Transparent intergenerational
risk-sharing for pension funds in the
Netherlands
Vincent Tiesinga
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: dhr. V. (Vincent) Tiesinga MSc
Student nr: 10126783
Email: v.tiesinga@gmail.com
Date: August 26, 2016
Supervisor: dhr. J. (Jitze) Hooijsma MSc
This document is written by Vincent Tiesinga who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Transparent intergenerational risk-sharing — Vincent Tiesinga iii
Abstract
In this thesis, several adjustments to the financial policy of Dutch pension funds are discussed and analyzed, to deal with the declining social support of intergenerational risk-sharing. The main reasons of the declining social support are low interest rates and low investment returns in combination with intransparent forms of intergenerational risk-sharing. The objective is to design a financial policy with either zero-mean intergenerational risk-sharing properties, or a more explicit and transparent form of intergenerational risk-sharing, without com-promising in indexation quality of the pension result. By zero-mean intergenerational risk-sharing it is meant that although risks are shared between generations, the average result of all possible scenarios is such that there is no net transfer of economic value.
It is shown that alternative policies with zero-mean intergenerational risk-sharing offer similar pension results on average, but more volatility is unavoidable. Better results are obtained for financial policies with more transparent intergenerational risk-sharing. A new pension fund ambition is introduced, such that the pension result is at least 50% of a fully compensated pension for price inflation, with a 97.5% probability. It can be concluded that this ambition is of a more transparent na-ture, offers better protection in bad scenarios while maintaining good pension results in other scenarios. The order of magnitude of intergen-erational value transfer is comparable to the current financial policy in the Netherlands, while the more transparent nature of the pension fund ambition can help to increase the much needed social support in bad economic times.
Keywords FTK, Intergenerational risk-sharing, Intergenerational value transfer, Generational accounting, Value-based ALM
Preface vi
1 Introduction 1
2 Dutch pension system 3
2.1 Pension pillars . . . 3
2.2 Pension deals . . . 3
2.3 Financial Assessment Framework . . . 4
3 Intergenerational risk-sharing and transparency 5 3.1 Solidarity in pension deals . . . 5
3.2 National pension dialogue . . . 6
3.2.1 Less support for accumulation of pension benefits where contri-butions are unequal to the economic cost . . . 6
3.2.2 A call for more transparency on risk-sharing. . . 6
3.3 Intergenerational risk-sharing . . . 6
3.3.1 Financial policy A: Benchmark policy . . . 7
3.3.2 Financial policy B: Zero-mean intergenerational risk-sharing. . . 7
3.3.3 Financial policy C: Transparent intergenerational risk-sharing . . 7
4 Literature review 9 4.1 Intergenerational risk-sharing in pension deals . . . 9
4.2 Value-based generational accounting . . . 9
5 Method 11 5.1 Pension fund characteristics . . . 11
5.1.1 Pension salary . . . 11 5.1.2 Pension benefits . . . 12 5.1.3 Payments . . . 12 5.1.4 Liabilities . . . 12 5.1.5 Assets . . . 13 5.1.6 Funding ratio . . . 13
5.2 Economic scenario generator. . . 13
5.3 Intergenerational value transfer . . . 16
5.4 Simulation. . . 17
5.4.1 Generating scenarios . . . 17
5.4.2 Initial pension salaries and benefits. . . 18
5.4.3 Calculating annuities. . . 18
5.4.4 Developing the pension fund . . . 18
5.4.5 Calculate results . . . 18
Transparent intergenerational risk-sharing — Vincent Tiesinga v
6 Financial policies 19
6.1 Financial policy A: Benchmark . . . 19
6.1.1 Minimum capital requirement (MVEV) . . . 19
6.1.2 Long term recovery plan . . . 20
6.1.3 Future-proof indexation . . . 20
6.1.4 Missed indexation . . . 21
6.1.5 Contributions . . . 21
6.2 Financial policy B: Zero-mean risk-sharing. . . 21
6.2.1 Adjustment B.I: No residue option . . . 22
6.2.2 Adjustment B.II: No net benefit option . . . 22
6.2.3 Adjustment B.III.a: Time diversification . . . 23
6.2.4 Adjustment B.III.b: Age diversification . . . 23
6.3 Financial policy C: Transparent risk-sharing . . . 24
6.3.1 Adjustment C.I: Zero net benefit . . . 24
6.3.2 Adjustment C.II: Transparent risk-sharing. . . 24
6.3.3 Adjustment C.III: Maximum residue . . . 24
7 Results 26 7.1 Financial policy A: Benchmark . . . 26
7.2 Adjustment B.I: No residue option . . . 28
7.3 Adjustment B.II: No net benefit option . . . 29
7.4 Adjustment B.III.a: Time diversification . . . 30
7.5 Adjustment B.III.b: Age diversification . . . 31
7.6 Adjustment C.I: Zero net benefit . . . 32
7.7 Adjustment C.II: Transparent risk-sharing . . . 32
7.8 Adjustment C.III: Maximum residue . . . 33
8 Conclusions and future research 35 8.1 Benchmark policy . . . 35
8.2 Zero-mean intergenerational risk-sharing . . . 35
8.3 Transparent intergenerational risk-sharing . . . 36
8.4 Future research . . . 36
First, I would like to thank my supervisors at the University of Amster-dam Jitze Hooijsma and Roger Laeven for their feedback and general guidance during the course of this thesis. Furthermore I would like to thank my parents for their unconditional love and support. Last but not least I would like to thank my girlfriend for always encouraging and believing in me.
Chapter 1
Introduction
The pension system of the Netherlands is considered as one of the best in the world (Mercer, 2015). In the past years it has become clear however, that social support for intergenerational risk-sharing is decreasing. Many pension funds have seen their funding ratios drop below hundred percent. The consequence is that pension benefits have not been compensated for price and/or wage inflation. In many cases pension benefits were also cut. In order to keep the pension system as one of the best in the world, it needs to adjust to the changing environment. On the first of Janaury 2015, changes were im-plemented in the regulation framework for Dutch pension funds. The changes consisted of improvements in governance, communication and financial sustainability.
Despite the recent changes, there is a broad consensus that the Dutch pension sys-tem needs further improvements. A national debate has been initiated by the Dutch government called ”De nationale pensioendialoog”, translated as the national pension dialogue. An important outcome of this dialogue is that there is a call for more trans-parency in the risks that are shared between generations. Another important outcome is that there is less support for contributions which are unequal to the economic cost of the liabilities for new pension benefits. The outcome is discussed in more detail in chapter three. As a contribution to the dialogue, the social economic council (Sociaal Economische Raad, SER) is also exploring different variants for adjustments in the pen-sion system.
In this thesis, several adjustments to the financial policy of Dutch pension funds are discussed and analyzed, to deal with the above mentioned problems of intransparent in-tergenerational risk-sharing. The factors contributing to inin-tergenerational risk-sharing are distinguished and more explicit and transparent alternatives are described. The objective is to design a financial policy with either:
• zero-mean intergenerational risk-sharing properties, or
• a more explicit and transparent form of intergenerational risk-sharing,
without compromising in indexation quality of the pension result. By zero-mean inter-generational risk-sharing it is meant that although risks are shared between generations, the average result of all possible scenarios is such that there is no net transfer of economic value, also called intergenerational value transfer. For a specific scenario the outcome can be either positive or negative, but in advance the expectation is such that all gen-erations share equal risks.
The effects of the adjustments on the funding ratio and pension results will be an-alyzed by a classical ALM-model. The classical ALM-model uses economic scenarios, produced by an economic scenario generator, to generate distributions of variables like
the funding ratio and pension result or indexation quality. Furthermore, the intergener-ational value transfers are calculated by a combination of generintergener-ational accounting and value-based ALM.
