## rules in a funded pension with intergenerational risk sharing

### E.C.Mersmann

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: E.C.Mersmann

Student nr: 12815608

Email: ellenmersmann@gmail.com

Date: July 9, 2021

Supervisor: dr. T.J. Boonen Second reader: dr. S. van Bilsen

This document is written by Student Ellen Corien Mersmann who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

### Abstract

This study examines different risk allocation rules in order to determine the welfare effect for participants in a funded pension scheme. The focus of this paper is on intergenerational risk sharing of the systematic investment risks in pensions schemes. The allocation rules that are considered have benefits and/or pension contributions depending on the funding ratio of the fund.

The allocation methods are either an exponential function of the funding ratio or a plateau function. The latter implies that contributions and or benefits are constant if the funding ratio is between a certain limit and threshold and change at a linear rate for funding ratios outside these values.

The constant relative risk aversion (CRRA) utility function is used in order to compare the allocation rules. The utility function gave the possibility to establish the certainty equivalents for the lifetime consumption levels of the participants, where the consumption levels depend on the contribution rate before retirement and on the benefit levels after the retirement age is reached. By analyzing different allocation rules, it followed that a hybrid allocation rule, where both the contribution and benefit levels are a function of the funding surplus, are optimal from a welfare perspective.

Besides the certainty equivalents, the funding ratio dynamics are taken into account in the comparison of the allocation methods. A funding ratio con- straint of 80% for the 5 percent worst scenarios is implemented in order to make the pension scheme sustainable for future generations. Comparing the exponential and plateau allocation rules with the linear allocation rule found in Cui et al. (2011), showed that both methods are not an improvement of the linear allocation rule in terms of funding ratio dynamics or certainty equivalents. However, a benefit of the plateau allocation are the relative con- stant contribution and benefit levels, which will in turn lead to more constant consumption levels. Therefore, from a practical perspective the plateau allo- cation rule might be preferred, as it allows the participants to make financial decisions based on a more stable future income level.

Keywords Intergenerational risk sharing funded pension system, Dutch 3 pillar system, certainty equivalent

### Contents

Preface vi

1 Introduction 1

2 Dutch 3 pillar pension system 2

2.1 Intergenerational risk sharing . . . 3

2.2 Recovery plan . . . 3

3 Model setup 4 3.1 Economy and overlapping generations . . . 4

3.2 Pension schemes with intergenerational risk sharing. . . 4

3.2.1 Fund dynamics . . . 4

3.3 Risk allocation rules . . . 6

3.3.1 Contribution and benefit rates . . . 6

3.3.2 Social implications . . . 9

3.4 Optimal pension policies . . . 10

3.4.1 Optimal collective strategy . . . 10

3.4.2 Certainty equivalent . . . 10

4 Simulation 10 4.1 Parameter specifications . . . 11

4.2 Simulation basic parameters . . . 11

4.3 Optimization parameters. . . 12

4.3.1 Optimization method . . . 12

4.3.2 Optimization exponential parameters. . . 13

4.3.3 Optimization plateau parameters . . . 14

4.3.4 Optimal parameter values with funding ratio restrictions. . . 15

4.4 Simulation different age groups . . . 16

4.5 Optimal investment strategy collective pension . . . 18

4.6 Investment strategy functions . . . 18

5 Linear allocation rule 20 5.1 The linear allocation rule . . . 20

5.2 Comparison certainty equivalents . . . 21

5.3 Comparison funding ratio dynamics . . . 21

6 Sensitivity analysis 22 6.1 Sensitivity towards the initial funding ratio . . . 22

6.2 Sensitivity towards the rate of return on equity . . . 23

6.3 Sensitivity towards the risk aversion coefficient . . . 23

7 Optimization social welfare 23 7.1 Objective . . . 24

7.2 Optimal parameter values for social optimum . . . 24

7.3 Funding ratio dynamics . . . 25

7.4 Comparison consumption levels . . . 28

8 Conclusion 29 9 Suggestions for further research 29 References 32 10 Appendix 34 10.1 Simulation consumption exponential allocation methods . . . 34

10.2 Simulation consumption plateau allocation methods . . . 36

10.3 Optimization allocation parameters. . . 39

10.3.1 Funding ratio dynamics for optimal parameter values age 25 . . . 40

10.4 Sensitivity analysis . . . 41

10.4.1 Sensitivity towards initial funding ratio . . . 41

10.4.2 Sensitivity towards the rate of return on equity . . . 41

10.4.3 Sensitivity towards the risk aversion coefficient . . . 42 10.5 Linear allocation rule funding ratio dynamics . . . 43 10.6 Comparison consumption distribution . . . 44

### Preface

With this master thesis, my academic journey has come to an end. I enjoyed my student period to the fullest and during my bachelor’s and master’s degree, I was able to develop myself on both a personal, and a professional level. I appreciated all the opportunities that came along and realize that gaining new knowledge and skills is something that can be scaring at first, but it is amazing once you have tried.

The past year has been different from all years before due to the corona pandemic. It has been a strange year with all the corona restrictions and I have never thought that I would get a master’s degree without being inside the university building once. Despite all these restrictions, I still enjoyed gathering more knowledge in the field of actuarial sciences and I am looking forward to continue learning from the professionals in the working field.

I would like to thank my supervisor dr. Tim Boonen, for his online advice, while writing my master thesis. Furthermore, I would like to express my appreciation to my parents and brother for their ongoing support in the past years.

To new beginnings!

Ellen Mersmann Groningen July 9, 2021

### 1 Introduction

The current Dutch pension system is one of the best worldwide. It successfully combines the first pillar pension for all citizens with a second pillar that is labor related and a third pillar which includes voluntary savings (Kemna et al., 2011). One of the key features in the second pillar is the risk sharing between non-overlapping generations, which is seen as a major benefit of the collective system (Bovenberg et al.,2007). However, in the Dutch pension plan it is not specified which generation bears the risk in case of a funding deficit. This is seen as the major shortcoming of the Dutch pension plan (Kortleve and Ponds,2009). Especially, due to the recent financial position of the pension funds, which has deteriorates due to the financial crisis in com- bination with low interest rates and population aging. Therefore, the pension arrangements are under increasing pressure (Beetsma et al.,2015), and the criticism on the Dutch pension system is increasing. Older participants face a missed indexation of their pension benefits, while future participants face low funding ratios. Therefore, it is important to decide which generations bears which part of the recovery burden, such that new generations are still voluntarily prepared to enter the system and can still face the benefits of the Dutch pension system. One should note that entry of new generations is essential in order for the Dutch pension system to remain stable.

