Do people with higher SES live longer? An analysis of the relationship between socioeconomic status and mortality and its implications for Dutch pension funds

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Do people with higher SES live longer?

An analysis of the relationship between socioeconomic status

and mortality and its implications for Dutch pension funds

Ashik Anwar Ali (Jibon) s1873989 Master’s thesis Actuarial Sciences

Under supervision of prof. R.H. Koning and dr. J. van der Ploeg University of Groningen

TKP Pensioen January 16, 2014

Abstract

This paper investigates the statistical relationship between socioeconomic status (SES) and mortality for members of Dutch pension funds. By using SES groups based on zip codes as an indicator for SES, we found for men a negative and diminishing relationship between SES and mortality. For women this relationship behaved negatively and erratically. In terms of life expectancies, we found for a newborn girl and boy differences equal to 2.4 and 1.6 years between the lowest and highest SES groups. For retired individuals these differences were 1.8 and 1.6 years, respectively for men and women. Furthermore, we found evidence that the effect of SES on life expectancy becomes smaller as age goes up, this phenomenon was especially the case with retired individuals. Moreover, we present statistical evidence that mortality and life expectancy between our population and the whole Dutch population differs significantly. As a result of this, pension funds should be careful when using merely the basic CBS life table to compute pension liabilities. Finally, our standardized mortality ratio (SMR) curves for men can be used by pen-sion funds in order to compute consistent penpen-sion liabilities. Robustness of our curves are determined by comparing them to those from an external party.

JEL codes:G22, H55, I15.

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Acknowledgments

This thesis is a research project performed in order to obtain a Master of Science degree in Econo-metrics, Operational Research and Actuarial Sciences, specialization Actuarial Sciences, from the University of Groningen. This research was combined with an internship at TKP Pensioen B.V. in Groningen from July 2013 to January 2014. First of all, I would like to express my gratefulness to prof. R.H. Koning and dr. J van der Ploeg for their guidance and support on this thesis. Secondly, I am indebted to the actuarial department of TKP Pensioen for providing me a nice working place for the past months and giving me the opportunity to have a behind the scenes look in the life of actuaries. I would also like to thank my friends Dennis Prak and Ilse Stubbe for proofreading my thesis and providing me useful comments. At last, I want to express my gratitude to my dearest mother and father, who supported me in every step in life.

Jibon Ali

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1 Introduction

Do people with higher SES live longer is a question with no simple answer. Some believe life and death is something divine which we cannot influence, others believe that wealth could have a relationship with mortality. The existence of a relationship between wealth and life expectancy is widely accepted among scientists, and it is commonly referred to as “the wealth gradient in mortality”. Much academic research has been conducted to investigate the relationship between socioeconomic status (SES) and mortality. For instance, Leigh and Jencks (2006) investigated and concluded that changes in economic inequality affects mortality in rich countries. Also Attanasio and Emmerson (2003) investigated this relationship and concluded that wealth is an important determinant for mortality by using data from the British Retirement Survey. The outcomes of the above research are also in line with the paper of Kawachi and Kennedy (1997) who presented similar results for the United States. Although all these papers use different tools and models to measure wealth or SES and its relationship with mortality, in the general sense the conclusion remains the same.

The relationship between SES and mortality is also important for the actuarial practice and pen-sion funds to construct penpen-sion liabilities. In the actuarial practice mortality fundamentals for Dutch pension funds are based on two components, namely the basic life table from the Dutch Actuarieel Genootschap (AG) and pension fund specific factors. The second component is neces-sary in order to correct for the fact that mortality rates between the pension fund population and the whole Dutch population may differ significantly. At the same instance this second component is very interesting for actuaries and causes an intense debate in the pension world. As a result of this debate, on 12 September 2012 De Nederlandsche Bank (DNB) introduced the “Good Practice Gebruik Fondsspecifieke Ervaringssterfte” for pension funds in order to compute their technical pension reserves in a proper way. As a consequence, the AG presented the “Concept Leidraad Er-varingssterfte” in order to guide pension funds in this process. One should notice that in the latter document no such thing as “the best” method is presented. However, it does provide important points of interest and quality standards for the computations that pension funds should take into account. For instance, the AG suggests to choose from three different data sources to determine pension fund specific factors, namely own data, a reference population or characteristics of the whole Dutch population. With respect to the referencing populations, the “Concept Leidraad Er-varingssterfte” explicitly refers to the existing literature, for instance Knoops and van den Brakel (2010) that describes the relationship between SES and mortality. As a result of this, it has become common actuarial practice to use information on the relationship between SES and mortality in the construction of fund specific factors. Amongst others, both income and zip codes are utilized in this context to capture SES. From an academic point of view, a qualitative relationship between SES and mortality is in itself noteworthy. On the other hand, from an actuarial perspective, a mere qualitative relationship is far from adequate. In order to use this relationship to establish appro-priate technical provisions, the relation needs to be quantified as well.

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the retirement age of 65 in the Netherlands. One of their key findings is that, for both men and women, low income individuals have approximately 2.5 years lower life expectancy than high income individuals. On the other hand, we have the contributions of Smith et al. (1996) and Shaw et al. (2008). Their paper/book uses a different approach to measure SES, namely zip codes in the United States and the United Kingdom. Both research present evidence that there exists a signif-icant relationship between SES and mortality. Continuing, one could also state that the sector an individual works in could be an accurate measure for his SES to investigate the relationship with mortality. The latter is also very interesting to investigate for pension funds since it administers often the pensions of people who work in the same sector. For instance, Cambois et al. (2001) used profession as a measure for SES and presented that for France the difference between life expectancy among managers and manual workers for 35 years old individuals differed 5.4 years in 1991.

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we have determined our optimal mortality models, we will examine whether we can find evi-dence that the mortality of our population differs from the whole Dutch population. Next, we will translate our mortality models in terms of life expectancies to compare our findings with those from the literature. Based on our models we will construct (weighted) standardized mor-tality ratio (SMR) curves which can be used to compute pension liabilities. In order to test the robustness of our SMR curves, that are weighted with individuals’ established pension rights, we will compare them with those provided by an external party. To sum up, we want to answer the following research questions in this paper.

• What is the relationship between SES and mortality? • Does mortality differ among pension funds?

• Do these relationships differ between men and women?

• How are these relationships in terms of gains in life expectancy?

• Are there differences in mortality between our population and the whole Dutch population? • Are our weighted SMR curves in line with those from the external party?

The paper is structured as follows: we start the next section by providing a literature review. Then in section 3 we explain the relevance of our paper for the actuarial practice. In section 4 we present the data set we used and explain in detail the variables we utilize in our analysis. Moreover, also the descriptive statistics of our data along with prima facie evidence on the relationship between SES and mortality will be presented in this section. In the subsequent two sections, section 5 and 6, we present our theoretical and empirical models along with our empirical results. Finally, the paper ends with a discussion and a conclusion where we present the answers on our research questions.

2 Literature review and our hypotheses

2.1 Knoops and van den Brakel (2010)

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also the influence on life expectancy in good health.

Their results in terms of life expectancy for different income groups is given in an abbreviated version in table 1. Notice that ∆ represents the difference in life expectancy between the High and Low income groups.

