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P E N S I O N F U N D S P E C I F I C E X P E R I E N C E M O RTA L I T Y

Corrections on Mortality Rates due to Socioeconomic Differences and

the Impact on the Liabilities of a Pension Fund

l au r a m e n t i n g 1 0 2 1 9 4 3 9

A Master’s Thesis to obtain the degree in

ac t u a r i a l s c i e n c e a n d m at h e m at i c a l f i n a n c e Carried out at

Supervisor: Mr. Z. li MPhil Second reader: dr. S. van Bilsen

In-company supervisor: drs. I. Hauet AAG RBA August 9, 2016 – Final version

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iii

Statement of Originality

This document is written by Student Laura Menting 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 completion of the work, not for the contents.

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A B S T R A C T

The mortality rates of participants in the portfolio of a pension fund could deviate from the mortality rates of the average Dutch population. This deviation in mortality, known as experience mortality, is attributed to differences in socioeconomic status. In this thesis, socioeconomic correc-tions on average Dutch mortality rates are estimated. These correccorrec-tions adjust the mortality rates to fit a pension fund specific portfolio. The corrections are estimated, with methods of generalized linear models, by using mortality data from the Dutch Central Bureau of Statistics on socioeconomic indicators. These indicators measure the socioeconomic status of an individual by the level of education and income class. Due to overdispersion in the data, a negative binomial regression is performed.

The results of this regression show a difference between the corrections on mortality rates. A higher level of education or income results in lower mortality rates. By analyzing the impact on the liabilities of a pension fund, the optimal method to correct the mortality rates for a pension fund is determined. According to the results, a pension fund should ana-lyze the proportional division of the level of education of its participants and weight the estimated corrections on education respectively.

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A C K N O W L E D G E M E N T S

I would like to take this opportunity to show my special thanks to the people who have supported and helped me writing my Master’s thesis. Firstly, I would like to express my sincere gratitude to my supervisor Merrick Zhen Li for his insightful comments, motivation and support. His guidance helped in all time of writing this thesis. It was a pleasure working with him.

Furthermore, I would like to show my special thanks to Ivo Hauet, my supervisor at Sprenkels en Verschuren, for his guidance. His support and our fun conversations helped me through this period. I thank Marieke Klein for providing me with material and for our discussions on the sub-ject matter. Additionally, i would like to thank my other colleagues for their encouragement to successfully complete this thesis.

I would like to show my sincere gratitude to my parents, Frans and Nicasia Menting, for their unconditional support and patience

And last, I would like to thank my friends for their loving support. Today, after an intensive period of four months, I proudly present my Master’s thesis to obtain the Master’s degree in Actuarial Science and Financial Mathematics.

Laura Menting

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C O N T E N T S

1 i n t ro d u c t i o n 1

2 l i f e e x p e c ta n c y a f f e c t i n g p e n s i o n f u n d s 3

2.1 Impact of longevity on pension funds . . . 3

2.1.1 Longevity in the Netherlands . . . 3

2.1.2 AG mortality table . . . 5

2.1.3 Expected versus realized mortality . . . 5

2.1.4 Regulation . . . 6

2.1.5 Pension provision . . . 6

2.2 Corrections on mortality rates . . . 9

2.2.1 Socioeconomic difference in life expectancy . . . . 9

2.2.2 Portfolio-specific corrections . . . 11

3 m o d e l i n g m o rta l i t y 13 3.1 Exponential mortality forecasting models . . . 13

3.2 Portfolio-specific corrections . . . 14

3.3 Generalized Linear Models . . . 14

4 t h e data f r a m e wo r k 17 4.1 Choice of data . . . 17

4.2 Level of education . . . 18

4.3 Income class . . . 20

5 t h e d e s i g n o f t h e m o d e l 23 5.1 Generalized Linear Model . . . 23

5.2 Distribution of the data . . . 25

5.3 Fitting the regression model . . . 26

5.4 Goodness of fit . . . 27

5.5 Corrections on mortality rates . . . 27

6 t h e r e s u lt i n g c o r r e c t i o n s 29 6.1 Level of education . . . 29

6.2 Income class . . . 34

7 i m pac t o f e x p e r i e n c e m o rta l i t y o n p e n s i o n f u n d s 37 7.1 Data of a pension fund . . . 37

7.2 Impact on pension funds . . . 42

8 c o n c l u s i o n 49 b i b l i o g r a p h y 51 a ac t u a r i a l t h e o ry 55 a.1 Mortality and survival probabilities . . . 55

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x Contents

a.2 Old-age pension . . . 56

b e x p o n e n t i a l f a m i ly 59 b.1 Gamma-Poisson Mixture . . . 59

b.2 Mean and Variance Functions . . . 60

c i t e r at i v e ly r e - w e i g t h e d l e a s t s q u a r e s 63 d ta b l e s o f r e s u lt s 67 d.1 The parameter estimates . . . 67

d.2 Education mortality corrections . . . 69

d.3 Income mortality corrections . . . 71

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1

I N T R O D U C T I O N

Recently pension funds have had rough times regarding their financial condition. According to the De Nederlandse Bank (2013), 68 pension funds in the Netherlands cut the pension rights of their participants in 2013. Approximately 5.6 million participants of pension funds suffer from these cuts. Due to the decline of their accumulated pension rights, many people lost trust in the pension funds which weakened the reputation of these funds. This weakened reputation caused a decrease in the number of active participants and a drop in the number of pension funds in the Netherlands. One of the reasons for these rough financial times for pension funds is the longevity of its participants.

According to data from the Dutch Central Bureau of Statistics (CBS,

2016b), the Dutch population is getting older. The life expectancy is increasing which causes a longer period of retirement. The pension has to be paid out for more years, causing financial problems for most pension funds. The government is deferring the retirement age to decrease the pension payout years. However, this deference of the retirement age is not an immediate solution to the longevity problem. It is crucial for pension funds to use correct predictions of the life expectancy to make a proper estimation its liabilities.

The Dutch Actuarial Association, in Dutch named Actuarieel Genootschap (AG,2014), publishes a mortality table for financial institutions to base

their calculations on. This table specifies future mortality rates for the average Dutch population. Pension funds use this mortality table to es-timate their liabilities. However, the mortality rates of an individual can differ from the mortality rates of the average Dutch population. This dif-ference is known as experience mortality. Pension funds have to adjust the average mortality rates to match their specific portfolio.

Mortality rates of the AG are adjusted by making corrections on the mortality rates due to differences in socioeconomic status. The level of education, income class, occupational sector and geographical area are indicators for the socioeconomic status. For example, an individual who

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2 i n t ro d u c t i o n

achieved a higher level of education is expected to live longer than an individual with a low level of education. Income class is positively corre-lated with life expectancy. People with a heavy physical job are expected to die younger than people working in an office job. Living in a big city is expected to have a negative impact on a person’s health. Corrections on mortality rates are estimated based on these socioeconomic indicators.

This study is concentrated on the main question:

“What is the optimal method to estimate pension fund specific corrections on mortality?"

