Intergenerational effects of different reduction methods within recovery plans in the Netherlands

(1)

Intergenerational Effects of different Reduction

Methods within Recovery Plans in the

Netherlands

drs. ir. B.J.J. van de Laar

Supervisors:

dr. T.J. Boonen (University of Amsterdam)

drs. ir. R.G.C. Sebregts AAG (Towers Watson)

Second reader:

prof. dr. ir. M.H. Vellekoop (University of Amsterdam)

Eindhoven, 2015

(2)

Abstract

The current applied reduction method under the nFTK is characterized by being a closed reduction system, indicating that only accrued pension rights up until the start of the recovery plan are to be reduced. The system is also a conditional reduction system, meaning that although it is allowed to use a spread period for the total necessary reduction only a fraction of this has to actually be applied by the pension fund neglecting the remaining reduction for future years. In this thesis we investigate both the change in the financial position of the pension fund and the expected reductions using Classic ALM and the intergenerational transfer values using a combination of generational accounting and Value-based ALM when implementing different reduction methods.

When changing to an unconditional reduction method or shorten the spread period within the reduction system we conclude that the average financial position of the pension fund in terms of decision funding ratio improves towards the end of the prognosis. This is mainly due to higher expected necessary reductions conditionally on having to apply a reduction. Furthermore the probability of having to apply a reduction in the future decreases.

In the current situation retirees benefit from the conditional character of the reduction method in com-bination with the allowed spread period. Middle-age cohorts have to contribute to overcome the current deficit. Changing to an unconditional reduction system or shorten the spread period leads to significant in-tergenerational value transfers from middle-age cohorts and retirees to future generations. In these systems the deficit is stalled less and is sooner caught up by these participants. The future generations benefit from the improved financial position of the pension fund at future time points.

(3)

Acknowledgments

I would like to thank my supervisor Tim Boonen from the University of Amsterdam for his contributions to this thesis and the useful and valuable meetings throughout the whole period. Furthermore I am grateful for the freedom and flexibility he gave me to be able to combine this research next to my job and for express-ing his trust in my ability.

Next I would like to thank Ron Sebregts from Towers Watson for him taking considerable amount of time throughout the whole thesis for sharing thoughts and guiding me in interpreting results. Furthermore I want to thank Dani¨elle van Dalen, who performed her internship at Towers Watson during my thesis, for our useful discussions about value-based ALM and for providing valuable information concerning risk neu-tral valuation. I would like to thank my other colleagues at the office of Eindhoven for taking over several projects so I had time to work in this thesis. In addition I would like to thank Harold van Heijst from Tow-ers Watson for all useful information and insights he gave me, especially at the beginning of the project. Because of him I was to make a very good start with my project. I would like to wish Harold and his family all the best and much strength in the upcoming period.

Last but no least I would like to thank my girlfriend for her unconditional love and support and for adapt-ing herself to my situation duradapt-ing the weekends and all public holidays.

Bart van de Laar Eindhoven, June 2015

(4)

Chapter

1

Introduction

In this chapter an introduction to the research problem, a brief description of the research problem and finally the thesis outline is stated.

1.1 Towards reducing pension entitlements

Pension funds in the Netherlands have had a very turbulent period in the last few years. As from the financial crisis in 2008 up until now pension funds have to deal with a lot of changes to both exogenous and endogenous factors.

Interest rates, which are used to calculate the liabilities have dropped tremendously in the last few years. Where in 2007 pension funds had to calculate the liabilities using a fixed interest rate of 4%, nowadays the present value of the liabilities have to be determined on a term structure of interest rates (in Dutch Rentetermijnstructuur (RTS)), corresponding to a fixed rate of around 1.8% up until 2.0%1 _{(the fixed rate}

corresponding to the RTS depends on the characteristics of the pension fund and could therefore be different for each pension fund).

Another market-related change is the decrease on investment returns. The return on equity has dropped in comparison to a decade ago. Due to regulations pension funds are obligated to ensure more certain pension entitlements and therefore are obligated to invest more in risk-less investments such as bonds. The downside of taken less risk is the lower but more certain investment returns.

One other important change is the increase of the life expectancy over the last years. Because people live longer than they used to, they receive benefit payments from pension entitlements for a longer period of time.

Due to all changes lots of pension funds had to develop a long-term recovery plan at the end of 2008, in which they had to show whether the financial position of the pension fund could increase to a certain specified sufficient level within 15 years. If this was not the case, they had to take additional measures by changing their policies, such that their financial position at least satisfied the sufficient level at the end of the recovery plan. For example, in time of recovery, the contribution payment by employers and/or employees could be increased or the indexation that was granted could be decreased. After these additional measures pension funds in theory were able to reach the acquired level.

Each year the recovery progress had to be evaluated. If it turned out during the evaluation of the re-covery plan at the end of the year that the pension fund was not be able to reach the prescribed sufficient level at the end of the recovery plan period, again additional measures had to be implemented. At the end

(7)

CHAPTER 1. INTRODUCTION

of 2013 (when the absolute deadline of the short-term recovery plan was reached) for the first time in the history of the Netherlands several pension funds, which were still in a funding deficit at that moment, had to reduce participants’ accrued pension rights. By reducing the accrued rights, the liabilities of the pension fund drop. The reduction was calculated in such a way that the pension fund was out of the funding deficit per December 31, 2013.

At December 31, 2014 still lots of pension funds did not have a sufficient level of the financial position. Furthermore the life expectancy increases and the interest rates even dropped further in 2014. In the past it was unthinkable for pension funds having to reduce the participants’ accrued pension rights. Nowadays it is part of reality and no longer an extreme exception.

1.2 Research description

As per January 1, 2015 a new financial assessment framework (in Dutch ‘Financieel Toetsingskader’) has become effective within the Netherlands. The current applied reduction method under the nFTK is charac-terized by being a closed reduction system, indicating that only accrued pension rights up until the start of the recovery plan are to be reduced, and a conditional reduction system, meaning that although it is allowed to use a spread period for the total necessary reduction only a fraction of this has to actually be applied by the pension fund neglecting the remaining reduction for future years.

In this thesis both the changes in the financial position of the pension fund and the potential intergenera-tional value transfers that occur when changing the applied reduction method are investigated. Interesting characteristics of the reduction method are therefore; an open versus a closed system (only accrued pension rights up until moment of recovery plan are to be reduced), the number of years a reduction can be spread, a conditional versus an unconditional reduction system (where in the latter the announced reduction in future times always takes place independent of the financial position of the pension fund) and in what way can you choose the reduction path over the number of years.

The financial position is investigated using Classic ALM resulting in probability distributions for vari-ables of interest like the decision funding ratio.

To evaluate the intergenerational value transfers a combination of generational accounting and Value-based ALM is used. It is investigated if and to what extent intergenerational value transfers occur when adjusting the reduction method.

The goal of the thesis is not to judge which reduction method should be applied, but to get insight in the intergenerational value transfers of applying different reduction methods and to be able to quantify those transfers.