Men

Age Low Lower middle Middle Higher middle High ∆

0 73.9 77.5 78.9 80.1 81.1 7.2

30 45.1 48.4 49.7 50.7 51.8 6.7 65 15.1 17.0 17.8 18.4 19.1 4.0

Women

Age Low Lower middle Middle Higher middle High ∆

0 78.8 84.1 84.5 84.2 85.5 6.7

30 49.8 54.6 55.0 54.8 55.8 6.0 65 18.6 22.4 22.6 22.1 22.6 4.0

Table 1: Life expectancy per income group

The key finding of the paper is that Dutch newborn boys from the lowest income class have life expectancy equal to 73.9 years, whereas for high income boys this is 81.1 years. Simultaneously, for newborn girls the difference between these income groups is approximately 6.7 years. Con-tinuing, for men we see a positive and decreasing relationship between income groups and life expectancy, whereas for women this relationship is positive, but behaves erratically. We also ob-serve that as the age goes up the difference between life expectancies of the lowest and highest income groups becomes smaller; this is the case for both men and women. Furthermore, for life expectancy in good health the differences are even larger (notice that we do not present this table with results): for males 17.8 years and females 17.6 years. Hence, the conclusion of this paper is straightforward: higher income correlates with higher life expectancy. In addition, the paper also concludes that income is an important determinant for SES, however, one should be aware of scenarios that income could fluctuate substantially from year to year due to economic or personal circumstances. With respect to our analysis, we will use the results from Knoops and van den Brakel (2010) to compare them with the ones we obtain. We want to make this comparison since their analysis is also conducted for the Netherlands. Therefore, it will provide us insights on whether the mortality of our population differs from the whole Dutch population.

2.2 Smith et al. (1996) and Shaw et al. (2008)

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Trial between 1973 and 1975. They investigated the socioeconomic differentials in mortality risk among white men who had been previously screened for a multiple risk factor intervention trial. By using zip codes as an indicator for SES, the paper found a strong relationship between high SES and lower mortality rates. In addition, we have Shaw et al. (2008) who conducted their research in the United Kingdom where a unique zip code represents only fifteen households. Hence, it is an very accurate measure for the SES in a neighborhood. An important contribution of Shaw et al. (2008) is that the paper provides a useful atlas of mortality by cause of death by geographical area in the United Kingdom. One can conclude from this atlas that there are significant differences in overall mortality and by cause of death in different geographical areas. For our analysis we also use zip codes as a measure for SES. Our choice is strengthened by the fact that both Smith et al. (1996) and Shaw et al. (2008) have shown in their research that it is a good indicator for SES in Western countries comparable with the Netherlands. It is such a good indicator since it does not only focus on the income component, but also on the area someone lives in.

2.3 Cambois et al. (2001)

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2.4 Kalwij et al. (2012)

Another paper that investigates the relationship between income and life expectancy is Kalwij et al. (2012). This article quantifies the association between individual’s income and remaining life expectancy at the retirement age of 65 years in the Netherlands. An important contribution of their paper is their distinction between individual’s income and spouse’s income. This is a rather interesting contribution, since one could think of scenarios where mortality risk is perhaps more likely to be negatively related to the spouse’s income than the own income. To be more precise, this is especially the case when women leave the labor force at the time of marriage or the birth of their first child. Another interesting aspect of this paper is that it uses empirical models that con-trol for unobserved individual-specific characteristics. The data sets used for this paper are gained from the CBS Income Panel Study 1996-2007 and the CBS Causes Of Death 1997-2007. Their panel consists of 19,258 individuals that provide 141,725 observations which is a representative sample for the Dutch population aged 65 and older over the period 1996-2007. Moreover, the analysis of this paper is based on the independent variables of gender, age, marital status and income. On the other hand, we have the dependent variable mortality which is defined as being deceased before January 1st of the next year.

Next, this paper presents a mortality risk model, where they consider the following latent variable model that relates the next year’s mortality status, at age (a + 1), to individual characteristics at age a

Ha+1= −αa− Xaβ − Λ − εa,

(

Ma+1= 1 if Ha+1< 0

Ma+1= 0 otherwise.

Here the latent variable Ha+1is an individual’s stock of health, which indicates that the

individ-ual is deceased if it falls in the next year below a given threshold. Moreover, the variable Ma+1

denotes observed mortality at age (a + 1), and this variable is equal to 1 if an individual became a years old and died at the of age (a+1) and 0 otherwise. Furthermore, αarepresents an age-specific

intercept, Xais a vector of length k of the individual’s observed characteristics with a

correspond-ing parameter vector β of length k. Λ stands for an individual’s unobserved characteristics and it is assumed to be constant over time and independent of the covariates. Simultaneously, it is as-sumed to be normally distributed with mean equal to 0 and a variance of σ2

Λ. Moreover, this paper

makes the assumption that the error term εais independently distributed across individuals and

follows a logistic distribution with mean 0 and variance normalized to π₃2.

This paper also corrects for dynamic selection, for technical details see Kalwij et al. (2012). This boils down to the following notion

Λ = ˜Xτγ + θ,

where ˜Xτ = ((1, Xτ)×(τ −65), (1, Xτ)×(τ −65)2), with a corresponding parameter vector γ with

length (2k + 2). θ is a random effect that is assumed to be independent of ˜Xτ and normally

dis-tributed with mean equal to 0 and variance σ2. They state that not accounting for such dynamic selection may yield inconsistent estimates of αa and β as described in Cameron and Heckman

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risk in which the correlation between Λ and income becomes increasingly positive with age up to a certain age and then decreases from that point as the sample becomes more homogenous with respect to income and Λ. The model takes into account dynamic selection when all individuals are observed from the age of 65.

To sum up, they estimate the following model by means of Maximum Likelihood Estimation P(Ma+1= 1|Xa, ˜Xτ, θ, Ma= 0) = F (αa+ Xaβ + ˜Xτγ + θ),

where F (·) is the logistic cumulative distribution function and the condition Ma = 0 represents

the fact that all respondents in the population at risk are alive at age a.

By using the random effects logistic models that accounts for dynamic selection with mortality risk as the dependent variable, they conclude that conditional on marital status for both Dutch men and women older than 65, own income is about equally strong and negatively related with mortality risk (p-values 0.00 and 0.00). Moreover, spouse’s income is only weakly associated with mortality risk for women (p-value 0.09). Also, for men significant evidence is found for the pres-ence of random effects, however, for women they are less significant (p-value 0.03). Also the standard deviations of the random effects are for both men and women relatively small compared to the standard deviations of the error term. Thirdly, using Markov Chain Monte Carlo simula-tions, for both men and women the paper quantifies the remaining life expectancy at age 65 for low-income respondents as approximately 2.5 years lower than that of high income respondents. The theoretical models we present in our paper will be to a large extent in line with those from Kalwij et al. (2012), except that we do not correct for dynamic selection. A comparison between our results and the results of this paper will also be done since both researches are performed for the Netherlands.

2.5 Our hypotheses and expected findings

The main hypothesis of our paper is that individuals with a higher SES tend to live longer than lower SES individuals. This phenomenon could have multiple explanations. To begin with, this could be due to individuals with lower SES having only access to limited health services, because they cannot afford an extensive health insurance, see Attanasio and Emmerson (2003). Lower SES individuals also tend to have an unhealthier lifestyle than high SES individuals. For instance, they smoke more often or consume more alcohol. This phenomenon is also investigated in Huisman et al. (2005) and Macintyre (1997). Simultaneously, lower SES people usually have a lower level of education, see for instance Wodtke et al. (2011). As a result of this, they have no choice but to work in the heavy labor industries, which does not require higher education, however it does come with higher risk. Fourthly, demographics could also play a significant role when modeling mortality risks. For example, lower SES individuals generally live in less developed areas with higher crime rates or more pollution. On the whole, all of these explanations could have a significant effect on the life expectancy or mortality of individuals. However, one should notice that we do not know to what extent these explanations are representative for the Netherlands.

