Freedom of choice and longevity risk:
how they affect the level of pension
benefits in defined contribution pension
schemes in the Dutch pension system
Ruth Oude Avenhuis
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: Ruth Oude Avenhuis
Student nr: 11829613
Email: ruth.oude.avenhuis@willistowerswatson.com
Date: December 14, 2018
Supervisor: Dr. T. Boonen Second reader: Dr. S. van Bilsen Supervisor: drs. W. Hoekert AAG
This document is written by Student Ruth Oude Avenhuis who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are 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.
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuisiii
Abstract
This study examines the effect of longevity risk in the decumulation phase of pension on different pension variants based on freedom of choice in DC plans in the Dutch pension system. This longevity risk is split up between macro longevity risk and micro longevity risk and the evaluation of it is based on the participant’s perspective. The pension variants that are evaluated are: a standard pension, an early or deferred pension, a high/low construction, an early pension combined with a high/low construction and a variable pension.
A reference person is used for making the calculation. First fu-ture mortality is modelled using a Lee Carter model. Then the formulas for calculating the pension capital accrual, returns on investment, the annuity price and the pension benefit for all pension variants are given. With these elements, the present value of future cashflows can be determined. We calculate the actuarial present value (APV) of future pension benefits evaluated at time t = 0 with different pension benefits and different life expectancies included to find the APV with investment risk and longevity risk, the APV with only micro and macro longevity risk and the APV with only micro longevity risk. Pension variant specific formulas for the APV are given.
The analysis shows that the highest APV is attained with a variable pension. However, longevity risk is found to be highest for a variable pension as well. Furthermore, the results suggest that the impact of micro longevity risk solely on a variable pension is lower than on a standard pension. Finally, micro (and macro) longevity risk is found to be higher for higher pensionable ages.
The conclusion states that both macro and micro longevity risk influences the best choice to make regarding freedom of choice in DC plans. One has to decide for him-/herself how much risk he/she is willing to take. This thesis offers help by making the right decision regarding longevity risk.
Keywords Defined Contribution plans, Freedom of Choice, Macro longevity risk, Micro longevity risk, Investment risk, Early/deferred pension, High/low construction, Variable pension, Lee Carter, Survival probabilities, APV, Life-cycle
Preface vi
1 Introduction 1
1.1 Problem formulation . . . 1
1.2 The Dutch pension system . . . 2
1.2.1 First Pillar . . . 2
1.2.2 Second Pillar . . . 2
1.2.3 Third Pillar . . . 3
1.3 Defined Contribution plans . . . 3
1.3.1 Risks involved in DC plans . . . 3
1.3.2 Freedom of choice in DC plans . . . 5
2 Methodology 9 2.1 Lee Carter model . . . 9
2.1.1 Maximum Likelihood Estimation . . . 9
2.1.2 Forecasting future mortality. . . 10
2.1.3 Bootstrapping the model . . . 10
2.1.4 Survival probabilities. . . 11
2.1.5 Linear extrapolation of survival probabilities . . . 12
2.2 Pension benefit modeling . . . 12
2.2.1 Capital accrual . . . 13
2.2.2 Returns on investment . . . 13
2.2.3 Annuity price . . . 14
2.2.4 Pension benefits . . . 14
2.3 Valuation of pension benefits . . . 15
2.3.1 APV . . . 15
3 Different variants 17 3.1 Early/deferred pension . . . 17
3.2 High/low construction . . . 18
3.3 Early retirement with high/low construction . . . 18
3.4 Variable pension . . . 19
4 Data and assumptions 20 4.1 Data . . . 20 4.2 Reference person . . . 20 4.3 Assumptions . . . 20 5 Results 23 5.1 Lee Carter. . . 23 5.2 Pension capital . . . 23 5.3 Annuity price . . . 26 5.4 Pension benefits. . . 27 5.5 Life expectancy . . . 30 iv
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuisv 5.6 APV . . . 31 6 Conclusion 38 7 Discussion 39 Appendix 40 References 42
After 4,5 years of studying of which 1,5 years concerning the Master ASMF, this period of studying has come to an end. I would like to thank my supervisor Tim Boonen for his support, help and advice during the process of writing my thesis. Moreover, I would like to thank Willis Towers Watson for offering me space and time to write the thesis and of course my supervisor at Willis Towers Watson Wichert Hoekert for helping me finding an interesting subject and assisting me where needed.
Chapter 1
Introduction
1.1
Problem formulation
During the last several years, the Dutch pension system has changed a lot. Probably the largest change is that Defined Contribution pensions plans are rapidly making their entrance. Whereas a decade ago most pension plans were Defined Benefit (DB) plans, now many are switched towards Defined Contribution (DC) plans. The need to keep these DC plans up to date is found to be high (Klijnsma, 2016). Several extra options to one’s pension are already offered to many participants in pension plans, for instance going for an early or deferred retirement, choosing to retire part-time or choosing for the so-called ’high/low construction’. Some countries even offer the option to choose for a (partial) lump-sum at retirement in stead of buying a life-long annuity. On Septem-ber 1st 2016, the ’Wet verbeterde premieregeling’ entered into force. This law makes it possible to invest your pension capital in the decumulation phase as well as during your working-phase. However, naturally the pension benefits are variable when you choose for this option since the investment returns are uncertain. So today people are facing several options regarding their pension.
Next to possibilities, pensions are facing uncertainties and risks. Your pension capital at retirement depends on investment returns and on interest. Furthermore, the value of your pension benefit depends on your life expectancy. For pension funds and insurers, longevity risk is likely to be the toughest risk to face. Longevity is the risk that people are getting older and older and therefore need benefits for a longer period of time. This is very costly for pension funds and insurers. Therefore, valuation of longevity risk is needed.
In the decumulation phase of pensions, longevity risk can be split up into micro longevity risk and macro longevity risk. Micro longevity risk is the risk that an individual be-comes older than his/her life expectancy. Macro longevity risk is the risk that the life expectancy of the whole population changes.
In this thesis, the effect of longevity risk in the decumulation phase of pensions on pension benefits from the participant’s perspective is estimated for different pension variants with a Defined Contribution scheme. The pension variants evaluated in this thesis are based on the offered freedom of choice in pensions in the Netherlands. This thesis is an addition to existing literature where freedom of choice in pension is evalu-ated, for instance inVan Ewijk et al. (2017)andHinrichs (2004), in a way that it focuses mainly on longevity risk in the decumulation phase in the different pension variants.
Research Question:
What is the effect of freedom of choice and longevity risk in the decumulation phase
of Defined Contribution pensions schemes in the Dutch pension system on the level of pension benefits?
In the remainder of Chapter 1 an explanation of the Dutch pension system, DC plans and freedom of choice is given. Chapter 2 explains the standard models used in this research and Chapter 3 gives the variant specific models for the evaluation of different options regarding pensions. To be able to answer the research question, we need data, a reference person and we need to make assumption. These are given and explained in Chapter 4. The results are presented and interpreted in Chapter 4. We will draw con-clusions based on the findings in Chapter 5 and in Chapter 6 the research is discussed.
1.2
The Dutch pension system
For a long time, the Netherlands is seen as the champion in pensions (Bovenberg, 2014). However, three years ago, Denmark and Australia surpassed us. To win back the title of pension champion, the Dutch pension system is revised with success.
