Application of the Poisson log-bilinear models with multiple populations to the Dutch mortality

Application of the Poisson log-bilinear

models with multiple populations to the

Dutch mortality

Master’s Thesis Econometrics, Operations Research and Actuarial Studies

Rijksuniversiteit Groningen

Author:

Joost Wilbers

s1780891

Supervisor:

prof. dr. L. Spierdijk

Co-assessor:

prof. dr. R.H. Koning

July 16, 2015

Contents

1 Introduction 1

1.1 Objective and Outline . . . 3

2 Methodology 4 2.1 Notation . . . 4

2.2 Lee-Carter model . . . 4

2.3 Poisson log-bilinear model . . . 5

2.4 Age-Period-Cohort model . . . 5

2.5 Model estimation . . . 5

2.6 Wilmoth-Valkonen model . . . 6

2.7 (Augmented) Common Factor model . . . 7

2.8 Common Factor Cohort model . . . 8

2.9 Projecting mortality . . . 8 2.10 Comparison . . . 8 2.11 Data . . . 9 3 Results 10 3.1 Benchmark Groups . . . 10 3.2 Benchmark Statistics . . . 10 3.3 Test-Groups . . . 14 3.4 Test-Group estimation . . . 18 3.5 Final Groups . . . 19 4 Analysis 23 5 Conclusion 26 Appendices 28 A Fitting Methodology 28 A.1 Lee-Carter Fitting Algorithm . . . 28

A.2 Age-Period-Cohort Fitting Algorithm . . . 30

1

Introduction and Motivation

The persisting increase in life expectancy beyond previously expected limits emphasizes the im-portance of proper mortality forecasts. Increased living standards, better health care and better medication are some of the key factors driving the decrease of death rates at all ages. Although this is mainly a positive development, it also creates serious difficulties. Public retirement sys-tems, health care systems and the annuity business need to be adjusted to account for these changes, producing longevity risk (Brouhns, Denuit, and Vermunt, 2002b; Denuit and Goderni-aux, 2005). Reliable mortality forecast are thus crucial in these areas.

The problems caused by the increasing number of deaths around the average age at moment of death (rectangularization) and increasing life expectancy have received a lot of attention in recent years. An example of these problems are the pension schemes. These schemes are under a lot of pressure in several countries due to the combination of increasing life expectancy and decreasing interest rates. Pension eligibility ages are rising, but the uncertainty about future pensions still remains.

Stochastic mortality models have gained popularity over the years, giving rise to numerous new methods. Methods within the field of mortality forecasting are typically part of one of the three main families of approaches: expectation, extrapolation and explanation (Booth and Tickle, 2008). Expectation is mortality forecasting based on (subjective) expert opinions. Extrapola-tion is the most commonly used approach. It looks for regularities in age patterns and time trends and projects these past trends in mortality into the future. And finally, the explanatory methods predict future mortality based on known and measurable causes of death.

A fundamental model in the development of mortality modelling is the Lee-Carter method (Lee and Carter, 1992). The Lee-Carter method involves a bilinear two-factor model, which only depends on factors related to age and year and is fitted to historical data and is thus part of the family of extrapolation methods. It combines a parsimonious demographic model with statistical time analysis The method has been praised for its simplicity, robustness and elegance. Subse-quently several modifications and extensions of the Lee-Carter method have been developed. Lee (2000) and later on Booth, Hyndman, Tickle, and de Jong (2006) summarized the current extensions and applications.

the improvement of the fit outweighs the downside of complicating the model. The age-period-cohort models can be of help when modelling populations that have had a period in their history that significantly influenced specific cohorts.

To be able to compare (sub)populations, Wilmoth and Valkonen (2002) proposed a generali-sation of the Poisson variation of the LC model, adding covariates. Using the Wilmoth-Valkonen model to compare mortality of the G5 countries, Delwarde, Denuit, Guill´en, and Vidiella-I-Anguera (2006) demonstrated that this model allows for comparison between countries under the assumption that time trends of the mortality patterns of these counties are similar. Tul-japurkar, Li, and Boe (2000) had previously tried to compare the G7 countries with the use of the classic LC method, but Delwarde et al. (2006) showed that using the Wilmoth-Valkonen model provided better comparison opportunities and the possibility to identify shared charac-teristics.

Multi-population mortality models such as the Wilmoth-Valkonen model have recently gained more attention within the area of mortality forecasting. Since these models use bigger mortality datasets from different sources, this allows for more robust mortality modelling. Under the as-sumption that mortality patterns of closely related populations are converging, Lee and Li (2005) proposed an (augmented) common-factor model to analyse mortality patterns of a collection of populations with similar socio-economic conditions and close connections and use these similar-ities to improve the quality of the forecasts.

The models mentioned above are all extrapolative stochastic methods, assuming that observed recent to mid-term trends will extend into the future. Several authors criticized or questioned this approach (e.g. Gutterman and Vanderhoof (1998)), as it uses no information other than previous history. As the underlying mechanisms of mortality patterns are still to complex to be sufficiently understood, Wilmoth (2000) points out that the extrapolation of past trends is still the most reliable option for predicting the future development of mortality patterns.

Using similarities in mortality patterns of countries that have resembling conditions might en-hance the quality of the model, since bigger population group can be used, hence might also help to predict mortality patterns better. A more stable and reliable model can be obtained, eliminating outliers and shocks caused by country-specific one-time events and identifying shared characteristics. Especially relatively smaller countries can benefit from these types of models, as it enlarges the dataset.

1.1

Objective and Outline

The aim of this paper is to model mortality patterns in the Netherlands with the use of a multi-population Lee-Carter type model. Trying several options, we want to check which type of multi-population model achieves the best fit for the Dutch mortality data by comparing the likelihood-ratio statistics as in Brouhns et al. (2002a) and Booth, Maindonald, and Smith (2002) and explanation ratios proposed by Lee and Li (2005) (see section 2.10). The likelihood-ratio statistic will be used to test overall fit of a model to the data as it compares the model to the saturated model, whereas the explanation ratio illustrates the improvement of the fit of the model compared to the very simple model that only uses an age factor. The explanation ratio thus takes the complexity of the data into account, i.e. if the likelihood-ratio statistic indicates a good fit to the data, we can check using the explanation ratio if this fit is good because of the characteristics of that particular model or if the simple fit is already good and the extra variables added by the model do not improve the fit by much and vice versa. Whether a fit is good or bad is still up to subjective judgement, as Lee and Li (2005) note, but these statistics are very effective when comparing multiple models as they do indicate a better or worse fit, which is what we want to test. Furthermore, we want to determine which countries should be used to achieve this fit. To illustrate the economic impact of the differences between the models, we will also calculate the life expectancies and life annuities (as in Brouhns et al. (2002a) and Brouhns et al. (2002b), see Section 2.10) resulting from the estimates of the models. Hence these two economic indicators will be used to check if the differences in statistical indicators also translate to noticeable differences in life expectancies and life annuity premiums. The main research question is thus as follows:

• What is the optimal multi-population Lee-Carter type mortality model for the Netherlands? The following hypotheses and questions are formulated in order to answer the research question: • Which group of countries should be chosen to optimize the likelihood-ratio of the Dutch

mortality model?

• Is it possible to include a cohort term in the multi-population model and does this inclusion improve the fit of the model?

• Are global mortality patterns converging?

We expect that minor improvements can be achieved compared to the multi-population models used by Antonio et al. (2015) and Stoeldraijer et al. (2013) by investigating which group of coun-tries improves the fit of the model the most. Scaling the councoun-tries, such that they are of equal size in the group, could have a greater effect, especially since the Netherlands has a relatively small population compared to some other European countries. Less of the Dutch characteristics will be included in the group when the countries are not scaled, as with Antonio et al. (2015). Extending the model with cohort effects is expected to improve the fit of the model, but it remains to be seen if this improvement is significant enough to compensate for the increased complexity it generates. The economic indicators in combination with our statistical ratios can help with this decision as they not only show the contribution to the fit of the model, but also the size of the impact of this fit.

