Coale & Kisker approach

Transcription

1 Coale & Kisker approach Often actuaries need to extrapolate mortality at old ages. Many authors impose q120 =1but the latter constraint is not compatible with forces of mortality; here, we impose µ110 = { 0.8 for females 1 for males following COALE &KISKER (1990). The time t is henceforth dropped (because the CK method works transversally).

2 Coale & Kisker approach (Ctd) Assumption: steady decrease in the rate of increase in mortality with ages 80 years and above. Formally, for age x 66 let us introduce the annual increasing rates k66,k67,... as ln µx =lnµ65 + x y=66 ky µx = µ65 exp x y=66 ky. This is in contrast with the commonly used Gompertz model µx = δ exp(kx) leading to a constant rate of increase of µx with age x.

3 Coale & Kisker approach (Ctd) For most Western European countries, the curve x kx is unimodal with a mode around age 80; after that age, the kx s decrease approximately linearly. Based on the latter empirical evidence, we specify kx = k80 + s(x 80) for x 80 yielding s = ( ln µ 79 µ110 ) k80.

4 Implementation of the Coale & Kisker method In practice, the appropriateness of the CK assumption is investigated by fitting the curves ln µx = a + bx + cx 2 + error or ln µx+1 ln µx = A + Bx + error in which a significantly negative ĉ or B together with an inspection of the LS residuals support the underlying CK assumption; see Renshaw & Haberman (2003b). It is then possible to monitor the suitability of the constraint imposed on µ110.

5 Rough estimates µx(2000) and µ x(2000) obtained from the Coale & Kisker approach

6 Rough estimation of the µx(t) sfor Belgian women on

7 Application to Belgian mortality statistics

8 Application to Belgian mortality statistics (Ctd)

9 Application to Belgian mortality statistics (Ctd) Following the Box-Jenkins methodology yields 1κt = ϑt 1 + ϑt where ϑt iid N(0, σ =3.2675).

10 Application to Belgian mortality statistics (Ctd) Age x ex(2000) ax(2000) with i =4% INS Note that < 90.23!!! (this comes from the fact that the inequality e0(t) <x+ ex(t + x) always holds true whereas e0(t) <x+ ex(t) is not necessarily valid)

11 Mortality reduction factors The Lee-Carter model can be cast into µx(t) =µx(tmin) RF (x, t), t tmin, where RF (x, t) =exp ( βx(κt κtmin) ). RF (x, t) is called the mortality reduction factor (for age x and year t): it describes the reduction in mortality at age x over the period (tmin,t). Renshaw & Haberman (2001) suggested the alternative modelling µx(t) =µ (0) x RF (x, t) with the x =exp(α x (0) ) corresponding to some reference lifetable and RF (x, t) =exp(βxκt) with the modified constraint κ0 =0. µ (0)

12 Advantages of the Lee-Carter approach Easy interpretation of the parameters αx, βx and κt. The influence of the calendar time is summarized in a single index and it suffices to extrapolate the κt series in the future to get the projected lifetables.

13 Criticisms The errors ɛx(t) have been assumed to be homoskedastic unrealistic because the logarithm of the observed force of mortality is much more variable at older ages than at younger ages because of the much smaller absolute number of deaths at older ages. Because of SVD, a matrix of observed death rates is actually needed the mortality statistics may have to be completed before fitting the LC model.

14 Inspection of residuals SVD residuals are given by ɛx(t) =ln µx(t) ( αx + βx κt). They are standardized by dividing by 1 ν xmax x=xmin tmax t=tmin { ɛx(t)} 2 where ν = (xmax xmin)(tmax tmin 1) is the number of df s.

15 Standardized SVD residuals

16 Renshaw & Haberman (2003a) extension (LC2 model) Instead of including only the first order approximation from SVD, Renshaw & Haberman (2003a) enrich the LC model as follows: ln µx(t) =αx + β x (1) κ (1) t + β x (2) κ (2) t with the constraints t κ (1) t = t κ (2) t =0and x β x (1) = x β x (2) =1. to ensure identifiability.

17 Estimating the parameters of the LC2 model The αx s are still estimated by averaging the ln µx(t) over time. The κ (j) t s and β x (j) s, j =1, 2, areobtained from the first and second SVD components. Only the κ (1) t s are adjusted in order to reproduce the observed yearly total deaths. The bivariate time series {(κ (1) t,κ (2) t )} has to be considered and extrapolated to the future to generate mortality forecasts.

18 From Lee-Carter to Poisson log-bilinear LC method does not model the observed number of deaths but the logarithms of the force of mortality. For actuarial applications, the law of the number of deaths is very useful (e.g. for simulating the cash flows of a life insurance portfolio). Poisson model centres on the Dxt s and remedy to some of the drawbacks of the LC approach.

