Quantifying the uncertainty in
pensionable-age projections
S.N. van Westen
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: S.N. van Westen Student nr: 5813956
Email: snvanwesten@gmail.com Date: August 12, 2016
Supervisor: Prof. dr. ir. M.H. Vellekoop (University of Amsterdam) Second reader: dr. S. van Bilsen (University of Amsterdam)
Abstract
In this thesis we quantify the systematic, parameter and model uncertainty in the future projections of the AOW-age and the pensionable-age in the Netherlands. To quantify the systematic uncertainty, we use a number of stochastic mortality models. These are the Lee-Carter model and the AG2014 model. Furthermore, a semi-parametric boot-strap technique is applied to account for parameter uncertainty in these projections. A comparison between the projections resulting from both models, gives an idea about the model uncertainty.
We find that for the AOW-age and the pensionable-age a further increase in the retirement age is necessary in the future, regardless of which mortality model is used. Moreover, when we compare the two models we see that the expected increase in the AG2014 model is larger and that the expected values of both models diverge. The expected AOW- and pensionable-ages of the AG2014 model are very close to the pro-jections published by the CBS, while the expected ages of the Lee-Carter model are lower. When we compare the results of the pensionable-age with the projections of the AOW-age, we see that the year in which the pensionable-age reaches a certain age falls in an earlier calendar year than the AOW-age. This means it is likely that the AOW-age and the pensionable-age will no longer be the same in the future. However, the difference in the pensionable-age and the AOW-age is expected to decrease in the coming years. For the AG2014 model we expect it to increase however, due to an earlier expected increase of the pensionable-age.
Furthermore, we find that the systematic uncertainty in both the AOW-age and pensionable-age projections is very large for both models and this uncertainty increases with time. The AG2014 model exhibits more uncertainty. We note, however, that the uncertainty in the CBS projections is even larger. We conclude that the influence of parameter uncertainty is negligible in the projections, while model uncertainty can be substantial.
Keywords Stochastic mortality forecasts, Pensionable-age, Parameter uncertainty, Bootstrap techniques
ivS.N. van Westen — Quantifying the uncertainty in pensionable-age projections
Statement of Originality
This document is written by Student Sonny van Westen who declares to take full re-sponsibility 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.
Preface vii I Background information 1 1 Introduction 2 1.1 The problem . . . 2 1.2 Structural reforms . . . 2 1.3 Research question. . . 4
1.4 Outline of the thesis . . . 4
2 The pension landscape in the Netherlands 5 2.1 The four-pillar system . . . 5
2.1.1 First pillar . . . 5
2.1.2 Second pillar . . . 5
2.1.3 Third pillar . . . 8
2.1.4 Fourth pillar . . . 8
2.2 Adjustments in the AOW-age, the pensionable-age and the Witteveenkader 8 II Research methods and existing literature 13 3 Stochastic mortality models 14 3.1 Description of the data. . . 14
3.1.1 Data for the Netherlands . . . 14
3.1.2 Data for the group of Western European countries . . . 15
3.2 Assumptions . . . 15
3.2.1 Estimation for each gender . . . 15
3.2.2 Closing table for high ages. . . 15
3.2.3 Stochastic framework . . . 15
3.2.4 Piecewise constant hazard rates . . . 16
3.2.5 Remaining life expectancy . . . 16
3.2.6 Definition best estimate . . . 17
3.2.7 Definition confidence interval . . . 17
3.3 Analysis of the data . . . 17
3.4 Lee-Carter model . . . 20 3.4.1 Model specification . . . 20 3.4.2 Fitting procedures . . . 20 3.4.3 Forecast procedure . . . 24 3.4.4 Comments. . . 25 3.5 AG Projectiontable 2014 . . . 26 3.5.1 Model specification . . . 26 3.5.2 Fitting procedure. . . 28 3.5.3 Forecast procedure . . . 30 v
viS.N. van Westen — Quantifying the uncertainty in pensionable-age projections 3.5.4 Comments. . . 31 4 Bootstrap techniques 32 4.1 Motivation . . . 32 4.2 Parametric bootstrap. . . 33 4.3 Semi-parametric bootstrap. . . 34 4.4 Residual bootstrap . . . 35
III Results, analysis and conclusions 37 5 Results and Analysis 38 5.1 Parameter estimates . . . 38 5.1.1 Lee-Carter model . . . 38 5.1.2 AG2014 model . . . 40 5.2 Parameter uncertainty . . . 42 5.2.1 Lee-Carter model . . . 42 5.2.2 AG2014 model . . . 44
5.3 Forecast of the mortality rates . . . 47
5.3.1 Lee-Carter model . . . 47
5.3.2 AG2014 model . . . 48
5.3.3 Parameter and model uncertainty. . . 49
5.4 Forecast of the period and cohort life expectancy . . . 50
5.4.1 Lee-Carter model . . . 51
5.4.2 AG2014 model . . . 52
5.4.3 Parameter and model uncertainty. . . 54
5.4.4 Cohort life expectancy . . . 58
5.4.5 Sex-neutral life expectancy . . . 60
5.5 Forecast of the AOW-age . . . 63
5.5.1 AOW-age projections . . . 63
5.5.2 Distribution of the year of increase . . . 65
5.5.3 Distribution of the AOW-age for cohorts. . . 66
5.6 Forecast of the pensionable-age . . . 68
5.6.1 Pensionable-age projections . . . 68
5.6.2 Distribution of the year of increase . . . 69
5.6.3 Difference between pensionable-age and AOW-age . . . 70
6 Conclusion 73
IV Appendices 76
Appendix A: Logistic mortality law of Kannisto 77
First of all, I would like to thank my supervisor Michel Vellekoop for his guidance in writing this thesis. Our meetings were a real pleasure and his criticism and tips were of great value. Special thanks goes to my brother Dylan van Westen, who supported me during this project and gave valuable suggestions concerning the topic of this thesis. I also like to thank my friend Martin Jonk for his feedback in the final stage of writing this thesis.
Last but not least, I would like to thank my parents, my brothers and my friends for their immense patience and support.
Background information
Chapter 1
Introduction
1.1
The problem
In 2015 the total expenditures of the Dutch government on the state pension (AOW, Algemene Ouderdomswet ) are 35.8 billion euro, see CBS Statline (2016d). The total expenditures have increased with more than 35% since 2008, which is mainly caused by the ageing of the population in the Netherlands. Only a small part of this increase is due to indexation of the AOW-benefits (Bruil et al.(2015)). The total amount of AOW-premiums only covered 68% of the expenditures in 2015, see CBS Statline(2016c). The remaining part is paid by the Dutch government. This coverage ratio has been gradually decreasing since 2000. If this increase in expenditures and decrease in coverage ratio were to continue, the AOW would become unsustainable in the long run.
The increase in expenditures is caused by multiple demographic changes. The popu-lation in the Netherlands is ageing, which causes an increase in the number of retirees. In addition the life expectancy is increasing for all ages. Given a fixed age of retirement of 65, this results into a longer payment of the AOW-benefits on average. The decrease in the number of births since 1970 is a structural change, which causes a decreasing work-ing population and consequently a decline in the amount of AOW-premiums received by the government. The total effect is enhanced by the babyboom after the Second World War. As a comparison, in 1957 the ratio of retirees and the payers of AOW-premiums was 1 to 6. In 2015 this ratio has changed to 1 to 3 and it is expected to decrease even further to 1 to 2 in 2040, see CBS Statline (2015) and CBS Statline (2016f). These developments do not only concern the AOW: the workers and private pension systems, life insurers and medical health care are affected as well.
The consequences of these demographic changes on the government’s finances have been studied amongst others by the Bureau for Economic Policy Analysis (CPB, Cen-traal Planbureau) in van Ewijk et al. (2006), van der Horst et al. (2010) and Smid et al.(2014). The CPB concluded in 2010 that the expenditures of the government will increase faster than its income. Without any structural reforms the government deficit and public debt will continue to increase.