The outline of this thesis is as follows. In chapter two the background of the Dutch pension fund system is described. The outcome of the national pension dialogue is de-scribed in chapter three. A literature review of the existing theory of intergenerational risk-sharing and value-based generational accounting is described in chapter four. Chap-ter five describes the theory and financial model used in this thesis. The description of the financial policies, both the current pension system and the adjustments are de-scribed in chapter six. The results are discussed and analyzed in chapter seven. Finally, the conclusions and suggestions for further research are given in chapter eight.
Chapter 2
Dutch pension system
2.1
Pension pillars
The Dutch pension system consists of three pillars. The first pillar is the general old-age law (AOW, Algemene Ouderdomswet). The AOW provides a basic income for all inhabitants of the Netherlands, starting from the retirement age. The size of the AOW is dependent on the amount of years a person has been living in the Netherlands, from age 15 to 65. For every year of living in the Netherlands, 2 percent of the AOW is granted. The first pillar is based on a pay-as-you-go system. This means that no savings are made for the future payments. The payments for the current generation of retirees are financed by taxes. The size of the AOW is dependent on the minimum wage in the Netherlands.
The second pillar consists of pension benefits accrued by active service for an em-ployer as part of the labor contract. The second pillar pension is added to the first pillar AOW. Contrary to the first pillar, the pension liabilities are backed by assets which are accumulated during the working period. Contributions are paid from the salary of the employee and usually also partly by the employer. These contributions are managed by pension funds and are invested in assets to generate returns over the lifetime of the employee.
The third pillar is a voluntary pension, which employees can either arrange with in-surers or sometimes as optional products with the existing pension funds. Similar to the second pillar, the pension benefits are backed by assets. The contributions are paid by the employee.
The focus of this thesis is on the second pillar. The second pillar usually consists of different products apart from the retirement benefit, such as widow benefits or orphan benefits. Widow benefits are payments made to the widow, if the participant passed away leaving the widow behind. Other products which are available are disability pen-sions, in case the participant is unable to work after an accident.
2.2
Pension deals
The pension deals in the second pillar mainly consist of a Defined Benefit (DB) deal or a Defined Contribution (DC) deal. In a DB pension deal, pension benefits are accrued yearly based on a fixed percentage of the pension salary. In a DC pension deal the con-tribution rate is set first, after which pension benefits are accrued based on the economic value of new liabilities. A disadvantage of DB pension deals it that the illusion is given that pension benefits are implicitly guaranteed. Although pension benefits are not cut immediately when the pension fund is financially underfunded, eventually pension cuts
will have to be implemented if the situation of underfunding continues. The majority of pension deals in the second pillar are DB pension deals in the Netherlands. This will also be the benchmark policy in this thesis.
2.3
Financial Assessment Framework
The financial assessment framework for Dutch pension deals in the second pillar is called ”Financieel Toetsingskader” (FTK). Its sets mandatory rules for valuation and financial policy. The rules for indexation and pension cuts of the financial policy of DB pension deals under the FTK are described below.
• The policy funding ratio is calculated by discounting the liabilities with interest rates imposed by De Nederlandse Bank (DNB), which are assumed to be risk-free.
• Indexation takes place once a year, if the assets are higher than the level of the dis-counted value of the liabilities, including a residue of 10% of the nominal liabilities. If the residue is bigger, the allowed indexation will also increase.
• Full indexation (price or wage indexation) is given if the amount of capital available for indexation exceeds the capital needed for giving indexation of 2% for price inflation or 2,5% for wage inflation.
• Pension cuts take place if the pension fund cannot recover to a funding ratio equal to the discounted value of the liabilities, including a residue so that with 97,5 percent certainty, the funding ratio doesnt decrease below 100% within one year. This funding ratio is the required capital, or ”Vereist Eigen Vermogen” (VEV) in Dutch. The recovery time is allowed to be 10 years.
• Pension cuts will take place if the funding ratio is below 105% for 5 consecutive years. This level is the minimum required capital, or ”Minimaal Vereist Vermogen” (MVEV) in Dutch.
• Pension cuts are allowed to be spread out over 10 years.
• Pension benefits accumulation and the contributions are based on a uniform rate of the pension salary for all ages.
The financial benchmark policy as well as the adjustments which are being researched will be described mathematically in chapter six.
Chapter 3
Intergenerational risk-sharing
and transparency
3.1
Solidarity in pension deals
There are many forms of solidarity within the current pension deals. Kun´e (2006) dis-tinguishes different types as given below.
• Solidarity for individual risk. Some people live longer than others. For a pension fund this means that payments need to be given during a longer lifetime, which costs more. This risk is shared between all participants.
• Solidarity between younger and older active workers. Younger workers pay more contribution than the economic cost of the value of new liabilities, older workers pay less.
• Solidarity between active participants and non-active participants. If a pension fund is underfunded, indexation may not be given and contributions may increase. This results in solidarity between active workers who pay contributions and retirees who receive pension payments.
• Solidarity between different generations. Maintaining a residue results in solidarity between generations. The residue can be divided into parts for the different gen-erations. If new contributions or pension payments don’t contain a surplus for the available residue in the pension fund, value transfers occur between generations. • Solidarity between groups with different life expectancies. Certain groups of
par-ticipants are known to have a lower or higher life expectancy. Higher income or women in general have historically higher life expectancies, which results in more costs for the pension fund.
• Solidarity between active and disabled participants. Disabled participants still accrue new pension benefits every year. This is paid for by the active participants.
• Solidarity between participants with high and low career growth. Participants with lower career growth pay relatively less contribution in their older years. Due to the uniform contribution rate and pension accrual, this results in solidarity between different career paths. A higher growth in salary means that relatively more pension is accumulated in older years, where the contribution is less than the economic cost of the value of liabilities for new pension benefits.
The list above is not a complete list, but illustrates the many different forms of solidarity that exist in a pension fund. In this thesis, the main focus will be on the intergenerational solidarity resulting from maintaining a residue and from contributions which are unequal to the economic cost of the liabilities for new accrued pension benefits.
3.2
National pension dialogue
As mentioned in the introduction, there is a broad agreement that the system needs adjustments to a changing society. People are more insecure about the certainty of an adequate pension, since the funding ratios have been decreasing in recent years. In order to generate new ideas and discussion about the future of the Dutch pension system, there has been a national dialogue on pensions called ”De nationale pensioendialoog”. The outcome of the debate is summarized in a letter from the State Secretary of Social Affairs (Klijnsma,2015). In this letter, five trends are identified which call for a change in the Dutch pension system:
• a changing labor market;
• less support for accumulation of pension benefits where contributions are unequal to the economic cost;
• a call for more transparency on risk-sharing;
• the awareness that pension benefits cannot be guaranteed;
• increasing cultural diversity in the Netherlands, which calls for more personal choices.
This thesis aims to address the second and third bullet, by applying adjustments to the current financial policy in the FTK. These trends are described in more detail below.
3.2.1 Less support for accumulation of pension benefits where contri-butions are unequal to the economic cost
The contribution rate is the same for all generations in defined benefit (DB) pension schemes in the Netherlands. The price for accumulation of pension benefits on the other hands is the same for all generations. Every generation gets the same percentage of their pension salary. From an actuarial standpoint this is unfair. The support for this one-sided structural intergenerational value transfer from young to old is becoming less, because of the changing labor market as described in section 3.1.
