Modelling, Estimation and Hedging of Longevity Risk

Transcription

1 IA BE Summer School 2016, K. Antonio, UvA 1 / 50 Modelling, Estimation and Hedging of Longevity Risk Katrien Antonio KU Leuven and University of Amsterdam IA BE Summer School 2016, Leuven Module II: Fitting a Lee-Carter mortality model

2 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 2 / 50 Parametric models for mortality Ambition: (dates back to the 18th century with Abraham de Moivre, 1725) specify a parametric model for the death rate (m x ), the mortality rate (q x ) or the force of mortality (µ x ). A functional expression is suggested, depending on unknown parameters which should be estimated. When the parametric model is well chosen: very useful and elegant; dimensionality of problem is reduced. If it s not well chosen: completely wrong.

3 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 3 / 50 Parametric models for mortality Gompertz (1825) s law for µ x : µ x = θ 2 θ x 3, with θ 2 > 0 and θ 3 > 1. The law tries to describe the process of human ageing. Makeham (1860) adjusted the Gompertz law: µ x = θ 1 + θ 2 θ x 3, with θ 1 0, θ 2 > 0 and θ 3 > 1.

4 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 4 / 50 Parametric models for mortality The law of Heligman & Pollard (1980) is popular among anglo saxon actuaries (see McDonald et al., 1998). Heligman & Pollard: q x (x+θ 2)θ3 = θ 1 + θ 4 exp ( θ 5 (ln x ln θ 6 ) 2 ) + θ 7 θ p 8. x x Building blocks: - first term = mortality of infants; - second term = accident hump; - third term = Gompertz law for adult mortality.

5 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 5 / 50 Parametric models for mortality Several methods have been proposed to estimate the parameters in such parametric laws for mortality. Namely: - Frère (1968); - Ballegeer (1973); - De Vylder (1975).

6 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 6 / 50 Parametric models for mortality Several methods have been proposed to estimate the parameters in such parametric laws for mortality. Note various parameterizations of mortality laws, e.g. with Makeham µ x = θ 1 + θ 2 θ3 x ( then t p x = exp θ 1 t θ 2θ3 x ) (θ3 t 1) ln θ 3 or tp x = θ t 5 θ (θt 3 1)θx 3 6 or ln p x = θ 1 + θ 7 θ x 3.

7 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 7 / 50 Parametric models for mortality Example of fitting strategy: least-squares method of Ballegeer (1973): - use e.g. Makeham s law with p x = θ 5 θ (θ3 1)θx 3 6, then ln p x = ln (θ 5 ) + (θ 3 1) θ x 3 ln (θ 6); - minimize the following least squares problem x max x=x min (ln(θ 5 ) + (θ 3 1)θ x 3 ln (θ 6 ) ln p x ) 2, - numerical techniques, like Newton-Raphson, are available to solve this optimization problem (see further).

8 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 8 / 50 Parametric models for mortality Example of fitting strategy: ML method of De Vylder (1975): - consider the d x as realizations from a random variable D x following a Binomial law D x Bin(λ x, q x ), - use Maximum Likelihood (ML) to estimate θ 1, θ 3 and θ 7 from x max λ x! (λ x=x x d x )!d x! min dx pλx x qx dx, where ln (p x ) = θ 1 + θ 7 θ3 x. This problem reduces to maximizing x max x=x min ((λ x d x ) ln (p x ) + d x ln (q x )).

9 IA BE Summer School 2016, K. Antonio, UvA Parametric models for mortality 9 / 50 Parametric models for mortality Heligman-Pollard mortality law calibrated on Belgian male q x (from 2012): Belgian 2012 male log death probabilities log q x Observed Infant Accident hump Adult Full Age(x)

10 IA BE Summer School 2016, K. Antonio, UvA Mortality at old ages 10 / 50 Mortality at old ages Very often mortality data at old ages are of bad quality. Many techniques have been developed to extrapolate mortality at old ages i.e. the closing of mortality tables. We cover the approach proposed by Kannistö (1992).

