Actuarial Studies

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Actuarial Studies

Master’s Thesis

Intragenerational redistribution in Dutch

occupational pension schemes

April 11, 2014

Author: Fonkei Chan

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Abstract

Most pension funds in the Netherlands employ a uniform contribution rate. Every participant pays the same contribution rate for the accrual of pension rights regard-less of age, gender, level of education, health or other individual characteristics. In this thesis we investigate the redistribution of wealth in Dutch occupational pension schemes which occurs as a consequence of the uniform contribution system. We focus only on redistribution within a single cohort and examine the sensitivity of the results to the choice of mortality table. In the first part of the thesis, we fit the classic Lee-Carter model to Dutch mortality data and make a forecast of future mortality rates. In the second part we quantify and compare redistribution using the concept of net benefit with both the Lee-Carter and the AG mortality projection. The use of both mortality models results in value transfers from males to females, but also from participants with a low income to those with a high income. We find that using the Lee-Carter mortality forecast leads to a higher degree of redistribu-tion than with the AG forecast.

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2.3.2 Solidarity . . . 6 2.4 Regulations . . . 7 2.4.1 Pension Act . . . 7 2.5 Demographic changes . . . 8 3 Mortality forecasting 10 3.1 Data . . . 11 3.2 Lee-Carter model . . . 12 3.2.1 Model specification . . . 13 3.2.2 Model estimation . . . 13 3.2.3 Forecasting . . . 15 3.2.4 Backtesting . . . 24

3.3 Education specific mortality . . . 32

3.4 Summary . . . 35

4 Redistribution 36 4.1 Stylized pension fund . . . 36

4.2 Data . . . 38 4.3 Net benefit . . . 39 4.4 Sensitivity analysis . . . 42 4.4.1 Discount rate . . . 42 4.4.2 Retirement age . . . 43 4.5 Summary . . . 45 5 Conclusion 46 5.1 Recommendations for further research . . . 47

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Chapter 1 Introduction

There is a growing debate about the fairness of occupational pension schemes in the Netherlands. Participation in pension schemes is not mandatory by law, but it is required when they are part of the collective labor agreement. Collective pension schemes reduce costs for participants through economies of scale and enable the sharing of risk between participants. Most pension funds charge a uniform contri-bution rate for the accrual of new pension rights. Hence, every participant pays the same contribution rate regardless of age, gender, health, level of education and so on. As the actuarial costs for pension accrual is not the same for everyone, this system leads to inter- and intragenerational redistribution of wealth. The Dutch

Bu-reau for Economic Policy Analysis1 (CPB) recently published a paper (CPB, 2013)

discussing the advantages and disadvantages of the uniform contribution rate sys-tem in the Netherlands. Their research is based on the model by Bonenkamp (2013) which is used to examine lifetime redistribution in collective pension schemes. These papers provide insight into the fairness of the current Dutch occupational pension system and could therefore be interesting for policy makers discussing the future of the system. The main findings of the papers are that the uniform contribution system in occupational pensions schemes leads to immoral redistribution of wealth between participants. High educated participants receive a subsidy on their pension premiums at the cost of low educated participants, while the former group has a higher income than the latter on average. CPB (2013) suggests alternative systems for occupational pension schemes, such as increasing contribution rates or decreasing pension accrual over age. Switching to another system has large consequences for generations who have already built up a part of their pension in the uniform con-tribution system. CPB (2013) estimates that the costs, in the form of lost implicit

pension rights of current participants, amount to approximately AC100 billion.

1_{Centraal Plan Bureau}

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2 1.1. RESEARCH QUESTION

1.1 Research question

The Pension Act states that pension fund liabilities need to be valued using appro-priate and up-to-date mortality tables. This means that there is not just one correct mortality table for all pension funds. The two papers mentioned in the

introduc-tion base their calculaintroduc-tions on mortality projecintroduc-tions made by Statistics Netherlands2

(CBS). In practice, various mortality tables are being used by pension funds, so the question comes to mind whether the results of these papers are applicable for all Dutch pension funds. To this end, we want to examine redistribution under dif-ferent mortality assumptions using the model of Bonenkamp (2013). Our research question can therefore be summarized as follows: How sensitive is intragenerational redistribution in collective pension schemes to the choice of mortality table? We will compare the degrees of redistribution resulting from using the Dutch Actuarial

Society’s3 _{(AG) mortality table, the AG Prognosetafel 2012-2062 (AG, 2012) and}

from using mortality rates calculated with the Lee-Carter model. The AG mortality table has been specifically constructed for the Netherlands and is currently being used by many pension funds for the valuation of pension liabilities. The Lee-Carter model is a well known stochastic mortality model which is adopted by several coun-tries as the main tool for mortality forecasting, see (Stoeldraijer et al., 2013). Since not all data used in the papers by Bonenkamp (2013) and CPB (2013) is publicly available and our assumptions are slightly different, we cannot compare their results with our own. Instead, we will employ their model and base our conclusion on the comparison between the AG and the Lee-Carter mortality table. Furthermore, we will investigate how the results change when we adjust our model assumptions by means of a sensitivity analysis. Calculations will be performed in the statistical

computing software R which is available for free on the internet4_.

1.2 Thesis outline

This thesis is organized as follows. In Chapter 2, we explain pensions in general and the pension system in the Netherlands. In Chapter 3 we fit the Lee-Carter model to Dutch mortality data and quantify longevity risk. We also calculate education

specific mortality rates using the approach of H´ari et al. (2006). Chapter 4 will

be dedicated to examining redistribution in collective pension schemes following Bonenkamp (2013). Finally, we conclude this thesis in Chapter 5.

2_{Centraal Bureau voor de Statistiek} 3_{Actuarieel Genootschap}

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Chapter 2 Pensions

Pensions are a form of income that replaces labor income once a person retires. There are different ways in which pensions are financed. The most common form is the funded scheme where people save up a portion of their salary during their working life for future pensions. Another variant is the unfunded scheme, a so called pay-as-you-go system, in which current benefits are paid with current contributions. We will go into more detail below. This chapter is based largely on (Pensioenfederatie, 2010).

2.1 Unfunded scheme

The most common unfunded pension scheme is the state pension which starts paying benefits once a person reaches the state pension age. The setup of state pensions differs per country, but the level of benefit is usually linked to the statutory minimum wage and depends on the number of years a person has lived or worked in the country. The costs of the state pension benefits are paid by the labor force in the form of contributions and from government public funds. Consequently every citizen, whether or not retired, contributes to the state pension.

2.2 Funded schemes

Funded schemes come in many forms and sizes. We describe the three most common pension schemes.

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4 2.2. FUNDED SCHEMES

Defined Benefit

In Defined Benefit (DB) schemes the level of pension depends on the number of years worked and the salary level. For every year a person has worked and thus paid contribution, pension rights are built up. In recent years, most of the DB schemes have switched from final to average pay schemes. In a final pay scheme, participants accrue pension rights as a percentage of their final salary, whereas in average pay schemes pension rights are based on the average salary. To accommodate for inflation or wage increases, pension rights and benefit payments are revised annually which is known as indexation. This indexation is either unconditional or conditional on the pension fund’s funding ratio. Most pension funds have switched to conditional indexation. Since DB schemes are unconditional promises to pay out accrued pension rights in the future, this implies that even when a fund is in bad financial health, it still has to pay the same promised amount of benefit payments. In this situation, a pension fund can either raise contributions, require the sponsor/employer to make a one time deposit, postpone indexation or in the worst case reduce pension entitlements and rights.

