Welfare of a hybrid DB/PPR pension plan

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Welfare of a hybrid DB/PPR pension

plan

D. Ruiter

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: D. Ruiter Student number: 10003082

Email: d.ruiter92@gmail.com Date: July 15, 2017

Supervisor: dhr. dr. S. van Bilsen Second reader: S. van Stalborch

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Statement of Originality

This document is written by Dani¨elle Ruiter who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

The current Dutch pension system is under pressure due to increas-ing life expectancy and sensitivity to fluctuations in the market. Hence, there is a need for a new pension plan. In this thesis the welfare of a new pension contract has been investigated. This new pension contract is a hybrid pension contract that combines defined benefit (DB) and personal pensions with risk sharing (PPR). This hybrid pension contract offers policyholders more freedom and gives room for customization of the investment function. More specif-ically, the welfare of different options within the hybrid pension contract have been compared and investigated using the expected utility theory. In the hybrid pension plan a distinction has been made between the ambition and the required ambition. The am-bition indicates the percentage of the average wage a policyholder wants to receive after the retirement date. The required ambition is the minimum percentage of the average wage that a policyholder has to receive after the retirement date. The contribution needed to finance the required ambition is put into the defined benefit plan. The difference between the contribution needed for the pension am-bition and required amam-bition can partially or in whole be invested in a PPR. This choice lies with the policyholder. Any remaining premium is invested in the defined benefit plan. The results showed the expected consumption stream to be the highest when as much as possible of the premium is put into the DB pension plan. The policyholder would rather go for certainty than taking advantage of the equity risk premium. This changes if risk taking in the retire-ment phase is allowed. In that case the policyholder would rather put the premium in the PPR contract. This is because the DB ben-efit may increase in the retirement phase while the PPR benben-efit can only stay the same or decreases if risk taking in the retirement phase is not allowed.

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Introduction

The current Dutch pension system is under pressure. People are getting older and the current system is sensitive to fluctuations in the market due to a mismatch be-tween assets and the value of the liabilities (DNBulletin, 2013). The value of the liabilities heavily depend on the nominal interest rate, whereas the assets only par-tially depend on the nominal interest rate. As a result, assets and the value of the liabilities do not respond one to one, which causes the nominal funding rate to fluc-tuate heavily over time.

Currently, the Netherlands has an occupational defined benefit (DB) system. This means that an employer promises a predetermined nominal annuity based on the employee’s earnings history, tenure of service and age. The provider of such a pension is usually a pension fund. In order to guarantee such a benefit, a pension fund have to fully hedge interest rate risk. As a consequence, the policyholder is sub-jected to inflation risk (i.e. the real value decreases over the years). But policyholders want to be insured against inflation risk. Therefore, a pension fund has the ambition to index pension entitlements and benefits. To enable indexation they cannot fully hedge interest rate. If interest rate risk is fully hedged, the funding ratio will always be one and there is no money left for indexation. Therefore, when a pension fund aims to index the pension entitlements and benefits, there is a mismatch between assets and liabilities causing the nominal funding rate to fluctuate heavily over time. Dutch pension funds have to obey the Pension Act. This law includes the legal financial requirements that pension funds must adhere to. The law states that the funding ratio has to be at least 105%1. If the funding ratio is below 105% the pen-sion fund has to hand in a recovery plan. In a recovery plan a penpen-sion fund must set up a plan showing that they can have sufficient equity in ten years. As long as pension funds do not meet this threshold they submit a new recovery plan each year explaining how they will return the funding ratio to the required level within ten years. There are a number of things a pension fund can do to get the funding ratio again above 105%. For example, they can increase the contribution. Unfortunately, this is not very effective due to the fact that the value of total pension liabilities is large compared to total wages. Another possibility is to reduce the benefits, but the law demands that the pension reductions are capped at 7% per year (Rijksoverheid, 2012). Another problem is the eroding of the trust in pension funds since 2008. The financial crisis of 2008 caused pension funds to reduce the benefits. Most people were not aware of the fact that pension funds could do this, causing the people to lose trust in the pension funds (van Dalen et al., 2015). This lack of positive indexation caused pensioners to experience a backlog of inflation of more than 7% since 2008. Nowadays, the funding ratio still does not meet the financial requirements. There-fore, the benefits and entitlements will also probably not be indexed in the coming years and may even have to be cut further (Rabobank, 2016).