The economic crisis in 2008 resulted in an increase of the unemployment rate, which led to a reduction in tax income and an increase in the payments of the social security system. Together with the demographic changes, this resulted into a substantial deto-riation in the government’s finances. The economic crisis accelerated the discussion of structural reforms of these finances and the Dutch pension system.
1.2
Structural reforms
One of the many structural reforms the Dutch government introduced is a law which increases the retirement age of the AOW (AOW-age) and the retirement age for new pension rights (pensionable-age) in the second and third pillar (Wet Verhoging
en pensioenrichtleeftijd, VAP ). This bill was approved by the Dutch Senate (Eerste Kamer) on July 10, 2012, seeMinisterie van Sociale Zaken en Werkgelegenheid(2012). The AOW-age, which was fixed at the age of 65 since its introduction in 1957, will increase by a predetermined schedule from 2013 onwards. This results in an AOW-age of 66 years in 2019 and 67 years in 2023. From the year 2024 onwards, a further increase in the AOW-age will depend on the remaining period life expectancy of a 65-year old.
Next to an increase in the AOW-age, the ”Witteveenkader” is adjusted as of January 1, 2014. The ”Witteveenkader” fiscally limits the accrual of new pension rights in the Netherlands. As of January 1, 2014 the retirement age of new pension rights in the second and third pillar will also increase to 67 years. From the year 2015 onwards, a further increase in the retirement age will also depend on the remaining period life expectancy of a 65-year old. Other changes involve the lower maximum accrual rates for the final-pay (1.9%) and average-pay (2.15%) pension schemes.
As part of the coalition agreement ”Bruggen slaan”, which was introduced in October 2012, the AOW-age will increase at an even higher rate compared to the predetermined schedule in VAP, seeMinisterie van Sociale Zaken en Werkgelegenheid(2014). This bill was approved by the Dutch Senate on June 2, 2015. According to this new schedule, the AOW-age will increase with 3 months per year in the period 2016-2018, and with 4 months per year in the period 2019-2021. This will result in an AOW-age of 66 years in 2018 and 67 years in 2021. From the year 2022 onwards, a further increase in the AOW-age will depend on the remaining period life expectancy of a 65-year old. The rules concerning the retirement age of new pension rights remain unchanged in this bill. Another part of the coalition agreement was a further adjustment in the Witteveen-kader. As from January 1, 2015 the Witteveenkader is adjusted again (Ministerie van
Sociale Zaken en Werkgelegenheid (2013)). The changes involve the lower maximum
accrual rates for the final-pay (1.657%) and the average-pay (1.875%) pension schemes. Also the accrual of new pension rights is fiscally maximized to a wage of 100.000 euro and the arrangement of net-pensions is fiscally allowed. The net-pension allows for ac-crual of new pension rights above the wage of 100.000 euro. However the additional pension premiums paid are not exempt from tax, whereas the accumulated capital and the benefits are tax free.
These structural reforms can have positive effects (+) or negative effects (-) on the budget of the government (Centraal Planbureau (2009a)):
• The increase in the AOW-age results in a deferred payment of the AOW-benefits and the expected duration of these benefits is shorter (+);
• The deferred payment of AOW-benefits will be offset by longer payments in dis-ability and unemployment benefits (-);
• The increase in the AOW-age results in a larger potential working population, which causes an increase in AOW-premiums, since the working population retires at a later age (+);
• The increase in the pensionable-age results in a deferred payment of the pension-benefits. The pension-benefits are taxable and have a claim on premiums for health care. The deferred payment results in a loss of tax income and premiums. How-ever, the increase in the pensionable-age also results in a larger potential working population. Therefore the above effect will be offset by a larger tax income (+/-); • The decrease in the maximum accrual rates results in a smaller pension premium. Therefore the payer of the premium has a higher gross income, which results in a higher tax income, since the pension premiums itself are not taxable (+).
The CPB shows in their latest study, see Smid et al.(2014), that the reforms in the retirement age and health care lead to lower expenditures of the Dutch government and a
4
higher tax income. The income after the year 2023 will be higher than the expenditures, leading to a surplus. Therefore the structural reforms enhances the sustainability of the government’s finances.
1.3
Research question
As a consequence of the reforms, the formerly fixed AOW-age and pensionable-age have become stochastic, because they both depend on the future period life expectancy of a 65-year old. Since the AOW-age now depends on the date of birth, this means that the number of retirees in the future is also stochastic. This leads to a new definition of the old-age dependency ratio, which depends on the AOW-age.
The developments in the AOW-age, pensionable-age and the old-age dependency ratio will have major consequences for the government’s finances, the sustainability of the pension systems, life insurers and medical health care. Therefore it is an important task to analyse these developments.
To analyse the effects of the AOW-age, pensionable-age and old-age dependency ratio, a stochastic mortality model is needed. There are a number of reasons to prefer a stochastic model over a deterministic model. Firstly, the stochastic models have been used and examined extensively and are considered state-of-the-art for risk management purposes. Moreover they allow a formal statistic goodness-of-fit analysis. A second ad-vantage of stochastic models is that they provide insight into the uncertainty of future mortality developments.
However, a stochastic model, like any other simplication of the real world, also has its limitations. In the projections of stochastic models it is assumed that the size of the shocks in the past is representative for the size of the shocks in the future. Furthermore, the standard time series projections of the stochastic mortality models do not take into account the uncertainty with regard to parameters, nor uncertainty with regard to the model. This motivates the following research question:
How large is the systematic, parameter and model uncertainty in the future projec-tions of the AOW-age and the pensionable-age in the Netherlands?
1.4
Outline of the thesis
This thesis starts with a brief description of the Dutch pension landscape in Chapter2. In addition the changes in the AOW-age and pensionable-age are described in more detail. Next we discuss a number of stochastic mortality models in Chapter 3. The mortality models of Lee and Carter (1992a) and Koninklijk Actuarieel Genootschap
(2014b), are described in more detail. In Chapter4the bootstrap method is introduced, which allows us to quantify the parameter uncertainty. In Chapter 5 the results and analysis are presented and compared with the projections of the CBS. Finally in chapter 6 we present the conclusion.
The pension landscape in the
Netherlands
To motivate the relevance of the research question, we start with a brief description of the pension landscape in the Netherlands. Next we discuss the reforms of the Dutch government in the AOW-age, the pensionable-age and the Witteveenkader in more detail.
2.1
The four-pillar system
2.1.1 First pillar
The first pillar of the Dutch pension system is the state pension (AOW, Algemene Ouderdomswet ), a social insurance introduced in 1956 which is managed by the Dutch government. The state pension provides a basic income for the elderly population. This ensures a minimum standard of living and economic independence. The state pension is financed as a pay-as-you-go system: the AOW-benefits are paid directly from the public resources, no capital funding takes place. The AOW-benefits are financed by the active working population, who pay a fixed percentage of their gross income. When the received AOW-premiums are not sufficient to cover the AOW-benefits, the remaining part is paid by the Dutch government.
The AOW-benefits are independent of employment history, salary, other forms of income and the amount of capital. The AOW-benefits are accrued annually by every person who lives or works in the Netherlands between the age of 15 and 65. The accrual rate is 2% per year, therefore a full AOW-benefit is reached within 50 years. The size of the AOW-benefits depends on the net minimum wage. A single person receives an AOW-benefit equal to 70% of the net minimum wage, while a person with a partner receives an AOW-benefit of 50%. Traditionally, the payments of the benefits commences on the age of 65, which is the retirement age of the AOW (AOW-age). Starting from 2013, the AOW-age is gradually increasing in steps by a predetermined schedule until the age of 67 years is reached in 2021. From the year 2022 onwards, a further increase in the AOW-age will depend on the period life expectancy of a 65-year old.