This thesis will focus on the intergenerational risk sharing of systematic investment risks in pension schemes. The pension schemes that are analyzed have no sponsor, have mandatory participation and investment risk is borne by all generations of the pension fund. The surpluses or deficits of the pension fund are shared among future, young and old generations, by adjusting the contributions, benefit levels or both. The research question is whether this intergenerational risk sharing is desirable and what the optimal allocation rule is.

Previous literature has investigated the benefits of intergenerational risk sharing quit rig- orously. For example, Cui et al. (2011) and Gollier (2008) both optimize the asset allocation strategy with as main difference thatGollier (2008) introduces a guaranteed minimum return on pension savings. Like Cui et al. (2011), Bams et al. (2016) and Goecke (2013) also focus on whether the smoothing of capital market returns is advantageous for participants in the pension fund. However, where Cui et al. (2011) focuses on risk allocation rules that are a linear function of the funding surplus, Bams et al. (2016) assumes constant contribution and an indexation depending on the logarithm of the funding ratio plus the expected portfolio return.

Our model is based onCui et al.(2011), with as main difference that we allow for non-linear allocation rules. These allocation rules are a function of the funding surplus or deficit, and we will discuss the exponential and plateau allocation rule. The exponential allocation rule has contributions and benefits changing as an exponential function of the funding ratio. The plateau allocation rule has a constant contribution and or benefit level as long as the surplus value is between a certain limit and threshold. Outside these values, the contributions and/or benefits will absorb the funding surplus or deficit at an increasing rate. The comparison of the models is based on the certainty equivalent of the participants, which is the guaranteed income an individual would accept, rather than taking a change at a higher income. Furthermore, the funding ratio dynamics and average consumption distribution are taken into account.

Discussing the plateau function is an important extension of the current literature. This allocation rule allows for stable contributions and/or benefit levels in case the funding surplus or deficit is between certain values. This implies that the contributions and benefit levels are rel- atively stable during a participants live. This is different from the previous discussed literature.

For exampleCui et al.(2011) allow for constantly changing benefits and or contribution levels in their linear allocation method. In practice, it is not preferred to have constantly fluctuating consumption levels, because this gives less financial stability to a participant. Fluctuating con- sumption levels make it more difficult for consumers to meet their current financial obligations like mortgage payments, but it also makes it more difficult to make financial decisions regarding the future, as income levels are highly uncertain. Furthermore, from a political perspective it is not a popular decision to implement pension cuts. In the plateau allocation rule, these pension cuts happen less frequently, which is also beneficial from a political perspective. The plateau allocation method has major practical benefits and is therefore an important extension of the literature.

The main findings of this paper are the following. First of all, it follows that the hybrid

allocation rules, whereby both the contributions and the benefits change as a percentage of the funding surplus, perform best in terms of the certainty equivalent. This result holds for both the plateau and exponential method, and the linear method discussed in Cui et al.(2011). It is shown that the consumption levels for the plateau allocation methods are more often equal to the target level as the consumption levels for the exponential allocation method. This shows that the income level under the plateau allocation method is indeed more stable, which is a main advantage. Furthermore, the certainty equivalents of the linear and plateau allocation rules show approximately similar values in both the social optimum, as for different age cat- egories. The certainty equivalent for the linear method is slightly higher as for the plateau and exponential allocation method, but this difference is negligible. Furthermore, the funding ratio dynamics of all three allocation methods show a stable pattern. For the exponential and plateau allocation method the implementation of a funding ratio constraint was necessary in the optimization process. In conclusion, the plateau allocation is not performing significantly worse in terms of certainty equivalent or funding ratio dynamics as the exponential or linear alloca- tion methods. Whereas the plateau allocation method outperforms the exponential and linear allocation methods from a practical perspective. That is, the plateau allocation method has less fluctuations in consumption levels, and a lower chance at a pension cut. Therefore, the plateau allocation method is considered as an improvement of the models discussed in previous literature.

The thesis is structured as follows, we start with a discussion about the Dutch 3 pillar system, the principle of intergenerational risk sharing and the current regulations regarding the recovery plan of the pension funds. Section 3 starts with discussing the state of the economy together with the asset and fund dynamics. In subsection 3.3 we continue with the allocation rules and the section ends with an overview of the method used to compare the allocation rules. In section 4, the simulation results are discussed, where we start with a set of basic parameters and continue with the optimization of the allocation rules for different age categories. The section finishes with different investment strategies. Section 5 introduces the linear allocation rule discussed in Cui et al.(2011), Section 6 continues with a sensitivity analyses and section 7 discusses the social optimum for all three allocation methods. The thesis ends with the conclusion of the thesis and some final suggestions for further research.

### 2 Dutch 3 pillar pension system

The Dutch pension system consists of three pillars. The first pillar consists of a state pension, the so called AOW, and there is a mandatory participation for all Dutch citizens. The AOW is a Pay as You Go System with as main objective to prevent old-age poverty (Bovenberg and Nijman, 2018). There is both risk sharing between and across generations in the first pillar.

The risk sharing between generations occurs, as the current workers pay for the current retirees.

There is risk sharing within generations as the retirement benefits are equal for all retirees, while the contribution depends on the labor income.

In this paper, we will focus on the second pillar: the occupational pension with as main objective to maintain the standard of living. In general, there is mandatory participation and only the self-employed do not participate in the second pillar. Both the employee and employer contribute to this pension and the pension fund invests these contributions. At retirement, the accumulated assets are used to payout the pension benefits.

The second pillar could be a either a defined benefit contract (DB) or a defined contribution contract (DC). The benefits in both methods depend on the total contribution paid during the accumulation period. The main difference between both schemes is that workers accrue a promised monthly life annuity from the date of their retirement until death in case of a DB contract, whereas the benefits from the DC contract depend on the investment returns (Broadbent et al.,2006). Only a small part, about 10% in asset value, of the Dutch occupational pension is based on the DC contract. However, in the past years one could see a trend to shift DB to DC plans (Bovenberg and Nijman,2018). In the Netherlands, most occupational pensions are not a pure DB contract, but are more a hybrid system. This implies that if a fund gets in financial difficulties, contributions can be increased, indexation can be limited and if these measures are too limited, pension rights can be cut (Kortleve and Ponds, 2009). The financial shocks are shared between generations, which is called intergenerational risk sharing and discussed in full detail in the next section.

The third pillar is an individual supplementary pension in addition to the first and sec- ond pillar. This pillar is most important for the self-employed, as they do not participate in

the second pillar. Participation is voluntary but most often tax favored. The general plan is an individual DC plan, which implies that there is no risk sharing between or within the generations.

### 2.1 Intergenerational risk sharing

The second pillar enables risk sharing between non-overlapping generations, which is seen as a major benefit of the collective system (Bovenberg et al., 2007). Intergenerational risk sharing, with mandatory participation, allows to smooth consumption over time and could improve welfare, especially if we deal with systematic risks like for example inflation shocks (Cui et al., 2011). Furthermore, future generations can already invest their human capital in the stock market by borrowing from older generations (Teulings and De Vries,2006). This gives younger generations a larger investment horizon and financial shocks can be distributed among more generations, which is welfare improving for both generations.