1_{Pension Act, Art. 28}

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Hence, there is a need for a new pension plan. The Social and Economic Council came up with four variants in order to strengthen the current system (Rijksoverheid, n.d.c). Van Riel, B. (2016) states that the variant Personal Pension with Risk Sharing (PPR), introduced by Bovenberg and Nijman (2015), is the most interesting variant. PPR is a pension plan that combines the strengths of DB and defined contribution (DC) pension plans. This pension plan integrates income adequacy, affordability and security in transparent and flexible ways, while tailoring pensions to the preferences of employees. This is done by unbundling the functions of a pension plan (i.e., in-vestment, draw-down and insurance) and customizing it to different stages of life. The investment function is a strength of the DC pension plan, where every poli-cyholder has personal ownership of their financial assets. This allows tailoring the investment policy to the specific needs of a policyholder. The drawdown function allows the policyholder to customize the withdrawal of their assets. The risk sharing function is a strength of the DB pension plan, which completes the financial market by trading systematic risks, such as macro longevity risk, with future contributors. In this thesis we study the welfare of a hybrid pension plan that combines de-fined benefit (DB) and personal pensions with risk sharing (PPR), and compare this with the current DB pension plan. In this hybrid pension plan a distinction is made between pension ambition and required ambition. In the case of premium agreements, there is usually an age-dependent contribution table. This contribution table has been grafted to achieve a certain average wage pension. Then the pension ambition is the percentage ambition that a policyholder wants to receive after the retirement date as a percentage of the average wage. The required ambition is the minimum percentage of the average wage that a policyholder has to receive after the retirement date. The contribution that is needed to achieve the required ambition is put into a DB pension plan. The difference in contribution between the contribu-tion needed for the pension ambicontribu-tion and required ambicontribu-tion can partially or all be invested in a PPR. This choice lies with the policyholder. The remaining premium will also be invested in a DB pension plan.

1.1 Thesis outline

In Chapter 2 an overview of different pension plans is provided. Section 2.1 discusses the Dutch three pillar pension system and Section 2.2 describes the personal pen-sions with risk sharing introduced by Bovenberg and Nijman (2015). In Section 2.3 the hybrid pension plan investigated in this thesis will be described qualitatively. Chapter 3 will laid out the model of the economy and models the hybrid pension plan. Section 3.1 will discuss the assumptions that are made that were necessary to set up the model. Section 3.2 will describe the financial market and Section 3.3 will describe mathematically the model of the new pension contract. Section 3.4 will describe mathematically the expected lifetime utility. In Chapter 4 the results of different possibilities of the hybrid plan will be given. Section 4.1 will show the results of the current DB system. Section 4.2 will show the results of the DB system underlying the hybrid pension plan. Then, in Section 4.3, the results will be showed of the hybrid pension plan. In Section 4.4 the results of the expected lifetime utility is given. Finally, Section 4.5 will examine the extent to which the results are influ-enced by some assumptions that are made. In Chapter 5 the conclusion of the thesis will be given. Finally, Chapter 6 will discuss the limitations of this thesis and will give some suggestions for future work.

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Chapter 2

Pension plans

This chapter gives an overview of different pension plans. First, the three pillar system of the Netherlands and its current problems will be discussed. Next, the variant personal pension with risk sharing is described. Finally, the hybrid pension plan consisting of a defined benefit (DB) and a personal pension with risk sharing (PPR) investigated in this thesis is presented.

2.1 The Dutch pension system

In the Netherlands the pension system consists of three pillars (Rijksoverheid, n.d.b). This section discusses these pillars, where the second pillar is especially important in this thesis. The first pillar is the state pension and is discussed in Section 2.1.1. Section 2.1.2 describes the second pillar, also called occupational pension. Finally, section 2.1.3 discusses the third pillar which is the private pension.