3.2.2 A call for more transparency on risk-sharing
One of the major points of criticism of the current system is the amount of implicit intergenerational solidarity. When funding ratios are high and the economy is doing well this is not as much a concern. Since the financial crisis, this has changed. The interest rates have dropped and investment returns are lower, leading to lower funding ratios and a higher concern about the financial status of the pension fund. Also the housing prices have been decreasing and the economy has been growing less, which adds to the concerns. Because of this uncertainty the support for solidarity decreases. Older people have emotions of feeling robbed of their pension entitlements, while younger participants dont have the confidence there will be enough money left for them at retirement.
3.3
Intergenerational risk-sharing
As mentioned in section 3.1, this thesis will focus on the intergenerational risk-sharing resulting from maintaining a residue and from contributions which are unequal to the economic cost of liabilities for new accrued pension benefits. In the following sections the adjustments and expected effect on the intergenerational risk-sharing will be described.
Transparent intergenerational risk-sharing — Vincent Tiesinga 7
3.3.1 Financial policy A: Benchmark policy
The benchmark policy consists of a number rules for indexation, pension cuts and contributions which result in intergenerational risk-sharing. The ambition is to provide a retirement pension which is equal to 70% of the average pension salary if the participant has worked for 40 years. Furthermore, the ambition is to compensate the pension benefits for price inflation over the lifetime of the participant. The financial policy is not directly linked to this ambition. The uniform rate of contribution and uniform accumulation of pension benefits do not necessarily contribute to this ambition, but it does lead to intergenerational value transfer. Also, it is not transparent how the required capital levels MVEV and VEV contribute to the ambition. The MVEV sets a minimum level on the funding ratio, but it is allowed to temporarily not meet this level. The VEV is based on a 97.5% chance of not becoming underfunded within a period of one year. The intransparent nature of the link between the financial policy and the pension fund ambition leads to participants questioning which intergenerational risks are shared and for what reason.
3.3.2 Financial policy B: Zero-mean intergenerational risk-sharing
The goal of the adjustments in financial policy B is to define an alternative pension deal with zero-mean intergenerational risk-sharing. This means that intergenerational value transfer due to maintaining a residue and accumulation of pension benefits which are unequal to their economic cost, are not allowed. Other forms of the desired intergenera-tional risk-sharing features of the benchmark pension fund are maintained. This consists of spreading investment risks, and in reality also other forms of solidarity as mentioned in section 3.1, although these are not in the scope of this thesis. First, the residue is always kept at 100% by indexation or pension cuts, which eliminates this part of the intergenerational value transfer. Realized investment losses or profits are immediately shared between generations, because indexations or pension cuts bring the funding ratio back to 100% immediately. Second, the economic cost of the liabilities of accumulation of pension benefits will be set equal to the contribution for all generations. Maintain-ing the fundMaintain-ing ratio at 100% will lead to a more volatile pension payment, since the investment risks are shared immediately. As a third adjustment to counter this effect, without introducing structural one-sided intergenerational value transfer, a delay in the indexation is implemented. This does introduce a temporary residue, either positive or negative, but the mean result is that intergenerational value transfer does not occur. As an alternative to share the investment risk by a delay of 10 years, another adjustment is analyzed which exposes the younger generations to higher investment risk. The idea is that the younger generations have more time to let the investment returns revert to their mean level over time.
3.3.3 Financial policy C: Transparent intergenerational risk-sharing
As an alternative to financial policy B, where there is no structural one-sided inter-generational value transfer from young to old or vice versa, in financial policy C this is allowed. The goal here is to allow a residue, but in a more transparent and explicit manner compared to the benchmark policy. First, as in policy B, the contribution for new accrued pension benefits are set equal to the economic cost of new accrued liabil-ities. The only difference is that in the economic cost of the liabilities, a surplus for the residue of the pension fund is added. This way the intergenerational value transfer resulting from accrual of pension benefits is still zero, even though the pension fund allows a residue. Second, a pension fund residue is allowed, to protect pension results in bad scenarios. The goal is to do this in a transparent and explicit manner. This is done by a financial policy which allows indexation only when the amount of missed indexation is below a certain level. In this case the level is set at 50%. It will frequently
occur that only older generations will receive catch-up indexation. This is not a problem however, if it is agreed upon by all stakeholders of the pension fund that the ambition of the pension fund is to protect the level of indexation quality at this level. It is a more transparent form of risk sharing, even though in most of the cases the given indexation is unequal for different generations in a certain year. Also the financial policy in this case is directly in line with the pension fund ambition which helps understand why risks are being shared and increases social support. Third, to not allow the residue to get too high, which leads to intergenerational value transfers from old to young, the maximum funding ratio is set at 200% in this adjustment. This level can be agreed upon by all stakeholders of the pension fund, as a maximum residue to protect pension results for bad scenarios. Above this level indexation will be given to bring the funding ratio back to 200%.
Chapter 4
Literature review
4.1
Intergenerational risk-sharing in pension deals
Intergenerational risk-sharing in pension deals are an important part of the financial policies of pension funds. Cui et al. (2005) developed a framework wherein pension funds can be evaluated in economic value terms as well as in utility terms. They show that pension schemes that provide safer and smoother consumption streams are ranked higher in utility terms. Therefore intergenerational risk-sharing can be welfare enhanc-ing. Even initially underfunded pension funds may still provide higher utilities than individual pension deals. Furthermore, according to Gollier (2008) there is also a wel-fare gain because of the increased return on contributions.
After the pension crisis in the early 2000s, many employers wanted to reduce the risks they were exposed to by compensating for bad investment results. There was however opposition to transform the traditional DB pension deals to pure DC pension deals by members of the pension funds. Most importantly, the fixed benefits in the traditional pension deals were unsustainable as the solvency of the pension funds needed to im-prove. As a result, flexible indexation rules were introduced and the financial assessment framework ”Financieel Toetsingskader” was implemented. Also, the method of fair value accounting was introduced, resulting in varying interest rates based on market pricing.
After the financial crisis in 2008/2009, funding ratios declined significantly. This re-sulted in less social support for intransparent intergenerational risk-sharing. The Cen-traal Planbureau (CPB, 2012) stated that the intergenerational value transfer in the current pension deals is not explicitly stated. Also, the policy rules are not complete and the chosen asset mix influences the amount of value transfer between generations. This underscores the need for clear policy rules.
The Sociaal Economische Raad (SER,2015) explored the feasibility of a collective DC pension deal with intergenerational risk-sharing. This has some similarities with the adjustments proposed in this research. They used best estimates of expected returns to determine the pension payment and defining a residue to absorb volatility in investment returns. Also, in their proposal, pension benefits are in the form an personal account with a cash balance. In this research the pension benefits remain in the form of the expected pension payments as in a DB pension deal.
4.2
Value-based generational accounting
To determine the amount of intergenerational value transfer, the concept of value-based generational accounting is used. Value-based generational accounting to analyze the economic value of a Dutch pension fund is introduced by Ponds (2003): ”We combine
the value-based approach and generational accounting in a model to analyze a pension fund in terms of the economic value of the stakes of the different generations and the issue of who gains and who loses (transfers of value between generations) from alternative funding and indexation policies”. The method shows that a pension fund is a zero-sum game between the different stakeholders. Furthermore the article showed that:
• policy implications of value-based generational accounting contrast sharply with those of the traditional actuarial approach;
• the base contribution rate must reflect the economic cost price of new accrued liabilities in order to avoid transfers of value to workers at the expense of the other stakeholders;
• a pension fund policy can be said to be ex ante fair for future generations when the economic value of the funding residue remains unchanged.
Hoevenaars and Ponds (2008) stated that value-based generational accounting for pen-sion funds is a useful tool to aid decipen-sion-making in financial policy. They also separate the intergenerational value transfer into two parts, a net benefit option and a net residue option. The net benefit option results from the change in fair value of the liabilities, contributions and pension payments. The net residue option results from the change in fair value of the residue. A more detailed description is given in section 5.3.