The aim of the pension system is to provide elderly of stable and adequate income after retirement to maintain the standards of living. The Dutch pension system consists of three pillars. Due to this three-pillar design the Dutch pensions system is internation-ally seen as a good system (Van Ewijk et al., 2014). The first pillar is the state pension provided by the government, also called the AOW. Employees can accrue pension via their employer in the second pillar. This pillar is mandatory for many employees. Lastly, the third pillar pension is a fully voluntary pension in which people can insure addi-tional pension at insurers. These three pillars together form the pension entitlements for workers in the Netherlands.
1.2.1 First Pillar
Every person living or having lived for some period in the Netherlands is insured for a state pension. For every year as of the age of 15 that someone lives in the Netherlands, 2% of state pension is accrued. You receive a full state pension if you have lived for 50 or more years in the Netherlands. This state pension is also called Social Security (in Dutch AOW). You receive Social Security (SS) as of your SS-age. This age was 65 but is increasing every year since January 1st, 2013 and as of 2022 the SS-age is linked to the average life expectancy. This is the solution to the problem that more and more people need SS benefits and, due to the increasing life expectancy, need benefits for a longer period of time. The amount of SS is linked to the statutory minimal wage and someone’s marital status. Married couples or couples living together receive 50% of the statutory minimal wage each and someone living alone receives 70% of the minimal wage as SS pension. The first pillar is financed as a pay-as-you-go system meaning that the employed pay for the retired in the form of contributions.
1.2.2 Second Pillar
Contrary to the first pillar, the second pillar is a private pension system were employees accrue pension via their employer. For many employees this type of pension is included in the conditions of employment and is therefore mandatory. However, not all employers are obliged to offer their workers pension. Three types of pension funds exist in the Netherlands: industry-wide pension funds, company pension funds and occupational pension funds. Companies not belonging to any of these funds can offer workers a pension plan at an insurer. Basically, two types of pension plans can be offered namely a defined benefit (DB) plan and a defined contribution (DC) plan. In the former plan, as the name already suggests, the benefits are fixed meaning that accrual of pension is
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis3
such that the pension benefit is equal to a fixed percentage of the salary and in the latter case the contributions are fixed meaning that premiums are paid in the accumulation phase which will be invested and the value of the benefits depend on the sum of those premiums together with the investment returns.
1.2.3 Third Pillar
The third pillar constitutes a personal private pension on voluntary basis. It offers the opportunity to accrue additional pension on top of the first and second pillar pension in the form of a pension insurance or bank savings product. This option is often chosen in the case the first and second pillar pension is insufficient for maintenance of the lifestyle in the working-phase or it is chosen if no second pillar pension is accrued via the employer. Insurers are the only institutions that offer third pillar pension.
1.3
Defined Contribution plans
As stated before, there are mainly two types of pension plans: defined benefit (DB) plans and defined contribution (DC) plans. In DB plans, the benefits an employee re-ceives after retirement is a fixed amount. To get this amount, the employee accrues a percentage of his/her salary and the employer pays premiums. The pension capital is invested in the accumulation phase. When returns on investment are low, the premium the employer is paying goes up. In case of good returns on investment, the employer can pay less premium. This way, employers face risks in DB plans whereas for employees DB plans are risk-free as their entitlements are guaranteed. Contrarily, in DC plans, it is the employee who faces risk. He/she is the owner of his/her pension capital. Every year, a premium paid by the employee is added to this pension capital. The pension capital is invested. At retirement, a life-long annuity can be bought from the total pension capital at that point consisting of premiums paid and returns on investment. Therefore, the value of the benefits is not fixed in advance but it depends on the return on investment.
In the last couple of years, many pension plans are shifted from DB to DC. In 2010, from all pension fund members, 93 per cent belonged to a DB scheme, only 4.5 per cent belonged to a DC scheme and 2.3 per cent had a combination of DB and DC plan (Cannon et al., 2015). The years that followed show a sharply shift towards DC plans. In the past 7 years the number of employees with DC plans has doubled whereas the number of employees with DB plans has steadily decreased (Treur, 2018). So employers seem to shift the risks from themselves to the employee. This may sound unfavourable for employees but DC plans are in some ways also advantageous for them. In this thesis, the focus is on DC plans in second pillar pensions in the Netherlands. Therefore, the remainder of this thesis will primarily be about DC plans.
1.3.1 Risks involved in DC plans
In DC plans, the worker is the holder of his/her own pension capital during the accu-mulation phase unlike with DB plans where pension funds or insurers own the pension capital. This implies that workers can make their own decisions about their pension cap-ital during the accumulation phase in DC plans. For instance, employees can chose how they would like to have their pension capital invested: with much risk, rather risk-free or somewhere in between. Providers of DC plans mostly offer their participants investment options. It differs per provider how many options are offered but most of the time three standard options are given: the default option, the offensive option and the defensive option. The amount of pension capital at retirement then depends on, next to other things, the investment return on the pension capital and therefore on the investment strategy chosen by the employee. This way, employees can manage their pension capital
to some extent. Naturally, this means that investment risk is shifted fully to employees in DC plans whereas, as stated before, in DB plans employers have to supplement the pension capital of their workers in case of bad investment returns.
Another risk that can be seen as part of investment risk is interest rate risk. Inter-est rate risk is the risk that the interInter-est is low during the accumulation phase and at retirement. During the accumulation phase of pensions part of the pension capital is invested risk free meaning that the return is equal to the interest rate. As a consequence, when the interest rate is low during this phase, the return on pension capital is low so the pension capital grows less. Interest rate risk also plays a role at retirement. In DC plans (and also in DB plans), a life-long annuity should be bought at retirement. The amount of pension benefit you receive during retirement depends on the pension capital accrued up to the retirement date and the price of an annuity at that date. The annuity price is based on the life expectancy of the population and on the interest rate. When the interest rate is low it means that an insurer will need more money to pay the pen-sion benefit to the retiree than with a high interest rate. Therefore, with a low interest rate the price of an annuity is higher. Hence, interest rate constitutes a risk at retire-ment. Since the retirement date is fixed and it is a one-time event, interest rate risk is substantial. However, there exist many ways to hedge interest rate risk, for instance us-ing swaps. For simplicity, we assume in this thesis that the interest rate is flat over time.
Another risk participants in pension plans face is inflation risk. Inflation is the well known phenomenon that prices of goods and services rise so money becomes worth less. When your pension capital is not supplemented according inflation, you can buy less pension at retirement. However, in many cases, pension funds and insurers will increase your pension capital with the inflation rate. This is called indexation. Indexation will only take place when the coverage ratio (ratio between assets and liabilities) of the fund of insurer is high enough. This boundary is fiscally managed and is equal to 110% in the Netherlands (Pensioenwet, artikel 137). Indexation is not considered in this thesis for simplicity.
A potentially more challenging risk is longevity risk. Longevity risk is the risk that people are getting older and older and consequently, pension funds and insurers have to pay out more benefits without receiving more premiums. When an individual outlives his/her life expectancy, it means that he/she needs additional years of pension bene-fits. The pension fund or insurer is therefore making a loss for this individual. When outliving life expectancy happens only for a small part of the participants of a fund or insurer, it can most of the time be compensated by the ’wins’ of people dying earlier than expected. However, when the life expectancy of the whole population is increasing, it might form a problem for pension funds and insurers. Longevity risk is seen as one of the most dangerous risks for pension funds and insurers. Hedging longevity risk is diffi-cult as the trend of people getting older and older itself is not the biggest problem but the uncertainty of the future development of mortality is hard to capture (De Weage-naere et al., 2010). Until a decade ago, the only way to hedge longevity risk was via insurance. Pension liabilities were sold to insurers and subsequently the insurers bought reinsurance. Today, so called longevity swaps exist (Coughlan et al., 2011). The idea of a longevity swap is exchanging longevity risk for a premium. As explained by Van Delft (2012), the buyer of a swap (pension fund of insurer) pays predetermined fixed payments depending on future mortality and receives floating payment reflecting the realized number of survivors in a population. So, several ways of mitigating longevity risk exist but they are complicated.