2.1.1 State pension

The first pillar is called the state pension (AOW). The main objective of this pillar is to prevent old-age poverty. This is done by giving a flat-rate payment, which is related to the poverty line, to all residents. Therefore, the benefit is not very high but the poverty under elderly is very low (OECD, 2015). It is financed on a pay-as-you-go (PAYG) basis, which means that the current workers and tax payers pay for the pensions of the elderly. This creates a redistribution within and across genera-tions. There is a redistribution within generations because the rich pay for the poor (i.e. the rich have contributed more into the system when they were active), which is called income-specific taxes and transfers. However, a maximum contribution is set by law. There is also a redistribution across generations because the young pay tax and the elderly get subsidy, this is called age-specific taxes and transfers.

Over the years, the AOW has become very expensive due to increasing life ex-pectancy and the decreasing ratio of workers to retirees. Therefore, the retirement age will increase to 67 years in 2021. In 2022, the retirement age is set at 67 years and 3 months. From 2022 and onwards, the retirement age depends on the life ex-pectancy of people. An increase in the life exex-pectancy automatically results in a rise of the retirement age with three months. Furthermore, the first pillar in the Netherlands is originally a defined benefit (DB) system. However, nowadays it is more a defined contribution (DC) system where you pay a fixed contribution but the outcome may vary.

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2.1.2 Occupational pension

The second pillar is called occupational pension. This pension contract is agreed upon by social partners in an industry or company with the main objective to main-tain the standard of living. This pension contract is mandatory for all employees in these industries or companies that offer such a pension contract.

Traditionally, the occupational pension is a defined benefit (DB) plan where the employer beares the risk of the plan. Most of these pension plans are average wage schemes. In the current situation an employee accrues maximally 1.875% of his wage each year, which results in a benefit of 75% of his average wage (Rijksoverheid, n.d.c). Furthermore, it is financed by financial assets. This means that the employees pay contributions. These contributions are invested into, e.g., the stock market, in-frastructure and bonds. As a consequence, each employee invest in the same assets. When an employee reaches the retirement age they receive a predetermined nomi-nal annuity. Contributions can be actuarially determined by discounting the annual accrual of benefits, but there is an uniform contribution rate and uniform accrual rate. This means that the benefits are backloaded: young employees pay too much and the older employees too little. This implies a redistribution across generations. As mentioned before, the law requires that the funding ratio of the pension fund is always at least 105%. If the funding ratio is less than 105%, the pension fund has to hand in a recovery plan. The pension fund has several policy instruments to get the funding ratio again above 105%, such as the indexation policy, premium policy and investment policy. The pension fund has the ambition to increase the entitle-ments and benefits with respectively the wage or price inflation. To what extent this is possible depends on the funding ratio. The funding ratio at the end of a year determines whether or not to index in the following year. The indexation policy in-dicates what the minimum funding ratio should at least be for indexation and what the funding ratio is for full indexation. Therefore, indexation is conditional. It is also possible that indexation is negative which means that the benefits are reduced. An-other policy instrument is the premium policy. The premium policy indicates that the premium can be increased and decreased if necessary. When the funding ratio is too low, premiums can be raised and vice versa. The last policy instrument, the investment policy, indicates how much is invested in the stock market, infrastructure or bonds respectively.

Nowadays, a lot of employers are withdrawing as a sponsor because of the in-creasing risk and expenses (Bovenberg et al., 2014). This causes a shift from DB to DC pension plans. In a defined contribution (DC) pension plan the policyholder is committed to a specific premium and can choose the investment strategy. By this freedom of choice the investment risk in the period before the retirement date lies with the policyholder itself. The pension benefit is derived from the returns on the premiums. Therefore, the pension benefit is not fixed until the retirement date. At the retirement age the total amount saved is converted into an immediate lifelong old-age pension.

Since September 1, 2016 there is new legislation which allows risk taking in the retirement phase and allows choice options on how much to decumulate in a given year (Rijksoverheid, n.d.c). This withdrawal policy depends on the Assumed Interest Rate (AIR). The law requires the AIR to be less than the expected return on the in-vestment portfolio. The law also requires the equity risk exposure in the inin-vestment portfolio that is used to get a higher AIR to be capped at 35%. Otherwise there is a risk there might be nothing left when the policyholder is old.