Chapter 5
Method
5.1
Pension fund characteristics
The population of the pension fund in the initial stage is based on data from a large pension fund in the Netherlands. The variables of the population are the pension salary and the pension benefits. It is assumed that participants enter the fund at age 20 and retire at age 67. Pension benefits are in the form of deferred annuities at retirement age and deferred annuities for the partners of the participants. The relative pension salary and initial liabilities of the population are shown in figure 5.1.
age (x) 20 30 40 50 60 0% 1% 2% 3% 4% PSx,0PS0 age (x) 20 30 40 50 60 70 80 90 0% 1% 2% 3% 4% 5% Lx,0L0
Figure 5.1: Relative pension salary and liabilities at t=0.
The sum of the relative salaries and liabilities over all the age cohorts is equal to 1. Liabilities increase with age due to the past accumulation of benefits from active service. After the retirement age the liabilities decrease because of mortality.
5.1.1 Pension salary
As the model runs through time, younger generations will enter the fund and older generations will reach the retirement age. In reality the salary is also adjusted for mor-tality. It is assumed that these developments will not alter the relative distribution of the pension salary. This means that mortality and retirement are compensated by new inflow of participants. The only adjustment made is that the pension salaries for each age increase with the level of price inflation. Mathematically this is expressed by the
following formula:
PSx,t= PSx,t−1(1 + π). (5.1)
Where PSx,t is the total pension salary for all the participants in age cohort x at time
t and π is the yearly price inflation. There are two main advantages of this method. The first advantage is that assumptions on the distribution and size of inflow of pension salary from new participants are avoided. The second advantage is that although the model runs multiple years in time, the results represent the fixed population of the pension fund for that moment in time.
5.1.2 Pension benefits
Pension benefits at time t are modeled based on the benefits from the previous year t -1. First, the total benefits of a certain age cohort are decreased by mortality of a part of the participants, according to the mortality rate. Second, new benefits will be added for a year of active service by the participants through a uniform accrual rate. Mathematically this is expressed as:
PBretx,t,s= px,tPBretx−1,t−1,s+ acretPSx,t,
PBwidx,t,s= pxy,tPBwidx−1,t−1,s+ acwidPSx,t.
PBwidy,t,s = py,tPBwidy−1,t−1,s+ qx,tpy,tPBwidx−1,t−1,s,
(5.2)
Here PBretx,t,sis the total retirement benefit and PBwidx,t,sis the total widows benefit for age cohort x, at time t and scenario s. PBwidy,t,s is the widow benefit for the partner, after the participant died. The variables acret and acwid are the accrual rates for retirement and widow pension, used for accrual of new pension benefits due to one year of active service. The variables px,t and py,tare the one year survival probabilities of the participants and
the partner respectively. The variable pxy,t is the probability that the participant and
partner both survive one year and qx,t is the one year mortality probability.
5.1.3 Payments
The payments follow from the pension benefits. For the retirement payments, the pay-ments are equal to the benefits from the age cohort of 67 (the retirement age). The payments for the widows follow from the widow benefits for the partners of the partic-ipants. The mathematical expressions are given by:
Px,t,sret =
PBretx,t,s, if x > 67,
0, if x < 67, (5.3)
Py,t,swid = PBwidy,t,s, (5.4)
Px,t,s= Px,t,sret + Py,t,swid. (5.5)
5.1.4 Liabilities
To calculate the liabilities, expressions for the discounted value of one euro of pension benefit are determined. The present value of the (deferred) annuities for the retirement and widow benefits are calculated by equations 5.6 and 5.7 respectively:
Transparent intergenerational risk-sharing — Vincent Tiesinga 13 m|¨ax,t,s= 120−x X i=m (e−R(t,i)i·ipx ,t) s, (5.6) ¨ ax|y,t,s= 120−y X i=1 (e−R(t,i)i·iqx ,t ·ipy,t) s. (5.7)
Here m stands for the time before the first payment takes place and R(t, i) stands for the spot rate determined at time t for maturity i, given a scenario s. The maximum age of participants is 120. The variableiqx,t is the probability of death between time t and
t +i and the variableipx,t stands for the probability of survival between time t and t +i.
The total nominal liabilities for age cohort x at time t for scenario s then follow by:
Lx,t,s= PBretx,t,s·max(67−x,0)|¨ax,t,s+ PBwidx,t,s· ¨ax|y,t,s+ PBwidy,t,s·0|a¨y,t,s, (5.8)
The sum of the liabilities for a certain age cohort gives the total liabilities of the pension fund: Lt,s= 120 X x=20 Lx,t,s. (5.9) 5.1.5 Assets
The amount of assets of the pension fund is dependent on the premiums received, the payments paid and the return on investments. The contributions and payments are done at the beginning of the year. The assets and pension fund residue at time t +1 are calculated by:
At+1,s = (At,s+ Ct,s− Pt,s)(1 + rt,sinv), (5.10)
where Ct,s is the received contribution, Pt,s is the total payment and rinvt,s is the return
on investments at time t and for scenario s. The pension fund residue is determined by the difference between the assets and liabilities:
RSt,s= At,s− Lt,s, (5.11)
5.1.6 Funding ratio
The funding ratio is calculated by dividing the assets by the nominal liabilities:
FRt,s =
At,s
Lt,s
. (5.12)
5.2
Economic scenario generator
The risk-free interest rates are modeled by the Vasicek model (Vasicek, 1977). The Vasicek model describes the nominal short rate as a function of time. The term structure of the interest rate is fully determined by the short rate. The formula for the Vasicek model is given by:
drt= α(µr− rt)dt + σrdWtrP, (5.13)
where rtis the nominal short rate, α is the speed of reversion, µr is the long-term mean
of the short rate, σr is the short rate volatility and dWtrP is a Wiener process. The Vasicek model shows mean reversion behavior, which is appropriate for interest rates. This is because interest rates tend to stay within a certain range of some long-term
time 0 20 40 60 −2% 0% 2% 4% 6% 8% 10% 12% rt R(0,t) VaR2.5%/VaR97.5%
Figure 5.2: A stochastically generated scenario of the short rate with the Vasicek model.
mean, which in our case is µr. An example of a generated scenario is given in figure 5.2.
At time t =0, the spot rate curve R(0, t) is equal to:
R(0, t) = −logP (0, t)
t . (5.14)
where P (0, t) is the price of a zero-coupon bond with maturity t, which follows from the solution of the Vasicek model:
P (0, t) = e−A(0,t)r0+D(0,t), (5.15)
where A(0, t) is equal to:
A(0, t) = 1 − e−αt
α , (5.16)
and D(0, t) is equal to:
D(0, t) = (µr−(σ
r)2
2α2 )[A(0, t) − t] −
(σr)2A(0, t)2
4α . (5.17)
This is an advantage in the simulation process, because for every time step in the ALM-model, the spot rate needs to be determined to calculate the present value of the liabilities. Because these expected values are obtained through equation 5.14, only one scenario of interest rates needs to be generated for the calculation of one scenario in the model.
A disadvantage of the Vasicek model is that interest rates can also become negative. This is actually not so unrealistic anymore since recently interest have actually become negative. It is still desirable however that interest rates do not become too large in negative territory. Therefore the combination of a high variance and low initial value or long-term mean should be avoided in the model.
The stock prices are modeled by the Black-Scholes equation (Hull,2012):
dSt= (rt+ µS)Stdt + σSdWrSP, (5.18)
Transparent intergenerational risk-sharing — Vincent Tiesinga 15
WtSP a Wiener process. The combination of the stock price formula with the Vasicek interest rate creates a Black-Scholes stock price movement with a stochastic interest rate and a constant equity risk premium. The equity risk premium is greater than zero to generate a higher expected return, as compensation for the higher volatility of the stock price returns.