2

Methodology

2.1

Notation

We study the force of mortality as a function of age and time. We have data per gender on population sizes and death rates per integer age x and calendar year t. Key parameters in this setting are:

µx,t : the force of mortality for age x in year t

ex,t : the expected remaining lifetime of a person reaching age x in year t

px,t : the survival probability of an individual of age x in year t (px= exp(−µx,t))

Dx,t : for the recorded number of deaths at age x during year t

Ex,t : the exposure-to-risk of the population at age x in year t (i.e. the total amount people of

age x alive at the start of year t) ˆ

mx,t : the empirical force of mortality ( ˆmx,t= Dx,t Ex,t)

2.2

Lee-Carter model

The base of all models used in this paper is the Lee-Carter model structure, proposed by Lee and Carter (1992). Their bilinear model is given by:

ln µx,t= αx+ βxκt+ x,t (1)

where µx,t is the the force of mortality for age x in year t. The error term (x,t) represents

age-specific historical effects not captured by the model. In this model the force of mortality µx,t hence only depends on age- and year related factors. The meaning of the parameters is as

follows:

αx represents the main age effects, averaged over time, x = x1, x2, . . . , xk

κt is the time trend of the force of mortality, t = t1, t2, . . . , tn

βx represents the age-specific response to variations in the time trend (κt)

x,t is the error term, x,t∼ N (0, σ2)

We thus model k different ages in n different years. Note that the structure of model (1) is invariant to the following transformations:

{αx, βx, κt} 7→ {αx,

βx

c , cκt} {αx, βx, κt} 7→ {αx− cβx, βx, κt+ c}

Hence to ensure model identification, these parameters have to satisfy the constraints X x βx= 1 and tn X t=t1 κt= 0 (2)

which imply the least squares estimator:

2.3

Poisson log-bilinear model

The OLS estimation of standard Lee-Carter model uses singular value decomposition (SVD). SVD requires the assumption that the errors are homoskedastic. This does not seem to be a realistic assumption (Alho, 2000), since the force of mortality is a lot more variable at older ages than it is at young ages. As an alternative, Brouhns et al. (2002a) proposed Poisson modelling for the number of deaths, hence assuming heteroskedastic errors. So, instead of

X x Dx,t= X x Ex,texp(αx+ βxκt)

as is the case for the original Lee-Carter model (1), we now have for the number of deaths Dx,t∼ Poisson(Ex,tµx,t) with µx,t= exp(αx+ βxκt) (4)

Instead of by SVD, these parameters are determined using maximum likelihood estimation.

2.4

Age-Period-Cohort model

For several countries, the residuals of the Age-Period Lee-Carter models exhibited clear patterns that indicated age-cohort effects. Renshaw and Haberman (2006) opted for an extension of the Lee-Carter model, including a bilinear term containing the cohort effect into the Lee-Carter methodology. The generalized version of the model is given by:

ln µx,t= αx+ βx(0)ιt−x+ βx(1)κt+ x,t (5)

where βx(0)ιt−x represents the cohort effects. Cohorts are grouped per year-of-birth z = t − x.

The identification constraints of the Age-Period Lee-Carter model (2) change to: X x βx(0)= X x βx(1)= 1 and κ1= 0 (6)

We will try to combine this extension with the multi-population in section 2.7 to see if the inclusion of the cohort term significantly improves the fit and the forecasts of the model.

2.5

Model estimation

Let us first formulate the model in a Generalized Linear Model (GLM) form, to be able to estimate the model using maximum likelihood methods. We write the number of deaths at age x in year t as Yx,t. The numbers of deaths (age-, sex- and period-specific) are assumed to be

independent response variables of a quasi-Poisson GLM with parameters:

Yx,t= Dx,t, E[Yx,t] = Ex,tµx,t= Ex,texp(αx+ βxκt), V ar[Yx,t] = φE[Yx,t] (7)

where φ > 0 is a dispersion parameter to allow for heterogeneity. The response variable Yx,tnow

corresponds to a GLM model with log-link and non-linear predictor:

ηx,t= log(ˆyx,t) = log Ex,t+ ˆαx+ ˆβxˆκt (8)

where ˆyx,t is the predicted number of deaths at age x in year t. The above model is still subject

to the constraints (2), to ensure model identification. During fitting log Ex,t is treated as an

Since some observations are missing or population is zero for certain age-year combinations, weights (ωx,t) are introduced. Non-empty data cells have ωx,t = 1 and empty data cells have

ωx,t= 0 assigned to them. The total deviance is now given by:

D(yx,t, ˆyx,t) = X x,t dev(x, t) =X x,t 2ωx,t  yx,tlog yx,t ˆ yx,t − (yx,t− ˆyx,t)  (9) where dev(x, t) are the deviance residuals, which are dependent on the weights.

However, the bilinear term (βxκt) in the model (4) prevents the model from being implemented

as a GLM. Because of this, Brouhns et al. (2002a) applied the fitting methodology proposed by Goodman (1979) to estimate then parameters of the Poisson log-bilinear model. In this paper will use use an extended version of the method of Brouhns et al. (2002a) proposed by Renshaw and Haberman (2006). This iterative method is based on the Newton Raphson minimisation method which is given by:

u(ˆθ) = ˆθ − ∂

2_D(θ)

∂θ∂θ0

−1_∂D(θ)

∂θ (10)

for a given parameter θ, in our case α, β an γ. To maximize the quasi-Poisson likelihood, we need to minimize the total deviance. The deviance funtions and updating algorithm that we will use to estimate our models is given in Appendix A. To ensure model identification for the Age-Period-Cohort model, Renshaw and Haberman (2006) proposed a two-stage fitting algorithm, different from the algorithm used by Brouhns et al. (2002a). Different from the basic Lee-Carter fitting algorithm, first the αxis estimated, using the SVD as in the original Lee-Carter approach

and in the second stage, these αx are treated as an offset and held constant, while the other

parameters are estimated using the iterative process described in Appendix A.2

2.6

Wilmoth-Valkonen model

In an attempt to extend the Lee-Carter model to a model that distinguishes between different population groups, Wilmoth and Valkonen (2002) proposed a generalized version of the Poisson log-bilinear model (4) using covariates. The model Wilmoth and Valkonen (2002) proposed is given by: mijk1...kp = exp{β + β (1) k1 + . . . + β (p) kp + αi+ α (1) ik1+ . . . + α (p) ikp+  γi+ γ (1) ik1+ . . . + γ (p) ikp  δj} (11)

This model uses p groups of covariates, with group i consisting of ki covariates. The β’s are the

group mortality averages (not included in the normal Poisson log-bilinear model (4)). The α’s are represent the main age deviations of the group averages, hence together with the β’s have the function of the α’s in the Poisson log-bilinear model, the γ’s represent the age-specific sensitivity to variations in the time index (similar to the β’s in the original model) and δj denotes the

overall time trend, hence the κtin 4. As δj represents the overall time trend of the total group,

Delwarde et al. (2006) showed that this model can be used to compare mortality in different countries by setting each country as a covariate. A condition for the chosen countries is that the κt’s of the countries follow a similar pattern, just like in the Wilmoth-Valkonen model (11).

Males and females are modelled separately, due to the significant differences between the two genders as shown by Lee and Carter (1992). The model (11) now simplifies to:

µk_x,t= exp α + αk+ αx+ αkx+ (βx+ βkx)κt
(12) Dk_x,t∼ Poisson Ek x,tµ k x,t

where k indicates the specific country.

The constraints to ensure model identification in this setting are X x αx= X x αk_x=X x βx= 0 for all k (13) X k αk=X k αk_x=X k βk_x= 0 for all x (14) X t κt= 0 (15)

In this setting this model can then be used to compare the main age patterns of mortality per country and age-specific sensitivity to changes in the time index. Since the time pattern is held constant across the different countries, the model isolates the country-specific differences in main age patterns and the age-specific responses to variations in the time trend.