A disadvantage of a DC pension plan is the focus on wealth accumulation while the focus should be on old-age insurance. Another disadvantage is people making poor investment decisions resulting in suboptimal investment strategies (Lusardi et al., 2011).

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The hybrid DB/PPR pension plan — D. Ruiter 7

2.1.3 Private pension

The third pillar is the private or personal pension. The main objective of this pillar is to taylor pensions to specific individual circumstances. Individuals can establish their own life insurance contract or DC scheme at an insurance company. This is used by individuals who are not covered by the second pillar or individuals wanting extra pension income. The private pension is funded and actuarial fair. The contribution is mostly invested in an investment fund. This type of pension is tax favoured and has no link with the employer.

2.2 Personal pension with risk sharing

Bovenberg and Nijman (2015) introduced the Personal Pension with Risk shar-ing (PPR). This pension contract unbundles the three main functions of pension contracts (i.e. investment, withdrawal and insurance) and customizes these various functions to the different stages of life. In particular, the investment and withdrawal function are individualized and agreements are made about risk sharing. Therefore, it is a flexible contract that takes into account the preferences of policyholders.

In a PPR the policyholder has a personal investment account with easy-to-value financial assets. This implies objective valuation, because it is clear how much money the policyholder owns. This also implies personal risk management. The investment portfolio can be subdivided into an intertemporal hedging portfolio and a specu-lative portfolio. The hedge portfolio is the portfolio that hedges interest rate risk in the annuity factor and produces a return equal to the risk-free interest rate in order to achieve a stable income stream after retirement. The speculative portfolio is the portfolio where a policyholder can take advantage of the bond and equity risk premium.

Furthermore, PPR allows for customized withdrawal of the individual assets in the retirement phase. The withdrawal is defined as the individual assets divided by the conversion factor, where the conversion factor depends on the AIR. In other words, the withdrawal is the amount deducted from the individual assets used for consumption and the AIR is the discount rate where a larger AIR implies a higher withdrawal. In order to be able to customize the withdrawal, the AIR must be cus-tomized.

Finally, pension plans are a combination of investment and insurance. In a PPR the policyholder is insured for idiosyncratic longevity risk by generating biometric returns when a policyholder lives longer than average. When a policyholder passes away the remaining assets are left to the insurer rather than to their heirs and distributed among the policyholders. The risk sharing function of a PPR allows to exchange systematic risks that are not traded on financial markets by internal swap agreements. Lastly, there is no counterparty risk from the insurer in a PPR.

2.3 Hybrid pension plan

A possibility for a new occupational pension plan is a hybrid pension plan that combines a DB with a PPR pension plan. In this pension plan there is a pension ambition that indicates the percentage of the average wage a policyholder wants to receive after the retirement date. On the other hand, there is a required ambi-tion. The required ambition is the minimum percentage of the average wage that a policyholder has to receive after the retirement date. The contribution needed to finance the required ambition is put into the DB plan. The difference between the contribution needed for the pension ambition and required ambition can partially or

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in whole be invested in a PPR. This choice lies with the policyholder. Any remaining premium is invested in the DB plan. For example, suppose that full utilization of the fiscal area of 75% average wage results in a premium of 16% of the pensionable salary. And suppose a premium rate of 13% is sufficient to achieve 65% average-wage pension. Then the policyholder has the choice to invest a part or all of this 10% average wage, i.e. 3% premium, in the PPR pension plan.

This pension contract offers policyholders more freedom and gives room for cus-tomization of the investment function. Another advantage a contract with a lower ambition, the required ambition, offers is it can be better communicated. This is because the ambition that is communicated to the policyholder is the required am-bition and the probability that the pension on the retirement date will be higher than this communicated ambition is large. The challenge will be in putting the right defaults to avoid choice stress and suboptimal choices.