Correlation between the short rate and the stock price return is set to be equal to ρ by the following relationship:
dWtSP= ρdWtrP+p1 − ρ2dWSP
t , (5.19)
where WtrP and WtSP are independent.
The Vasicek deflator can be defined by:
dΛrt = −rtΛrtdt − Λrλr(rt)dWtrP (5.20)
with the solution:
Λrt = e− Rt 0rudu− Rt 0 1 2(λ r)2du−Rt 0λ rdWrP u , (5.21) = e−R(0,t)t−12(λ r)2t−λrr√t , (5.22)
so that the deflator for Black-Scholes with Vasicek becomes:
Λt= e−R(0,t)t− 1 2(λ r)2t−λrr√t−1 2(λ S)2t−λSS√t , (5.23)
where λr is equal to the market price of risk:
λr= r0− µ
r,i
σr,i . (5.24)
The variables µr,i and σr,i stand for the mean and volatility of a bond with maturity i, as determined by the market at t = 0. λS is equal to the equity risk premium divided by stock price volatility:
λS= µ
S
σS, (5.25)
so that the discounted value of initial assets of investment returns between t =0 and t is the equal to the initial assets:
V0[A0(1 + rt,sinv)] = E[ΛtA0(1 + rinvt,s)] = A0. (5.26)
α 0.08 speed of reversion
rs(0) 1% initial nominal short rate
µr 3.2% long term mean of the short rate σr 0.5% standard deviation of the short rate µS 3.55% equity risk premium
σS 20% standard deviation of the stock price return
ρ 0% correlation between short rate and stock price return λr 0 market price of risk for the Vasicek model
Table 5.1: Parameters for the short rate and stock return scenario generator.
The speed of reversion, initial short rate and long term mean of the short rate are chosen such that a good fit is found with the actual spot rate as of 31st of December, 2015 as published by De Nederlandse Bank (DNB,2015). The standard deviation of the short rate and of the stock price return, as well as the correlation are obtained from the Commissie Parameters (Langejan, 2014), which sets parameters for pension funds in the Netherlands. The equity risk premium is chosen such that the mean stock price return converges to the maximum allowed return of 6.75% (sum of long term mean of the short rate and equity risk premium), which is also prescribed by the Commissie Pa-rameters. The market price of risk for the Vasicek model is assumed to be zero, because in this thesis the main interest is comparing financial policies relative to each other. This means that the implicit assumption is made that the market doesn’t adjust the price for increased volatility in the instantaneous rate of return for of a bond with a certain higher maturity.
5.3
Intergenerational value transfer
To calculate the intergenerational value transfer it is first shown that the pension fund as a whole is of a zero-sum nature. The relationship between the pension fund assets at time t and t +1 is given in equation 5.10. If we substitute the assets as the sum of the liabilities and the residue we get the following expressions:
At+1,s= (At,s+ Ct,s− Pt)(1 + rt+1,sinv ),
Vt[At+1,s] = (At,s+ Ct,s− Pt),
Vt[Lt+1,s] + Vt[Rt+1,s] = Lt,s+ Rt,s+ Ct,s− Pt,s.
(5.27)
The equation can be rearranged to:
Vt[Lt+1,s] − Lt,s− Ct,s+ Pt,s+ Vt[Rt+1,s] − Rt,s= 0. (5.28)
The expression shows that the change in the value of the liabilities is equal to the sum of the change in value of the residue and the contributions and pension payments. Although this formula holds true for the pension fund as a whole, this is not necessarily true for the different age cohorts:
Transparent intergenerational risk-sharing — Vincent Tiesinga 17
The variable ∆GAx,t+1,s is called the generational account option for age cohort x, in
year t +1 for scenario s. The residue for a specific cohort Rx is based on the ratio of the
liabilities of that cohort to the total liabilities:
Rx,t,s=
Lx,t,s
Lt,s
Rt,s. (5.30)
The total generation value transfer for a certain age cohort x is the present value of all the generational account options between t =1 and 20 years:
∆GAx,s= 20
X
i=1
V0[∆GAx,i,s]. (5.31)
The generational account option can be separated in two parts, a net benefit option ∆N B and a residue option ∆R:
∆GAx,t+1,s= Vt[Lx,t+1,s] − Lx,t,s− Cc,t,s+ Pc,t,s | {z } N Bx,t+1,s + Vt[Rx,t+1,s] − Rx,t,s | {z } ∆Rx,t+1,s (5.32)
The net benefit option consists of the change in fair value of the liabilities including pension payments and contributions. The residue option consists of the change in fair value of the claim on the residue. Finally, by averaging over all the scenarios, the mean of the generational account option is obtained:
∆GAx = 1000 P s=1 ∆GAx,s 1000 (5.33)
5.4
Simulation
The model is run by the following steps.
1. Generate 1,000 economic scenarios.
2. Determine the initial pension salaries, pension benefits and mortality rates.
3. Calculate annuities for every time t, age x and scenario s.
4. Develop the pension fund through time.
(a) Determine pension salaries and benefits at time t. (b) Determine payments and contributions at time t.
(c) Determine liabilities, assets and residue at time t.
(d) Apply indexation or pension cuts based on the funding ratio.
5. Calculate results.
5.4.1 Generating scenarios
The scenarios are generated with a Monte Carlo simulation. The resulting set of 1,000 nominal short rate term structures and the return on investments are stored for all the scenarios. The return on investments is based on the percentage of assets in stocks and the percentage of assets in bonds.
5.4.2 Initial pension salaries and benefits
The second step consists of reading and storing the initial pension salaries and benefits. Real data from an existing pension fund is used. Also, the mortality rates are set, which are obtained from the Actuarieel Genootschap (AG2012-2062).
5.4.3 Calculating annuities
The third step consists of calculating annuities for the retirement benefits and widow benefits and payments. These annuities are calculated and stored in a separate step, so that the development of the pension funds in the next step consists of a single chronological loop over time. The annuities are calculated by using the nominal risk-free interest rate expectation for every moment in time according to equation 5.15. One time step later, the annuities are calculated again, with a new expectation because there is a new realization for the short rate from the scenario generator.
5.4.4 Developing the pension fund
The fifth step consists of the main loop over time. The model starts at time t =0 and runs through time t =70. The loop starts with the development of the salaries and benefits according to equations 5.1 and 5.2. After this the payments and contributions are determined and subtracted and added from the assets respectively. The liabilities for the new time t and the pension fund residue are calculated as well at this point in the loop. Based on the preliminary funding ratio, indexation or cuts of the benefits are applied. The exact description of the financial policies will be described in chapter six.
5.4.5 Calculate results
The final step in the simulation is the calculation of the results. This consists of the classical ALM-results, such as mean funding ratio or indexation rate, but also the in-tergenerational value transfer.
Chapter 6
Financial policies
In this chapter the financial policies for the different variants which are analyzed are given. Section 6.1 starts with the benchmark policy, which is the current policy under the FTK. Section 6.2 describes the different adjustments for the zero-mean intergenera-tional risk-sharing. Section 6.3 describes the financial policy where structural one-sided intergenerational value transfer is allowed, but in a more explicit and transparent man-ner compared to the benchmark policy.
6.1
Financial policy A: Benchmark
The benchmark pension fund consists of a defined benefit pension deal under the fi-nancial assessment framework (FTK) which came into effect on January 1st 2015 in the Netherlands. The pension fund has the following characteristics for all the financial policies:
• the ambition of the pension fund is to give yearly indexation of the benefits ac-cording to price indexation, which is assumed to be 2% per year;
• the initial funding ratio is 100%;
• the minimum required funding ratio (MVEV) is 105%;
• the required funding ratio (VEV) is 126%;
• initial benefits are fully compensated for price inflation in the past;
• the percentage of yearly accrual of benefits is 1.75% for the retirement benefits (acret) and 1.25% for the widow benefits (acwid);
• the investment mix consists of 50% stocks and 50% bonds, which is rebalanced every year.