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis5
risk and macro longevity risk. Micro longevity risk is the risk that an individual is out-living his/her life expectancy. When this risk is not shared within a pool of people, the participant faces the risk that his pension pot is getting empty and he/she will not get enough benefits or the benefits will be lower. Macro longevity risk is the risk that the life expectancy of the whole population is changing. Life expectancy can in doing so increase as well as decrease. Principally, macro longevity risk appears when mortality tables are changed or with a change of the experience mortality.
1.3.2 Freedom of choice in DC plans
Uncertainty due to risks is usually unfavorable though sometimes inevitable. However, in some cases risks are seeked. For instance in the case of the in 2016 implemented option of a variable pension, where pension capital in DC plans is invested in the decumulation phase as well as in the accumulation phase only. In the last couple of years, more options regarding your pension are given. Pension innovation in the Dutch pension system is needed to fill the weak spots in our system and preserve a strong pension environment (Bovenberg, 2014). Freedom of choice in pensions is a main element of this pension in-novation and it comes in many forms. You can choose to vary the height of your pension benefits with the use of the so-called high/low construction. With this option you can get high benefits in the first years of your retirement en lower benefits in the later years or visa versa. Another option is an early, deferred or part-time pension. As the name suggests it implies that you can choose to retire earlier with the pension benefits getting lower due to a longer period of time of payments and a lower accrued pension capital, you can choose to delay your retirement for some years due to which your benefits will be higher or you can choose to retire part-time and receive benefits proportionally. A third option is a (partial) lump-sum at retirement. This option is not offered in the Netherlands yet but many other countries do. A (partial) lump-sum implies that you can have (a part of) your pension capital disbursed as a one-time payment at your retire-ment date. This option is very popular in the US (Mottola and Utkus, 2007) but also in other countries the option is taken frequently (Dominquez-Barrero and Lopez-Baborda, 2011). In the Netherlands, retirees are required to annuitize their pension capital. How-ever, this is likely to change in the coming years (Klijnsma, 2016). Lastly, one has the option to invest pension capital in DC plans also in the decumulation phase. This is a so-called variable pension. The law that introduced a variable pension in the Nether-lands, ’Wet verbeterde premieregeling’, entered into force on September 1st 2016. Even though freedom of choice in pension may seem desirable, it is often doubted whether people are able to make the right choices. Christopher Daykin (1998), Director of the UK Government Actuary’s Department, once properly qualified this hesitation about freedom of choice:
”It is difficult to be against choice, but the essential factor with pensions is to ensure that the consumer has adequate safeguards since the issues are rather too complicated for most people to grasp fully the nature of the choices with which they are faced. Although it may sound paternalistic, it is sometimes better to limit the number of choices, in order to ensure that everyone receives a reasonable level of pension.”.
In order to get a better understanding of the choices in pensions, the different choices will be explained in more detail below.
The high/low construction is an option offered by many pension funds and insurers. However, funds and insurers are not obliged to offer this option. Retirees can choose in advance whether they want to make use of the option. The option consists of two forms: first a high pension and later a low pension, and first a low pension and later a high pension. The former is the most frequently chosen form of the high/low construc-tion. Dutch law regulates that the ratio between high-low is at most 100 : 75 meaning that the lowest benefit is at most 75% of the highest benefit. As an example, you can
choose to have high pension benefits in the first 5 or 10 years of your retirement and low benefits in the later years. You can also choose to let the height of your pension benefits decrease every year by a fixed percentage. The basis of a high/low construction is that the actuarial present value of the benefits in a high/low construction is equal to the actuarial present value of benefits in a regular annuity. Several reasons may entice retirees to go for a higher pension in the first years of retirement. Dietvorst and Visser (2012) mention some of them: retirees may need some extra money in the first years of retirement to be able to make a long-distance travels or even start an own business. Also retirees who are expecting not to live long anymore may have financial advantage going for a high pension in the first years. Next to these grounds, a main reason for applying a high/low construction on your pension is as compensation for the SS. Since the SS age is increasing, it can happen that you receive SS later than your retirement date. This implies that in the years of your retirement until your SS age, you receive only second pillar pension. A higher second pillar pension in the first years, so implementing the high/low construction, compensates for this. Also people who choose to go for an early pension, and hence retire before their SS age, will have a small pension in those years before they receive SS pension. To compensate for this, the high/low construction is suitable.
Similar to the choice of a high/low construction, many pension funds and insurers offer, however are not obliged to offer, the choice of an early, deferred or part-time pension. The default option is that a participant at a fund or insurer retires at the determined pensionable age. This pension age used to be 65 in the Netherlands, similar to the SS age, yet this SS age is increasing with the life expectancy and therefore the pensionable age is adjusted as well. In 2017, most funds and insurers handled a pensionable age of 67. Advancing or deferring the pension date is in most cases possible and the ranges for this are usually negotiated with the executor of the pension scheme. For instance, a fund offers the possibility to advance and defer the pension date 5 years. An early re-tirement though implies a lower pension. The cause is that when retiring early, pension is accrued for fewer years and benefits have to be paid for more years. For a deferred pension reversed reasoning applies: it leads to a higher pension. Most pension funds and insurers apply factors to early and deferred pensions. Pension benefits are decreased by a fixed percentage every year the participant is retiring earlier and increased by a fixed percentage every year the participant is deferring his/her pension. This way going for an early retirement is discouraged. A more recent form of flexibility in the pension date is the part-time pension. As the name suggests, the participant receives part of his/her pension and continues to work part-time. For people who are reluctant about working longer due to the increased pensionable age but are not willing to give up much of their pension by going for an early retirement, this option is a suitable solution. Pension ben-efits will be proportional to the part-time rate. The option of a part-time pension is not evaluated in this thesis. Even though a flexible retirement age may seem appealing, the idea can be overused. In Sweden, a statutory retirement age was almost abolished since older individuals were expected to rationally choose an exit age balancing their suffer-ing from work and income needs dursuffer-ing retirement (Hinrichs, 2004). As a consequence, income needs were underestimated and too early pension date was chosen.
A extensively negotiated but not (yet) implemented choice in Dutch pensions is the (partial) lump-sum at retirement. The option to receive (part of) your pension capital as a one-time payment at retirement sounds appealing to many. Participants in pension schemes may want to use a part of their pension capital for mortgage repayment, ad-justment of their homes or as a buffer for unforeseen (medical) costs (Van Ewijk et al., 2014). However, offering the option of a lump-sum at retirement entails a substantial risk: outliving of pension. When participants underestimate their pension needs and
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis7
their life expectancy but do choose for a lump-sum at retirement, they face the risk that they run out of money and are not able to make a living anymore at some age. This is a reason why annuitizing pension capital is obliged in some countries including the Netherlands. Despite the fact that an annuity guarantees a life-long income stream, participants choose lump-sums on large scale. International number show that only a minority of the participants voluntarily purchase an annuity, which is called the ’annuity puzzle’ (B¨utler and Teppa, 2007). For the NetherlandsVan Ewijk et al. (2017)conclude that a lump-sum payment is a realistic option provided that it is bounded to for instance 10% or 20% of the pension capital. The residual of the pension capital can be harnessed for the purchase of an annuity. Lump-sum payments entail another risk which is present to a smaller extent in a high/low construction as well: the risk of strategic behaviour. Participants with a low life expectancy will tend to take a lump-sum rather than an annuity. Sharing of micro longevity risk will be undermined due to this. We will not evaluate the option of a (partial) lump-sum in this research since it is not (yet) offered in the Netherlands.
