Increasing retirement age and its effect on different social classes

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Faculty of Economics and Business Section Quantitative Economics

Increasing retirement age and its effect on

different social classes

J.C. Schilder

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: J.C. Schilder Student nr: 6074510

Email: jackbok@gmail.com Date: March 20, 2018

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Abstract

Increasing longevity and decreasing birth rates have urged the Dutch government to increase both AOW-age and pension-age in response, as pension entitlement expenditures would soon become unaffordable otherwise. The increase in retirement age in the future is determined by increases in remaining life expectancy after the age of 65. Since these increases need to be signaled well in advance, it is necessary to make future forecasts. In this thesis, we do this by using the Lee-Carter stochastic mortality model. We find life expectancy set to increase in the future, taking into account the uncertainty of the model. We also observe that the difference between AOW- and pension-age follows a decaying pattern, which is likely to result in unnec-essary costs (also among pension funds and governments) and uncertainty among participants. Furthermore, we divide the population into three groups based on education level and com-pare future increases in remaining life expectancies, as well as the difference in the amount of entitlements one can expect to receive. We observe that these differences are fairly significant.

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Statement of Originality

This document is written by Student Jack Schilder who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of comple-tion of the work, not for the contents.

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Chapter 1 Introduction

In 1956, the state pension law (AOW, Algemene Ouderdomswet) came into effect, and in 1957 the first state pension retirement payments were payed to retirees. The retirement age was fixed at 65 years without much thought about sustainability in the long term. At that time there was no reason to worry about such things, since the birthwave after the Second World War would ensure that the workforce was always large enough to pay the premiums for the retirees. Indeed, in 1957 the ratio of retirees (receiving entitlements) and workers (paying premiums) was 1 to 6. This rose-coloured view of a perpetually sustained old-age pension slowly but surely became undermined by 2 important factors as time went on.

The first factor was the overall increase in life expectancy of the Dutch population, which meant entitlements had to be paid over longer periods of time than had been assumed at the time of conception of the AOW back in the Fifties. The second factor was a decline in birth rates from the Seventies onward, as a result of changing views on society, birth control and many other drivers. This meant that in the future a declining workforce would have to pay premiums to cover expenditures on pension payments for an ever-increasing group of retirees.

By 2016 the total yearly AOW-expenditures had risen to over 36.9 billion euros, seeCBS Statline(2017b) and only about 69% of these expenditures were covered by AOW-premiums, seeCBS Statline(2017a). Meanwhile, the ratio of retirees versus workers has changed from 1:6 to about 1:3 at this point, seeCBS Statline(2017c) and is expected to look like 1:2 by 2040, as judged fromCBS Statline(2016a). The effects of these changes in demographics is explored in more detail in for example Smid et al.(2014) and also invan der Horst et al.(2010), where it is found that government expenditures on AOW-entitlements will keep on increasing relative to income from premiums. It should come as no surprise that these are very concerning prospects not only for the Dutch government but for private pension funds and life insurers as well, since they will all have to deal with an ageing population in their own way while rules for forecasting provisions are all set by law and assume a retirement age of 65. Structural reforms were neces-sary and were enforced in their current final form by the bill Ministerie van Sociale Zaken en Werkgelegenheid(2014), which the Dutch Senate approved in June of 2015.

This structural reform is comprised of an increase in the AOW-age by a predetermined schedule, followed by a further increase based on remaining life expectancy forecasts. Prior to the current mechanism of linking the pension-age and pension entitlements to increases in life expectancy being established, research has been done and suggestions for implementation have been made on provisional increase mechanisms, for example inDe Waegenaere et al. (2012), where a different incremental increase of the retirement age was assumed, based on a different Starting from 2013, the AOW-age increases at an accelerating rate until it reaches 67 years and

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three months in 2023. Increases in the AOW-age will be dependent on changes in remaining life expectancy hereafter. A somewhat similar schedule was decided upon for the pension-age for pension rights accrued in the second as well as the third pillar. A first increase in 2014 of two years to 67, and a further increase from 2024 onward dependent on changes in remaining life expectancy.

In this thesis we want to look at what these scheduled changes to retirement age might look. More in particular we will look at how differents social classes benefit from or are disadvantaged by such a scheduled increase when future remaining life expectancies are forecasted based only on data of the total Dutch population, without a further breakdown into groups. Over the past decades there have been several studies that have asked the question of whether or not educational or occupational social class is a significant indicator for mortality risk, for example

Davey Smith et al.(1998). Another indicator of social class would be income level, as explored for the Dutch population inKnoops and Brakel(2010). In this thesis we will use education level as an identifier, because from a relative perspective it is the only identifier of social class with enough data points to make it usable, even though the available data is actually very scarce. As mentioned, the scheduled increase of the retirement age after 2023 is determined by changes in remaining life expectancy. We need to make forecasts of future remaining life expectancy and for this, we need a stochastic mortality model. Such a model allows for proper analysis and also gives an insight into uncertainty of developments in mortality going into the future. For this purpose, we will be using the Lee-Carter model (Lee and Carter,1992). This model works by taking as input a time-ordered matrix of age-dependent mortality rates and then parameterizing the logarithms of these mortality rates in terms of two age dimension and one time dimension. From there it is possible to make future forecasts using statistical time series methods.

While quite revolutionary at the time, the Lee-Carter model is relatively elementary com-pared to later developed models such as the RH model (Renshaw and Haberman, 2006) and the BMS model (Booth et al., 2002), both of which are extensions of the Lee-Carter model. Furthermore, the accuracy of the Lee-Carter model must not be overstated. Because stochastic mortality models are generally fitted with historical data, future projections are represented by past shocks and do not take into account any future developments outside of historical changes in mortality. However, despite the above we choose the Lee-Carter model because the aim of this thesis is not to find the most accurate mortality model for our population, but to look at fur-ther applications to the forecasts found using this model. This approach of using a more basic mortality model when the modeling itself is not the main goal is commonly taken,Boonen et al.

(2017) being one recent example.

What we find after establishing our mortality model and making our forecasts is that AOW-and pension-age increase gradually going into the future, with continuous fluctuations over time in the difference between starting ages of AOW- and pension-age, which can be expected to complicate things for individuals closing in on retirement. Furthermore, we find that education level has a signifanct effect on the entitlements one can expect to receive after retirement.

This thesis will proceed by first giving an overview of the Dutch pension landscape and describe in more detail the changes to the retirement age, along with a description of our data and assumptions in Chapter 2. In Chapter 3, we look at a detailed description of the stochastic mortality model we use for our purpose. We analyze the results of our forecasts in Chapter 4 and we look at differences in remaining life expectancy after retirement for groups with differ-ent education levels. Finally, in Chapter 5 we draw our conclusion based on the results in the preceding chapter.

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Chapter 2 Data, assumptions and the Dutch pension

landscape

2.1 Dutch pension landscape overview

The current Dutch pension system can be desrcibed as a four-pillar system. The first pillar is the state pension as managed by the Dutch government. As a form of social security, this pension provides a basic income that covers the minimum standard of living for retirees. Between the age of 15 and 65 all Dutch residents, whether they are working or just residing legally in the Netherlands accrue annual state pension benefits. Over a period of 50 years, full benefits are accrued, and for an individual these benefits are equal to 70% of the net minimum wage, as determined by the government. The AOW-benefits received by retirees are financed directly by the working population through taxation. Should it happen that the AOW-premiums received are insufficient to pay all outstanding AOW-beneftis, then the Dutch government guarantees the remainder part. Since its inception, the AOW assumed a fixed retirement age at 65, but the pressure on entitlement payments due to increasing life expectancy and declining birth rates has prompted the government to increase the retirement by a predetermined schedule from 2013 until 2023, then from 2024 have it increase dependent on further increases in the period life expectancy from the original retirement age (at 65 years old).