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for men and women. Translating our mortality models in terms of life expectancy, we expect to find a positive relationship between SES and life expectancy, and we expect this relationship to be diminishing. Furthermore, since everyone from our population is part of a pension fund, and thus participates in the labor force, we expect to observe lower mortality in our population than the whole Dutch population. As a consequence, we expect that pension funds should use different life tables than those from the CBS when computing pension liabilities. After we have constructed SMR curves, we expect to provide graphical evidence that there is a negative relationship between SES and mortality. Using these graphs, we also anticipate that our population experiences lower mortality rates than the whole Dutch population. Finally, we expect that our weighted SMR curves will be in line with those from the external party, and thus can be used by pension funds in order to compute pension liabilities.

3 Relevance for the actuarial practice

The past couple of years most Dutch pension funds had to deal with turbulent economic times due to the financial crisis. As a result of the low interest rates, the pension liabilities increased whereas the inflow of pension premium by working individuals remained relatively steady. As a consequence of this, the coverage ratio decreased, and on top of that the situation even worsened due to the fact that the life expectancy of the pensioners increased significantly. In case of a low coverage ratio, the pension fund has to make a couple of decisions in order to become solvent again. Firstly, they could think of a better investment strategy, however, there is no guarantee that the returns on investments are going to be higher. Another, and perhaps more obvious decision could be to cut the pensions of the retired people or increase the pension premiums for the work-ing people. However, this could cause an intense protest from both parties. Due to the fact that the Dutch pension system is based on solidarity within a pension fund, no clear cut solution could be found until now.

Because of all these developments, the pension funds with an inadequate coverage ratio had to submit a report to the DNB with a recovery plan. When submitting such a plan it is of importance to use the adequate interest rate forecasts and expected mortality rates for computing the cover-age ratio. As mentioned, in this paper we focus on the latter component. In order to compute the coverage ratio, the pension funds need to compute the pension liabilities first. And for this computation, one needs proper pension fund specific factors, which correct for the fact that the mortality between the pension fund population and the whole Dutch population could differ sig-nificantly. In the past the AG advised pension funds to compute the pension fund specific factors by means of age demotion from the basic CBS life table. However, these days such methods are outdated and usage of more advanced mortality research is stimulated. Given that most pension funds have a couple of thousand participants, a different expected life expectancy or mortality has an enormous effect on the pension liabilities, and thus the coverage ratio.

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mentioned populations. As a next step, we will construct weighted SMR curves that take into account the established pension rights. Pension funds could use these curves to determine their pension liabilities. Therefore, it is of importance to test the robustness of these curves. Similar weighted SMR curves are built by an external party, hence in order to examine the robustness of our curves we will compare these with each other. On the whole, after investigating the rela-tionship between SES and mortality, we can provide confirmation that pension funds should use different life tables than those from the CBS for their pension liabilities computation. Additionally, from our mortality models we can construct weighted SMR curves. These curves can contribute to the debate on how to compute consistent pension liabilities, and thus coverage ratios for Dutch pension funds.

4 Data

This paper uses data provided by one of the largest pension administrators of the Netherlands that has more than 25 Dutch pension funds in its portfolio, which leads to a customer database with approximately 1.7 million individuals. We obtained raw data for nine pension funds from the internal database of the pension administrator and edited the data into our desired format. The edited data set contains information on the individual’s personal and demographic characteristics, like date of birth/death and zip code, as well as financial information like the established pension rights. The data on the birth and death dates are very accurate since the internal database is directly linked to the GBA of the municipality where the individual is part of. Moreover, we excluded all observations where the zip code of the individual was unknown, these were about 3.9% of the total observations. On the whole we are left with a final sample from 2007 up to 2012 of 249,313 individuals that provide an unbalanced panel of 1,268,677 observations.

4.1 Mortality

The most important variable of our analysis is mortality, since it is our outcome variable. For each respondent and the corresponding observation year we have the date on which he or she died. We transformed this variable into a binary variable, which takes on the value 1 in case the respondent died in the corresponding observation year and 0 otherwise. Notice that an individual that died in observation year t is excluded from the subsequent observation years and that we do not observe individuals that died before 2007 or after 2012. Moreover, in case an individual is married/partnered and dies in year t, then starting from year t the partner will be included in the data as a new observation. Notice that in a scenario where the partner is already a member of a pension fund, then he is already included in the data.

4.2 Covariates of interest

4.2.1 Age, gender, role and pension fund

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2 participants et cetera. Furthermore, we have a small number of individuals that were part of two pension funds, for these individuals we gave them for the second pension fund a new unique identity. Moreover, we transform later on this variable into nine dummy variables, where pension fund 1 is equal to 1 for individuals part of pension fund 1, pension fund 2 is equal to 1 for pension fund 2 participants et cetera. Pension fund 1 is the reference category in our analysis. This choice is reasonable since this fund provides the most observations for our data set. Furthermore, we have to be cautious when interpreting later on the estimates for the pension fund dummies from our model. We have pension fund 1 as our reference category, however, this does not represent an ordinal relationship.

4.2.2 SES groups

In 2012 the Sociaal Cultureel Planbureau (SCP) of the Netherlands presented a document where they provide for each four digits zip codes an SES score. These social status scores are derived from the characteristics of inhabitants like their education level, income and occupation. More-over, zip codes with only industries or zip codes with less than 100 households are excluded from the survey. For each unique zip code the SCP conducted an interview with a so called informant to gain information about the SES of the neighborhood. This was done for a period of four years in order to get a complete view for the whole country.

The SES scores consist of four components, namely the average income in the neighborhood, the percentage individuals with a low income, the percentage individuals with a low education and the percentage of individuals that are unemployed. By means of a Principal Component Analysis (PCA) performed by the SCP, these four characteristics are combined into one component: the SES score. The interpretation of the scores are fairly simple: a high score corresponds to a high SES of the neighborhood and a low score to a low SES. In our data set we have the complete last known zip code for each individual, id est both the first four digits as the last two letters, however, we only use the first four digits in our analysis. We translate the SES scores into five equally large SES groups. We are able to make these groups equally large since we know for each four digits zip code the number of inhabitants. For each individual we link their last known zip code to an SES score and stratify it into the corresponding SES group. These SES groups are presented in table 2. Moreover, for our analysis we transform this variable SES group into five dummy variable, where SES 1 equal to 1 indicates that someone is part of the lower SES class, SES 2 represents individuals from the lower middle SES class et cetera.

SES group Lower limit SES score Upper limit SES score Name SES group

1 -7.25 -0.83 Lower SES class

2 -0.83 -0.10 Lower middle SES class

3 -0.10 0.46 Middle SES class

4 0.46 0.99 Upper middle SES class

5 0.99 3.19 Upper SES class

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4.2.3 Established pension rights

The variable established pension rights is somewhat cumbersome, this is mainly caused by the complex construction of the Dutch pension system. There are two major components in the pen-sion system: general retirement penpen-sion and penpen-sion for the surviving relatives. We choose to define a variable established pension rights which is the sum of the general retirement pension and the widow pension for the partner. The choice of the general retirement pension is evident since it is the largest liability for a pension fund. We included the widow pension for the following reason. In case a married/partnered participant dies in year t, then we observe his widow starting from that year with a widow pension which must be paid out by the pension fund starting from that year. Hence, this is also an important component of the liability for a pension fund. One should notice that these established pension rights are expressed as pension payments per year. We are interested in such a variable established pension rights since in the actuarial practice it is required to not merely look at the number of deaths when computing the pension liabilities, but also take into account the corresponding established pension rights. In our analysis we will construct SMR curves for two of our largest pension funds where we take the established pension rights of the individuals into account, and test their accuracy by comparing them to those from the external party.