Defined Contribution

Contrary to DB schemes, no promise regarding the future level of pension is made in Defined Contribution (DC) schemes. The amount of pension a participant will receive during retirement depends on the contributions paid during the accumulation period and the realized return on investments. The capital is used to purchase an annuity on or before the retirement date. Depending on accumulated contribution payments and investment return, the level of pension can be either sufficient or insufficient. Hence, the risk is completely borne by employees.

Collective Defined Contribution

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CHAPTER 2. PENSIONS 5

Funding ratio

The funding ratio of a pension fund is the present value of its assets divided by the value of its liabilities. This ratio provides a general measure of pension fund health. A funding ratio of 100% implies that a pension fund has exactly enough money to pay its current and future liabilities. In most countries, regulators require pension funds to maintain a funding ratio of more than 100%, since pension funds are exposed to different types of risk. Pension fund liabilities have a particularly long time horizon. The present value of these liabilities is calculated by discounting the future cash flows using a certain discount rate.

2.3 The Dutch pension system

According to the Melbourne Mercer Global Pension Index in 2012, the Dutch pension system ranked second best in the world. The Dutch pension system consists of three pillars: the state pension, the supplementary collective pension and the individual pension, see Figure 2.1. The combination of these three pillars determines the level of pension a person will receive during retirement. We will go into more detail on the pillars below.

Old age Pension

1st Pillar State pension 2nd Pillar Occupational pension 3rd Pillar Individual pension

Figure 2.1: The three pillar pension system in the Netherlands

2.3.1 Three pillar structure

The first pillar is the state pension, also known as the General Old Age Pension1

(AOW). The state pension is a pay-as-you-go system as described in the previous section. For every year lived or worked in the Netherlands, a person builds up a right to 2% of the full AOW up till a maximum of 100%. The state pension is linked to the

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6 2.3. THE DUTCH PENSION SYSTEM

statutory minimum wage. Pensioners living alone receive 70% of the minimum wage, whereas two pensioners living together receive 100% of the minimum wage in total. Historically, the state pension age has been fixed at 65. However, due to increasing life expectancy the state pension age will be steadily increased to 66 in 2018, 67 in 2021 and afterwards it will be linked to the development of life expectancy.

The second pillar is the occupational pension. Workers are not obliged to build up pension in the second pillar unless there is a collective pension scheme available in their corporation, sector or industry. Approximately 90% of all workers participate in pension funds. This allows pension funds to take advantage of economies of scale and reduce costs to a minimum. Pension funds are non-profit organizations. As a result, only 3.5% (2004) of pension fund contributions is used for other things besides building up pension rights as opposed to insurance companies where this amounts to 25.7% (2004) of the paid premiums (Van der Lecq & Steenbeek, 2006). The majority of the laborforce participates in average pay DB schemes. The second pillar retirement age has historically been fixed at the age of 65, but is increased to 67 as of 2014 and will be linked to the development of life expectancy afterwards.

The third pillar is the individual pension which is mainly used by self-employed and workers without collective pension schemes. People can increase their pension by purchasing annuities, while profiting from tax benefits.

2.3.2 Solidarity

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CHAPTER 2. PENSIONS 7

time of introduction of the current system, it was common for workers to stay at the same employer for their entire career. Nowadays, workers switch jobs or industry more often. Beside intergenerational distribution, the uniform contribution rate also redistributes wealth within generations. Participants with a high life expectancy are subsidized by those with a low life expectancy. More specifically, Bonenkamp (2013) found that highly educated women are subsidized the most by other participants whereas the opposite sex with a low education pays the most. In Chapter 4 we will investigate the magnitude of this redistribution.

2.4 Regulations

Pension funds are subject to the Pension Act2. Compliance with the Pension Act is

monitored by the Dutch Central Bank3 (DNB) and the Authority for the Financial

Markets4 (AFM). The DNB assesses the financial health of pension funds and

inter-venes when they expect that a fund might not be able to fulfil its future obligations. The AFM monitors pension fund behaviour towards participants. We summarize some of the key points of the Pension Act below.

2.4.1 Pension Act

This section is mostly based on information from the DNB5. The Pension Act fully

came into force on 1 January 2007. The Act lists all conditions and requirements with respect to pensions. An important element of the Act is that pensions are seen as an employment condition. Another important element is the Financial

Assess-ment Framework6 _{(FTK) for pension funds.}

The FTK is a framework for the assessment of the financial health of pension funds. An essential change has been made with respect to the valuation of liabilities. Under the FTK, market valuation is used for for both assets and liabilities. Pension funds are required to have at least enough assets to cover their liabilities. In addi-tion, pension funds must hold regulatory own funds, the size of which depends on the pension fund’s risk profile. The FTK states these conditions in terms of funding ratios. The minimum funding ratio is 105% and the required funding ratio which depends on a fund’s risk profile often lies around 125%. When a pension fund’s

2_Pensioenwet

3_{De Nederlandsche Bank} 4_{Autoriteit Financi¨}_{ele Markten} 5_{http://www.dnb.nl}

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8 2.5. DEMOGRAPHIC CHANGES

funding ratio falls below the required funding ratio we speak of a reserve deficit. The fund is required to submit a long-term recovery plan to the DNB in which they describe how the fund will regain the required funding ratio within a period of fifteen years. A funding shortfall occurs when the funding ratio drops below the minimum funding ratio. In case of a funding shortfall, pension funds are required to submit a short-term recovery plan to the DNB in which they propose measures in order to regain the minimum funding ratio within three years.

2.5 Demographic changes

Life expectancy has increased tremendously over the last decades. The increases are strongest for younger ages and tend to weaken for older ages, see Figure 2.2. Where a newborn in the Netherlands in 1900 was expected to reach the age of 48, this rose to the age of 78 in 2000. The increase of life expectancy for a 65 year old

is less dramatic, i.e. from 76 to 82.5, albeit still a significant increase7_{. The extreme}

rise in life expectancy for newborns can be attributed to the substantial reduction of infant mortality. Better hygiene and increased welfare have contributed a great deal to the overall increase in life expectancy.

Other important developments are the temporary spike in birth rates after the Second World War coupled with declining birth rates, see Figure 2.3. As a result, the average age of the population is shifting upward. This demographic change has a large impact on the pension system, in particular on state pensions where benefits for the older generation are financed by the younger generation.

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CHAPTER 2. PENSIONS 9 1950 1960 1970 1980 1990 2000 2010 75 80 85 90 Males Year T otal lif e e xpectancy Age 85 Age 65 Age 45 Age 25 1950 1960 1970 1980 1990 2000 2010 75 80 85 90 Females Year T otal lif e e xpectancy Age 85 Age 65 Age 45 Age 25

Figure 2.2: Development of total life expectancy in the Netherlands. Source: CBS.