The second pillar applies to workers in paid employment and basically covers all occu-pational pension schemes. These are accrued through the employer, usually on a compulsory basis. These occupational pensions are usually paid on an annual basis by the employer and managed by a pensions fund. The idea is that the premium payments over many years are in-vested and that the combination of capital inin-vested and capital gains is enough to provide the employee with an income after retirement that has the same purchasing power as his income when employed and that this income can be paid until death.

The third pillar covers the additional possibility to have a private pension. This is mainly aimed at self-employed workers who wish to arrange for pension payments in addition to AOW-payments, although in general an employee already accrueing pension rights through his em-ployer is also allowed to accrue additional pension rights in the third pillar and the premiums paid are also exempt for taxation, provided that these do not exceed fiscal boundaries. The most common way one can receive third pillar-benefits is by purchasing a life annuity.

The fourth and final pillar is not subject to changes in pension law nor is it of interest for this thesis, but for completeness we mention it regardless. Basically this pillar comprises all of an individuals private assets. Think of real estate (onwing a house), investments and savings, which can be used to finance life after retirement. These are generally not tax exempt.

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In 2012, the Dutch government proposed a law (Ministerie van Sociale Zaken en Werkgele-genheid,2012) that would provide in a means of increasing AOW-age and pension-age, as well as limit the accrual of pension rights in the second and third pillar in an effort to decrease the fi-nancial load of an ageing society and keep pensions affordable for both the working population and with a view on goverment expenditures.

According to the latest schedule as proposed in updated shape in Ministerie van Sociale Zaken en Werkgelegenheid(2014) and approved by the Dutch Senate as of November 2017, the AOW-age will increase by three months each year in the period 2016-2018, and by 4 months in the period 2019-2021. This means the AOW-age will be 66 years by 2018 and 67 years by 2021. Starting from 2024 onwards, a further increase in the AOW-age depends on forecasts of a 65-year old’s remaining period life expectancy. An individual nearing the AOW-age will have to take the following into account:

Increase in the AOW-age as per November 2017

AOW-age Born during Year

65 years ← - 31-12-1947 2012 65 years + 1 m 01-01-1948 - 30-11-1948 2013 65 years + 2 m 01-12-1948 - 31-10-1949 2014 65 years + 3 m 01-11-1949 - 30-09-1950 2015 65 years + 6 m 01-10-1950 - 30-06-1951 2016 65 years + 9 m 01-07-1951 - 31-03-1952 2017 66 years 01-04-1952 - 31-12-1952 2018 66 years + 4 m 01-01-1953 - 31-08-1953 2019 66 years + 8 m 01-09-1953 - 30-04-1954 2020 67 years 01-05-1954 - 31-12-1954 2021 67 years + 3 m 01-01-1955 - 30-09-1955 2022 67 years + 3 m 01-10-1955 - 30-09-1956 2023 – – –

Table 2.1: Current schedule of the AOW-age, with the period in which individuals who this increase applies to were born and the year in which this increase will take place.

Further increases in the AOW-age starting from 2024 are to be estimated using a stochastic mortality model and will be based on the best estimate of the remaining life expectancy fore-casts. Whether or not an increase in the AOW-age is deemed justified, depends on the result of the formulas used to determine whether an increase is necessary. The increase in retirement age can simply be given as Pt= Pt−1+Vt, with the formulas for the incremental increases Vt (based

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V AOW_t = ( 0.25, if (Lt− 18.26) − (Pt−1− 65) ≥ 0.25 0, otherwise V_tpension= ( 1, if (Lt+10− 18.26) − (Pt−1− 65) ≥ 1 0, otherwise

respectively for the AOW- and pension-age. Here t stands for the year of interest where we look at whether an increase in AOW- or pension-age is necessary or not, Vt stands for the

incremental step with which AOW-/pension-age is increased, measured in periods of years and L_t is the best estimate of a 65-year old’s remaining life expectancy in calendar year t, based on period life expectancy. This estimate will be determined by the CBS at t − 5 for the AOW age. Pt−1 is the previous AOW- or pension-age. Finally, the ’18.26’ in the formula is the

gender-neutral, macro average remaining life expectancy of a 65-year old as observed in reference period 2000-2009.

The intended purpose of the formula would seem to be to keep remaining life expectancy after retirement the same as during the reference period 2000-2009, while continuously account-ing for an ’original retirement age’ of 65, regardless of a (much) higher retirement age in the future. One could argue that by 2030, for example, it will not make much sense to refer back to an estimate of remaining life expectancy made two decades in the past corresponding to a much lower retirement age. Instead, it might have been better to have the formulas only refer to the previous retirement age and forecasted remaining life expectancy.

As a minimum increase in V AOW_t in one year that prompts an increase in the AOW-age, a value of 0.25 (or three months) is maintained. For any increase below 0.25 (or possibly even a decrease), the AOW-age in that year remains unchanged.

Similarly for the pension-age, for any increase in V_tpension in a given year greater than or equal to 1, the pension-age increases with 1 year, while any increase below 1 or any decrease leaves the pension-age unchanged in that year.

Again, these linear transformations are made based on the assumption that we should always refer back to a retirement age at 65 with the accompanying remaining life expectancy, while in the future it would be more sensible to only refer back to the retirement age and remaining life expectancy in the previous year.

The incremental increases with three months for the AOW-age and one year for the pension-age described above will inevitably cause a mismatch between AOW and pension and the payout of entitlements. While everyone would agree this is undesirable, there is not really a straight-forward solution: Having the pension-age increase in steps of three months would undoubtedly result in large amounts of additional administrative work and costs associated with that, which will ultimately be passed on to the individuals participating in the pension plan. On the other hand, an increase of the AOW-age in stepts of one year would disadvantage individuals very close to retirement to a great extent, since there is usually already a set date for retirement. In-creasing the AOW-age by a year will in that case result in such an individual having to bridge a 1-year financial gap with no income and no, or insufficient, pension entitlements.

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2.2 Data

We study the mortality in the Netherlands. For this, we first work out some of the notation we will be using later on. If we let x be the age of a person and t is the year that this person is x years old, then we need the exposures-to-risk and the number of deaths, denoted asDx,t andEx,t, to

construct the mortality model.Dx,t then gives the observed number of deaths for individuals of

the age [x, x + 1) during time interval [t,t + 1). In other words, these are the people who do not live until year t + 1 and die at age x in year t without reaching age x + 1. And in a similar way, Ex,t is the observed number of individuals in the population aged [x, x + 1) exposed to the risk of

death during interval [t,t + 1), based on annual population estimates. All estimations, forecasts and further computations in this thesis are based on Dutch data of death rates and exposure-to-risk, as made available by the Royal Dutch Actuarial Association (Koninklijk Actuariaal Genootschap), see Koninklijk Actuarieel Genootschap (2016a). This dataset contains 1-year age- and calendar year-observations with ages x ∈X := {0,1,...,90} and years t ∈ T := {1970, 1971, . . . , 2015}.