4.2.4 CBS mortality rates

It is important to compare the mortality rates of our population with the ones from the whole Dutch population and to include them in our mortality model. For this purpose we use the CBS 2007-2012 life tables. In this table the mortality rate qk, which is the probability of dying before

reaching age k + 1 for a k years old person, is computed for each observation year in the following way

qk=

# of individuals that died and would be k years old on 31/12 # of individuals that would be k years old on 31/12 .

To illustrate an example, for a respondent that dies at age 41.3 years in observation year 2007, the corresponding mortality rate from the CBS is the one for a 42 years old in 2007. Furthermore, the CBS life table ends after age 99, and we have individuals that are older than 99. In order to obtain a CBS mortality rate for these individuals, we use for the ages 100 to 120 years the CBS prognosis life table 2012. In contrast to the ages 1-99, where the mortality rates decrease as the observation year goes up, we assume that for individuals in the age interval 100-120 the mortality rate does not improve as observation year goes up. Instead it remains constant. For each individual, his given age and the observation year, we computed his corresponding qk from the CBS life table.

We refer to his variable as the CBS mortality rate.

4.2.5 Variable income

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analysis income was included as a component. Thus from our point of view, income is already captured for a large extent in the variable SES group. Considering all of the above mentioned issues, we have chosen not to use income as a covariate for our mortality models.

4.3 Descriptive statistics and prima facie evidence

In figure 1 and 2 we provide box plots for the variables age and established pension rights by SES groups separately for men and women. One should notice that for these box plots we have removed the outliers. We used this approach because the variable established pension rights has large outliers that resulted in boxplots where information on the median, minimum and maximum was not clearly visible anymore. The box plots should be read as follows, the ends of the whiskers are the lowest datum still within 1.5 Interquartile range (IQR) of the lower quartile, and the highest datum still within 1.5 IQR of the upper quartile. The thick dark horizontal segment inside the box represents the median and 50% of the data occurs between the lower and upper edges of the box, namely between the first and third quartile.

Figure 1: Box plots men

Let us first investigate figure 1. We find that the median age among the men is roughly around 52 years old and the median of age becomes smaller as the SES group goes up. Secondly, the es-tablished pension rights increases as the SES group goes up until SES group 4 and this variable is right skewed. Furthermore, from figure 2 we find for women the median age decreases as the SES group goes up and the median of established pension rights increases slightly as the SES group goes up. Simultaneously, the latter variable is right skewed. Comparing men and women, we observe that women are a little bit younger and have lower established pension rights.

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Figure 2: Box plots women

on how the SES groups are distributed among pension funds. For these purposes we present in figures 3a and 3b mosaic plots containing this information. From figure 3a, we see that pension

(a) (b)

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fund 1, 2 and 3 are the largest funds. Furthermore, the portion women is higher in pension funds 3 and 4, whereas for the other pension funds the portion of men is higher. Next, we have figure 3b, for which we first investigate the men. For pension fund 1 the individuals are almost equally distributed among the SES groups. Pension funds 2, 3, 4, 5, 6, 8 and 9 have more individuals in the higher SES groups, whereas in pension fund 7 most individuals are part of the lowest SES group. For women, we find for pension funds 1 and 3 that the individuals are roughly equally distributed among SES groups. Moreover, pension funds 2, 4, 5, 6, 8 and 9 have more individuals that are part of higher SES groups. Finally, in line with the male population, for women in pension fund 7 a large portion of the individuals are part of the lowest SES group.

Continuing, in table 3 we compute mortality rates for nine different age groups conditioned on SES groups. We split the variable age up in nine groups. The first one is for respondents between 0 and 20 years, the second one for 21 to 30 years old, the subsequent group for 31 to 40 years old et cetera. The mortality rates are computed by dividing the total number of persons that died at the end of the period by the total number of people that were alive at the beginning of the period within the same age and SES group. Moreover, in the last column of table 3 we calculated the mean of the variable CBS mortality rate for each age group. In this way we can compare the mortality rates of our population with those from the whole Dutch population.

Men

Age SES 1 SES 2 SES 3 SES 4 SES 5 All CBS m.r. 0-20 - - - 0.00059 0.00100 0.00034 0.00035 21-30 0.00030 0.00046 - 0.00035 0.00036 0.00030 0.00047 31-40 0.00093 0.00046 0.00040 0.00104 0.00077 0.00074 0.00082 41-50 0.00194 0.00196 0.00180 0.00131 0.00152 0.00167 0.00200 51-60 0.00520 0.00459 0.00407 0.00390 0.00401 0.00432 0.00552 61-70 0.01468 0.01235 0.01093 0.01070 0.00867 0.01144 0.01394 71-80 0.03833 0.03513 0.03326 0.02902 0.02968 0.03332 0.03931 81-90 0.11104 0.08651 0.10017 0.09015 0.08975 0.09637 0.10795 90+ 0.24691 0.21739 0.25974 0.21429 0.26804 0.24528 0.25878 Women

Age SES 1 SES 2 SES 3 SES 4 SES 5 All CBS m.r.

0-20 - - - - 0.00072 0.00016 0.00015 21-30 0.00008 0.00010 0.00021 0.00032 - 0.00013 0.00025 31-40 0.00065 0.00055 0.00053 0.00039 0.00047 0.00051 0.00056 41-50 0.00173 0.00147 0.00163 0.00098 0.00146 0.00145 0.00157 51-60 0.00453 0.00496 0.00399 0.00343 0.00337 0.00400 0.00402 61-70 0.01054 0.00765 0.00906 0.00805 0.00752 0.00857 0.00862 71-80 0.02481 0.02349 0.02274 0.01979 0.02488 0.02327 0.02430 81-90 0.07330 0.07269 0.06969 0.07842 0.06844 0.07239 0.08177 90+ 0.25287 0.28902 0.22099 0.20465 0.17797 0.22702 0.21975

Table 3: Mortality rates of age groups by SES groups

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respondent in the age group 51-60 from SES group 5 is 22.9% smaller compared to his peer from the lowest SES group. For age group 61-70, this difference in mortality between the SES groups 1 and 5 is 40.9%. Continuing, for women the mortality rates among almost all age groups behave very erratic. From the last column, in line with what we see for men, we observe for most age groups (except for 0-20 and 90+), that the mortality rate of the whole Dutch female population is higher than our female population. To strengthen this statement we provide in figure 4 an SMR curve, which is the ratio of the observed numbers of deaths and the expected number of deaths for each age group.

Figure 4: SMR curves of age groups

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5 Logistic regression model

5.1 Theoretical model

Let us say that we randomly pick an individual i with age a in observation year t from the whole Dutch population. Next, the probability that this individual dies before his next birthday is equal to the mortality rate given by the CBS life table (CBS mortality rate), so we have

P[Mit= 1] = qa(i,t),tfor i = 1, ..., N and t = 2007, ..., 2012.