In this study, mortality data from the Dutch Central Bureau of Statis-tics (CBS) of the Dutch population is used, concerning level of education and income class. This data, with a distribution of the exponential fam-ily, is used to model mortality with methods of generalized linear models. From the results, the corrections on mortality rates due to income class and level of education are estimated and the impact of these corrections on the liabilities of a specific pension fund is analyzed.

This thesis is structured as follows. Firstly, Chapter2provides insight into the life expectancy of the Dutch population and the importance to estimate this life expectancy correctly for pension funds. In Chapter 3, studies on modeling mortality rates are discussed. Chapter4describes the mortality data of the CBS. Chapter 5focuses on describing the method of solving the generalized linear model and estimating the corrections. In Chapter6, the resulting corrections are analyzed. Chapter7 analyzes the impact on the liabilities of a pension fund. Chapter8 concludes this thesis and gives a suggestion for further research.

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2

L I F E E X P E C TA N C Y A F F E C T I N G P E N S I O N F U N D S

As mentioned in the introduction, it is crucial for pension funds to use accurate mortality predictions of its participants for the estimation of the liabilities. For the prediction of mortality, pension funds should take the increasing life expectancy and socioeconomic influences on the life expectancy into account. Section2.1elaborates on how the increasing life expectancy influences pension funds. Section2.2focuses on differences in mortality rates due to socioeconomic status.

2.1 i m pac t o f l o n g e v i t y o n p e n s i o n f u n d s

Unexpected longevity is problematic for both the pension fund and its participants. There are several factors contributing to the increasing life expectancy of the Dutch population.

2.1.1 Longevity in the Netherlands

Historic data provides information about the development of the life ex-pectancy in the Netherlands. According to data from the CBS (2016b), men born between 1861 and 1866 had a life expectancy of 36 years and women, born in this period, were expected to live 38 years. This life ex-pectancy has increased significantly over time. Men and women born in 2014, respectively, have a life expectancy of 78 and 83 years.Beer(2006) studied the influences on the development of the life expectancy of the Dutch population. According to his study, the increase in life expectancy can be explained by changes in environmental factors, a decrease of fatal diseases, improved health care, less accidental affairs, less malnutrition, improved dinking-water supplies and improved sewer systems.

According to data of the CBS, the increasing life expectancy is different for men and women. Figure 2.1a is based on the dataset of the CBS

(2016b) and visualizes the life expectancy at birth for men and women for the period of 1950 to 2014. Based on the same dataset, Figure 2.1b

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4 l i f e e x p e c ta n c y a f f e c t i n g p e n s i o n f u n d s

shows the one-year mortality rate for men and women at the age of 50. According to these figures, women have a higher chance of reaching a higher age than men. Reasons for this difference are, for instance, that men have a higher expectation to suffer from heart diseases, women have a stronger immune system and women do less dangerous actions than men. Figure 2.1 shows that the difference in life expectancy and mortality rates between men and women increased in the years after 1950. Beer

(2006) suggested that this increasing difference after 1950 is caused by the smoking behavior of men. After 1970, the percentage of smoking men dropped and the difference between the life expectancy of men and women declined. However, women were and still are expected to live longer than men.

(a)Life expectancy at birth

(b) One-year mortality rate at age 50

Figure 2.1. (a) The life expectancy at birth and (b) the one-year mortality

rate at age 50 for the period of 1950 to 2014 for men and women. The data is obtained fromCBS(2016b).

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2.1 impact of longevity on pension funds 5

It is certain that during the past decade the life expectancy of the Dutch population has increased excessively. However, the increasing trend has not been smooth and it is uncertain to what limit this increase will extend. The AG provides a mortality table in which they try to make an accurate prediction of this increasing trend.

2.1.2 AG mortality table

Every two years the AG publishes a new mortality table based on the Dutch population. This mortality table is published to inform the fi-nancial sector about new developments in mortality rates. The AG uses data of the CBS and the Human Mortality Database (HMD). This data contains information about mortality of the Dutch population and com-parable European countries. The most recent table, called AG2014 (AG,

2014), gives the best estimates of the one-year mortality rates for men and women separately. The rates in the table are specified for ages of 0 to 120 with a horizon of the year 2014 to 2184. Although this table is thought to provide the best predictions on mortality, these predictions are no guarantees.

2.1.3 Expected versus realized mortality

The realized mortality rates, hence the life expectancies, could deviate form the AG estimates. For example, the predicted life expectancies in a former table of the AG (2010) is different from the realized life ex-pectancies. Table2.1 shows the deviance in life expectancy at birth for the period 2004, 2005 and 20061.

The deviance between the life expectancies is positive. The realized life expectancy turned out higher than predicted. Pension funds, which based their calculation on the AG predictions, have to pay out the pension for a longer period than expected.

Men Women

Year AG Realized Deviance AG Realized Deviance 2004 76.42 76.60 0.18 81.03 81.16 0.13 2005 76.59 76.98 0.39 81.09 81.47 0.38 2006 76.75 77.38 0.63 81.16 81.77 0.61

Table 2.1. The AG predicted and the realized life expectancy at birth for the

Dutch population, (AG, 2010).

1 The life expectancy of the AG is calculated from the one-year predicted mortality rates and the realized life expectancy from the observed one-year mortality rates. Ap-pendixA.1describes how the life expectancy is calculated.

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6 l i f e e x p e c ta n c y a f f e c t i n g p e n s i o n f u n d s

2.1.4 Regulation

When the observed life expectancy of the Dutch population is higher than predicted, problems arise for both the pension fund and its participants. To protect both the pension fund and its participants from cuts and from the fund going bankrupt, all pension funds are monitored by a regulator. The DNB, De Nederlandse Bank, monitors the Dutch pension funds. The Netherlands is set up with several requirements for pension funds regard-ing their financial condition. These requirements are documented in the nFTK. The nFTK is the new financial assessment framework for pension funds in the Netherlands. According to the nFTK, (Overheid, 2016a), pension funds have to use accurate mortality predictions for the estima-tion of their liabilities. Therefore, pension funds use the most recently published mortality table of the AG for their calculations. The liabilities of a pension fund is the sum of all the accumulated pensions of the par-ticipants. The connection between the assets and liabilities of a pension fund is the funding ratio. The funding ratio is a measurement of the financial condition of a pension fund. The funding ratio is defined by:

F R = Total assets

Total liabilities. (2.1)

According to the DNB(2016), there is a minimal required funding ratio for pension funds. The regulator wants pension funds to be able to pay out the pensions to their participants and to hold a buffer in case of a finan-cial depression. If the funding ratio is lower than the required minimal, the pension fund is required to reestablish its financial condition. The reestablishing of the financial condition could lead to cutting the pension rights of the participants. When monitoring the financial condition of a pension fund, the regulator checks if the liabilities of the pension funds are calculated using the correct interest rates and mortality assumptions.

2.1.5 Pension provision

A pension fund holds a provision to fulfill the pension payouts for all the participants. If a pension is accumulated, a provision should be built. The sum of the provisions is a liability on the balance sheet of a pension fund. Different kinds of pension are that have to be taken into account when calculating the provision of a fund. These types are:

• Old-age pension: starts nowadays usually at an age of 67 and is paid out from the moment the participant stops working.

• Survivors pension: a payment to the partner and orphans starting after the death of the participant.

• Occupational disability pension: starts when the participant be-comes disabled to work.