The financial policy of the benchmark pension fund consists of indexation or cuts in the benefits, depending on the funding ratio. The different measures are described below.
6.1.1 Minimum capital requirement (MVEV)
If the pension fund has a funding ratio below 105% for five consecutive years, a pension cut which will bring the funding ratio back to 105% is mandatory. Although there is a possibility under the FTK to spread the cut over a maximum of 10 years, it is still an unconditional pension cut. This means that no matter how the funding ratio develops in the years after the first cut, the remaining cuts still have to be implemented. This also means that if the pension cut is spread out, all the future cuts will have to be incorporated in the liabilities immediately. Because of this unconditional property of
the pension cuts, it is assumed that the pension cut is not spread out. This results in a bigger cut at once, but we are mainly interested in cumulative pension indexation quality in the ALM-results, which remain visible if the cut is not spread out. Mathematically the pension cut is expressed as follows:
cutt,s=
1 −F Rt,s
105%, if FRt,s < 105% for all t in[t − 4, t]
0, otherwise (6.1)
6.1.2 Long term recovery plan
If the funding ratio drops below the level of the required capital VEV, which is 126% in our case, a long term recovery plan has to be submitted to the financial regulator. The pension fund needs to show that the funding ratio can reach a level of 126% within 10 years, under the assumption that a certain return on investments can be made. We use an equity risk premium of 3.55%, according to table 5.1, which is the maximum allowed expectation for the rate of return. This means the funding ratio can increase with e((1+0.0355/2)·10)− 1 = 19% in ten year. The equity risk premium is divided by 2 because only 50% of the asset mix is invested in stocks. The critical level of the funding ratio before cuts are taken place is therefore equal to 126%/119% = 106%. If the funding ratio drops below this level, a pension cut needs to be applied to bring it back to 106%. This cut can be spread out over a maximum of 10 years, conditionally. This means that 1/10th of the difference between the actual funding ratio and 106% has to be applied. Furthermore, the maximum recovery time of 10 years is allowed to remain constant so that for future years the same cut size is applied, irrespective of the number of consecutive years the funding ratio is below 126%. Mathematically the cut is expressed as follows: cutt,s= (1 −F Rt,s 106%)/10, if FRt,s < 106% 0, otherwise (6.2) 6.1.3 Future-proof indexation
If the funding ratio is above 110%, it is allowed to give indexation with the surplus capital above this threshold. The amount of indexation given is however capped by a maximum of the price or wage indexation level for that year, depending on whether the pension fund ambition is to compensate the benefits by price or wage inflation. Besides the maximum level of indexation, the amount of indexation allowed has to be such that the indexation can be given for all years to come. This is offset by the rule that the cashflows resulting from this continuous indexation may be discounted with the maximum allowed return on stocks, which is 6.75%. The cost of giving the future proof indexation for the retirement benefit is determined by:
Lret,indx,t,s = PBretx,t,s 120−x X i=max(67−x,0) ipx,t(e(indt,s)i− 1) e(6,75%)i , = PBretx,t,s 120−x X i=max(67−x,0) (ipx,te (indt,s)i e(6,75%)i − ipx,t e(6,75%)i), = PBretx,t,s 120−x X i=max(67−x,0) ( ipx,t e(6,75%−indt,si) − ipx,t e(6,75%)i), = Lx,t,s ¯ zs=6.75%−indt,s − Lx,t,s ¯ zs=6.75% (6.3)
The equations for the widow benefits follow a similar derivation. In the model the indexation level is found numerically by starting at 0% to the maximum indexation of
Transparent intergenerational risk-sharing — Vincent Tiesinga 21
2%. The indexation level is found if the surplus capital available for indexation matches the cost:
Lindt,s = At,s− 1.1Lt,s (6.4)
6.1.4 Missed indexation
If the funding ratio is above the level of required capital (VEV), it is also allowed to give extra indexation above the price indexation level. The missed indexation can be given as long as there is any cumulative missed indexation of previous years. This missed indexation can be either from previous pension cuts, or from previous years where the given indexation was below the price indexation. The maximum amount available for indexation is one fifth of the surplus capital above the required VEV. The indexation level is found if the available surplus capital matches the cost of indexation. Because the missed indexation is dependent on the history of the pension fund, older generations in general will have a higher level of missed cumulative indexation. The available surplus capital on the other hand is available for all the participants. In the model, this is solved by numerically determining the missed indexation level for every age first, which is the maximum indexation allowed: indmaxx,t,s. The level of missed indexation is determined by dividing the current pension benefits by the benefits if full indexation would have been given every year. Mathematically the expression for the given missed indexation in a year is given by:
indx,t,s= ( min((At,s−1.26Lt,s) 5Lt,s , ind max x,t,s), if FRt,s> 126% 0, otherwise (6.5) 6.1.5 Contributions
In the FTK it is allowed for the pension fund to set the contribution rate, but it has to be equal to the present value of the liabilities of the newly accrued benefits, discounted with an expected maximum return on investments and a surplus for the VEV-level. For the benchmark pension fund, contributions summed over all the ages are set to be equal to the present value of the liabilities of the newly accrued benefits, but discounted with the risk-free interest rate. This is much higher than the maximum allowed interest rate, so this shouldn’t be a problem from a regulatory perspective. First the contribution rate CRt is determined according to:
CRt,s= 120 P x=20 1.75% · PSx,t·max(67−x,0)|¨ax,t,s+ 1.25% · PSx,t· ¨ax|y,t,s) 120 P x=20 PSx,t (6.6)
The contribution for age x is then determined by:
Cx,t,s= CRt,s· PSx,t (6.7)
6.2
Financial policy B: Zero-mean risk-sharing
In this section, the adjustments for an alternative pension deal with zero-mean intergen-erational risk-sharing are described. The goal is to maintain the desired intergenintergen-erational risk-sharing features of the benchmark pension fund, which consists of spreading invest-ment risks, but not the structural one-sided solidarity from young to old or vice versa. The pension fund characteristics are initially the same as for the benchmark pension fund. The following subsections will describe the different adjustments in detail.