1850 1900 1950 2000 15 20 25 30 35 Birth rates Year Bir ths per 1000 inhabitants 1850 1900 1950 2000 28 30 32 34 36 38

Average population age

Year

Age

Figure 2.3: Birthrates and average population age in the Netherlands. Source:

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Chapter 3 Mortality forecasting

Pension funds provide participants with a monthly income during retirement for as long as they live. The value of a pension fund’s liabilities is thus evidently highly dependent on mortality rates. When pension funds misjudge future mortality rates, the value of the liabilities could become higher than the value of the assets, imply-ing that there is not enough money to fulfill all pension obligations. An accurate prediction of future mortality allows pension funds to charge the correct premium level which greatly reduces the risk of underfunding in the future. Various models have been constructed for this purpose, each with its own strengths and limitations. Nevertheless, history has shown that mortality rates are quite difficult to predict accurately as can be seen in the almost systematic underestimation in the past (De Waegenaere et al., 2011). In this chapter we describe the Lee-Carter model which forms the foundation for many other mortality models and compare the re-sulting mortality projections with the mortality projections published by the AG which is currently used by many pension funds in the Netherlands. The model used by the AG is deterministic, that is to say, it only gives best estimates of future mor-tality rates. The Lee-Carter model is stochastic and is therefore able to quantify longevity risk by providing prediction intervals. At the end of this chapter we will

follow the approach of H´ari et al. (2006) in order to construct education specific

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CHAPTER 3. MORTALITY FORECASTING 11

3.1 Data

There are two main sources for Dutch mortality data on the internet. The HMD1

provides smoothed annual death rates and life tables for ages 0 till 110 for various countries. These are calculated based on raw demographic data provided to them by the various countries. We refer the interested reader to HMD’s website for their precise calculation methodology. The HMD has published mortality data for the

period 1850-2009. The other source is Statistics Netherlands2 _{(CBS), an institute}

responsible for collecting and processing data in order to publish statistics to be used in practice, by policymakers and for scientific research. CBS has published

annual raw 1-year probabilities of death qx,t for ages 0 till 99 for the period

1950-2012. For comparison purposes later in this paper we will make use of the raw CBS data. Notice that there seem to be structural breaks around 1973, 1988 and 2002, see Figure 2.2. The slope of the trend in male life expectancy switches from negative to positive around 1973. We choose the fitting period to be 1973-2011 excluding the structural break in 1973. This greatly improves the fit of the Lee-Carter model. Following the approach of AG (2012), some adjustments will be

made to the raw data first. The qx,t are smoothed by applying the Van Broekhoven

algorithm (Van Broekhoven, 2002). The resulting smoothed rates are then used to estimate rates for high ages using Kannisto’s mortality law (Kannisto, 1992). The

purpose of this last step is to correct for possible errors in the estimation of qx,t due

to the limited availability of data for high ages.

Now, let us first introduce the quantities of interest. The central death rate mx,t is

defined as

mx,t =

Dx,t

Ex,t

, (3.1)

where Dx,t is the number of deaths and Ex,t the average number of people living

aged x ∈ [0, ω] at time t ∈ [1973, 2011], respectively. Following Bonenkamp (2013), we set the highest attainable age ω equal to 99. Let X denote the number of age groups and let T denote the number of observations. Following the method of the

HMD3 _{we transform the 1-year probabilities of death from CBS into central death}

rates as follows:

mx,t ≈

qx,t

1 − (1 − cx)qx,t

, (3.2)

1_{Human Mortality Database.} _{University of California,} _{Berkeley (USA), and Max}

Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de

2_{Centraal Bureau voor de Statistiek}

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12 3.2. LEE-CARTER MODEL

where cx is the average number of years lived within the age interval [x, x + 1) for

people dying at that age and we assume that cx = 0.5 for x > 0. Throughout this

paper we will refer to the central death rate as the mortality rate. Since the highest

attainable age is 99, it follows that q99,t= 1. The complement of qx,t is the one-year

probability of survival px,t = 1 − qx,t. The probability of a person aged x at time t

surviving k ∈ N years is given by

kpx,t = k−1

Y

j=0

px+j,t. (3.3)

Summing Equation (3.3) over all k and adding 1₂ yields the remaining life expectancy

of a person aged x, ex,t = 99−x X k=1 kpx,t+ 1 2, (3.4)

Note that the 1

2 is included in Equation (3.4) to account for the assumption that

people on average die in the middle of the year. The log mortality rates for the Dutch population are plotted in Figure 3.1.

Age 40 60 80 Year 1980 1990 2000 2010 log centr al death r ate −6 −4 −2 Males Age 40 60 80 Year 1980 1990 2000 2010 log centr al death r ate −8 −6 −4 −2 Females

Figure 3.1: Graphical representation of the data

3.2 Lee-Carter model

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field of mortality modelling. Countless variants have been developed based on this model attempting to improve goodness of fit, forecasting capability and practical usability. In this section we will specify the classic Lee-Carter model and apply it to Dutch mortality data.

3.2.1 Model specification

The Lee-Carter model decomposes mortality into three components: the age-specific

pattern of mortality ax, the sensitivity of an age group to changes in mortality trend

bx and the trend of mortality kt. Mathematically, this is formulated as

mx,t = exp(ax+ bxkt+ x,t), (3.5)

or equivalently as

log mx,t = ax+ bxkt+ x,t, (3.6)

where x,t ∼ N (0, σ2) is the error term for each age-year combination. Due to the

product term in the equation, Lee & Carter (1992) added the following restrictions to ensure identifiability of the parameters.

X x bx= 1, (3.7) X t kt= 0. (3.8)

Note that (3.8) and x,t ∼ N (0, σ2) imply that an unbiased estimator for ax is given

by: X t log mx,t = X t ax+ bx X t kt+ X t x,t, (3.9) X t log mx,t = T ax, (3.10) 1 T X t log mx,t = ˆax. (3.11)

3.2.2 Model estimation

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Least Squares. Let

Z = log mx,t− ˆax (3.12)

= ˆbxˆkt, (3.13)

denote the centered log mortality rates. Using SVD the centered log mortality rates can be decomposed into

Z = UΣV0 (3.14) = r X i=1 siuiv0i (3.15)

where r = rank(Z), U = [u1 u2· · · uX] is an (X × X) orthogonal matrix, Σ is an

(X × T ) diagonal matrix containing the singular values s1 ≥ s2 ≥ . . . ≥ 0 of Z in

descending order and V0 denotes the conjugate transpose of the (T × T ) orthogonal

matrix V = [v1 v2· · · vT].

Lee & Carter (1992) use only the first singular value to approximate Z arguing that it accounts for almost all the variance over time. Girosi & King (2007) state that the Lee-Carter model is more likely to work well when at least 90% of the variance is explained by the first singular value. Using Equation 3.16 we find that approximately 98.6% and 97.5% of the variance of respectively Dutch males and females is explained by the first singular value.

% − variance explained by the first singular value = s

2 1

Pr

i=1s2i

(3.16)

Hence Z can be approximated using only its first singular value which greatly reduces the dimensionality of the data. That is,

Z = ˆbxkˆt≈ s1u1v01, (3.17)

resulting in the following estimators for bx and kt,

ˆ_b_x _{= u}₁_, _(3.18)

ˆ

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Finally, we need to make a small adjustment in order to satisfy constraint (3.7).

ˆ_b_x₌ ˆbx P xˆbx , (3.20) ˆ kt= s1v01 X x ˆ_b_x_. _(3.21)

Since ktis fitted using log mortality rates rather than the original mortality rates, the

predicted number of deaths will generally not match the actual number of deaths. This is because the low death rates of younger ages receive the same weight as the high death rates of older ages, even though they contribute far less to the total

deaths. Lee & Carter (1992) reestimate kt while keeping other parameters fixed

in a second step such that the implied number of deaths equal the actual number of deaths. However, this approach requires an additional source of data, namely

population data. Lee & Miller (2001) propose another approach in which kt is

rees-timated while keeping other parameters fixed such that fitted life expectancies at

birth e0 equal observed life expectancies at birth. We will perform the second stage

estimation of kt using e25 instead, since that is the lowest age in our age range.