With the implementation of the ’Wet verbeterde premieregeling’ (Wvp) on Septem-ber 1st, 2016 the idea of a variable pension in the Dutch pension system became a fact. As the name suggests, a variable pension implies that pension benefits are not fixed but vary periodically. The idea is that when the option of a variable pension is chosen, pen-sion capital will be invested after retirement as well as only during the working phase which is the case with a fixed pension. Since, naturally, the returns on investment are not certain, the height of the pension capital with which an annuity can be purchased varies. And therefore, the annuity amount varies. Hence, choosing for a variable pension entails investment risk. The main reason for a variable pension is that higher pension benefits are expected. This is found by KATOCO (2014) who concludes that invest-ing after retirement leads to a higher expected pension with mainly the risk of yearly pension adjustments. Variable pensions can be executed in several ways. An important distinction is a collective execution of the variable pension or an individual execution of it. In the collective version, one pot of money is managed for the entire allocation pool. Results on investment of the collective portfolio are passed on to the participants by increasing or decreasing the pension benefits once a year. Actuarial results, for in-stance the biometric return, are passed on to the participants in the same way. In the individual version of execution of a variable pension every individual has his/her own pension pot. So the amount of pension benefits depends on the returns on investment of this individual portfolio. An advantage of this individual version is that more trans-parency is possible. However, individual execution of a variable pension is more costly since for every participant an appropriate interest rate hedge should be realized which is complex. Next to the choice of collective or individual execution, there is an option to choose between a capital approach or a benefit approach for a variable pension in the Wvp. The former implies that a capital is held after retirement date, similar to the accumulation phase. Yearly (possibly more frequent) results are added to this cap-ital and with this new capcap-ital the amount of pension benefit is determined. In stead of adding premiums to the capital as in the accumulation phase, in the decumulation phase benefits are subtracted from the pension capital. The benefit approach implies that at retirement date, pension capital is converted into a pension right. So after retirement, not a pension capital but a benefit amount is managed which will be adjusted yearly (or more frequently) based on the results. This is the more obvious variant in case of a collective execution. The Wvp also offers the option to smooth positive or negative shocks regarding investment returns and actuarial results over a period of maximal 10 years. The idea behind this option is that pension benefits are less fluctuating since the excess returns (the difference between the risk-free rate and the financial return) are smoothed over time. This smoothing however does not influence the total result of the
pension benefits even though it influences the benefits in certain years. Furthermore, the Wvp has implemented the option of a fixed increase or decrease (high/low construc-tion) in pension benefits. Micro and macro longevity risk are accounted for in the Wvp. In the case of micro longevity risk, money that becomes available due to decease of a participant is added to the pension capital of the other participants in the risk-pool. This is called the biometric return. To estimate the biometric return to be allocated, the right mortality tables and experience mortality should be consulted. Different from micro longevity risk, macro longevity risk does not depend on decease of single partic-ipants in a risk-pool but on the change in mortality for the whole pool so it does not influence the pension benefits only at individual level but for everyone. Adjustments of mortality tables influence the price of annuities and hence influence the annuity amount at retirement. Hence, the Wvp offers many options regarding variable pensions. Exist-ing literature confirms that choosExist-ing a variable pension is valuable. Steenkamp (2016)
found evidence for it in his research and concluded that investing pension capital in the decumulation phase with limited risk provides welfare gains.
Chapter 2
Methodology
2.1
Lee Carter model
Lee and Carter (1992) found a model for estimating mortality: the Lee Carter model. The discrete-time Lee Carter model estimates the logarithmic mortality rate mx,t, where
mx,t is the fraction of Deaths over Exposure, i.e.
mx,t=
Dx,t
Ex,t
, (2.1)
with Dx,tthe number of deaths recorded at age x during period t and Ex,tthe
exposure-to-risk, i.e. Extis the average number of people living during the calender year t (
Spedi-cato et al.). The logarithmic mortality rate of someone with age x in year t is given by
log(mx,t) = αx+ βxκt+ x,t, (2.2)
where αxand βx are time-invariant parameters and κtis often called the mortality index
and is independent of age. It is assumed that x,t is i.i.d. with zero mean.
As explained byBeutner et al. (2016)model (2.2) is over-parametrized. To deal with this problem, Lee and Carter proposed a solution of imposing constraints to the parameters to ensure identifiability. The constraints are given by:
X x βx= 1 and (2.3) X t κt= 0. (2.4)
2.1.1 Maximum Likelihood Estimation
Let Dxt and Ext be data about the number of deaths recorded at age x during period
t and about the exposure-to-risk, respectively.
Now let
Dxt∼ P oisson(Extµxt) with µxt= exp(αx+ βxκt).
Then
P(Dxt= dxt) =
(Extµxt)dxtexp(−Extµxt)
dxt!
and L(Dxt, Ext; θ) = Y x,t (Extµxt)Dxtexp(−Extµxt) Dxt! = (Extµxt) P x,tDxtexp(−P x,tExtµxt) Q x,tDxt! ,
where θ is the parameter space given by θ = (µxt). Now taking the logarithm of the
above, we get ln L(Dxt, Ext; θ) = l(θ) = X x,t (Dxtln Extµxt− Extµxt− ln Dxt!) =X x,t (Dxtln Ext+ Dxtln µxt− Extµxt− ln Dxt!) =X x,t (Dxtln µxt− Extµxt) + C (1) = X x,t (Dxt(αx+ βxκt) − Extexp(αx+ βxκt)) + C,
where C is a constant and (1) follows from substituting µxt = exp(αx+ βxκt). Then,
maximizing ln L(Dxt, Ext; θ) with respect to θ gives best estimates for ˆαx, ˆβx and ˆκt.
2.1.2 Forecasting future mortality
In order to forecast future mortality, an ARIMA(0,1,0)-model is determined for ˆκtusing
the Box-Jenkins methodology (Box and Jenkins, 1970). Lee and Carter (1992)explain that a random walk with drift is the simplest model that describes κ reasonably well. Therefore, the model for ˆκ used in this research is given by
ˆ
κt= ˆκt−1+ φ + t, (2.5)
where t i.i.d.
∼ N (0, σ2), σ is a constant standard deviation and φ is a constant drift.
2.1.3 Bootstrapping the model
We want to find confidence intervals around the estimates of ˆαx, ˆβx and ˆκt.
Bootstrap-ping is a resampling technique widely used for estimating parameters. Therefore, the bootstrapping method described by Keijzer (2014) is applied to find these confidence intervals. The following algorithm is runned N times:
1. Estimate the Lee Carter model using ˆαx, ˆβx and ˆκt and derive the Pearson
resid-uals rx,t. Here rx,t is given by:
rx,t = Dx,t− ˆDx,t q ˆ Dx,t , (2.6)
where ˆDx,t is the estimated number of deaths following from the model.
2. To ensure that the expectation of the residuals ˜rx,t is equal to zero, let ˜rx,t =
rx,t− ¯r. Then draw with replacement from the group of residuals.