2.1.2 Second pillar
The second pillar of the Dutch pension system consists of the occupational pension, which is provided by the employer. In addition to the AOW-benefits of the state pension, employees can accrue additional pension benefits in the second pillar. This occupational pension is capital funded: each year a premium is paid by the employer (or partly by the employee), which is invested and managed by a pension fund or a life insurer. The total contributions and the returns on investments over time should be sufficient to cover the pension benefits paid for life. The goal is to provide a pension with the same purchasing power as during the working life of the employee.
6
Pension providers
The occupational pensions in the second pillar can either be administrated by a pension fund or a life insurer. The funds of these financial institutions and the sponsoring com-pany are strictly separated by the Dutch Pension Act (Pensioenwet ). This is to prevent cuts in the pensions of employees in case the sponsoring company goes bankrupt. The differences between the pension providers is discussed below:
• Industry-wide pension fund (BPF): The employees of a company may be obligated to participate in an industry-wide pension fund. In a BPF the pensions of several companies, which are active in a certain industry or sector, are managed by the same pension fund. The Dutch government decides if an industry-wide pension fund is applicable in an industry;
• Profession pension fund: For some professions, such as dentists or general practitioners, the pensions are managed by the same pension fund. Participation is often mandatory;
• Company pension fund (OPF): If a company is not required to enter into an industry-wide or profession pension fund, it may start its own pension fund. Large companies such as Shell and Unilever have their own company pension fund; • Life insurer: In addition to pension funds, the pensions of a company may also
be administrated by a life insurer. In contrast to a pension fund, a life insurer has to guarantee the payments of the nominal benefits. A pension fund may decrease the pension rights in case of financial hardship, depending on the funding ratio (ultimum remedium). To realize the guarantee, the buffer requirements for life insurers are higher in comparison with pension funds.
Pension schemes
The occupational pensions in the second pillar are arranged in a pension scheme. The type of the pension scheme defines the nature of the pension benefits. These can either be risky or guaranteed. Furthermore, it determines how the pension benefits are financed and how the risks in the contract are shared between the pension provider, the employer, the employee and the retiree. The Dutch Pension Act distinguishes the following pension schemes:
• Defined Benefit (DB): In a Defined Benefit scheme the nominal pension bene-fits are guaranteed. The premiums paid by the employer and employee may vary, depending on the funding ratio of the pension fund. If the funding ratio is low, additional premiums may be needed to recover, while a high funding ratio may result in premium cuts. The risks in a Defined Benefit scheme are borne by the employer, the employee, the retiree and the pension fund. The size of the nominal pension benefits depends on the type of plan within the Defined Benefit scheme. The most common plans are the final pay plan (Eindloonregeling) and the aver-age waver-age plan (Middelloonregeling). In a final pay plan the pension benefits at retirement depends on the final wage of the employee. Whenever the salary of the employee is increased, the existing pension rights are raised with the difference in the pensionable earnings multiplied by the number of years of service and the accrual rate (Backservice). Because of this unconditional backservice, the final pay plan is quite expensive. In an average wage plan the accrual of new pension rights depends on the pensionable earnings in the year of accrual multiplied by the accrual rate and a part-time factor. The existing pension rights are not adjusted for a raise in the salary of the employee. In an average wage plan the pension rights and benefits are usually adjusted for price or wage inflation to maintain the
same level of purchasing power. This indexation might be unconditional or de-pendent on the funding ratio of the pension fund. After the dotcom crisis in 2000 most pension funds changed their final pay plan into an average wage plan. The average wage plan has become the most common pension plan in the Netherlands. Although the nominal pension benefits are guaranteed, exceptional circumstances have led to a cut in the pension rights and benefits of multiple pension funds in 2013. The short-term plans of these pension funds were not able to recover from the sharp decline in the funding ratio, caused by the economic crisis in 2008; • Defined Contribution (DC): In a Defined Contribution scheme, the employer
and employee pay a fixed premium. This premium is usually a percentage of the salary or the pensionable earnings and is dependent on the age of the employee. The premiums are invested until the pensionable-age. Therefore the total amount of capital available at the pensionable-age depends on the total premiums paid and the returns on the investments. At retirement this capital is used to buy a life annuity. The price of this life annuity depends on the interest rates and mortality rates at the time of retirement. This means the pension benefits are not guaranteed, since the benefits are exposed to investment risk, interest rate risk and longevity risk. These risks are completely borne by the employee. After the capital is used to buy a life annuity, these risks are transferred from the employee to the life insurer. The risks borne by the employer are very small, since the pension premiums are fixed. The only risk the employer faces is that the amount of capital at retirement might be unsufficient for an employee to retire;
• Capital Agreement: In a Capital Agreement scheme, the employer and em-ployee pay a premium to a life insurer. This premium is used to save for a guar-anteed capital at the pensionable-age, usually adjusted for profit sharing. It is mandatory to convert this guaranteed capital at retirement into a life annuity. During the period of accrual the investment risk is borne by the life insurer. Be-cause the guaranteed capital is converted to a life annuity, the interest rate risk and the longevity risk are borne by the employee. This type of pension scheme is very rare in the Netherlands.
Types of pension
In the second pillar there is a distinction between different types of pension. The most common types of pension are discussed below.
• Old-age pension: The most common type of pension is the old-age pension. It provides employees with an additional income, on top of the AOW-benefits, after retirement. Each year the employee can accrue pension rights, usually dependent on salary, for which the employer and/or the employee must pay premiums. In return the employee receives a benefit after retirement. These benefits are only paid if the employee is alive and after the employee reaches the retirement age (pensionable-age). The accrual of pension rights in the Netherlands is fiscally stim-ulated. The pension premiums, as well as the accumulated capital are exempt from tax. Only the pension payouts are taxable (Omkeerregeling ). To manage these tax advantages, fiscal boundaries are set in the ”Witteveenkader” in the form of max-imum accrual rates, the pensionable-age, the minmax-imum social security offset and other rules. The actual accrual rate, pensionable-age, etcetera are established by the social partners and are fixed in the pension contract. Traditionally, the pay-ments of the benefits commences on the age of 65, the same age as the AOW-age. However, starting from 2014 the pensionable-age for the accrual of new pension rights is 67 years. A further increase in the pensionable-age will depend on the period life expectancy of a 65-year old.
8
• Partner pension: In case the employee dies during the period of accrual or after retirement, a financial loss occurs for the relatives of the employee. The income of the relatives of the employee is diminished. To account for this financial loss, a partner pension may be arranged. It provides the relatives of the employee with an income when the employee dies. The size of the benefits is usually equal to 70% of the old-age pension that would haven been accrued, if the employee had lived to the pensionable-age. The payments of the partner pension benefits are dependent on the type of contract, the moment of death and possibly on the resignation of the employee.
• Disability pension: If the employee becomes disabled during the working pe-riod, a reduced income results for the employee. Moreover, the accrual of new pension rights is not/partly possible, depending on the disability of the employee. A disability pension provides a benefit in case of disability of the employee. These benefits are on top of the social security benefits of the government (WIA). The payments of the disability pension starts after two years of disability, since the employer is obligated to pay for the first two years. It also allows for the further accrual of new pension rights during disability, depending on the pension contract.
2.1.3 Third pillar
The third pillar of the Dutch pension system consists of private pensions. It fiscally allows an individual to accrue additional pension rights in addition to their accrual of pension in the second pillar. Moreover, individuals without a pension agreement, such as the growing group of self-employed individuals, can use the third pillar to arrange their own pension savings. The pension benefits in the third pillar usually are paid in the form of a life annuity, purchased at an insurance company. The paid premiums in the third pillar, as well as the accumulated savings are exempt from tax, as long as the fiscal boundaries are not exceeded.