Jump-off bias

The way the Lee-Carter model is fitted to the data leads to a discontinuity between the last observation and the start of the forecast, also known as the jump-off bias. Lee & Carter (1992) mentioned this in their paper, but chose to accept the discon-tinuity, arguing that it only affects the rates that are absolutely low in any event and have little influence on life expectancy. However, Lee & Miller (2001) show that correcting for the jump-off bias improves the forecast. The jump-off bias can be taken into account by setting

ˆ

ax = log mx,tj, (3.22)

ˆ

ktj = 0, (3.23)

where tj is the jump-off date, i.e. the year of the last observation. This adjustment

does not affect constraint (3.7). The estimated Lee-Carter parameters for Dutch males and females in the period 1973-2011 are displayed in Figure 3.2 and Figure 3.3, respectively.

3.2.3 Forecasting

Lee & Carter (1992) assume that axand bxremain constant over time. In the original

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16 3.2. LEE-CARTER MODEL 40 60 80 100 −8 −6 −4 −2 Age ax 40 60 80 100 0.000 0.005 0.010 0.015 0.020 Age bx 1980 1990 2000 2010 0 10 20 30 40 Year kt

Figure 3.2: Estimated Lee-Carter parameters for Dutch males

40 60 80 100 −8 −6 −4 −2 Age ax 40 60 80 100 0.005 0.010 0.015 0.020 Age bx 1980 1990 2000 2010 0 10 20 30 Year kt

Figure 3.3: Estimated Lee-Carter parameters for Dutch females

i.e. an ARIMA(0,1,0) process. We turn to a general ARIMA specification for the Dutch data. An ARIMA(p, d, q) model is defined as

∆dkt= δ + p X i=1 φi∆dkt−i+ q X j=1 θjt−j+ t, (3.24)

where ∆ is the difference operator, that is, ∆kt = kt − kt−1, δ is a drift term,

t∼ N (0, σ2) is the error term, φi and θj are the model parameters.

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CHAPTER 3. MORTALITY FORECASTING 17 0 5 10 15 20 25 30 −0.4 0.2 0.8 Lag A CF Males 0 5 10 15 20 25 30 −0.4 0.2 0.8 Lag A CF Females 0 5 10 15 20 25 30 −0.2 0.4 0.8 Lag P ar tial A CF 0 5 10 15 20 25 30 −0.2 0.4 0.8 Lag P ar tial A CF Figure 3.4: Correlogram of kt

level. We can see from Figure 3.4 that the PACF behaves nicely, but the ACF does not go to zero as the number of lags increases. Hence, we conclude that the original series is not stationary. Figure 3.5 displays the correlogram of the first differences

of kt. Aside from a few spikes, nearly all values are close to zero indicating that the

first differences of kt are stationary, see Figure 3.6.

We select a model for the data at hand based on the Bayesian Information Criterion (BIC), a score that reflects the trade-off between goodness of fit and number of parameters. Models are rewarded for good fit quality and penalized for overfitting:

BIC = k ln T − 2 ln L, (3.25)

where k is the number of parameters and L is the likelihood. Hence, we select models with the lowest BIC scores. We proceed by fitting ARIMA(p, 1, q) models to the data for different combinations of p, q ∈ {0, 1, 2, 3, 4} and calculating the BIC for each model. We find that a random walk with drift as suggested by Lee & Carter (1992) is most parsimonious according to the BIC for both males and females.

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18 3.2. LEE-CARTER MODEL 0 5 10 15 20 25 30 −0.2 0.4 1.0 Lag A CF Males 0 5 10 15 20 25 30 −0.2 0.4 1.0 Lag A CF Females 0 5 10 15 20 25 30 −0.3 0.0 0.2 Lag P ar tial A CF 0 5 10 15 20 25 30 −0.3 0.0 0.3 Lag P ar tial A CF Figure 3.5: Correlogram of ∆kt

noise. We have plotted the correlogram of the fitted models as specified in Table 3.1 in Figure 3.7. The correlogram of the residuals show that there is no serial correlation, implying that the residuals are in fact white noise. We also apply the Ljung-Box test to test the nullhypothesis of no autocorrelation in the residuals.

Parameters Males ARIMA(0,1,0) Females ARIMA(0,1,0) δ -1.1647 (0.2188) -0.9814 (0.2244) BIC 137.84 139.76

Table 3.1: Selected models for the mortality indices based on the lowest BIC score. Standard errors are shown between brackets.

Q = T (T + 2) k X i=1 ˆ ρ2 i T − i ∼ χ 2 k, (3.26)

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CHAPTER 3. MORTALITY FORECASTING 19 1980 1990 2000 2010 −3 −2 −1 0 1 2 Males Year kt − kt− 1 1980 1990 2000 2010 −3 −2 −1 0 1 2 Females Year kt − kt−1

Figure 3.6: First differences of kt

The Ljung-Box test fails to reject the nullhypothesis of no autocorrelation in the residuals of the fitted models at a 95% significance level for all lag lengths up to thirty, see Figure 3.8. Hence, we conclude that the model is correctly specified and turn to forecasting the mortality indices with the ARIMA specifications as given

in Table 3.1. Holding ax and bx constant and using the projected mortality indices

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of the forecast horizon. The confidence intervals are a measure for longevity risk. In theory, the realizations should lie between the dashed lines 95% of the times. As life expectancy has been systematically underestimated in the past, we are mainly interested in the lower confidence bound of the mortality rates or equivalently the upper confidence bound of life expectancy.

Lee-Carter AG Males e25,t e45,t e65,t e25,t e45,t e65,t 2028 57.3 (56.0; 58.5) 37.8 (36.5; 38.9) 19.7 (18.7; 20.5) 58.8 39.2 20.9 2045 59.4 (57.8; 60.8) 39.8 (38.3; 41.2) 21.2 (20.0; 22.3) 61.0 41.3 22.5 2062 61.2 (59.5; 62.7) 41.5 (39.9; 42.9) 22.6 (21.3; 23.7) 62.0 42.2 23.1 Females e25,t e45,t e65,t e25,t e45,t e65,t 2028 60.1 (58.9; 61.3) 40.5 (39.3; 41.6) 22.4 (21.3; 23.3) 60.9 41.2 22.9 2045 61.8 (60.2; 63.3) 42.1 (40.6; 43.5) 23.8 (22.5; 25.0) 62.0 42.3 23.9 2062 63.4 (61.6; 65.0) 43.6 (41.9; 45.1) 25.1 (23.6; 26.4) 62.4 42.6 24.1

Table 3.2: Projected life expectancies. The values between brackets represent a 95% confidence interval for the point forecasts.