A drawback of intergenerational risk sharing is that it removes the direct link between the individual contributions and benefits. In most pension plans, the contributions are related to the labor income. Therefore, the labor supply decision is distorted due to the disconnection between the contributions and the benefits, which might lead to considerable welfare loss (Bonenkamp et al.,2014). Furthermore, in the Dutch pension plan it is not specified which generation bears the risk in case of a funding deficit. This is as discussed before, a major shortcoming of the Dutch pension plan (Kortleve and Ponds,2009).

### 2.2 Recovery plan

In the past years, the funding ratio of many Dutch pension funds deteriorated. The Dutch Central Bank requires the pension fund to submit a recovery plan, if the policy funding ratio is below the required policy funding ratio (RPFR). The required ratio differs per fund and depends mainly on the financial risks of the pension fund. The policy funding ratio is the average funding ratio from the past 12 months.

In the recovery plan, the fund has to show that it recovers from the funding deficit within 10 years. The recovery of the pension fund could take place by increasing contributions, limiting indexations or cutting existing pension rights. According to Ponds and Van Riel (2009), the effectiveness of the contribution as a method to recover is decreasing in the coming period.

Mainly due to the fact that the ratio between liabilities and total wages will rise from 2.5 to 4.5 in the year 2030 (Ponds and Van Riel,2009).

Another requirement of the pension funds is that the policy funding ratio is above the
minimum required funding ratio. The minimum required funding ratio is 104.3% for the years
2019/2020.^{1} The pension fund is forced to cut the pension rights, if the policy funding ratio is
below the minimum required funding ratio for five consecutive years.

Furthermore, Dutch DB plans have conditional indexation, which implies that pension funds
evaluates annually whether the financial position of the fund allows for indexation of the accrued
pension rights. The fund requirements and indexation ambitions for the year 2020 of the two
largest pension funds in the Netherlands are summarized in Table 1.^{2} The table shows the
requirements for the funding ratio in case of indexation and it shows the required policy funding
ratio.

Pension Fund RPFR No Indexation Partial Indexation Full Indexation

ABP 126% <110% 110%-123% >123%

PFWZ 121.4% <110% 110%-125% >125%

Table 1: Funding ratio requirements

Since the financial crisis of 2007, most pension funds are facing funding deficits. In June 2007, the average funding ratio was 150% for the Dutch pension funds. In 2008, the return on real investments of pension funds was -17% and funding ratios dropped to 120% by the end of quarter 3 in 2008. The fall in the funding ratio was entirely driven by losses on assets in the portfolio. In 2009, the interest rate dropped from 5% to 2% which led to another strong decline in the funding ratio (Beetsma et al., 2015). Since 2007, most pension funds were not

1Source DNB

2Source ABP and PFWZ

able to index their pension rights and participants of for example ABP missed an indexation of
a maximum of 19.95%.^{3}

### 3 Model setup

The aim of this chapter is to discuss the model that will be used in order to calculate the optimal contribution rate and payout rate of the pension funds. The model is based on the model described in Cui et al. (2011), with as main difference that our model also allows for non linear changes in the contribution rates and benefit payments. This section starts with the general outline of the economy and the generations, after which it describes the fund dynamics.

Thirdly, it discusses the allocation rules as a function of the funding ratio and it concludes with a description of the method to compare the pension schemes.

### 3.1 Economy and overlapping generations

We assume that the investment fund of the pension fund consists of two asset classes, the risk- free assets and the risky equities. For simplicity, deterministic interest rates are used and we make use of a discrete time environment in the remainder of the paper. The dynamics of the total asset portfolio are given by:

Vt+1= ((µ − rf)πt+ rf)Vt+ C − B + σπtVtt, (1) where µ and σ are respectively the mean return and the volatility of the stocks, and t is a standard normal random variable. Furthermore, C represents the total contributions, B the total benefits in each period and πt is the ratio invested in equities. It is assumed that 0 ≤ π(·) ≤ 1, which implies that short-selling is not allowed.

The default values of our model parameters are in line with the assumptions used in the
report of Dijsselbloem et al. (2019). That is, µ =5.8% and σ=20%. Furthermore, we assume
that the real risk free interest rate is constant and equal to r_{f} = 2%, which implies that the
equity premium is equal to 3.8%.

In this economy, we assume a population with 60 overlapping generations, with age ranging from 25 to 85. We assume that everyone starts working at age 25 (t = 0), retires at age 65 (t = R = 40) and deceases at age 85 (t = D = 60). This implies that we have an active working phase of 40 years, and a retirement phase of 20 years. Similar to the methods described inCui et al. (2011) and Teulings and De Vries (2006), we assume that all generations have an equal size, and labor supply is fixed. Furthermore, we assume that retirees have no other income than retirement benefits and that the working generations earn a flat labor income which is normalized to 1. The preferences of all generations are homogeneous and we assume a Constant Relative Risk Aversion (CRRA) utility function given by:

u(x) =
(_{x}1−γ

1−γ if γ 6= 1,

log(x) if γ = 1, (2)

for all x > 0. This utility function is in line with previous literature in life cycle optimization of for exampleGollier(2008) andBovenberg et al.(2007). In this utility function, γ represents the risk aversion parameter, whereby a higher value for γ corresponds to a more risk averse participant.

### 3.2 Pension schemes with intergenerational risk sharing

In this section, we will discuss the dynamics of the pension funds assets and liabilities, whereby the fund allows for intergenerational risk sharing. This implies that our model allows for a mismatch between the assets and liabilities and the mismatch is not necessarily fully covered by the current generations. In line with the Dutch pension law, we assume mandatory participation for all current and future generations in the pension fund.

3.2.1 Fund dynamics

The fund allows for intergenerational risk sharing in such a way that the contributions of each individual are not directly converted in an individual pension right. Sometimes part of the

3Source ABP

contributions are used to absorb financial shocks of other generations. The liabilities of a DB fund with intergenerational risk sharing can therefore not be calculated with only knowing the past contributions. Furthermore, benefit levels are also changing each year, which implies the liabilities are not easy to project anymore. Therefore, we first introduce the concept of target contribution levels, target benefit levels and target liabilities for each age cohort.

In a pension fund without intergenerational risk sharing, it is true that for every age cohort, the market value of the contributions are equal to the market value of the benefits. This is also known as the actuarial fair principle (Queisser and Whitehouse, 2007). This implies that for each age cohort, the target contribution C and benefit levels B are given by:

R

X

t=0

Ce^{−rt}=

D

X

t=R

Be^{−rt}, (3)

where r is the interest rate used for liability valuation, which might differ from the risk free rate.

