orphan’s pensions
M.W.L. Bos
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: M.W.L. Bos
Student nr: 10682368
Email: maartenbos87@gmail.com
Date: May 8, 2017
Supervisor: Prof. dr. M.H. Vellekoop Second reader: dr. S. van Bilsen
Statement of Originality
This document is written by Maarten Bos 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.
Abstract
In this thesis a model is constructed to calculate the pension pro-vision for deferred orphan’s pensions. This model is based on data for education, mortality and birth rates, largely collected from Statis-tics Netherlands (CBS). The collected information is used to create a model that determines a best estimate provision per participant based on gender, age and education level. Furthermore the model is checked for robustness towards certain underlying assumptions. The expected final age of the orphan’s pension as well as the interest rate are as-sumptions that influence the outcome substantially.
Currently, it is customary to set the pension provision for deferred or-phan’s pension equal to a percentage (5%) of the provision of deferred spouse pensions. It is checked whether this percentage, which is often outdated, is a good indicator of the best estimate provision. The out-come of the model shows that the 5% increment is over prudent and that the actual best estimate is roughly 0.5%, depending on the pop-ulation and level of education. Next to the best estimate, the variety of possible outcomes is checked using a stochastic model for mortality rates.
Preface vi 1 Introduction 1 1.1 Orphan’s Pension . . . 1 1.2 Pension Provision . . . 1 1.3 Research Goal . . . 2 1.4 Organisation . . . 2 2 Current situation 3 2.1 Provision . . . 3
2.2 Provision for Orphan’s Pensions . . . 3
3 Relevant Factors 5 3.1 Mortality . . . 5
3.1.1 Basic Mortality . . . 5
3.1.2 Experience Mortality . . . 6
3.2 Expected Number of Children . . . 7
3.2.1 Population Average . . . 8
3.2.2 Educational Effect . . . 8
3.3 Expected Age of Children . . . 9
3.3.1 Population Average . . . 9
3.3.2 Educational Effect . . . 9
3.4 Educational Level of Children . . . 10
3.4.1 Maximum Age for Orphan’s Pension . . . 10
3.5 Interest Rates . . . 12
4 Deterministic Model 13 4.1 Modelling Experience Mortality . . . 13
4.1.1 Data . . . 13
4.1.2 Experience Mortality by Education . . . 14
4.2 Modelling the Expected Number of Children . . . 18
4.2.1 Data . . . 18
4.2.2 Modelling Expected Number of Children by Education . . . 20
4.3 Orphan’s Pension Provision . . . 23
4.3.1 Model of Orphan’s Pension . . . 23
4.3.2 Orphan’s Pension as Increment of Spouse Pension . . . 25
5 Stochastic Model 27 5.1 Best Estimates in Stochastic Mortality . . . 27
5.2 Stochastic Mortality . . . 29 iv
6 Results 30
6.1 Deterministic Model . . . 30
6.2 Sensitivity Analysis . . . 31
6.2.1 Orphan’s Benefit Length . . . 31
6.2.2 Interest Rates . . . 33 6.3 Stochastic Longevity . . . 34 6.4 Stochastic Mortality . . . 35 7 Conclusions 39 References 40 Appendix A 42 Appendix B 46
Firstly I would like to thank Prof. dr. Michel Vellekoop, my supervisor of the faculty of Economics and Business at the University of Ams-terdam. With his advice he steered me in the right direction and he made sure the schedule was met in the end.
Secondly I want to thank Bert Hogendoorn, my supervisor of Willis Towers Watson who helped me with his expertise on this subject. I also would like to thank all other Willis Towers Watson colleagues at the Apeldoorn office for all support and advice, not always asked for. Last but not least I want to thank all other persons who supported me to complete this research.
Introduction
Anyone who lives in the Netherlands is entitled to a payment after retirement. This payment is called the AOW-payment. The amount of AOW-payment the retiree receives depends on the number of years he has lived in the Netherlands. It is not relevant whether he was employed or not. Besides the AOW-payment most former employed retirees have an additional pension payment. This is due to the fact that in most cases an employee is obliged to take part in the pension scheme offered by his employer. A large proportion of the employees is participating in a pension scheme. At the moment over 90% of all Dutch employees are taking part in a pension scheme [6].
When taking part in a pension plan a portion of the current salary is converted into income after retirement. Participating in a pension scheme is also a form of insurance, for example in case of disability during employment. Furthermore, in case of death of the main insured during or after active employment, a spouse can be entitled to payments. This is the case for children of the main insured as well, in which case the payment is called orphan’s pension.
1.1
Orphan’s Pension
The conditions for claiming an orphan’s pension are described in the pension regulations of the pension plan the deceased main insured participated in. It is not mandatory for an employer to accomodate a pension plan that includes an orphan’s pension. However, in practice almost all pension plans contain regulation on orphans. In detail the charac-teristics can differ between pension schemes, broadly they are similar most of the time. The pension provider pays out an orphan’s pension to the child when the deceased main insured was at the time either:
• An active employee,
• A former employee with deferred orphan’s pension rights,
• A retiree receiving a benefit payment with deferred orphan’s pension rights. After death of the main insured the orphan’s pension starts immediately. The pay-ment will stop when the orphan reaches the maximal age as described in the pension scheme. In most cases this maximal age equals 18 or 21 years with an increased maximal age for students.
1.2
Pension Provision
When participating in a pension scheme all accrued pension rights are managed by a pension provider. This can be an insurer or a pension fund. The pension provider needs to make sure that there are enough resources available to pay out all accrued pension
rights, today and in the future. Therefore the provider calculates a pension provision for its portfolio of pensions. This provision is the best estimate for expected future payments, possibly including a risk margin. For old age pension and spouse pension the provision is calculated using the expected future cashflows. For deferred orphan’s pension this is generally not the case. This provision is often estimated based on the level of the provision for the spouse pension. This is due to pragmatic considerations. The provision for deferred orphan’s pension is relatively small. Moreover, determining the cashflow for deferred orphan’s pension is more complicated than for old age pension or spouse pension. Most pension providers determine the provision for orphan’s pension by adding a fixed percentage to the provision for deferred spouse pension. The increment is often assessed once, without being recalibrated after that.
1.3
Research Goal
Because the added percentage for orphan’s pension is often unaltered and outdated, the corresponding provision may be very prudent. This leads to the main research question of this thesis:
How can the estimation for the provision for deferred orphan’s pensions be improved? To resolve the main research question some research questions are formulated: 1. How are the provisions for orphan’s pensions currently determined?
2. Which extra factors can be taken into account when calculating a provision for deferred orphan’s pensions?
3. How can the chosen factors be used to determine a better provision for deferred orphan’s pensions?
1.4
Organisation
To examine the main research question it is required to investigate the current situation. Chapter 2 starts with an overview that gives more insight in orphan’s pensions in general. In this section the provision for orphan’s pensions is explained and how it is currently determined. In chapter 3 all relevant factors that can influence the liability of orphan’s pensions are discussed. This is done based on aggregated data and scientific publications. Using all factors that affect orphan’s pensions a deterministic model is constructed in chapter 4. In chapter 5 the robustness of the deterministic model is checked by simulating a stochastic model. The outcome of the model will be compared to the current situation in chapter 6.