2.3 Assumptions

Here we describe the notation used and the assumptions made for further derivation of formulas, formulation of models and estimation of parameters and variables.

2.3.1 Estimation for total population

It is common practice to make separate estimations for males and females when using stochastic mortality models, but AOW- and pension ages are ultimately based on estimations for the total population, since discrimination based on gender is not allowed (even though gender has signif-icant effect on the remaining life expectancy). Taking this into consideration, we only focus on modeling mortality for the total population, with exception of some visualization of mortality rates for both sexes.

2.3.2 Survival probabilities and hazard rates

First we let the 1-year probability of death qx,t be the probability that an individual aged x is

alive at the first day of year t, with t an integer, but dies somewhere in the interval (t,t + 1). Taking this one step further, we definekqx,t, with k ≥ 0, as being the probability of death for

an individual aged x at time t in the interval [t,t + k). From this it immediately follows that the survival probability of the same individual in the same interval can be defined as_kp_x,t:= 1−_kq_x,t and kqx,t is assumed to be a differentiable function of k. With these functions it is possible to

express death in the next moment of time conditional on survival up to age x, by a conditional density function. We call this the force of mortality µx,t, with t real-valued, which is defined as:

µx,t := 1 kpx,t ∂ qx,t ∂ k = − 1 kpx,t ∂ px,t ∂ k , (2.1)

and it is well-known that this can be written as:

kpx,t = exp − Z k 0 µx+u,t+udu . (2.2)

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2.3.3 Piece-wise constant hazard rates

To ultimately be able to do calculations, we need approximations of the 1-year mortality rate qx,t. If we assume µx+k,t+k = µx,t for k ∈ [0, 1) (i.e. µx,t is a piece-wise constant hazard rate),

we may write: qx,t = 1 − px,t = 1 − exp − Z 1 0 µx+u,t+udu = 1 − exp(−µx,t). (2.3)

2.3.4 Remaining life expectancy

We define remaining life expectancy of a person aged x at time t as the average number of years that he or she can expect to live beyond time t. This we calculate in two different ways, the first being based on the assumption that the 1-year mortality rate does not change over time, notated as qx,t = qx,t+s, for all s > 0. This would give us the complete remaining period life expectancy,

given as: eperiod_x,t := 120−x

∑

k=0 k

∏

`=0 (1 − qx+`,t) +1 2. (2.4)

To get to this equation, we assume that ex is the present value of a life annuity that pays 1 per

year, beginning at time t = 1, with constant interest rate 0 and Uniform Distribution of Deaths (UDD). The 1₂is there to account for the fact that an individual most likely lives through part of his year of death, which under UDD would mean half a year on average. For more detail, one can refer toDavid Promislow(2006).

The other way in which we calculate the remaining life expectancy, is by taking into account all developments in mortality rates, both positive and negative, in the future:

ecohort_x,t := 120−x

∑

k=0 k

∏

`=0 (1 − qx+`,t+`) +1 2, (2.5)

which is defined as the complete remaining cohort life expectancy, with the mortality rate in year t+` for an individual aged x + ` denoted as qx+`,t+`. Although remaining life expectancy

in the past was commonly calculated based on periods (likely due to insufficient data and/or computational limitations), it makes much more sense to use the cohort-method, given that an individual experiences mortality rates over consecutive years, which one can visualize as moving diagonally through a table of mortality years for given ages. In this thesis, we will use both methods for further calculations.

2.3.5 Life tables for the higher ages

The dataset used in this thesis only considers the ages X := {0,...,90}, but considering the fact that longevity is the red thread running through this thesis, the higher ages certainly de-serve attention and for estimating and forecasting cohort life expectancy, including these higher ages even becomes a necessity. The agesXold:= {91, . . . , 120} are likely excluded because the

corresponding mortality rates are very volatile, caused by a low number of observations and small exposure-to risk. To close the life table, we can use the logistic mortality law ofKannistö

(1992). Life tables as established by the Koninklijk Actuarieel Genootschap (2016a) use this method for closing the lifetables, as well. Kannistö’s method works as follows.

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The force of mortality µx,t for x ∈Xold is modeled as:

µx,t =

φ1,t· eφ2,t·x

1 + φ1,t· eφ2,t·x

, (2.6)

where these parameters φ1,t and φ2,t can be estimated for any t by first transforming the

mortality rates into hazard rates, under the assumption that within the age-time interval, these hazard rates are constant:

µx,t = − ln(1 − qx,t). (2.7)

This yields from (2.5) that:

logit(µx,t) = ln( µx,t 1 − µx,t ) = ln(φ1,t) + φ2,t· x. (2.8)

The parameters φ1,t and φ1,t are then estimated by applying Ordinary Least Squares (OLS)

to the agesXf it := {80, . . . , 90}: X := 1 1 1 . . . 1 80 81 82 . . . 90 0 , φt := ln(φ1,t) φ2,t , (2.9)

and the OLS estimate ˆφt is then found by matrix multiplication:

ˆ

φt := (X0X)−1X0Yt, (2.10)

where Yt= {logit(µx,t)}x∈Xf it. The estimates ˆφ1,t and ˆφ2,t that are found this way, enable the

extrapolation of the mortality rates for higher ages:

qold_x,t := 1 − e − φ1,t ·eˆ ˆ φ2,t ·x 1+ ˆ_{φ1,t ·e}φ2,t ·xˆ ! , ∀x ∈Xold. (2.11)

Ultimately, we can use the force of mortality found from the above equation to determine the mortality rates for each year t:

qx,t = 1 − px,t = 1 − exp − Z 1 0 µx+u,t+udu = 1 − exp(−µx,t)). (2.12)

2.3.6 Best estimate and confidence interval

In this thesis, as inKoninklijk Actuarieel Genootschap(2016c), we define the best estimate as the most likely outcome under uncertainty in the projections of our stochastic mortality model. For qx,t (our mortality probabilities), we obtain this best estimate by setting the error terms in

the stochastic time series equal to zero. The remaining life expectancy calculated from the best estimate of the median realisations of these qx,t then becomes a best estimate of the remaining

life expectancy of its own.

The above-mentioned best estimate follows from a number of simulations N of the mortality projections, with realizations X1:N, . . . , XN:N. For a certain confidence level α, we define the

confidence level as [X_bα

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of the simulations that lie within the lower- and upper bound for the (α

2)-th percentile and

(1 −α

2)-th percentile, respectively. These estimates for the upper and lower bounds become

more accurate as the number of realizations N is increased.