The liabilities of the pension fund is the total of all accrued pension rights from all genera- tions. For an age cohort with age x the total accrued pension rights, and therefore the liability of the pension funds, are given by:

L_{x}=

(PD−1

t=R Be^{−r(t−x)}−PR−1

t=x Ce^{−r(t−x)} for x ≤ R,
PD−1

t=x Be^{−r(t−x)} for R < x < D. (4)

The first line describes the accrued pension rights of an cohort with an age before retirement.

This is the difference between the market value of lifetime retirement payments and the market value of the contributions it has to make from age x till retirement. The total liability of the fund can be calculated as the sum of the liability from each age cohort:

L =

85

X

x=25

Lx. (5)

As discussed in the previous section, we make use of homogeneous age cohorts. This implies that
target contributions, benefits and liabilities do not change over time. Therefore, the dynamics
of the funding ratio F R_{t}which is defined as:

F Rt=At

L , (6)

where Atrepresents the funds total assets and L represents the funds total target liability, are completely determined by the dynamics of the assets. Furthermore, this also implies that at t=0 the assets are given by:

A0= L · F R0, (7)

where F R0is the funding ratio at t=0 and determines whether the fund is over or underfunded at the start of the period.

The economy consists of 40 working cohorts and 20 retired cohorts. This implies that 40 age cohorts make contributions to the pension fund and 20 age cohorts receive payments from the pension fund. Therefore, the asset dynamics are given by:

A_{t+1}= A_{t}((µ − r_{f})π_{t}+ r_{f}) + 40C_{t}− 20Bt+ σπ_{t}A_{t}_{t}, (8)
where π_{t}is the fraction invested in risky assets, _{t}is a standard normal random variable and C_{t}
and B_{t}are the actual contribution and benefit levels, which will be discussed in more detail in
the next section. The first three terms of the asset the dynamics, which contain the expected
return, total contribution and benefits, show that the expected funding ratio increases, if the
actual contributions and return on assets outweigh the actual benefit payments. That is, if the
current working generation contributes more than the actuarial fair principle would suggest, the
funding ratio is expected to increase. Furthermore, if the benefits are higher as the actuarial
fair principle states, the funding ratio is expected to decrease. In the first part of this thesis, the
fraction πtis assumed to be constant over time. In subsection4.5, a dynamic portfolio strategy
is proposed.

### 3.3 Risk allocation rules

As discussed in the previous section, pension funds invest in risky assets in order to increase the rate of return on the investments. Due to these risky investments, pension funds might become underfunded or overfunded. This implies that there is a possibility that the liabilities do not longer match the assets of the pension fund. In order to allocate the mismatch, the contribution rate (Ct) and benefit payments (Bt) need to be adjusted as a function of the surplus/deficit of the fund. In Cui et al. (2011), three stylized risk allocations rules are discussed. All of these allocation rules have in common that the change in the contributions and/or benefits are a linear function of the funding residual.

In this paper, we create six options for the risk allocation rules, where three allocation rules are a so called plateau function and the other three allocation rules are exponential functions of the funding ratio. For all six allocation rules, it holds that in case of significant underfunding or overfunding, the contribution rates and/or benefits will absorb a high percentage of the funding mismatch. The plateau function allows for constant (the plateau) contribution and benefit levels, if the funding ratio is between a certain limit and threshold. This makes the consumption level of participants more stable as the exponential allocation rule, which makes participants better able to meet their financial obligations and make financial decisions.

A result of both methods is that the degree of intergenerational risk sharing will be limited to some extend. That is, the current working generation and retirees are forced to absorb a larger fraction of the mismatch in case of a large funding deficit or surplus. This method is in line with the criticism of for exampleBohn et al.(2003), who states that governments are likely to shift too little risk to the retirees. Hence, this method will limit the transfers from the current working or retired generation to the future generations in case of negative financial shocks. This might decrease the potential resistance of future generations to participate in the scheme after a financial shock (Gollier, 2008).

3.3.1 Contribution and benefit rates

The funding surplus or deficit Stis defined as the difference between the total assets and target liabilities:

St= At− L,

where the target liabilities are again constant. The dynamics of the surplus are solely determined by the asset dynamics of the fund. We define the following three allocation rules, where each allocation rule has two possible formats:

1. Defined benefit with contribution adjustment (DBc)

In this allocation rule, all the funding surplus or deficit S_{t}is borne by the working cohorts
represented by W . The amount the current working cohorts bear is a function of the
funding ratio. The higher the funding mismatch, the larger the change in the contribution
rate of the current working cohorts will be. In Equation (9) and (10), one could find 2
possible allocation rules for the DB_{c} method.

The exponential defined benefit contract (DB^{e}_{c}) is defined as:

Ct= C − f1St/W =

min(C − ^{B}^{F Rt−1.1}^{cn}_{B} ^{−1}

cn−1
S_{t}

W; 1) for F Rt− 1.1 < 0,
max(C − ^{B}^{F Rt−1.1}^{cp}_{B} ^{−1}

cp−1
S_{t}

W; 0) for F R_{t}− 1.1 ≥ 0. (9)
It shows that if the funding ratio is exactly equal to 110%, the contribution ratio is
equal to the target contribution level. The 110% is chosen in particularly, since the two
largest pension funds start the partial indexation from a funding ratio of 110%, as could
be seen in Table 1. In the DB^{e}_{c} allocation rule, benefits are fixed, which implies that
only contributions will exponentially change if the funding ratio deviates from 110%. The
function f_{1} could be seen as a buffering factor. The higher the value of f_{1}, the more the
current working generation will absorb the either positive or negative financial shock in
the funding ratio. One should note that the level of B_{cn}and B_{cp}measures the curvature of
the buffering factor in case of a negative or positive shock. The lower the levels of B_{cn}and
Bcp, the closer the curve is to a straight line. The level of Bcn and Bcpcan be different,
which implies that the effect of overfunding on the contribution rate is different as the
effect of underfunding. Finally, the contribution level should be positive, therefore in case

of overfunding the minimum contribution level equals 0. Furthermore, it is assumed that the contribution level cannot be more than 1, which is the income level.