Current situation
The Dutch pension system consist of three pillars. The first pillar pension is a general pension benefit (AOW), as elaborated on in the introduction. In the second pillar, additional pension is accrued by most employees in a pension plan. These two benefits combined should be sufficient for retirees to maintain a reasonable standard of living. The third pillar consist of private pension savings, provided by a bank or an insurer.
Pensions in the second pillar are accrued by participating in a pension scheme offered by an employer. The Dutch government encouraged to participate in a pension scheme by practising a delayed tax system. The premiums paid for pension in the second pillar are tax free (up until a certain accrual rate) and benefits are taxable. Although the employer offers the pension plan, the pension capital is placed outside the company that offers the plan. This pension capital is transferred to and managed by a pension provider.
A pension provider is an institution that administrates and executes the pension agreement between an employer and employee. The employer is obliged to pay contri-butions for all his employees according to the agreement between the pension provider and the employer. Most of the time a part of these contributions is paid by the em-ployees through withholding of salary. In exchange, the provider is obliged to make benefit payments from retirement age onwards. To make sure that these benefit pay-ments can be made, the pension provider calculates a provision for all possible future benefit payments.
2.1
Provision
A pension provision can be seen as the present value of all expected future benefit payments. These payments include, next to the most common old age pension benefits, often spouse pension benefits, benefits in case of disability and benefits for orphans. A provision is determined based on the amount of pension rights, the expected mortality rates and the interest rates, which are used to discount the expected future payments. The pension provision should be backed up by the assets of the pension provider. It is important that the pension provision represents the expected value of the future benefits at all times to be able to monitor the current status of the pension provider. That is why the provision is under constant scrutiny and needs to be reported to The Dutch Central Bank (DNB) on a monthly basis.
2.2
Provision for Orphan’s Pensions
The provision for old age benefits and spouse benefits are determined as described above; the provision is a best estimate of the present value of future payments. For orphan’s pensions this is often not the case. Due to the complexity of the calculation and the relative small amount of the provision, the provision for an orphan’s pension is not
determined as the best estimate of future benefits. It is common that the provision is fixed as a percentage of the provision for deferred spouse pensions. Currently it can be seen in the market that the provision is often determined as 5% of the pension provision for the deferred spouse pensions. In most cases this amount is not based on (recent) calculations and this percentage is kept constant for many years as a result of pragmatic considerations. This leads to the question if this percentage is a good estimation and how the best estimate provision can be improved. In the following chapter all relevant factors that should be taken into account in a model for deferred orphan’s pension are described.
Relevant Factors
The previous chapter elaborated on orphan’s pensions and how it currently is calcu-lated. The research goal is to develop a model which can be used to determine the pension provision for deferred orphan’s pension. Before this model can be constructed the underlying factors that could affect the provision are investigated.
3.1
Mortality
It is key for a pension provider to have an accurate estimation of the expected amount of payments it has to pay out to its participants. This is largely dependent on the date the insured deceases as this is the date the annuity for old age ends (if it was already started) as well as the date when possible benefits for the spouse pension starts. The payment for an orphan’s pension also starts when the main insured dies, if there are children of the main insured within the age range determined by the pension regulation. Upfront the exact date of death is unknown of course. Therefore historic data is used to estimate mortality rates. These mortality rates are based on, among others, the date of birth and sex of the main insured.
3.1.1 Basic Mortality
Until a decade ago the expected mortality used to determine pension provisions was based solely on the most recent best estimate mortality rates. These rates were used to construct the entire future cashflow stream. The underlying assumption that an insured of age x at time t would have the same expected mortality rate as some other insured of age x at time t + 1 turned out to be not realistic. A publication by Van Duin [24] showed that the life expectancy in the Netherlands rapidly increased over the last decades and was likely to continue to grow in the near future. Nowadays most pension funds use mortality rates which encorporate a mortality trend. This mortality trend indicates that the mortality rates for future times are lower, which leads to higher life expectancies. The most recent study of the Royal Actuarial Association of the Netherlands [13] on expected mortality rates includes a mortality trend, and resulted in the publication of the Prognosetafel AG2016. These mortality rates are based on the entire population of the Netherlands and are used as the basis mortality rates of our model.
Figure 3.1: Period life expectancy over time.
Figure 3.1 is part of the publication of the Prognosetafel AG2016 and shows the evo-lution of period life expectation over the last decades for numerous European countries. The period life expectation is calculated using mortality rates corresponding to a single year for all ages. The period life expectation is lower than the cohort life expectation in which the estimated mortality rates for future years are used. It shows quite clearly that the estimate for future mortality rates should incorporate the given trend.
3.1.2 Experience Mortality
The Prognosetafel AG2016 represents the expected mortality rates for the entire Dutch population. However, a pension provider only has to determine a pension provision for it’s own participants. Due to the fact that the participants in a pension scheme often work in the same line of business, their characteristics tend to deviate from the entire population that the Prognosetafel is based on. The standard mortality rates do not suffice in that case.
It is common for pension providers to apply experience mortality rates when calcu-lating the pension provision. Experience mortality rates are adjustments to the standard mortality table that clarify the differences in mortality rates of a specific group compared to the entire population as a whole. These differences can be caused by a number of reasons. Well studied reasons are level of income, unemployment and level of education. These are discussed in this section based on a number of researches.
Knoops and Van den Brakel [7] have studied the effect of income on the probability of longevity. The goal of this research is to investigate social economical differences for the Dutch population. In this research, mortality data divided into different levels of income is used over the period 2004-2007. For this study five classes of income are defined. Using this data the life expectancy per age cohort, gender and each of the five income levels is determined. The outcome of this study gives insight in the difference in remaining life expectancy between income levels. It shows that men with a high level of income can live up to seven years longer than men with a low level of income. For women the difference is slightly less than seven years. At higher ages these differences increase in percentage terms. At birth the additional life expectancy due to the effect of a high income is equal to 9.8% for men, at the age of 65 this effect is 26.4%. For women these percentages are 8.6% at birth and 21.3% at the age of 65. This research shows that the level of income is a good indicator of the life expectancy and therefore could be used as input to model experience mortality rates.
et al [11]. One objective of this research is to examine what the impact of unemployment is to the physical well-being of an individual. This study is based on meta-analysis. This is a statistical analysis that uses the results of multiple studies. By aggregation of the results of diverse unemployment studies, it is possible to get a better estimation or a higher statistical power can be achieved. This meta-analysis uses 104 empirical studies from multiple countries. In total the well being of 6,684 unemployed and 15.988 employed individuals are compared. The outcome of this analysis is that the unemployed individuals had lower mental health and physical health than employed individuals. The study does not report on estimated differences in life expectancies between the employed and unemployed population.
Lleras Muney [10] have studied the relationship between level of education and mortality rates. In this research, data for the years 1960, 1970 and 1980 is used for part of the US population (circa 1% of it). This data includes the level of education per individual. For groups of people with different levels of education it is checked what portion died during the two decades. Using a regression model, it is checked whether or not education affects health. The conclusion of the study is that there is a large causal effect of education on health and mortality. An additional year of education for an individual of 35 years old result in a 1.7 year higher remaining life expectancy.