2.4 Mortality rates

The maximum likelihood estimator of the mortality rates, as defined previously, is equal to the number of deaths at age x in year t denominated by the corresponding exposure-to-risk. This follows from the assumption that the number of deaths Dx,t has a Poi(Ex,t· µx,t)-distribution,

with its mean proportional to the population size, as inBrillinger(1986). The likelihood function Lin this case looks like:

L :=

_∏

x∈Xt∈T

∏

(Ex,t· µx,t)Dx,t· e−Ex,t·µx,t Dx,t! . (2.13)

The log-likelihood is then derived from this as:

ln(L) =

_∑

x∈Xt∈T

∑

[Dx,t· ln(Ex,t· µx,t) −Ex,t· µx,t− ln(Dx,t!)] , (2.14)

and from this, we maximize this log-likelihood by taking the partial derivative with respect to µx,t: ∂ ln(L) ∂ µx,t = Dx,t ˆ µx,t −Ex,t = 0. (2.15)

When we rewrite this we end up with the equation introduced at the beginning of this sub-section, which is the estimator of the hazard rate ˆµx,t, for which the log-likelihood is maximized:

ˆ

µx,t = Dx,t

Ex,t

:= mx,t. (2.16)

This ˆµx,t stands for the number of people that died during an (x,t)-interval, divided by the

number of people that could have died during the same interval, and this estimator is known as the central death rate which is denoted, as in the above equation, as mx,t. This ratio allows for

visualizing using the available datasets. Figure2.1shows the pattern of logarithmic death rates for ages 0 to 90 for both males, females and the total population, respectively:

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Figure 2.1: Logarithm of death rates (ln(mx,t)) for Dutch males, females and total population for ages 0 to 90 and

years 1970, 1985, 2000 and 2015.

Notice the rates declining for children as they age, with the ’accident bump’ around age 20, signifying an increase in risk-seeking behaviour as adolescents turn into adults and becoming independent. A quick glance at the male-, female- and total rates suffices to see that this bump is also much more pronounced for the male rates.

If we take a look at the log central death rates relative to time, we get different insights, as displayed in Figure2.2:

Figure 2.2: Estimates of the log force of mortality (ln(mx,t)) for Dutch males, females and total population for the

period 1970 to 2015 and ages 0, 25, 65, and 85.

This figure shows that mortality in the Dutch population is falling rapidly for the younger ages, but not all that much for the highest ages. One thing to note is that log death rates for in-fants were quite a bit higher than those of 25-year olds in the recent past, and the differences for the total population are actually quite small at present. Comparing this for males and females, we are again reminded of the accident hump (i.e. more reckless behaviour in early adulthood in men) as we see log death rates for 25-year old women consistently below those of infants, while

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Figure 2.3: Estimates of the log force of mortality for ages 0 to 90 and years 1970 to 2015 for the total Dutch population

for males, infant mortality appears to be decreasing faster than the mortality of 25-year olds. Finally, we can create a three-dimensional plot for the unisex log death rates, plotting them against both time and age, resulting in Figure2.3above.

This figure shows consistenly higher log central death rates for the earlier years in the data set, as well as a more pronounced accident bump. Later years show a more smooth progression as the age increases and lower values for the log central death rates, supporting the accepted and obvious view that going into the future, longevity is increasing.

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Chapter 3 Stochastic Mortality Model

This chapter briefly discusses the Lee-Carter model itself as well as the time-series model which we need later on for forecasting.

3.1 The Lee-Carter Model

Human mortality modeling is done using the Lee-Carter model, defined as:

ln(mx,t) := αx+ βxκt+ εx,t, (3.1)

with mx,t defined previously in equation (2.16). Here αx and βx are age-dependent parameters

which do not change over time, while κt is a time-varying parameter which is invariant of the

age x and εx,t is a zero-mean error term. More specifically:

• we impose the normalization restrictions that ∑

x∈X

ˆ

βx= 1 and ∑

t∈T κˆt = 0;

• αxis the average of the logarithm of the central death rates (ln(mx,t)) over time;

• βx is the rate of change of the central death rates for age x given a change in the

time-varying parameter κt;

• κt is defined as the time-varying index of the level of mortality.

This way, the model separates age- and period effects. The error term is assumed to be an independent identically distributed Gaussian variable for all pairs (x,t). To fit the model, there are three fitting procedures that are most commonly associated with the Lee-Carter model. The first option is through singular value composition, where one would have a least squares criterion minimizing the sum of squared residuals as below:

SSR:= min αx,βx,κt_x∈

∑

_X_t∈

∑

_T ln D x,t Ex,t − αx− βx· κt 2 . (3.2)

The problem with this approach is that in the event that the number of deaths Dx,t or

exposure-to-riskEx,t equals zero, it results in an error. Sticking with singular value

decomposi-tion, this can only be avoided by excluding the conflicting age(s) x or year(s) t, which becomes problematic when we want to use the full dataset.

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A second method of fitting then, is to use weighted least squares, where the Dx,t act as

weights of the observations:

W SSR:= min αx,βx,κt_x∈

∑

_X_t∈

∑

_T Dx,t· ln D x,t Ex,t − αx− βx· κt 2 (3.3) Although this method avoids the problem of errors caused by taking the logarithm of zero, both these methods share the drawback that errors are assumed to be normally distributed and homoskedastic. Considering that the smaller (absolute) death count at the higher ages makes the logarithm of the central death rates much more variable for these ages, these seem like unrealistic assumptions. The third option available for the fitting procedure, which does not suffer from the problems mentioned above, is maximum likelihood estimation (MLE), which comes down to maximizing the likelihood of a Poisson distribution, explained in much greater detail in Brouhns et al. (2002), among others. If we let Dx,t ∼ Pois(Ex,t· µx,t), and take the

hazard rates to be µx,t := eαx+βxκt, this makes it so that the natural logarithm of the hazard rates

is now specified like the Lee-Carter model. We can now define the likelihood function L:

L=

_∏

x∈Xt∈

∏

T Dx,t· (Ex,t· eαx+βxκt)Dx,t Dx,t! · e−Ex,t·eαx+βxκt ! , with log-likelihood: ln(L) =

_∑

x∈Xt∈T

∑

Dx,t · (ln(Ex,t) + αx+ βxκt) −Ex,t· eαx+βxκt− ln(Dx,t!) =

_∑

x∈Xt∈T

∑

D x,t· (αx+ βxκt) −Ex,t· eαx+βxκt +C,

with C a constant. To maximize the log-likelihood, we start by partially differentiating with respect to αx, βxand κt: ∂ ln(L) ∂ αx =

_∑

t∈T D x,t−Ex,t· eαx+βxκt = 0, ∀x ∈X . ∂ ln(L) ∂ βx =

_∑

t∈T D x,t· κt−Ex,t· κt· eαx+βxκt = 0, ∀x ∈X . ∂ ln(L) ∂ κt =

_∑

x∈X D x,t· βx−Ex,t· βx· eαx+βxκt = 0, ∀t ∈T .

This system of equations cannot be solved analytically, but it is possible to find an ap-proximation through an iterative procedure using the Newton-Raphson method. This procedure for estimating log-linear models with bilinear terms was first suggested by Goodman(1979), and later used inBrouhns et al.(2002) for estimating the parameters of the Lee-Carter model through MLE, instead of Singular Value Decomposition as inLee and Carter(1992). This pro-cedure involves updating a single set of parameters in an iteration step ν + 1, while keeping the other parameters fixed at their present estimates according to the updating scheme below:

ˆ

ϑ(ν+1)= ˆϑ(v)− ∂ L

ν_{/∂ ϑ}

∂2Lν/∂ ϑ2, (3.4)

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Applying this to the system of equations we found earlier, we derive: ˆ αx,n+1 := αˆx,n− ∑ t∈T Dx,t−Ex,t· e ˆ αx,n+ ˆβx,nκˆt,n − ∑ t∈T Ex,t· e ˆ αx,n+ ˆβx,nκˆt,n , ∀x ∈X . ˆ βx,n+1 := βˆx,n− ∑ t∈T κˆt,n+1·Dx,t−Ex,t· e ˆ αx,n+1+ ˆβx,nκˆt,n+1 − ∑ t∈T κˆ 2 t,n+1·Ex,t· eαˆx,n+1+ ˆβx,nκˆt,n+1 , ∀x ∈X . ˆ κx,n+1 := κˆx,n− ∑ t∈T ˆ βx,n·Dx,t−Ex,t· eαˆx,n+1+ ˆβx,nκˆt,n − ∑ x∈X ˆ β_x,n2 ·Ex,t· eαˆx,n+1+ ˆβx,nκˆt,n , ∀t ∈T .