The plateau defined benefit contract (DB^{p}_{c}) is defined in Equation (10) and given by:

Ct= C − f2St/W

=

min(1; C − min(_{N B}^{F R}^{t}^{−1.1}

n−C_{i} −_{N B}^{C}^{i}

n−C_{i}; 1)_{W}^{S}^{t}) for F Rt− 1.1 < Ci,

C for C_{i}≤ F Rt− 1.1 ≤ Cd,

max(0; C − min(_{N B}^{F R}^{t}^{−1.1}

p−Cd−_{N B}^{C}^{d}

p−Cd; 1)_{W}^{S}^{t}) for F Rt− 1.1 > Cd,

(10)

where the function f_{2} could also be seen as a buffering factor. The higher the value of f_{2}
the less gradual the absorption of the financial shock will be. It determines how much of
the financial shock, the so called surplus or deficit, is absorbed by the contribution rate in
period t. We assume that the target funding ratio is 110%, so in case the funding ratio is
above or below 110%, we consider it as a funding mismatch. f2 measures the percentage
of the shock that is absorbed in the current period. If the percentage point funding mis-
match is between Ci and Cd, we neither increase nor decrease the contribution rate. In
this case the contribution rate is equal to the target contribution. If the percentage point
funding surplus is larger than Cd, the contribution rate for the current working generation
will decrease. The higher the funding surplus, the larger the buffering factor f2, and the
smaller the effect of the financial shock on future generations will be. If the percentage
point funding surplus is higher than N B_{p}, there is no buffering of the positive financial
shock anymore. This implies that the current working generation fully uses the fund-
ing surplus to decrease its contributions, where we assume that the contributions cannot
be negative and the contribution cannot exceed income. A similar explanation could be
used to explain the trend in the contribution rate if there is a funding ratio smaller than
Ci+ 1.1. An example of the DB_{c}^{e} and DB_{c}^{p} allocation rule, is given in the Figure1. The
target contribution and liability levels are equal to C=2000 and L=100000 in both graphs.

Figure 1: Contribution rates as a function of the funding ratio

The vertical line at the funding ratio of 1.1 shows the target funding ratio. The main differences between the graphs is that the graph corresponding to the plateau contribution rate contains a plateau. That is, the contribution rate is equal to the target contribution rate as long as the funding ratio is between Ci+ 1.1 and Cd+ 1.1. Where Ciand Cdare the percentage point funding mismatch we should have in order to start increasing/decreasing contributions. The addition of the plateau is beneficial, because it allows the contribution level to be equal to its target in case of small financial shocks. This allows for more stability in the contribution rate in relatively stable financial times.

Furthermore, we could observe that both graphs are convex for funding ratios smaller than 1.1 and concave for funding ratios larger than 1.1. This implies that in case of for example a deficit, we have an increasing marginal contribution rate. So for small deficits the contribution rate is relatively constant and the change in contribution rate is increasing in percentage points in case of high deficits. This makes sure that only in case of severe deficits, the contributions are significantly increasing.

2. Collective Pension with benefit adjustments (CP_{b})

In this allocation rule, all the funding mismatch S_{t} is completely borne by the retired
cohorts, which are represented by D − W . The amount the current retired cohorts bear
is again a function of the funding ratio. The higher the funding mismatch, the larger
the change in the benefit levels will be. In Equation (11) and (12), one could find the
exponential and plateau methods for the CPb allocation rule.

The CP_{b}^{e}is given by:

B_{t}= B + f_{1}S_{t}/(D − W ) =

max(0; B +^{B}^{cn}^{F Rt−1.1}_{B} ^{−1}

cn−1 St

D−W) for F Rt− 1.1 < 0,
B +^{B}^{cp}^{F Rt−1.1}_{B} ^{−1}

cp−1 St

D−W for F Rt− 1.1 ≥ 0. (11)

The idea of this allocation rule is similar to Equation (9). One could see that the target benefit level is reached if the funding ratio is 110%. If the funding ratio is below/above 110%, the benefit level will exponentially decrease/increase. Furthermore, the benefit levels cannot be negative, since retirees are never making a payment to the fund. As before, the parameters Bcn and Bcp measure the curvature of the graph in case of a negative or positive shock.

Equation (12) is known as the CP_{b}^{p} method and is given by:

Bt= B + f2St/(D − W )

=

max(0; B + min(_{N B}^{F R}^{t}^{−1.1}

n−Bd−_{N B}^{B}^{d}

n−Bd; 1)_{D−W}^{S}^{t} ) for F Rt− 1.1 < Bd,

B for Bd≤ F Rt− 1.1 ≤ Bi,

B + min(1;_{N B}^{F R}^{t}^{−1.1}

p−Bi −_{N B}^{B}^{i}

p−Bi)_{D−W}^{S}^{t} for F R_{t}− 1.1 > B_{i}.

(12)

The benefit level is equal to the target benefit, if the percentage point funding mismatch
is between Bd and Bi. If the percentage point funding mismatch is smaller or larger as
respectively B_{d}and B_{i}, the benefits of the current retirees will start to decrease or increase.

As mentioned before, there is a funding mismatch in case of a funding ratio higher or lower
than 110%. Note that f_{2}could again be interpreted as a buffering factor. The higher the
value of f_{2}, the higher the percentage of the funding mismatch that is absorbed by the
current retirees. If the percentage point funding mismatch is either below N B_{n} or above
N Bp, the current retirees absorb the complete financial shock. Again we assume that the
benefit levels cannot be negative.

An example of these two allocation rules are given in the Figure 2. The target benefit
and liability levels are equal to B=2000 and L=100000 in both graphs. The vertical line
represents that target funding ratio of 1.1. The main difference between the two graphs
is that the second graph, corresponding to the CP_{b}^{p} method, allows for a constant benefit
level, as long as the financial shock is such that the funding ratio is between B_{d}+1.1 and
B_{i}+1.1.

3. Collective Pension with both benefit and contribution adjustments (CP_{h})

In this allocation rule, the funding imbalance can be shared among current workers, current
retirees and future participants. According toPonds and Van Riel(2009), this hybrid plan
performs particularity well due to this broad risk sharing capacity. The exponential hybrid
plan (CP_{h}^{e}) is defined as:

B_{t}= B + f_{1} S_{t}
D − W,
C_{t}= C − f_{1}St

W.

(13)

Figure 2: Benefit rates as a function of the funding ratio

The plateau hybrid plan (CP_{h}^{p}) is defined as:

B_{t}= B + f_{2} S_{t}
D − W,
C_{t}= C − f_{2}St

W.

(14)

It follows that Equation (13) and (14) are a combination of the previously described DBb

and CPb methods. In the remainder of the thesis, the allocation methods corresponding to the above equations will be referred to as described in Table 2.

Equation (9) Equation (10) Equation (11) Equation (12) Equation (13) Equation (14)

DB_{c}^{e} DB^{p}_{c} CP_{b}^{e} CP_{b}^{p} CP_{h}^{e} CP_{h}^{p}

Table 2: Name allocation rules

3.3.2 Social implications

The Dutch pension regulations force Dutch pension funds to create a recovery plan in case of underfunding. In this recovery plan, they have to show how they recover from the financial shock within 10 years. They plan to absorb 10% of the financial shock each year, even in case of severe underfunding. The benefit of this system is that the individuals have time to adjust their spending behavior, and do not face high fluctuations in their income levels. The drawback of this system is that it might be difficult to resolve the problem of severe underfunding in case multiple financial shocks happen in a row, which might in turn make future generations reluctant to join the system.