Hardarson [5] shares this conclusion in an Iceland study. The goal of this study is to estimate the relationship between educational level and mortality. A total of 18,912 people were followed over different time periods of 4 up to 30 years. This resulted in data that includes both sexes and different ages of 33 up to 81 years. This data was split in four groups based on the level of education. A multiple Cox regression analysis was used to determine the relationship between mortality and education level. The results of this study show a significant relationship between mortality and level of education.
Doornbos and Kromhout [3] study the effect of education on mortality in the Nether-lands by following approximately 80,000 men in military service. All men were born in 1932 and conscripted for military service in 1950/1951. These men were followed for a total of 32 years. Survival analysis was used to analyse the data resulting these 32 years. The conclusion of this study is that educational level and survival are negatively correlated. A study by Kunst and Mackenbach [9] uses the data of the Doornbos and Kromhout study to compare the effect of education on mortality for different countries. For the Netherlands, the mortality rates of men with a low level of education of age 38 turn out to be 72% higher than the mortality rates for men in the same age range with a high level of education. For other countries the effect is larger. The US tops the list with an average effect of 262% in the age group of 35 up to 44 years old.
These studies show that mortality rates can differ between certain groups of the population. Level of income, unemployment and level of education can be used to im-prove estimates for these mortality rate differences. The experience mortality rates in this study will be based on the education level of the main insured. It is preferred over the level of income and unemployment rate due to the fact that the education level is also used in other parts of the model (which will be discussed further on in this chapter). The model to determine orphan’s pensions will use the Prognosetafel AG2016 for the basic mortality and a set of experience mortality rates based on the characteristics of the specific population. This experience mortality will be derived from data published by CBS (Statistics Netherlands) on differences in life expectancies by education level.
3.2
Expected Number of Children
When the main insured deceases it is possible the pension provider is obliged to pay orphan’s pension. This is only the case if the decedent had any children younger than the given maximum age of receiving orphan’s pension, as described in the pension regulations. The number of children any participant of a pension scheme has is unknown
up until the death of the main insured. The number of children is often not administrated by the pension provider. To be able to determine an accurate provision for orphan’s pension, an estimation for the number of children is needed.
3.2.1 Population Average
The number of children born per person (or fertility rate) differs greatly from country to country. In countries where children are more often part of the work force or where the mortality rates are higher, the fertality rate is higher as well. This is usually the case in developing countries. The higher the development, the lower the fertility rate. This is shown by Myrskyl¨a et al [12] in which it is also concluded that in case of ’very high development’ the fertility rates increase slightly.
Next to economic influences there are political aspects that can alter the fertility rate. An example of a political measure that caused a decrease in fertility rates is the one child policy that was in effect in China from 1980 until 2015. During this period of 35 years the fertility rate dropped from 3.0 in 1980 to 1.6 in 2015 [22]. Part of the decrease can be ascribed to the improved economical situation. Another example of a political nature that influenced the fertility rate is an unbalanced political situation like in times of war. For instance, in the years following World War II the number of births radically increased [4].
Research by Van Duin and Stoeldraijer [25] published by CBS show that in the Netherlands the total fertility rate is expected to stay at the same level as it is at the moment. In contrast to the life expectancy, there appears to be no long term trend. Therefore, in our model, the expected number of children per age will be held constant over time.
3.2.2 Educational Effect
As explained earlier, the total fertility rate of the population is expected to be constant over time. However this does not mean that the fertility rate is the same for everyone in this population. Education plays an important role in making a distinction between groups with different fertility rates. Currie and Moretti [2] conclude that female col-lege graduates have less children than high school graduates. This effect seems only noticeable among women and not among men. A possible reason for this is that having children tends to influence the career of men less than it does those of women. The educational effect on the average number of children is marginal for men. A Norwegian study, Kravdal [8], even finds a small positive effect on fertility for men with a high level of education.
These conclusions also follow the CBS [15] study for the average fertility rate in the Netherlands. In this study the results are divided in three groups of education level. The outcome of this study is stated in table 3.1.
Fertility rate by education Low Med High Male 1945-1949 1.95 1.90 1.94 1950-1954 1.85 1.86 1.87 1955-1959 1.79 1.77 1.81 1960-1964 1.67 1.65 1.73 Female 1945-1949 2.06 1.87 1.60 1950-1954 2.02 1.88 1.64 1955-1959 2.02 1.88 1.65 1960-1964 1.99 1.84 1.63
Table 3.1: Level of education has an effect on the average number of children. When modelling the expected cashflows of orphan’s pensions the expected number of children for the main insured is taken into account. Also the education level of the main insured will be included as this affects the expected fertility rate. The expected number of children per level of education is derived in chapter 4.
3.3
Expected Age of Children
The pension provider is obliged to pay out orphan’s pension as long as the orphan satisfies the regulatory definition as stated in the pension regulations. One of the con-ditions is the age of the orphan. Once the child reaches the maximum age as described in the pension regulations the orphan’s pension will automatically stop and the pension provider is no longer obliged to pay. The expected value of an orphan’s pension hence depends on the current age of the children. The date of birth of the main insured’s children is often not part of the pension provider’s administration. This means that the age of the children is unknown when the main insured dies.
3.3.1 Population Average
The number of children born in the Netherlands is published annually by the CBS [18]. Also the corresponding age of the parents is available. Using this data, the expected age of the main insured’s children can be derived. These are expected values for the entire population, before adjusting for the characteristics of the pension provider’s population.
3.3.2 Educational Effect
As stated before, research shows that education of the parents (predominantly the ed-ucation of the mother) impacts the expected number of children. It not only affects the number of children however: also the age at which the children are born differs when dividing the parents by education. A study by Van Agtmaal-Wobma and Van Huis [23] shows that the age at which the first child is born is positively related to the education level of the parents. This effect is present for both father and mother. That this affects boths sexes (in contrary to the number of children) is probably due to the fact that in a relationship the education level of the male and female are related to each other as well. A highly educated man is more likely to be in a relationship with a highly educated woman. Hence the average age at which the first child is born increases for both men and women with the level of education.
To model the expected cashflows of orphan’s pensions, the education of the main insured is taken into account; this affects the expected number of children as well as their age.
3.4
Educational Level of Children
Orphan’s pension starts when the main insured dies. The entitlement continues as long as the orphan is alive and fills the conditions as stated in the pension regulations. In most pension schemes the age at which orphan’s pension ends is 18. However an exception is made in case the orphan is still following education. The entitlement then ends whenever the following happens: the education is completed or aborted or the age of 27 is reached. This means that the expected duration of orphan’s pension depends on the education the child follows or is going to follow after the age of 18. Children with lower education are most likely to enter the labor market directly after high school. The orphan’s pension will be stopped by the pension provider in this case. Children who go to college after high school are still entitled to their orphan’s pension. As with the number and age of children, the education level of the children is unknown. Research by Behrman et al [1] shows that the education level of the parents affect the education level of the children. This is also shown in a study by CBS [13] in the Netherlands. These results are showed in table 3.2.