2.1.4 Fourth pillar
In addition to the above three pillar system, individuals can utilise private assets to fund their pension. This so-called fourth pillar consists of personal savings, investments, company revenues and the ownership of a house or other buildings.
2.2
Adjustments in the AOW-age, the pensionable-age and
the Witteveenkader
On July 10, 2012 a law was approved by the Dutch Senate which increases the retirement age of the AOW and the retirement age for new pension rights in the second and third pillar (Wet verhoging AOW- en pensioenrichtleeftijd, VAP ), seeMinisterie van Sociale Zaken en Werkgelegenheid(2012).
The AOW-age, which was fixed at the age of 65 since its introduction in 1957, will increase by a predetermined schedule from 2013 onwards, see table 2.1. This results in an AOW-age of 66 years in 2019 and 67 years in 2023. As part of the coalition agreement ”Bruggen slaan”, the AOW-age will increase at an even higher rate (as of 2016) compared to the predetermined schedule in VAP, seeMinisterie van Sociale Zaken
en Werkgelegenheid (2014). According to this new schedule, the AOW-age will be 66
years in 2018 and 67 years in 2021. A comparison between these schedules is illustrated in table 2.1. After the AOW-age reaches the age of 67 years, a further increase in the AOW-age will depend on the remaining period life expectancy of a 65-year old.
Table 2.1: The increase of the AOW-age in a predetermined schedule, before and after the coalition agreement. Source:Ministerie van Sociale Zaken en Werkgelegenheid(2012) and Ministerie van Sociale Zaken en Werkgelegenheid(2014).
VAP (2012) Coalition Agreement (2014)
Year AOW-age Cohorts born during AOW-age Cohorts born during
2012 65 years ←− – 31-12-1947 65 years ←− – 31-12-1947 2013 65 years + 1 m 01-01-1948 – 30-11-1948 65 years + 1 m 01-01-1948 – 30-11-1948 2014 65 years + 2 m 01-12-1949 – 31-10-1949 65 years + 2 m 01-12-1948 – 31-10-1949 2015 65 years + 3 m 01-11-1949 – 30-09-1950 65 years + 3 m 01-11-1949 – 30-09-1950 2016 65 years + 5 m 01-10-1950 – 31-07-1951 65 years + 6 m 01-10-1950 – 30-06-1951 2017 65 years + 7 m 01-08-1951 – 31-05-1952 65 years + 9 m 01-07-1951 – 31-03-1952 2018 65 years + 9 m 01-06-1952 – 31-03-1953 66 years 01-04-1952 – 31-12-1952 2019 66 years 01-04-1953 – 31-12-1953 66 years + 4 m 01-01-1953 – 31-08-1953 2020 66 years + 3 m 01-01-1954 – 30-09-1954 66 years + 8 m 01-09-1953 – 30-04-1954 2021 66 years + 6 m 01-10-1954 – 30-06-1955 67 years 01-05-1954 – 31-12-1954 2022 66 years + 9 m 01-07-1955 – 31-03-1956 — — 2023 67 years 01-04-1956 – 31-12-1956 — — 2024 — — — —
A further increase in the AOW-age is announced five years in advance. Therefore the possible increase in the AOW-age for the year 2022 is determined on January 1, 2017. This increase is determined with the following formula (art. 7a AOW ):
Vt := (Lt− 18.26) − (Pt−1− 65). (2.1)
Here the variables are defined as follows:
• t is the year in which a possible increase in the AOW-age will happen.
• Vtdetermines the increase in the AOW-age and the initial age of accrual, measured in periods of years.
• Ltis the best estimate of the sex-neutral remaining period life expectancy of a
65-year old, for the 65-year t. This estimate is determined by the Dutch Central Bureau of Statistics (CBS) at time t − 5.
• Pt−1 is the previous AOW-age (i.e. in the year before the year of a possible
in-crease).
If the increase Vt is greater or equal than 0.25, the AOW-age is raised with three
months. If the increase Vtis smaller than 0.25 or possibly negative, the AOW-age remains
the same. This means the AOW-age is adjusted for an increase in the remaining life expectancy of a 65-year old in steps of three months, while a decrease has no effect on the AOW-age.
The increase Vt in formula (2.1) consists of two parts. The first part, (Lt− 18.26),
accounts for the difference in the remaining period life expectancy of a 65-year old of two different periods. The number 18.26 is determined by the CBS as the average of the sex-neutral remaining period life expectancy of a 65-year old in the period 2000-2009, see van Duin and Stoeldraijer (2012). The second part, (Pt−1 − 65), is the difference
between the current AOW-age and the traditionally fixed AOW-age of 65. It accounts for the increments in the AOW-age made in the past.
Figure 2.1 shows the projections of the AOW-age based on VAP and the coalition agreement. These projections result from the population forecasts published by the CBS, see van Duin and Stoeldraijer(2014). The figure shows the expected paths of the AOW-age and the 67% and 95% confidence intervals of the AOW-age as proposed in
10
the coalition agreement. The projection shows an average increase in the AOW-age of three months per year. This results in an expected AOW-age of 71 years and 6 months in 2060. Compared to the previous projection, the differences are small.
Figure 2.1: The projections of the AOW-age based on VAP and the coalition agreement. Source:van Duin and Stoeldraijer (2014).
Next to an increase in the AOW-age, the ”Witteveenkader” is adjusted as of January 1, 2014. The Witteveenkader fiscally limits the accrual of new pension rights in the Netherlands. As of January 1, 2014 the retirement age of new pension rights in the second and third pillar will increase to 67 years. From the year 2015 onwards a further increase in the retirement age will also depend on the remaining period life expectancy of a 65-year old. Other changes involve the lower maximum accrual rates for the final pay (1.9%) and average wage (2.15%) pension schemes.
Another part of the coalition agreement ”Bruggen slaan” was a further adjustment in the Witteveenkader. As from January 1, 2015 the Witteveenkader is adjusted again (Ministerie van Sociale Zaken en Werkgelegenheid,2013). The changes involve the lower maximum accrual rates for the final-pay (1.657%) and the average wage (1.875%) pen-sion schemes. Also the accrual of new penpen-sion rights is fiscally bounded by a wage of 100.000 euro and the arrangement of net-pensions is fiscally allowed. The net-pension allows for accrual of new pension rights above the wage of 100.000 euro. However the additional pension premiums paid are not exempt from tax, whereas the accumulated capital and the benefits are tax free. No changes were made to the pensionable-age.
A further increase in the pensionable-age of new pension rights is announced one year in advance. Therefore the possible increase in the pensionable age for the year 2015 is determined on January 1, 2014. This increase is determined with the following formula (art. 11 Wet op de loonbelasting 1964 ):
Vt := (Lt+10− 18.26) − (Pt−1− 65). (2.2)
Here the variables are defined as follows:
• t is the year in which a possible increase in the pensionable-age will happen. • Vtdetermines the increase in the pensionable-age, measured in whole years.
• Lt+10 is the best estimate of the sex-neutral remaining period life expectancy of a 65-year old 10 years after year t. This estimate is determined by the Dutch Central Bureau of Statistics (CBS) at time t − 1.
• Pt−1 is the previous pensionable-age (i.e. in the year before the year of a possible increase).
If the increase Vt is greater or equal than 1, the pensionable-age is raised with
one year. If the increase Vt is smaller than 1 or possibly negative, the pensionable-age
remains the same.