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2.1 impact of longevity on pension funds 7

This thesis focuses on the type old-age pension. An old-age pension provision is the present value of the expected old-age pension payout based on the accumulated pension rights of the participants. The pension payout is usually a monthly periodic payment, series of these periodic payments are called an annuity. An old-age pension provision is calculated using actuarial annuity factors. In this thesis, the assumption is made that the first half of the yearly pension is paid out at the beginning of a year and the second half at the end of the year2. Furthermore, the

assumption is made that all participants retire at the age of 67. To explain the calculation of the provision, the following symbols are defined:

r is the interest rate

vt = (1+1r)t is the t year discount rate

x is the age of a man y is the age of a woman

px is the one-year survival probability of a man at age x

tpx = t−1

P

k=0

px+k is the probability a man lives at least another t years

qx = 1 − px is the probability a man with age x dies within 1 year

tqx = 1 −tpx = t

P

k=0

kpx· qx+k is the probability a man with age x dies

within t years.

¨ax = ∞

P

k=0

vk·kpx is a direct entering lifelong annuity for a man with

constant payments at the beginning of the year.

˜ax = ∞

P

k=0

(12 ·kpx· vk+ 12 ·k+1px· vk+1) is a direct entering lifelong

annuity for a man with constant payments every half year.

For example, the old-age pension provision of a 50-year-old man with a yearly pension entitlement of C is calculated by :

O P50 = ∞ X k=12 C ·(1 2 ·kpx· vk+ 12 ·k+1p50· vk+1) = C ·12p50· v12· ˜a67. (2.2)

Here, the product of the annuity factor and the yearly pension entitle-ment is multiplied by12p50· v12. The probability that a 50-year-old man

survives up to the pension age is 12p50 and v12 is included to discount

the entitlements to the present. The general formulation for calculating

2 Considering a monthly pension payout, this choice is better than assuming yearly pen-sion is paid out once a year, considering the discounting of the penpen-sion. Furthermore, this method it is not too complex.

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8 l i f e e x p e c ta n c y a f f e c t i n g p e n s i o n f u n d s

the old-age provision is denoted by Equation (2.3). This is the formula for calculating the present value of the old-age pension for a male with age x and pension age 67, receiving every year C euro from the age of 67.3

O Px = C ·67−xpx · v67−x· ˜a67. (2.3)

The impact of the longevity on the provision of a pension fund is cal-culated and presented in Example1.

Example 1 The old-age pension of a man with age 67 and

a pension entitlement of e30,000 is calculated using the mor-tality rates of 2000, 2010, 2020, 2030 and 2040. The mormor-tality rates are derived from previous and recent AG tables. The interest rate is assumed to be 3%.

Mortality rates Provision Increase

2000 e345,070

-2010 e369,560 e24,490

2020 e391,364 e21,804

2030 e413,544 e22,180

2040 e433,911 e20,367

Table 2.2. The old-age pension provision for 67 year old man, with pension

entitlement of e30,000, based on different mortality rates.

Table2.2 shows that the provision of the old-age pension in-creases due to lower mortality rates every 10 years. Due to longevity, pension funds have to hold more provision each year. The right column of Table2.2shows the increase of the provision compared to the provision based on mortality rates of 10 years before. The increase is on average around e22,000 every 10 years.

As shown in Table2.2, due to longevity, the pension funds have to hold a higher provision every year. In this example, the provision increases ev-ery 10 years with approximately e22,000. This is the increase for only one person, whereas a pension fund could have over thousands of par-ticipants in its portfolio. This increase shows the impact of longevity on the provision of a pension fund. If a fund does not hold enough provi-sion, because of mortality rates that have been predicted too high, the funding ratio will decrease. This decrease occurs because the fund has to take the new developments of the mortality rates into account and

3 More on actuarial notation and calculating the discounted value of the old-age pension is described in AppendixA.2.

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2.2 corrections on mortality rates 9

has to adjust its provision accordingly. As the provision will increase, the liabilities of a fund increase and consequently the funding ratio decreases (see Equation (2.1)).

The mortality rates should be accurate to calculate the pension pro-vision, hence the liabilities, correctly. The mortality rates of the partici-pants of a pension fund could deviate from the average Dutch mortality rates in the AG mortality table.

2.2 c o r r e c t i o n s o n m o rta l i t y r at e s

The mortality table of the AG provides recently updated mortality rates of the Dutch population by age and gender. Besides age and gender, the socioeconomic status of an individual provides information about his or her life expectancy.

2.2.1 Socioeconomic difference in life expectancy

Socioeconomic status is the position of a person according to his or her rank on the social ladder. The difference in rank is explained by knowl-edge, possessions and employment. According toGalobardes et al.(2006), there is not a definite way to measure the socioeconomic status. However, there are several indicators to approximate the socioeconomic status of an individual. The most used indicators are the level of education, income class, occupational sector and geographical area. These indicators could help to estimate the life expectancy of an individual more accurately.

Education

According toGalobardes et al.(2006), the level of education can be used as a benchmark for the socioeconomic status of an individual. The level of education affects a person’s socioeconomic status. Firstly, education contains information about future job and income. Secondly, education gives a reflection of the change from childhood to adulthood. Furthermore, the cognitive functioning of people is affected by the level of education. People who enjoyed a higher education have, on average, more knowledge on how to live a healthier life.

The effect of education on the life expectancy of the Dutch population is shown by Bruggink (2012). He used data from CBS for the period 1997-2000 and 2007-2010. With this data, he showed the difference in life expectancy between people with the highest level of education and the lowest level of education in the Netherlands. Bruggink (2012) showed a gap of nearly seven years in life expectancy between high educated and low educated people. The gap stays the same for the period of 1997-2000 and the period 2007-2010. This proves that the level of education has an impact on the life expectancy of an individual in the Netherlands.

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10 l i f e e x p e c ta n c y a f f e c t i n g p e n s i o n f u n d s

Income

Another often used indicator is the level of a person’s income. On average, people with a lower income have a worse health than people with a higher income.Kippersluis et al.(2008) studied the relationship between health and income and concluded that there is a correlation between income and health. People with a lower income have a double disadvantage on health. Firstly, a low income can disable someone from good health-care. Secondly, a person with a bad health has a lower labor force participation and consequently earns less income. Concluding, people with a lower income have a higher probability of a bad health, hence a higher mortality risk.

This correlation between mortality and income is also shown byKnoop and Brakel(2010). In this study, data from the CBS is used to study the effect of income on the life expectancy of the Dutch population. Averagely, in 2007, the Dutch population with a high income has lived nearly six years longer than people with an income below the poverty level. Their study showed a positive correlation between the level of a person’s income and his or her life expectancy in the Netherlands.

Occupational sector

The socioeconomic status, and thus the life expectancy of an individual, is influenced by their occupation. People working a physically intensive job have a higher mortality risk than people with an office job. Dangerous work decreases a person’s life expectancy because this person has a higher chance of getting involved in an accident. Furthermore, working in a bad environment may have a negative influence on a person’s health. This shows that the occupational sector may influence the life expectancy of an individual.