Education Father Education Mother
Low Med High Low Med High
Educ. Son Low 53.2% 35.5% 10.1% 63.2% 30.1% 5.9%
Med 37.9% 44.0% 17.5% 50.8% 38.0% 10.6%
High 24.5% 44.7% 30.3% 42.1% 40.6% 16.9%
Total 43.3% 40.3% 15.5% 55.6% 34.6% 9.1%
Educ. Daughter Low 55.5% 33.9% 9.4% 67.2% 27.2% 4.6%
Med 39.8% 43.1% 16.1% 52.9% 36.8% 9.8%
High 26.6% 44.8% 28.3% 38.8% 46.0% 14.5%
Total 43.1% 40.3% 15.7% 56.0% 34.7% 8.6%
Table 3.2: The levels of education for parents and children are related.
This CBS study shows that the education of the parents and children are related. For example, whenever the son has a high level of education the probability of that his father also has a high level education is roughly three times the probability of having a father with a low level of education. This effect is going to be taken into account when developing the deferred orphan’s pension model.
3.4.1 Maximum Age for Orphan’s Pension
As described in the introduction, in most pension regulations the orphan’s pension benefit stops after reaching 18 years. An exception is made for orphan’s attending education. The maximum age of receiving the benefit is then increased to 27 years. These ages can differ between pension regulations. In the model for orphan’s pensions the age at which the orphan’s benefits end will be based on data of students graduating at certain levels of education.
Age MBO HBO / University 15 230 0 16 3 104 0 17 12 499 0 18 26 874 9 19 31 589 366 20 25 548 10 107 21 16 283 19 131 22 10 603 22 067 23 7 083 21 950 24 4 747 18 417 25 3 460 13 262 26 2 520 8 833 27 2 040 5 794
Table 3.3: Number of graduated students per age and education level.
The figures in table 3.3 are published by CBS [19], [20]. It shows the total number of students that graduate at a certain age and education level. This is the most recent data available on this topic and correspond to the academic year 2014/2015. Based on this data the final ages of the medium and high level of education are determined. This age is set equal to the average graduating age per education class. For the MBO students (medium level of education) this is 20 years and for the HBO/University students (high level of education) this is 23 years. For the low level of education it is assumed the orphan’s pension ends when reaching the age of 18. These final ages, subject to the education level, will be referred to as S(s).
The final age of the orphan’s pension benefits is estimated based on the education level of the orphan. As this education level is not known, probably not even when dying of the main insured takes place, it also needs to be estimated. As touched upon earlier, research by Behrman et al [1] shows that the education level of the parents affect the education level of the children. This also clearly shows when looking at the CBS data on this (table 3.2). This data is published with the children as baseline (i.e. the row sums equal 100%). For the purpose of this research the parents need to be the baseline. Therefore the data from this table is adjusted. At first the data is averaged over both child genders. Secondly, the probabilities are derived such that the parents are the baseline. This is done by using Bayes formula on conditional probabilities:
P (A|B) = P (B|A)P (A) P (B)
In this case A represents the educational level of the child and B the educational level of the parent. For instance the probability that a child has a high level of education given that the parent has a high level of education is equal to the probibaility that the parent has a high level of education given the son has a high level of education, times the unconditional probability that a child has a high level of education, divided by the unconditional probability that the parent has a high educational level. Using Bayes formula the probabilities in table 3.2 are adjusted. The adjusted probability factors are called ps(g,r) and consist of the following variables:
g : Gender of the parent,
r : Education level of the parent, s : Education level of the child,
The derived probability factors are showed in table 3.4.
Father Mother
Low Med High Low Med High
Education child Low 48.5% 33.2% 24.0% 45.2% 31.8% 22.6%
Med 44.3% 53.3% 53.1% 46.0% 53.1% 56.2%
High 7.2% 13.5% 22.8% 8.8% 15.1% 21.3%
Table 3.4: Probabilities ps(g,r) of child’s education level given the education level and gender of parent.
For instance, the probability that a male parent with a high level of education, phigh(M,high) has a child with a high level of education is equal to 48.5%.
3.5
Interest Rates
The exact future benefits that the pension provider is obliged to pay are unknown. Whenever there is a possibility that these benefits need to be paid, the pension provider should maintain a provision for it. This provision is the value of the best estimate of these future payments. Because the possible benefits lie in the future, the pension provider applies discounting to these benefits. This incorporates the expected risk free yield the pension provider makes on the current provision. When constructing the or-phan’s pension model, interest rates need to be used for discounting the expected future cashflows. The interest rates used in the model are according to the term structure for pension funds as per December 31, 2016 as published by DNB [21].
Deterministic Model
To achieve the research goal a model is developed to determine the provision of deferred orphan’s pension. In Chapter 3 the relevant factors are treated that will incorporated in the model. This section will describe how these factors can be included in a deterministic model.
4.1
Modelling Experience Mortality
As stated in chapter 3 there are numerous studies that show the effects of education on the mortality rate and life expectancy. The higher the education level, the higher the life expectancy of the participant. This means that a distinction must be made in mortality rates between participant groups with different education levels. This is done using different experience mortality rates for these groups. In this section these experience mortality rates are derived.
4.1.1 Data
The data used to model the different experience mortality rates is published by CBS [17]. This publication includes period life expectancy data split by gender, age and education level. Figures 4.1 and 4.2 show the published life expectancies split by education level. In figure 4.3 can be seen what the difference is between the highest and lowest education group.
Figure 4.1: Life expectancy Male by education level.
Figure 4.2: Life expectancy Female by education level.
Figure 4.3: Differences in life expectancies for both genders, comparing the highest and lowest education level.
As figure 4.3 shows there is a major difference in life expectancies between the highest and lowest education level. When comparing these differences to the results of the study of Knoops and Van den Brakel [7] it can be seen that these figures are very similar to the results as published in the mentioned research. Knoops and Van den Brakel noticed a seven year lower life expectancy at birth for men with a low income compared to men with high income. At an age of 65 this difference is equal to four years, which also is similar to the results in figure 4.3. This is an indication that high income and a high level of education have the same effect on life expectancy. The differences in life expectancy as seen in figure 4.3 are taken into account by adjusting the mortality rates per education level. This is done using experience mortality rates which are derived in the following section.
4.1.2 Experience Mortality by Education
As the published data for the effect of education on fertitily rates are given in three categories (low, medium and high), the experience mortality is going to be modeled including this same division. In the mentioned publication of CBS the education level of ’Basisschool’ and ’VMBO’ are taken together as low education level. For the experience mortality rates the life expectancies for these two groups are combined.
Experience mortality is the factor between expected mortality for a certain group and the expected mortality for the population as a whole (defined as basic mortality in chapter 3). Therefore, to compute experience mortality the life expectancy of this basic
mortality needs to be determined. As the life expectancies published by CBS are period life expectancies, and not cohort life expectancies, the period life expectancies of the whole population are derived in order to construct the experience mortality rates. The following variables are used:
x : age of main insured in years, q(x) : expected mortality rate for age x, E(x) : life expectancy for age x.
The period life expectancy of the whole population is derived as follows: E(x) = 0.5q(x) +1 − q(x)1 + E(x + 1), x < 120,
E(x) = 0.5, x = 120.