1.3 Thesis outline

In this chapter a short introduction to the research problem is stated. In Chapter 2 relevant aspects of the Financial assessment framework are discussed. In Chapter 3 literature concerning generational accounting, (Value-based) ALM, market consistent valuation and finally what is written about reduction (methods) are discussed. In Chapter 4 the used model, the characteristics of the benchmark pension fund and its0_policies

used in this thesis and the reduction methods of interest are described. In Chapter 5 both the Classic ALM-and Value-based ALM-results are discussed. Chapter 6 elaborates on the sensitivity on the value transfers of particular initial pension fund characteristics. Finally in Chapter 7 conclusions and recommendations for further research are given.

(8)

Chapter

2

Financial assessment framework

In this chapter we will briefly discuss the different aspects within the Financial assessment framework (in Dutch ‘Financieel Toetsingskader’) applicable up until December 31, 2014 (FTK) and the (new) Financial assessment framework applicable as per January 1, 2015 (nFTK) that are important for the purpose of this thesis.

2.1 FTK

The FTK has been established on January 1, 2007. It mainly focuses on protection of nominal accrued pen-sion rights. In the sequel of this section we discuss the most important aspects regarding the purpose of this thesis.

Valuation of liabilities and assetsThe valuation of both the assets and liabilities is on a fair-value basis. Assets are therefore valued using the daily market prices. The liabilities are discounted with the current nominal term structure of interest rates (in Dutch ‘Rentetermijnstructuur’ (RTS)). The RTS is published by DNB on a monthly basis. To make the RTS less volatile, as per September 2012 the ultimate forward rate (UFR) has been introduced. Furthermore a three-month averaging of the interest rates has been applied for the first 20 maturities.

Financial position of the pension fundThe measure for the financial position of the pension fund is the actual funding ratio, which is the ratio between the assets and the liabilities. Under the FTK two capital requirements are defined.

A pension fund is said to be in a funding deficit if the actual funding ratio is under the Minimal Required Solvency Ratio (MRSR). This minimal requirement depends on having an investment risk and the risk cap-ital resulting out of the pension plan, e.g. partners- and disability pension. For pension funds within the Netherlands this leads to a MRSR of approximately 104%.

A pension fund is said to be in a reserve deficit if the actual funding ratio is between the MRSR and the Required Solvency Ratio (RSR). The level of the RSR is such that the probability that a pension fund can meet its nominal liabilities one year from now starting from that level is at least 97.5%. It depends on the amount of risk a pension fund takes and therefore on the investment mix of the pension fund. The RSR is on average approximately 21.7% (varying between 10% and 30%).

IndexationIn order to compensate for inflation pension funds have the ambition to increase the accrued pension rights with the same rate. Most pension funds use an indexation ladder, which is depending on actual funding ratio of the pension fund. The lower bound in this indexation ladder is under the FTK set

(9)

CHAPTER 2. FINANCIAL ASSESSMENT FRAMEWORK

equal to the MRSR of the pension fund. If the funding ratio is lower than the lower bound, no indexation is granted.

The upper bound of the indexation ladder, being the boundary at which pension funds grant full index-ation if the funding ratio is above this value, is set equal to the RSR of the pension fund. If the funding ratio is between the lower and upper bound, the indexation is granted proportional.

Cost-covering contributionThe contribution payments in a year should cover the costs for that particular year of the newly accrued rights. A pension fund is allowed to use the RTS (as in the calculation of the liabilities), either a 10-years moving average of the RTS or of the expected return on the assets (resulting in a so-called smoothed cost-covering contribution). Furthermore it is required that the premium contributes to recovery.

Recovery PlanIf a pension fund is in a reserve deficit, it has to develop a long-term recovery plan, possibly by implementing changes to the current contribution policy, indexation policy and other instruments during recovery and assuming prescribed parameters such as interest rates and investment returns. In the long-term recovery plan the pension fund has to show that it is no longer in a reserve deficit over 15 years from now. If a pension fund is in a funding deficit, it has to show in addition how it is no longer in a funding deficit within 3 years. Every year the recovery plan has to be evaluated by the pension fund to look whether they are still on track of recovery or not. If the pension fund is no longer on track, and it has already used all methods described in the recovery plan, the final solution is to reduce the pension rights.

Under the FTK it was possible to stall an actual reduction up until the end of the short-term recovery plan (deadline). This could therefore lead to sudden and high reductions at the end of the short-term recovery plan. It was not allowed to spread the reduction2_.

2.2 nFTK

As per January 1, 2015 the nFTK has become effective. It again focuses on protection of nominal accrued pension rights. In general you can state that due to the nFTK the pension contract will become a more complete contract. Furthermore it should result in a more stable financial position of the pension fund, less dependent on the market conditions and other exogenous factors. In the sequel of this section we discuss the different aspects (as in the previous section). The information in this section is mainly based on chapter 4 in Memorie van toelichting [16].

Valuation of liabilities and assetsUnder the nFTK the valuation of both the assets and liabilities is again based on a fair-value basis. The published RTS by DNB is still using the UFR-methodology. However, the three-month averaging of the interest rates is abolished.

Financial position of the pension fundWithin the nFTK the new measure to evaluate the financial posi-tion of the pension fund is the average of the actual funding ratio, the so-called decision funding ratio (or policy funding ratio). It is averaged over the preceding eleven months and the actual funding ratio at the calculation date.

Under the nFTK a pension fund is said to be in a reserve deficit if the decision funding ratio is below the RSR. The level of the RSR is increased with approximately 5% in comparison to the RSR under the FTK. For an average fund the RSR under nFTK equals approximately 26.6% (see page 29 in Memorie van toelichting [16]).

2_{In case the applied reduction percentage was higher than 7% at the end of December 31, 2013, then it was allowed, only once, to}

maximize the reduction as per 1 April 2014 at 7%. The remainder than should be effectuated as per 1 April 2015 only applied on the accrued right up until December 31, 2013.

(10)

2.2. NFTK

IndexationIn contrast to the indexation ladder in the FTK, the lower bound for granting indexation is fixed and the same for all pension funds and chosen equal to 110%.

The upper bound of the indexation ladder is determined, such that the pension fund is able to grant the indexation ambition for all upcoming years in the future (so not only at this moment). As a rule-of-thumb this comes down to approximately an indexation of 1% for every 10% above the lower bound of 110% (see page 21 in Memorie van toelichting [16]). Because of fiscal limitations in the Witteveenkader the full indexation is maximized on the wage or price inflation.

Catch-up indexation is granting indexation that has not been granted in the past due to the insufficient financial position of the pension fund at that time. The lower bound for granting catch-up indexation is equal to the maximum of the upper bound of the indexation ladder and the RSR. In case the decision fund-ing ratio is above this maximum, the pension fund is allowed to use a total of 1

10th of the surplus in own

funds (calculated using the difference in the decision funding ratio and the maximum) for granting catch-up indexation.

ContributionAs under the FTK it is still possible to smooth the cost-covering contribution by using a 10-years moving average of the RTS or an expected return on assets. An additional requirement is to take into account a decrease of the discount rate for the costs of indexation and an increase of the discount rate for the risk in the investment mix, resulting in general in a lower discount rate. The requirement that the con-tribution should contribute to recovery is dismissed.