2.7

(Augmented) Common Factor model

Lee and Li (2005) introduce a common factor model and augmented common factor model to take advantage of similarities in historical experience and age patterns of populations. This is the main model for our tests to try and find the best group of countries in combination with the Netherlands. The Common Factor model they proposed is the following:

ln µk_x,t= αk_x+ BxKt (16)

Here the Bx and Kt represent age-specific sensitivity to the time trend and the time trend

respectively and is thus not taken country-specific in this model, hence BxKt is the common

factor. In this model the mortality group of countries is expected to evolve in the exact same way and the age-specific differences (the α’s) between countries do not change over time. Since these are quite strong assumptions to make, Lee and Li (2005) also proposed an augmented common factor model, which does take into account the differences in levels, age patterns and trends between countries. This Augmented Common Factor model is given by:

ln µk_x,t= αk_x+ BxKt+ βxkκ k

t+ 

k

x,t (17)

where last term denotes the modelling error. κk

t has to tend to some constant level over time,

such that βk

xκkt describes the short- or medium-term deviations of population k from the common

trend. For the Common Factor model (16) Bx and Kthave to be estimated in first, using the

group sample and αk

x is estimated as in (3) using the country-specific data. Using the country

specific residuals of the common factor model, the Augmented Common Factor model estimates βk

2.8

Common Factor Cohort model

We propose an extension to the Common Factor of Lee and Li (2005) that we want to test in this paper. Combining the Age-Period-Cohort model of Renshaw and Haberman (2006) with the Common Factor model, we propose the following model:

ln µk_x,t= αk_x+ B_x(0)It−x+ B(0)x Kt (18)

where Bx(0)It−xrepresents the cohort effects of the group. As the countries used in the Common

Factor model is based on similarities between populations, the countries in the group are also likely to exhibit similar cohort effects if these are present. The model we propose takes this into account with the use of the overall cohort term. Constraints 6 of the APC model still apply.

2.9

Projecting mortality

As the time factor κ is viewed as a stochastic process, the forecasting of the Lee-Carter model is done by extrapolation future trends in κtusing a ARIMA time series model, which is

character-ized by three parameters (p, d, p). We will use a ARIMA(0,1,0), a random walk with drift, also used by e.g. Booth et al. (2006) and Butt and Haberman (2010), which Lee and Miller (2001) showed to be a good preforming model when forecasting mortality. The ARIMA(0,1,0) model is given by

κt= δ + κt−1+ t (19)

where δ is the drift or time trend and t denotes a white noise process. The ARIMA(0,1,0) is

also used for the projection of the cohort factor ιt−xof the APC model.

2.10

Comparison

To compare the different variations of the Lee-Carter model we will use both statistical compar-ison and actuarial comparcompar-ison. For the statistical comparcompar-ison we will use both the explanation ratio (R) used by Lee and Li (2005) and the likelihood-ratio statistic (L2_{) that is used by Brouhns}

et al. (2002a) and Booth et al. (2002).

The explanation ratios used by Lee and Li (2005) measure how well certain models work by comparing its performance to the most simplified model (ln µx,t = αx). The explanation ratio

for the standard Lee-Carter (RS) is given by:

RS = 1 − P t P x(ln( ˆmx,t) − αx− βxκx,t) 2 P t P x(ln( ˆmx,t) − αx) 2 (20)

Hence the model being tested is in the numerator and the simple model is in the denominator. The explanation ratio thus shows improvement of the fit compared to the simple model and can be used to check if complicating the model enhances the fit significantly, although the decision itself is still arbitrary.

L2= 2X t X x Dx,tln Dx,t ˆ Dx,t − (Dx,t− ˆDx,t) ! (21) The use of these two statistical measures makes it possible to analyse which model and which group of countries produces the best fit.

A clear indicator of the implications of the forecasted development of mortality patterns is the change in life expectancy. It is a measure often used by demographers to analyse the evolution of mortality. As we are particularly interested in the economic and actuarial implications of the differences in forecasted mortality patterns between the different models, we will use this life expectancy to calculate the value of a life annuity for an individual at age x in year t. The differences in life annuity resulting from the estimates of the models are a good indication of the size of the differences between the implications of the models. We will use the definitions of life expectancy and life annuity as used by Brouhns et al. (2002a) and Brouhns et al. (2002b). Life expectancy (denoted by ex,t) is given by:

ex,t= 110−x X k=0    k Y j=0 px+j,t+j    (22)

And the corresponding value of an immediate life annuity at age x in yeart (ax,t) is obtained by

discounting (22): ax,t= 110−x X k=0    k Y j=0 px+j,t+j    vk+1= ex,tvk+1 (23)

where v is the discount factor. By means of these two economic indicators we will compare the implications of the differences between models that we test. Significant differences in economic indicators between models suggest sizeable impact on predictions and scenarios of companies and governments. This will help decide if a more complex fit with better statistical indicator values also has a substantial impact on the economic outcomes and thus is worth the extra complexity. Furthermore, these economic indicators can give insight in what developments we can expect in the future.

2.11

Data

3

Results

3.1

Benchmark Groups

We start by reproducing the groups used in the recent papers on Dutch mortality of Stoeldraijer et al. (2013), Antonio et al. (2015), Koninklijk Actuarieel Genootschap (2014) and Wilbers (2014). These groups can serve as benchmarks for our research and it is interesting to see how these groups perform in relation to each other. The countries used in the models in these papers are given in Table 1. Note that even big national institutions like the Dutch Central Bureau of Statistics (CBS) and the Royal Dutch Actuarial Association (KAG) do not only differ in the amount of countries used in their group, but also in which countries they use. The KAG selected countries based on their GDP, whereas the CBS says to have selected the countries based on geographic location (Western-Europe). Wilbers used a combination of geographic location and similarity in patterns. CBS KAG Wilbers the Netherlands X X X Belgium X X X Denmark X X X Norway X X X Sweden X X X Finland X X X France X X 0 Italy X 0 0 Spain X 0 0 Switzerland X X 0 England&Wales X X 0 West Germany X X 0 Austria 0 X 0 Luxembourg 0 X 0 Iceland 0 X 0 Ireland 0 X 0

Table 1: Countries included in the models

As both Antonio et al. (2015) and Stoeldraijer et al. (2013) do not mention scaling countries used in the group, we will henceforth assume that this was not tested in either of the researches. In our research, we will test both scaled and not-scaled groups and thus will also produce scaled versions of the CBS and KAG groups, as to have a more complete benchmark set. We scale the countries by changing the exposure at all ages to 1,000,000 and the corresponding deaths accordingly. This makes sure that the group death rates at all ages are influenced equally by all countries used in the group.

3.2

Benchmark Statistics

the smallest group (the unscaled group used by Wilbers (2014)) and Wilmoth-Valkonen (WV) models using the group using both the scaled and unscaled group used by Wilbers (2014) to check the fit to historical data. In the estimation process of all models, we group data above age 100 as is common in practice because of the small number of cases above age 100 (see Brouhns et al. (2002a), Antonio et al. (2015) and Danesi (2014)). We start by testing the fit of the estimated benchmark models to the Dutch mortality data. The values of the indicators (see 2.10) can be found in Tables 2 and 3.