In this fomula the mortality rates q(x) are interpolated for non integer ages x. For this purpose (constructing experience mortality rates), the mortality rates do not depend on the time parameter t as the period life expectancy is calculated (and not the cohort life expectancy). In the orphan’s pension provision model, as constructed later on, the mortality rates do depend on the time parameter t. The life expectancy is computed by iteration. Starting with a life expectancy of 0.5 at age 120 the life expectancies for the other ages can be determined by using the mortality rates. In the iteration there are two possibilities for age x:
• The person dies at age x with probability q(x) and has a remaining life expectancy of 0.5. This is due to the assumption that the event will happen on average halfway during the year.
• The person survives with probabiliy 1 − q(x) and has a remaining life expectancy of a person that is of age x + 1 plus 1, because of surviving age x.
Combining these two possibilities will give the iteration stated above. Given the best estimates of the Prognosetafel AG2016 [12], the period life expectancies of the basic mortality for both genders can be derived. The mortality rates used correspond to the year 2016 of the mortality table. The period life expectancies are shown in figure 4.4.
Figure 4.4: Period life expectancy in 2016 for both genders derived from the probabilities of the Prognosetafel AG2016.
The life expectancies as shown in figure 4.4 are used as input to derive the experience mortality rates. The experience mortality rates are modelled in such a way that the
deviation of the life expectancy (figures 4.1 and 4.2) between all three education levels and the basic mortality are translated to three sets of experience mortality rates. The following additional variables are used:
r : education level (high, medium or low),
cr(x) : experience mortality correction factor based on education level r for age x,
Er(x) : life expectancy based on education level r for age x.
The experience mortality correction factors are derived using:
Er(x) = 0.5cr(x)q(x) + 1 − cr(x)q(x) 1 + Er(x + 1) , x < 120, Er(x) = 0.5, x = 120.
The iteration stated above is almost equal to the iteration stated earlier for the basic mortality. Only the mortality rates are changed using the correction factors cr(x). The
purpose of making this iteration differs from the iteration for the basic mortality. In case of the basic mortality the mortality rates were known and the life expectancies were computed. In this iteration the mortality rates, or rather the correction factors, are computed. This can be done because the life expectancies per education level are known (figures 4.1 and 4.2). Hence the correction factor can be derived from these equations. This is done using the Goal Seek function of Excel, setting the differences in life expectancies Er(x)−E(x) equal to the differences in life expectancy of the CBS data
(figures 4.1 and 4.2) and the computed life expectancy of the basic mortality. By doing this top down (starting with the highest age) mortality experience correction factors are determined. As the data of life expectancies by education level is only published for ages rounded to 5 years, the differences in life expectancy for all, including intermediate, ages is determined by using a parabola. The resulting differences in life expectancies and the corresponding parabola for all groups are showed by gender in the following figures1.
Figure 4.5: Male life expectancy differences and the corresponding trendline to determine the differences for all ages.
1
The published data show differences in life expectancy already at ages below 5 years old. It is unclear to us how the level of education was determined at that age.
Figure 4.6: Female life expectancy differences and the corresponding trendline to deter-mine the differences for all ages.
The experience mortality rates are chosen to make the mortality rates of the basic mortality times the experience mortality rates result in the life expectancies correspond-ing to the education level. Figures 4.7 and 4.8 show these resultcorrespond-ing experience mortality rates.
Figure 4.7: Male experience mortality rates cr(x) for all three education levels.
An experience mortality correction factor of 1 means that the mortality rate of that given education level does not differ from the total population. A correction factor lower than 1 gives a lower mortality rate and therefore a higher life expectancy. These figures show that the life expectancy for a medium level of education is close to the life expectancy of the population as a whole. For a high education level the mortality rates are lower than those for the total population and vice versa for the lower educated group. This is as expected and in line with the life expectancies as showed in figures 4.1 and 4.2. As discussed in section 3.1.2, the data of the Doornbos and Kromhout study [3] show that for men with a low level of education of age 38 the mortality rates are 72% higher compared with a high level of education. In figure 4.7, for the age of 38, the experience mortality factors are 1.59 and 0.55 for a low and high level of education respectively. This results in an increase of 289% in mortality rates. We do not know for sure what causes this increased difference. A possible explanation is that over the last 35 years since the study of Doornbos and Kromhout the differences between high and low education level in the Netherlands increased to a level that already was noticeble in the US, as elaborated on before. The experience mortality rates of figures 4.7 and 4.8 are used in the model to derive a provision for orphan’s pension.
4.2
Modelling the Expected Number of Children
4.2.1 Data
As elaborated on in section 3.2 the number of the children entitled to an orphan’s pension benefit when the main insured dies is unknown upfront. But there is some data available on the main insured in the pension provider administration:
• Age, • Gender,
• Education level.
The latter of the three is possibly not always known. However, due to the nature of the Dutch pension system, the population of a pension provider is often from the same workfield. Based on this, the education level of the population can be estimated. For instance, it can be expected that the pension plan for medical specialists contains only participants with a high level of education.
These three parameters are used as input to model the expected number of chil-dren. To effectuate this, data is required on how these parameters affect the number of children. Herewith the following is taken into account:
• Population average - average number of children per gender and age, • Educational effect on number of children,
• Educational effect on age of having children.
The historical data that is used to derive the population average number of children is published annually by CBS [18]. Each year the number of children born per age and gender is disclosed. By combining this data with the total population number of the same age, a function can be derived. For both genders and all ages the number of children born is divided by the total population number of that age. For instance the total number of women of age 20 equals 1,000. The total number of women given birth at age 20 equal 10. This gives that the expected number of children born by a woman of age 20 is equal to 1%. This can be done for both genders and all ages and results in a function that shows the expected number of children born over time; the fertility rate per age. This is called b(g,x), with gender g and age x.
Figure 4.9: Average number of children born per gender and age of the parent b(g,x).
Figure 4.9 shows the fertility rate per gender and age. What stands out is that the average age of having children differs between genders. Women tend do have children earlier than men. The age difference according to this graph is about 3 years. This is not surprising, as this is also the average age difference between a man and woman in a relationship that pension providers account for when determining spouse pensions. The function in figure 4.9 is the average number of children born regardless of the education level of the main insured.
The education of the main insured is related to the number of children. This is already discussed in section 3.2.2. The data to model this effect can be found in table 3.1. This leaves only the effect of the education on the age of having children to analyse. The data used for this is published by CBS [15] which is the age of the parent when, on average, the first child was born. This data is showed in the following table:
Average age first child Low Med High
born by education Male 1945-1949 26.8 27.4 29.0 1950-1954 27.5 28.6 30.7 1955-1959 28.6 30.0 31.9 1960-1964 29.3 30.9 32.6 Female 1945-1949 23.8 25.4 27.6 1950-1954 24.1 26.0 28.8 1955-1959 25.0 27.2 30.1 1960-1964 25.5 28.3 31.0
Table 4.1: Average age of birth first child per gender and education level. In this table the data is subdivided by birth cohort of the parents. It can be seen that the age at which the first child is born is related to the education level of the parents. This effect is stronger for mothers than for fathers.