As starting values we take S0:= (αx,0 = 0, βx,0 = 1, κt,0 = 0), and following through the

iterative procedure, the parameters can be solved numerically by setting a remainder term and continuining the procedure of computing and adjusting the parameters until successive compu-tations result in a change that is no larger than the specified remainder term.

3.2 Forecasting

Once fitted with historical data, this model yields a set of parameter estimatesX = ( ˆαx, ˆβx, ˆκt).

The time factor ˆκt is a stochastic process for all t after today. We need an ARIMA time series

model to forecast this time factor, and for this we proceed in the same manner as Lee and Carter did for the U.S. population, where they used a random walk with drift model (ARIMA(0,1,0)), the details of which can be found inBox and Jenkins(1970). We can define this ARIMA-model as:

κt+1:= κt+ θ + σ · εt, (3.5)

where θ denotes the drift term and εt is a Gaussian white noise with zero mean, with σ the

standard deviation of this white noise, meaning that ε has a constant variance σ2ε . To get unbiased estimates ˆθ and ˆσ , we consider the following:

∆κt := κt−1− κt = θ + σ · εt.

E[∆κt] := E[θ + σ · εt] = θ + σ · E[εˆ t] = θ .ˆ Var[∆κt] := Var[θ + σ · εt] = σˆ2· E[εt] = σˆ2.

(3.6)

The estimates ˆθ and ˆσ that we obtain from this along with the last value of ˆκt generate

mortality-level forecasts, which are signified as ˜κt. If we assume αx and βx to remain constant

over time, these ˜κt-forecasts enable us to make forefasts of future mortality rates (which, for

our model, start from the year 2015): ˆ

µx,t = mx,t = eαˆx+ ˆβxκˆt, for t≥ 2015,

ˆ

q_x,t = 1 − e− ˆµx,t ₌ _{1 − e}−eαx+ ˆˆ βx ˆκt_, _for _t_{≥ 2015.}

(3.7) Lee and Carter went with this time series model because of its relative simplicity and good

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fit, but as mentioned, they concerned themselves with forecasting ˆκt for the U.S. population,

which means it might not necessarily be the best choice for the Dutch population. Regardless, the ease of using ARIMA(0,1,0) compared to more complex time series models is too good to pass up and as such, forecasting of ˆκt for the Dutch data will be based on this time series model.

It should be noted that forecasts based on the Lee-Carter method are nothing more than extrapolations of historical trends. The model does not take into account any improvements or deteriorations in mortality due to external factors, such as medical innovations, disasters, war, socio-economic changes.

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Chapter 4 Results and Analysis

The previous section covered the different components of the stochastic mortality model used. In this section we will look at and discuss the parameters as estimated and fitted using the mortality model. Using this, we find estimations of the remaining life expectations and make a distinction for groups of different education leve/income. Forecasts are done starting frome the year 2015 (as this is the last year for which data is available). Programming and estimations are mainly done in R, while Excel is used for the remaining part.

4.1 Parameter estimates

We fit the Lee-Carter model with historical mortality data, which we obtain from the datasets provided by Koninklijk Actuarieel Genootschap(2016a). The data of importance are DNL_x,t and E_x,tNL, which are the observed number of deaths in the Netherlands for each calendar year and the exposures-to-risk for the Netherlands in each calendar year during the period 1970-2015. One of the assumptions underlying the formulas for AOW- and pension-age increases, is that the forecasts are to be gender-neutral, thus we only consider the data for the total population, which is limited to ages x = 0, 1, . . . , 90. All the ages above need to be extrapolated using the method by Kannistö as explained in subsection2.3.5.

It was found earlier that the maximum likelihood estimation method was best suited for our fitting procedure with Lee-Carter and the estimated parameters ( ˆαx, ˆβx, ˆκt) are shown below:

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Figure 4.1: Estimated parameters of the Lee-Carter model

Here ˆαxis the effect of age x, or the average of the log central death rates mx,tover time. The

figure for ˆαxclearly shows how mortality decreases rapidly from infancy to (early) adolescence,

with the accident bump somewhere around the age of 20 and then increasing with age after that. The estimates ˆκt stand for the effect of calendar year t, capturing the time-variation in the

level of mortality. The central death rates change for all ages in accordance with the time-variation of κt decreasing linearly over time and they do so at a constant exponential rate, as

can be seen in the figure. The values of ˆκt are decreasing approximately linearly for the total

population.

Finally, ˆβx is the effect of a specific age responding with a particular calendar year, or how

m_x,t decreases for age x as a result of a change in ˆκt. From the figure we see that the values of ˆβx

are decreasing and positive, with higher values at the low ages and a hump near the retirement age, suggesting that developments in the central death rates are strongest for both lowest highest ages. We cross-reference the values of our parameter estimates with the best estimates provided inKoninklijk Actuarieel Genootschap(2016b), to make sure our estimations are correct.

The mortality-level index ˆκt is a stochastic process, for t into the future. As described in the

previous chapter, we use an appopriate ARIMA time series model to forecast this time factor ˆ

κt using a random walk with drift (RWD) model in the same way Lee and Carter did for the

U.S. population. This model is not necessarily the most appropriate choice for the Netherlands, as mentioned before, but for the sake of parsimony we will use it regardless. After computa-tion, this results in the RWD-model parameter estimates ( ˆθ , ˆσ ) for the total Dutch population as shown in the following table.

Drift Term Standard error

Population θˆ σˆ

Total -1.905928 2.368908

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4.2 Forecasting the remaining life expectancy

Reiterating our assumptions that remaining life expectancy is estimated based on both period-and cohort methods period-and that the best estimate is the most likely outcome under uncertainty in the projections of our stochastic mortality model, combined with Kannistö’s method for closing the life tables, we are able to produce projections of the remaining life expectancy for 0-year old, 25-year old, 65-year old and 85-year old individuals based on 10,000 simulations of mortality rates projections, which are then transformed into life expectancies. These projections are shown for period life expectancy in the following figure:

Remaining life expectancy of a 0-year old Remaining life expectancy of a 25-year old

Remaining life expectancy of a 65-year old Remaining life expectancy of an 85-year old

Figure 4.2: Projections of the remaining period life expectancies for the total (unisex) Dutch population based on 10,000 simulations

What becomes visible from these projections is that remaining life expectancy for all ages is increasing as we go into the future, and that the rate at which remaining life expectancy increases, is decelerating. The projected life expectancy increasing, results directly from the decreasing projected mortality rates, as per our assumptions about the stochastic framework of the used model. While the (admittedly poor) choice of units on the y-axis makes it so that it’s not directly clear, the figures suggest that improvements in life expectancy are on the one hand larger for the lower ages compared to the higher ages, while on the other hand the higher ages have considerably less systematic uncertainty in contrast with these lower ages, as expressed by the confidence intervals.