Geographical area

A less obvious indicator of socioeconomic status is the geographical area. However, there are regional differences in socioeconomic status and life expectancy. For example, a male in Delft is expected to live 10 years longer than a male in Kerkrade. Figure 2.2 is published by the Dutch national institute for public health and environment, (RIVM,2014). This figure shows a difference in life expectancy between regions. People in the North-East of the Netherlands and in the big cities have a lower life expectancy. People living in the middle and the west of the Netherlands have a higher life expectancy, with an exception for people living in big cities.

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2.2 corrections on mortality rates 11

Figure 2.2. Life expectancy per region in the Netherlands in the period

2009-2012, (RIVM, 2014).

People living in big cities have a lower life expectancy compared to the average population because of more traffic accidents, pollution and crime. People living close to an airport or highway have a lower life expectancy because of air pollution by smog. According toMackenbach et al.(2012), regions, where most people have a certain religion, can also influence the life expectancy. Religion influences a person’s behavior, hence has an impact on a person’s lifestyle and health. For example, smoking is common among the roman catholic community in the Netherlands. These reasons give an explanation for the 10 years difference in life expectancy between the man living in Delft and a man living in Kerkrade.

2.2.2 Portfolio-specific corrections

The socioeconomic status contributes to making a more accurate estima-tion of a person’s life expectancy. Therefore, socioeconomic status of the participants of a pension fund should be investigated. Consequently, if the socioeconomic status of the participants deviates from the average Dutch population, the mortality rates of the AG table should be adjusted.

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12 l i f e e x p e c ta n c y a f f e c t i n g p e n s i o n f u n d s

For example, the portfolio of a pension fund for dentists includes highly educated participants. Furthermore, dentists earn a high income. The so-cioeconomic status of these participants is higher than the average Dutch population. If a pension fund for dentists would be using the mortality rates of the average Dutch population to calculate their liabilities, these liabilities will turn out too low. Hence, the mortality rates need to be adjusted.

The mortality rates of the average Dutch population can be adjusted by making portfolio-specific corrections on mortality. These corrections are determined by the indicators influencing a person’s socioeconomic status. Pension funds can decide how the corrections are estimated and on which indicators these corrections are based. However, the corrections need to be explainable to the regulator and their participants. Currently, there is no tight method to specify the corrections, so these corrections are subject to discussion.

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3

M O D E L I N G M O RTA L I T Y

This thesis, on estimating corrections on mortality rates, also known as experience mortality, is based on previous studies on modeling the mor-tality of a population. In this chapter, the contributing parts of these previous studies on modeling mortality are discussed. The main focus lies on the exponential character of mortality and on using methods of a generalized linear model. The focus of Section3.1lies on a mortality fore-casting model and the exponential character of mortality. In Section3.2, a study on specifying portfolio-specific factors is discussed. Section 3.3

provides insight into former studies on generalized linear models.

3.1 e x p o n e n t i a l m o rta l i t y f o r e c a s t i n g m o d e l s

Mortality models often assume mortality to have an exponential charac-ter. Several famous models on forecasting mortality have been and are still used in predicting the future mortality of a population. These models forecast the force of mortality, µx. The force of mortality can be defined

as the instantaneous rate of mortality at age x measured on a yearly ba-sis.1 A famous model to forecast the force of mortality is the Gompertz

model. The Gompertz model, depending on age x, is given by:

log(µx) =a+B · x with B > 0 and x ≥ 0. (3.1)

Gompertz(1825) expected the force of mortality to increase exponentially with age over most of an adult’s lifetime. This exponential increase is shown by writing the Gompertz model as:

µx =ea+B·x. (3.2)

1 In AppendixA.1, the force of mortality is derived from mortality and survival proba-bilities.

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14 m o d e l i n g m o rta l i t y

The Gompertz model assumes mortality to have an exponential char-acter. Considering the exponential character of mortality, a generalized linear model is convenient in modeling mortality because of the link func-tion, which is explained in Section 3.3. To estimate the corrections on mortality rates using a GLM, the corrections are viewed as factors.

3.2 p o rt f o l i o - s p e c i f i c c o r r e c t i o n s

Vellekoop et al.(2015) did research on portfolio-specific mortality, taking into account the importance for insurers and pension funds to value their liabilities correctly. According toVellekoop et al.(2015), portfolio-specific factors are a common way to estimate the portfolio-specific mortality. According toPlat (2009), these portfolio-specific factors are specified as:

Pt,x =

qt,xA

qpopt,x . (3.3)

Here, x is the age, t is the calendar year, qA

t,x is the portfolio-specific

mortality rate for age x and calendar year t and qpop

t,x is the mortality rate

of the average population. Pt,x is the portfolio-specific factor for age x

and calendar year t. In this thesis, these factors are called corrections. To estimate portfolio-specific corrections, Vellekoop et al.(2015) suggested the usage of a generalized linear model.

3.3 g e n e r a l i z e d l i n e a r m o d e l s

A generalized linear model (GLM) has convenient properties for model-ing mortality. The GLM is introduced byNelder and Wedderburn(1972). The GLM is a generalization of an ordinary linear regression. The re-sponse variable of the GLM is allowed to have errors with another distri-bution than the normal distridistri-bution. Another characteristic of the GLM is that the response variable is connected to the linear model by a link func-tion. This characteristic is convenient considering the Gompertz model in Equation (3.1). For example, the response variable µx is related to the

linear model a+B · x by a log-link function g(µx):

g(µx) =log(µx) =a+B · x. (3.4)

A GLM has a stochastic component, defined in this thesis as the num-ber of deaths, with a distribution of the exponential family. According to Vellekoop et al. (2015), it is better to define the stochastic compo-nent as the observed deaths rather than the portfolio-specific corrections. There are fewer uncertainties in modeling observed deaths compared to observed portfolio specific factors, because within a calendar year these factors could be very volatile. Therefore, the stochastic component is defined as the number of deaths.

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3.3 generalized linear models 15

Concerning count data on the number of deaths, the stochastic compo-nent in the GLM is likely to have an overdispersed Poisson distribution. Count data is often assumed to have a Poisson distribution. However, ac-cording toCurrie(2014), the Poisson distribution could be problematic in studies on observed data. The Poisson distribution has the characteristic of an equal mean and variance which may not be realistic. Usually, the variance of observed data is greater than its mean, which causes overdis-persion. To deal with overdispersion in the data, a negative binomial distribution could be considered according to Douglas Stirling (1984). He studied a Poisson process containing groups with different means. He explained how to clarify the differences in the means by assuming the mean to have a random distribution itself. The mean is proportional to a Gamma distribution. The mixture of this Poisson-Gamma distribution results in a negative binomial distribution.

Douglas Stirling(1984) extended his research by solving a GLM with an iteratively re-weighted least squared (IRLS) technique introduced by

Nelder and Wedderburn (1972). The IRLS method is also suggested by

Breslow (1984), who shows the ease of implementing the IRLS method in the free software R. He discussed the usefulness of the IRLS method to solve GLM’s and modeling mortality. The software R is also used by

Richards(1984) to model mortality. He researched different methods to include socioeconomic differences in a statistical model to estimate mor-tality. He states that the simplest model assumes an individual to have an exponentially distributed lifetime. This exponential distribution can be transformed into a more practical model for actuaries using the software R.