3. Create new realizations Dx,ts given by
Dx,ts = ˆDx,t+
q ˆ
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis11
4. Using the bootstrap sample dsx,t, reestimate the model to obtain new ˆαsx, ˆβxs and ˆ
κs
t. Again, use these to determine the residuals.
5. Estimate the times series for ˆκst using model (2.5).
6. Repeat steps 2 until 5 for s = 1, ..., N .
Now, N sets of estimations of ˆαsx, ˆβxs and ˆκst are found. These estimators are used to find N sets of mortality rates by applying formula (2.2).
2.1.4 Survival probabilities
Having calculated the mortality rate mx,t, we can calculate the survival probabilities.
The probability that someone aged x at time t will survive another year is given by
px,t= exp (−mx,t), (2.8)
assuming mx+s,t+s= mx,t where s ∈ [0, 1) and x and t are integer. Then multiple-year
survival probabilities, the probability that an x-year-old at time t will survive another i years, is given by ipx,t = i−1 Y j=0 px+j,t+j. (2.9)
And then the probabilities of dying are given by qx,t= 1 − px,t andiqx,t= 1 −ipx,t.
As explained in Section2.1.3, we have created N sets of mortality rates by running the bootstrapping algorithm N times. So for these N sets of mortality rate, we can calculate the survival probabilities using formula (2.8). Let’s denote for simulation s ∈ {1, ..., N },
psx,t= exp (−msx,t)
and then the multiple-year survival probabilities
ipsx,t = i−1
Y
j=0
psx+j,t+j.
Using the survival probabilities, we are able to calculate the remaining life expectancy ex,t. The remaining life expectancy gives the time an individual of age x at time t
is expected to life. We define the remaining life expectancy, just as The Royal Dutch Actuarial Association (2018) did, as follows
esx,t = 0.5 + ∞ X i=0 i Y k=0 (1 − qx+k,t+ks ) (2.10) which is equal to esx,t= 0.5 + ∞ X i=0 i Y k=0 (psx+k,t+k) (2.11)
for all s ∈ {0, ..., N } and where esx,t is given in years. It is assumed that decease happens on average at the half of the year.
2.1.5 Linear extrapolation of survival probabilities
When calculating the multiple-year survival probability ipx,t as explained in Section
2.1.4 for determining the annuity price, we multiply future one-year survival probabil-ities. These one-year probabilities are based on the forecasts of the κ’s as in equation (2.5). However, when calculating the multiple-year survival probability at time t, we cannot use the forecasts of the κ’s for later times since these are based on a stochastic process and thus yet unobserved and uncertain whereas the annuity price should be deterministic. Therefore, we need to make an estimation of the κ’s for these later times. This is done by linear extrapolation. Linear extrapolation is the process of estimating the value of a variable beyond the observation range. This estimation is given by
¯ κi,t = ˆκt+ ψt· i, (2.12) where ψt= ˆ κt− ˆκ0 t .
Here i is the future year for which κ is estimated at time t. This method of estimation is applied to all N scenario’s (see Section 2.1.3). So we get for every s ∈ {1, ..., N }
¯ κsi,t= ˆκst+ ψts· i, (2.13) where ψts= ˆκ s t− ˆκs0 t .
Then for calculating the survival probabilities used to determine the price of an annuity, we substitute ¯κsi,t into equation (2.2) together with the ˆαsx and ˆβxs found by the boot-strapping method explained in Section2.1.3to get the mortality rates ¯msx,t. We calculate the corresponding survival probabilities, ¯psx,t, and multiple-year survival probabilities,
ip¯sx,t using equations (2.8) and (2.9). That is,
log( ¯msx,t) = ˆαsx+ ˆβxsκ¯st, ¯ psx,t= exp (− ¯msx,t) and ip¯sx,t = i−1 Y j=0 ¯ psx+j,t+j.
These multiple-year survival probabilitiesip¯sx,t are used for calculating the annuity price
as given in equation (2.17).
Note that for calculating the actuarial present value of the future benefits (see Section
2.3), we can make use of the ’normal’ multiple-year survival probabilities ipsx,t where
the forecasted κ’s as given in equation (2.5) are being used except for the variant with only micro longevity risk.
2.2
Pension benefit modeling
We set the starting date of the calculations at 1-1-2017 and call this t = 0, and we as-sume that the reference person has an age of 62 (x0 = 62) at this date (see Section4.2).
The pensionable age is called xop and is set to 67 for the standard variant. Note that in
the variant of an early/deferred pension, this xop can take on any value between 62 and
72. The pensionable age of 67 is the age most pension funds and insurers applied in 2017.
Note that t from this point on, where t = 0 is the starting date of the calculation (1-1-2017), is different from the t in Section 2.1, where t = 0 was equal to the first year of the mortality data.
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis13
2.2.1 Capital accrual
Since at the starting date of the calculations, the reference person has not yet reached the pensionable age, we need to calculate the pension capital in the years from age 62 until age xop. The capital at time t = 0 is known. The following equation gives the
estimated capital for these years
Ct+1= (Ct+ Pt) · (1 + αxtrt+ (1 − αxt)r
f) for t = 1, ..., t
op , (2.14)
where top is the retirement date, rt is the return on stocks at time t (see Section2.2.2),
rf is the (constant) risk-free rate, αxt is the fraction of capital invested in stocks at age
xt and Pt is the premium paid at time t. In a DC plan, these premiums are based on
age-dependent premium rates. The premium at time t is then equal to the pensionable salary times the premium rate corresponding to the age at time t where the pensionable salary is equal to the yearly salary minus a so-called offset (O). This offset represents the part of the salary over which the participant receives social security (AOW).
Note that due to the uncertain return on stocks, the capital in years t = 1, ..., top is
uncertain. So we have Ct+1s = (Cts+ Pt) · (1 + αxtr s t + (1 − αxt)r f) for t = 1, ..., t op. (2.15) for all s ∈ {1, ..., N } 2.2.2 Returns on investment
It is commonly assumed that stock returns are normally distributed (Aas, 2004). There-fore, we generate stock returns rtusing a normal distribution with mean µrand standard
deviation σr given by 7.0% and 20% respectively which is advised by the Commissie
parameters (2014). The returns can be calculated by the following formula
rt= µr+ ρ · σr· t−1+ (1 − ρ) · σr· t, (2.16)
where ρ is the autocorrelation parameter in the model and t is the error term which
we assume to be i.i.d. and standard normally distributed (N(0,1)).
The Dutch Central Bank (DNB) calculated the autocorrelation of stocks in the period 1988-2006. Stocks in mature markets are found to have an autocorrelation of 1%, and private equity and stocks in emerging markets are found to have an autocorrelation of 19%. For simplicity, we assume the autocorrelation to be equal to 13%.
We simulate the normally distributed stock returns N times, so we get for every simu-lation s ∈ {1, ..., N }
rst = µr+ ρ · σr· st−1+ (1 − ρ) · σr· st.
We assume that the risk-free rate (rf) is fixed over time. So, a constant rf can be used over the entire investment horizon.
Note that the simulation of stock returns is independent from the simulation of the mortality rates. This means that the s for which s ∈ {1, ..., N } belonging to the invest-ment return simulations is a different one from the s for which s ∈ {1, ..., N } belonging to the bootstrap simulations for the calculation of the N mortality rate sets as explained in Section2.1.4.