Recovery PlanIf the decision funding ratio of the pension fund is below the RSR, it has to develop a recov-ery plan. The nFTK prescribes a duration of the recovrecov-ery plan on the long run of 10 years3_{. The pension}

fund has to show that the decision funding ratio at the end of the recovery plan is above the RSR. If this is not the case the pension fund has to take additional measures. Pension funds, for instance, can temporarily increase the participant0_{s contribution to obtain higher assets and therefore a better financial position in the}

upcoming years or decrease the indexation potential within the recovery plan. When it turns out that at the end of the pension funds0 recovery plan the decision funding ratio is still lower than the RSR, the final remaining measure is reducing the participants’ accrued pension rights.

The applicable reduction method under the nFTK can be characterized by the combination of:

• a closed reduction system, meaning that only accrued pension rights at the beginning of the recovery plan are to be affected in the calculation of the total necessary reduction;

• a conditional reduction system, meaning that although you are allowed to spread the total necessary reduction over a spread period, only the first reduction is actually applied. The applied spread period under the nFTK is 10 years.

In case the pension fund’s decision funding ratio has been below the MRSR for five consecutive years, additional reductions have to be announced by the pension fund. These additional reductions should also be applied within a closed reduction system. However, in case the pension fund chooses to spread these additional reductions, they are unconditional and therefore are effected in the future for sure. Within this thesis these additional reductions are not under consideration.

3_{DNB has pronounced that it is allowed (not obligatory) to use a recovery plan period of 12 years for 2015 and a recovery plan time}

(11)

Chapter

3

Literature

In this chapter the relevant literature for this thesis is presented. First generational accounting as a way to investigate intergenerational transfer values is described. Then, because of the dependence of for instance the indexation policy at the financial position of the fund, Asset-Liability-Management (ALM) is discussed. Value-based ALM turns out to be an excellent way to evaluate transfer values in both direction and mag-nitude. Then market consistent valuation is elaborated, which is applied in Value-based ALM and which makes it possible to calculate intergenerational value transfers.

The approach of using Value-based ALM in the context of Dutch pension plan redesign to gain insight in generational value transfers has been conducted in several other studies (the list is non-exhaustive):

• Kortleve et al. (2006) [17] have applied it to investigate pension deals ranging from pure defined benefit to pure defined contribution and to asset allocations of 100% in equities versus 100% in bonds; • Hoevenaars et al. (2008) [12] have applied it to the plan redesign of collective pension schemes and

changes in funding policy and risk sharing rules;

• Heijst (2014) [10] has applied it to gain insight in the generational value transfer of changing from the (old) financial assessment framework and the in 2013 proposed new financial assessment framework in the Netherlands. He mainly focused on the generational value transfers of the ‘Adjustment mech-anism for Financial Shocks’ (AFS) and the ‘Longevity Adjustment Mechmech-anism’ (LAM). Both mecha-nisms did not make it to the final elaboration of the new financial assessment framework;

• Chin (2014) [7] has applied it to gain insight in the generational value transfer of changing from the (old) financial assessment framework to the new (and accepted) financial assessment framework in the Netherlands that is effective as per January 1, 2015.

(12)

3.1. GENERATIONAL ACCOUNTING

3.1 Generational accounting

Generational accounting is defined as the difference between benefits to be received and contributions to be paid by a specific age cohort. It has first been developed within public finance by Auerbach et al. (1991) [3]. It was used to gain insight in what current and future generations are projected to pay to the govern-ment now and in the future, to be able to say something about the fiscal burden that current generations are placing at future generations. The government’s inter-temporal budget constraint states that the current net wealth plus the present value of the all net receipts from all current and future generations must be sufficient to pay for the present value of the government’s current and future consumption. In short this means that from the government0s point of view it is required that either current or future generations pay the government’s bills (Auerbach et al. (1998) [4]). The instrument for the government, to make sure the constraint is met, is changing the height of the tax. Generational accounting reveals the zero-sum feature of government finance, i.e. what some generations receive as an increase in net lifetime income must be paid for by other generations, who will experience a decrease in net lifetime income.

Ponds (2003) [23] points out the similarities between applying generational accounting in public finance and applying it to the financial side within pension funds. Pension funds also face an inter-temporal budget constraint; the promised benefits to participants have to be paid out of the sum of the current and future contributions and the returns over the paid contributions. Secondly, pension funds have also instruments, i.e. their contribution and indexation policies, to make sure the constraint is satisfied. Because of these sim-ilarities the zero-sum feature also applies for pension funds, implying that changes in a funding strategy or in risk allocation rules will lead to intergenerational redistribution.

Hoevenaars and Ponds (2008) [12] are the first to apply the methodology of generational accounting to real existing pension funds with intergenerational risk sharing. This is done by rewriting the balance sheet of a pension fund, using the theory about generational accounting, in terms of embedded generational options. In this way they are able to explore intergenerational value transfers that result from policy changes within the pension fund.

Both the contribution- and the indexation policy of a pension fund can depend on the financial position of the pension fund. The financial position is defined as the ratio of the assets and the liabilities. Therefore in the next section ALM is discussed.

3.2 Asset Liability Management (ALM)

The activa of the balance-sheet of a pension fund consist of the market value of the asset portfolio. The passiva of the balance-sheet of a pension fund consist of the liabilities and the own funds, where the latter is equal to the assets minus the liabilities. The own funds can therefore also be negative.

ALM can be used to simulate both the liabilities as the assets in a stochastic framework at the same time. It is used to gain insight in the correlation between the liabilities and the assets and the sensitivities of the financial position of the pension fund. Furthermore new strategies concerning for example contribution, indexation or investment can be examined using ALM.

(13)

CHAPTER 3. LITERATURE

3.2.1 Classic ALM

In Classic ALM a lot of macro-economical scenarios, e.g. combination of different wage increases, price inflations, investment returns and interest rates, are examined and for each macro-economical scenario the development of the financial position is produced. The outcome can be used to provide insight in the distribution of future possible results, such as the probability of over- or underfunding and the probability of granting a certain level of indexation or no indexation at all. It gives policy makers the opportunity to understand the outcomes of their policies better and to get some idea about the sustainability of the policies in the long run.

3.2.2 Value-based ALM

Value-based ALM adds additional information to Classic ALM by showing the present value (or economic value) of all decisions about the funding strategy, indexation policy and investment strategy (Kortleve et al (2006) [17]). Value-based ALM essentially uses the same output out of Classic ALM scenario analysis, however the future outcomes, such as the cash flow of contributions and indexations, are discounted back to the present with an appropriate risk adjusted discount rate.

The shift to Value-based ALM leads to two new insights (Kortleve et al. (2006) [17]). First of all it gives insight in the value the market at this moment attaches to future cash flows. Because in general people are risk averse, people will value the case of having no short-cuts but also no indexation (where the expectation of the total received indexation is equal to 0) more than the case of having a probability of short-cuts but also having a higher probability of getting indexation (where the expectation of the total received indexation is higher than 0).

Secondly, combining the above with the methodology of generational accounting, Value-based ALM gives as a useful methodology to investigate the different results for different age cohorts.