The studies, discussed in this chapter, contribute to this thesis on es-timating the portfolio-specific corrections on mortality. According to the exponential character of mortality, a GLM with a log-link function, is used to model mortality and to estimate the corrections on mortality. The stochastic component of the GLM is assumed to be Poisson or nega-tive binomial distributed depending on overdispersion in the count data on the number of deaths. A GLM in the software R and the IRLS method is used to fit the model and to estimate the parameters. These estimates lead to the estimation of the corrections. The next chapter, Chapter 4, describes the mortality data of the Dutch population, which is used in modeling mortality.

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4

T H E D ATA F R A M E W O R K

To estimate the corrections on mortality due to socioeconomic status, a regression is performed on mortality data of the Dutch population. There is not a tight method to estimate the socioeconomic status of an individ-ual. Therefore, there is not exact data showing the influence of socioeco-nomic status on mortality in the past. However, the level of education, income class, occupational sector and geographical area are benchmarks for the socioeconomic status. In this chapter, the data used to estimate the corrections on mortality rates is described. To estimate corrections on mortality for pension funds, only mortality data on the level of ed-ucation and income class is used. The reason for not including data on occupation sector and geographical area is explained in Section4.1. Sec-tion4.2focuses on the mortality data on the level of education. Data on mortality and income is explained in Section4.3.

4.1 c h o i c e o f data

The data on mortality needs to satisfy criteria to be functional for pension funds for calculating the corrections on mortality. First of all, the data should be valid. Secondly, the cost aspect of achieving the data needs to be investigated. In addition, the study on the data needs to be explainable to the public, namely the regulator and the participants. For example, there are parties claiming to have mortality data about the population on a 6-digit postal code level. They want to sell their data at a high price, however, basing the corrections on mortality on a person’s postal code may seem as a vague approach by the regulator and pension fund’s participants. They might question if this party has validly obtained the data.

The indicators geographical area and occupational sector are not cluded in estimating the corrections because mortality data on these in-dicators is not satisfying the criteria. The factor geographical area is not included because there could be heterogeneity between the population

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18 t h e data f r a m e wo r k

living in a region. Additionally, valid data on mortality of the population on postal code level is also costly and not easy to achieve. Furthermore, basing corrections on geographical area is difficult to explain to partici-pants: is it proper to assign a certain life expectancy based on the region a person is living in? Next to the indicator geographical area, the indicator occupational sector is also not included in this study. The mortality data on occupational sector is not useful. Although the data is not costly, there is no data with a proper division between physical intensive occupational sectors and non-physical intensive sectors.

In this thesis, separate datasets on mortality for education and income class are used. There is no valid mortality data available with informa-tion about both income and level of educainforma-tion of the Dutch populainforma-tion. According to the CBS, this data is not available because of privacy re-quirements. Therefore the level of education and income are analyzed separately.

4.2 l e v e l o f e d u c at i o n

The data of the Dutch population is obtained from theCBS(2016a). The data is for the period of the year 2007 to 2010. The CBS published the life expectancy of the Dutch population for different educational levels. The four different levels of education in the data are1:

• University and HBO (uni) • VWO, HAVO and MBO (vhm) • VMBO (vmbo)

• Primary school (prim)

The data is transformed to obtain the number of deaths. From the life expectancies for different ages, gender and level of education, the mortality rates are derived. Figure4.1shows the differences in mortality rates and life expectancies for men with different levels of education. To approximate the number of deaths, these mortality rates are multiplied by the numbers of the Dutch population for age, gender and highest achieved level of education. These are the numbers on 24 September 2010 and obtained from a database of the CBS, (Graham and Lalta,

2013). This transformation of the data, to obtain the number of deaths, makes the data more practical for modeling.

1 The Dutch abbreviation of the educational levels are used because of the difference between education levels in the English language.

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4.2 level of education 19

(a)Mortality rates

(b) Life expectancy

Figure 4.1.(a) The mortality rates and (b) the life expectancies of men by age

for different levels of education for the period 2007-2010, (CBS,2016a).

Data of the Dutch population for ages from 21 until 90 are used in this study. The law of pensions states an age boundary for pension partici-pants, (Overheid,2016b). According to this law, accumulation of pension rights for an employee starts at last at the age of 21 or on a later date when the employee first begins to work. Therefore, the mortality data of the population from an age of 21 is chosen. The data of the Dutch popula-tion is used until the age of 90 because there are not enough observapopula-tions for higher ages. This makes the data invalid for higher ages. Furthermore, the socioeconomic differences in the life expectancy decrease at high ages. Next to the data on education, the mortality data on income class is also of the Dutch population from the age from 21 until 90.

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20 t h e data f r a m e wo r k

4.3 i n c o m e c l a s s

The data on income and mortality is obtained from the CBS, (Muller and Geertjes,2014), (de Mooij et al.,2012). Two datasets provide numbers of the Dutch population for the period 2007-2010. These numbers contain the number of deaths and the number of people within different income classes. The number of deaths are given for a 4-year period. The number of deaths in these 4-years is divided by 4 to correspond to the average number of deaths in one year. Corresponding to the average number of deaths, the average of the number of people in this 4-year period is taken. Figure4.2 shows the difference in mortality rates for the income classes. These mortality rates are calculated by dividing the number of deaths by the number of people. The height of the income in the data is the sum of wage, profit and pension. The income classes are:

• no income or unknown (i0) • less than e15,000 (i1)

• between e15,000 and e25,000 (i2) • between e25,000 and e35,000 (i3) • between e35,000 and e45,000 (i4) • between e45,000 and e55,000 (i5) • more than e55,000 (i6)

Figure 4.2. The mortality rates for men according to income class for the

period 2007-2010, (Muller and Geertjes, 2014), (de Mooij et al.,2012).

The data is adjusted to deal with uncertainties in the data. The data shows an error from the age of 65 in the class ‘no income or unknown income’ (i0 in the legend of Figure 4.2). In the period 2007-2010, peo-ple in the Netherlands received the social security contribution from the age of 65. The people with an unknown or no income started to gain a known amount of income due to the social security contribution. To deal

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4.3 income class 21

with this uncertainty in the data, the number of people in this group is assumed to decrease exponentially. The number of deaths is calculated by multiplying the mortality rates in the original data of the ’no income or unknown’ group by the adjusted number of people.

The data of the CBS on the level of education and income class is used to estimate the corrections on mortality. Chapter 5 describes the model and the method to estimate these corrections.

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5

T H E D E S I G N O F T H E M O D E L

A generalized linear model (GLM) is used to model the mortality data on the level of education and income class. The modeling of mortality is performed in the software R, (R Core Team,2014).

In Section 5.1, the GLM is described and the stochastic component of the GLM is analyzed in Section 5.2. This analysis of the stochastic component is performed by investigating the distribution of the observed deaths. Section5.3 focuses on solving the model. The method to derive the corrections from the model parameter estimates is described in Sec-tion5.4.

5.1 g e n e r a l i z e d l i n e a r m o d e l

A GLM is a functional model for a regression of the exponential family. As described byKaas et al.(2008), a GLM consists of three components, the stochastic component, the systematic component and the link function.