2.2.3 Annuity price
As mentioned in Section 1.3.2, in the Netherlands people are obliged to annuitize their pension capital. An annuity is a fixed life-long yearly benefit from the moment of pur-chasing the annuity on. To be able to estimate the amount of benefit receivable during retirement, annuity prices are calculated. The annuity price is the actuarial fair price of a benefit stream equal to e1 during the entire resting life. In this case, the benefits are made immediately at the beginning of each subsequent year the participant aged xt
survives. As explained in Section 2.1.5we need linear extrapolation for calculating the multiple-year survival probabilities since the annuity price should be deterministic and cannot be based on a stochastic process. The price of purchasing an annuity at time top
can be calculated by the following formulas
¨ astop = 1 + Tmax−xtop X i=1 ip¯sxtop,top+i−1 (1 + rf)i , (2.17) with ip¯sxtop,top+i−1 = i−1 Y j=0 ¯ psxtop+j,top+i−1+j (2.18)
for every s ∈ {1, ..., N } where Tmax is an age an individual is assumed to be deceased
with probability 1 and rf is the risk-free rate. Note that the price of an annuity at time t > 0 is uncertain due to uncertain survival probabilities. Furthermore we assume that the participant pool consists of men only so the survival probabilities implemented in equation (2.17) are based on men.
2.2.4 Pension benefits
At retirement date, top, the pension capital accrued up to that time (Ctsop) is converted
into an annuity. The amount of pension benefits corresponding to an annuity purchased at time top, Btsop, can now be calculated by
Btsop = C
s top
¨
astop (2.19)
for all capital scenario’s s ∈ {1, ..., N }. Therefore, when retiring at time top, the amount
of benefit to receive yearly during the rest of your life is equal to Btsop. The amount of benefit receivable from time top is uncertain at time t = 0 because of the uncertain
accrued capital and the uncertain annuity price. The average pension benefit receivable from time top onwards over all scenario’s s ∈ {1, ..., N } is given by
Btop = 1 N N X s=1 Btsop. (2.20)
In the Netherlands, the first pillar pension consists of a Social Security pension (in Dutch the AOW). Because the total amount of pension the participant receives after retirement is equal to the sum of the first and second pillar pension, it is relevant to take into account Social Security (SS). Note that the SS pension is receivable as of the age of 67 for a participant of age 62 in 2017, so tSS = 5. Let BSS be the yearly SS payment
assumed that this is a fixed payment over the entire benefit horizon. Since the reference person in this research is aged 62 in 2017 (see Section4.2) it holds that top= tSS and
we then get
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis15
for an uncertain pension benefit as given in equation (2.19) and
Bttotalop = Btop+ B
SS (2.22)
for an average pension benefit as given in equation (2.20).
2.3
Valuation of pension benefits
The aim of this thesis is to compare different variants of pensions taking into account longevity risk from the participant’s perspective. To be able to estimate longevity risk, we need to calculate the present value of the cashflows in the decumulation phase of pensions, i.e. the present value of the future pension benefits. We will look at the actu-arial present value (APV) of the future pension benefits in different pension variants to evaluate micro and macro longevity risk in these variants. The APV of future pension benefits is evaluated at time t = 0. In the evaluation, SS benefits are taken into account. Note that the second pillar pension benefits are receivable as of top which is in most
variants unequal to 0 and the SS benefits are receivable as of the SS-age, thus at time tSS, which is in for our reference person (see Section4.2) 67. In the standard variant the
first and second pillar pension are receivable from the same moment on namely t = 5.
2.3.1 APV
For the evaluation of the different pension variants regarding micro and macro longevity risk we calculate the APV of future pension benefits at time t = 0. Note again that the pension benefits (first and second pillar) are receivable as of time top= tSS in a standard
variant. The following model is used for calculating the APV in the standard pension variant AP V0s(Btotal,stop ) = g(Txs0) = Tmax−x 0 X i=top 1 (1 + rf)−i · B total,s top · 1Tx0s ≥i , (2.23) with 1Ts
x0≥i the indicator function given by
1Ts x0≥i=
1 if Txs0 ≥ i 0 if Txs0 < i ,
where Txs0 is moment of death of an individual with age x0 for scenario s ∈ {1, ..., N }.
This moment can be determined using formula (2.11). Note that Txs0 ∈ {0, ..., Tmax−x 0}.
We have that AP V0s(Btsop) is a vector of length N with APV’s of future pension benefits at time t = 0 for all scenarios s. We can calculate the expected APV by
AP V0(Bttotal,1op , ..., B total,N top ) = 1 N N X s=1 AP V0s(Bttotal,sop ) (2.24)
and the standard deviation by
σ0 = s PN s=1(AP V0s(B total,s top ) − AP V0(B total,s top )) 2 N − 1 . (2.25)
The standard deviation of the APV, σ0micro, gives the uncertainty in the APV at time t = 0. This uncertainty is caused by the uncertainty in pension benefits due to investment risk and due to micro and macro longevity risk which manifests in the uncertainty in remaining life expectancy (which may be caused by for instance an acute deadly disease). Substituting Bttotalop in stead of Btotal,stop into equations (2.23)-(2.25) gives the standard deviation which gives the uncertainty in the APV which is caused by micro and macro
longevity risk only. In this case investment risk is thus excluded. The APV of future certain pension benefits is then given by
AP V0longevity(Bttotalop ) = g(Txs0) = Tmax−x 0 X i=top 1 (1 + rf)−i · B total top · 1Tx0s ≥i . (2.26)
To look purely at the uncertainty caused by micro longevity risk we calculate the mo-ment of death of an individual using
¯ esx,t= 0.5 + ∞ X i=0 i Y k=0 (¯psx+k,t+k) (2.27)
where the survival probabilities ¯psx+k,t+k are the forecasted probabilities as given in equation (2.14). This gives new moments of death ¯Ts
x0. Substituting these new moments
of death into equation (2.26) gives the APV regarding only micro longevity risk, i.e. AP V0micro(Bttotalop ). So in this variant, both investment risk and macro longevity risk are excluded. This APV is given by
AP V0s(Btotal,stop ) = g( ¯Txs0) = Tmax−x 0 X i=top 1 (1 + rf)−i · B total,s top · 1T¯x0s ≥i . (2.28)
Chapter 3
Different variants
As stated before, the main objective of this thesis is to compare longevity risk in different pension variants. We evaluate longevity risk from the participant’s perspective instead of from a pension fund’s or insurer’s perspective. For a standard pension, the standard variant, the formulas for the calculation of the annuity price, the pension benefits and the APV of the pension benefits are given in Chapter 2. The formulas for the variants with a early/deferred pension, high/low construction and a variable pension are given below. We also evaluate the combination of an early pension with a high/low construction. Again the annuity price and the pension benefits are calculated at time topand the APV
of future pension benefits is evaluated at time t = 0. The different options regarding pensions are explained in more detail in Section 1.3.2.
3.1
Early/deferred pension
Many insurers and pension funds offer their participants the option to retire earlier or later than the pensionable age. The option to go for an early pension implies fewer years of pension accrual and more years of pension benefits. Opposite reasoning holds for a deferred pension. We assume that going for an early pension is at most 5 years before pensionable age and going for a deferred pension at most 5 years after pension-able age. Note that t = 0 corresponds to an age of 62 in this research and thus the earliest moment of retiring. Again let xop be the pensionable age, however this time
this age may be chosen, i.e. xop ∈ {62, ..., 72}. And then similar let top be the chosen
retirement date, where top∈ {0, ..., 10}. We again want to evaluate APV of future
pen-sion benefits of both first and second pillar penpen-sion at time t = 0. The annuity price is calculated in the same way as given in equation (2.17) but now with a different top. The
pension benefits including SS pension is however calculated differently from equation (2.21) since we have that top 6= tSS if the chosen pensionable age is unequal to 67,
so top 6= 5. This means that the total receivable pension benefits is not flat over the
years. Therefore, we include them separately into the equations for calculating the APV.