11 IA BE Summer School 2016, K. Antonio, UvA Mortality at old ages 11 / 50 Mortality at old ages Use Kannistö to close the mortality table for old ages, say x {91, 92,..., 120}: - estimate (φ 1, φ 2 ) using the relation (see Doray, 2008) logit(µ x ) = log (φ 1 ) + φ 2 x, with OLS on the ages - say - x {80, 81,..., 90}; - use these ( ˆφ 1, ˆφ 2 ) in the parametric law for µ x as proposed by Kannistö: µ x = φ 1 exp (φ 2 x) 1 + φ 1 exp (φ 2 x), for ages x > 90; - p x follows from the connection: p x = exp with piecewise constant µ x (see further). ( ) 1 0 µ x+r dr = exp ( µ x )

12 IA BE Summer School 2016, K. Antonio, UvA Mortality at old ages 12 / 50 Mortality at old ages Mortality at old ages: Kannistö applied to Belgian male data (from 2013): Belgian 2013 male death rates Kannistö regression for old ages source: ADSEI Belgian males 2013 data, source: ADSEI m x 1e 04 1e 03 1e 02 1e 01 1e+00? logit m x Age(x) Age(x) Closed mortality rate Belgian males 2013 data, source: ADSEI m x 1e 04 1e 03 1e 02 1e 01 1e Age(x)

13 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: set-up 13 / 50 Forecasting mortality: set-up Allowing for future mortality trends (+uncertainty?) is required in a number of actuarial calculations: - pensions, life annuities, long term care ( LTC ) covers, whole life sickness products; - survival probabilities over a long time horizon are required!. To avoid underestimation of the relevant liabilities - use appropriate forecasts of future mortality!

14 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: set-up 14 / 50 Forecasting mortality: set-up Consider mortality as a function of both age (x) and calendar year (or: period) (t). A dynamic mortality model is a function, say Γ(x, t), of x and t. E.g. consider one year probabilities of death, mortality odds, force of mortality,... The projected mortality model is Γ(x, t) t > t, with t the year for which the most recent (reliable) period life table is available.

15 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: set-up 15 / 50 Forecasting mortality: set-up q x,t is the probability that a person who is alive at January 1st of year t and who was born on January 1st of year t x will be death on January 1st of year t + 1. Thus, q x,t is the probability that an individual aged exactly x at exact time t will die between t and t + 1. In scientific literature: - we model q x,t ( mortality rate ) OR µ x,t ( force of mortality ); - we assume (see further) µ x+s (t + s) = µ x (t) for all 0 s < 1, and then q x,t = 1 exp ( µ x,t ). This expression allows straightforward transition from q x,t to µ x,t and vice versa.

16 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: set-up 16 / 50 Forecasting mortality: set-up A mortality/life table can be read as follows: (1) vertical arrangement: q 0 (t), q 1 (t),..., q x (t),... period life tables... t t + 1 t q 0(t) q 0(t + 1) q 0(t + 2) q 1(t) q 1(t + 1) q 1(t + 2) x... q x(t) q x(t + 1) q x(t + 2)... x q x+1(t) q x+1(t + 1) q x+1(t + 2) ω 1... q ω 1(t) q ω 1(t + 1) q ω 1(t + 2)...

17 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: set-up 17 / 50 Forecasting mortality: set-up A mortality/life table can be read as follows: (2) diagonal arrangement: q 0 (t), q 1 (t + 1),..., q x (t + x),... cohort life tables.... t t + 1 t q 0(t) q 0(t + 1) q 0(t + 2) q 1(t) q 1(t + 1) q 1(t + 2) q 2(t) q 2(t + 1) q 2(t + 2) x... q x(t) q x(t + 1) q x(t + 2)... x q x+1(t) q x+1(t + 1) q x+1(t + 2) ω 1... q ω 1(t) q ω 1(t + 1) q ω 1(t + 2)......

18 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: cohort and period life expectancy 18 / 50 Forecasting mortality: period and cohort life expectancy The period life expectancy for an x year old in year t is ex per (t) = 1 exp ( µ x,t) µ x,t + k 1 exp ( µ x+j,t ) 1 exp ( µ x+k,t). µ x+k,t k 1 j=0 The cohort life expectancy for an x year old in year t is ex coh (t) = 1 exp ( µ x,t) µ x,t + k 1 exp ( µ x+j,t+j ) 1 exp ( µ x+k,t+k). µ x+k,t+k k 1 j=0

19 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: cohort and period life expectancy 19 / 50 Forecasting mortality: period and cohort life expectancy Assume piecewise constant µ x (t) then ξ p x (t) = (p x (t)) ξ and: e per x (t) = ξ 0 = k 0 ξp x (t)dξ kp x (t) 1 0 ξp x+k (t)dξ = 1 exp ( µ x(t)) µ x (t) + k 1 exp ( µ x+j (t)) 1 exp ( µ x+k(t)), µ x+k (t) k 1 j=0 where we use (e.g.) 1 0 ξ p x (t)dξ = 1 0 (p x(t)) ξ dξ = px (t) 1 ln p x (t) = 1 exp (µx (t)) µ x (t).