Therefore, we proposed a new method, where contributions and benefits are more flexible.

This allows the pension fund to return relatively quick to a healthy funding ratio, depending on the parameters we choose. A disadvantage of the exponential allocation method is that contri- butions and or benefit levels are constantly fluctuating. This makes it difficult for consumers to smooth consumption during lifetime and this is important in order for consumers to meet their financial obligations. The plateau allocation method tries to resolve this problem, by making sure the contributions and benefit levels are constant between certain levels of the funding ratio.

Historical funding ratio data of the Dutch pension funds show that from 2007 to 2008, there was a severe drop in the funding ratio of approximately 40 percentage points. The funding ratio dropped from 140% till 100%, with as main reason the introduction of the market-based interest rate term structure for discounting in combination with the financial crisis (Bikker et al.,2011).

In our parameter specifications, we have to take the importance of income stability into account in case of such severe financial shocks. In the remainder of this paper, we will therefore assume that contributions are maximal 50% of the income level. Furthermore, the pension benefits have a minimum value of 0.5 and a maximum value of 2, which is respectively a half and twice the income level during the working phase.

### 3.4 Optimal pension policies

In this section, the method to compare the optimal pension policies of the collective pension scheme will be discussed. The collective pension scheme adapts the contributions and benefit levels to the levels prescribed in section3.3.1.

3.4.1 Optimal collective strategy

The pension fund optimizes the utility of the participants. The optimization problem of the pension fund, assuming γ 6= 1, becomes:

U = max E

^{D}
X

t=1

e^{−δt}c^{1−γ}_{t}
1 − γ

,

with ct= 1 − Ct t < R, (15)

c_{t}= B_{t} R ≤ t ≤ D,

which is subject to the asset dynamics described in Equation (8). The actual contributions and benefits follow directly from the risk allocation rules for different pension schemes, described in section 3.3.1. Here γ and δ represent the risk aversion parameter and the subjective discount rate. Literature suggests that the value of γ is typically between 1 and 10 (Teulings and De Vries, 2006). We benchmark our calculations using γ =4. The time preference δ is set at 2% per year, which is in line with Cui et al.(2011). In Section 6, the sensitivity of our results with respect to the risk aversion coefficient will be computed.

3.4.2 Certainty equivalent

In order to compare the strategies with each other, we will make use of a so called certainty equivalent. In previous literature, for example in Cui et al. (2011), Gollier (2008) and Romp and Beetsma(2020), this method is used to compare pension schemes with each other. We will make use of the certainty equivalent, because although we can select the optimal strategy for an individual by looking at the expected utility, the expected utility does not allow for an intuitive comparison between two strategies (van der Lecq et al., 2016). The certainty equivalent is the guaranteed yearly income a participant would accept, rather than taking a chance at a higher but uncertain income. The certainty equivalent for a period of 60 years, again assuming γ 6= 1, is calculated as follows:

CE = U^{−1}

1 1000

P1000 s=1

PD
t=1e^{−δt c}

1−γ t,s

1−γ

P60
t=1e^{−δt}

!

, (16)

where the numerator represents the expected utility from consumption for the coming 60 years.

We simulate 1000 scenarios and c_{t,s}represents the consumption at time t for scenario s, param-
eter γ is the risk aversion coefficient and e^{−δt}is the discount factor of the utility function. The
CE allows for a monetary comparison between two strategies.

### 4 Simulation

This section starts with discussing the base case parameter specifications. These are mainly based on the pension policy of the two largest Dutch pension funds: the ABP and PFWZ, whereby one should note that we assume that income is normalized to 1. Secondly, the sim- ulations of the real contributions and benefit levels, given the base case parameters, for an individual aged 25 are discussed. The section continues with the optimization of the parameters for the exponential and plateau allocation methods for someone aged 25. The first optimization results are without funding ratio constraints and in the second part funding ratio constraints are imposed. The fifth part of this section discusses the parameter optimization for participants aged 45 and 65. Finally, different investment strategies are described and the optimal investment strategy is chosen.

### 4.1 Parameter specifications

It is common to have an accrual rate of 1.875% in the Netherlands. This implies that the total retirement benefits are given by 75% of the average income, assuming the worker has been active for 40 years. This 75% consists of both the occupational pension and the first pillar franchise.

However, in this paper we will only focus on the occupational pension and assume that the target benefits B are equal to 0.75. Using the actuarial fair principle, a risk free interest rate of 2% and Equation (3), it follows that the target contribution is given by C=0.202. The liabilities at the t = 0 can be calculated using Equation (4) and (5), and are given by 349.96. Furthermore, Table1 shows that the two largest pension funds in the Netherlands have no indexation if the funding ratio is lower than 110%, and there is full indexation from a funding ratio of 123% or 125%.

The plateau values in the basic parameter specification are chosen such that Bi equals 0.2.

This implies the fund starts the indexation of the pension rights, if the funding ratio is 130%

or higher. Furthermore, the fund starts to decrease the contributions from a funding ratio of
140% or higher, so C_{d}=0.3. The minimum required funding ratio is 104.3%. Pension funds are
forced to cut the pension rights, if the funding ratio is below this minimum for 5 consecutive
years. However, from a theoretical perspective, it is known that losses hurt more. Therefore,
the parameter values are given by: Ci=-0.3 and Bd=-0.4. This is a rather conservative method,
which implies we only increase contributions if the funding ratio is lower than 80%, and we
decrease benefits if the funding ratio is below 70%.

For the exponential allocation method, we chose basic parameters of B_{cn}^{b} = 20, B^{c}_{cn} =20,
B_{cp}^{b} =10 and B_{cn}^{c} =10. This implies that positive shocks are absorbed to a higher extent by the
current generation as negative shocks. The parameter specification is summarized in Table 3.

In the next section, we will discuss the dynamics of consumption and benefit levels with the given parameters.

C B L Ci Bi Bd Cd N Bp N Bn B_{cn}^{b} B_{cn}^{c} B_{cp}^{b} B_{cn}^{c} r rf π δ γ

0.202 0.75 349.96 -0.3 0.2 -0.4 0.3 1 -1 20 20 10 10 0.02 0.02 0.4 2% 4

Table 3: Parameter specifications

### 4.2 Simulation basic parameters

In the appendix section 10.1 in Figure 7, one could find the 90% confidence interval for the
consumption levels of the DB_{c}^{e}, CP_{b}^{e} and CP_{h}^{e} methods. As could be seen from the figures,
the workers consumption is between approximately 0.7 and 1 in both the DB_{c}^{e}method and the
CP_{h}^{e} method. Whereas the consumption level of the retirees is between 0.5 and 2 in case of
the CP_{b}^{e}method and between 0.6 and 2 in case of the CP_{h}^{e}method. The confidence interval of
the benefits has a wider range as the confidence interval of the contributions. This makes the
benefits more uncertain as the contributions for the exponential allocation method.