The APV of total pension benefits is now equal to
AP V0s(Btsop, BSS) = Tmax−x 0 X i=top 1 (1 + rf)−1 · B s top· 1Tx0s ≥i + Tmax−x 0 X i=5 1 (1 + rf)−i · B SS · 1 Ts x0≥i . (3.1)
Note that tSS = 5 is the date the first SS benefits is received. In a similar way as in
Chapter 2, we can use a fixed second pillar pension benefit Btop (average over all B
s top)
to calculate the APV of pension benefits including SS with only longevity risk and we can use equation (2.27) to calculate the moments of death ¯Txs0 for calculating the APV regarding micro longevity risk only.
3.2
High/low construction
In the high/low construction the participant can choose to receive high (low) benefits in the first 5 or 10 years of retirement and low (high) benefits afterwards. I evaluate the case of a high benefit in the first 5 years and low benefits thereafter with a high low factor defined fhl. The annuity price for this variant is given by
¨ ahl,stop = 1 + 4 X i=1 ip¯sxtop,top+i−1 (1 + rf)i + Tmax−x top X i=5 ip¯sxtop,top+i−1· fhl (1 + rf)i . (3.2)
Using this actuarial fair annuity price, we can calculate the pension benefits. The (high) benefits to be received during the first 5 years (calculated at time top) are given by
Bth,sop = C
s top
¨
ahl,stop (3.3)
and the receivable (low) benefits after these 5 years (calculated at time top) until decease
can be written as a fraction of the high benefits, i.e.
Btl,sop = Bh,stop · fhl. (3.4) So at time top the high and low benefits to be received during the first five years of
retirement and the years thereafter, respectively, are determined. These high and low benefits are flat benefits. However, due to uncertain capital accrual caused by invest-ment risk and uncertain annuity prices Bth,sop and Btl,sop are uncertain. The total benefits Bttotal,h,sop , Btotal,l,stop are calculated by equation (2.21) and the benefits without investment risk Bttotal,hop , Bttotal,lop by equation (2.22). Note that these equations can be used since in this variant it is assumed that top= tSS.
The APV of the pension benefits in a high/low variant evaluated at time t = 0 is given by AP V0s(Bttotal,h,sop ) = top+4 X i=top 1 (1 + rf)−i · B total,h,s top · 1Tx0s ≥i + Tmax−x 0 X i=top+5 1 (1 + rf)−i · B total,h,s top · f hl· 1 Ts x0≥i . (3.5)
We can calculate the expectation and standard deviation of AP V0s(Bttotal,h,sop ) in the same way as explained in Section 2.3 Note that uncertainty in this APV is caused by investment risk and longevity risk.
When we want to look purely at the effect of uncertainty in the APV due to macro and micro longevity risk, we substitute Bttotal,hop into equation (3.5). And for the cal-culation of the APV with only micro longevity risk, we again use equation (2.27) to calculate the moments of death ¯Txs0 and substitute these into equation (3.5).
3.3
Early retirement with high/low construction
In Section 1.3.2 the phenomenon of a SS-gap when retiring early is explained and the high/low construction is introduced as being a solution to this gap. That means that when you choose to retire before the SS age, i.e. in this research xop = 62, ..., 66, the
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis19
benefits are high enough to maintain the standards of living and the total pension benefits (first pillar and second pillar) will not fluctuate too much over the years. The formulas given in Section3.1and3.2are combined to find the (expectation and standard deviation of the) APV of pension benefits including SS in the variant where an early retirement at pensionable age 62 is combined with the high/low construction.
3.4
Variable pension
A variable pension gives the option to invest your pension capital after retirement as well as before retirement. In this research, the individual version of execution of a variable pension with a capital approach is investigated meaning that the individual is assumed to have an own pension pot and capital is hold and adjusted yearly after retirement based on results.
The progression of the pension capital after retirement is calculated by
Ct+1s = (Cts− Bs t) · (1 + αxt∗ r s t + (1 − αxt)r f) · rbio,s t , (3.6)
for t ≥ top and for all s ∈ {1, ..., N }. We have that rbio,st = 1−q1s x+t,t =
1 ps
x+t,t is the
biometric return added to the pension capital. The biometric return is the amount of capital release due to decease of participants in a certain year in a participant pool (see Section1.3.2).
In the case of a variable pension with a capital approach, every year the amount of benefit to receive that year is calculated based on the annuity price at that time and the (resting) pension capital. The annuity price at time t where t ≥ topis calculated by
¨ ast = 1 + Tmax−x t X i=1 ip¯sxt,t+i−1 (1 + rf)i, (3.7)
and we assume that the pension benefit at time t where t ≥ top is equal to
Bts= C s t ¨ as t . (3.8)
So the pension capital to start with is equal to Ctop and a corresponding benefit B
s top
is calculated. Then for all subsequent periods, the pension capital and corresponding benefits are calculated using equations (3.6)-(3.8). Note that the pension benefits are thus variable due to the uncertain returns. The total pension benefit at time t, Bttotal,s, is calculated by equation (2.21) and without investment risk, Bttotal, by equation (2.22).
The APV’s of the pension benefits including SS at time t = 0 are calculated using equa-tion (2.23) only the benefits substituted are different in this case. In Section 2.3these benefits were a flat amount whereas in this case they variate for every t ≥ top. Then,
we can take the expectation and standard deviation of the APV’s over all s ∈ {1, ..., N } (like explained in Section2.3) to be able to compare the results with the other variants. Note that we can substitute Bttotal into equation (2.26) to look purely at longevity risk and we can substitute ¯Txs0 into equation (2.23) together with the benefits Bttotal to look purely at micro longevity risk.
Data and assumptions
4.1
Data
The mortality data used in the Lee Carter model and thus for calculating the survival probabilities are obtained from theHuman Mortality Database (HMD). This data con-sist of numbers of deaths and exposures-to-risk of the Dutch population between 1850 and 2016. The number of deaths, Dx,t, is equal to the number of people of age x that
deceased during period t. The exposure-to-risk, Ext, is the average number of people
living during the calender year t. The data provides number of deaths and exposures for ages 0-110. Since the data for ages above 100 is impractical to work with, it is assumed that the maximum age an individual is getting is equal to 100. Furthermore, for more reliable survival probabilities data from before 1970 is deleted from the dataset.
4.2
Reference person
To be able to investigate pension benefits for different survival probabilities we need a reference person to work with. The reference person is assumed to be an unmarried man with age 62 now (at time t = 0). Furthermore, we assume that his salary is e37,000 (at t = 0), which is the average income by the CPB in 2017, and is flat until retire-ment. This salary is used for calculating the premiums paid in the accumulation phase. The reference person is assumed to work full-time. He faces several option regarding his pension: standard pension, high/low construction, early/deferred pension and a variable pension. His aim is to select the option that maximizes the actuarial present value of the pension benefits keeping in mind longevity risk. Since the option of an early pension is assumed to be at most 5 years before the standard pensionable age, so at an age of 62, today is the best date to reflect the options for the reference person. The actuarial present value of future benefits of all variants will be evaluated at t = 0 so that they can be fairly compared to each other.