L2 RS e0 e65 Poisson LC 5,577.2 0.7762 82.64 20.82 APC 4,515.3 0.8192 82.60 20.78 CBS 35,637.6 0.6653 83.95 21.55 CBS scaled 37,840.3 0.7193 83.59 21.35 KAG 34,628.0 0.6818 83.87 21.52 KAG scaled 37,805.7 0.7174 83.66 21.39 Wilbers 12,242.3 0.7733 83.08 21.08 Wilbers scaled 14,162.0 0.7575 83.10 21.09 Wilbers Cohort 16,594.4 0.6818 81.61 21.04 Wilbers WV 10,280.2 0.7852 82.85 20.95 Wilbers scaled WV 10,652.2 0.7835 82.85 20.95 Table 2: Benchmark Statistics Females

L2 _R S e0 e65 Poisson LC 12,394.6 0.8211 78.53 16.98 APC 5,145.9 0.8801 78.24 17.40 CBS 45,458.9 0.7922 78.29 17.93 CBS scaled 32,010.0 0.8047 78.27 17.68 KAG 43,750.3 0.7916 78.18 17.95 KAG scaled 32,977.6 0.8015 78.13 17.72 Wilbers 18,927.2 0.8182 78.00 17.34 Wilbers scaled 19,900.9 0.8086 77.81 17.34 Wilbers Cohort 14,600.1 0.6036 77.83 17.58 Wilbers WV 13,644.5 0.8198 78.12 16.75 Wilbers scaled WV 13,805.2 0.8131 78.07 16.72 Table 3: Benchmark Statistics Males

have a slightly better fit than the Common Factor models as they also use the β’s of the Dutch mortality. Furthermore, the estimated life expectancy when using the original Dutch data is 82.65 for newborn girls and 20.76 for women age 65. For men we have e0=78.53 an e65=17.40.

The Poisson Lee-Carter thus estimates the life expectancy of a newborn nearly identical to the original data for both sexes, while being slightly further off for older ages. The other benchmark models estimated higher life expectancy at both younger and older ages for the female population and for the male population they all have lower life expectancy at age zero than the original data along with both lower and higher estimates at age 65.

L2 RS e0 e65 Poisson LC 36,900.0 0.2520 81.73 19.70 APC 133,934.7 -0.6434 78.46 16.33 CBS 30,993.5 0.2353 81.96 20.01 CBS scaled 33,133.8 0.2090 81.80 19.92 KAG 31,508.1 0.2271 81.92 20.00 KAG scaled 33,233.5 0.2255 81.84 19.90 Wilbers 31,885.7 0.3040 82.06 19.90 Wilbers scaled 31,977.4 0.3040 82.05 19.89 Wilbers Cohort 469,819.9 -0.4739 78.87 16.24 Wilbers WV 33,777.2 0.2791 81.79 19.57 Wilbers scaled WV 33,837.2 0.2783 81.78 19.57 Table 4: Benchmark Forecasts Statistics Females

L2 RS e0 e65 Poisson LC 79,616.3 0.2918 76.25 16.03 APC 603,944.2 0.4842 76.16 15.33 CBS 62,139.6 0.3276 76.96 16.62 CBS scaled 65,704.1 0.3224 76.81 16.49 KAG 62,119.2 0.3342 76.96 16.61 KAG scaled 65,243.4 0.3254 76.83 16.49 Wilbers 70,509.9 0.3160 76.60 16.31 Wilbers scaled 70,566.4 0.3154 76.60 16.32 Wilbers Cohort 270,688.6 0.1175 75.13 14.24 Wilbers WV 78,921.9 0.2974 76.39 16.07 Wilbers scaled WV 78,801.6 0.2990 76.38 16.06 Table 5: Benchmark Forecasts Statistics Males

both sexes all benchmark models underestimate the increase in life expectancy during the period of 2000-2009 at both younger and older ages.

0 20 40 60 80 100 −10 −8 −6 −4 −2

Observed death rates 2000−2009 Females

Age Log death r ate ● ● ● ● ● ● ● ● ● ● 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 0 20 40 60 80 100 −10 −8 −6 −4 −2 0

Observed death rates 2000−2009 Males

Age Log death r ate ● ● ● ● ● ● ● ● ● ● 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Figure 1: Observed death rates 2000-2009

0 20 40 60 80 100 −10 −8 −6 −4 −2 0

APC death rates versus Observed death rates Females

Age Log death r ate ● ● Observed APC 0 20 40 60 80 100 −10 −8 −6 −4 −2 0

APC death rates versus Observed death rates Males

Age Log death r ate ● ● Observed APC

The bad fit of the APC model compared to the other models really stands out in Tables 4 and 5. Although the APC model has the best explanation ratio for the male sample of all models, it also has by far the worst likelihood ratio. For the female sample, these statistics are even worse. The negative explanation ratio indicates that even the simplest model, which is constant over time, outperforms the forecast of the APC model for the female sample. Furthermore, the very high likelihood ratios indicate that the fits of the forecasts of the APC model are especially far of at ages with high numbers of deaths. To illustrate this, we plotted the observed log death rates between 2000 and 2009 Figure 1 and added a plot with the predicted log death rates of the APC models in Figure 2. As can be seen, the forecasted female death rates are particularly far off after age 60 and getter worse with higher ages, hence the ages at which the largest number of deaths occur. The forecasted male death rates also clearly overestimate the number of deaths at the important ages, more so than all other models we tested, whereas the overall fit to the data is relatively good.

3.3

Test-Groups

Delwarde et al. (2006) noted that a condition for grouping countries in the Wilmoth-Valkonen model (12) was that their time trends (κt’s) were similar. The Common Factor model of Lee and

Li (2005) (16) also uses the age-specific sensitivities to this time trend (the β’s) of the grouped countries. Brouhns et al. (2002b) showed that there is relatively little uncertainty in the β esti-mates for countries with population sizes comparable to the Netherlands or bigger when using data of more than 50 years. A reasonable condition on which to chose countries for our group therefore seems to be the combination of similar κ’s and similar β’s with the base country, in our case the Netherlands. For illustrative purposes, the κ’s and β’s of the different countries our plotted in Figures 3 and 4.

Females Year κ NLD DENM NORW FINL SWED BELG LUX ITA SPA SWI UK−W UKR POL AT FRA WGER ICE IRE POR CAN JAP USA AU RUS 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 −80 −60 −40 −20 0 20 40 60 80 Males Year κ NLD DENM NORW FINL SWED BELG LUX ITA SPA SWI UK−W UKR POL AT FRA WGER ICE IRE POR CAN JAP USA AU RUS 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 −60 −40 −20 0 20 40 60

Figure 3: Time trends (κt) of country-specific mortality patterns

0 20 40 60 80 100 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Females Age β NLD DENM NORW FINL SWED BELG LUX ITA SPA UK−W SWI UKR POL AT FRA WGER ICE IRE POR CAN JAP USA AU RUS 0 20 40 60 80 100 −0.02 0.00 0.02 0.04 0.06 Males Age β NLD DENM NORW FINL SWED BELG LUX ITA SPA UK−W SWI UKR POL AT FRA WGER ICE IRE POR CAN JAP USA AU RUS

Figure 4: Age-specific sensitivities (βx) to the time trend of country-specific mortality patterns

combined correlations. We only used data until 1999 for these test-groups, as we will then be able to check the performance of our groups in the period 1999-2009. For comparative purposes we also created a group, which we named Test Kappa, based only on the similarity of the κ’s. The groups are displayed in Tables 2 and 3. Although the statistics of the forecasts of the Age-Period-Cohort model in Tables 4 and 5 created serious doubts on the usefulness of the APC model in forecasts, we decided to also estimate a combination of the Common Factor model and the Age-Period-Cohort model using the countries of our test group to study the performance of this new model when using our criteria. We decided not to include a Wilmoth-Valkonen version, as the Common Factor model is an advanced version of this model and proved to be better in our tests.

Test Group Test Kappa

the Netherlands X X Belgium X X Denmark X X Norway X X Sweden X X Finland X X France 0 X Italy X 0 Spain 0 0 Switzerland 0 X England&Wales X X West Germany X X Austria X 0 Luxembourg 0 0 Iceland 0 0 Ireland 0 X Portugal X X

Table 6: Countries included in the test-groups for Female sample

Test Group Test Kappa

the Netherlands X X Belgium X X Denmark X X Norway X X Sweden X X Finland X X France 0 X Italy X 0 Spain 0 0 Switzerland X X England&Wales X X West Germany X X Austria X 0 Luxembourg 0 0 Iceland 0 0 Ireland 0 X Portugal 0 X

Table 7: Countries included in the test-groups for Male sample

Note that these groups all differ from the benchmark groups. We included several new countries in our research, such as Portugal, Poland and Ukraine. We found that the Portuguese popu-lation is a good match for the female popupopu-lation, whereas for the male popupopu-lation the κ’s are similar, but the β’s differ too much for Portugal to be used in our male test-group based on both parameters. Other new countries appeared to be no good candidates for inclusion, but Ukraine is helpful in showing that even within Europe mortality patterns can differ significantly as for Ukraine we see an upward sloping κ, indicating that life expectancy is decreasing in Ukraine. Russia seems to have very similar characteristics to Ukraine, exhibiting an upward sloping time trend and relatively high values for β between age 20 and 60. Notice that the negative values for β until age 20 in combination with the positive κ indicate that death rates at these ages are declining like the other countries.