In the next section we discuss the method to calculate the present value of a cash flow using market consis-tent valuation.

3.3 Market consistent valuation

A pension plan is a financial contract whose pay-off depends on market contingencies (Lekniute et al. (2014) [20]). For instance; in the Dutch market it is very often the case that the indexation policy of a pension fund depends on the funding ratio. The return on the pension scheme depends on the plan specifics as well as the market conditions. Because the cash flows of contributions and indexation are linked to cash flows of financial titles like equities and bonds (Kortleve et al. 2006 [17]), a pension contract can be seen as a combination of contingent claims. These claims can be valued using market consistent valuation.

Hibbert et al. (2006) [11] describe the importance of market consistent valuation by looking at the dif-ference between the market value of the pension asset, i.e. the cost of replicating the benefits today using government bonds, and the actuarial funding value of the benefits today (how much should a pension fund have to be able to pay the benefits). When a person is offered an amount of money in exchange for given up rights, it seems fair to give the market value rather than the actuarial funding value, because having rights gives protection to risks where an amount of money gives no protection. Because in general people are risk-averse, it seems fair that bearing risks should be compensated.

In the literature mainly two methods are described that can be used for market consistent valuation; risk-neutral valuation and State price deflator approach. The former is used in this thesis and is therefore elabo-rated further. For more information about the latter we refer to Hibbert et al. (2006) [11].

(14)

3.4. REDUCTION OF ACCRUED PENSION RIGHTS

Applying the technique of risk-neutral valuation is introduced by Black and Scholes (1973) [5]. For this purpose they show that if you can replicate the pay-off for each possible outcome of a contingent claim using a combination of a risky and a risk-free asset, under the assumption of no-arbitrage opportunities, the price of the contingent claim should equal the price of the replicating portfolio (Hibbert et al. (2006) [11]). Furthermore in the risk-neutral world all individuals are indifferent to risk. Investors require no compensa-tion for risk and the expected return on all assets is the risk-free interest rate (Hull (2009) [13]).

The risk neutral pricing formula for market consistent valuation is as follows: Ct Nt = E Q t[ CT NT ], (3.1)

where Ctis the value of the derivative at time t, Ntthe value of the num´eraire, i.e. an asset with a positive

price which price is used for expressing the relative price of other assets, and EQ

t indicating taking

expecta-tion under the probability measure Q. This means that the uncertain future value of the derivative at time T can be discounted with the future value of the num´eraire asset.

In this thesis we use a risk neutral set provided by Towers Watson (as was used by Heijst (2014) [10]) in which the value of the num´eraire asset is given. The value is based on the money market account (Tow-ers Watson (2011) [24]). In subsection 4.1.2 we briefly discuss the construction of the variables within the economic set.

3.4 Reduction of accrued pension rights

In this section a brief overview of relevant subjects concerning the reduction of accrued pension rights is discussed. This section is mainly based on the report of KPS (2011) [18]. First the history of having to apply reduction within the ‘Pensioenwet’ (PW) and the ‘Pensioen- en Spaarfondsenwet’ (PSW) is discussed. Then some permitted and prohibited (theoretical) reducing methods according to the Dutch regulations are described.

3.4.1 PW and PSW

The PW has become effective in the Netherlands as per January 1, 2007. It replaces the PSW, which has been used as from 1952. It is not within the scope of this thesis to fully describe the differences and similarities between the two laws. Within this thesis we only focus on what is written about reduction of rights. In article 134 of the PW [2] it is stated that a pension fund is allowed to reduce accrued rights if and only if the funding ratio is below the Minimal Required Solvency Ratio (MRSR) and the pension fund is not able to recover above the MRSR within a small period without unfairly harming a particular group of stake-holders, i.e. active participants, deferred participants, pensioners and employers, more in comparison to another group of stakeholders. We denote the latter by having a balanced stakeholders0_{interest. Examples}

of unbalanced stakeholders0_{interest methods: for a long period of time granting no indexation (more unfair}

for pensioners), asking far too high premiums (more unfair for active participants) or asking for too high ad-ditional employer contribution (more unfair for employers). Within the PSW it was also possible to reduce rights, although there were no defined regulations which described the circumstances in which a pension fund was allowed to reduce the rights. In the PW the employer is also explicitly mentioned as a stakeholder, where in the PSW this was not the case.

Another goal of the introduction of the PW was to improve the communication and transparency towards participants around retirement. In article 35 paragraph 2 of the PW it is stated that the reduction method should be included in the pension plan (contract between employee and the pension fund). Although,

(15)

CHAPTER 3. LITERATURE

based on article 35 paragraph 1 of the PW, the pension plan should be consistent with the contract between the employer and the pension fund (in Dutch ‘uitvoeringsovereenkomst’) and the contract between the employee and the employer (in Dutch ‘pensioenovereenkomst’), in the legal regulations nothing explicit has been included for the latter two contracts. In the Actuarial and Technical Business Memorandum (in Dutch ‘Actuarie¨ele en bedrijfstechnische nota’ (ABTN)) a description should be included about the way a pension fund deals with article 134 of the PW, although the amount of detail is not clear.

3.4.2 Reducing methods

A pension fund should always ensure that it acts in accordance to the balanced stakeholders0interest. A par-ticular group of stakeholders should not be harmed more in an unfair way than other groups. The question arises whether it is allowed to differentiate by different groups when applying reduction methods. When during a recovery plan all the additional measures have only a negative impact for active participants (e.g. only the employee contribution is increased), than it sounds fair in some sense to not reduce or to less reduce the rights of active participants. Out of the parliament debate [15] it can be concluded that it is possible to differentiate between different stakeholders (responsibility lies by Social Partners in the contract between the employer and the pension fund). The final judgment whether the reduction method is fair for all stake-holders lies by the Dutch National Bank (DNB).

Also the ‘Wet gelijke behandeling op grond van leeftijd bij de arbeid’ (WGBL [25]) (law for equal treatment based on age in case of labor) does not prohibit pension funds to take measures which effect only one of the stakeholders in case of reducing methods. If, for example, only the pension rights of pensioners are reduced, only older generations are harmed. The WGBL does not prohibit this measure if and only if the pension rights of all pensioners are reduced. The WGBL therefore does not cover for the difference between active participants and pensioners. It is however not allowed to differentiate within a group of stakeholders based on age.

As mentioned in the report of KPS 2011 [18] the following differentiations within applying or granting re-duction are possible: amongst different pension plans, amongst different stakeholders within a specific pen-sion plan, amongst different type of penpen-sion rights (old-age penpen-sion, partners0_{pension, and so on) within a}

specific pension plan and the period on which you spread the reduction.

It is however prohibited to differentiate based on: participants0 income within a specific pension plan, participants0gender in all situations (based on ‘Wet gelijke behandeling van mannen en vrouwen’ (WGBMV) [26]) and amongst employees associated with the same ‘Bedrijfstakpensioenfonds’ (BPF).

(16)

Chapter

4

Model description

In this chapter the model that is used in this thesis to gain insight in the intergenerational value transfers is described. In Section 4.2 the pension fund characteristics and the policy characteristics of the benchmark pension fund are discussed. In Section 4.3 the reduction methods of interest are described. Finally some key characteristics of the implemented model in R are stated in Section 4.4.