6.2.1 Adjustment B.I: No residue option
The benchmark pension fund has four rules for pension cuts or indexation as described in sections 6.1.1 through 6.1.4. In this adjustment, these rules are replaced by applying a single pension cut or indexation, which brings back the funding ratio to 100% after every year. This eliminates the intergenerational value transfer by the residue option, as given by equation 5.27. Also, realized losses or profits are immediately shared between generations, because indexations or pension cuts bring the funding ratio back to 100% immediately. In our model only investment risk is modeled, but in reality this results in a desirable risk sharing of all risks that may occur, for example mortality risk. The indexation, which can also become negative, to bring the funding ratio back to 100% is determined by:
indt,s= FRt,s− 1 (6.8)
6.2.2 Adjustment B.II: No net benefit option
In this adjustment, the contribution rate and the contributions according to equations 6.6 and 6.7 remain the same. The accumulation of benefits acret and acwidx however, are such that the economic cost of the liabilities are equal to the contribution for every cohort. The accumulation of pension benefits are then determined by the following equations:
PBretx,t,s PBwidx,t,s =
1.75%
1.25% (6.9)
CRt,s· PSx,t= ·PBretx,t,s·(67−x)|¨ax,t,s+ PBwid· ¨ax|y,t,s (6.10)
Solving the equations for PBretx,t,sand PBwidx,t,sby substituting 6.9 in 6.10 gives:
PBretx,t,s= CRt,s· PSx,t (67−x)|¨ax,t,s+1.25%1.75%a¨x|y,t,s (6.11) PBwidx,t,s= 1.75% CRt,s· PSx,t 1.25% (67−x)|a¨x,t,s+ ¨ax|y,t,s (6.12)
In the benchmark pension fund, the contribution rate follows from the accrual rates of the pension benefits. After this adjustment, only the ratio of widow pension accrual to retirement pension accrual is kept at the same level. The contribution rate set at the mean of the contribution rate of the benchmark pension fund for equal comparison. Therefore the contribution rate is set at a fixed rate of 20%:
CRt,s= 20% (6.13)
Substituting the fixed contribution rate in equations 6.11 and 6.12 results in:
PBretx,t,s= 20% · PSx,t (67−x)|¨ax,t,s+1.25%1.75%a¨x|y,t,s (6.14) PBwidx,t,s= 1.75% 20% · PSx,t 1.25% (67−x)|a¨x,t,s+ ¨ax|y,t,s (6.15)
Transparent intergenerational risk-sharing — Vincent Tiesinga 23
6.2.3 Adjustment B.III.a: Time diversification
Adjustment B.I will lead to a more volatile pension payment, since the investment risks are shared immediately, also for retirees. To counter this effect, without introducing structural one-sided intergenerational value transfer, a delay in the indexation is im-plemented. A delay of 10 years is chosen here, so that the indexation size becomes:
indt,s= (FRt,s− 1)/10 (6.16)
6.2.4 Adjustment B.III.b: Age diversification
As an alternative to share the investment risk by a delay of 10 years, it is also possible to give the younger generations a higher exposure to investment risk. The idea is that the younger generations have more time to let the investment returns revert to their mean level over time. To get an idea of the maximum loss that can occur for a 100% investment in stocks the VaR97.5% is estimated. This stands for the maximum Value at
Risk in 97.5% of the scenarios. Based on a stock return with a normal distribution with a mean of 6.75% and a standard deviation of 20%, the VaR97.5% gives a maximum loss
of:
V aR97.5%[Li] = −iµS+
√
iσSΦ−1(0.975), (6.17)
where Φ is the standard Gaussian cumulative distribution function and i is the invest-ment horizon. The Loss function is shown in figure 6.1.
time (t) 1 3 5 7 9 11 13 15 17 19 0% 20% 40% 60% 80% Lt
Figure 6.1: Maximum investment loss Lt with a 97.5% probability as a function of
investment horizon.
Based on figure 6.1, it would be necessary to decrease the investment losses for older generations by around a factor 5 to 6, to be 97.5% certain that the investment loss doesn’t exceed approximately 10%. In practice this would result in too much invest-ment risk for the younger generations. More realistically, increasing the investinvest-ment risk for younger generations by a factor of more than 2 seems to be unrealistic. This would already result in investment loss greater than 100% with a 2.5% probability. This won’t happen in our results, because the asset mix consists of 50% stocks and 50% bonds. The adjustment is therefore assumed to be that any indexation given is multiplied by a factor such that the total indexed liabilities are equal in value, but the relative indexa-tion percentage for younger generaindexa-tions compared to older generaindexa-tions decreases from a factor of 2 for age 20 to a factor of 1 for age 67. This gives:
indx,t,s ind67,t,s = 2 −x−2047 , if x 6 67 1, otherwise (6.18)
6.3
Financial policy C: Transparent risk-sharing
As an alternative to financial policy B, where there is no structural one-sided inter-generational value transfer from young to old or vice versa, in financial policy C this is allowed. The goal here is to allow a residue, but in a more transparent and explicit manner compared to the benchmark policy. The different adjustments will be explained in detail in the following sections.
6.3.1 Adjustment C.I: Zero net benefit
In this adjustment the contribution for new accrued pension benefits are set equal to the present value of new accrued liabilities, as in adjustment B.II. The only difference is that in the economic cost of the liabilities, a surplus for the residue of the pension fund is added. This way the benefit from the net residue option which is gained by new liabilities is offset by a loss in the net benefit option, because the same contribution is paid for less liabilities from new pension benefits. The contribution rate is fixed at 20%, as in adjustment B.II. The accrued pension benefits are equal to the equations 6.14 and 6.15, but corrected by the funding ratio:
PBretx,t,s= 20% · PSx,t· FR −1 t,s (67 −x )|¨ax,t,s+ 1.25%1.75%¨ax|y,t,s (6.19) PBwidx,t,s= 20% · PSx,t· FR −1 t,s 1.75% 1.25% (67−x)|a¨x,t,s+ ¨ax|y,t,s (6.20)
6.3.2 Adjustment C.II: Transparent risk-sharing
In this adjustment a residue is allowed, to protect pension results in bad scenarios. The goal is to do this in a transparent and explicit manner. This is done by a financial policy which allows indexation only when the amount of missed indexation is below a certain level. In this case the level is set at 50%. It will frequently occur that only older generations will receive catch-up indexation. This is not a problem however, if it is agreed upon by all stakeholders of the pension fund that the ambition of the pension fund is to protect the level of indexation quality at this level. It is a more transparent form of risk sharing, even though in most of the cases the given indexation is unequal for different generations in a certain year.
indx,t,s= min((At,s−Lt,s) Lt,s , 0.5PBret,f ullx,t,s PBret x,t,s ), if FRt,s> 100% 0, otherwise (6.21)
The term PBret,f ullx,t,s represents the retirement benefits if full indexation has been given for all years in the past.
6.3.3 Adjustment C.III: Maximum residue
In adjustment C.II only catch-up indexation is given if the cumulative indexation level falls below 50% of full indexation. To not allow the residue to get too high, which leads to intergenerational value transfers from old to young, the maximum funding ratio is set at 200% in this adjustment. This level can be agreed upon by all stakeholders of the pension fund, as a maximum residue to protect pension results for bad scenarios.
Transparent intergenerational risk-sharing — Vincent Tiesinga 25
Above this level indexation will be given to bring the funding ratio back to 200%. Mathematically this can be expressed as:
indx,t,s=
( (A
t,s−2Lt,s)
Lt,s , if FRt,s> 200%
Results
7.1
Financial policy A: Benchmark
The ALM-results for the benchmark policy are shown in table 7.1.
Percentile (2.5th) (50th) (97.5th)
FR 89% 123% 261%
CR 17% 20% 22%
PR67 52% 100% 100%
PR87 31% 100% 100%
Table 7.1: ALM-results for the benchmark policy.
The first row shows the nominal funding ratio percentiles at the end of the 70-year sim-ulation period. It shows that the funding ratio median increases from the initial funding ratio of 100% to 123%. In good scenarios, a large residue is accumulated, which is a re-sult of the maximum allowed cumulative indexation, which is equal the price indexation at 2% per year. In bad scenarios, the funding ratio is well protected by lower indexation and pension cuts. The second row shows the contribution rate for accrual of new pension benefits, calculated as the mean over the 70-year simulation period. The fluctuation in the contribution rate results from volatility in the interest rates. The last two rows show the pension result for a person with age 67 and 87 at the end of the 70-year simulation period. A percentage of 100% indicates that the maximum price indexation of 2% is given each year, with new accrual of pension benefits according to the financial policy of the benchmark pension fund. The results show that the cumulative indexation doesn’t exceed 100% as expected, because no indexation is given above this level. The median cumulative indexation is also 100% showing that in more than half of the scenarios a fully indexed pension is given. In the 2.5th percentile however, retirees of age 67 and 87 have experienced severe pension cuts. The pension result is the lowest for age 87, at 31% of the fully indexed pension result during the 70-year period. This shows that although a large residue is accumulated during good scenarios, the ambition of a fully indexed pension result does impose significant risks and can lead to very low pension results in bad scenarios.