• The stochastic component

This stochastic component Y corresponds to the number of deaths. This component assumes the number of deaths to be an indepen-dent random variable with a distribution of the exponential family. Count data often has a Poisson distribution, however, the mean and variance of this observed data will probably not be equal. The data will most likely be overdispersed, which is analyzed in Section 5.2. • The systematic component

The systematic component ascribes a linear predictor to every ob-servation. The linear predictor is given by ηi =Pjxijβj. The linear

predictor is linear in the parameters β1, β2, ...., βp. Here, xij are the

covariates, i indicates a particular group1 and j is the number of

parameters .

1 Each group has its own unique profile for age, gender and income class or level of education.

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24 t h e d e s i g n o f t h e m o d e l

• The link function

The link function is the link between the expected value µi of Yi

and the linear predictor ηi. The link function is g(·), so ηi=g(µi).

Mortality is assumed to increase exponentially. Therefore, the linear predictor is connected to the expected value by a log-link function:

ηi=g(µi) =log(µi).

The general regression model is defined by:

ηi=log(µi) =β0+xi1β1+xi2β2+...+xipβp. (5.1)

The regression model, on the mortality data of level of education, is defined by:

log(Yi) =β0+agei· β1+genderi· β2+unii· β3+vhmi· β4

+vmboi· β5+log(exposurei).

(5.2)

Here,

idenotes the particular group Y = the number of deaths age= the age

gender = 1 if female, 0 otherwise

uni= 1 if the level of education is university or HBO, 0 otherwise vhm= 1 if the level of education is VWO, HAVO or MBO, 0 otherwise vmbo= 1 if the level of education is VMBO, 0 otherwise

exposure= the size of the group

The variable primi, corresponding to primary school, is not in the

re-gression because of collinearity. The explanatory variables uni, vhm, vmbo, and prim are dummy variables. For every observation i, the sum of these variables equals 1. To deal with collinearity, one of these dummy variables is left out of the regression. The variable prim is corresponding to the coefficient β0.

The term log(exposurei)is included in the regression to correct for the

different sizes of each group. This term is called the offset. For example, there are 5000 women with the age of 32 with an educational level of primary school, while 43000 women with the age of 32 have an educa-tional level of VWO, HAVO or MBO. More women are exposed to death in the bigger group. To correct for different group sizes, the offset term is included in the regression.

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5.2 distribution of the data 25

The regression model, on the mortality data on income class, is defined by:

log(Yi) =β0+agei· β1+genderi· β2+i1i· β3+i2i· β4+i3i· β5

+i4i· β6+i5i· β7+i6i· β8+log(exposurei).

(5.3)

Here,

idenotes the particular group Y = the number of deaths age= the age

gender = 1 if female, 0 otherwise

i1 = 1 if the income class is <e15,000, 0 otherwise i2 = 1 if the income class is e15,000-e25,000, 0 otherwise i3 = 1 if the income class is e25,000-e35,000, 0 otherwise i4 = 1 if the income class is e35,000-e45,000, 0 otherwise i5 =1 if the income class is e45,000-e55,000, 0 otherwise i6 = 1 if the income class is >e55,000, 0 otherwise exposure= the size of the group

The variable i0, corresponding to no income or unknown income, is not included in the regression to correct for collinearity. This variable is corresponding to the coefficient β0.

To perform the regressions, the distribution of the data is analyzed to define the distribution of the stochastic component Y of the GLM.

5.2 d i s t r i b u t i o n o f t h e data

As discussed in Section 3.3, the distribution of observed data will most likely be overdispersed. To confirm the overdispersion in the data, a test on overdispersion is performed in R by using the AER package, (Kleiber and Zeileis, 2008). Suppose E(Yi) = µ is the expected value of Yi and

var(Yi)its variance. The distribution is overdispersed if α, in the

equa-tion var(Y) = (1+α)∗ µ, is significantly bigger than zero. The null hypothesis of the test states that there is existence of equidispersion, which means the mean and variance are equal. This hypothesis is tested against the alternative of overdispersion. These hypotheses are defined by:

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26 t h e d e s i g n o f t h e m o d e l

This test is performed on the two mortality data on education and income. The test on overdispersion is performed starting with a Pois-son regression. The coefficient α is estimated by an auxiliary ordinary least squares (OLS) regression. This coefficient is tested by the z-statistic which is asymptotically standard normal distributed under the null hy-pothesis. The results of the overdispersion test are shown in Table5.1.

Data Dispersion z-value p-value Education 1.606 4.182 1.4e-05 Income 11.438 14.287 2.2e-16

Table 5.1.The result of the test on dispersion.

The z-values of the tests exceed the critical value 1.645, the 5% signifi-cance level of the standard normal distribution. The p-values are smaller than 5%, so the null hypothesis of both tests can be rejected against the alternative. Concluding, in both data-sets there is existence of overdis-persion.

To deal with the overdispersion in the data, a mixture of the Poisson distribution is considered. The negative binomial distribution is a Poisson-Gamma mixture. The negative binomial model is suggested to deal with overdispersion, it gives a better fit on the data. This Poisson-Gamma mixture is explained as follows:

Suppose Y is the number of deaths and the conditional distribution

Y on λ has a Poisson distribution, Y |λ ∼ P oisson(λ). The mean of the Poisson distribution, λ, is a random variable with a Gamma distribution,

λ ∼ Gamma(θ, k). Then the unconditional variable Y is negative bino-mial distributed. The probability density function of Y can be derived from the probability density functions of a mixture of the Poisson and the Gamma distribution. This derivation is given in AppendixB.1. The result of the derivation of the probability density function is defined as:

f(y; k, θ) = Γ(θ+y) Γ(y+1)Γ(θ)( k 1+k) θ·( 1 1+k) y, (5.4) with θ > 0 and k 1+k <1.

Here, y has only values of non-negative integers. The mean of the proba-bility density function is µ= θk.

To conclude, the number of deaths is assumed to have a negative bi-nomial distribution of the form Y ∼ NB(θ,1+kk). Therefore, to fit the regression model, a negative binomial regression on the data is performed.

5.3 f i t t i n g t h e r e g r e s s i o n m o d e l

The negative binomial regression model is now defined as:

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5.4 goodness of fit 27

with µ= kθ, and Y ∼ NB(θ,1+kk).

The negative binomial regression is performed to find estimates for the unknown parameter θ and the parameters β1, β2, ...., βp. To fit the

regres-sion model, the maximum likelihood method is used to find estimates for

β given θ by iteration. Within each iteration, an estimate for θ is found,

given β. This technique leads to estimates for the parameters. There is a tight formulation, a general solution method, to find the estimates for the parameters. This method is explained by the iteratively re-weighted least squares algorithm. The algorithm is described in AppendixC.

With the MASS package from Venables and Ripley (2002), the fitting of the model is performed in R. The solution leads to the parameter estimates. After fitting the model, a test is performed to analyze if the model has an adequate fit on the data.