4.1 Value-based generational accounting

Hoevenaars and Ponds (2008) [12] are the first to apply the methodology of Value-based generational ac-counting to real existing pension funds with intergenerational risk sharing. A generational account looks at a certain age cohort and records every contribution paid and every benefit received for this age cohort. These contributions and benefits are to be paid/received in the future. We calculate the economic value being the market consistent present value of all those future cash flows.

In subsection 4.1.1 we give a description of the approach of Hoevenaars and Ponds (2008) for generational accounting options. In subsection 4.1.2 we elaborate on the risk-neutral scenario set that has been provided by Towers Watson.

4.1.1 Generational account options

The balance sheet of a pension fund consists of on the one hand the value of the assets, denoted by At, and

on the other hand the pension fund liabilities, denoted by Lt, plus the own funds (also called the residue),

Rt. It should therefore be true that:

At= Lt+ Rt. (4.1)

We define the function Vt(x)as the economic value of x at time t. In this thesis we use rtto denote the return

on the assets during period (t, t + 1), Btto refer to the benefits paid in the period (t, t + 1) and Ctto refer

to the contributions paid during period (t, t + 1). The economic value at time t of the balance sheet at time (t + 1)is equal to:

Vt[At(1 + rt)] + Vt[Ct] − Vt[Bt] = Vt[Lt+1] + Vt[Rt+1]. (4.2)

Due to market consistent valuation the economic value of the assets at time t increased by rtis equal to the

(17)

CHAPTER 4. MODEL DESCRIPTION

the liabilities at time (t + 1), which is equal to the accrued liabilities up until time (t + 1). The terms Vt[Ct],

Vt[Bt]and Vt[Rt+1]are the economic value at time t of these amounts.

Rearranging the terms in equation (4.2) and using Vt[At(1 + rt)] = Attogether with equation (4.1) gives

Vt[Lt+1] − Lt+ Vt[Bt] − Vt[Ct] + Vt[Rt+1] − Rt= 0. (4.3)

This expression shows the zero-sum game of a pension fund for all participants together. We now split up the above expression per age cohort x, by defining the term ∆GAx

t+1as the generational account option of

age cohort x, resulting in

∆GAx_t+1= Vt[Lxt+1] − L x t + Vt[Btx] − Vt[Ctx] + Vt[Rxt+1] − R x t. (4.4) ∆GAx

t+1is the economic value at time t of the change in the generational account of age cohort x during

(t, t + 1). For a particular age cohort x equation (4.4) is in general not equal to 0. However, because of the zero-sum game of the pension fund, the sum over all age cohorts should equal 0.

Another way of interpreting equation (4.4) is stating that the generational account option of age cohort xis equal to the change in the economic value of the liabilities, i.e. Vt[Lxt+1] − Lxt, plus the economic value of

all benefit payments in the period (t, t + 1) minus the economic value of all contributions paid in the period (t, t + 1).

We would like to calculate the change in the generational account of age cohort x over a time horizon T at time t = 0. Using the above we get:

∆GAx_T = V0[LxT] − L x 0+ T −1 X t=0 V0[Bxt] − T −1 X t=0 V0[Ctx] + V0[RxT] − R x 0. (4.5)

As assumed in Hoevenaars and Pond (2008) [12] we assume that the own funds of the pension fund at time T is allocated amongst the age cohorts equal to the ratio between the cohort’s liability and the pension fund’s total liability at time T , i.e.

Rx_T = L

x T

LT

· RT. (4.6)

The economic value for each age cohort x is calculated by calculating the average over all 2000 simulations out of the economic set of Towers Watson (which is more discussed in the next subsection).

It will be convenient in discussing the results to divide equation (4.5) into two parts: the net-benefit op-tion, i.e. the change in the economic value of the liabilities, the collection of all benefit payments and all contributions payments, denoted by

∆N B_Tx = V0[LxT] − L x 0+ T −1 X t=0 V0[Bxt] − T −1 X t=0 V0[Ctx], (4.7)

and the residue option, i.e. the change in the economic value of the residue, denoted by ∆Rx_T = V0[RxT] − R

x

0. (4.8)

To be able to determine the value transfers between generations, the change in the generational account option due to a change in the pension fund’s policy, in this thesis focusing on a change of the reduction method, is of interest. Therefore we will compare the generational account options using the benchmark reduction method with the generational account options using alternative reduction methods.

(18)

4.1. VALUE-BASED GENERATIONAL ACCOUNTING

4.1.2 Risk-neutral valuation of generational account options

For the purpose of this thesis we need (lots of) economic scenarios. Because we would like to use Value-based ALM in combination with market consistent valuation, risk-neutral scenarios are needed. Describing the characteristics of an economic scenario generator, and implementing those characteristics to obtain risk-neutral scenarios, is outside the scope of this thesis. For more detailed information about economic scenario generators we refer to Plomp (2013) [22].

In this thesis we use a risk neutral set provided by Towers Watson (as was used by Heijst (2014) [10]). The set contains 2000 simulations of the nominal term structure of interest rates, the equity-index, the CPI-index and corresponding value of the num´eraire asset (based on the money market account) for maturities from 0 to 60. In the sequel of this subsection we briefly discuss the provided information and the underlying methodology in the set of Towers Watson using Towers Watson (2011) [24] and Heijst (2014) [10].

Nominal term structure of interest rates

The underlying model for the nominal term structure of interest rates used in the economic set is the short-rate model known as the one-factor Hull-White-model model [14] (or Extended Vasicek-model), which is defined by the following:

dr(t) = a ˆθ(t) − r(t)dt + σ(t)dWr(t), (4.9) where a is the mean reversion parameter, which determines how strong the mean-reverting effect is, and Wr_(t)_{the Wiener process for the short rate, modeling the random market risk. The Hull-White model is}

fitted to the initial term structure of interest rates.

The parameter ˆθ(t)is derived from the following equation: ˆ θ(t) = f (0, t) + 1 a ∂f (0, t) ∂t + k X i=1 σ2 iC 2a2

e−2a(t−ti)_{− e}−2a(t−ti−1)_. _(4.10)

If ˆθ(t) > r(t), the interest rate r(t) increases on average. If ˆθ(t) < r(t), the interest rate r(t) decreases on average.

The forward rate f (0, t) in (4.10) is deducted from the initial term structure. The volatility σ(t) is piecewise constant and equals a term structure of volatilities:

σ(t) =          σ1 if 0 ≤ t < t1, σ2 if t1≤ t < t2, .. . ... σn if tn−1≤ t < tn. (4.11)

The volatility is calibrated to swaptions and reflects the time dependent volatility of the short rate.

The initial term structure of interest rates (nominal yield curve) used in the construction of the set and the corresponding forward rates are plotted in the next figure:

Equity-indices

The equity-indices in the economic set are modeled using a geometric Brownian motion with the stochastic interest rate from equation (4.9) and piecewise constant volatility parameters, η(t), using the following:

dS(t) = S(t)r(t)dt + η(t)dWS(t). (4.12)

(19)

CHAPTER 4. MODEL DESCRIPTION 0 5 10 15 20 25 30 35 40 45 50 55 60 0 1 2 3 4 5 Maturity (years) Rate (%)

Nominal yield curve Forward rates

Figure 4.1: Initial term structure of interest rates in Hull-White model.