0 5 10 15 20 25 30 −0.2 0.4 1.0 Lag A CF Males 0 5 10 15 20 25 30 −0.2 0.4 1.0 Lag A CF Females 0 5 10 15 20 25 30 −0.3 0.0 0.2 Lag P ar tial A CF 0 5 10 15 20 25 30 −0.3 0.0 0.2 Lag P ar tial A CF

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CHAPTER 3. MORTALITY FORECASTING 21 ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ●● 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 Males Lag length p−v alue ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 Females Lag length p−v alue

Figure 3.8: p-values of the Ljung-Box test

1980 2000 2020 2040 2060 −80 −60 −40 −20 0 20 40 Males Year kt Best estimate 95% Confidence interval 1980 2000 2020 2040 2060 −60 −40 −20 0 20 40 Females Year kt Best estimate 95% Confidence interval

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22 3.2. LEE-CARTER MODEL ● ● ● ●●●● ● ● ●●●● ● ● ● ●_●● ● ●● ● ● ●●● ●● ●● ● ●● ●● ● ●● 1980 2000 2020 2040 2060 −10.0 −9.5 −9.0 −8.5 −8.0 −7.5 −7.0 Males Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval AG projection ● ● ● ●● ●● ●●●● ● ●●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● 1980 2000 2020 2040 2060 −10.0 −9.5 −9.0 −8.5 −8.0 −7.5 −7.0 Females Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval AG projection ●●●● ●● ●● ● ● ● ●●●●● ●●_{●●●●●●} ●●●●●●● ● ● ● ● ●● ●● 1980 2000 2020 2040 2060 −7.5 −7.0 −6.5 −6.0 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval AG projection ● ●●●●● ● ● ●●●● ● ●● ● ● ●●●●●●●●●● ● ●●●● ● ● ●_●● ●● 1980 2000 2020 2040 2060 −7.5 −7.0 −6.5 −6.0 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval AG projection ●●●● ●●●●●●_●●●● ●●● ●●●●_●●● ●● ●● ●● ● ●● ●● ●●● ● 1980 2000 2020 2040 2060 −5.5 −5.0 −4.5 −4.0 −3.5 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval AG projection ●●●● ● ●● ●●●●● ●●●●●●●●●●●● ●●●●●● ● ●●● ●●●●● 1980 2000 2020 2040 2060 −5.5 −5.0 −4.5 −4.0 −3.5 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval AG projection

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CHAPTER 3. MORTALITY FORECASTING 23 ●●●●●● ●●●●●●●● ●●●●● ● ● ●●● ●●● ●●●● ●● ●● ●●● ● 1980 2000 2020 2040 2060 50 55 60 65 Males Year e25 , t ● _Observations Best estimate 95% Confidence Interval AG projection ● ●●● ●●● ●●●●●●●●●●●● ● ●●●●●●●●●● ● ●● ●●● ●●● 1980 2000 2020 2040 2060 50 55 60 65 Females Year e25 , t ● _Observations Best estimate 95% Confidence Interval AG projection ●●●● ●●●●●●●●●● ●●●●● ● ● ●●● ●●●● ●●● ●● ●● ●●● ● 1980 2000 2020 2040 2060 30 35 40 45 Year e45 , t ● _Observations Best estimate 95% Confidence Interval AG projection ●●● ● ●●● ●●●●●●● ●●●●●● ●●●●● ●●●●●● ●●● ●●●●● 1980 2000 2020 2040 2060 30 35 40 45 Year e45 , t ● _Observations Best estimate 95% Confidence Interval AG projection ●●●● ● ●●●●●●●●● ●●●●●●● ●●●●●● ●●●● ●●● ●● ●● ● 1980 2000 2020 2040 2060 14 16 18 20 22 24 26 Year e65 , t ● _Observations Best estimate 95% Confidence Interval AG projection ●●● ● ●●●● ●●●●●● ●●●●●● ● ●●●●●●●●●● ●●● ●● ●●● 1980 2000 2020 2040 2060 14 16 18 20 22 24 26 Year e65 , t ● _Observations Best estimate 95% Confidence Interval AG projection

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24 3.2. LEE-CARTER MODEL 2010 2020 2030 2040 2050 2060 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Lee−Carter best estimate

Year

Diff

erence in gender lif

e e xpectancy Age 25 Age 45 Age 65 2010 2020 2030 2040 2050 2060 0.5 1.0 1.5 2.0 2.5 3.0 3.5 AG best estimate Year Diff

e e

xpectancy

Age 25 Age 45 Age 65

Figure 3.12: Difference between male and female life expectancy with the Lee-Carter model and AG model

2010 2020 2030 2040 2050 2060 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Lee−Carter upper confidence bound

Year

Diff

e e xpectancy Age 25 Age 45 Age 65 2010 2020 2030 2040 2050 2060 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Lee−Carter lower confidence bound

Year

Diff

e e

xpectancy

Age 25 Age 45 Age 65

Figure 3.13: Difference between male and female life expectancy with the Lee-Carter upper and lower confidence bounds

3.2.4 Backtesting

In this section we assess the forecast performance of the Lee-Carter model by means of a backtest. We leave out the last observations and attempt to predict these values with a reestimated Lee-Carter model while maintaining the same ARIMA

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26 3.2. LEE-CARTER MODEL ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −8.5 −8.0 −7.5 −7.0 Males Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −8.5 −8.0 −7.5 −7.0 Females Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −6.6 −6.4 −6.2 −6.0 −5.8 −5.6 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● 1980 1990 2000 2010 −6.6 −6.4 −6.2 −6.0 −5.8 −5.6 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval ● ●● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −5.0 −4.6 −4.2 −3.8 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval ● ● ●● ● ● ● ●●● ● ● ● ●● ●●●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● 1980 1990 2000 2010 −5.0 −4.6 −4.2 −3.8 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval

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CHAPTER 3. MORTALITY FORECASTING 27 ● ● ● ● ●● ● ●● ●● ● ● ● ●● ●● ●●● ● ● ● ● ●● ● ●●● ● ● ● ●● ●● ● 1980 1990 2000 2010 48 50 52 54 56 58 Males Year e25 , t ● _Observations Best estimate 95% Confidence Interval ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● 1980 1990 2000 2010 48 50 52 54 56 58 Females Year e25 , t ● _Observations Best estimate 95% Confidence Interval ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ●● ● 1980 1990 2000 2010 30 32 34 36 38 Year e45 , t ● _Observations Best estimate 95% Confidence Interval ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ●● 1980 1990 2000 2010 30 32 34 36 38 Year e45 , t ● _Observations Best estimate 95% Confidence Interval ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ●● ● ● 1980 1990 2000 2010 14 16 18 20 Year e65 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●● 1980 1990 2000 2010 14 16 18 20 Year e65 , t ● _Observations Best estimate 95% Confidence Interval

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28 3.2. LEE-CARTER MODEL ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● 1980 1990 2000 2010 −8.5 −8.0 −7.5 −7.0 Males Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −8.5 −8.0 −7.5 −7.0 Females Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −6.6 −6.4 −6.2 −6.0 −5.8 −5.6 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● 1980 1990 2000 2010 −6.6 −6.4 −6.2 −6.0 −5.8 −5.6 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval ● ●● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −4.8 −4.4 −4.0 −3.6 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval ● ● ●● ● ● ● ●● ●● ● ● ● ● ●●●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● 1980 1990 2000 2010 −4.8 −4.4 −4.0 −3.6 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval

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CHAPTER 3. MORTALITY FORECASTING 29 ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ●● ● 1980 1990 2000 2010 48 50 52 54 56 58 Males Year e25 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●● 1980 1990 2000 2010 48 50 52 54 56 58 Females Year e25 , t ● _Observations Best estimate 95% Confidence Interval ●● ● ● ● ● ● ●● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ●● ● 1980 1990 2000 2010 30 32 34 36 38 Year e45 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●● 1980 1990 2000 2010 30 32 34 36 38 Year e45 , t ● _Observations Best estimate 95% Confidence Interval ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● 1980 1990 2000 2010 14 16 18 20 Year e65 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ●● ●●● ● ● ● ● ●●● ●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●● 1980 1990 2000 2010 14 16 18 20 Year e65 , t ● _Observations Best estimate 95% Confidence Interval