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Chapter 3

Modelling

This chapter describes a model for the hybrid pension contract in order to investi-gate the consumption stream and the impact of decision making by the policyholder. Section 3.1 specifies the assumptions that are made regarding the starting situation and the policyholders of the pension fund. Section 3.2 describes the financial mar-ket. Section 3.3 introduces the new pension contract, where Section 3.3.1 and 3.3.2 describe the model for DB and PPR, respectively. Finally, Section 3.4 discusses the theory used to compare the consumption streams.

3.1 Assumptions

In this section the economic assumptions are provided. Assumptions have been made regarding the starting situation and the policyholders of the pension fund.

The pension fund starts January 1, 2017, which is stated to be t = 0. Policy-holders are born on January 1st. Entering the pension fund and retiring only occurs at the change of a year. The age at which a policyholder enters the pension fund x0 is assumed to be 25. The retirement age R is 67 and we assume there is no partial retirement. According to the CBS1, the current life expectancy for someone who is born right now is 81.54 years. Dowd, J.B., and Hamoudi, A., (2015) state that higher educated people tend to live longer and because there are less lower educated people in the pension fund than the CBS took into account in their calculation, the life expectancy D is rounded up to 85 years. Furthermore, there is no micro longevity risk due to the law of large numbers and we abstract away from macro-longevity risk.

As a result, the pension fund has 60 policyholders, one for each year. Each pol-icyholder had an initial labor income S₀0 of 20.000 at the age of x0. Subsequently, this labor income has grown over the years according to an assumed career path cx for every x < R. The assumed career path is defined as follows:

cx= (

0.01, for x0 ≤ x < x0+ 20; 0.005, for x0+ 20 ≤ x < R.

(3.1)

Furthermore, we suppose that on January 1st the annual labor income over the coming year is known. Also, at any time t the variable S_tx0 is defined as the annual salary of a policyholder at age x. The labor income S_tx0 of policyholders after t = 0 is also adjusted for wage inflation πwage. Consequently, the labor income of the policyholder with age x < R at t1 will be higher than the labor income of the policyholder with the same age x < R at t0 < t1:

S_tx₁0 = S_tx₀0(1 + πwage) > S_tx₀0. (3.2)

1_{CBS is the Dutch Central Agency for Statistics}

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It is also assumed that the labor income is risk-free, which means that the policy-holder always has a labor income during the working phase.

Not the entire labor income of the policyholder is used, but only the pension-providing salary. Occupational pension schemes are considered to be supplementary to the AOW. Therefore, AOW is included when calculating the occupational pension benefit. AOW and the occupational pension benefit together should be 75% of the average wage. The part of the income over which no additional pension entitlements are accrued, because it is expected to be covered by the AOW, is known as the offset (Rijksoverheid, 2013). The offset at time 0, which we call Offset0, is 8.000 and is adjusted for the price inflation πprice after that time. Then the pensionable income S_tx is the labor income S_tx0 minus the social security offset Offsett. In summary, the pensionable income at time t ≥ 0 for a policyholder with age x ≥ x0can be described as: S_tx=      S₀25(1 + πwage)t· x−1 Y i=25

(1 + ci) − Offset0(1 + πprice)t, for x < R;

0, for R ≤ x < D.

(3.3)

Finally, the wage and price inflation are deterministic and assumed to be 3% and 2% respectively (Rijksoverheid, 2014). Table 3.1 provides a summary of the values of the parameters mentioned in this section.

Variable Symbol Value

Entering age x0 25

Retirement age R 70

Time of death D 84

Initial salary S₀0 20.000 Offset in year 0 Offset0 8.000 Careerpath x0–(x0+20) cx 0.01 Careerpath (x0+20)–R cx 0.005 Wage inflation πwage 0.03 Price inflation πprice 0.02 Table 3.1: Summary economic values

3.2 Financial market model

This section describes a model of the financial market. We assume that the interest rate is fixed and there is one risk factor, namely the stock price index St. Further-more, we assume that the pension fund invests in a Black-Scholes-Merton financial market. There are several assumptions underlying the Black-Scholes model, which can be found in Appendix C. This financial market consists of two assets: a risk-free asset and a risky asset. The value of the risk-free asset at time t is given by Bt and can be described as the price evolution of a bank account with a rate of return rf. We assume that Bt satisfies the following dynamic equation:

dBt Bt

= rfdt, (3.4)

where rf denotes the risk-free interest rate and B0 the value of the bank account at time 0 is given. We assume B0 to be equal to 1.

