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CAIRNS Workshop Presentation 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 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. www.actuaries.org.uk/ARC 3 Actuarial Research Centre (ARC) Current research programmes (2016-2021) • 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 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 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 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 England and Wales males mortality (log scale) 7 Age = 25 Age = 45 Mortality rate Mortality rate 0.005 0.010 0.020 0.001 0.002 0.004 1900 1940 1980 1900 1940 1980 Year Year Age = 65 Age = 85 0.05 0.10 Mortality rate Mortality rate 0.2 0.4 0.8 1900 1940 1980 1900 1940 1980 Year Year 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 STOCHASTIC MORTALITY n lives, probability p of survival, N survivors • Unsystematic mortality risk: ) Njp ∼ Binomial(n; p) ) risk is diversifiable, N=n ! p as n ! 1 • Systematic mortality risk: ) p is uncertain ) risk associated with p is not diversifiable • Longevity Risk: the risk that in aggregate people live longer than anticipated. 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 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 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 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 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 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 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 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 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 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 The Lee-Carter (1992) model (1) (2) log m(t; x) = βx + βx κt (1) Component 1: βx • Age effect • Baseline log-mortality curve (κt = 0) 21 (1) (2) log m(t; x) = βx + βx κt (2) Component 2: βx κt • Age-period component • κt: period effect – changes with time, t ) mortality improvements • (2) βx : age effect – dictates relative rates of improvement at different ages 22 The Lee-Carter (1992) model (1) (2) log m(t; x) = βx + βx κt • Time series model for κt (e.g. random walk) • Single κt for all ages • (2) × Future T : St.Dev.[log m(T; x)] = βx St.Dev.[κT ] 23 Comparison of Eight Models Cairns, et al (2009) North American Actuarial Journal • 8 models • Historical data • Backtesting • Plausibility of forecasts 24 General class of models (1) (1) (1) (N) (N) (N) log m(t; x) = βx κt γt−x + ::: + βx κt γt−x OR (1) (1) (1) (N) (N) (N) logit q(t; x) = βx κt γt−x + ::: + βx κt γt−x • (k) βx = age effect for component k (k) • κt = period effect for component k (k) • γt−x = cohort effect for component k 25 Lee-Carter family (1) (1) (1) (N) (N) (N) log m(t; x) = βx κt γt−x + ::: + βx κt γt−x • (k) βx = non-parametric age effects not smooth (can be smoothed) (k) (k) • κt and γt−x = random period and cohort effects 26 M1: Lee-Carter (1992) model (LC) (1) (2) (2) log m(t; x) = βx + βx κt • N = 2 components • (1) (2) βx , βx age effects (2) • κt single random period effect (1) • κt ≡ 1 • # parameters = 2 × nages + nyears 27 Cohort Effects (e.g. Willetts, 2004) Annual mortality improvement rates (Engl. & Wales, males) 4% 3% 2% 1% Age 0% −1% 20 40 60 80 Annual improvement rate (%) −2% 1970 1980 1990 2000 Year 28 M2: Renshaw-Haberman (2006) model (RH) (1) (2) (2) (3) (3) log m(t; x) = βx + βx κt + βx γt−x • N = 3 components • (1) (2) (3) βx , βx , βx age effects (2) • κt single random period effect (3) • γt−x single cohort effect 29 M3: Age-Period-Cohort model (APC) (1) (2) (3) log m(t; x) = βx + κt + γt−x • N = 3 components • Special case of R-H model • (1) (2) (3) βx age effect; βx = βx = 1 (2) • κt single random period effect (3) • γt−x single random cohort effect 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 M4: P-splines family Age-Period models X (k) (l) (k;l) log m(t; x) = βx κt γt−x k;l where • (k) (l) βx and κt are B-spline basis functions (k;l) • γt−x are constant in t − x for each (k; l) 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 CBD family q(t; x) logit q(t; x) = log 1 − q(t; x) (1) (1) (1) (N) (N) (N) = βx κt γt−x + ::: + βx κt γt−x • (k) βx = parametric age effects pre-specified, e.g. constant, linear, quadratic in x (k) (k) • κt and γt−x = random period and cohort effects 34 M5: Cairns-Blake-Dowd (2006) model (CBD-1) X2 (1) (2) − (i) (i) (i) logit q(t; x) = κt + κt (x x¯) = βx κt γt−x i=1 • N = 2 components • (1) (2) − βx = 1, βx = (x x¯) age effects (1) (2) • κt , κt correlated random period effects (1) (2) • γt−x = γt−x ≡ 1 (model has no cohort effect) 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 Case study: England and Wales males log q_y/(1−q_y) −5 −4 −3 −2 −1 0 60 65 70 75 80 85 90 95 Age of cohort at the start of 2002 qy = mortality rate at age y in 2002 Data suggests logit qy = log qy=(1 − qy) is linear 37 M6-M8: Cohort-effect extensions to CBD-1 • M6: (1) (2) (3) logit q(t; x) = κt + κt (x − x¯) + γt−x • M7: (1) (2) logit q(t; x) = κt + κt (x − x¯) n o (3) − 2 − 2 (4) +κt (x x¯) σx + γt−x • M8: (1) (2) (3) logit q(t; x) = κt + κt (x − x¯) + γt−x(xc − x) 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 Past and Present: Modelling Genealogy Currie/Richards (M4) - Multi- 1 2-D P-splines population Eilers/Marx - - DDE P-splines Hyndman et al.
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