Comparing the formulas (2.1) and (2.2) we notice the increase in the AOW-age and the pensionable-age differ in terms of the remaining life expectancy. The life expectancy used for the increase in the pensionable-age, uses a calendar year which is 10 years higher compared to the increase in the AOW-age. Furthermore, the pensionable-age is allowed to increase in steps of one year, whereas the AOW-age is allowed to increase in steps of three months. This means the AOW-age and the pensionable-age will no longer match each other in the future. Possible reasons for these differences are the following: • An increase of the pensionable-age for new pension rights with three months, would result in high administration costs. By increasing the pensionable-age in steps of one year, the number of different pension rights is reduced;
• An increase of the AOW-age in steps of one year is not desirable, since someone who is nearly retired might not anticipate a sudden increase in the AOW-age. This could result in a loss of income, if the employee has already agreed to retire at a certain date. It would be hard to compensate for this loss, since it is difficult to find a new job for someone who is nearly retired.
The above problems can be avoided if the increase is announced well in advance for both arrangements. Moreover, it is preferable to use the same definition of life expectancy in the above formulas. After these changes, both the AOW-age and the pensionable-age may increase in steps of one year. These changes would allow the AOW-age and the pensionable-age to converge in the near future, which reduces the complexity of the system and lowers the administration costs.
Although the pensionable-age has been increased to 67 years, it is still possible for pension funds to accrue new pension rights with a pensionable-age of 65 (art. 18d Wet op de loonbelasting 1964 ). However the accrual of new pension rights under the lower pensionable-age must be actuarially equivalent to the new pension rights under the pensionable-age of 67 years. In practice this results in a reduced maximum accrual rate when the pensionable-age is lowered. These maximum accrual rates are published by the tax collectors office (Belastingdienst ) for a variety of pensionable-ages, see Belast-ingdienst(2014).
The increase in the pensionable-age only considers the accrual of new pension rights. The pension entitlements accrued before January 1, 2014 are still based on the pensionable-age of 65 years. Since the accrual of new pension rights is not restricted to the age of 67, the following options are available after January 1, 2014:
• The accrual of new pension rights is still based on a pensionable-age of 65, but with a lower maximum accrual rate. The pension entitlements remain under the pensionable-age of 65;
• The accrual of new pension rights is based on a pensionable-age of 67. The pension entitlements remain under the pensionable-age of 65.
Another option would be to collectively convert the already accrued pension entitle-ments to a pensionable-age of 67. However, this option is not possible under the current Dutch Pension Act, since the pension entitlements are unconditional. No changes are
12
allowed without the explicit permission of the participants. This conversion of pension entitlements would make the system less complex, more transparant to its participants and it reduces the administration costs. Other effects and (financial) consequences are discussed in a study by Laan(2013).
Research methods and existing
literature
Chapter 3
Stochastic mortality models
The objective of our analysis is to quantify the systematic, parameter and model uncer-tainty in the future projections of the AOW-age and the pensionable-age in the Nether-lands. To quantify these different types of uncertainty, a stochastic mortality model is required. In the last decades numerous mortality models were proposed and extensively scrutinised in academic research. In this chapter we will discuss some of these mortality models and the historical data which is used to fit these models.
3.1
Description of the data
The input of the stochastic mortality models considered in this thesis, consists of the number of deaths Dx,t and the exposures-to-risk Ex,t, where x denotes the completed
age rounded off to the nearest lower integer, and t denotes the calendar year of the observation. The number of deaths are the recorded number of deaths for a certain age x in a given year t. The exposures-to-risk Ex,t are estimates of the average population
exposed to the risk of death during some age-time interval.
The datasets used in this thesis are the number of deaths and the exposures-to-risk for the Netherlands and a group of Western European countries as published by the Royal Actuarial Society (AG), see Koninklijk Actuarieel Genootschap (2014a). These datasets are constructed with data originally published by the Dutch Central Bureau of Statistics (CBS) and the Human Mortality Database (HMDB). A description of both datasets is given below.
3.1.1 Data for the Netherlands
The dataset for the Netherlands consists of the number of deaths Dx,tNLand the exposures-to-risk ENL
x,t which are obtained from the HMDB, seeHuman Mortality Database(2016).
The number of deaths are available by completed age, calendar year of death and year of birth, for both females, males as well as the total population. The structure of the exposures-to-risk is the same as that of the number of deaths.
The dataset used corresponds to observations of 1 year in age and 1 year in time (1 × 1), with ages x ∈ X0 := {0, 1, ..., 90} and years t ∈ T0 := {1970, 1971, ..., 2009}.
The dataset for the Netherlands is extended with data published by the CBS. The CBS provides population estimates Px,t and estimates for the number of deaths Cx,t,
with ages x ∈ X0 := {0, 1, ..., 90} and years t ∈ T1 := {2010, 2011, ..., 2013}, see CBS Statline (2016a) and CBS Statline (2016e).
• Px,t denotes the estimate of the population on January 1 of year t with ages
between x and x + 1.
• Cx,tdenotes the estimate of the number of deaths in year t, where the people who die during the given year have ages between x and x + 1.
To be consistent with the dataset obtained from the HMDB, the data published by the CBS is converted to number of deaths and exposures-to-risk using the protocol of the Human Mortality Database, seeWilmoth et al. (2007). The deaths and exposures-to-risk are obtained using the following method:
DNLx,t := ( C0,t+12· C1,t for x = 0 1 2 · (Cx,t+ Cx+1,t) for x = 1, 2, ..., 90. (3.1) Ex,tNL := ( 1 2 · (P0,t+ P0,t+1) + 1 6 · C0,t− 1 2 · C1,t for x = 0 1 2 · (Px,t+ Px,t+1) + 1 6· 1 2 · Cx,t− 1 2 · Cx+1,t for x = 1, 2, ..., 90.(3.2)
3.1.2 Data for the group of Western European countries
The group of Western European countries consists of 14 countries, for which the Gross Domestic Product (GDP) is above the average European GDP, seeKoninklijk Actuar-ieel Genootschap(2014b). The following countries are included in the dataset: Austria, Belgium, Denmark, England & Wales, Finland, France, Germany, Iceland, Ireland, Lux-embourg, the Netherlands, Norway, Sweden and Switzerland.
The dataset for the group of Western European countries consists of the number of deaths DEUx,t and the exposures-to-risk Ex,tEU which are obtained from the HMDB, see
Human Mortality Database (2016). The dataset used for each country corresponds to
observations of 1 year in age and 1 year in time (1 × 1), with ages x ∈ X0:= {0, 1, ..., 90}
and years t ∈ T0:= {1970, 1971, ..., 2009}. The deaths DEUx,t and exposures-to-risk Ex,tEUof
the group of European countries are found by summing the deaths Dx,ti and exposures-to-risk Ex,ti respectively over all 14 countries i (including the Netherlands).
3.2
Assumptions
In order to use the stochastic mortality models described in this chapter, a number of assumptions and definitions are necessary. In this section we explain the assumptions used in this chapter and we present a number of definitions.
3.2.1 Estimation for each gender
It is usual to make a separate estimation for females and males in the stochastic mortality models. We will pursue this standard approach in this thesis.
3.2.2 Closing table for high ages
The observed central death rates of the oldest-old population are very volatile, because the number of observations for higher ages decreases. This may result in undesired uncer-tainty in the estimates and forecasts of the modelled mortality probabilities. Therefore the observed central death rates for the high ages are not included in the estimation of the stochastic mortality models. Instead the mortality probabilities for the high ages (above 90) are generated by the extrapolation method of Kannisto. This method is described in more detail in Appendix A.
3.2.3 Stochastic framework
Consider the random variable Tx,t, which is the remaining time till death for a person
with exact age x on January 1 of year t. We will refer to the event of death as ’failure’. At any time before failure we will say we are in a state of ’survival’. Now the survival distribution of Tx,t for an individual aged x in year t is defined as:
16
This is the probability that someone who is alive on January 1 of year t and who was born on January 1 of year t − x will be alive on January 1 of year t + k.
The cumulative distribution function of Tx,t for an individual aged x in year t is
defined as:
Fx,t(k) := kqx,t := P [Tx,t ≤ k], 1qx,t := qx,t. (3.4)
This is the probability that someone who is alive on January 1 of year t and who was born on January 1 of year t − x will be dead before January 1 of year t + k.