CPI-indices

The CPI-indices are also modeled using a geometric Brownian motion. The average over all simulations and time points of the CPI-index in the set is equal to 1.9%. Looking at the history of the CPI in the Netherlands over the last decades4_{a long-term average CPI of around 1.9% is accurate.}

Correlation

In the economic set correlation between the nominal term structure of interest rates, the equity-index and the CPI-index is taken into account according to the correlations in Table 4.1

Nominal interest rate Equity-index CPI-index Nominal interest rate 1 0.3 0.4

Equity-index 0.3 1 0.25

CPI-index 0.4 0.25 1

Table 4.1: Correlation matrix used in the construction of the risk-neutral economic set.

4_{http://nl.inflation.eu/inflatiecijfers/nederland/historische-inflatie/cpi-inflatie-nederland.}

(20)

4.2. BENCHMARK PENSION FUND

4.2 Benchmark Pension Fund

In this section the pension fund characteristics and the policy characteristics of the benchmark pension fund are described.

4.2.1 Pension fund characteristics

The characteristics of the pension fund under consideration are as follows: • Pension Scheme

We consider a career-average pension plan, which is a defined benefit plan (DB-plan). The accrual percentage is set equal to 1.875%, which is the current maximal allowed accrual percentage for a career-average pension plan in the Netherlands.

The applicable maximum salary and the offset in the first year is set equal toe100,000 and e12,5525

respectively. The applicable maximum salary and the offset are yearly adapted to the (non-negative) price inflation, denoted by π. The price inflation (or CPI-index) follows from the economic set and differs per time point and per scenario. It is assumed that the applicable maximum salary and the offset do not decrease in case of a negative price inflation.

• Salary development

The development of the salary of participants consists of two components:

– The first component is an age-independent general wage inflation, which is assumed to be equal to the (non-negative) price inflation. The price inflation follows from the economic set. It is assumed that the salary does not decrease due to a negative price inflation;

– The second component is an age-dependent salary increase; 3% for ages 25 to 34, 2% for ages 35 to 44, 1% for ages 45 to 54 and 0% otherwise.

• Initial participants characteristics

– Age distribution

The distribution of the pension fund participants is based on the age distribution of the total Dutch population per January 1, 20146. The minimal age of participants is chosen equal to 25 years. The assumed retirement age is equal to 67 years.

– Gender distribution

The male ratio is based on the gender distribution of the working population in the Netherlands per January 1, 20147_{. This results in a ratio of male participants equal to 55%.}

– Number of participants

The total number of participants in the pension fund is chosen equal to 2,000 participants. Be-cause we look to relative changes in the generational account, another number of participation will not lead to other results.

– Salary distribution

The gender-independent salary for a 25-year old participant is set equal toe30,0008 _{in the first}

year. For older ages (until retirement age of 67 years) the initial salary equalse30,000 increased by the cumulative age-dependent salary increase (no general wage inflation) prior to the age of the participant, e.g. the salary of a 27-year old participant is equal toe30,000 increased by the age-dependent salary increase of a 25 year- and 26 year-old resulting ine31,827.

5_{Fiscal minimal allowed offset in 2015}

6_CBS-Statline _{http://statline.cbs.nl/StatWeb/publication/?DM=SLNL&PA=7461BEV&D1=0&D2=1-2&D3=0-100&}

D4=0,10,20,30,40,50,l&HDR=T,G3&STB=G1,G2&VW=T

7_CBS-Statline _{http://statline.cbs.nl/Statweb/publication/?DM=SLNL&PA=71802NED&D1=1&D2=0-7&D3=66,l&}

HDR=T,G2&STB=G1&VW=T

(21)

– Initial pension rights

The initial pension rights are calculated as if the participants have accrued in the pension fund from the beginning (starting at 25 years old with a salary ofe30,000) using an offset equal to e12,552 and the accrual rate of 1.875%. For retired participants the initial accrued pension rights are calculated by yearly increasing the initial accrued pension rights up until 67 years by an inflation rate of 2%.

• Development of participants

We assume that the pension fund is an open pension fund, meaning that new participants enter the pension scheme each year. It is assumed that participants only leave the pension fund by mortality. This implies a participant is either active (age smaller than 67) or retired (age greater than or equal to 67) (disabled or deferred participants are not under consideration).

It is assumed that the total number of active participants stays constant throughout the prognosis, which is common in recently performed ALM-studies at Towers Watson. Furthermore it is assumed that new participants have an age of 25 years. The initial salary of the new participants is equal to the (non-negative) price inflation adapted initial salary ofe30,000, i.e. in accordance with the develop-ment of the offset throughout the prognosis.

• Other initial values

– The initial funding ratio is 100%9_{, which is approximately the average of the actual funding ratios}

per December 31, 2014 within the Netherlands.

– The investment portfolio consists of 40% stocks and 60% bonds.

– The assumed investment return within the recovery plan equals 4.0% in the first year linear in-creasing to 4.9% in the last year10_.

– The Required Solvency Ratio (RSR) is equal to 125%, which corresponds in order of magnitude with the number mentioned in Section 2.2.

– The upper bound of the indexation ladder is set equal to 130%. This is based on the rule-of-thumb that for an indexation of 1% the lower bound of 110% should be increased by approximately 10%. In the following figure the initial age and gender distribution is illustrated:

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Age at t=0 Fund participants (%) _FemaleMale

Figure 4.2: Age and gender distribution of initial pension fund.

9_{http://www.pensioenthermometer.nl/index.php?option=com_content&view=article&id=307:}

dekkingsgraad-december-2014&catid=11:opinie-opinie&Itemid=9

(22)

4.2. BENCHMARK PENSION FUND

The age distribution of the pension fund participants, the initial funding ratio, the investment portfolio and the assumed investment return within the recovery plan are adapted in Chapter 6 in order to get insight of the sensitivity on the results of the initial value of these characteristics.

4.2.2 Policy characteristics

In this subsection the contribution-, (regular) indexation- and catch-up indexation-policy of the benchmark pension fund are described. The reduction policy in the benchmark case is discussed in Subsection 4.3.

Contribution policy

Some pension funds in the Netherlands use a contribution ladder to determine the height of the contribu-tion as a percentage of the salary sum of the active participants. In this contribucontribu-tion ladder the height of the percentage is depending on the funding ratio of the pension fund and therefore on the financial position of the pension fund.

In this thesis it is assumed that the contribution paid in a particular year is equal to the present value of the new accrued rights in that particular year and that the contribution is no steering mechanism. The contribution therefore does not depend on the financial position of the pension fund.

Not using the contribution as a steering mechanism is in line with recent developments. As concluded in Memorie van toelichting [16] it should no longer be the case that the volatility of the contribution increases because of the Financial assessment framework (FTK). Furthermore, as mentioned in Goudswaard et al (2010) [9], the price of new accrual is high and should increase sharply to retain the current ambition of accrual. They conclude that the contribution is at the highest desirable point and should not increase more. Another reason is that the number of active participants (due to the change in age distribution) decreases and adapting the contribution would not have a big impact on the financial position of the pension fund.