Note that the life-cycle pattern (see Section 4.3) is determined as of 15 years before retirement date. The life-cycles for a variable pension is different from the life-cycle for a standard pension. So at the age of 62 it should already be clear if he has chosen the variable pension or not. At time t = 0 we can only compare the APV of the variable pension with the other variants and tell if it was a ’good choice’ to go for a variable pension.
4.3
Assumptions
The parameter assumptions made in this research are given in Table4.1.
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis21
Table 4.1: Parameter assumptions used in this thesis.
Parameter Value Description
Starting year 1-1-2017 Starting date of calculations
BSS e1,091 The monthly net SS amount with tax credit in January 2017 for
an unmarried individual in the Netherlands C0 200,000 Pension capital at the age of 62
rf 1.0% Risk-free rate
µr 7.0% Mean of stock returns
σr 20.0% Standard deviation of stock returns
ρ 13.0% Autocorrelation in the stock return model
Tmax 100 The maximum age an individual is assumed to get
fhl 0.75 The factor corresponding to the high/low construction
O e19,191 Offset for unmarried in 2017 in the Netherlands N 500 Number of simulations/scenarios
x0 62 Age at starting date of calculations
xop 67 The pensionable age
tSS 5 The date corresponding to the SS-age
We set the starting year of the calculations equal to 2017 because the mortality data is up to and including 2016. Therefore, the first year that is forecasted is 2017. The rela-tive risk aversion parameter is set equal to 5 since that value belongs to a conservarela-tive investor who is relatively risk-averse which is likely for pensions (Steenkamp, 2016). The parameters in Table4.1are assumed to be constant over time. Furthermore, we assume that there is no indexation.
The premium rates, published by the Dutch tax authority (2017), are given in Ta-ble7.1. The percentages in the table are based on a 1.875% accrual rate for an average salary DB plan. The middle column gives the percentages corresponding to an applied actuarial interest of 3% and the right column gives the ones for a 4% actuarial interest. The premium percentages are based on old-age pension only and not on partner pension. In this thesis, the premiums based on 3% actuarial interest are used. Since the reference person is already 62 years old, only the last two rows from Table7.1 are applicable for this research. The premium rate for the ages 60-64 is equal to 23.0% and for the ages 65-66 it is 26.0%. Note that in the case of a deferred pension, the additional premium payments (for ages x ∈ {67, ..., xop}) are based on a premium rate of 26.0%.
Many insurers and pension funds offer so called life-cycle patterns. Life-cycle patterns are investment strategies in which the fraction of the pension capital invested in stocks is reduced as the retirement date is approaching. Typically, this reduction starts 15 or 20 years before retirement. In this research, we use a reduction starting 15 years be-fore retirement. In the case of a variable pension, pension capital is invested during the decumulation phase as well. So, it can be chosen that after retirement pension capital is invested in stocks as well as before retirement. The participant can choose which life-cycle pattern the insurer or pension funds should apply to his/her pension capital depending on the amount of risk the participant is willing to take. What is generally seen is that the more risk is taken, the higher the return. However, the higher the risk the higher the chance of a negative return. Therefore, the participant should be aware of the advantages and disadvantages of taking risk when making the choice which life-cycle pattern to apply. Most insurers and pension funds offer their participants three base options: the default life-cycle, the defensive life-cycle and the offensive life-cycle. Figure 4.1 gives the life-cycle pattern used in this research for the fixed pension vari-ants (standard, early/deferred, high/low). The life-cycle patterns which are used for the variable pension variant are given in Figures 7.1-7.3.
Figure 4.1: Graphical representation of the life-cycle investment pattern corresponding to the fixed pension variants where pension capital is invested only in the accumulation phase. The blue area represents the part of the pension capital invested in stocks.
Chapter 5
Results
5.1
Lee Carter
We used the Lee Carter model described in Section 2.1 to model mortality. In Figure
5.1 the results of the estimates of ˆαx, ˆβx and ˆκt are shown. The Lee Carter model is
estimated only for men. The results are used for calculating the annuity price and the APV of future pension benefits.
Figure 5.1: Graphical representation of the results of the parameter estimates ˆαx, ˆβx
and ˆκt of the Lee Carter model for Dutch men.
From the figures we see that the age dependent parameter ˆαx is decreasing for ages 0
until ± 10 and increasing thereafter with a small peak at age ± 20. This peak is in line with literature which explains that this peak for men is due to the more risky and wild behaviour of men around the age of 20. Furthermore, the time dependent parameter ˆκt
is decreasing which is also according literature that mortality rates have declined over the years.
5.2
Pension capital
In Section 2.2.1 we explained that in the years between the starting date and the re-tirement date, the reference person accrues pension capital. The capital is invested in risky stocks and risk-free bonds. In the variants different from the variable pension, we use the life-cycle given in Figure4.1. Since the returns on stocks are uncertain, the amount of pension capital is uncertain. Figure 5.2shows the progression of the pension capital (ine) for the ages 62 (which is the age at the starting date of the calculations) until 67 (which is the standard pensionable age). Note that this period belongs to the accumulation phase. The capital at t = 0 (x0 = 62) is assumed to be equal toe200,000.
The black line gives the average pension progression over all s ∈ {1, ..., N } investment
return scenarios. The green and red line give the area of possible pension capital within 1 standard deviation.
Figure 5.2: Graphical representation of the pension capital progression Cts (see equation (2.15)) as of the age of the reference person today (62) in the accumulation phase of a standard pension with pensionable age 67 with a starting pension capital of e200,000. The black lines gives the expected pension capital over all s ∈ {1, ..., N } for all ages 62 ≤ x ≤ 67 and the green and red line give the area within 1 standard deviation of this expected capital.
The average pension capital at age 67 is e245,120 and the standard deviation of the pension capital at age 67 is e17,794.
In the case of a variable pension, the pension capital is recalculated every year after re-tirement as well as before rere-tirement. The capital progression depends on the life-cycle chosen. Figure 5.3 gives the capital progression for the three different life-cycles: the black line gives the average pension benefit at every age and the green and red line give the area of possible pension capital within 1 standard deviation. The pensionable age is 67 for this variant.
Freedom of choice and longevity risk in DC plans — Ruth Oude Avenhuis25
Figure 5.3: Graphical representation of the pension capital progression Cts as of the age of the reference person today (62) in the accumulation and decumulation phase of a variable pension with pensionable age 67 for different life-cycle patterns with a starting pension capital of e200,000. The black lines gives the expected pension capital for all ages 62 ≤ x ≤ 67 and the green and red line give the area within 1 standard deviation of this expected capital.
(a) Pension capital progression with a default life-cycle pattern.
(b) Pension capital progression with a defensive life-cycle pattern. (c) Pension capital progression with an offensive life-cycle pattern.
From the figures above we can see that the capital is increasing in all three life-cycles until the age of 67 which is the pensionable age. This is because the ages 62-67 belong to the accumulation phase for which the pension capital is increased every year with premiums. Note that in the offensive life-cycle, the average pension capital is increasing even until the age of 70 even though the decumulation phase has already started. Reason for this is that the investment returns in a offensive life-cycle (which increase the pension capital) are higher than the amount of pension benefit (which decrease the pension capital) in the years between the age of 67 and 70. The pension capital from age 100 on is set equal to 0 since people are assumed to become no older than 100. Moreover, from the figures we see that the average pension capital for every age is highest in the variant with an offensive life-cycle and lowest for the variant with a defensive life-cycle. However, the uncertainty in this capital is also highest for the variant with an offensive