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30 3.2. LEE-CARTER MODEL ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ● 1980 1990 2000 2010 −8.5 −8.0 −7.5 −7.0 Males Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −8.5 −8.0 −7.5 −7.0 Females Year lo g m25 , t ● _Observations Best estimate 95% Confidence Interval ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 1980 1990 2000 2010 −6.6 −6.4 −6.2 −6.0 −5.8 −5.6 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● 1980 1990 2000 2010 −6.6 −6.4 −6.2 −6.0 −5.8 −5.6 Year lo g m45 , t ● _Observations Best estimate 95% Confidence Interval ● ●● ● ●●●● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● 1980 1990 2000 2010 −4.8 −4.6 −4.4 −4.2 −4.0 −3.8 −3.6 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●●● ●_{● ● ● ●} ● ● ● ●● ●● ● ● ●● ● ● ● ●● 1980 1990 2000 2010 −4.8 −4.6 −4.4 −4.2 −4.0 −3.8 −3.6 Year lo g m65 , t ● _Observations Best estimate 95% Confidence Interval

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CHAPTER 3. MORTALITY FORECASTING 31 ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ●● ●● ●● ● ● ● ●● ●● ● 1980 1990 2000 2010 48 50 52 54 56 58 Males Year e25 , t ● _Observations Best estimate 95% Confidence Interval ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● 1980 1990 2000 2010 48 50 52 54 56 58 Females Year e25 , t ● _Observations Best estimate 95% Confidence Interval ●●● ● ● ● ● ●● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ●● ●● ● ● ● ●● ●● ● 1980 1990 2000 2010 30 32 34 36 38 Year e45 , t ● _Observations Best estimate 95% Confidence Interval ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● 1980 1990 2000 2010 30 32 34 36 38 Year e45 , t ● _Observations Best estimate 95% Confidence Interval ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ●● ● ● 1980 1990 2000 2010 14 16 18 20 Year e65 , t ● _Observations Best estimate 95% Confidence Interval ● ● ●● ● ● ●● ● ●● ● ● ● ● ●●● ●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●● 1980 1990 2000 2010 14 16 18 20 Year e65 , t ● _Observations Best estimate 95% Confidence Interval

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32 3.3. EDUCATION SPECIFIC MORTALITY

3.3 Education specific mortality

This section is mainly based on H´ari et al. (2006). In order to examine redistribution

between different levels of education, we need to have education specific mortality

rates. Since these rates are not publicly available, we need to estimate them. H´ari

et al. (2006) argue that relative discrepancies between mortality rates of the different levels of education and the average level of mortality rates should be comparable for neighbouring countries such as The Netherlands and Belgium. Deboosere & Gadeyne (2002) published education specific mortality rates for Belgium for the

period 1991-1996 for ages 25-90. We can use these data to estimate education

specific mortality in the Netherlands. This approach implicitly assumes that relative discrepancies in mortality rates due to education level are equal for the Dutch and Belgian population and that these relative discrepancies remain constant over time. Mackenbach et al. (2003) show that socioeconomic inequalities in mortality have been widening in the late 20th century. The results in the next chapter therefore might underestimate the degree of redistribution.

Deboosere & Gadeyne (2002) distinguish four education levels which we denote by i = {L, LS, HS, H}. In the Netherlands this translates to:

• Low education (L): Basisonderwijs • Lower secundary education (LS): Vmbo

• Higher secundary education (HS): Mbo / Havo / Vwo • High education (H): Hbo / Wo

The relative discrepancy for Dutch education specific mortality rates is defined as

η_xi = qˆ i x,t ˆ qx,t , (3.27)

where ˆq_x,ti is the Belgian education specific mortality rate and

ˆ qx,t = P iNi,tqˆ i x,t P iNi,t (3.28)

is the weighted average of all Belgian mortality rates, where the weighting occurs

using Ni,t which indicates the number of people with age x present in socioeconomic

group i in the Netherlands in 2012. Before calculating Dutch education specific

mortality rates, H´ari et al. (2006) first fit a cubic spline to η_xi to smooth the

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CHAPTER 3. MORTALITY FORECASTING 33 30 40 50 60 70 80 90 0.0 0.5 1.0 1.5 2.0 Males Age Relativ e discrepancy L LS HS H 30 40 50 60 70 80 90 0.0 0.5 1.0 1.5 2.0 Females Age Relativ e discrepancy L LS HS H

Figure 3.20: Relative discrepancy from average Dutch mortality rates due to edu-cation level

education specific mortality rates by multiplying the mortality rates for the Dutch

population by ηi

x. The smoothed relative discrepancies are displayed in Figure 3.20.

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34 3.3. EDUCATION SPECIFIC MORTALITY

Males Females

e25 e45 e65 e25 e45 e65

Low education 55.1 36.0 18.4 58.5 39.0 21.2

Lower secundary education 56.5 37.1 19.2 60.1 40.5 22.4

Higher secundary education 57.1 37.6 19.5 60.4 40.8 22.6

High education 58.8 39.1 20.6 60.9 41.2 22.9

Difference between high and low education 3.7 3.1 2.2 2.4 2.2 1.7

Table 3.3: Projected Lee-Carter life expectancies per education level in 2028

Males Females

e25 e45 e65 e25 e45 e65

Low education 57.6 38.2 20.1 60.4 40.8 22.8

Low secondary education 58.7 39.2 20.8 61.8 42.2 23.8

High secondary education 59.2 39.7 21.1 62.1 42.4 24.0

High education 60.7 40.9 22.1 62.5 42.8 24.3

Table 3.4: Projected Lee-Carter life expectancies per education level in 2045

Males Females

e25 e45 e65 e25 e45 e65

Low education 59.7 40.2 21.6 62.1 42.5 24.2

Low secondary education 60.6 41.0 22.2 63.4 43.6 25.1

High secondary education 61.1 41.4 22.5 63.6 43.8 25.3

High education 62.3 42.5 23.4 64.0 44.2 25.5

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3.4 Summary

In this chapter we have fitted the Lee-Carter model to Dutch mortality data and made a comparison between the Lee-Carter forecast and the widely used AG

mor-tality projection. The two projections are constructed in different ways. The

Lee-Carter model is an extrapolative stochastic model involving minimal subjec-tive judgement (Stoeldraijer et al., 2013) (Booth & Tickle, 2008), whereas the AG model is based on directly extrapolating a short and long term trend (AG, 2012). At the moment we cannot determine whether one model is superior to the other, be-cause it is not possible to backtest the AG model due to its construction. We have evaluated the forecasting performance of the Lee-Carter model. The Lee-Carter model performed relatively well in the short term backtest, but when a structural break occurs during the forecast period, the quality of the forecast drops dramat-ically. This was well seen in the longer horizon backtests. The Lee-Carter model predicts lower life expectancies for males and higher life expectancies for females in the long run compared to the AG model, although the AG predictions lie well within the 95% confidence intervals of the Lee-Carter model at the end of the forecast hori-zon. We will use the confidence intervals in the next chapter to represent a good and bad scenario with a 95% confidence level with respect to future development of mortality. We also observe that differences in life expectancy between males and females diminish at a slower pace with the Lee-Carter model than with the AG model. The realization of the upper confidence bound of the Lee-Carter model will result in slightly smaller differences in gender life expectancy, whereas the opposite occurs when the lower confidence bound is realized. The larger difference in gender life expectancy in the Lee-Carter forecast will likely result in a higher degree of redistribution between men and women which we will examine in more detail in the next chapter.