Following Kalwij et al. (2012), let us assume that this probability can be translated to a latent health Hit. And in case Hitfalls below 0, then the individual dies before his next birthday. Let us assume

Hithas the following form

Hit= −log(

1

q_a(i,t),t − 1) + εit, where we refer to (_q 1

a(i,t),t − 1) as the CBS mortality component from now on in this paper. It

follows from this form that in case someone has a lower mortality rate, then he will have a higher latent health than someone with a higher mortality rate. We do not observe a latent health of an individual. Instead we observe

(

Mit = 1 if Hit < 0

Mit = 0 otherwise.

The interpretation is as follows: Mitis equal to 1 in case individual i with age a dies somewhere in

observation year t, because his latent health fell below 0. Continuing, let us turn back to this latent health and assume that we know more about this randomly picked individual. For instance, we know that he participates in a pension fund. We assume that this information has an influence on Hitand this influence is independent from the age of the respondent and the observation year.

We make this statement because we assume that this age and observation year effect is already captured in log(_q 1

a(i,t),t − 1), since qa(i,t),tis different for all ages and observation years. Now let us

write

Hit= −λlog(˜˜

1

q_a(i,t),t − 1) + εit.

We included a parameter for log of the CBS mortality component in order to test later whether this component has indeed a significant effect on the latent health, and whether there is difference between mortality of our pension fund participants and the whole Dutch population. Continuing, from our data we have more information about the individual. For instance, we know his role, SES group and in which pension fund he participates in. Now we write Hitin the following form

Hit= −˜λlog(

1

q_a(i,t),t − 1) − ˜β1role − ˜β2SES1 − ... − ˜β6SES5 − ˜β7PF1 − ... − ˜β14PF9 + εit.

Alternatively, introducing an intercept α, we may write Hit= −α − λlog(

1 qa(i,t),t

− 1) − β1role − β2SES2 − ... − β5SES5

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The interpretation of the above equation is as follows, for instance, in case the individual is part of a better SES group, then this will have a positive or negative effect on his latent health. One should notice that the SES does not vary over time since it is based on the last known zip code of the individual which is the same for all observation years. For notional reasons, we introduce the vector Xit, where the intercept and all the covariates are included. We now write

Hit= −Xitθ + εit,

where θ = (α, λ, β1, ..., β13)is the parameter vector. Translating this to Mit, which we can observe,

we want to estimate the model

P[Mit= 1|Xit] =P[Hit< 0]

=P[−Xitθ + εit< 0]

=P[εit< Xitθ]

= F [Xitθ].

Next, in line with Kalwij et al. (2012), we assume that εit is logistically distributed with mean

equal to 0 and varianceπ₃2. At the same time, we assume it is independent across individuals and observation years. The cumulative distribution function of the logistic distribution is given by

F [x] = P [X ≤ x] = e

x

1 +ex.

As a result of this we can write

P [Mit = 1|Xit] = e Xitθ 1 +eXitθ = e Xitθ 1 +eXitθ · e−Xitθ e−Xitθ = 1 1 +e−Xitθ,

where the parameter estimates ˆθ = ( ˆα, ˆλ, ˆβ1, ..., ˆβ13) are obtained by Maximum Likelihood

Esti-mation as described in Cameron and Trivedi (2005). Moreover, in a scenario were ˆλis equal to −1 and statistically significant and the other parameter estimates are equal to 0 and insignificant, then one could state that for our population the mortality is completely explained by the CBS life table. Let us illustrate this mathematically, in the just mentioned scenario we have a latent health equal to

Hit = 1 ·log(

1

q_a(i,t),t − 1) + εit, as a next step we write

P[Mit= 1|Xit] = 1 1 +elog( 1 qa(i,t),t−1) = 1 1 +_q 1 a(i,t),t − 1 = qa(i,t),t,

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5.2 Dynamic selection

In our data set we observe an individual in subsequent years somewhere between 2007 and 2012. In this interval, observations could be added to or removed from our data. In table 4 we present the most important cases of these dynamic selection issues together with how we deal with this in our data. Notice that for simplicity we assume in this table that individuals were present either in the beginning of 2007 or until the end of 2012.

Case Issue

1 Individual dies in year t∗.

→ We observe him between 2007 and t∗.

→ In case he is married/partnered, we observe his partner between t∗ _{and 2012.}

2 Individual goes from active worker status to retirement status in year t∗. → We observe him between 2007 and 2012.

3 Individual goes from known PF to another known PF in year t∗.

→ We observe him as a new identity between t∗and 2012 and the old identity between 2007 and 2012.

4 Individual goes from known PF to unknown PF in year t∗. → We observe him between 2007 and 2012.

5 Individual comes from unknown PF to known PF in year t∗. → We observe him between t∗ and 2012

6 Individual does value transfer (pension paid out before retirement) at t∗. → We observe him between 2007 and t∗.

Table 4: Dynamic selection issues

We could think of scenarios where multiple of these cases should be combined. To illustrate, let us take an individual that comes from an unknown pension fund to a known pension fund in 2008, moves to an unknown pension fund in 2011 and does a value transfer at the same instance. In this scenario we observe him only between 2008 and 2011.

For our mortality models we do not take into account dynamic selection correction terms. The first reason is that we do not have the tools or data to include dynamic selection correction terms in our models. The second justification is based on Kalwij et al. (2012). In their paper they state that in case one includes random effects in a mortality model, then one should take into account dynamic selection. In their mortality model with random effects and dynamic selection correction terms, they assumed that the random effects are normally distributed with mean 0 and variance σ2

Λ.

Af-ter estimating this model, they find evidence that the standard deviations of the random effects are relatively small compared with the standard deviation of the error term, which was equal to pπ2_{/3 ≈ 1.814. Furthermore, in their paper models with and without random effects correcting}

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increased or decreased with εitwhere E[εit] = 0. We expect that this CBS mortality component is

an important, if not the most important component of someone’s latent health. Now for 2007 his latent health can be written as

Hit= −α − λlog(

1

q_{a(i,2007),2007} − 1) − β1role − β2SES2 − ... − β5SES5 − β6PF2 − ... − β13PF9 + εit.

Then for the next year, the CBS mortality component of this individual’s latent health is not any-more based on the CBS life table of 2007. Instead it is based on the life table of 2008 which is not dependent on the one from 2007. For 2008 his latent health can be written as

Hit= −α − λlog(

1 qa(i,2008),2008

− 1) − β1role − β2SES2 − ... − β5SES5

− β₆PF2 − ... − β13PF9 + εit.

So, in case an individual has a “low” (”high”) latent health in 2007, then this does not imply that his latent health in 2008 will be low (high) again. This is due to the fact that the latter is computed from a different life table that is not dependent on the one from 2007. One can think of a scenario here like new game, new chances for every year. Due to this phenomenon, we expect that all ob-servations are independent over observation years.

6 Empirical analysis and results

6.1 Addressing the gender issue

We first want to investigate whether we can find statistical evidence that we need to perform our analysis separately for men and women. We proceed in the following way; we have a variable gender that is equal to 1 in case the individual is a male and 0 if female. In order to test whether the relationship between SES and mortality differs, we first write for the latent health

Hit= −Xitθ − γgender_it− fXitθ + εe _it.

Here fXitis a matrix of the independent variables log of the CBS mortality component, role,

dum-mies for SES groups and dumdum-mies for pension funds multiplied by the dummy variable gender. And eθis the corresponding parameter vector. Notice that Xit is the same matrix of independent

variables and the intercept as described in section 5.1. Next, we estimate the following model P [Mit= 1|gender_it, Xit, fXit] =

1

1 +e−(γgenderit+Xitθ+ fXitθ)e

.