The risky asset is the stock price index and can be described by a geometric Brownian motion. We assume that St satisfies the following dynamic equation:

dSt St

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where µ denotes the expected rate of return on the stock price index and σ the volatility of the stock returns, Z = {Zt|t ≥ 0} represents a standard Brownian motion and S0 > 0, the value of the stock at time 0, is given. The solution of these differential equations can be found by using Ito’s Lemma which is shown in Appendix E. Furthermore, µ has to be greater than rf. Otherwise it is not interesting to invest in the risky asset. According to the Commission parameters, µ is equal to 7% and σ equals 20% (Rijksoverheid, n.d.a). Furthermore, we assume S0 to be equal to 1.

3.3 New pension contract

This section describes the model of the hybrid pension contract. In this pension con-tract, a premium π of the pensionable salary is paid at the beginning of every year. Part of this premium π1 is intended for the required ambition. Then a premium of π - π1 remains to be partially or all be invested in a PPR. The percentage of this premium that is invested in a PPR is denoted by γ. The policyholder chooses this percentage when entering the pension fund and is assumed to be the same for every policyholder. The contribution paid in a DB and PPR by a policyholder with age x < R at time t can be described as follows:

cx_t = (

π1Stx+ (1 − γ)(π − π1)Stx, in DB; γ(π − π1)Stx, in PPR.

(3.6)

3.3.1 Defined benefit with conditional indexation

In a defined benefit plan a predetermined nominal annuity is promised based on the employee’s earnings history, tenure of service and age. These defined benefit plans can be subdivided into average wage, final wage and fixed amount plans. At an average wage scheme, the accrued entitlements for retirement grow each year with an accrual percentage, ap, times the annual pensionable salary. In this thesis we assume an average-wage defined benefit, because almost 98% of the policyholders in defined benefit schemes have an average wage scheme (OECD, 2015). The accrual percentage is calculated as follows:

ap% = β1+ (1 − γ) · (β − β1) dmax

, (3.7)

where dmax is the maximum service time to be achieved at the given retirement age, β is the ambition of the pension plan and β1 is the required ambition of the pension plan.

At an average wage scheme the accrued pension entitlements are heavily sub-jected to inflation. By indexing for inflation the accrued pension entitlement keep their value, this is called a defined benefit pension plan with conditional indexation. For indexation of the entitlements we use the symbol ϕ. We assume that indexation is granted only once a year at the beginning of the year. The indexation policy is assumed as follows: ϕt=        0%, for Ft ≤ Fmin; Ft−Fmin Fmax−Fmin

· 100%, for Fmin < Ft < Fmax; 100%, for Ft ≥ Fmax,

(3.8)

where Fmin and Fmax are the policy instruments of the pension fund and Ft the funding ratio. The funding ratio is calculated by dividing the assets by the liabilities postnumerando:

Ft= At−1 Lt−1

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The policyholders at time 0 have accrued benefits for dx years, where dx is the service time at age x. We assume there is no indexation up to and including t = 0 for the active policyholders and full indexation for price inflation for the retirees. Hence, the accrued benefits at time 0 are:

bx,DB₀ =          ap% · x−1 P i=x0 Si₀, if x < R; ap% · R−1 P i=x0 Si₀· (1 + πprice₎x−R_, _{if R ≤ x < D.} (3.10)

In the years t > 0, the policyholder accrues ap% of the pensionable labor income. Subsequently, the aggregate accrual is, depending on the real funding ratio, indexed for wage inflation before retirement and for price inflation after retirement. Given the funding ratio and the indexation policy the benefit at t > 0 can be described as follows:

bx,DB_t = (

bx−1_t−1(1 + ϕt· πtwage) + ap% · Stx, for x < R; bx−1_t−1(1 + ϕt· πtprice), for R ≤ x < D.