Finally, we have estimated Dutch education specific mortality rates following

H´ari et al. (2006) by using Belgian education specific mortality rates. We have

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Chapter 4 Redistribution

In the Netherlands, nearly all pension funds charge a uniform contribution rate for their DB schemes. Inevitably, some participants pay too much and others too little for their pension accrual. Participants accumulate and pay premiums for new pen-sion rights annually of which the actuarially fair costs increase with age. Premium payments made by young participants accumulate interest for a longer period of time than that of old participants and the probability of reaching the retirement age is higher for old than for young participants. Also, the actuarial cost of pension accrual is higher for a participant with a high life expectancy than for someone with a low life expectancy. The degree of redistribution depends on a multitude of fac-tors such as mortality, age composition of the participants, interest rates, pension scheme characteristics, career paths, and so on and so forth. We focus only on intra-generational redistribution due to gender and level of education, so we assume that the pension fund does not face funding shortfalls or surpluses. Hence, participants receive exactly their accrued pension during retirement. We also abstract from in-flation and indexation which are connected to the uncertain purchasing power of pensions. These simplifications can be justified by arguing that we are only in-terested in ex ante redistribution, i.e. redistribution that occurs regardless of the financial situation of the fund or changes in macro economic variables.

4.1 Stylized pension fund

We consider a stylized pension fund that resembles a common Dutch average-pay

defined benefit pension scheme. Workers enter the pension fund at age xe = 25,

retire at age xr = 67 and we assume that they do not survive past xh = 99. We

make use of average wages per gender and level of education in 2008 (CBS, 2010) in order to calculate the pension accrual and contributions for a single cohort of

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CHAPTER 4. REDISTRIBUTION 37

participants aged 25 in 2008 over their entire lifetime. The different combinations of gender and level of education as specified in the previous chapter yield eight groups of participants. We take the distribution of participants over these groups to be

equal to the countrywide distribution1 _{of the workforce. Calculations are based on}

the pension base yh,i,x, which is defined as

yh,i,x= wh,i,x− F, (4.1)

where h ∈ {m, f } denotes gender, i ∈ {L, LS, HS, H} denotes level of education,

x ∈ [xe, xr] denotes age, wh,i,xis the annual salary and F is the state pension offset2.

Participants do not need to accumulate pension over their entire income, since they will receive a state pension as well. Individual pension accrual is given by

ah,i,x= αyh,i,xλh,i,x, (4.2)

where α is the accrual rate and λh,i,x is the labour force participation rate in 2009 as

given in Bonenkamp et al. (2013). The unit cost price of pension rights is calculated as δh,i,x= xr−xe Y j=x−xe+1 vh,i,j 1 + xh−xr+1 X k=1 vk xr−xe+k Y l=xr−xe+1 h,i,l ! , (4.3)

where v = (1 + r)−1 is the discount factor with r the risk-free rate of return and h,i,j

is the dynamic survival probability which corresponds to the j-th diagonal entry

of the survival probabilities ph,i,x,t of group (h, i). The term between brackets in

(4.3) is the actuarial present value of the life annuity at retirement age and the first term discounts that value to the year in which the pension rights are accrued. The actuarially fair contribution rate is then given by

πF(h, i, x) =

ah,i,xδh,i,x

yh,i,xλh,i,x

. (4.4)

As we have mentioned in the introduction of this chapter, participants do not pay

the actuarially fair contribution rate. Instead, a uniform contribution rate πU is

calculated every year and charged to all participants

πU(x) =

P

h

P

ifh,iah,i,xδh,i,x

P

h

P

ifh,iyh,i,xλh,i,x

, (4.5)

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38 4.2. DATA

where fh,i denotes the fraction of group (h, i) in the Dutch workforce. Our main

assumptions are summarized in Table 4.1.

Average-pay defined benefit scheme

Accrual rate (α) 2%

State pension offset (F ) AC12,209

Entry age (xe) 25

Retirement age (xr) 67

Highest attainable age (xh) 99

Real discount rate (r) 3%

Table 4.1: Summary of the pension fund characteristics

4.2 Data

CBS (2010) provides hourly wages by age, gender and level of education for people working in the private and public sector. Using the fractions of people working in each sector we have computed a weighted average of wages in the Netherlands. By lack of additional data, we assume that workers above the age of 64 earn the same wage as the (55-64) age group. Table 4.2 displays the annual wages.

Males Females Age L LS HS H L LS HS H 23-34 28,267 29,883 33,278 45,020 24,170 26,306 30,219 40,868 35-44 31,408 36,371 42,816 67,185 25,958 29,249 35,673 53,626 45-54 34,524 39,443 46,710 76,737 27,369 30,382 36,392 55,271 55-64 35,024 39,968 47,304 77,704 27,664 30,776 37,659 54,051

Table 4.2: Annual wages in 2008 in euros calculated from income data from (CBS, 2010). Males Females Age L LS HS H L LS HS H 25-34 72.6% 82.0% 85.2% 87.3% 24.0% 40.9% 58.8% 72.6% 35-44 73.5% 84.2% 89.8% 90.4% 29.8% 40.3% 50.7% 62.6% 45-54 71.1% 83.3% 86.2% 87.6% 27.5% 36.4% 48.1% 61.5% 55-64 44.3% 53.3% 54.6% 60.0% 12.1% 18.2% 25.5% 37.6% 65-74 3.7% 6.0% 6.5% 6.5% 1.1% 1.4% 2.1% 3.2%

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Males Females

L LS HS H L LS HS H

Workers in 000’s 314 826 1,864 1,483 365 989 1,831 1,276

% of workforce 3.5% 9.2% 20.8% 16.6% 4.1% 11.1% 20.5% 14.3%

Table 4.4: Number of workers by gender and level of education in the Netherlands in 2008, source: CBS.

Participation rates by level of education are usually not publicly available. Bo-nenkamp et al. (2013) calculated these using data from a national survey amongst the workforce (EBB), see Table 4.3. Using these participation rates, we can com-pute the average wage for each of the eight groups. Table 4.4 shows the distribution of the Dutch workforce which we use to determine the uniform contribution rate. As for mortality rates, we augment the mortality projections given in the previous chapter with observed mortality rates from CBS for the years 2008-2011.

4.3 Net benefit

Bonenkamp (2013) quantifies lifetime redistribution as net benefit (N B)

N B(h, i) = P Vb(h, i) − P Vc(h, i), (4.6)

which is the difference between the present values of future pension benefits and contributions. More specifically, this is as follows

P Vb(h, i) = xr−1 X j=xe ah,i,jδh,i,jΨ(h, i, j), (4.7) P Vc(h, i) = xr−1 X j=xe

πU(j)yh,i,jλh,i,jΨ(h, i, j), (4.8)

Ψ(h, i, x) =        1 if x = xe, x−xe Y j=1 vh,i,j if xe < x < xr. (4.9)

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40 4.3. NET BENEFIT

(2013) emphasizes that net benefit is a rather narrow concept for redistribution, since it does not take into account welfare effects derived from sharing risk among pension fund participants. Nonetheless, it does allow us to get an idea of the degree of redistribution and how it is affected by various factors. Although the absolute net benefit as described in (4.6) is denoted in euros, it should not be interpreted too literally. It is the actuarial present value of future cashflows which is subject to assumptions concerning income, participation rates, interest rates and so on. In the next tables we also present the relative net benefit, which is the net benefit as a percentage of the present value of pension benefits, a more tangible concept of redis-tribution. Nevertheless, we still present the absolute net benefit for the comparison of the degrees of redistribution.