1989-1999 and 1979-1989. This increase is not only in relation to the Netherlands, but also in relation to each other. On average, the period 1999-2009 has more than 10 percent higher correlations between these different continents than the periods before.

Females Year κ ● ● ● ● ● ● ● ● ● ● ● ● Real NL−PLC CBS CBSsc KAG KAGsc WILB WILBsc Test Group Test Groupsc Test Kt Test Ktsc 1960 1970 1980 1990 2000 2010 −80 −60 −40 −20 0 20 Males Year κ ● ● ● ● ● ● ● ● ● ● ● ● Real NL−PLC CBS CBSsc KAG KAGsc WILB WILBsc Test Group Test Groupsc Test Kt Test Ktsc 1960 1970 1980 1990 2000 2010 −60 −40 −20 0 20

Figure 5: Time trends (κt) and forecasts of all groups using data up to 1999

0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 Females Age β ● ● ● ● ● ● ● ● ● ● ● NLD CBS CBSsc KAG KAGsc WILB WILBsc Test Group Test Groupsc Test Kt Test Ktsc 0 20 40 60 80 100 0.00 0.01 0.02 0.03 0.04 Males Age β ● ● ● ● ● ● ● ● ● ● ● NLD CBS CBSsc KAG KAGsc WILB WILBsc Test Group Test Groupsc Test Kt Test Ktsc

3.4

Test-Group estimation

Using the data of the period 1960-1999, we estimated the groups presented in Tables 2 and 3 and forecasted a period of 20 years. Since we have the Dutch mortality rates until 2009, we can thus check the performance of the different models is the period 2000-2009. Figures 5 and 6 show the estimated and forecasted κt’s of the different groups and their estimated βx’s together with the

actual time trend between 2000 and 2009.

The forecasted time trends of the different groups, plotted in Figure 5 illustrate the under-estimation of the development of the κ’s. The trends also show that the normal Test Group (normal in the sense that the countries are not scaled) seems to forecast the decreasing time trend better than the other groups, especially in the longer run. The plots of the β’s in Figure 6 show that, just as in the plots of Figure 4, the Dutch women have low β values between age 20 and 65 in comparison to the other countries and Dutch males have very low value for β between age 60 and 100. This causes the estimated β’s of the groups to deviate from the Dutch β at those ages as is clearly visible in Figure 6. As we try to model the Dutch mortality, this may not seem as a positive finding, but the intuition behind the Common Factor approach of Lee and Li (2005) is that these patterns are converging and the Dutch β’s are hence expected to evolve towards the average β’s of their peers.

L2 _R

S e0 e65

Best Benchmark 30,993.5 0.3040 82.06 20.01

Test Group 29,735.7 0.3825 82.33 19.99

Test Group scaled 31,380.9 0.3533 82.18 19.89

Test Group Cohort 518,987.2 -0.4249 79.62 16.98

Test Group Cohort scaled 596,602.5 -0.8414 79.37 16.55

Kt Group 30,345.3 0.3374 82.24 19.93

Kt Group scaled 31,970.6 0.3192 82.11 19.87

Table 8: Forecasts Statistics Females Best statistic in bold

L2 _R

S e0 e65

Best Benchmark 62,119.2 0.4842 76.96 16.62

Test Group 60,300.2 0.3514 77.08 16.59

Test Group scaled 63,678.3 0.3361 76.94 16.50

Test Group Cohort 675,361.2 -1.4402 76.25 15.47

Test Group Cohort scaled 244,778.6 -0.8228 74.44 14.58

Kt Group 64,715.7 0.2947 76.80 16.56

Kt Group scaled 68,463.3 0.3011 76.67 16.41

Table 9: Forecasts Statistics Males Best statistic in bold

ratios and explanation ratios of the normal Test Group are better for both sexes. The total life expectancy is also the highest for this group, as was to be expected after seeing the time trends in Figure5. Life expectancy after age 65 however is a bit lower than some of the benchmark models. Furthermore, we see that the Common Factor Cohort models perform very poorly.

3.5

Final Groups

In the same way that we selected the test-groups, using correlations of both the κ’s and the β’s, we picked the groups to use to forecast the Dutch mortality between 2010 and 2029. These final groups are presented in Table 10. The groups for females and males are very similar, the inclusion of Portugal for the female group being the only difference between the two. These groups again differ from all benchmark groups, although they are of similar size as the CBS and KAG groups. After seeing the performance statistics of the scaled versions of all Common Factor models, we concluded that scaling the model does not enhance the model and therefore we did not make a scaled version of the final groups. The performance of the Common Factor Cohort models were also clearly worse than the Common Factor models, hence we did not estimate a Common Factor Cohort model version of our final groups.

Female Group Male Group

the Netherlands X X Belgium X X Denmark X X Norway X X Sweden X X Finland X X France X X Italy X X Spain 0 0 Switzerland X X England&Wales X X West Germany X X Austria X X Luxembourg 0 0 Iceland 0 0 Ireland 0 0 Portugal X 0

To illustrate the impact of the forecasts, we computed the life expectancies of newborns and 65 year old persons in 2029 according to the forecasted mortality rates. Tables 11 and 12 display the results. The life expectancies forecasted by the model using our final group are together with the normal CBS model the highest for both age-groups and both sexes. The standard Poisson Lee-Carter model that only uses the Dutch mortality data has remarkably lower estimates, as do both models of Wilbers (2014), which use a relatively small group of countries.

Females Year κ ● ● ● ● ● ● ● ● NL−PLC CBS CBSsc KAG KAGsc WILB WILBsc Final Group 1960 1970 1980 1990 2000 2010 2020 −80 −60 −40 −20 0 20 40 Males Year κ ● ● ● ● ● ● ● ● NL−PLC CBS CBSsc KAG KAGsc WILB WILBsc Final Group 1960 1970 1980 1990 2000 2010 2020 −80 −60 −40 −20 0 20

Figure 7: Time trends (κt) and forecasts of all groups

0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 Females Age β ● ● ● ● ● ● ● ● NLD CBS CBSsc KAG KAGsc WILB WILBsc Final Group 0 20 40 60 80 100 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Males Age β ● ● ● ● ● ● ● ● NLD CBS CBSsc KAG KAGsc WILB WILBsc Final Group

e0 e65 κ2029− κ2009 95% CI Spread κ2029 Poisson LC 84.97 22.61 -30.21 48.93 CBS 86.80 23.69 -40.05 44.72 CBS scaled 86.34 23.42 -36.99 37.31 KAG 86.70 23.64 -39.04 45.07 KAG scaled 86.38 23.42 -36.65 37.42 Wilbers 85.67 23.05 -32.84 36.67 Wilbers scaled 85.68 23.05 -32.66 36.16 Final Group 86.77 23.67 -39.82 42.91

Table 11: Forecasted Life Expectancies 2029 for Females and forecasted κ2029’s with confidence

interval spreads e0 e65 κ2029− κ2009 95% CI Spread κ2029 Poisson LC 80.56 18.15 -32.76 52.09 CBS 82.17 19.99 -36.71 36.42 CBS scaled 81.70 19.52 -35.26 33.91 KAG 82.21 19.98 -36.69 37.68 KAG scaled 81.82 19.54 -35.38 34.08 Wilbers 81.07 18.88 -33.09 36.67 Wilbers scaled 81.02 18.88 -32.64 35.55 Final Group 82.22 20.01 -36.89 35.95

