, x) ˆm(t, x)e(t, x) If the data are not i.i.d.: What do the patterns tell us?

47 47 Are standardised residuals i.i.d.? LC and RH models Model M Model M Black Z(t, x) <

48 48 APC and P-splines models Model M Model M

49 49 CBD-1 and CBD-2 models Model M Model M

50 50 CBD-3 and CBD-4 models Model M Model M

51 51 Are the standardised residuals i.i.d.? More graphical diagnostics: Scatterplots of residuals versus Age Year of observation Year of birth

52 52 M1: LC model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

53 53 M2: RH model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

54 54 M3: APC model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

55 55 M4: P-splines model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

56 56 M5: CBD-1 model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

57 57 M6: CBD-2 model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

58 58 M7: CBD-3 model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

59 59 M8: CBD-4 model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

60 60 Robustness Want to see stability in parameter estimates Extra years of data Extra ages Within model hierarchy

61 61 M7 (CBD-3): (a) 1961 to 2004 (dots) or (b) 1981 to 2004 (solid lines) Kappa_1(t) Kappa_2(t) Year Year 1 e 03 4 e 04 2 e 04 Kappa_3(t) Gamma_4(t x) Year Year of birth

62 62 RECAP: M5: CBD-1 model Standardised residuals Standardised residuals Standardised residuals Year of Observation Age Year of Birth

63 63 M2 (RH): (a) 1961 to 2004 (dots) or (b) 1981 to 2004 (solid lines) Beta_1(x) Beta_2(x) Kappa_2(t) Age Age Year Beta_3(x) Gamma_3(t x) Age Year of birth

64 64 Robustness Parameter estimates should not be too sensitive to the choice of range of ages and years. M2 has a possible problem β (3) x age effect seems to be qualitatively different for the versus

65 65 Qualitative criteria or issues Forecast reasonableness More on robustness

66 66 Simulation models Up to now: historical fit only Forecasting requires a stochastic model ARIMA time series models to simulate future period and cohort effects Process Risk or Stochastic Risk Later: parameter risk and model risk

67 67 Simulation models Examples: M1: Lee-Carter model period effect, κ (2) t M7: CBD-3 model = random walk with drift (κ (1) t, κ (2) t, κ (3) t ) = multivariate random walk with drift γ (4) c = AR(1) model ARIMA(1,0,0)

68 68 Mortality Fan Charts + A plausible set of forecasts Mortality rate, q(t,x) Model CBD 1 Fan Chart AGE 85 AGE 75 AGE Year, t

69 69 Model risk Mortality rate, q(t,x) Model CBD 1 Fan Chart AGE 85 AGE 75 AGE Year, t

70 70 Model risk Mortality rate, q(t,x) Combined CBD 1, CBD 3 Fan Chart AGE 85 AGE 75 AGE Year, t

71 71 Model risk Mortality rate, q(t,x) Combined CBD 1, CBD 3, CBD 4 Fan Chart AGE 85 AGE 75 AGE Year, t

72 72 Plausibility of forecasts Defining Plausible is impossible! Visually: given the forecast are you reasonably comfortable? slightly uncomfortable? fan chart is clearly unreasonable?

73 73 US males : M8 unreasonable forecasts Mortality Rate x = 84 x = 75 x=65 M8A

74 74 Robustness of Forecasts Forecasts Set 1: Data from β (k) x Use full set of β (k) x Forecasts Set 3: Data from β (k) x Use full set of β (k) x, κ(k) t, γ (k) t x, κ(k) t, γ (k) t x to make forecasts (30), κ (k) t, κ(k) t, γ (k) t x (24), γ (k) t x (45) to make forecasts

75 75 Robustness of Forecasts Forecasts Set 2: Data from β (k) x To make forecasts:, κ(k) t, γ (k) t x Use all 30 β (k) x Use the last 24 κ (k) t only (out of 44) Use the last 45 γ (k) t x only (out of 65) i.e. as if

76 76 Perfect model + large population Forecast sets 2 and 3: Same β (i) x, κ(i) t Same forecasts Good robust model, γ (i) t x Forecast sets 2 and 3: Similar β (i) x, κ(i) t Similar forecasts, γ (i) t x

77 77 Robustness: e.g. M3 - Age-Period-Cohort model APC Model Age 75 Mortality Rates Mortality rate data: APC full data: APC limited data: APC Year, t

78 78 Robustness: e.g. M7 - CBD-3 model Model M7 Age 65 Mortality Rates gamma Mortality rate Age 75 Mortality Rates Age 85 Mortality Rates Mortality rate Mortality rate

79 79 Not all models are robust: Renshaw-Haberman model Model R H (ARIMA(1,1,0)) projections Cohort effect x=85 x=75 x= data: R H full data: R H limited data: R H Year, t

80 80 Robustness Problem Likely reason: Likelihood function has multiple maxima Consequences: Lack of robustness within sample Lack of robustness in forecasts central trajectory prediction intervals Some sample periods implausible forecasts

81 81 Parameter Uncertainty: CBD model M5 example factor model: Kappa_1(t)= Year, t factor model: Kappa_2(t) Year, t

82 82 κ t = (κ (1) t, κ (2) t ) Model: Random walk with drift µ = (µ 1, µ 2 ) = drift κ t+1 κ t = µ + CZ(t + 1) V = CC = variance-covariance matrix Estimate µ and V Quantify parameter uncertainty in µ and V Quantify the impact of parameter uncertainty