A more detailed view of the contribution and benefit dynamics could be found in Figure8.

This Figure shows the distribution of the benefit and contribution levels after we runned the
1000 simulations for 30 years. One could see that for the DB_{c}^{e}method, the contribution is close
to 0 (between 0 and 0.05) in approximately 50% of the scenarios. In 30% of the scenarios, the
contribution is close to the target level contribution of 0.202 (between 0.15-0.25). A similar
observation could be made for the contributions in the CP_{h}^{e} method. In this case, around
60% of the scenarios have a contribution level close to the target contribution and 25% have a
contribution close to 0. For the benefit dynamics of the CP_{b}^{e}and CP_{h}^{e}method, one could see a
peak in the frequency at a benefit level close to the target benefit. Furthermore, one could see
a peak at a benefit level of 2, which is highest for the CP_{h}^{e}method. This is due to the fact that
pension benefits are limited till a value of 2 in our model setting.

In the appendix section 10.2, one could find an example of the simulation of the allocation
methods DB^{p}_{c}, CP_{b}^{p}and CP_{h}^{p}and the 90% confidence interval for the consumption level of both
workers and retirees for the three equations. The confidence intervals show that for the DB_{c}^{p}
method, the consumption for workers is between approximately 0.7 and 1, whereas the CP_{h}^{p}
method has a consumption of workers between 0.8 and 1. For the retirees, one could see that
the CP_{b}^{p} method has consumption values ranging from 0.7 to 2 and the CP_{h}^{p} has consumption
values ranging from 0.5 to 2.

Comparing the confidence intervals of the exponential allocation method and plateau al-
location method, it follows that both lead to more or less similar confidence intervals for the
consumption levels. Furthermore, if we compare the histograms of Figure 9with Figure 8, one
could clearly see the effect of the plateau. In the histogram corresponding to the DB_{c}^{p}and CP_{h}^{p}
allocation methods, it follows that the contribution rate is either equal to 0 or equal to the
target contribution of 0.202. Furthermore, the benefit level for the CP_{b}^{p} and CP_{h}^{p} method, is
equal to the target benefit level in more than 50% of the scenarios, and equal to 2 in 25% of the
scenarios. It follows that benefit and contribution levels that are unequal to their target level or
their limit (0 in case of contribution and 2 in case of benefits), happen much more frequent in
the exponential methods. This shows that under the plateau allocation rule, contributions and
benefits are more stable.

In order to show the value of the plateau function into more detail, we show 10 random simulation streams of 60 years for the plateau allocation methods, in the appendix section10.2.

For the DB^{p}_{c} allocation method, we focused on the consumption of the workers, for the CP_{b}^{p}
method we focused on the consumption of retirees, and for the CP_{h}^{p}method we simulated both
the consumption of workers and retirees.

In Figure10, one could find the consumption for workers corresponding to the DB_{c}^{p}allocation
method . As could be seen, the consumption in most simulations fluctuates around 0.8, which is
1 minus the target contribution. In some scenarios, the consumption is most frequently around
1, this implies that there are no contributions to the fund and the return on the stock market
is enough to payout the benefit for the retirees. On average, positive shocks in the consumption
happen more often as negative shocks. This has mainly to do with our conservative estimation
of the target benefits and target contributions. On average, the return on our portfolio will be
higher as the interest rate used for the liability valuation. The minimum consumption level is
equal to 0.5, which implies that the worker contributes half of is income to the pension fund.

In Figure 11, one could find 10 scenarios for the consumption of retirees belonging to the
CP_{b}^{p} allocation method. The maximum consumption for the retirees is limited till 2, which is
twice the income they were receiving during the working period. The minimum consumption
level is again 0.5. The consumption level of the retirees is fluctuating around 0.75, which is
equal to the target benefit level. Again one could see that there are more positive shock in
consumption than negative shocks.

Finally, in Figure12and Figure13, one could find a simulation of 60 years for 10 consump-
tion streams of both the workers and retirees. The simulation is based on the CP_{h}^{p} allocation
method. It leads to similar results as previously discussed. The consumption of the workers
is again fluctuating around 0.8 with 0.5 as minimum and 1 as maximum. Whereas the con-
sumption of the retirees is fluctuating around 0.75 with 0.5 as minimum and 2 as maximum.

In conclusion, the plateau allocation methods lead to contribution and benefit levels which are most frequently equal to the target levels.

### 4.3 Optimization parameters

In this section, the optimization of the allocation methods will be discussed. The section starts with a description of the optimization method and will be followed by the optimization of the exponential allocation rules described in section 3.3.1. In the third part of this section, the plateau allocation rules described in section 3.3.1 are discussed. This will be followed by a discussion about the funding ratio dynamics, given the optimal parameter values. Finally, we will compare both allocation methods.

4.3.1 Optimization method

The optimization of the allocation methods is based on the highest CE for a specific age category.

That is, in section 4.3.2and4.3.3, the allocation methods are optimized such that it results in the highest CE for an individual aged 25, whereas in section 4.4 the methods are optimized such that it results in the highest CE for either age 45 or age 65.

For both allocation methods, it follows that there exists no interior point. Therefore, the
optimal allocation parameters are approximated using the Monte Carlo technique with a nu-
merical grid search. For the exponential allocation method, it follows that higher values of the
parameters, lead to a higher degree of intergenerational risk sharing. For a participant aged
25, it is therefore expected that it prefers the highest possible values for the B_{cn}^{c} and B_{cn}^{b} pa-
rameters, as this implies that in case of a negative shock, this shock is spread over multiple

generations. In the optimization for age 25, it is expected that it is optimal that B_{cn}^{c} and B_{cn}^{b}
go to infinity. In order to be able to solve this problem numerically, boundary conditions on the
parameter values are established and are given by:

B_{cn}^{c} ≤ 500, B^{b}_{cn}≤ 500, B_{cp}^{c} ≤ 500, and B^{b}_{cp}≤ 500. (17)
The parameters are optimized using a numerical grid search, these grid values can be found in
the appendix section10.3. The CE of all combinations of these grid values are calculated based
on 1000 simulations. The optimal parameter values corresponds to the method that results in
the maximum CE. For the plateau allocation method, a similar optimization method is applied.

In this case the chosen boundary conditions on the parameter values are given by:

− 0.95 ≤ Ci≤ 0, 0 ≤ Cd≤ 0.95, −0.95 ≤ Bd ≤ 0, and 0 ≤ Bi≤ 0.95. (18) 4.3.2 Optimization exponential parameters

In the Table4, the optimal, second best and third best parameters for the exponential allocation methods can be found, for an individual aged 25. The optimal parameters are chosen such that it results in the highest CE for an individual aged 25 and is based on 1000 simulations of the utility function. One should note that we assume that the initial funding ratio is 100%. The results can be found in Table4.