Table 12: Forecasted Life Expectancies 2029 for Males and forecasted κ2029’s with confidence

interval spreads

Another important aspect of the Common Factor model is that grouping the countries and thus enlarging the dataset that will be used for modelling, increases the robustness of the model. Tables 11 and 12 and Figure 9 illustrates this aspect by also displaying the spread of the 95 % confidence interval around the forecasted κ2029’s (the spreads in Tables 11 and 12 display the

upper bound of the confidence interval of the forecasted κ2029 minus the lower bound). The

differences in confidence intervals of the forecasted κ2029 between the Common Factor model

Poisson Lee−Carter Females 1960 1970 1980 1990 2000 2010 2020 2030 −60 −20 0 20 40 60 80 Year κ

Poisson Lee−Carter Males

1960 1970 1980 1990 2000 2010 2020 2030 −60 −20 0 20 40 60 80 Year κ

Final Group Females

1960 1970 1980 1990 2000 2010 2020 2030 −60 −20 0 20 40 60 80 Year κ

Final Group Males

1960 1970 1980 1990 2000 2010 2020 2030 −60 −20 0 20 40 60 80 Year κ

Figure 9: Time trends (κt) and forecasts of Final Group models and Poisson Lee-Carter models

with CI (95 %) a65 95% CI Poisson LC 15.81 [14.89,16.69] CBS 16.40 [15.62,17.16] CBS scaled 16.25 [15.58,16.91] KAG 16.36 [15.57,17.14] KAG scaled 16.25 [15.58,16.93] Wilbers 16.05 [15.36,16.73] Wilbers scaled 16.05 [15.36,16.72] Final Group 16.39 [15.62,17.13] Table 13: Forecasted Annuity Values 2029 for Females a65 95% CI Poisson LC 13.77 [13.13,14.37] CBS 14.30 [13.70,14.91] CBS scaled 14.04 [13.51,14.56] KAG 14.30 [13.68,14.92] KAG scaled 14.04 [13.52,14.57] Wilbers 13.98 [13.47,14.48] Wilbers scaled 13.98 [13.48,14.47] Final Group 14.30 [13.70,14.90] Table 14: Forecasted Annuity Values 2029 for Males

4

Analysis

Seeing the differences in groups used to model the Dutch mortality and the way these groups were selected, it was clear that there is room for improvement. The plots in Figures 3 and 4 show that some of these choices seem strange, as countries that clearly differ in mortality patterns from the Netherlands should not be used to model the Dutch mortality.

Looking at the mortality patterns (Figures 3 and 4), we found strong similarities between the Netherlands and developed countries in other continents. We found that these similarities grow stronger through the years, suggesting that global mortality patterns are converging. As we only have data on a few, very developed, countries, it is hard to tell if this is really the case. We did find some relatively big differences between mortality patterns within Europe, as we even saw increasing time trends for Ukraine and Russia, hence declining life expectancy and decreasing correlations between these two countries and Western Europe. Seeing the trends of all other countries in our research, it seems that especially mortality patterns of the highly developed countries across the globe are converging, despite their climatic differences, and that former members of the Soviet Union are diverging from these countries as life expectancy is stagnating or declining. As we do not have reliable data on developing countries, we cannot say anything about the convergence or divergence of the mortality patterns of those countries. The evidence of convergence of the mortality patterns of developed countries and the divergence of former Soviet Union members that we found is in line with the findings of Wilson (2001). Despite these findings, we feel that including countries like Japan and the USA in our model is still not advisable, since at this moment socio-economic conditions differ too much in our view. If the mortality patterns keep converging, this might become an option in the future.

The performance statistics (Tables 2 and 3) showed that although Common Factor models with small groups of countries, the Poisson Lee-Carter and the Age-Period-Cohort models clearly perform better when modelling the data, which is obvious as these model stay closer to the historical Dutch mortality, the Common Factor models outperform the Poisson Lee-Carter when forecasting (Tables 4 and 5). As the mortality patterns of the male populations of the countries used are more alike than the female populations, Common Factor models did specifically well for males. The bigger differences in mortality patterns between some of the female populations are likely to be the explanation for the slightly better performance of the smaller group used by Wilbers (2014) for the female sample than the bigger groups used by Antonio et al. (2015) and Stoeldraijer et al. (2013), also because some of the countries used in these bigger groups have noticeably different female mortality patterns than the Netherlands.

We expected that scaling the countries would have a positive effect on the fit of the model and the quality of the forecast as it doesn’t favour the bigger countries, but gives all countries used an equal influence on the model. This however did not prove to be the case as we find that the scaled models do perform better than the normal models in the fit to the data (Tables 2 and 3), but in the out-of-sample tests of the forecasts do not perform as well as the non-scaled models (Tables 4 and 5). Based on these findings we decided to propose only non-scaled groups for our final models as our main objective is improving the quality of the forecasts rather than the fit to the current data.

showed that this difference with the other models is caused by the bad fit of the forecasts at the ages with high death counts (see Figure 2). Furthermore, the explanation ratios (Tables 4 and 5) showed that the contrast between the performance of the forecasts of the APC models to the different sexes is significant. The overall fit of the forecasted force of mortality to the female sample data is even worse than the simple one-factor model with only an age factor, whereas the overall fit to the data of the male sample of the forecasted force of mortality using the APC model was the best of all benchmark models according to the explanation ratio. This shows the difficulties with this type of model. The multicollinearity of the three factors of the model makes it very difficult to separate the age, period and cohort effects. To be able to identify the effects, additional constraints have to be imposed which effect the estimation results. The presence of interaction effects thus makes these APC models less useful for forecasting, as arbi-trary assumptions about future changes have to be made (see Tabeau, van den Berg Jeths, and Heathcote (2001)). Currie (2012) states that any forecasts of the cohort effect ιt−xcannot ignore

information on κt because of the dependence between parameters. The dependence makes it

very difficult to produce sensible forecasts. It shows that a good fit to the historical data does not guarantee a sensible forecast. Cairns, Blake, Dowd, Coughlan, Epstein, and Khalaf-Allah (2011) found that the forecasts of an APC model are not robust compared to the amount of historical data that is used and that adding an extra 10 years of data to the beginning of the dataset can seriously change the parameter estimate and have major impacts on the forecasts. After experiencing these pitfalls of the APC model ourselves and due to the fact that it takes substantially longer to estimate, it was clear that the APC model is not the type of model to use to forecast the Dutch mortality.

The Common Factor Cohort model, that we proposed in this paper, proved to be as bad in forecasting as the APC model (Tables 8 and 9), because it suffers from the same multicollinear-ity weaknesses as the APC model when forecasting. The fit to the historical data is a lot worse than the APC model, since the purpose of adding other countries to the model is to enhance the forecast, but this impairs the fit to historical data. Hence, as the forecasts and the fit of the model are clearly worse than other models tested in this paper, we can conclude that the Common Factor Cohort model is no addition to the existing models in the demographic literature. As the correlations of both the time trends (κ’s) and the age-specific sensitivities (β’s) to the time trends were the criteria we proposed for the selection of countries for our test groups and final groups, we used the strong points that we saw of the groups that were tested, while removing the disturbing factors. These criteria seem more sensible than using only the geographic location or the GDP of countries, used by the CBS and KAG respectively, as the criteria we proposed are based on the mortality patterns themselves.

On of the main beliefs behind the Common Factor model of Lee and Li (2005) is that coun-tries with similar socio-economic and climates are likely to have similar mortality patterns in the long run and that divergence from these patterns are due to country-specific events and small datasets. It is credible that these underlying ”real” patterns do not differ a lot from on age to the next (except for the very young ages). Including similar β’s as a criterion for the selection of countries seems to have achieved this smoothed pattern. As all countries used in the models based on this criterion have closely related mortality patterns, country-specific jumps between ages are filtered out and a smoothed mortality pattern is achieved.