19 Poisson modelling According to BRILLINGER (1986), the Poisson assumption appears to be plausible for the number of deaths Dxt. The assumption that the mortality rates are constant within squares in the Lexis diagram, is compatible with Poisson modelling for death numbers. Indeed, let us focus on a particular couple age x - year t: we observe Dxt deaths among Nxt individuals aged x on January 1. We assume that the remaining lifetimes of these individuals are iid.

20 Poisson modelling (Ctd) To each of the Nxt individuals, we associate a variable δi indicating whether the person dies or not, i.e. δi = { 1 if person i dies at age x 0 otherwise, i =1, 2,...,Nxt. Wealsodefine τi as the time lived by individual i. We assume that we have at our disposal iid observations (δi,τi) for each of the Nxt individuals.

21 Poisson modelling (Ctd) The contribution of individual i to the likelihood writes 1. if he survives (δi =0, τi =1) px(t) =exp( µx(t)); 2. if he dies (δi =1, τi < 1) τi px(t)µx+τi(t + τi) = exp( τiµx(t))µx(t). Therefore, the contribution of individual i can be cast into exp( τiµx(t)){µx(t)} δ i.

22 Poisson modelling (Ctd) Invoking the mutual independence of the remaining lifetimes yields the likelihood L ( µx(t) ) = Nxt i=1 exp( τiµx(t)){µx(t)} δ i = exp( µx(t)τ ){µx(t)} δ where τ = Nxt i=1 τi = Lxt and δ = Nxt i=1 δi = Dxt.

23 Poisson modelling (Ctd) Clearly, L ( µx(t) ) ( L Poisson µ x(t) ) where the ( Poisson likelihood L Poisson µ x(t) ) is given by t,x exp ( Lxtµx(t) ) ( Lxtµx(t) Dxt! ) D xt based on the distributional assumption Dxt Poisson ( ) Lxtµx(t). Therefore, it is equivalent to work on the basis of the true" likelihood or on the Poisson likelihood.

24 Poisson log-bilinear model We now consider that Dxt Poisson ( ) Lxtµx(t) with t µx(t) = exp(αx + βxκt) where the parameters fulfill κt =0and x βx =1. The meaning of the αx, βx, and κt parameters is essentially the same as in the classical LC model.

25 Estimation of the parameters The parameters αx, βx and κt are estimated by maximizing the Poisson log-likelihood L(α, β, κ) = x,t { Dxt(αx + βxκt) Lxt exp(αx + βxκt) } +constant. Commercial software with Poisson regression cannot fit the model because of the bilinear term βxκt.

26 Goodman s algorithm GOODMAN (1979) suggested uni-dimensional or elementary Newton method to solve the likelihood equations. In iteration step ν +1,asingle set of parameters is updated fixing the other parameters at their current estimates using the following updating scheme θ(ν+1) = θ(ν) L (ν) / θ 2 L (ν) / θ 2 where L (ν) = L (ν) ( θ (ν) ).

27 Estimation of the parameters (Ctd) Specifically, the algorithm is α (ν+1) x = α (ν) x κ (ν+1) t = κ (ν) t β(ν+1) x = (ν) β x t ( t ( x x ( t t Dxt Lxt exp( α (ν) x + ( Lxt exp( α (ν) x + Dxt Lxt exp( α (ν+1) x + ( Lxt exp( α (ν+1) x + Dxt Lxt exp( α (ν+1) x + ( Lxt exp( α (ν+1) x + (ν) β x κ (ν) t ) (ν) β x κ (ν) t ) ) ) (ν) β x κ (ν) (ν) β x κ (ν) t ) t ) ) β(ν) x )( β(ν) x (ν) β x κ (ν+1) t ) (ν) β x κ (ν+1) t ) )( ) ) 2 κ (ν+1) t κ (ν+1) t ) 2. starting from the Lee-Carter ( α (0), β (0), κ (0) ).

28 Estimation of the parameters (Ctd) In the Lee-Carter approach, the κt s were reestimated in order to reconstitute the total numbers of deaths observed each year. In the Poisson-LB model, the likelihood equations ensure that the model reconstitutes the total observed number of deaths at each age across the observation period, i.e. L(α, β, κ) =0 Dxt = Lxt exp(αx + βxκt) αx t t there is thus no need for a readjustment of the κt s.

29 The LC-constraints The ML estimations of the parameters have to be adapted in order to fulfill the Lee-Carter constraints: specifically, κ t =( κt κ) x βx, βx = βx βx x and α x = αx + βxκ α x,β x and κ t fulfill the constraints and are such that αx + βx κt = α x + β xκ t. ARIMA models are used to project the κt s.

30 Advantages of the Poisson model on its LC counterpart The Poisson model recognizes the integer nature of Dx,t, contrarily to the Lee-Carter model. The Poisson model does not impose homoskedasticity on the random structure and recognizes the higher variability of the µx(t) s at older ages The Poisson model does not rely on SVD so that the mortality statistics do not have to be previously completed. In the Poisson approach, the κt s do not need to be reestimated.