Indexation policy

The benchmark pension fund has the ambition to fully compensate the pension rights of the participants by the (non-negative) price inflation. The price inflation follows from the economic set and differs per time point and per scenario under consideration. We will use the notation πs

t for the time- and

scenario-dependent price inflation deducted from the economic set. In case πs

t < 0the pension rights are not adapted

independent of the financial position of the fund. In case πs

t > 0the pension fund uses an indexation ladder.

Under the nFTK the measure, on which pension funds have to base their decisions, is the decision funding ratio, denoted by FRdect . As described in Section 2.2 the indexation ladder should consist of a lower bound,

LBind, and an upper bound, UBind. Using the previous notations we get the following indexation ladder for

(regular) indexation used in the benchmark pension fund to be granted at time t in scenario s:

is_t=      0 if FRdec_t ≤ LBind, πts· FRdec t −LBind

UBind−LBind if LBind< FR

dec t < UBind, πs t if FR dec t ≥ UBind.

(23)

Catch-up indexation policy

Catch-up indexation is granted when the financial position is sufficient to do so. As described in Section 2.2 catch-up indexation is allowed to be granted in case the decision funding ratio is greater than the maximum of the Required Solvency Ratio (RSR) and the upper bound for full indexation, UBind. We assume a uniform

catch-up indexation over all generations. We therefore do not keep track of the cohort-dependent catch-up indexation. We denote the total catch-up indexation up until time t in scenario s by catchs,totalt . The granted

uniform catch-up indexation at time t, denoted by is,catcht , equals:

is,catcht =

0 if FRdec_t < max(UBind, RSR) + πts,

min(₁₀1(FRdec_t − (max(UBind.RSR) + πts)), catch s,total

t ) if FR

dec

t ≥ max(UBind, RSR) + πts.

4.3 Reduction methods of interest

In this thesis it is assumed that the indexation policy (as discussed in subsection 4.2.2) and the reduction method are steering mechanisms. The contribution is equal to the present value of the accrued rights in a particular year and is independent of the financial position of the fund.

In this section we discuss the reduction methods of interest for the purpose of this thesis. First we define terminology used in the sequel of this thesis regarding the characteristics of the reduction method, i.e. an open versus a closed reduction system, the applicable spreading period and an unconditional versus a conditional reduction system. Finally we give an overview of the reduction methods of interest within this thesis, based on the information from Section 3.4.

4.3.1 Open versus closed reduction system

In this thesis an open reduction system indicates that also pension rights that are accrued in the future are taken into account in determining the height of the granted reduction. In case of a closed reduction system only pension rights accrued up until the start of the recovery plan are reduced.

4.3.2 Applying a spreading period

A pension fund has to make a choice in whether they want to apply the total required reduction, i.e. the reduction needed to have a decision funding ratio above the RSR at the end of the recovery plan, at once (spread period of 1) or spreading it over a period (with a maximal duration equal to the maximal duration of the recovery plan period). Depending on the choice of the pension fund different effects for different age cohorts are observed. The height of the granted reduction in the first year of the recovery plan highly depends on the choice of the duration of the spreading period.

Unconditional versus conditional reduction system

In case the pension fund uses a spreading period within the recovery plan, it calculates reduction percent-ages over the chosen period at time points occurring in the future. In an unconditional reduction system these reduction percentages in the nearby future will actually take place on the particular time points in the future independent of the actual financial position of the pension fund at those particular time points in the future. In a conditional system the reduction percentages in the nearby future will not be applied. Only the reduction percentage in the first year actually will be applied. In the second year (and further) at each time point a completely new recovery plan is developed (if applicable) and a new calculation determines a potential new reduction only applied at that certain time point.

(24)

4.3. REDUCTION METHODS OF INTEREST

Spread method

In case the pension fund uses a spreading period within the recovery plan, it calculates reduction percent-ages over the chosen period at time points occurring in the future. The height of the reduction percentage at particular time points depends on the assigned weights to these time points. Under the nFTK the recovery rate at the beginning of the recovery plan should at least equal the recovery rate at the end of the recovery plan (see page 18 of Memorie van toelichting [16]). Translated to applying reduction methods this implies that the applicable weight function should be non-increasing.

In this thesis four spread methods are investigated: flat (i.e. the weight function is a flat line), linear (i.e. the weight function is a linear decreasing line), geometric (i.e. the weight function is a geometric line) and no spread (i.e. the weight function equals 1 for the first time point and 0 otherwise).

A weight function in general has to sum up to 1. Because we have discrete time points the sum over all values of the weight function should sum up to 1 (rather than the integral over all time points in case of continuous time points). Normalizing the weight function for the four methods above results in the following weights (in the figures the spread period is set equal to the maximal allowed spread period equal to the maximal duration of the recovery plan period equal to 10):

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1

(a) Spread method: Flat

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1

(b) Spread method: Linear

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1

(c) Spread method: Geometric

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1

(d) Spread method: No spread

Figure 4.3: Weight at each time point for different spread methods.

4.3.3 Overview of reduction methods of interest

Using the above characteristics of the spreading period and the reduction methods we define the benchmark reduction method and alternative reduction methods investigated in this thesis in Table 4.2.

Note that for the alternative with no spread period it does not make a difference whether it is called a closed/open or conditional/unconditional reduction method.

(25)

Reduction method Open/closed Spread period Unconditional/conditional Spread method

Benchmark Closed 10 Conditional flat

Alternative 1 N/A 1 N/A no

Alternative 2 Open 10 Conditional flat

Alternative 3 Closed 10 Unconditional flat

- Alternative 3a Closed 10 Unconditional linear

- Alternative 3b Closed 10 Unconditional geometric

Alternative 4 Open 10 Unconditional flat

- Alternative 4a Open 10 Unconditional linear

- Alternative 4b Open 10 Unconditional geometric

Alternative 5a Closed 7 Conditional flat

Alternative 5b Closed 4 Conditional flat

Table 4.2: Reduction methods of interest.

4.4 Pension fund in ALM-model

In Appendix A it is described how the different characteristics or variables of interest evolve over time. In this section some key characteristics of the implemented model in R are discussed.

4.4.1 Process at each time point

In each simulation at every time point the following has to be processed:

1. If the decision funding ratio at the time point under consideration is above the Required Solvency Ratio (RSR), then no recovery plan has to be developed and the model progresses to the next time point.

2. If the decision funding ratio at the time point under consideration is below the RSR, then a recovery plan has to be developed. Using the indexation policy and contribution policy in normal circum-stances, the decision funding ratio at the end of the recovery plan is calculated.

(a) If the decision funding ratio at the end of the recovery plan period is above the RSR, the index-ation according to the indexindex-ation policy is granted and the model progresses to the next time point.