5.4 g o o d n e s s o f f i t

The analysis of deviance is a method to test whether the model has an adequate fit to the data. To perform the test, the scaled deviance is defined. According to Kaas et al. (2008), the scaled deviance is written as: D φ =−2 log ˆ L ˜ L. (5.6)

Here, D is the deviance, ˆL is the likelihood of current model and ˜L is the likelihood of the full model. The full model is a model with as many parameters as observations. This scaled deviance has approximately a χ2

distribution with the degrees of freedom equal to the number of obser-vations minus the number of estimated parameters in the current model. According to the analysis of deviance test, the model has an adequate fit if the p-value is above the critical value. With this test, the goodness of fit of the model is analyzed. If there is an adequate fit to the data, the corrections are calculated from the parameter estimates.

5.5 c o r r e c t i o n s o n m o rta l i t y r at e s

The calculation of the corrections on mortality starts with calculating mortality rates. The mortality rates for the different groups are calculated from the parameter estimates of β1, β2, ...., βp. The estimated one-year

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28 t h e d e s i g n o f t h e m o d e l

in a year by the total number of people. The number of people is the exposure per group. The one-year mortality rate for group i is given by:

qi= Yi exposurei = e β0+xi1β1+xi2β2+...+xipβp+log(exposurei) exposurei = exposurei· e β0+xi1β1+xi2β2+...+xipβp exposurei =0+xi1β1+xi2β2+...+xipβp. (5.7)

The correction is calculated by dividing the mortality rate of group i by the average mortality rate of the Dutch population. Take qav as the

average one-year mortality rate, for the corresponding gender and age of group i, for the same period as the other mortality data, the period 2007-2010. The correction for group i is defined as:

Cori=

qi

qav. (5.8)

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6

T H E R E S U LT I N G C O R R E C T I O N S

In this chapter, the results from modeling the mortality data are ana-lyzed. Due to the fact that two different datasets are used, two separate regressions are performed to calculate the corrections. The adequacy of the fit of the model on the two datasets is analyzed and the resulting corrections are calculated and visualized. In Section6.1the results of the regression on the level of education are analyzed and Section6.2analyzes the results of the regression on income class.

6.1 l e v e l o f e d u c at i o n

This section analyzes the results of fitting the regression model to the data on level of education. Firstly, the adequacy of the model is analyzed. Secondly, the parameter estimates are interpreted and the corrections are calculated.

The regression model for level of education is defined by:

log(Yi) =β0+agei· β1+genderi· β2+unii· β3+vhmi· β4

+vmboi· β5+log(exposurei),

(6.1)

with µ= kθ, and Y ∼ NB(θ,1+kk).

The estimates for the parameters β0, β1, ..β5 and θ are found by fitting

the regression. Table 6.1 shows the estimates for the parameters β, the standard error, the exponentiated estimates, the z-value and the p-value

1. The estimate for θ is 138 with standard error 9.4.

1 AppendixD.1presents the 95% confidence level of the estimates.

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30 t h e r e s u lt i n g c o r r e c t i o n s

Coefficients β Standard error eβ z-value p-value

Intercept -10.232 0.044 3.601e-05 -234.89 <2e-16

age 0.098 0.014 0.594 -36.74 <2e-16

gender -0.522 0.001 1.103 174.51 <2e-16

uni -0.617 0.020 0.540 -31.01 <2e-16

vhm -0.469 0.020 0.626 -24.05 <2e-16

vmbo -0.338 0.019 0.713 -17.66 <2e-16

Table 6.1. Summary of the resulting coefficient estimates from the regression

on education.

Adequacy of the model

According to the p-values of the estimates, all coefficients are significant. The p-values are smaller than the critical value of 0.01. The z-value is the test statistic of the Wald-test. Under the null hypothesis of the Wald-test, the parameter βj is zero:

H0 : βj =0 Ha: βj 6=0.

The z-value of the estimates are all larger than the 95% critical value of 1.96. Therefore, the null hypothesis is rejected. Concluding, the pa-rameters are significantly deviating from zero.

The model fits the data adequately. This conclusion follows from ana-lyzing the residual deviance of the model. This residual deviance provides insight into the response of the model when the parameter estimates are included in the model. The residual deviance tests whether the model has an adequate fit under the null hypothesis. The p-value of this test is 0.85, which is larger than the critical value of 0.05. This p-value supports the null hypothesis, which is not rejected. The model has an adequate fit and the estimates of the coefficients are significantly deviating from zero.

Interpretation of the estimates

The parameter estimates are interpreted. The exponential value of the intercept, corresponding to primary school education, is taken as refer-ence. The general formulation of calculating the mortality rates is given by Equation (5.7). With the parameter estimates substituted in the equa-tion, this results in:

qi=e−10.232+agei·0.098+genderi·−0.522+unii·−0.617+vhmi·−0.469+vmboi·−0.338.

(6.2) The following examples explain how to interpret Equation (6.2). The one-year mortality rate of a male with age 75 and primary school as highest achieved level of education is estimated by:

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6.1 level of education 31

qmale,75,prim=0+75·β1+0·β2+0·β3+0·β4+0·β5

≈ e−10.232+75∗0.098≈0.0560

The estimation of the one-year mortality rate of a 36-year-old female with university as the highest achieved level of education is calculated by:

qf emale,36,uni=0+36·β1+1·β2+1·β3+0·β4+0·β5

≈ e−10.232+36·0.098−0.522−0.617≈0.000392

The corrections are calculated by dividing the one-year mortality rates for a specific group by the average mortality rates of the whole Dutch populations as described in Section5.4. These corrections are displayed in Figure6.12.

Figure 6.1a shows the corrections on mortality for men due to the level of education. If the correction is above 1, the one-year mortality rate for a person with the corresponding educational level is higher than the mortality rate for the average Dutch population. The opposite holds for a correction below 1. The corrections increase until the age of 42 and decrease a little thereafter. The correction for primary school is, from the age of 26, above 1 and tips the 1.8 level at the age of 42. The correction for VMBO is above 1 from the age of 33 till the age of 76, thereafter it decreases to 0.9. The correction for VWO, HAVO, and MBO is almost always below 1 except for the age 35 till 50. The correction for university and HBO is always below 1.

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32 t h e r e s u lt i n g c o r r e c t i o n s

(a) Men

(b)Women

Figure 6.1.The corrections on mortality due to the level of education for (a)

men and (b) women for the age 21 until 90.

Figure 6.1b visualizes the corrections for women due to the level of education. In the figure, there are two bumps, one at the age of 35 and one at the age of 71, hereafter the corrections decrease. The correction for primary school is, from the age of 25, above 1 and decreases after the last bump to 1. The correction for VMBO fluctuates around 1. The correction for VWO, HAVO and MBO is below 1 except for the bump at the age of 71. The correction for university and HBO is always below 1. The resulting corrections are in line with the expectations. The highest level of education has the lowest correction on the average mortality, whereas the lowest level of education has the highest correction rate. The one-year mortality rates for people with the lowest level of education are higher than the average mortality rate of the Dutch population. Whereas

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6.1 level of education 33

the mortality rates for people with the highest level of education is lower than the average mortality rates. According to these result, people with a higher level of education have a higher life expectancy. This is consistent with former studies on differences in life expectancy due to the level of education, as described in Section2.1.

The mortality rates for the level of education are derived from the data of the CBS on the life expectancy. A test is performed to analyze if the calculated life expectancy is close to the life expectancy of the CBS. The calculated life expectancy is based on the estimated corrections. As shown in AppendixD.4, the calculated life expectancy is close to the life expectancy of the CBS. Concluding, the model gives good estimates of the corrections on mortality due to educational differences.