The hazard rate or force of mortality of an individual aged x in year t is defined as: µx,t := lim
k→0+
P [Tx,t ≤ k]
k . (3.5)
The hazard rate is the continuous density function for failure at time t given survival up to that point. It is the probability that someone who is alive on January 1 of year t and who was born on January 1 of year t − x will die in the next instant of time.
3.2.4 Piecewise constant hazard rates
In this thesis we assume that the age-specific hazard rates are constant within regions of age and time, but allowed to vary from one region to the next. Following Brouhns et al.(2002a), given any integer age x and calendar year t, we assume that:
µx+ξ,t+τ := µx,t for any 0 ≤ (ξ, τ ) < 1. (3.6)
As a result of this assumption, the following approximation applies: qx,t := 1 − e−
R1
0 µx+k,t+kdk ≈ 1 − e−µx,t (3.7)
Every stochastic mortality model described in terms of the hazard rates can therefore be rewritten in terms of one-year probabilities of death (and vice versa) using this formula.
3.2.5 Remaining life expectancy
In most publications, the concept of life expectancy denotes the expected remaining lifetime of a person at birth. However we will refer to the remaining life expectancy, which is the average number of years an individual will survive, which is applicable to all ages.
If we determine the remaining life expectancy of an individual who is alive on Jan-uary 1 of year t and who was born on JanJan-uary 1 of year t − x, under the assumption that people who die during a calendar year still live for half of that calendar year on average, then the remaining cohort life expectancy is defined as follows:
ecohortx,t := ∞ X k=0 " k Y s=0 (1 − qx+s,t+s) # +1 2, ∀ x ∈ X . (3.8) This means that the remaining cohort life expectancy is based on 1-year mortality rates from the projection table corresponding to the observation years in which the respective ages are attained. This definition reflects the actual remaining lifetime of someone with a particular year of birth. The remaining cohort life expectancy incorpo-rates the effects of future improvements or deteriorations in the mortality incorpo-rates. This effect is not included in the remaining period life expectancy which is defined as:
eperiodx,t := ∞ X k=0 " k Y s=0 (1 − qx+s,t) # +1 2, ∀ x ∈ X . (3.9)
In the definition of the remaining period life expectancy it is assumed that the 1-year mortality rates are fixed and will no longer change over time. This implies that the mortality rates in a given calendar year t are used to calculate the remaining life ex-pectancy for all ages x. This definition results in an incorrect impression of the remaining life expectancy.
3.2.6 Definition best estimate
As in Koninklijk Actuarieel Genootschap (2014b), the best estimate is defined as the most likely outcome of a variable which is subject to the uncertainty in the projections of a stochastic mortality model. The best estimate of the projection table of mortality probabilities qx,t is obtained by setting all the error terms in the stochastic time
se-ries at zero. If the best estimate of the projection table is used to calculate the value of a remaining life expectancy, this results in the best estimate of the remaining life expectancy. The same does not apply to the average, the median or the quantiles.
3.2.7 Definition confidence interval
Let us have N realizations, X1, ..., XN of a random variable X, which itself is a general
function of the projection table of mortality probabilities. To determine the confidence interval of such a random variable with confidence level α, quantiles are used to estimate the upper and lower bounds. The confidence interval is defined as [bΦ(α2), bΦ(1−
α
2)], where
b
Φ(α2) is the lower bound estimated for the (α
2)-th percentile, and bΦ
(1−α2) is the upper
bound estimated for the (1 −α2)-th percentile. The same realizations, X1, ..., XN, can be
used to estimate the average, median, standard deviation and other relevant statistics. These estimates will be more accurate if the number of realizations N is large.
3.3
Analysis of the data
As a first step we want to estimate the hazard rates µx,t from the data we have for
different ages x and times t. The distribution of the number of deaths, given the popu-lation size, can be modelled appropriately by a Poisson distribution with a mean which is proportional to the population size as stated in Brillinger(1986).
Assuming that the number of deaths observed Dx,t has a Poisson distribution with
mean Ex,t· µx,t, we can find the estimator of µx,t which maximizes the log likelihood.
The likelihood function, L, is defined as follows:
L := Y x∈X Y t∈T (Ex,t· µx,t)Dx,t· e−Ex,t·µx,t Dx,t! . (3.10)
Then the log-likelihood, ln (L), is defined as:
ln (L) := X
x∈X
X
t∈T
[Dx,t· log (Ex,t· µx,t) − Ex,t· µx,t− log (Dx,t!)] . (3.11)
To maximize the log-likelihood we find the critical numbers by partial differentiation with respect to µx,t. The partial derivative must either be equal to zero or must not
exist for it to be a critical number. Therefore: ∂ ln (L)
∂µx,t
:= Dx,t µx,t
− Ex,t = 0. (3.12)
We observe that for every pair (x, t), the estimator of the hazard rate which maximizes the log-likelihood is equal to:
b
µx,t :=
Dx,t
Ex,t
18
The estimator is a local maximum of the log-likelihood. This simple estimator is the number of people who died during the age-time interval divided by the number of people who could have died during the age-time interval. This ratio is also known as the central death rate denoted as mx,t.
The figures below show the central death rates mx,t of females and males in the
years 1970 and 2009 for the Netherlands and the group of Western European countries on a log-scale.
0 20 40 60 80
1e−05
1e−03
1e−01
Observed central death rates of Dutch females
Age (Years) Centr al Death Rates NL Females 1970 NL Females 2009 0 20 40 60 80 1e−05 1e−03 1e−01
Observed central death rates of Dutch males
Age (Years) Centr al Death Rates NL Males 1970 NL Males 2009 0 20 40 60 80 1e−05 1e−03 1e−01
Observed central death rates of Western European females
Age (Years) Centr al Death Rates EU Females 1970 EU Females 2009 0 20 40 60 80 1e−05 1e−03 1e−01
Observed central death rates of Western European males
Age (Years)
Centr
al Death Rates
EU Males 1970 EU Males 2009
Figure 3.1: The observed central death rates mx,t of both females and males, in the
years 1970 and 2009 for the Netherlands and a group of Western European countries on a log-scale. Source:Koninklijk Actuarieel Genootschap (2014a).
Looking at the general shape of the curves, we notice the relative high central death rates at birth. We further notice the ”accident hump” in the central death rates of males is observable in both populations and that such a hump is not present in the central death rates of females. After this accident hump the central death rates are gradually increasing.
The observed central death rates in 2009 have decreased since 1970 for almost all ages. This observation is true for both populations and both genders. This decrease in central death rates is due to a better hygiene, medical innovations, better labour conditions, less smoking and an overall growth in welfare. The consequence of a decrease in the central death rates is that people will live a longer life, that is their remaining life expectancies will increase.
A comparison of the central death rates in the years 1970 and 2009 reveals that the amount by which these rates have decreased is not the same for all ages. The decrease is more pronounced at the young ages and near the retirement age. Comparing the two populations shows that the developments in these central death rates is very similar.
A more complete picture in the evolution of the central death rates for the Nether-lands and the group of Western European countries can be shown in a three-dimensional graph. The figures below show the logarithm of the observed central death rates of fe-males and fe-males for all ages and years available in the datasets.
Notice that the logarithm of the observed central death rates for the group of Western European countries is smoother in comparison with the Netherlands. This is because the exposures-at-risk are larger, which means the statistical noise is smaller due to a greater number of observations.
Age (Y ears) 0 20 40 60 80 Year 1970 1980 1990 2000 2010 Log Centr al Death Rates −10 −8 −6 −4 −2
Log central death rates of Dutch females Age (Y ears) 0 20 40 60 80 Year 1970 1980 1990 2000 2010 Log Centr al Death Rates −10 −8 −6 −4 −2
Log central death rates of Dutch males
Figure 3.2: The natural logarithm of the observed central death rates ln (mx,t) of both
females and males, in the years 1970-2013 for the Netherlands. Source: Koninklijk Ac-tuarieel Genootschap (2014a).