20 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: deterministic techniques 20 / 50 Forecasting mortality: deterministic techniques Using (graphical) extrapolation procedures: - assume trend observed in past years can be graduated (or smoothed); - suppose the trend will continue in future years; - future mortality is estimated by extrapolating the trend itself; - an example: AG projection used in The Netherlands.

21 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: deterministic techniques 21 / 50 Forecasting mortality: deterministic techniques Projection by graphical extrapolation of annual probabilities of death:

22 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: deterministic techniques 22 / 50 Forecasting mortality: deterministic techniques How to use mortality laws in a dynamic context? - The age-pattern of mortality is summarized by (small number of) parameters; - Apply projection procedure to parameters (instead of age-specific probabilities).

23 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: deterministic techniques 23 / 50 Forecasting mortality: deterministic techniques Projection in a parametric framework with mortality laws:

24 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: stochastic models 24 / 50 Forecasting mortality: stochastic models More rigorous approach requires stochastic assumptions about mortality. As a result of the projection procedures we obtain: - point estimates; - interval estimates.

25 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: stochastic models 25 / 50 Forecasting mortality: impact on valuation of life annuities Scenario A: deterministic scenario.

26 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: stochastic models 26 / 50 Forecasting mortality: impact on valuation of life annuities Scenario B: discrete setting.

27 IA BE Summer School 2016, K. Antonio, UvA Forecasting mortality: stochastic models 27 / 50 Forecasting mortality: impact on valuation of life annuities Scenario C: continuous setting.

28 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 28 / 50 Lee Carter model Lee & Carter (1992, JASA) specify for m x,t (the central death rate): ln m x,t = β (1) x + β (2) x κ t + ɛ x,t. The dependence on age is governed by the sequences of β x (1) s and β x (2) s, where x {1,..., X }. The dependence on time by the κ t s where t {1,..., T }. The error terms ɛ x,t (with mean 0 and variance σ 2 ɛ ) reflect influences not captured by the model. Reading list: - Lee-Carter model; - Girosi & King, Understanding the Lee Carter mortality forecasting method, 2007.

29 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 29 / 50 Lee Carter model Identifiability issue? (See Nielsen & Nielsen, 2010.) Consider the set of parameters θ = (β (1) 1,..., β(1) X, β(2) 1,..., β(2) X, κ 1,..., κ T ). These are not identified without additional constraints as for any scalar c and d 0 ln m x,t = β x (1) + β x (2) κ t = (β x (1) β x (2) = (1) (2) β x + β x κ t. c) + β(2) x d {d(κ t + c)} Thus, parametrization θ and θ are equivalent.

30 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 30 / 50 Lee Carter model Solution (from literature on stochastic mortality)? Impose an identification scheme (or: constraints) which solve the estimation problem. For Lee-Carter: (for example) x β (2) x = 1 κ t = 0. t

31 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 31 / 50 Lee Carter model Interpretation of the parameters: (using the set of constraints on previous sheet) (1) β (1) x is β (1) x = 1 T ln m x,t. t (2) β (2) x : d dt ln m x,t = β x (2) d dt κ t, indicates the sensitivity of the logarithm of the death rate at age x to variations in the time index κ t. (3) κ t : evolution over time of mortality ( ) ln m x,t β x (1) x = κ t.

32 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 32 / 50 Lee Carter: ln m x,t 0 log m xt = 3 log m xt Age (x)

33 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 33 / 50 Lee Carter: ln m x,t = β (1) x 0 log m xt = β x (1) 3 log m xt Age (x)

34 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 34 / 50 Lee Carter: ln m x,t = β (1) x + κ t 0 log m xt = β x (1) + κ t 3 log m xt Age (x)

35 IA BE Summer School 2016, K. Antonio, UvA Lee-Carter model: specification 35 / 50 Lee Carter: ln m x,t = β x (1) + β x (2) κ t 0 log m xt = (1) (2) β x + β x κ t 3 log m xt 6 Range Age (x)

36 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 36 / 50 Lee Carter: calibration A least squares approach. Our objective function: ( β (1), β (2), ˆκ) = arg min β (1),β (2),κ ( ) ln m x,t β x (1) β x (2) 2 κ t = arg min β (1),β (2),κ O LS(β (1), β (2), κ). x t We find parameter estimates β (1), β (2), ˆκ by applying - a singular value decomposition; - or Newton Raphson; to the objective function.

37 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 37 / 50 Lee Carter: calibration If the estimates for β x (1), β x (2) and κ t do not satisfy the identifiability constraints: with: β (1) x β (1) x + (2) β x κ κ t ( κ t κ) β (2) β (2) x β (2) x / β (2), - β (2) the sum of the (2) β x s (from SVD or Newton Raphson); - κ the average of the ˆκ t s.