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34 t h e r e s u lt i n g c o r r e c t i o n s

6.2 i n c o m e c l a s s

In this section, the results of solving the regression model on the data on income class are analyzed. Firstly, the result of testing the adequacy of the model is described. Secondly, the parameter estimates are interpreted and the corrections are calculated and analyzed.

The regression model for income class is defined by:

log(Yi) =β0+agei· β1+genderi· β2+i1i· β3+i2i· β4+i3i· β5

+i4i· β6+i5i· β7+i6i· β8+log(exposurei)

(6.3)

with µ= θk, and Y ∼ NB(θ,1+kk).

The parameter of β0, β1, ..β8 and θ are estimated by fitting the model.

The estimates for the parameters β, the standard error, the exponentiated estimates, the z-value and the p-value are displayed in Table 6.23. The

estimate of θ is 14 with a standard error of 0.88.

Coefficients β Standard error eβ z-value p-value

intercept -9.798 0.0447 5.554E-05 -219.07 <2e-16

age 0.103 0.0217 1.109 172.26 <2e-16 gender -0.663 0.000599 0.515 -30.57 <2e-16 i1 -0.200 0.0360 0.819 -5.56 2.75E-08 i2 -0.771 0.0366 0.462 -21.08 <2e-16 i3 -0.975 0.0371 0.377 -26.27 <2e-16 i4 -1.099 0.0390 0.333 -28.15 <2e-16 i5 -1.158 0.0425 0.314 -27.26 <2e-16 i6 -1.266 0.0422 0.282 -29.98 <2e-16

Table 6.2. Summary of the resulting coefficient estimates from the regression

on income.

Adequacy of the model

The model fits the data on income adequately. This conclusion follows from analyzing the adequacy of the model, as explained in Section 6.1

for the level of education. The p-values of the coefficient estimates are all smaller than the critical value of 0.01. Therefore, the parameter estimates are significant. The z-values are above the 95% critical value of 1.96. The null hypothesis that the parameters are zero, is rejected. The estimates of the parameters are, according to the p-value and the z-value, significantly deviating from zero.

The adequacy of the model is tested by analyzing the residual deviance. The p-value of the residual deviance test is equal to 0.57. The p-value is larger than the critical value of 0.05, so the null hypothesis, that the model has an adequate fit, is not rejected. Concluding, the model has an adequate fit on the data on income and the estimates are interpreted.

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6.2 income class 35

Interpretation of the estimates

The estimates are interpreted in a similar way as the estimates for the level of education. The intercept is corresponding to the reference income class i0, corresponding to ’no income or unknown income’. Equation (6.4) calculates the one-year mortality rates for different income classes.

qi =0+agei·β1+genderi·β2+i1i·β3+i2i·β4+i3i·β5+i4i·β6+i5i·β7+i6i·β8

≈ e−9.80+agei·0.10+genderi·−0.66+i1i·−0.20+i2i·−0.77+i3i·−0.98+i4i·−1.01+i5i·−1.16+i6i·−1.27.

(6.4)

In Equation (6.4) the following values for a particular group are in-serted to calculate the one-year mortality rates: the age, the gender (0 for male, 1 for female), 1 for the corresponding income level and 0 for the other income levels. From the one-year mortality rates, the corrections on mortality due to income class are calculated and displayed in Figure6.2 4.

Figure6.2avisualizes the corrections for men due to income class. The corrections increase until the age of 40, hereafter the corrections stay around the same. The corrections for the higher income classes are closer to each other than for the lower income classes. The corrections for the three higher income classes are always below the level 1. The correction for income class i3, ’e25,000-e35,000’, is around 1 after the age of 40.

Figure6.2bshows the corrections for women due to income class. Com-pared to the corrections for men, there is less fluctuation in the correc-tions at low ages but more at higher ages. The correccorrec-tions for higher income classes are, as for men, closer to each other than the corrections for low-income classes. The three highest income classes stay below the level 1, except for income class i4 at the age of 70. The income class i3 is below 1 until the age of 60 and above 1 from 60 till 80.

The results of the corrections are in line with the expectations. The fig-ures show that the corrections for the low-income levels are higher than the corrections for the higher income levels. Thus, the one-year mortal-ity rates for lower income classes are higher compared to high-income classes. This is consistent with former studies on mortality and income, as described in Section 2.1. Both for men and women, the corrections for the income class i0, ’unknown and no income’, are very high with respect to the other income classes. These corrections are not used by pension funds. Pension funds can always derive the income of a partici-pant. When the income of a participant is unknown, pension funds derive the income from the accumulated pension of this participant. In addition, When a participant has no income, the income class is derived from the accumulated pension or from information about previous income.

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36 t h e r e s u lt i n g c o r r e c t i o n s

(a) Men

(b) Woman

Figure 6.2.The corrections on mortality due to income class for (a) men and

(b) women for ages 21 until 90.

The optimal method for a pension fund, to correct the mortality rates for its specific portfolio, is analyzed by investigating the impact on the provision of a pension fund. The impact of correcting the average Dutch mortality rates for a pension fund is analyzed in Chapter7.

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7

I M PA C T O F E X P E R I E N C E M O RTA L I T Y O N

P E N S I O N F U N D S

In this chapter, the impact of experience mortality, by using the estimated corrections, on the provision of a pension fund is analyzed. Followed by investigating the optimal method to correct the average Dutch mortality rates. The impact is analyzed for a pension fund in the Netherlands, the portfolio of this pension fund is described in Section7.1. In Section 7.2, the impact of using corrections on the provision of the Dutch pension fund is analyzed.

7.1 data o f a p e n s i o n f u n d

The data of the Dutch pension fund contains information about the in-come, the level of education and the accumulated pension rights of the participants in the portfolio of the fund. The data is obtained from the database of the participants on the date 31/12/2013. The database con-tains the 1651 active participants who were included in the portfolio on 31/12/2013. An active participant is a person who is currently building up a pension. The data only includes active participants of the fund be-cause these are the participants from whom the level of education and income class are both known.

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38 i m pac t o f e x p e r i e n c e m o rta l i t y o n p e n s i o n f u n d s

Figure 7.1. The percentage of men and women in the pension fund.

The percentage of men and women in the fund is shown in Figure7.1. The portfolio is dominated by male participants by 80%, only 20% of the participants are female. The male to female ratio is 4 to 1. This data shows that the occupational sector this pension fund corresponds to, which is the corn industry, is dominated by men.

The level of education and the income class of the participants are analyzed by gender. In Figure 7.2, male and female participants are di-vided according to their level of education. Figure 7.2ashows most male participants have the level of education VWO, HAVO and MBO. The percentage of the male participants with an educational level of univer-sity and HBO is only a little smaller. The smallest percentage, around one sixth of the male participants, have an educational level of VMBO. This is in contrast to female participants, around a third of the female participants have an educational level of VMBO. This is shown in Fig-ure 7.2b. Compared to the male participants, there are proportionally fewer women with an educational level of university and HBO. All the participants had a higher education than solely primary school.

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