r = 3% xr= 67 Males Females

L LS HS H L LS HS H

AG best estimate -3,127 -2,422 -2,069 2,055 -434 158 548 2,220

LC best estimate -4,273 -3,700 -3,593 154 -1 903 1,885 5,167

Table 4.5: Absolute net benefit for the baseline scenario

r = 3% xr = 67 Males Females

L LS HS H L LS HS H

AG best estimate -7.5% -4.0% -2.6% 1.3% -4.0% 0.8% 1.5% 2.8%

LC best estimate -10.7% -6.3% -4.6% 0.1% 0.0% 4.4% 5.1% 6.3%

Table 4.6: Relative net benefit for the baseline scenario

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Forecast uncertainty

It is rather naive to assume that future mortality rates will exactly follow our fore-casts. It is therefore better to account for the possibility that future mortality might deviate from our predictions. The Lee-Carter model allows us to sketch possible sce-narios using the computed confidence intervals. We take the upper 95% confidence bound of predicted life expectancy as a possible ’bad’ scenario and the lower 95% confidence bound as a possible ’good’ scenario as seen from a pension fund’s point of view. The effects of the realization of the two scenarios are displayed in Table 4.7 and Table 4.8.

L LS HS H L LS HS H

LC upper 95% bound -3,824 -3,407 -3,356 -353 86 926 1,892 5,044

LC lower 95% bound -4,758 -4,024 -3,867 673 -86 890 1,896 5,331

Table 4.7: Absolute net benefit for a possible bad and good scenario

L LS HS H L LS HS H

LC upper 95% bound -9.0% -5.5% -4.1% -0.2% 0.7% 4.3% 4.9% 5.9%

LC lower 95% bound -13.0% -7.4% -5.4% 0.5% -0.8% 4.6% 5.5% 6.9%

Table 4.8: Relative net benefit for a possible bad and good scenario

The results can be explained by a number of effects. Education specific life ex-pectancy as described in the previous chapter is greatly influenced by the order of magnitude of mortality rates. This is inherent to the methodology suggested by

H´ari et al. (2006) in which education specific mortality is found by scaling average

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42 4.4. SENSITIVITY ANALYSIS -15% -10% -5% 0% 5% 10% ML MLS MHS MH FL FLS FHS FH AG best estimate LC best estimate LC 95% upper bound LC 95% lower bound

Figure 4.1: Relative net benefit for the baseline scenario

4.4 Sensitivity analysis

Our calculations are subject to various assumptions which, if changed, could greatly alter our outcomes. It is therefore interesting to see what happens to our results when we change some assumptions. We change one assumption at a time while keeping other parameters constant, i.e. ceteris paribus.

4.4.1 Discount rate

The discount rate plays an important role in determining the present value of future pension benefits and contributions. We therefore investigate the effects of different discount rates on the degree of redistribution. The results are summarized in Tables 4.9 - 4.12.

L LS HS H L LS HS H

AG best estimate -2,422 -1,971 -1,730 1,280 -282 202 573 2,039

LC best estimate -3,296 -2,966 -2,927 -277 62 797 1,638 4,385

Table 4.9: Change in absolute net benefit due to a 1% lower discount rate compared to the baseline scenario

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L LS HS H L LS HS H

AG best estimate -0.5% -0.4% -0.3% 0.0% 0.0% 0.3% 0.3% 0.4%

LC best estimate -0.8% -0.6% -0.4% -0.2% 0.3% 0.5% 0.6% 0.6%

Table 4.10: Change in relative net benefit due to a 1% lower discount rate compared to the baseline scenario

L LS HS H L LS HS H

AG best estimate 1,345 1,072 930 -785 169 -93 -283 -1,056

LC best estimate 1,833 1,623 1,591 60 -20 -420 -868 -2,344

Table 4.11: Change in absolute net benefit due to a 1% higher discount rate com-pared to the baseline scenario

L LS HS H L LS HS H

AG best estimate 0.5% 0.3% 0.2% 0.0% 0.0% -0.3% -0.3% -0.4%

LC best estimate 0.7% 0.5% 0.4% 0.1% -0.3% -0.5% -0.5% -0.6%

Table 4.12: Change in relative net benefit due to a 1% higher discount rate compared to the baseline scenario

are affected more by the change in discount rate compared to the other groups, since the duration of their pensions is longer. Consequently, this increases the degree of redistribution. Analogously, we can argue that a higher discount rate leads to a lower degree of redistribution. Lowering the discount rate with 1% increases the gap by 83% with the AG model and 81% with the Lee-Carter model, whereas increasing the discount rate by 1% reduces the gap by 45% and 44% for respectively the AG model and Lee-Carter model. We observe that for the Lee-Carter model, high educated males become net contributors with the lower discount rate, even though they were previously net receivers. This is likely due to the strong cross-gender redistribution present in the Lee-Carter model.

4.4.2 Retirement age

In view of the increasing trend in life expectancy, both the state pension and second pillar retirement age will be linked to the development of life expectancy in the future. We examine how redistribution changes when the second pillar retirement age is increased. The results are summarized in Table 4.13 and Table 4.14.

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44 4.4. SENSITIVITY ANALYSIS

L LS HS H L LS HS H

AG best estimate 168 71 37 -271 51 37 53 53

LC best estimate 226 117 86 -254 43 19 21 -20

Table 4.13: Change in absolute net benefit due to a higher retirement age compared to the baseline scenario

L LS HS H L LS HS H

AG best estimate -1.2% -0.7% -0.5% 0.1% -0.3% 0.4% 0.5% 0.6%

LC best estimate -1.8% -1.2% -0.9% -0.2% 0.5% 1.0% 1.1% 1.2%

Table 4.14: Change in relative net benefit due to a higher retirement age compared to the baseline scenario

years less to enjoy their retirement. In relative terms, however, the impact is larger for the groups with the lowest life expectancy. This can easily be seen in Table 4.14 where only the groups with the lowest life expectancy are worse off due to the higher retirement age. The gap between the largest tax and subsidy is slightly decreased by 2% and 3% for respectively the AG and the Lee-Carter model. The results of the sensitivity analysis are summarized in Figure 4.2.

-2% -1% 0% 1% ML MLS MHS MH FL FLS FHS FH AG 2% LC 2% AG 4% LC 4% AG 70 LC 70

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4.5 Summary

In this chapter we investigated the ex ante redistribution in our stylized pension fund. Following the method of Bonenkamp (2013) we have computed both absolute and relative net benefits for the eight different groups of participants using two different mortality tables. We found that using the Lee-Carter forecast increased the gap between the largest net contributor and net receiver by 77% compared to using the AG forecast. Nevertheless, the overall picture remains the same. Low educated females and all males with the exception of the high educated are worse off by participating in the pension fund, whereas the other groups benefit from participating in the fund. The higher degree of redistribution resulting from using the Lee-Carter forecast stems from the relatively large differences between male and female life expectancy. Cross-gender redistribution is much less pronounced in the AG forecast. Considering that forecasts are never really certain, we have constructed prediction intervals with the Lee-Carter model to get an idea of possible future scenarios. We have used the upper and lower confidence bounds of the Lee-Carter forecast of life expectancy as a possible bad and good scenario, respectively. In the bad scenario the degree of redistribution decreased slightly, because of smaller differences in both intereducational and gender life expectancy. We observed the opposite effect for the good scenario.

Actuarial Studies