The intergenerational value transfer for the benchmark policy is shown in figure 7.1. The horizontal axis shows the different age cohorts starting at age 0 at the beginning of the 20-year simulation period. In the left graph the generational account options are divided by the total liabilities at t=0, to get an idea of the size of the value transfer for each cohort compared to the total liabilities. The net benefit option shows the uniform
Transparent intergenerational risk-sharing — Vincent Tiesinga 27 age (x) 0 20 40 60 80 100 −3.0% −2.25% −1.5% −0.75% 0% 0.75% 1.5% 2.25% 3.0% DGx NBx DRx age (x) 0 20 40 60 80 100 −100% −75% −50% −25% 0% 25% 50% 75% 100% DGx NBx DRx
Figure 7.1: The left graph shows the generational account options for the benchmark policy, as a percentage of the total liabilities of the pension fund at t=0. The right graph shows the generational account options as a percentage of the age-specific liabilities at t=0, including the present value of contributions during the 20-year period.
contribution rate of the benchmark policy. Lower age cohorts pay more contribution compared to the economic cost of new liabilities, while the opposite is true for older cohorts. The residue option shows that cohorts who are increasing their relative size of liabilities compared to the total pension fund liabilities, get a bigger share of the residue. As their liabilities decline because of payments for retirement, their share of residue also decreases. This leads to intergenerational value transfers because the increasing residue for younger cohorts is not part of the contribution rate and the decreasing residue for older cohorts is not compensated in the pension payments either.
The right graph of figure 7.1 also shows the intergenerational value transfer, but in this case divided by the liabilities of each specific cohort. Also, the liabilities include pension accumulation during the 20-year simulation period. It shows that for cohorts of age 20 at the start of the simulation period, around 30% of the value of their liabil-ities at t =0 plus the contributions are being transferred to older generations. For ages around 40 to 50 the generational account option is the highest, the total value received by other generations results in an increase of around 20%. Furthermore, retirees at the start of the simulation pay around 10%-15% of the value of their liabilities to younger generations. This results from the benchmark policy which pays pensioners their pension benefit while leaving the surplus residue for the pension fund behind.
7.2
Adjustment B.I: No residue option
In this adjustment the intergenerational value transfer from maintaining a residue is set to zero, by not allowing any residue to exist. At the end of every year, the funding ratio is set to 100% of the expected liabilities by indexation or pension cuts, as described in section 6.2.1. The effects on the ALM-results are shown and compared with the benchmark policy in table 7.2.
Variant A A A B.I B.I B.I
Percentile (2.5th) (50th) (97.5th) (2.5th) (50th) (97.5th)
FR 89% 123% 261% 100% 100% 100%
CR 17% 20% 22% 17% 20% 22%
PR67 52% 100% 100% 48% 90% 201%
PR87 31% 100% 100% 28% 78% 271%
Table 7.2: ALM-results for financial policy B with adjustment B.I, compared with the benchmark policy.
The funding ratio stays at 100% as expected and the contribution rate is unchanged. The pension results show that the median indexation quality is lower, which is offset by very high indexation quality in the higher percentiles. This occurs because in the benchmark policy the residue is used as a protection for bad scenarios, whereas in adjustment B.I any positive residue is immediately used for indexation of pension benefits. This also has an effect on the worst case scenarios, as seen in the 2.5th percentile, although the effect is smaller here. Failure to protect the indexation quality during bad scenarios is therefore still a big disadvantage in this adjustment as well as in the benchmark policy. The results for the intergenerational value transfer are shown in figure 7.2.
age (x) 0 20 40 60 80 100 −3.0% −2.25% −1.5% −0.75% 0% 0.75% 1.5% 2.25% 3.0% DGx NBx DRx age (x) 0 20 40 60 80 100 −100% −75% −50% −25% 0% 25% 50% 75% 100% DGx NBx DRx
Figure 7.2: The left graph shows the generational account options for financial policy B with adjustment B.I, as a percentage of the total liabilities of the pension fund at t=0. The right graph shows the generational account options as a percentage of the age-specific liabilities at t=0, including the present value of contributions during the 20-year period.
Transparent intergenerational risk-sharing — Vincent Tiesinga 29
at any point in the simulation. There only exists a net benefit option, resulting from the contribution policy in the benchmark pension fund.
7.3
Adjustment B.II: No net benefit option
In this adjustment the intergenerational value transfer from the net benefit option is eliminated on top of adjustment B.I. This is done by setting the contribution equal to the economic cost of the liabilities of new pension benefits, as described in section 6.2.2. The effects on the ALM-results are shown in table 7.3.
Variant B.I B.I B.I B.II B.II B.II
Percentile (2.5th) (50th) (97.5th) (2.5th) (50th) (97.5th)
FR 100% 100% 100% 100% 100% 100%
CR 17% 20% 22% 20% 20% 20%
PRinfl67 48% 90% 201% 46% 101% 260%
PRinfl87 28% 78% 271% 27% 86% 304%
Table 7.3: ALM-results for financial policy B with adjustment B.I, compared with cu-mulative adjustments B.I-B.II.
The funding ratios are equal to adjustment B.I, since the funding ratio is kept constant at 100%. The contribution rate is fixed at 20%. The median level of pension results is increased compared to adjustment I. This is because during scenarios with lower interest rates, higher benefits are accumulated during the working period. The pension results in bad scenarios are slightly worse compared to adjustment B.I, but not significantly. The result for the intergenerational value transfer is shown in figure 7.3.
age (x) 0 20 40 60 80 100 −3.0% −2.25% −1.5% −0.75% 0% 0.75% 1.5% 2.25% 3.0% DGx NBx DRx age (x) 0 20 40 60 80 100 −100% −75% −50% −25% 0% 25% 50% 75% 100% DGx NBx DRx
Figure 7.3: The left graph shows the generational account options for financial policy B with cumulative adjustments B.I-B.II, as a percentage of the total liabilities of the pension fund at t=0. The right graph shows the generational account options as a per-centage of the age-specific liabilities at t=0, including the present value of contributions during the 20-year period.
The figure shows that on average there is no significant intergenerational value transfer among the participants of the pension fund as expected.
7.4
Adjustment B.III.a: Time diversification
In this adjustment the aim is to protect pension results from bad scenarios, but without structural one-sided intergenerational value transfers, as described in section 6.2.3. Table 7.4 shows the ALM-results when indexation or pension cuts are spread out over 10 years.
Variant B.II B.II B.II B.III.a B.III.a B.III.a
Percentile (2.5th) (50th) (97.5th) (2.5th) (50th) (97.5th)
FR 100% 100% 100% 69% 113% 172%
CR 20% 20% 20% 20% 20% 20%
PRinfl67 46% 101% 260% 55% 99% 256%
PRinfl87 27% 86% 304% 28% 93% 402%
Table 7.4: ALM-results for financial policy B with cumulative adjustments B.I-B.II, compared with cumulative adjustments B.I-B.II and B.III.a.
The results show that the funding ratio at the end of the simulation period becomes volatile. This is expected since it is not kept constant at 100%. The volatility is quite significant, leading to a funding ratio of 69% in the 2.5th percentile and 172% in the 97.5th percentile. The pension results for 67 year old retirees are more protected for bad scenarios, here time diversification seems to have a positive effect. For the 87 year old retirees the pension results in bad scenarios increases only by 1 percent to 28%. Scenarios with many cumulative years of bad investment results and low interest rates still lead to severe pension cuts in the long term.
The results for the intergenerational value transfer are similar to figure 7.3 and are therefore not shown separately.