The forecasted immediate life annuities (Tables 13 and 14) show the size of the impact of the differences between the models. The confidence intervals show that the differences are not sig-nificant. But, as the confidence intervals are up to 19% larger for the female sample using the Poisson Lee-Carter model, which is very undesirable as bigger uncertainty can be very costly in the annuity business, we see that the Common Factor models are more robust. For the male population, the lower β’s of the estimated Poisson Lee-Carter model cancel out most of the sig-nificantly larger spread of the confidence interval of the forecasted κt’s (Table 12) compared to

5

Conclusion

In this paper we investigate the performance of several mortality models on the Dutch mortality. We propose a new model, the Common Factor Cohort model, which is a combination of the Age-Period-Cohort model of Renshaw and Haberman (2006) and the Common Factor model of Lee and Li (2005). Our research shows that this combination is not useful, as the forecasts are of very poor quality due to the included cohort term. The Age-Period-Cohort model also has this drawback which is caused by the dependence between the parameters. The Age-Period Cohort model has the benefit that it fits historical data better than the other models, although it requires more complex computations. The Common Factor Cohort model doesn’t have this benefit as it is aimed at using the similarities of countries to improve the forecast, not the historical fit. We thus conclude that the Common Factor Cohort model can be discarded as it performs far worse than the Common Factor model in forecasting and performs worse than the Age-Period Cohort model in modelling historical data.

The main model in our research is the Common Factor model of Lee and Li (2005). This multi-population model uses similarities between countries to increase the size of the data set and improve the quality of the forecast as it assumes that mortality countries with similar socio-economic conditions and similar mortality patterns converges. Recent use of these multi-population models in research and official forecasts of national statistic bureaus shows that this type of model is becoming more popular in the area of mortality modelling. In the case of the Dutch mortality, we noticed that there is no consent on the countries to be used together with the Netherlands in this model. As the arguments for the choice of countries of official forecasts of the Dutch Central Bureau of Statistics (CBS) and the Royal Dutch Actuarial Association (KAG), geographic location and GDP respectively, were both not strong in our opinion, as they could result in the inclusion of countries with clearly different mortality patterns than the base country, we decided to opt for different criteria. As the model is based on converging similar-ities between the mortality patterns of countries, we proposed to take the combination of the correlations between the time trends (κt) and correlations between age-specific sensitivities (βx)

of the countries and the Netherlands as selection criteria for countries to be used for the Dutch Common Factor model.

Our research shows that there is some room for improvement for the official forecasts. The test groups outperformed the models of the CBS and KAG in out-of-sample testing, although the the differences in predicted life expectancies were negligible. The forecasted life expectancies and values of life annuities of our final model are very similar to those obtained using the group of the CBS and to a lesser extent also those obtained using the group of the KAG, hence when using data up to 2009 the impact of choosing countries for the Dutch Common Factor model based on our criteria instead of the criteria of the CBS or KAG is small, as in the case of the Netherlands these criteria result in very similar groups. Since the differences in countries used in the groups are small between our final group and the CBS and KAG groups and as mortality patterns are converging for all of these countries, the small difference in outcome in this case does not really come as a surprise. Still, the patterns of the βx’s estimated by these other models seem

less smoothed than the βx’s of our model, which intuitively seem to be more realistic. Looking

at the country-specific time trends and age-specific sensitivities, this group also seems to make more sense as it excludes some of the outliers.

dis-played in Table 10, as the Common Factor model clearly outperformed the other model tested in our research. Although the differences between the bigger groups in our research were very small, the groups in Table 10 gave the smoothest β estimates and groups based on the same criteria outperformed the other groups in out-of-sample test. It is possible to include a cohort term in this model, but this drastically worsens the fit and the quality of the forecast and is thus not recommended. We found that among highly developed countries across the world mortality patterns are converging and that the mortality patterns of former Soviet Union members are diverging from these developed countries. Due to the lack of data on developing countries, we could not test the convergence of these countries. The convergence of mortality patterns that we found among highly developed countries suggests that the advantages of using the Common Factor model will increase further in the future.

Appendices

A

Fitting Methodology

The deviance function (9) is given by::

∂D(θ) ∂θ = X∂dev ∂θ =      P∂dev ∂αx = P t2ωx,t(ˆy − y) P∂dev ∂βx = P t2ωx,tκt(ˆy − y) P∂dev ∂κt = P x2ωx,tβx(ˆy − y) (24) ∂2_D(θ) ∂θ∂θ0 = X∂2dev ∂θ∂θ0 =        P ∂2_dev ∂αx∂α0x = P t2ωx,tyˆ P ∂2_dev ∂βx∂βx0 = P t2ωx,tκ 2 tyˆ P ∂2dev ∂κt∂κ0 t =P x2ωx,tβx2yˆ (25)

Hence, by substituting (24) and (25) into (10), we get the following updating scheme for αx:

u( ˆαx) = ˆαx− P t2ωx,t(ˆy − y) P t2ωx,tyˆ = ˆαx+ P t2ωx,t(y − ˆy) P t2ωx,tyˆ (26) Similarly we get the following updating schemes for βx and κt:

u( ˆβx) = ˆβx+ P t2ωx,tκt(y − ˆy) P t2ωx,tκ2tyˆ (27) u(ˆκt) = ˆκt+ P t2ωx,tβx(y − ˆy) P t2ωx,tβx2yˆ (28) For the Age-Period-cohort model (5), we get the following updating schemes for the two addi-tional terms (βx(0) and ιz):

u( ˆβx(0)) = ˆβx(0)+ P t2ωx,tιz(y − ˆy) P t2ωx,tι2zyˆ (29) u(ˆιz) = ˆιz+ P t2ωx,tβ (0) x (y − ˆy) P t2ωx,t(β (0) x )2yˆ (30)

A.1

Lee-Carter Fitting Algorithm

The updating scheme of the fitting algorithm for the Poisson Lee-Carter modelling framework (4) is now as follows:

1. Obtain initial values for ˆαx, ˆβx and ˆκt:

ˆ

αx=_n1Ptlog ˆmx,t (the SVD estimate);

ˆ

βx= 1_k ( =⇒ Pxβˆx= 1);

ˆ κt= 0

2. Update ˆαx: ˆ αx= ˆαx+ P t2ωx,t(y − ˆy) P t2ωx,tyˆ → Compute ˆy( ˆαx, ˆβx, ˆκt). 3. Update ˆκt: ˆ κt= ˆκt+ P t2ωx,tβx(y − ˆy) P t2ωx,tβ2xyˆ

Adjust such thatPtn

t=t1κt= 0(ˆκt= ˆκt− ¯ˆκ) 1 → Compute ˆy( ˆαx, ˆβx, ˆκt) → Compute deviance Du(yx,t, ˆyx,t). 4. Update ˆβx: ˆ βx= ˆβx+ P t2ωx,tκt(y − ˆy) P t2ωx,tκ2tyˆ → Compute ˆy( ˆαx, ˆβx, ˆκt). → Compute deviance Du(yx,t, ˆyx,t). where yx,t= log ˆmx,t, yˆx,t= ˆαx+ ˆβxκˆt

5. Investigate deviance convergence:

∆D = D − Du

where Du is the updated deviance calculated at step 4.

• If ∆D > 10−6 _{=⇒ Repeat cycle from step 2.}

• If 0 < ∆D < 10−6 _{=⇒ Stop the process, take the fitted parameters as maximum}

likelihood estimators and go to step 6.

• If ∆D < 0 for 5 successive updating cycles =⇒ Stop the iteration process and take other starting values or declare the iterations non-convergent.

6. Change ˆκtand ˆβx: ˆ βx= ˆ βx P xβˆx , κˆt= ˆκt× X x ˆ βx

in order to satisfy the Lee-Carter constraints (2).