With this there is enough information to model the expected number of children using the known parameters. To summarise, the showed effects will be modelled using the following data:
• Educational effect on number of children - Table 3.1; • Educational effect on age of having children - Table 4.1.
4.2.2 Modelling Expected Number of Children by Education
The data as subscribed in the previous section can combined to derive a separate func-tion for all educafunc-tion levels and genders. This funcfunc-tion is referred to as b(g,r,x) and consists of variables:
g : Gender of the parent,
r : Education level of the parent, x : Age of the parent.
b(g,r,x) is constructed by starting with the function b(g,x), which is the number of births for the population as a whole (figure 4.9). When taking education of the parent into account this function is adjusted in two ways. At first the function is scaled up or down in such a way that it represents the number of children born corresponding to the education level. This scaling factor is called f(g,r). For the medium level of education
this scaling factor is equal to 1, for the low and high level of education this factor is equal to the fertility rate of the given education level divided by the fertility rate of the medium education level. The fertility rates used are based on the rates as stated earlier in table 3.1. The used rates are averaged over the published rates per birth cohort. This gives the following scaling factors:
Scaling factor Low Med High
Male 1.01 1.00 1.02
Female 1.08 1.00 0.87
Table 4.2: Scaling factors f(g,r) per gender and education level.
Secondly, the function is shifted due to the effect of education on the average age of having a first child. When the parent has a low level of education the function is shifted to correspond the earlier average age of having children. Vice versa this is done for a high level of education. The age difference with which the function is shifted is called a(g,r). These differences are based on the data in table 4.1. As with the scaling factor, the data is averaged over the birth cohorts. This gives the following age differences:
Age difference Low Med High
Male - 1.2 0.0 1.8
Female -2.1 0.0 2.7
Table 4.3: Age difference factors a(g,r) per gender and education level.
This gives all information needed to derive the expected number of children per education level. The function b(g,r,x) can be derived using
and is showed for both genders in figures 4.10 and 4.11
Figure 4.10: Births per education level for male b(M,r,x).
Figure 4.11: Births per education level for female b(F,r,x).
Figure 4.10 shows that the effect of education for male parents is most notable in the average age of having children. The number of children differs barely between the three education levels with education even having a positive correlation with number of children. This is consistent with the outcome of the Norwegian study by Kravdal [8]. The differences among women are much larger as figure 4.11 shows. Education and number of children as well as average age when having children are negatively correlated.
These figures show the number of children born divided by gender and education level. By summing the births and taking the expected mortality of children into account, figures on the expected number of children can be created. These expected numbers of children factors, given the age of the child, are referred to as n(g,r,x,y)(t). In addition to the variables mentioned earlier, the following variables are used:
y : Age of the child,
q(y, t) : Mortality rate of the child.
Using these variables the expected number of children function is derived as follows: n(g,r,x,y)(t) = n(g,r,x−1,y−1)(t − 1)1 − q(y − 1, t − 1), y > 0, n(g,r,x,0)(t) = b(g,r,x), y = 0.
The iteration above is clarified with the following example. Suppose the expected number of children born for a high educated male of 18 and 19 are on average 0.1 and 0.2 (and 0 for ages < 18). Also suppose that the mortality rate of a child of age 0 at time t − 1 is equal to 1%. The expected number of children a high educated male of age
19 at time t has is then equal to
n(M,high,19,0)(t) = b(M,high,19)= 0.2
children of age 0 and
n(M,high,18,0)(t − 1) = b(M,high,18)= 0.1
n(M,high,19,1)(t) = n(M,high,18,0)(t − 1)
1 − q(y − 1, t − 1) = 0.1 ∗ (1 − 0.01) = 0.099
children of age 1. The total expected number of children (summed over all ages of children y) is equal to 0.299.
The time parameter t is also influencing these factors by the mortality rates of the children, as these change over time due to the trend in the mortality table. In figures 4.12 and 4.13 these factors are showed when summed over all ages y of children (just as is done in the example).
Figure 4.12: Number of children per education level for male:X
y
n(M,r,x,y)(2016).
Figure 4.13: Number of children per education level for female:X
y
4.3
Orphan’s Pension Provision
Now all relevant factors as described in chapter 3 are modelled or based on available data. These comprise of:
• Basic mortality rates: best estimates of the Prognosetafel AG2016 - q(x, t), • Experience mortality rates - cr(x),
• Expected number of children - n(g,r,x,y)(t), • Probability of education level children - ps
(g,r),
• Expected ending age of orphan’s benefit - S(s).
As for the characteristics of the population of the pension provider, these are assumed to be known. This includes the education level of the participants. In real life this is probably not always the case. However this can often be estimated based on the occupational field of the population. Most of the time the population will be from the same work of field, due to the nature of the second pillar pension system in the Netherlands.
4.3.1 Model of Orphan’s Pension
The model for deferred orphan’s pensions is based on annuity factors which represent in pay orphan’s pension. These will be converted into deferred factors later on by taking into account the mortality of the main insured. The in pay factors will be adjusted for the expected number of children and the expected education of the children.
At first the in pay orphan’s pension factor is derived. These factor will be constructed using the following variables:
y(t) : Age of orphan at time t,
q(y, t) : Mortality rate of the orphan with age y at time t, s : Education level of the child,
S(s) : Age at which orphan’s pension ends given s, d(t) : Discount factor at time t.
For the mortality of the children no experience mortality is taken into account. It is questionable to assume that children who have a higher probability to attend a high education have got a lower mortality rate during the years before reaching the maximal age of orphan’s pension. Furthermore, the effect of experience mortality on children would be marginal. Based on the mentioned parameters the in pay orphan’s pension Ws can be computed using
Ws(y, t) = 0.5Ws+(y, t) + 0.5W − s (y, t)
in which the in pay orphan’s pension factor is calculated by averaging over the prenumerando factor Ws+ and postnumerando factor Ws−. By doing this it is assumed the payment will happen halfway during the year. The prenumerando factor is derived using iteration: Ws+(y, t) = 1 + 1 − q(y, t) Ws+(y, t + 1)d(t), y < S, Ws+(y, t) = 0, y ≥ S.
As long as the orphan is younger than the final age of the benefit S the orphan receives the benefit at the start of the year. This is the number 1 in the iteration. Because the benefit is paid out at the start of the year, mortality rates and discounting are not relevant for this benefit. The rest of the fomula consist of the possibility that the orphan survives times the discounted (in pay) orphan’s pension factor at t + 1 when the orphan is 1 year older. The postnumerando is constructed in the same way. The difference is that the benefit is assumed to be received at the end of the year. Hence the benefit is subject to mortality and discounting as well.
Ws−(y, t) = 1 − q(y, t) 1 + Ws−(y, t + 1) d(t), y < S, Ws−(y, t) = 0, y ≥ S.