Age (Y ears) 0 20 40 60 80 Year 1970 1980 1990 2000 Log Centr al Death Rates −8 −6 −4 −2
Log central death rates of Western European females
Age (Y ears) 0 20 40 60 80 Year 1970 1980 1990 2000 Log Centr al Death Rates −8 −6 −4 −2
Log central death rates of Western European males
Figure 3.3: The natural logarithm of the observed central death rates ln (mx,t) of both
females and males, in the years 1970-2009 for the group of Western European countries. Source:Koninklijk Actuarieel Genootschap (2014a).
20
3.4
Lee-Carter model
3.4.1 Model specification
In 1992 Lee and Carter proposed a model which became the cornerstone of human mortality modeling. Almost all stochastic mortality models that are used today can be seen as modifications of their original idea. The Lee-Carter model as described in
Lee and Carter (1992a) was originally developed to model and forecast the age-specific mortality patterns of the total population in the United States.
The main idea of the Lee-Carter model is to find a parsimonious model of the pattern of change over time of the natural logarithm of the central death rates. But in such a way that the variation in a single index parameter can generate the main characteristics of the observations across all ages. The Lee-Carter model is defined as follows:
ln (µbx,t) := ln
Dx,t
Ex,t
:= αx+ βxκt+ x,t. (3.14)
Here αx and βx are age-dependent parameters which do not change over time, while
κt is a time-varying parameter which does not depend on the age x. The model thus
separates age- and period-effects. The error term x,t is assumed to be an independent
identically distributed Gaussian variable, that is x,t∼ N (0, σ2) i.i.d. for all pairs (x, t).
The error term reflects particular age-specific historical events which are not captured by the model.
When the Lee-Carter model is fit by ordinary least squares (OLS) the interpretation of the parameters is as follows:
• αx: equals the average of the natural logarithm of the central death rates ln (mx,t)
over time, so eαx is the average shape across age of the mortality schedule.
• βx: represents the rate at which the natural logarithm of the central death rate for age x declines in response to a change in the mortality index κt, that is:
∆ ln (µx,t)
∆t := βx· ∆κt. This means that when κtis linear in time, the central death
rates at each age change at their own constant exponential rates.
• κt: is the time-varying index of the level of mortality. The central death rates for all ages change according to this overall level of mortality.
3.4.2 Fitting procedures
The Lee-Carter model cannot be fitted by ordinary least squares (OLS), because the right-hand side of the equation contains no known covariates. The parameters αx, βx and κt
are all unknown. The original fitting procedure in Lee and Carter (1992a) involves a least squares criterion and the use of singular value decomposition. Alternative fitting procedures are suggested byWilmoth (1993), Alho (2000) and Brouhns et al. (2002a). The fitting procedures of the Lee-Carter model are described below.
Singular value decomposition
To estimate the parameters of the Lee-Carter model for a given matrix of central death rates mx,t, Lee and Carter originally used the following least squares criterion which
minimizes the sum of the squared residuals:
SSR := min αx, βx, κt X x∈X X t∈T ln Dx,t Ex,t − αx− βxκt 2 . (3.15) The parameters b αx, bβx, bκt
that result from this minimization problem are not unique. The model is undetermined because of an identifiability problem. To see this,
suppose that
b
αx, bβx, bκt
is an unique solution of the minimization problem. Then for any scalar c, the following equations will result in the same solution:
b αx, bβx· c, b κt c =⇒ αbx+ b βx· c · b κt c = αbx+ bβx·bκt. (3.16) b αx− bβx· c, bβx, bκt+ c =⇒ αbx− bβx· c + bβx· (bκt+ c) = αbx+ bβx·bκt. (3.17) To have an unique specification of the parameters some constraints have to be im-posed. The following set of restrictions are required:
X x∈X b βx = 1 and X t∈T b κt = 0. (3.18)
The second constraint can be used to find an unique solution for αbx. To see this,
first find the critical numbers of the minimization problem by partial differentiation with respect to αx. Then use the second constraint to solve for αx:
∂SSR ∂αx := −2 ·X t∈T ln Dx,t Ex,t − αx− βxκt = 0. (3.19) X t∈T ln Dx,t Ex,t − αx·X t∈T 1 − βx· X t∈T κt = 0 X t∈T ln Dx,t Ex,t − αx· T − 0 = 0 b αx := P t∈T h lnDx,t Ex,t i T , ∀ x ∈ X . (3.20)
The resulting estimates of the parameters αbx are the average over time of ln (mx,t)
for each age x ∈ X . This corresponds to the average shape of the natural logarithm of the central death rates mx,t.
Substituting the estimates αbx in the minimization problem above we are left with
the reduced minimization problem:
SSR := min βx, κt X x∈X X t∈T [Mx,t− βxκt]2, where Mx,t := ln Dx,t Ex.t −αbx. (3.21)
In the reduced minimization problem, we want to approximate the matrix Mx,t
with a column vector βx, containing age-dependent parameters, times a row vector
κt, containing time-dependent parameters. The singular value decomposition procedure
(SVD) can be used to find a least squares solution of the matrix Mx,t. Singular value
decomposition is based on a theorem in linear algebra stating that a m × n matrix M has a factorization of the form M = U ΣV0. Here U is a m × m matrix containing the left singular vectors in each column, Σ is a m × n diagonal matrix where the diagonal entries are the singular values and V is a n × n matrix containing the right singular vectors in each column. Using only the first singular value, we can find bβx as the first
left singular vector of M and bκt as the first right singular vector of M times the first singular value. Finally the parameters (αbx, bβx, bκt) are adjusted to meet the required constraints (3.18).
22
Reestimation of kappa to match the total number of deaths
The parameters of the Lee-Carter model are originally estimated by minimizing the errors of the natural logarithm of the central death rates, rather than the central death rates themselves. As a consequence the fitted parameters will generally not lead to the actual numbers of deaths observed when applied to the given exposures-at-risk. Lee and Carter suggest to reestimate the parameters κt in a second step. Given the parameters
b
αx and bβx and the exposures-to-risk Ex,t, κt can be reestimated to match the total
number of observed deaths in a given year t ∈ T as follows:
D(t) := X x∈X Dx,t := X x∈X Ex,t· eαbx+ bβx·κt. (3.22)
The motivation of the authors is that the weights in the original fitting procedure are the same for each age, while the younger ages contribute far less to the total number of observed deaths. The reestimated parameters bκt are generally different from those
obtained by singular value decomposition. The reestimated bκt can be found using an iterative search. In the next sections two alternative fitting procedures are considered.
Weighted least squares
In the previous section the parameters of the Lee-Carter model were fitted, by minimiz-ing the sum of the squared errors (3.15). However it may be the case that for certain pairs (x, t) the number of deaths Dx,t, or the exposures-to-risk Ex,t are equal to zero.
We can avoid taking logarithms of zero by excluding the age x or year t for which this is the case. However if the goal is to use the full dataset this poses a problem. A solution to this problem is the use of weighted least squares as proposed by Wilmoth in his technical report, seeWilmoth (1993).
To estimate the parameters of the Lee-Carter model for a given matrix of central death rates mx,t, Wilmoth suggests to minimize the weighted sum of squared errors:
WSSR := min αx, βx, κt X x∈X X t∈T Dx,t· ln Dx,t Ex.t − αx− βx· κt 2 , (3.23)
using the number of deaths Dx,t as weights for each observation. Since these weights
are zero whenever the central death rates are zero or undefined, we can substitute an arbitrary non-zero value for the central death rates in the zero-cells. To have an unique specification of the parameters the same constraints defined in the previous section are used (3.18).