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For the cohort remaining life expectancy we move through the life table diagonally, taking the 1-year mortality rates of the years in which a certain age is attained, so it is necessary to make sure that for age x, there are just as many calendar years t (starting from 1970) as there are years after x, or (x − 120), since we take x = 120 to be the maximum attainable age. This approach is more realistic, but due to the increased amount of estimation and forecasting involved, overall there is more uncertainty, as well. Due to this, the confidence intervals appear earlier on compared to the period life expectancy, because they are partially forecasted even for calendar years for which estimates of the remaining life expectancy are available. This we visualize in the following figure:

Remaining life expectancy of a 0-year old Remaining life expectancy of a 25-year old

Remaining life expectancy of a 65-year old Remaining life expectancy of an 85-year old

Figure 4.3: Projections of the remaining cohort life expectancies for the total (unisex) Dutch population based on 10,000 simulations

From these figures, we can see that incorporating the effects of future improvements/deteri-orations in mortality into these projections results in a much more smooth increase.

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higher than for period life expectancy. Considering the fact that the cohort method takes future effects on mortality rates into account while the period method does not, leads to the conjecture that projections of the period life expectancy underestimate longevity.

A final thing we observe is that while there is more uncertainty overall with cohort life ex-pectancy, the confidence intervals for the later years are actually more narrow than with period life expectancy. This is due to the fact that the dynamics in later years as described before are not taken into account with the latter method. It also needs to be noted that the confidence intervals we find with both the period-based and cohort-based methods account for only the system-atic uncertaintyinherent to the stochastic model used, and the random errors in the stochastic process forecasts make up this systematic uncertainty. Anlayzing the accuracy of retirement age-projections is explored in much greater detail in for example van Westen (2016), where parameter uncertaintyis also taken into account. Uncertainty in the projections is however not the focus of this thesis, so we will not go into the additional bootstrapping techniques required to estimate this parameter uncertainty.

4.3 Forecasting the AOW-age

In the previous section, we looked at projections of the remaining life expectancy for both period- and cohort life expectancy for different ages. In this section, we focus on the life ex-pectancies of 65-year old individuals over time and we use these forecasts as the best estimates (Lt in the formula first introduced in Chapter 2:

V AOW_t = (

0.25, if (Lt− 18.26) − (Pt−1− 65) ≥ 0.25

0, otherwise

As described in Chapter 2, we only consider increases in remaining life expectancy after retirement of greater than or equal to 0.25, or three months. Applying this formula to the re-maining life expectancy projections as described in the previous section, results in the following figures:

Figure 4.4: AOW-age forecast for period life expectancy, with the Best Estimate equal to the median

of pension-ages following from the remaining life expectancy forecasts in Figure4.2.

Figure 4.5: AOW-age forecast for period life expectancy, with the Best Estimate equal to the median

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the future. There is a clear difference in the AOW-age projections for period- and cohort life ex-pectancy. This should come as no surprise, since we can see from the projections in Figure4.2 and Figure 4.3 that remaining life expectancy forecasts are consistently higher for the cohort-based model. As a result of the constraints imposed on the AOW-age increase, a clear staircase pattern emerges. We see that the maximum and minimum values for the AOW-age increase ap-pear from the year 2024 onwards. This is because the increase schedule up to the year 2023 is fixed (see Chapter 2).

Below we compare the best estimates of the increase in AOW-age as forecasted using the Lee-Carter model and period life expectancy with the predictions made by the CBS. From the table we can see that both approaches yield fairly similar results at the start, but the difference increases more and more over time, which finally results in a difference of 1.25 years in the best estimate of the AOW-age for the two different models. It is safe to assume that the Lee-Carter model, being much simpler than the models used by the CBS, underestimates increases in longevity and the associated AOW-age increases.

Model Best Estimate AOW-age per year (period)

2022 2025 2030 2035 2040 2045 2050 2055 2060

CBS (2014-2060) 67.25 67.50 68.00 68.50 69.25 69.75 70.25 71.00 71.50 Lee-Carter 67.25 67.25 67.75 68.25 68.50 69.00 69.50 69.75 70.25

Table 4.2: Best estimates of the AOW-age as published by the CBS and as forecasted with Lee-Carter (Period).

CBS Statline(2017d).

When we look at the difference between a cohort-based Lee-Carter forecast and the CBS forecast, the values of the different models turn out to be much closer. In fact, the Lee-Carter model with cohort life expectancy results in significantly higher values in the short- to midterm forecast compared to the CBS forecast. In the final years of the forecast however, it still under-estimates the AOW-age increase relative to the CBS forecast, although with the cohort-based method the difference is only 0.25 years.

Model Best Estimate AOW-age per year (cohort)

2022 2025 2030 2035 2040 2045 2050 2055 2060

CBS (2014-2060) 67.25 67.50 68.00 68.50 69.25 69.75 70.25 71.00 71.50 Lee-Carter 67.25 67.75 68.75 69.25 69.75 70.00 70.50 70.75 71.25

Table 4.3: Best estimates of the AOW-age as published by the CBS and as forecasted with Lee-Carter (Cohort).

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4.4 Forecasting the pension-age

Similar to the AOW-age, the pension-age increases based on the following formula first intro-duced in Chapter 2:

V_tpension= (

1, if (Lt+10− 18.26) − (Pt−1− 65) ≥ 1

0, otherwise

For the pension-age, we only consider increases V_tpension greater than or equal to 1 year. Any year-to-year increase in Vt less than 1 will leave the pension-age unchanged for that year.

Furthermore, the best estimate of the remaining life expectancy is taken from a calendar year that is 10 years later compared to the year of a possible pension-age increase. Because of this, AOW-age and pension-age will be different. Below are the projections of the pension-age using both period- and cohort life expectancy for the total Dutch population:

Figure 4.6: Pension-age forecast for period life expectancy, with the Best Estimate equal to the median

Figure 4.7: Pension-age forecast for period life expectancy, with the Best Estimate equal to the median

As with the AOW-age projections, we see the pension-age projections following a stair-case pattern as they are steadily increasing into the future. The same goes for the minimum and maximum values of the pension-age projections, as visualized by the blue squares. When comparing period- and cohort-based projections, we see that the latter gives higher values. This should not be surprising considering the different methods for calculating period- and cohort life expectancy. Because the cohort-method accounts for mortality dynamics in later years, the projection increases stronger yet more smooth (see Figure4.3), while the period-method only accounts for mortality rates in the year the forecasts are made, resulting in a projections that is less strongly increasing. Another observation we make is that the increase seems to be de-celerating: the horizontal levels between pension-age increases look like they get longer going into the future, the tables below also seem to suggest this (particularly the table for period life expectancy).

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Model Best Estimate pension-age per year (period)

2015 2020 2025 2030 2035 2040 2045 2050 2055 2060

CBS (2014-2060) 67 68 68 69 69 70 71 71 ... ...

Lee-Carter 67 67 68 68 69 69 69 70 70 70

Table 4.4: Best estimates of the AOW-age as published by the CBS and as forecasted with Lee-Carter (Period).

Model Best Estimate pension-age per year (cohort)

2015 2020 2025 2030 2035 2040 2045 2050 2055 2060

CBS (2014-2060) 67 68 68 69 69 70 71 71 ... ...

Lee-Carter 68 68 69 69 70 70 70 71 71 72

Table 4.5: Best estimates of the AOW-age as published by the CBS and as forecasted with Lee-Carter (Cohort).