These in pay orphan’s pensions factors Ws are given a certain education level of the
children s and for 1 child. They do not yet take into account the expected number of children n(g,r,x,y)(t). Therefore the in pay orphan’s pension factors Ws for 1 child and
a given education level are adjusted to in pay factors that encorporate the expected number of children Y(g,r,s)(x, t). Y(g,r,s)(x, t) = S X y=0 Ws(y, t)n(g,r,x,y)(t)
In this factor the in pay factors are multiplied with the expected number of children. These factors are summed over all possible ages of the children y < S, the final age given the level of education. This gives a best estimate for an in payment orphan’s pension Y given the gender of the main insured g, the education level of the parent r and the education level of the orphan s. This best estimate depends on the age of the(deceased) main insured x because this is part of the estimation of the number of children. Next to this the time parameter t is in order due to discounting using a term structure instead of a fixed rate, and also due to the mortality rated depending on the year of the mortality table.
The in pay factor Y can be used when the education level of the orphan is known. As mentioned before this is probably not the case. Therefore this factor needs to take into account the expected education level of the child given the known information of the parent. To determine the pension provision for deferred orphan’s pension, the in pay factor is adjusted to the deferred factor Z(g,r)(x, t) using the following iteration:
Z(g,r)(x, t) = cr(x)q(x, t) 3 X s=1 Y(g,r,s)(x, t)p s (g,r)+ 1 − cr(x)q(x, t) Z(g,r)(x + 1, t + 1)d(t), x < 100, Z(g,r)(x, t) = 0, x ≥ 100.
With probability cr(x)q(x, t), the main insured dies at time t. This probability consist
of the mortality rate corrected with the experience mortality factor for a given education level. If this happens, the orphan’s benefit will be paid out. This in pay orphan’s benefit for an estimated level of education of the orphan is equal to the in pay factor Y(g,r,s)(x, t) times the probabiliy ps
(g,r) that the orphan has a level of education s. Summing over
all three possible levels of education gives the best estimate in pay factor. Also with probability 1 − cr(x)q(x, t)
the main insured survives time t. In that case the value of the factor is equal to the discounted deferred factor at age x + 1 and time t + 1. The deferred factor is set to 0 for high enough ages, in this case for ages ≥ 100. The resulting factors are showed in chapter 6.
4.3.2 Orphan’s Pension as Increment of Spouse Pension
As elaborated in chapter 2, currently the provision for deferred orphan’s pension is pragmatically set equal to a fixed percentage of the provision for spouse pension. In most cases it is set to 5%. The presumption is that this way the provision for deferred orphan’s pension is established overprudent. Using the model as described in the previous section it is possible to check whether this is the case. This also requires actuarial factors for deferred spouse pension which will be derived using the following variables.
q(x, t) : Mortality rate for main insured of age x at time t, q(z, t) : Mortality rate of spouse of age z at time t,
d(t) : Discount factor at time t,
IP(g,r)(z, t) : In pay spouse pension factor with gender g and age z at time t,
for a died main insured with education level r,
SP(g,r)(x, t) : Deferred spouse pension factor for main insured of age x at time t.
The factor for deferred spouse pension is modelled in the same way as with the orphan’s pension factor. First the in pay factor for spouse pension is derived. Based on these in pay factors the deferred factor is constructed. As with the in pay orphan’s pension factor, the in pay spouse factor IP (z, t) is calculated by averaging over the prenumerando factor IP+(z, t) and postnumerando factor IP−(z, t). By doing this it is assumed the payment will happen halfway during the year.
IP(g,r)(z, t) = 0.5IP(g,r)+ (z, t) + 0.5IP(g,r)− (z, t)
The prenumerando factor is derived following:
IP(g,r)+ (z, t) =1 +1 − cr(z)q(z, t)
IP(g,r)+ (z + 1, t + 1)d(t), z < 120, IP(g,r)+ (z, t) =0.5, z ≥ 120. and the postnumernado factor is derived using:
IP(g,r)− (z, t) = 1 − cr(z)q(z, t) 1 + IP(g,r)− (z + 1, t + 1) d(t), z < 120, IP(g,r)− (z, t) =0.5, z ≥ 120. In the prenumerando factor the benefit is paid out at the start of the year. Hence the benefit is not subject to mortality and discounting. In the postnumerando factor the benefit is assumed to pay out at the end of the year and is subject to mortality and discounting during that year. As the mortality table ends at age 120, the spouse is assumed to die halfway at age 120 given that the spouse is alive.
The deferred spouse factor can be derived using a similar iteration based on the mortality of the main insured. The actuarial factors are based on the ’Bepaalde Partner’ spouse system. This means that it is assumed that the main insured has a partner at time t = 0. For t > 0 the probability of having a partner is equal to the surviving probabilty of the partner.
Once again the deferred spouse pension factor SP(g,r) is an average of the
prenu-merando factor SP(g,r)+ and the postnumerando factor SP(g,r)− . Furthermore it is assumed that the partner is of the opposite sex and and the man is three years older than the woman.
SP(g,r)(x, t) = 0.5SP(g,r)+ (x, t) + 0.5SP(g,r)− (x, t)
The prenumerando factor is derived following:
SP(g,r)+ (x, t) = 1 − cr(z)q(z, t) cr(x)q(x, t)IP(g,r)(z + 1, t + 1)+ 1 − cr(z)q(z, t) 1 − cr(x)q(x, t) SP(g,r)+ (x + 1, t + 1)d(t), x < 120, SP(g,r)+ (x, t) =0.5, x ≥ 120.
and the postnumernado factor is derived using:
SP(g,r)− (x, t) = 1 − cr(z)q(z, t) cr(x)q(x, t)IP(g,r)(z + 1, t + 1)d(t)+ 1 − cr(z)q(z, t) 1 − cr(x)q(x, t) SP(g,r)− (x + 1, t + 1)d(t), x < 120, SP(g,r)− (x, t) =0.5, x ≥ 120. For the prenumerando deferred spouse factor the benefit is equal to the in pay factor at time t if the main insured dies at time t with probability cr(x)q(x, t) and the partner
survives with probability 1 − cr(z)q(z, t). The in pay factor is not discounted because
it is assumed the events happen at the start of the year. In case both partner and main insured survive time t the factor is equal to the discounted factor for age x + 1 at time t + 1. If both partner and main insured die, the factor is equal to 0. This covers all possible outcomes. The postnumerando factor assumes that the event happen at the end of the year. Therefore in case the main insured dies the benefit is already discounted.
In chapter 6 these factors will be used to check whether the ratio deferred orphan’s pension divided by deferred spouse pension is close to the often used 5%.
Stochastic Model
The model as described in the previous chapter is based on a deterministic approach. For instance, it is assumed that each year a certain portion q(x, t) of the main insured participants will die and that this portion of orphan’s pension will be paid out. This gives the best estimate of the orphan’s pension provision. However it does not allow to conclude anything on the variety of outcomes possible. It could be that the real provision needed is within close range of the best estimate. It is also possible that there are some extreme cases in which the provision should be a lot higher to cover the orphan’s pension payments. To check whether this is the case the following parts of the deterministic model will be simulated using a stochastic model:
• Best estimates - simulating mortality rates including the trend in finding best estimates,
• Stochastic mortality - simulating mortality of the main insured instead of using deterministic mortality rates.
5.1
Best Estimates in Stochastic Mortality
In the deterministic model the mortality rates used are best estimates. However it is also possible to simulate mortality rates using the same model that the best estimates are based on. This model is described in the publication on the most recent Prognosetafel [13] by the Royal Actuarial Association of the Netherlands. All formulas in this section are taken from this publication.