To minimize the weighted sum of squared errors, the first step is to find the critical numbers by computing the partial derivatives with respect to αx, βx and κt and set
these equal to zero. This results in the following system of equations: ∂W SSR ∂αx := −2 · P t∈T Dx,t· h lnDx,t Ex,t − αx− βxκt i = 0, ∀ x ∈ X . ∂W SSR ∂βx := −2 · P t∈T κt· Dx,t· h ln Dx,t Ex,t − αx− βxκt i = 0, ∀ x ∈ X . ∂W SSR ∂κt := −2 · P x∈X βx· Dx,t· h lnDx,t Ex,t − αx− βxκt i = 0, ∀ t ∈ T . (3.24)
Solving for the required parameters results in three sets of normal equations: b αx := P t∈T Dx,t· h lnDx,t Ex,t − bβxbκt i P t∈T Dx,t , ∀ x ∈ X . b βx := P t∈T Dx,t·bκt· h ln Dx,t Ex,t −αbx i P t∈T Dx,t·κb 2 t , ∀ x ∈ X . b κt := P x∈X Dx,t· bβx· h ln Dx,t Ex,t −αbx i P x∈X Dx,t· bβt2 , ∀ t ∈ T . (3.25)
The parameters can be solved numerically using these normal equations in an it-erative procedure. Using a set of starting values, the normal equations are computed sequentially using the most recent set of parameter estimates available. This procedure continues until successive computations yields little to no change in the weighted sum of squared errors.
Maximum likelihood estimation
In the previous sections the parameters of the Lee-Carter model are fitted using a least squares or weighted least squares method. According to Alho (2000), the Lee-Carter model is not optimally implemented. The main drawback of the estimation using SVD is that the errors are assumed to be normally distributed and homoskedastic. This is quite unrealistic, since the logarithm of the observed central death rates is much more variable at older ages, because of the smaller absolute number of deaths at older ages, for example see figure A.1. An alternative approach is to maximize the likelihood of a Poisson distribution as proposed by Wilmoth (1993), Alho (2000), and Brouhns et al.
(2002a). This allows for an additive error term on the logarithm of central death rates, representing the Poisson variation. Another advantage of this method is that it allows a formal statistical goodness-of-fit analysis. Because of these advantages, the fitting procedure of the other stochastic mortality models will be based on the maximum likelihood approach.
Assume that the number of deaths observed Dx,t has a Poisson distribution with
mean Ex,t· µx,t. Now the hazard rates are required to have a specific structure, that is
µx,t := eαx+βxκt. This means that the natural logarithm of the hazard rates have the
specification of the Lee-Carter model. The objective is to find the estimatorsαbx, bβxand
b
κt which maximize the log-likelihood. The likelihood function, L, is defined as follows:
L := Y x∈X Y t∈T (Ex,t· eαx+βxκt)Dx,t Dx,t! · e−Ex,t·eαx+βxκt . (3.26)
Then the log-likelihood, ln (L), is defined as:
ln (L) := X x∈X X t∈T h Dx,t· (ln (Ex,t) + αx+ βxκt) − Ex,t· eαx+βxκt− ln (Dx,t!) i := X x∈X X t∈T h Dx,t· (αx+ βxκt) − Ex,t· eαx+βxκt i + C, where C is a constant.
24
To maximize the log-likelihood, the first step is to find the critical numbers by partial differentiation with respect to αx, βxand κt. The partial derivative must either be equal
to zero or must not exist for it to be a critical number. This gives the following system of equations: ∂ ln (L) ∂αx := P t∈T Dx,t− Ex,t· eαx+βxκt = 0, ∀ x ∈ X . ∂ ln (L) ∂βx := P t∈T Dx,t· κt− Ex,t· κt· eαx+βxκt = 0, ∀ x ∈ X . ∂ ln (L) ∂κt := P x∈X Dx,t· βx− Ex,t· βx· eαx+βxκt = 0, ∀ t ∈ T . (3.27)
There are no analytical solutions available, but an approximation of the roots can be found using the Newton-Raphson method in an iterative procedure. This iterative method for estimating log-linear models with bilinear terms was first proposed by Good-man (1979). Using the Newton-Raphson method for each parameter separately results in: b αx,n+1 := αbx,n− P t∈T h Dx,t−Ex,t·eαx,n+ bb βx,nbκt,n i −P t∈T h Ex,t·eαx,n+ bb βx,nbκt,n i , ∀ x ∈ X . b κt,n+1 := bκt,n− P x∈X b βx,n· h Dx,t−Ex,t·eαx,n+1+ bb βx,nbκt,n i − P x∈X b β2 x,n· h Ex,t·eαx,n+1+ bb βx,nbκt,n i , ∀ t ∈ T . b βx,n+1 := βbx,n− P t∈T b κt,n+1· h Dx,t−Ex,t·eαx,n+1+ bb βx,nbκt,n+1 i − P t∈T b κ2 t,n+1· h Ex,t·eαx,n+1+ bb βx,nbκt,n+1 i , ∀ x ∈ X . (3.28)
The parameters can be solved numerically with these equations in an iterative pro-cedure. Using a set of starting values, χ0 := (αx,0= 0, βx,0 = 1, κt,0 = 0), the equations
are computed sequentially using the most recent set of parameter estimates available. After finding a new approximation of either κt or βx, the parameters are adjusted
ac-cordingly to meet the constraints for unique parameters (3.18). This procedure continues untill successive computations yields little to no change in the log-likelihood.
3.4.3 Forecast procedure
Once the Lee-Carter model is fitted to historical data, we have a set of parameter estimates χ = (αbx, bβx, bκt). In the Lee-Carter methodology the time factor bκt is a stochastic process for these t after today. To forecast the time factorbκt, an appropriate
ARIMA(p, d, q) time series model is determined using standard Box-Jenkins techniques (Box and Jenkins (1976)). To forecastbκtfor the total U.S. population,Lee and Carter
(1992a) used a random walk with drift model (ARIMA(0, 1, 0)) because it gave a good fit and is a relative simple model. The random walk with drift model is defined as:
κt+1 := κt+ θ + σ · ζt. (3.29)
Here θ is the drift term, σ is the standard deviation of the random noise and ζt are
The random walk with drift model is appropriate for the total U.S. population and might not be the best ARIMA(p, d, q) model for the Dutch population. For the sake of parsimony we assume that the random walk with drift model is also appropriate for the Dutch population.
The estimates bθ and bσ are found by matching the first two moments of the first differences in the level of mortality index κt:
∆κt := κt+1− κt = θ + σ · ζt.
E[∆κt] := E[θ + σ · ζt] = θ + σ · E[ζt] = bθ. V ar[∆κt] := V ar[θ + σ · ζt] = σ2· V ar[ζt] = σb
2.
(3.30)
Given the estimates bθ and bσ and the latest value ofκbt, the random walk with drift model is used to generate forecasts of the level of mortality, denoted asκet. Assuming the
estimated age-dependent parametersαbx and bβxdo not change over time, these forecasts
of eκt allow us to generate projections of future mortality rates as follows: µx,t ≈ mx,t := eαbx+ bβxeκt. qx,t ≈ 1 − e−µx,t ≈ 1 − e−e b αx+ bβxeκt . (3.31) 3.4.4 Comments
It should be noted that the Lee-Carter method, as well as the other stochastic mortality models, are a mere extrapolation of past trends. This means that the model is unable to capture improvements in mortality due to advances in medical treatments, socio-economic and environmental changes. Similarly, deteriorations in mortality due to war, epidemics, new diseases, a decline in welfare, natural disasters, and global pollution are not taken into account.
The performance of the Lee-Carter method is investigated in Lee and Miller(2001). Various extensions, applications as well as advantages and disadvantages of the model are presented in Lee(2000).