We test by means of a Wald Test as described in Cameron and Trivedi (2005) whether γ = 0 and e

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6.2 Relationship between SES and mortality

We start by estimating a logistic regression model with mortality as the outcome variable and log of the CBS mortality component, role, dummy variables for SES groups and dummy variables for pension funds as the independent variables. Secondly, we estimate two restricted logistic regres-sion models: one without the dummies for SES groups and one without the dummies for penregres-sion funds as the independent variables. Using this approach we can determine which model fits our data the best by means of a Likelihood Ratio Test (LRT) as described in Cameron and Trivedi (2005). In this test the null hypothesis states that the restricted model is the optimal one. For men and women the results are presented separately in table 5 and (*), (**) and (***) indicate whether the coefficient estimate is significantly different from 0 at the 10%, 5% and 1% level.

Let us first determine for men the optimal model. Testing the unrestricted model against the re-stricted 1 model, we have the null hypothesis that the parameter estimates for the four SES dum-mies are all equal to 0. We compute a p-value of 0.00, thus we reject the null at a 1% level, so the unrestricted model is better. Next, testing the unrestricted model against the restricted 2 model we obtain a p-value equal to 0.00, thus we reject the null hypothesis that the eight pension fund dummies are all equal to 0 at a 1% level. As a consequence of these two tests, we can conclude that for men the unrestricted model is the optimal one. Using the same testing procedure for the women, we find statistical evidence that the model with SES dummies only is the optimal one. As a result of this, our interpretation will be based on these two optimal models.

For men we can clearly see a negative relationship between higher SES groups and mortality. To be more precise, in case an individual is part of SES group 2, then his mortality rate decreases with 10.8% (≈ 1 − e−0.114)compared to someone from SES group 1, ceteris paribus. And this estimate is statistically significant at a 1% level. Moreover, as the SES group improves the mortality rate keeps decreasing significantly. For respondents from the highest SES group we find evidence that their mortality rate is 23.0% lower than those from the lowest SES group, ceteris paribus. This result is statistically different from 0 at a 1% level. Moreover, in case the individual’s role is participant of one of the pension fund, then his mortality rate decreases with 17.6% compared to individuals that are partner, ceteris paribus. This estimate is significant at a 5% level. Furthermore, we find that only pension fund 7 has higher mortality than pension fund 1, however this coefficient is not sig-nificant. The other pension funds have lower mortality rates compared to pension fund 1. These effects are the largest for pension funds 8 and 9. Individuals from these pension funds experience respectively 50.2% and 51.7% lower mortality compared to pension fund 1, ceteris paribus. These estimates are statistically significant at a 1% and 5% level. Next, as one could expect, we also ob-serve that the CBS mortality component has a significant effect on mortality. An increases of one percent means that the odds of dying decreases statistically significant at a 1% level with factor 0.357, ceteris paribus. Translating this to the CBS mortality rate, this makes sense. Since in case the CBS mortality rate qitincreases, log of the CBS mortality component will decrease, which leaves

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Men

Unrestricted Restricted 1 Restricted 2 Estimate S.E. Estimate S.E. Estimate S.E. Constant 0.379∗∗∗ 0.103 0.260∗∗∗ 0.101 0.236∗∗ 0.099 Log CBS m.c. −1.030∗∗∗ 0.012 −1.036∗∗∗ 0.012 −1.017∗∗∗ 0.012 Role −0.194∗∗ _0.091 _−0.201∗∗ _0.091 _−0.188∗∗ _0.091 SES 2 −0.114∗∗∗ 0.043 −0.125∗∗∗ 0.043 SES 3 −0.186∗∗∗ 0.044 −0.199∗∗∗ 0.044 SES 4 −0.250∗∗∗ _0.045 _−0.272∗∗∗ _0.045 SES 5 −0.262∗∗∗ 0.045 −0.296∗∗∗ 0.045 PF 2 −0.156∗∗∗ 0.036 −0.168∗∗∗ 0.036 PF 3 −0.177∗∗∗ _0.040 _−0.195∗∗∗ _0.040 PF 4 −0.493∗∗∗ 0.108 −0.505∗∗∗ 0.107 PF 5 −0.188∗∗ 0.085 −0.227∗∗∗ 0.085 PF 6 −0.337∗∗ _0.132 _−0.352∗∗∗ _0.132 PF 7 0.039 0.089 0.054 0.089 PF 8 −0.698∗∗∗ 0.221 −0.759∗∗∗ 0.221 PF 9 −0.727∗∗ _0.356 _−0.789∗∗ _0.356 Number of obs. 716,538 716,538 716,538 AIC 50,301 50,340 50,350 Women

Unrestricted Restricted 1 Restricted 2 Estimate S.E. Estimate S.E. Estimate S.E. Constant 0.034 0.079 −0.038 0.073 0.047 0.061 Log CBS m.c. −0.990∗∗∗ 0.015 −0.993∗∗∗ 0.015 −0.991∗∗∗ 0.014 Role −0.034 0.047 −0.037 0.047 −0.051 0.046 SES 2 −0.043 0.060 −0.057 0.059 SES 3 −0.079 0.060 −0.090 0.060 SES 4 −0.192∗∗∗ _0.061 _−0.202∗∗∗ _0.061 SES 5 −0.164∗∗∗ 0.059 −0.172∗∗∗ 0.059 PF 2 −0.012 0.058 −0.020 0.057 PF 3 −0.014 0.052 −0.020 0.052 PF 4 −0.224∗∗∗ 0.082 −0.233∗∗∗ 0.082 PF 5 0.082 0.090 0.057 0.090 PF 6 0.107 0.187 0.099 0.187 PF 7 0.193 0.120 0.216∗ 0.119 PF 8 −0.037 0.506 −0.094 0.505 PF 9 −0.850 1.005 −0.892 1.005 Number of obs. 552,139 552,139 552,139 AIC 28,212 28,218 28,211

Table 5: Logit model, relationship between SES and mortality

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have P [Mit = 1|Xit] = 1 1 +e1.030·log( 1 qa(i,t),t−1) = 1 1 +e1.030₍ 1 qa(i,t),t − 1) < 1 1 + 1 · (_q 1 a(i,t),t − 1) = 1₁ qa(i,t),t = q_a(i,t),t,

where the third step is justified due to the fact that eψ _{is always bigger than 1 for all ψ ∈ (0, ∞).}

From this derivation we can conclude that for a randomly picked male from our population, whom are all participating in a pension fund, for whom we do not know their role, SES group or pension fund, his mortality is statistically lower than a male from the whole Dutch population. This finding is consistent with evidence found in the actuarial practice.

For women, the first thing that attracts our attention is that the estimates are less significant and smaller in magnitude than those for men. To be more precise, in case an individual is from SES group 5, then her mortality decreases with 15.8% compared to someone from the lowest SES group, ceteris paribus. And this effect is significant at a 1% level. Moreover, a somewhat striking result is that the mortality in SES group 4 is lower than the mortality of SES group 5. Whereas for men we found a decreasing relationship between SES groups and mortality, for women this relationship behaves erratically. Moreover, for women we find no statistical evidence that having as role participant has an influence on mortality. In contrast, for men we found that being partici-pant did have a negative influence on mortality. Furthermore, we find again evidence that the CBS mortality component has significant effect on mortality. In case it increases with one percent, then the odds of dying decrease with factor 0.371, ceteris paribus. The parameter estimate of ˆλwomen is equal to -0.991 and we perform a Wald Test in line with the one we performed for men. Thus, our null hypothesis is that λwomen = −1 and the other parameter estimates are equal to 0. We obtain a p-value equal to 0.00, hence we have to reject the null hypothesis at a 1% level. As we have done for men, we can derive mathematically that for a randomly picked female from our population, the mortality is lower than that of the whole Dutch female population.