(b) If the decision funding ratio at the end of the recovery plan period is below the RSR, then ad-ditional measures are needed. The decision funding ratio at the end of the recovery plan is cal-culated, granting no indexation throughout the recovery plan time period (independent of the financial position of the pension fund).

i. If in this case the decision funding ratio at the end of the recovery plan period is above the RSR, no additional measures are needed. No indexation is granted in the next year and the model progresses to the next time point.

ii. If in this case the decision funding ratio at the end of the recovery plan period is below the RSR, the accrued pension rights are reduced using the applicable reduction method as discussed in Section 4.3. Finally the model progresses to the next time point.

The model progresses first through all time points per simulation. If it reaches the last time point it pro-gresses to the next simulation until all simulations are processed.

(26)

4.4. PENSION FUND IN ALM-MODEL

4.4.2 Bisection method for calculating reduction percentage

In order to determine the necessary reduction to end up with a sufficient decision funding ratio at the end of the recovery plan, Algorithm 4.1 is developed. This algorithm uses the fact that the function that determines the decision funding ratio at the end of the recovery plan applying the reduction method of interest, denoted by f (x), is an increasing function in x, where x is the height of the total reduction. With the help of the bisection method the root of f (x)-RSR is calculated, which gives the total reduction x needed in order to get to the necessary level of the financial position at the end of the recovery plan. After the total reduction is determined by the algorithm, it is distributed according to the chosen reduction method using the weights in Figure 4.3. The algorithm is as follows:

Algorithm 4.1Algorithm to obtain the necessary total reduction using function f (x). 1. Initialization step:

(a) x1= 0

(b) x2= 99%

(c) i = 0

2. Set i = i + 1 and calculate αi= (x1+ x2)/2and f (αi)

if [f(αi) - RSR > 0]

x1= αi

else

x2= αi

3. Repeat step 2 until |f (αi)- RSR| < εalg, εalg> 0and return x = αi

In order to be able to use the above algorithm it should hold that:

1. f (0) − RSR < 0 in all situations. This is true, because when using the algorithm the pension fund is in a reserve deficit;

2. f (0.99) − RSR > 0 in all situations. This should be true, because if all rights have to be reduced by 99% the liabilities go to 0, whereas the assets should have an initial value at the beginning of the recovery plan that is sufficient to pay for liabilities going to 0;

3. f (x) is a continuous and increasing function in x for all situations, i.e. for all possible scenarios in-creasing the total reduction results in a higher decision funding ratio at the end of the recovery plan. It is hard to proof that f (x) is continuous and increasing because there is no simple closed expression to calculate the decision funding ratio at the end of a recovery plan. However intuitively it can be made clear that this should be the case. Note that the economic situation is the same at the start of the bisection method and that the only changing variable is the total reduction.

f (x)is continuous:

The function f (x) is not defined if and only if the liabilities are 0. This cannot be the case for x ∈ (0, 0.99), because you always end up with at least 1% of the liabilities.

(27)

f (x)is increasing:

There are several effects on the assets/liabilities that play a role:

(a) The decreasing benefit payments throughout the recovery plan, because of the reduction of pension rights, have a positive effect on the amount of assets at the end of the recovery plan. Applying a higher reduction percentage results in more assets at the end of the recovery plan.

(b) The contribution payments and the assumed investment return percentages throughout the recovery plan are not affected by the choice of the reduction system or by the height of the reduction.

(c) Accrued rights of participants are reduced by a small amount when applying a reduction. Because the liabilities equal the present value of the accrued rights and the economic variables are the same, i.e. the actuarial factors used at a given time point of the prognosis at a given time point within the recovery plan are the same going through the algorithm, the liabilities will be lower after a reduction is applied. Applying a higher reduction percentage results in even lower liabilities at the end of the recovery plan. Because newly accrual in an open reduction system is also reduced, the liabilities will even be lower in an open reduction system in comparison to the situation of applying a closed reduction system in case the total reduction percentages are the same (for same value of x).

Because the assets increase (by (a) and (b)) and the liabilities decrease in case x increases, the actual funding ratio and therefore the decision funding ratio at the end of the recovery plan increases. Hence, for both an open and closed reduction system, f (x) is an increasing function.

The bisection method used within Algorithm 4.1 is illustrated in Figure 4.4 for initial values x1 = 25%and

x2= 85%and RSR = 165%11: 10 20 30 40 50 60 70 80 90 −50 0 50 100 (x1, f (x1)) (x2, f (x2)) (α1, f (α1)) (α2, f (α2)) x =Reduction (%) f (x) − RSR(%)

Figure 4.4: Illustration of bisection method in benchmark model with RSR=165% at first time point in first simulation. The bisection method halves the interval where the root lies in every step. The method gives a lower and an upper bound of the root, rather than a single value. At every step the value of the middle of the interval is calculated and used as an input to calculate the decision funding ratio at the end of the recovery plan using this new value.

The εalg is set equal to 0.1% for both computational and practical reasons. The funding ratio as reported

to DNB has a accuracy of 0.1%. Therefore, in reality, a pension fund also has to calculate the necessary reduction percentage such that the decision funding ratio is above the RSR with this precision.

11_{Only for a good illustration of the bisection method the RSR is set equal to 165%. It is within this thesis and also in practice not a}

(28)

Chapter

5

Results

In this chapter both the Classic ALM-results and Value-based ALM-results using the different reduction methods are stated.

In the Classic ALM-results we focus on the distribution of the decision funding ratios, the probability of having to develop a recovery plan, the probability of having to apply a reduction and the expectation and standard deviation of the necessary reduction conditionally on having to apply a reduction. Note that the actual values are not solely of interest, more importantly the change of these values under different reductions methods compared to the benchmark reduction method.

In the Value-based ALM-results we look at the change in generational account-, net benefit- and residue option. In order to see the zero-sum game within a pension fund the values of the generational account should be displayed on an age cohort level, i.e. the total economic values of all participants in that par-ticular age cohort. For the purpose of this thesis it is more interesting to look at the value transfers in generational account per participant per age cohort. In the sequel of this chapter we therefore focus on the values per participant per age cohort. For this purpose we divide the values per age cohort by the number of participants within that age cohort.

The results of the benchmark reduction method (closed conditional reduction system) are discussed in Sec-tion 5.1. The remainder of the chapter uses the alternative methods as defined in Table 4.2. In SecSec-tion 5.2 a reduction system without a spread period is investigated. In Section 5.3 the results when changing to an open reduction system are shown. In Section 5.4 the effects of changing to an unconditional reduction sys-tem are investigated. In Section 5.5 changing to an open and unconditional reduction syssys-tem is discussed. Then in Section 5.6 the results of lowering the spread period within a closed conditional reduction system is described. Finally the results are summarized in Section 5.7.

5.1 Benchmark reduction method

The benchmark reduction method is set equal to the currently applied method within the nFTK. Pension funds under the nFTK are allowed to use a spread period of 10 years. Furthermore the reduction method should be applied according to a closed system, meaning that only the accrued pension rights up until the beginning of the recovery plan are to be affected by the reduction. It should be a conditional system, mean-ing that determined future reduction percentages all disappear after 1 year and the pension fund after 1 year, if applicable, again has to develop a new recovery plan neglecting calculated reduction percentages from the past.

Intergenerational effects of different reduction methods within recovery plans in the Netherlands