38 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 38 / 50 Lee Carter: SVD calibration Estimation of the parameters β (1) x : β (1) x since t κ t = 0 we get t O LS = 0 ln m x,t = T β x (1) + β x (2) β (1) x = 1 T ln m x,t. t κ t, t

39 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 39 / 50 Lee Carter: SVD calibration Estimation of the parameters β x (2) s and κ t s. Death rates can be combined into: m 1,1... m 1,T M =..... m X,1... m X,T, of dimension X T. Now create the matrix β (1) x Z = ln M ln m 1,1 =. ln m X,1 β (1) 1... ln m 1,T.... β (1) 1 (1) (1) β X... ln m X,T β X.

40 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 40 / 50 Lee Carter: SVD calibration The idea is to approximate Z with a product of a column and a row vector Z β (2) ˆκ, with β (2) = ( β (2) (2) 1,..., β X ) and ˆκ = (ˆκ 1,..., ˆκ T ). Therefore, we have to minimize (z x,t β x (2) κ t ) 2. Solution? Use SVD of Z. x t

41 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 41 / 50 Lee Carter: SVD calibration Find the (2) β x s and ˆκ t s that minimize: Õ LS (β (2), κ) = ( ) z x,t β (2) 2 x κ t. x t Define the square matrices Z Z and ZZ. - Let u1 be the eigenvector corresponding to the largest eigenvalue of Z Z. - Let v 1 be the eigenvector corresponding to the largest eigenvalue of Z Z. The best approximation of Z in the least squares sense is: Z Z = λ 1 v 1 u 1.

42 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 42 / 50 Lee Carter: SVD calibration The parameter estimates then become: β (2) = v 1 j v 1j ˆκ = λ 1 j v 1j u 1, satisfying the Lee Carter constraints, because u 1j = 0, since t z x,t = 0. j

43 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 43 / 50 Lee Carter: SVD calibration SVD only applicable to rectangular matrices. Missing values in the Z matrix will prevent the method from working. This is a rather old fashioned way to estimate parameters in a Lee Carter model (see further).

44 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 44 / 50 Lee Carter: OLS using Newton Raphson Update each parameter in turn using a univariate Newton Raphson recursive scheme. Take derivative of O LS (β (1), β (2), κ) wrt β x (1), β x (2) and κ t ( x, t). Solve a set of equations; each one is of the form f (ξ) = 0 (see next sheet). Starting from ξ (0) the (k + 1)th iteration gives ξ (k+1) from ξ (k) : ξ (k+1) = ξ (k) f (ξ(k) ) f (ξ (k) ). This alternative to SVD does not require a rectangular array of data.

45 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 45 / 50 Lee Carter: OLS using Newton Raphson Recall: O LS (β (1), β (2), κ) = x The system to solve is: t ( ln m x,t β (1) x β (2) x κ t ) 2. 0 = t (ln m x,t β (1) x β (2) x κ t ), for x X ; 0 = x β x (ln m x,t β (1) x β (2) x κ t ), for t T ; 0 = t κ t (ln m x,t β (1) x β (2) x κ t ), for x X.

46 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 46 / 50 Lee Carter: adjustment of ˆκ t Lee & Carter (1992) suggest an additional step: - adjust ˆκ t such that d xt x = x (1) (2) e xt exp ( ˆβ x + ˆβ x ˆκ t ), (or use another criterium, see R demo).

47 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 47 / 50 Lee Carter: calibration strategies Lee Carter model: calibration using SVD and Least Squares with Newton Raphson: Belgium Lee Carter fit Belgium Lee Carter fit β x (1) LS SVD β x (2) LS SVD Age(x) Age(x) Belgium Lee Carter fit κ t LS SVD SVD second stage Time(t)

48 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 48 / 50 Lee Carter: calibration strategies Lee Carter model: calibration using SVD and Least Squares with Newton Raphson: log q x observed q x LS fit Belgium 2012 male death probabilities Lee Carter fit SVD fit SVD second stage Age(x)

49 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 49 / 50 Demographics: computational illustration R scripts are available which: - reproduce the graphs and calculations in this module; - connect to HMD (through the demography package) and download data directly from this source. You will need an internet connection to run these scripts.

50 IA BE Summer School 2016, K. Antonio, UvA Lee Carter: calibration 50 / 50 Demographics: computational illustration R scripts are available which: - calibrate Makeham law (RdemoMakeham.R); - calibrate Heligman & Pollard law (RdemoHPlaw.R); - close for old age mortality using Kannistö (RdemoOldAgesKannisto); - calibrate Lee Carter using SVD and iterative least squares approach (RdemoLeeCarterSVDLS).