We can also test for each pension fund whether we can find statistical evidence that the mortality of their population differs from the whole Dutch population, and whether information like role and SES helps us to explain mortality. We divide our sample into nine sub samples: one for each pension fund. As a next step we estimate a similar model as described above, namely

Hit = −α − λlog(

1 qa(i,t),t

− 1) − β₁role − β2SES2 − ... − β5SES5 + εit,

for each pension fund sample separately. We use the Wald Test and have the following null hy-pothesis: α = 0, λ = −1, β1 = β2 = β3= β4 = β5 = 0. In table 6 we present for each pension fund,

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PF 1 PF 2 PF 3 PF 4 PF 5 PF 6 PF 7 PF 8 PF 9 Men 0.00 0.00 0.00 0.00 0.05 0.05 0.06 0.07 0.83 Women 0.15 0.77 0.09 0.02 0.94 0.07 0.45 0.26 1.00

Table 6: p-values Wald test mortality pension funds

For men we reject the null hypothesis at a 1% level for pension funds 1, 2, 3 and 4. Hence, for these pension funds we have obtained statistical evidence that their mortality differs from the whole Dutch population. On top of that, for these pension funds information like role and SES helps us to explain mortality. For pension funds 5 and 6, we reject the null at a 5% level, hence for these pension funds we can make the same statements as above at a lower significance level. Only for pension fund 9 we cannot reject the null hypothesis. Hence, for this pension fund we do not have statistical evidence its members’ mortality differs from the whole Dutch population. One should notice that this rejection of the null hypothesis could be due to the small sample for pension fund 9, see figure 3a. For women we immediately observe that for more than half of the pension funds we do not have evidence that the mortality differs from the whole Dutch popula-tion. Only for pension funds 3, 4 and 6 we can reject the null hypothesis at a 10% level. Hence, for these funds its members’ mortality differs from the whole Dutch population. Additionally, for these funds information on role and SES helps us to explain mortality. On the whole, our tests provide confirmation for pension funds that they have to use different life tables than those from the CBS, which are based on the whole population, when computing their mortality rates and pension liabilities.

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6.3 Translating mortality to life expectancy

In the literature often the results obtained from mortality models are translated to life expectancy. In order to proceed in the same line, let us first introduce some actuarial notation as described in Bowers et al. (1997). We first have (x) which is the notation for a life age x,tqxis the probability that

(x)dies within t years andtpxis the probability that (x) survives at least t years. Next, a discrete

random variable associated with the future lifetime is the number of future years completed by (x)prior to death. This is called the curtate-future-lifetime of (x) and is denoted by K(x). Now the expected value of K(x) is denoted by exand is called the curtate-expectation-of-life. In words,

this the expected number of complete years remaining to live or the expected number of birthdays that the person will celebrate. Following Bowers et al. (1997) we can write the following (for the technicalities of this derivation, see the appendix of Bowers et al. (1997))

ex =E[K] = ∞ X k=0 kkpxqx+k = ∞ X k=1 kpx = px+2px+3px+ ... = px+ px· px+1+ px· px+1· px+2+ ... = (1 − qx) + (1 − qx)(1 − qx+1) + (1 − qx)(1 − qx+1)(1 − qx+2) + ...

When using this equation we assume time constant probabilities as described in Lee and Carter (1992). From our mortality models we are able to compute for each (x) the corresponding qx, and

thus ex. Moreover, in practice often the assumption is made that on average people live a half

year in the year of death. In this case the complete expectation of the curtate-future-lifetime at (x)is equal to ex+1₂. Notice that our life expectancy table makes the following assumptions: it is

representative for 2012, individuals that have role participant, and we assume that after the life age of 121 the probability of dying next year is equal to 1. For men we only present life expectancies in table 7 for pension funds 1 and 2. For women the optimal model was without pension fund dummies, hence no distinction is made between pension funds for women. Moreover, ∆1 is the

difference between the life expectancy of SES group 1 and CBS 2012, ∆2represents the difference

between SES groups 2 and 1 et cetera. Furthermore, the reason we include life expectancies of a newborn is because in some scenarios he has an orphan pension that is relevant for a pension fund for computing pension liabilities.

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Men PF 1

Age CBS SES 1 SES 2 SES 3 SES 4 SES 5 ∆1 ∆2 ∆3 ∆4 ∆5

0 79.1 78.5 79.6 80.3 80.8 80.9 −0.6 1.0 0.7 0.6 0.1 20 60.2 59.6 60.6 61.2 61.8 61.9 −0.6 1.0 0.6 0.6 0.1 35 45.6 44.9 45.9 46.5 47.0 47.1 −0.6 1.0 0.6 0.5 0.1 50 31.2 30.6 31.5 32.1 32.6 32.7 −0.6 0.9 0.6 0.5 0.1 65 18.3 17.6 18.4 18.9 19.3 19.4 −0.6 0.8 0.5 0.4 0.1 80 8.0 7.4 7.9 8.2 8.5 8.6 −0.5 0.5 0.3 0.3 0.1 PF 2

0 79.1 80.0 81.0 81.7 82.2 82.3 0.8 1.0 0.6 0.6 0.1 20 60.2 60.9 61.9 62.5 63.1 63.2 0.8 1.0 0.6 0.6 0.1 35 45.6 46.3 47.2 47.8 48.4 48.5 0.7 1.0 0.6 0.5 0.1 50 31.2 31.8 32.7 33.3 33.8 33.9 0.6 0.9 0.6 0.5 0.1 65 18.3 18.7 19.4 19.9 20.4 20.5 0.4 0.8 0.5 0.4 0.1 80 8.0 8.1 8.6 8.9 9.2 9.3 0.1 0.5 0.3 0.3 0.1 Women

0 82.8 82.5 83.0 83.3 84.3 84.1 −0.3 0.5 0.3 1.0 −0.3 20 63.7 63.5 64.0 64.3 65.2 65.0 −0.3 0.5 0.3 1.0 −0.3 35 48.9 48.7 49.2 49.4 50.4 50.1 −0.3 0.5 0.3 0.9 −0.3 50 34.5 34.3 34.8 35.0 35.9 35.7 −0.2 0.5 0.3 0.9 −0.2 65 21.2 21.1 21.5 21.7 22.5 22.2 −0.2 0.4 0.2 0.8 −0.2 80 9.7 9.6 9.9 10.1 10.6 10.5 −0.1 0.3 0.2 0.6 −0.1

Table 7: SES and life expectancy

fund 2. Thus, although there are certainly differences in life expectancy between pension funds, the differences between SES groups are the same across pension funds. Fourthly, as the age goes up the differences between SES groups become smaller. This is especially the case for the older individuals. Our argument for this phenomenon is that for someone who already reached the respective age of 80, his own consciousness about health is a much more important determinant for his health, and thus life expectancy, than his SES. Furthermore, for men from pension fund 1 and SES group 1, we find that the life expectancy is lower than that of the whole Dutch male population for all ages. For pension fund 2, we observe for all SES groups higher life expectancies than the CBS.

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newborn girl the difference in life expectancy between the highest and lowest SES group is equal to 1.6 years. For a retired woman this difference is 1.1 years.