A.2

Age-Period-Cohort Fitting Algorithm

The two-stage fitting algorithm of the Age-Period-Cohort model by Renshaw and Haberman (2006) is given by:

1. Obtain values for ˆαx, and fix these:

ˆ αx= 1 n X t

log ˆmx,t (the SVD estimate)

2. Obtain initial values for ˆβx(0) and ˆβx(1):

ˆ

β_x(0)= ˆβ(1)_x = 1 k

Estimate the simplified period-cohort predictor to get initial values of ˆιz and ˆκt:

ηx,t= log(ˆyx,t) = (log Ex,t+ ˆαx) + ˆιz+ ˆκt → Compute ˆy( ˆαx, ˆβ (0) x , ˆβ (1) x , ˆιz, ˆκt) → Compute deviance Du(yx,t, ˆyx,t). 3. Update ˆιz: ˆιz= ˆιz+ P t2ωx,tβ (0) x (y − ˆy) P t2ωx,t(β (0) x )2yˆ

Adjust such that ˆιz= ˆιz− ˆι1

→ Compute ˆy( ˆαx, ˆβ (0) x , ˆβx(1), ˆιz, ˆκt) → Compute deviance Du(yx,t, ˆyx,t). 4. Update ˆβx(0): ˆ β(0)_x = ˆβ_x(0)+ P t2ωx,tιz(y − ˆy) P t2ωx,tι2zyˆ → Compute ˆy( ˆαx, ˆβ (0) x , ˆβ (1) x , ˆιz, ˆκt) → Compute deviance Du(yx,t, ˆyx,t). 5. Update ˆκt: ˆ κi+1_t = ˆκi_t+ P t2ωx,tβ (1) x (y − ˆy) P t2ωx,t(β (1) x )2yˆ

Adjust such that ˆκt1= 0 (ˆκt= ˆκt− ˆκt1)

6. Update ˆβx(1): ˆ βx(1)= ˆβx(1)+ P t2ωx,tκt(y − ˆy) P t2ωx,tκ 2 tyˆ → Compute ˆy( ˆαx, ˆβ (0) x , ˆβx(1), ˆιz, ˆκt) → Compute deviance Du(yx,t, ˆyx,t).

7. Investigate deviance convergence:

∆D = D − Du

where Du is the updated deviance calculated at step 6.

• If ∆D > 10−6 _{=⇒ Repeat cycle from step 3.}

• If 0 < ∆D < 10−6 _{=⇒ Stop the process, take the fitted parameters as maximum}

likelihood estimators and go to step 6.

• If ∆D < 0 for 5 successive updating cycles =⇒ Stop the iteration process and take other starting values or declare the iterations non-convergent.

8. Change ˆβx(0), ˆβ(1)x and ˆκt: ˆ β(0)x = ˆ βx(0) P xβˆ (0) x , βˆx(1)= ˆ βx(1) P xβˆ (1) x , ˆκt= ˆκt× X x ˆ βx(1)

to satisfy the Age-Period-Cohort constraints (P

xβˆ (0)

References

Alho, J.M. (2000). Discussion of Lee (2000). North American Actuarial Journal 4 (1), 91–93. Antonio, K., A. Bardoutsos, and W. Ouburg (2015). Bayesian Poisson log-bilinear models for

mortality projections with multiple populations. Internal reports, Research Center Insurance, Leuven.

Booth, H., R.J. Hyndman, L. Tickle, and P. de Jong (2006). Lee-Carter mortality forecasting: a multi-country comparison of variants and extensions. Demographic Research 15 (9), 289–310. Booth, H., J. Maindonald, and L. Smith (2002). Applying Lee-Carter under conditions of variable

mortality decline. Population Studies 56, 325–336.

Booth, H. and L. Tickle (2008). Mortality modelling and forecasting: a review of methods. Annals of Actuarial Science 3 (1-2), 3–43.

Brouhns, N., M. Denuit, and J.K. Vermunt (2002a). A Poisson log-bilinear regression approach to the construction of projected life-tables. Insurance: Mathematics and Economics 31 (3), 373–393.

Brouhns, N., M. Denuit, and J.K. Vermunt (2002b). Measuring the longevity risk in mortality projections. Bulletin of the Swiss Association of Actuaries (2), 105–130.

Butt, Z. and S. Haberman (2010). ilc: A collection of R functions for fitting a class of Lee-Carter mortality models using iterative fitting algorithms. Actuarial Research Paper 190, Cass Business School, City University of London.

Cairns, A.J.G., D. Blake, K. Dowd, G.D. Coughlan, D. Epstein, and M. Khalaf-Allah (2011). Mortality density forecasts: an analysis of six stochastic mortality models. Insurance: Math-ematics and Economics (48), 355–367.

Currie, I. (2012). Forecasting with the Age-Period-Cohort model? In International Workshop on Statistical Modelling. Prague, Czech Republic.

Danesi, I. (2014). Forecasting mortality in related populations using Lee-Carter type models. PhD thesis, Univesita degli Studi Padova.

Delwarde, A., M. Denuit, M. Guill´en, and A. Vidiella-I-Anguera (2006). Application of the Pois-son log-bilinear projection model to the G5 mortality experience. Belgian Actuarial Bulletin 6, 54–68.

Denuit, M. and A.C. Goderniaux (2005). Closing and projecting life tables using log-linear models. Bulletin of the Swiss Association of Actuaries (1), 29–49.

Eggleston, K.N. and V.R. Fuchs (2012). The new demographic transition: Most gains in life expectancy now realized late in life. Journal of Economic Perspectives 26 (3), 137–156. Goodman, L. (1979). Simple models for the analysis of association in cross-classifications having

ordered categories. Journal of the American Statistical Association 74 (367), 537–552. Gutterman, S. and I.T. Vanderhoof (1998). Forecasting changes in mortality: A search for a law

of causes and effects. North American Actuarial Journal 2 (4), 135–138.

Lee, R.D. (2000). The Lee-Carter method for forecasting mortality, with various extensions and applications (with discussion). North American Actuarial Journal 4 (1), 80–93.

Lee, R.D. and L.R. Carter (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association 87 (419), 659–671.

Lee, R.D. and N Li (2005). Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography 42 (3), 575–594.

Lee, R.D. and T Miller (2001). Evaluating the performance of the Lee-Carter method for fore-casting mortality. Demography 38 (4), 537–549.

Peltzman, S. (2009). Mortality inequality. Journal of Economic Perspectives 23 (4), 175–190. Renshaw, A.E. and S. Haberman (2003a). Lee-Carter mortality forecasting: a parallel generalized

linear modelling approach for England and Wales mortality projections. Journal of the Royal Statistical Society: Series C (Applied Statistics) 52 (1), 119–137.

Renshaw, A.E. and S. Haberman (2003b). Lee-Carter mortality forecasting with age specific enhancement. Insurance: Mathematics and Economics 33 (2), 255–272.

Renshaw, A.E. and S. Haberman (2006). A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics 38 (3), 556–570.

Stoeldraijer, L., C. van Duin, and F. Janssen (2013). Bevolkingsprognose 2012-2060: model en veronderstellingen betreffende de sterfte. Bevolkingstrends, July 1–27.

Tabeau, E., A van den Berg Jeths, and C. Heathcote (2001). Forecasting mortality in developed countries: insights from a statistical, demographic and epidemiological perspective.

Tuljapurkar, S., N. Li, and C. Boe (2000). A universal pattern of mortality decline in the G7 countries. Nature 405, 789–792.

Wilbers, J.A. (2014). Comparing Poisson log-bilinear model of mortality in the Netherlands with the Benelux and Scandinavia.

Willets, R.C. (2004). The cohort effect: insights and explanations. British Actuarial Journal 10, 833–877.

Wilmoth, J.R. (2000). Demography of longevity: past, present and future trends. Experimental Gerontology 35, 1111–1129.

Wilmoth, J.R. and T. Valkonen (2002). A parametric representation of mortality differentials over age and time. In Fifth seminar of the European Association for Population studies Working Group on Differentials in Health, Morbidity and Mortality in Europe. Pontignano, Italy. Wilson, C. (2001). On the scale of global demographic convergence 1950-2000. Population and