When we compare AOW-age increase and pension-age increase, we observe that the projec-tions of the the pension-age are higher than the AOW-age projecprojec-tions overall, which can in part be attributed to the fact that in the case of age projections, the formula for the pension-age increase takes the best estimate from a calendar year 10 years into the future, as opposed to the one for the AOW-age increase which takes the best estimate at the current time.

In the above tables, we also compare the best estimates of the pension-age as estimated by the Lee-Carter based forecasts and using the remaining life expectancy-forecasts as published in (CBS Statline, 2017d). As with the AOW-age earlier, the Lee-Carter model with period life expectancy underestimates the increase in pension-age compared to the CBS-model, while Lee-Carter with cohort life expectancy is much closer to the CBS forecasts.

The missing values for CBS in the years x = 2055 and x = 2060 are due to the fact that the best estimate of the remaining life expectancy is taken 10 years into the future, while the CBS only has data available up until the year 2060.

4.4.1 Difference between AOW- and pension-age

When we look at the projections for AOW- and pension-age, we see that the pension-age in-creases faster on average, but since this increase also looks like it is decelerating over time, we can expect to see that the difference between AOW- and pension-age will fluctuate less and less over time. This is visualized in the following figure:

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Figure 4.8: Best-estimate projection of the difference between AOW- and pension-age (Period)

Figure 4.9: Best-estimate projection of the difference between AOW- and pension-age (Cohort)

There is a clear pattern visible, where the difference decreases to 0 over the span of a few years as the AOW-age catches up to the increased pension-age, only for this difference to shoot up from one year to the next as the pension-age increases with 1 year in the new year.

We see that the first spike in difference is considerably higher than the other spikes, both for the period-based and the cohort-based estimation and forecasting method. This is due to the the pension-age increasing by 2 years starting from the year 2014. We see this as a decline from this spike as the AOW-age is also increased according to the increase schedule as provided in Chap-ter 2, then starting from 2024, any further increase in both AOW-age and pension-age becomes dependent on the best estimate of the remaining life expectancy at age 65, and the staircase pat-tern emerges. One observation we make for both these figures is that the difference on average decreases for each time the pattern is repeated: For period- and cohort life expectancy we see that the tops become lower and the decrease more stretched out. Interesting to note is that for both methods the difference also becomes negative in the final years of the forecast, meaning that sometimes the AOW-age will actually be larger than the pension-age.

4.5 Comparing remaining life expectancy and entitlements

for different education levels

In this next section we consider the effect of differences in social standing on the remaining life expectancy at the pension-age and AOW-age after both of these have been increased. Two variables that could possibly be used to divide the population into different social classes would be ’education’ and ’income’. Indeed, over the years multiple studies have been dedicated to researching the effect of the above two variables, for example Stam et al.(2008) looks at ed-ucation level while Knoops and Brakel (2010) focuses on income level. The CBS offers data on life expectancy and education level CBS Statline (2016c) and life expectancy and income level CBS Statline (2016b). Both of these have but a few datapoints that can be used for the specified purpose, with six datapoints for education level and three for income level. For this reason, we make the generalizations that education and income are strongly correlated, that any result stemming from the effect of education also applies to income and as such, that we can only look at the effect of education on remaining life expectancy and based on that generalize for social class (supposing that twice the amount of datapoints is likely to give a more accurate result).

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CBS Statline(2016c) gives the average remaining life expectancy at 65 for 6 periods for both males and females and for different education levels (which can be translated to ’elementary-’, ’lower secondary-’, ’higher secondary-’ and ’higher education’). Furthermore, for the sake of parsimony we only look at the remaining years after age 65 and we do not take into account whether these years are spent ’in good health’ or not. In line with the rest of this thesis, men and women are not considered separately, so the average of the remaining life expectancies of both is taken. Finally, the average life expectancy regardless of education level is taken to be the average of all four education levels. This we can visualize as in Figure4.10:

Figure 4.10: Remaining life expectancy at age 65 for different levels of education, including regression lines.

For the regression lines, we use a simple linear regression of the form yz= c + ρ · z, where z

refers to the time after calendar year 2000, and using Ordinary Least Squares, we estimate that for Elementary, Average and High these are given as:

yElementary_z = 15.278 + 0.1978z yAverage_z = 17.637 + 0.1756z yHigh_z = 19.405 + 0.1699z

As can be seen, the (scarce) data suggests that the effect of different education levels de-creases as time progresses, which seems like a plausible assumption, although the regression lines in this graph also suggest that an individual with only elementary education will eventu-ally outlive the average individual, who in turn will at some point outlive a person with higher education. This seems highly unlikely, but since our forecasts from the previous sections do not reach the years in which these regression lines cross, we can proceed without too much trouble. On further note, the first intuition would be to use something like a logarithmic or polynomial function instead of a linear function for the regression lines (based on the assumption that re-maining life expectancy will level off at some point), but because there is so little data, this results in a very exaggerated progression of the regression line that makes it impossible to work with.

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Based on the above equations, we can find discount functions that transform the remaining life expectancy projections of the (average) total Dutch into 3 different projections for our three groups. when we apply this to both the period- and cohort method, this gives the following vi-sual representations:

(a) (b)

Figure 4.11: Remaining life expectancy projections adjusted for education level (period+cohort)

In these figures we can see that cohort-based remaining life expectancy projections show a much smoother plotline in the years for which we have historical data available (and can make estimations) , since these are partially forecasted, as explained in an earlier section. As for the education-level effects, by discounting the average life expectancy projections, we find that there are serious differences in remaining life expectancy after the age of 65. Like this we end up with a considerably higher life expectation for ’Higher’ which continues on a minimally decelerating increase going into the future. For ’Elementary’, we see the opposite but in a much more pronounced manner, with an accelerated increase of the life expectancy going into the future. In fact, the figure almost seems to suggest that ’Elementary’ will catch up to ’Average’, but as mentioned earlier this is due to the lack of data we are forced to work with.

The remaining life expectanties after age 65 for ’Higher’ and ’Elementary’ are both found to differ by close to at least a year from the ’Average’ at any point in time and between one an-other this difference is roughly 4 years on average at least, according to our projections. That is not even considering that individuals with (much) higher-than-average education are generally healthier, while individuals with lower-than-average education perform worse on this front, as exemplified for Western countries in general inDavey Smith et al.(1998) and more specifically for the Netherlands inBijwaard et al.(2015). This means that if we were to further assume that worse health after retirement correlates with increased costs after retirement, the differences between our three groups would be even more pronounced.

Now we want to get an idea of what the above differences in life expectancy for individuals with different education levels means when we look at it from a financial point of view. We can get a good impression if we consider the life expectancy to be a life annuity that pays unit E every year with a certain amount of interest. Suppose we have a constant yearly interest of i%, with an E(> 0) unit yearly payout. We further assume that payments are made monthly. Now,

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from our forecasts of remaining life expectancies for different starting ages of the retirement we want to arrive at an annuity immediate that satisfies these conditions. If we write the forecast of remaining life expectancy at age x in year t as ˆex,t, we can express this as integer η months

by: ηx,t = 12 · b êx,tc + b{ êx,t} · 12c, where { êx,t} is the remainder term of the preceding floor

function. The present value of the AOW- and pension entitlements of a retired x-year old in year t is then given by:

PVx,t(E) = E 12· aηx,t₁₂i = E 12· ηx,t

∑

k≥1 1 + i 12 −k . (4.1)

Using this formula, we are able to show what the entitlements will be for each calendar year that sees either the AOW-age increasing to a full year or the pension-age increasing by a year. We assume that each year a total payout of E = 1 will be paid out, with payments made in each month, provided that the individual is still alive at the end of each month. This is illustrated for period life expectancy in the following figures:

AOW-age Year of increase Elementary Education Average Higher Education

65 2012 14.60036 16.10627 17.44014 66 2018 14.54115 15.93579 17.22148 67 2021 14.24358 15.53471 16.83533 68 2032 14.42241 15.59229 16.83533 69 2044 14.71851 15.70717 16.89077 70 2057 14.95364 15.70717 16.83533

Table 4.6: Annuities for different education levels and integer AOW-age in the year this age is first reached based on our forecasts (period), with yearly interest rate i = 2% and annuity payout E = 1 divided over 12 yearly

payments.