31 Application to Belgian mortality statistics

32 Projecting time trend Again, an MA(1) process is selected: 1κt = ϑt 1 + ϑt where ϑt iid N(0, σ =2.7326).

33 Forces of mortality by generations

34 Comparisons of ex(2000) We get for Belgian women ex(2000) Lee-Carter Poisson INS x

35 Comparisons of ax(2000) We get for Belgian women ax(2000) Lee-Carter Poisson INS x

36 Confidence intervals It is impossible to derive confidence intervals for µx(t), ex(t), ax(t), etc. analytically because 1. two very different sources of uncertainty have to be combined: sampling errors in the parameters of the Poisson model and forecast errors in the projected ARIMA parameters 2. the measures of interest are complicated non-linear functions of the Poisson parameters αx, βx, and κt and the ARIMA parameters.

37 Confidence intervals: three approaches Parametric bootstrap assumes that the Poisson log-bilinear specification is correct. Semi-parametric bootstrap assumes that the Poisson assumption is correct. Non-parametric bootstrap only needs the multinomial assumption.

38 Parametric bootstrap CI The mth sample in the Monte Carlo simulation is obtained by the following 4 steps: 1. Generate α x m, β x m, and κ m t from the appropriate MVN distribution. 2. Estimate the ARIMA model using the κ m t as data points. 3. Generate a projection of κ m t s using the ARIMA parameters. 4. Compute the quantities of interest on the basis of the α x m, β x m, and κ m t.

39 Semiparametric bootstrap CI Starting from the observations (Lxt,dxt), we create bootstrap samples (Lxt,d b xt ), b =1,...,B,where the d b xt s are realizations from the Poisson distribution with mean Lxt µx(t) =dxt. For each bootstrap sample, the αx s, βx s and κt s are estimated and the κt s are then projected on the basis of the reestimated ARIMA model. This yields B realizations α x, b β x, b κ b t and projected κ b t on the basis of which we compute the quantity of interest.

40 Nonparametric bootstrap CI We follow here a generation" (Lxt,dxt), (Lx+1 t+1,dx+1 t+1), (Lx+2 t+2,dx+2 t+2)... Death counts d b xt,db x+1 t+1,... are generated from a multinomial distribution with exponent d = k 0 dx+k t+k and parameters d xt d, d x+1 t+1 d,... We then proceed as for the semiparametric approach.

41 Confidence intervals: results for Belgian men 90% Confidence Intervals and Error Margins on a65(2000): [9.83, 11.51] [9.85, 11.60] [9.88, 11.53] 7.9% 8.3% 7.8%

42 Deviance residuals Since we work in a regression model, it is important to inspect residuals to detect a possible structure. If the residuals exhibit some regular pattern, this means that the model is not able to describe all the phenomena appropriately. Here, we use the deviance residuals, i.e. the signed squared root contribution to each observation (Lxt,Dxt) to the deviance D =2 tmax t=tmin xmax x=xmin { dxt ln d xt dxt (dxt dxt) }

43 Deviance residuals Those residuals are defined as rxt = 2sign(dxt dxt) dxt ln d xt dxt (dxt dxt). They monitor the quality of the fit. In practice, plotting t rxt at different ages x and discovering no structure in those graphs ensures that the time trends have been correctly captured by the model.

44 Standardized deviance residuals

45 Renshaw & Haberman (2003a) extension (PB2 model) In case residuals display some regular pattern, those authors proposed to extend the Poisson log-bilinear model as follows: ln µx(t) =αx + β x (1) κ (1) t + β x (2) κ (2) t with the constraints t κ (1) t = t κ (2) t =0and x β x (1) = x β x (2) =1. to ensure identifiability.

46 SOME OTHER APPROACHES

47 Jaumain model JAUMAIN (2001) suggested to model the death probability at age x during calendar year t as qx(t) = φ x +exp(αxt + βx) φx +exp(αxt + βx)+1 where φx controls the asymptotic value of qx(t). The model is then fitted to Belgian periodic lifetables to generate mortality forecasts. The restriction κt = t may be regarded as rather strong.

48 British CMI model The COMMITTEE FOR MORTALITY INVESTIGATION (CMI) proposed to introduce mortality reduction factors of the form RF (x, t) = q x(t) qx(0) q x(t) =RF (x, t)q0(t). According to CMIR (1990), RF (x, t) =αx +(1 αx)(0.4) t/20 where αx = 1 2 if x<60 x if x>110. if 60 x 110

49 British CMI model (Ctd) According to CMIR (1999), RF (x, t) =αx +(1 αx)(1 fx) t/20 where αx = c if x<60 1+(1 c) x if 60 x if x>110 and fx = h if x<60 (110 x)h+(x 60)k 50 if 60 x 110 k if x>110 with c =0.13, h =0.55 and k =0.29.

50 MANAGEMENT OF A PORTFOLIO OF LIFE ANNUITIES