Let us now compare the results we have found from our mortality models to those from the lit-erature. We found for a newborn boy the difference in life expectancy equal to 2.4 years between the highest and lowest SES group. For a newborn girl this difference was 1.6 years. Knoops and van den Brakel (2010) found these differences equal to 7.2 and 6.7 years, respectively for a new-born boy and girl. For a 65 years old individual we found the difference between the two extreme SES groups equal to 1.8 and 1.1 years, respectively for men and women. In contrast, Knoops and van den Brakel (2010) found these differences equal to 4.0 years for both men and women. Kalwij et al. (2012) present that these differences for a retired person are equal to 2.5 years for men and women. An explanation why our differences are so small could be due to the fact our newborns are part of a pension fund population. This means that they are already part of a higher SES group than the group that is indicated as the lowest SES group in the other two papers. We also found similarities between our results and the literature. We found a positive and diminishing relationship between SES and life expectancy for men. For women, this relationship was positive as well, however, it behaved erratically. This finding is perfectly in line with the results presented in Knoops and van den Brakel (2010).

6.4 SMR curves

Using the results from our analysis up to now, we are able to answer most of our research ques-tions. We also want to present our results in a graphical way, for this purpose we translate our mortality models to SMR curves. Moreover, these SMR curves can also be used by pension funds to compute pension liabilities.

For men we found evidence that the unrestricted model fits our data the best, whereas for women the model with the SES effects only was the optimal one. When translating our mortality models to SMR curves, we want to utilize our optimal models and present curves per SES group. Let us first present confidence intervals of our models. For men we present in figure 5a for pension fund 2 and SES group 2 the mortality curve together with the 95% confidence intervals of our model and the CBS mortality curve 2012. Notice that in this part of our analysis all curves are representative for individuals that have as role participant.

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(a) (b)

Figure 5: C.I. men (PF 2 and SES 2), women (SES 2)

We now present our main SMR curves for men and women. For men we present in figures 6a and 6b the SMR curves, respectively for pension funds 1 and 2. For both pension funds we observe that mortality decreases as the SES improves and that this effect is diminishing. Between SES groups 1 and 2 we observe a large difference in mortality, whereas for the 2 highest SES groups this effect is very small. Furthermore, for pension fund 1 starting from approximately 50 years, the mortality in SES group 1 is higher than the CBS. For SES groups 2 and 3 starting from respectively 85 and 100 years, the mortality is also higher than the CBS. For pension fund 2, we only observe for SES group 1, that the mortality starting from approximately 95 years is higher than the whole Dutch male population. To sum up, we have also found graphical evidence that the relationship between SES and mortality for men is negative and diminishing.

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(a)

(b)

Figure 6: SMR curves for men

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Figure 7: SMR curves for women compute their pension liabilities in an adequate way.

6.5 Taking established pension rights into account

The SMR curves we presented are not only useful to examine the relationship between SES and mortality. They also can be used by pension funds to compute pension liabilities. However, in the actuarial practice it is required to not merely investigate the number of deaths when comput-ing the pension liabilities, but also to take the established pension rights of the individuals into account. In our data we know the established pension rights of every individual we observe. So we can weigh our SMR curves with the pension rights. Before we explain our approach, we first state that this part of our analysis is based on observation year 2012, individuals that have as role participant and were alive at the end of 2012. The latter statement is made due to the fact that for a deceased individual we do not have any pension liability. We do have a pension liability for the partner, however she appears as a new observation in our data. Next, pension funds needs to be assured that our weighted SMR curves are robust and consistent. For this purpose we compare our curves with those from an external party. We have these external party curves only for pen-sion funds 1 and 2. As a result of this, we present our weighted SMR curves for these two penpen-sion funds only.

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individuals in fourteen different SES groups. So it is arguable that their model looks even more accurate into mortality at an individual level than ours. Finally, the external party also takes into account the established pension rights of the individuals when constructing SMR curves.

The approach to construct the weighted SMR curve is as follows, to begin with, for each individual we compute based on his characteristics and the optimal model the probability that he will not reach his next birthday: q_a(i)∗ . Secondly, we take all individuals with the same age together and compute a weighted qea based on their established pension rights. Let us illustrate this with a numerical example. Assume we have 3 individuals that are 20 years old: we have someone who is 20 years old, has pension rights equal to e100 and a mortality rate from our model equal to 0.0037. A second person is also 20 years old, has pension rights equal toe250 and a mortality rate equal to 0.0039. And thirdly, we have someone who is 20 years old, has pension rights equal to e1,000 and a mortality rate equal to 0.0027. Now we computeqe20in the following way

e q₂₀= 100 100 + 250 + 1, 000 · 0.0037 + 250 1, 350 · 0.0039 + 1000 1, 350· 0.0027 = 0.0030.

We perform this procedure for all ages that are present in our data and then compute the weighted SMR. In the next step, we fit a cubic spline as described in Harrell (2001) through these points. In figure 8a and 8b we present our weighted SMR curves along with the weighted SMR curves from the external party for men and women, respectively for pension funds 1 and 2. Notice that the ex-ternal party provides the curves starting from age 20 up to 120. In our data for both pension funds the youngest and the oldest individuals are 21 and 88 years old. So we only use the information from the interval 21 to 88 from the external party.

(a) (b)

Figure 8: Weighted SMR curves

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curve has the same shape as the external party. Starting between ages 20 and 40 both curves are fairly constant. Then starting from 40 years our curve starts to increase, whereas for the external party this increase starts after 50 years. Secondly, for women our curve behaves a bit differently than the external party. Especially the kink at approximately 57.5 years from which the external party curve starts to decrease is not present in our curve. Continuing, we investigate figure 8b for pension fund 2. For men we again find a curve that has only to some extent the same shape as the external party. Starting from approximately 52.5 to 88 years the curve of the external party increases steeper than ours. Moreover, since our curve is situated substantially above the exter-nal party curve, there are certainly differences between our curve and the one from the exterexter-nal party. Furthermore, for women, our curve behaves differently compared to the external curve. For instance, our curve shows a decreasing effect in the interval 20 to 50 years, while the external curve is constant in this interval. Next, from 50 to 88 years the external curve is increasing whereas ours remains fairly constant and even decreases at the end of the interval. Finally, for men from pension fund 1 and both men and women from pension fund 2, our curves are below the ones from the external party. This could be due to the different construction of the mortality models on which these curves are based. One should also notice that for our mortality models pension funds 1 and 2 provide the most observations. As a consequence, our mortality models will be bit biased towards the mortality of pension funds 1 and 2.

In order to perform a better comparison, we compute the correlation of our SMR and the external party’s SMR. We present in figure 9 our SMR on the vertical axis and the external party’s SMR on the horizontal axis. A 45◦line is also included in the figures to assist us to observe whether there is total positive correlation present. Furthermore, we present in each figure the Pearson correlation coefficient ρ. For men from pension fund 1 we find a high positive correlation, whereas for women for this fund we observe a small negative correlation. Secondly, for pension fund 2, we again find high positive correlation for men and a negative correlation for women. On the whole, based on our mortality models, we can construct weighted SMR curves which can be used by pension funds to compute pension liabilities. For men from pension fund 1, our curve is almost identical to the one provided by the external party. However, for men from pension fund 2 and women, pension funds should be careful when using our weighted SMR curves, since they were different from the external party.

6.6 Summarizing our empirical results

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(a) (b)

(c) (d)

Figure 9: Correlation our SMR vs. external party SMR

Do people with higher SES live longer? An analysis of the relationship between socioeconomic status and mortality and its implications for Dutch pension funds