Pension-age Year of increase Elementary Education Average Higher Education

65 2012 14.60036 16.10627 17.44014 67 2014 13.88324 15.24537 16.55677 68 2022 13.76233 15.01218 16.27591 69 2034 14.06385 15.12897 16.38853 70 2047 14.30329 15.24537 16.38853 71 2061 14.60036 15.30343 16.38853

Table 4.7: Annuities for different education levels and integer pension-age in the year this age is first reached based on our forecasts (period), with interest rate i = 2% and annuity payout E = 1 divided over 12 yearly

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What we see is that entitlements after retirement differ for different education-levels, as one would expect given the remaining life expectations. Another observation we make is that within each class, there is variation in the entitlements one can expect depending on the combination ’year of retirement’ and ’retirement age’. For all three groups it is the case that according to the table, the original AOW-age at 65 years old gave the highest entitlements after retirement.

When we look at the same table of values based on cohort life expectancy, we find a much more pronounced development of AOW- and pension entitlements going into the future. There is be more variation within the different classes themselves, as entitlements after retirement seem to be in decline regardless of education level, and differences among different education levels also differ significantly:

AOW-age Year of increase Elementary Education Average Higher Education

65 2012 15.30343 16.83533 18.24737 66 2018 15.18722 16.61267 17.98051 67 2021 14.83627 16.21945 17.54893 68 2026 14.60036 15.87878 17.16659 69 2033 14.54115 15.64978 16.89077 70 2043 14.60036 15.53471 16.77981 71 2056 14.77744 15.53471 16.66847

Table 4.8: Annuities for different education levels and integer AOW-age in the year this age is first reached based on our forecasts (cohort), with yearly interest rate i = 2% and annuity payout E = 1 divided over 12 yearly

payments.

Pension-age Year of increase Elementary Education Average Higher Education

65 2012 15.30343 16.83533 18.24737 67 2014 14.3629 15.76446 17.11161 68 2015 13.82283 15.24537 16.55677 69 2021 13.70173 15.01218 16.27591 70 2033 13.94354 15.07062 16.27591 71 2046 14.18377 15.07062 16.21945 72 2060 14.42241 15.12897 16.21945

Table 4.9: Annuities for different education levels and integer pension-age in the year this age is first reached based on our forecasts (cohort), with yearly interest rate i = 2% and annuity payout E = 1 divided over 12 yearly

payments.

The above tables shows well the intended purpose of the formulas for calculating AOW-and pension-age increases to keep the period during which pension entitlements are paid out

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more or less at the same levels they were in 2012, before any increase in retirement age. In that regard, this approach performs reasonably well, although the incremental steps of the AOW-age are not considered in the above tables, and we still see that the present value of entitlements after retirement decreases slowly but steadily as we move further away from a retirement age at 65 years old. And yet, from the point of view of the entity paying the entitlements, this method of calculating increases in the retirement age does not seem like the most ideal option. An alternative option would be to not continuously refer back to a retirement age at 65 and a best estimate for remaining life expectancy at age 65 of 18,26. Instead, the entity responsible for paying the entitlements would settle on a desired entitlement period, and would then match remaining life expectancy by forecasting for separate ages (and intervals within the years) and choosing the age that gives the closest fit of remaining life to entitlement period. This requires more computational work but would give greater flexibility to those responsible for paying the entitlements to keep it affordable.

Another point to make based on this is that the differences in entitlements remain signifi-cantly high even in far-future projections, which would mean that individuals who enjoyed less education are likely to receive less entitlements (while generally having worked longer, and this work being more physically demanding). And while state pension entitlements are the same for any individual (conditional on all things being equal), second and third pillar pension en-titlements generally depend on the salary earned before retirement, which further adds to the current system being skewed in favor of the better educated class. This is not even taking into account the fact that the ’higher class’ generally spend a much longer time after retirement ’in good health’ relative to the ’lower class’, while ’bad health’ also implies additional costs. The fact that the current system does not take these elements into consideration can be seen as a serious shortcoming, and a future improvement of the current Dutch pension system would ide-ally start with first addressing these points, along with the earlier mentioned situation where we continuously refer back to a pension age of 65 and its corresponding remaining life expectancy after retirement.

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Chapter 5 Conclusion

At the outset of this thesis we looked at the changes to the Dutch state pension age and re-tirement age for second and third pillar pension rights as a result of changes in Dutch pension law due to perceived longevity risks inherent to pension payments. To cope with this increasing longevity risk, the Dutch government introduced a new pension law in an effort to keep pension costs at sustainable levels. Part of this law is an increase of the AOW-age and pension-age based on improvements in life-expectancy, where remaining life expectancy after retirement roughly equals the remaining life expectancy as estimated from a reference period before any increase in AOW- or pension-age.

To this end, we discussed the stochastic mortality model by Lee-Carter, and based on this model we estimated life expectancy for the total Dutch population based on historical data and from there we computed projections for future life expectancy. We made our forecasts based on both period- and cohort life expectancy. The period-based approach is the one actually being used by the government, but the cohort-based method gives a better estimate of remaining lifetime, so we feel it should not be omitted. The results of these projections show that life expectancy can be expected to see decelerated growth and that both AOW-age and pension-age will follow suit. The forecasted AOW- and pension-ages as found in this thesis differ somehwat from the values found by the CBS, particularly for dates in the late future.

When we look at the difference between the AOW-age and pension-age in the future, we notice it follows a staircase pattern, but it looks like the average difference is decreasing each time the pattern repeats itself. This situation where AOW-age and pension-age are not aligned most of the time is bound to create additional complexity as well as increased costs in the pension system, while it is also much harder to communicate this to participants in a pension fund. One wonders if most of these problems could not be avoided by using the same best estimate of the life expectancy in the formulas for further increase of AOW- and pension-age, as well as having the AOW-age increase in steps of one year as opposed to the current three-month steps. In addition to this, it would make more sense to decide upon a fixed period of entitlements (within a certain bandwidth) and find the appropriate retirement age by matching the length of this entitlement period with a life expectancy projection of the same length. This would mean less complexity and less costs, as well as more transparency for participants.

The forecasts from which we draw the above conclusions were all based on data for the total Dutch population and as such do not tell the whole story. To account for differences in social class, we also compared the life expectancy projections and pension entitlements for individuals with different education levels. The results are in line with past studies. While life excpectancies for the three different groups look like they slowly converge as time goes on, the differences remain significantly high even in 50+-year forecasts, and this is without taking

Increasing retirement age and its effect on different social classes