83 83 Application: cohort survivorship Cohort: Age x at time t = 0 S(t, x) = survivor index at t proportion surviving from time 0 to time t S(t, x) = (1 q(0, x)) (1 q(1, x + 1)) (1 q(t 1, x + t 1))

84 84 90% Confidence Interval (CI) for Cohort Survivorship Proportion surviving, S(x) E[S(x)] with param. uncertainty CI without param. uncertainty CI with param. uncertainty Data from Age

85 85 Cohort Survivorship: General Conclusions Less than 10 years: Systematic risk not significant Over 10 years Systematic risk becomes more and more significant over time Over 20 years Parameter risk begins to dominate (+ model risk)

86 86 Part 1: Concluding remarks Range of models to choose from Quantitative criteria are only the starting point Additional criteria Some models pass Some models fail Focus here on mortality data at higher ages Wider age range CBD models less good

87 Applications A Taster 87

88 88 Applications Scenario Generation Example: the Lee Carter Model m(t, x) = β (1) (x) + β (2) (x)κ(t) Choose a time series model for κ(t) Calibrate the time series parameters using data up to the current time (time 0) Generate j = 1,..., N stochastic scenarios of κ(t) κ 1 (t),..., κ N (t)

89 89 Generate N scenarios for the future m(t, x) m j (t, x) for j = 1,..., N, t = 0, 1, 2,..., x = x 0,..., x 1 Generate N scenarios for the survivor index, S j (t, x) Calculate financial functions + variations for some financial applications.

90 90 Period Effect: One Scenario Period Effect, kappa(t) Historical Simulated κ(t): Generate scenario 1 Time

91 91 Period Effect: Multiple Scenarios Period Effect, kappa(t) Historical Simulated Multiple scenarios Time

92 92 Period Effect: Fan Chart Fan chart Period Effect, kappa(t) Historical Simulated Time

93 93 Death Rates, Age 65: One Scenario Death Rate (log scale) Time

94 94 Death Rates, Age 65: Multiple Scenarios Death Rate (log scale) Time

95 95 Death Rates, Age 65: Fan Chart Death Rate (log scale) Time

96 96 Age Extract Cohort Death Rates, m(t,x+t 1) Time Annuity valuation follow cohorts m(0, x) m(1, x + 1) m(2, x + 2)...

97 97 Cohort Death Rates From Age 65: One Scenario Death Rate (log scale) Cohort Age Annuity valuation follow cohorts m(0, x) m(1, x + 1) m(2, x + 2)...

98 98 Cohort Death Rates From Age 65: Multpiple Scenarios Death Rate (log scale) Cohort Age

99 99 Cohort Death Rates From Age 65: Fan Chart Death Rate (log scale) Cohort Age

100 100 Survivor Index (log scale) Survivorship From Age 65: One Scenario Cohort Age Cohort death rates cohort survivorship

101 101 Survivor Index (log scale) Survivorship From Age 65: Multiple Scenarios Cohort Age

102 102 Survivor Index (log scale) Survivorship From Age 65: Fan Chart Cohort Age

103 103 Cohort Life Expectancy from Age 65 Cohort Life Expectancy from Age 65 Frequency Cumulative Probability Life Expectancy From Age Life Expectancy From Age 65 Cohort survivorship ex post cohort life expectancy Equivalent to a continuous annuity with 0% interest

104 104 Present Value of Annuity from Age 65 Present Value of Annuity from Age 65 Frequency Cumulative Probability Present Value ofannuity From Age Present Value ofannuity From Age 65 Annuity of 1 per annum payable annual in arrears Interest rate: 2%

105 105 Present Value of Annuity from Age 65 Present Value of Annuity from Age 65 Frequency Mean +8% 99.5% Va Cumulative Probability Present Value ofannuity From Age Present Value ofannuity From Age 65 Mean = $15.17 per $1 annuity; BUT Need $16.38 to be 99.5% sure of covering all liabilities

106 106 Age Extract Period Death Rates, m(t,x+t 1) Time Period life expectancy and related quantities

107 107 Period Life Expectancy Period Life Expectancy From Age 65 By Calendar Year Historical Forecast Time

108 108 Death Rates, Age 65: One Scenario Death Rate (log scale) Historical Simulated Simulated Recalibration+ Central Forecast Recalibration Valuation at time 10 Time Recalibrate model and parameters central forecast Updated liability value at time 10

109 109 Death Rates, Age 65: Multiple Scenarios Death Rate (log scale) Historical Simulated Recalibration+ Central Forecast Time Recalibration

110 110 Cumulative Probability Present Value of Annuity from Age 65 PV: Full Runoff PV: Valuation at Time 10 PV: Valuation at Time Present Value ofannuity From Age 65 Applications: Hedging longevity risk

111 111 Part 2: Concluding Remarks Here: Lee-Carter m(t, x) application Modular code Model X m(t, x); m(t, x) application Applications Development of simple stress tests Reserving Longevity risk transfer Multi-population models

112 112 References Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A., and Balevich, I. (2009) A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal 13(1): Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G., Epstein, D., and Khalaf-Allah M. (2011) The Plausibility of Mortality Density Forecasts: An Analysis of Six Stochastic Mortality Models. Insurance: Mathematics and Economics, 48: Cairns, A.J.G., Kallestrup-Lamb, M., Rosenskjold, C.P.T., Blake, D., and Dowd, K., (2017) Modelling Socio-Economic Differences in the Mortality of Danish Males Using a New Affluence Index. Preprint. andrewc/papers/ajgc73.pdf