The mortality rates qx(t) can be derived when the force of mortality µtx is known:
qx(t) = 1 − e−µx(t).
The force of mortality is modelled using the Li-Lee model for ages until 90 years and both genders g. This is done using the following formulas:
lnµgx(t)= lnµg,EUx (t)+ ln µg,N Lx (t) ln µg,EUx (t) = Agx+ BxgKtg lnµg,N Lx (t)= αgx+ βxgκgt
in which for both genders, age x ≤ 90 and year t ≥ 2016 the time series K and κ are given by: Ktg = Kt−1g + θg+ gt κgt = agκg t−1+ δ g t 27
In this model µgx(t) equals the force of mortality for the Dutch population, µg,EUx (t)
equals the force of mortality for a certain group of Western European countries and µg,N Lx (t) is the deviation of the Dutch force of mortality from the Western European
values. The stochastic variables Zt= (Mt , δtM, Ft, δtM) are i.i.d. and have a four
dimen-sional normal distribution with covariancematrix C.
C = 2.035241381 0.273354741 2.238406941 −0.507043787 0.273354741 0.180446685 0.284892711 0.315588744 2.238406941 0.284892711 2.920278366 −0.454564892 −0.507043787 0.315588744 −0.454564892 1.674923636
The force of mortality for ages x > 90 are modeled using the Kannist¨o model (which will not be elaborated on in this research).
Before using this stochastic model, the A, B, α, β, κ, θ, a and C are calibrated using mortality data from The Netherlands and the Western European countries. This is described in more detail in the publication of the Prognosetafel AG2016. The calibrated parameters as published are used to model stochastic mortality. This will result in a range of possible mortality tables with different trends for longevity. With a simulation pool of 10,000 simulations, figures 5.1 and 5.2 show the life expectancy (simulated using the mortality rates of the model described above) for newly born children for the coming years. The best estimate as published is also plotted as a solid line.
Figure 5.2: Simulated female life expectancy.
The effect of these different mortality rates on the orphan’s pensions provision is shown in chapter 6.
5.2
Stochastic Mortality
In the previous section the mortality table including the long term trend is simulated using a stochastic model. This gives some support for the outcome of the deterministic model with the best estimate mortality rates. However this simulation is still using a deterministic approach to derive the orphan’s pensions provision. To check whether our best estimate provision is a good indication of reality and how many times the best estimate provision is actually enough to cover the payments, the mortality itself should be simulated. In this model the mortality of the main insured, spouse and children are simulated. The stochastic variables are using the best estimate mortality rates as input. For instance if the best estimate mortality rate of the main insured is 0.1% then the stochastic variable is equal to 1 with a probability of 0.1% and equal to 0 with a probability of 99.9%. When the stochastic variable is equal to 1 the main insured dies and the orphan’s pension payment starts. That is if the child is still alive at that time, which is also simulated. The mortality of the spouse is simulated such that the ratio orphan’s pension divided by spouse pension of the stochastic model can be compared with the best estimate. By simulating a large set, results on the distribution can be derived and compared with the best estimate. This is done in chapter 6.
Results
In this section the results are shared from the deterministic model presented in chapter 4. The deterministic case is checked on robustness to the assumed length of the orphan’s pension as well as the used interest rate. Finally the results are shown of the stochastic approach.
6.1
Deterministic Model
In chapter 4 a deterministic model is constructed to derive the best estimate deferred orphan’s pension factor in order the calculate the according pension provision. The resulting actuarial factors are included in appendix A. As described in chapter 2, in the current situation the provision for orphan’s pensions is often not calculated exactly but pragmatically set equal to 5% of the provision for deferred spouse pension. As both the factors for deferred orphan’s pension and deferred spouse pension are derived, it can be tested whether this value is a good approximation. The figures below show the orphan’s pensions provision as a percentage of the deferred spouse pension provision: Z(g,r)(x, t)/SP(g,r)(x, t).
Figure 6.1: Orphan’s pensions provision for male main isured, as percentage of spouse pension.
Figure 6.2: Orphan’s pensions provision for female main isured, as percentage of spouse pension.
Both figures 6.1 and 6.2 show that a high level of education gives a more spread out provision over all ages. This is due to the higher average age when having children. This contributes to a lower provision for younger ages and a higher provision later on. The increased length of orphan’s pension for higher educated children also affects this. According to these best estimates, the currently used increment of 5% on top of the spouse pension provision is significantly over prudent.
6.2
Sensitivity Analysis
In this part it is checked what the influence is of some of these assumptions on the outcome of the model. The parameters subject to this analysis are the following:
• Length of orphan’s pension benefit, • Interest rates.
6.2.1 Orphan’s Benefit Length
The best estimate factors are based on the benefit for orphan’s pension ending when reaching the age of 18, 20 or 23 respectively for an education level low, medium or high. These ages are set equal to historical data on the average age when graduating certain school types. To gain insight in how much effect this parameter has on the outcome, the model will be adjusted in such a way that the ending age is increased with 1 year for all education levels. The resulting effect on the orphan’s pension provision as a ratio of the spouse pension provision is showed in figures 6.3 - 6.5.
Figure 6.3: Orphan’s pensions provision for male with low education as percentage of spouse pension: 1 year increased length.
Figure 6.4: Orphan’s pensions provision for male with medium education as percentage of spouse pension: 1 year increased length.
Figure 6.5: Orphan’s pensions provision for male with high education as percentage of spouse pension: 1 year increased length.
The effect is only shown for male main insured as the figures for females show the same effects. It can be seen that the increased length for the benefit with low education has the most impact. This is due to the fact that 1 additional year for this group is a bigger increase in total length than for the other groups.
6.2.2 Interest Rates
Pension provisions in general are strongly dependent on interest rate changes. This is due to the high duration of most pension obligations. The further the (expected) benefits are in the future, the more discounted the benefit payments are today. The results presented in this research are based on the interest term structure for pension funds published by DNB as per December 31, 2016. This interest rate is historically low. In this analysis it is checked what the effect is of using a higher interest rate for discounted the pensions benefits. A fixed interest rate of 4% will be used. The figures 6.6 and 6.7 show again the ratio deferred orphan’s pension divided by deferred spouse pension for men and women.
Figure 6.6: Orphan’s pensions provision for male as percentage of spouse pension: 4% interest rate.
Figure 6.7: Orphan’s pensions provision for female as percentage of spouse pension: 4% interest rate.
When comparing these figures with 6.1 and 6.2 it can be seen that, as a ratio of spouse pension, the orphan’s pension provision increases quite a bit. As touched upon before this is due to the sensitivity of both types of pension benefits towards interest rate changes. The average duration of deferred spouse pension is higher than the duration of deferred orphan’s pension. Therefore the increased interest rate causes the provision of deferred spouse pension to decrease more than the provision for deferred orphan’s pension. This result shows that if a pension provider decides to set the provision for deferred orphan’s pension equal to a fixed percentage of the provision for deferred spouse pension, one should reconsider this value in case of (significant) interest rate changes.