Simulation DB^{e}_{c} Optimal parameter values CE
Optimal parameter B_{cn}^{c} =500, B_{cp}^{c} =0 0.8375
Second best Parameters B_{cn}^{c} =450, B_{cp}^{c} =0 0.8375
Third best Parameters B_{cn}^{c} =400, B_{cp}^{c} =0 0.8369

Simulation CP_{b}^{e}

Optimal parameter B^{b}_{cn}=500, B_{cp}^{b} =500 0.8151
Second best Parameters B^{b}_{cn}=450, B_{cp}^{b} =500 0.8150
Third best Parameters B^{b}_{cn}=500, B_{cp}^{b} =450 0.8150

Simulation CP_{h}^{e}

Optimal parameter B^{c}_{cn}=500, B^{c}_{cp}=0, B_{cn}^{b} =500, B_{cp}^{b} =500 0.8459
Second best Parameters B^{c}_{cn}=400, B^{c}_{cp}=0, B_{cn}^{b} =500, B_{cp}^{b} =500 0.8458
Third best parameters B^{c}_{cn}=500, B^{c}_{cp}=0, B_{cn}^{b} =500, B_{cp}^{b} =400 0.8458
Table 4: Optimal parameters values for allocation method: DB_{c}^{e}, CP_{b}^{e}and CP_{h}^{e}
The lower the value of the parameters, the closer the allocation principle is to a linear
function. For high parameter values, the allocation method is more curved, which implies that
the change in either the contribution levels or benefit levels are minimal for relatively small
positive or negative financial shocks. These changes become larger for large financial shocks.

For the DB^{e}_{c} method, it follows that is it optimal to have a large curve in the function in
case of a negative financial shock, as B_{cn}^{c} =500 is optimal. It has no curve in case of a positive
financial shock and a possible funding surplus is distributed among the participants as a linear
function of the funding surplus. For the CP_{b}^{e}method, it follows that it is optimal to have a large
curve in the allocation function in case of both a positive and a negative financial shock. Finally,
for the CP_{h}^{e}method, it follows that this method is a combination of the optimal parameters for
the CP_{b}^{e} and DB_{c}^{e}method.

If we compare the optimal CE of all three methods with each other, it follows that the hybrid
method leads to the largest CE. The optimal value is 3.8% higher as the CP_{b}^{e}method and 1.0%

higher as the DB_{c}^{e}method. Finally, if we look at the funding ratio dynamics shown in Table5,
one could see that for the optimal parameter values, the funding ratio after 60 years (F R60) is
between 65% and 70.9% in the worst 5% of the scenarios. For completeness, the funding ratio
dynamics for the coming 60 years are shown in Figure15a,15cand15e, which could be found
in the appendix section10.3.1

Allocation method 5% Quantile F R60 mean F R60 95% Quantile F R60

Funding ratio for DB_{c}^{e}method 70% 196% 500.1%

Funding ratio for CP_{b}^{e} method 70.9% 148.1% 232.9%

Funding ratio for CP_{h}^{e} method 65.1% 107,6% 158,7%

Table 5: Funding ratio values after 60 years for the optimal parameters of the optimization results in Table4

4.3.3 Optimization plateau parameters

In Table6, one could find the certainty equivalent and the optimal, second and third best plateau parameter values. The simulation is based on an individual aged 25 and 1000 simulations are computed. One should note that in this simulation we assume that the funding ratio at the start of the period is 100%. The grid we used in the optimization of the plateau parameter values is shown in section10.3of the appendix. The optimization results can be found Table6.

Simulation DB_{c}^{p} Optimal parameter values CE

Optimal plateau Cd=0.0, Ci=-0.95 0.8322

Second best plateau Cd=0.0, Ci =-0.8375 0.8318 third best plateau Cd =0.0 , Ci = -0.725 0.8311

Simulation CP_{b}^{p}

Optimal plateau Bi=0.8, Bd=-0.9 0.8113

Second best plateau Bi=0.725, Bd= -0.9 0.8113 third best plateau Bi=0.875, Bd=-0.9 0.8112

Simulation CP_{h}^{p}

Optimal plateau Cd= 0 Ci= -0.9, Bi= 0.8, Bd = -0.95 0.8433
Second best plateau Cd= 0 Ci= -0.9, Bi= 0.85, Bd = -0.95 0.8432
third best plateau C_{d}= 0 C_{i}= -0.9, B_{i}= 0.9, B_{d} = -0.95 0.8432
Table 6: Optimal plateau value for allocation method DB^{p}_{c} CP_{b}^{p} and CP_{h}^{p}

From the optimization results of the DB_{c}^{p} allocation method, it is shown that the highest
certainty equivalent is reached, if the contribution decreases immediately by a funding ratio that
is above its target level of 110%. It only increases if the funding ratio is close to 0%. Although
on average this may lead to the optimal CE, there is a relatively large probability that the
funding ratio deteriorates to non preferable values. Figure15bin the appendix shows that this
allocation method implies that in the worst 5% of the scenarios, the funding ratio is lower than
35% after 60 years. Besides, the slope of the 5% quantile is still negative between 50 and 60
years, so it is expected that after 60 years the funding ratio will become even lower. Such an
allocation method is not sustainable for future generations and the balance between the optimal
CE and funding ratio dynamics should be taken into account.

From the optimization results of the CP_{b}^{p} method, one could see that it is optimal if the
benefits increase by a funding ratio close to 200% and decrease if the funding ratio is extremely
low, around 20%. In other words, it is optimal if the benefits are equal to the target benefits and
only change in extreme financial times. The reason for this result, is that we are looking at an
individual who is 25 years old. In the CP_{b}^{p} method, only the benefits are changing. Therefore,
for an individual aged 25, it is beneficial if the funding ratio is as high as possible after 40 years,
as this is the moment the participant retires and could benefit from changes in the benefit levels.

From the optimization results of the CP_{h}^{p} method, one could see that the optimal plateau
values are a combination of the optimal parameter values for the CP_{b}^{p} and DB_{c}^{p} allocation
methods. It is optimal to immediately decrease the contribution in case of a positive financial
shock, while an increase in the benefits or contribution and a decrease in the benefits only
happen if the funding ratio is close to 200% or 0%. Figure15fin the appendix shows that the
optimization result leads to a funding ratio close to 40% in the worst 5% of the scenarios. This
implies that although this plateau value is optimal, one could argue that it is not the preferred
plateau value, as it leads to an unsustainable pension scheme. This will be discussed into more
detail in the next section.