(3.11) In the defined benefit pension plan with conditional indexation the benefits will be paid from the collective assets. If the assets are not sufficient to compensate for the current and future benefits, indexation can be negative. This implies an increase in the contribution. This is the case if the funding ratio is lower than a certain threshold which we assume to be 85%. If this is the case, the surplus SPt will be smaller than 0 and is defined as follows:

SPt= At−1− Lt−1, (3.12)

where the assets and the value of the liabilities are postnumerando. This deficit will be divided among the active policyholders, where a maximal increase of the contribution Imax is set at 1. In summary, the contribution paid in the DB pension plan by a policyholder at time t can be described as follows:

cx_t = ( π1Stx+ (1 − γ)(π − π1)Stx− max(Qnt, −1 · Imax· c x t), for x < R; 0, for R ≤ x < D,(3.13)

where n is the number of active policyholders.

The pension fund collects premium and invests in a Black-Scholes-Merton finan-cial market, with a fraction α in the risky asset. As a result, the assets of the pension fund at the beginning of time t will be the assets at time t − 1 plus return, plus all contributions paid by the active policyholders, minus all benefits paid to the retired policyholders. In summary, the dynamics of the assets of the pension fund can be described as follows: dAt= (rf+ α(µ − rf))Atdt + ασAtdZt+ R−1 X x=x0 cx_tdt − D−1 X x=R bx_tdt. (3.14) Furthermore, at time 0 we assume a funding ratio of 100%. This implies that the value of the liabilities are equal to the assets at time 0 (i.e. A0 = L0). The liabilities at time t are defined as the sum of all the individual liabilities:

Lt= D−1

X

x=x0

Lx_t, (3.15)

where Lx_t is the value of the liability of a policyholder with age x at time t and is defined as follows:

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where dx_t is the discount factor defined as:

dx_t = (

R−x|¨axt, for x0 ≤ x < R; ¨

ax_t, for R ≤ x < D. (3.17) Due to the fixed life expectancy D, this is the sum of a geometric progression. Therefore, it can be written in closed form. The derivation can be found in Appendix D.

3.3.2 Personal pensions with risk sharing

As mentioned before, the percentage of the premium π - π1 that is invested in a PPR is denoted by γ. The policyholder chooses this percentage when entering the pension fund and is assumed to be the same for every policyholder. The contribution that is put into the PPR by a policyholder with age x < R at time t can be described as follows:

cx_t = (

γ(π − π1)Stx, for x < R;

0, for R ≤ x < D. (3.18)

This contribution can be seen as wealth accumulation. Subsequently, the policy-holder invests his wealth in a risky stock by a certain percentage αx, which depends on the age of the policyholder. This fraction is based on the life-cycle investment strategy (Blake et al., 2013). This strategy implies αx _{to be high when the} policy-holder is young and declines until retirement age is reached, after which it is zero. We assume that a x0-year-old policyholder invests 70% of his wealth in a risky stock and this percentage decreases linearly until the pensionable age. We also assume that the total wealth in PPR satisfies the following dynamic equation:

dW_tx = (rf+ rxb + αx(µ − rf))Wtxdt + αxWtxdZt+ cx,PPRt dt − b x,PPR

t dt (3.19) where r_bx ≥ 0 represents the biometric return of a policyholder at age x, and bx,PPR_t denotes the annuity payment at time t to a policyholder at age x. We assume rx_b to be deterministic and equal to 1%.

At retirement age, an individual has to convert the accrued wealth into an an-nuity with actuarial anan-nuity factor ¨ax_t. This factor is calculated as follows:

¨ ax_t = D−x X t=0 tpx (1 + rf)t , (3.20)

wheretpx is deterministic and equal to 1 for all values except when x + t is equal to D. If x + t is equal to D thantpxis equal to 0, which means that the probability the policyholder survives that year is zero. The amount that eventually can be drawn down after retirement depends on the draw-down function in the pension contract and is defined as follows:

bx,PPR_t = ( 0, for x < R; Wx t ¨ axt , for R ≤ x < D. (3.21)

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3.4 Welfare analysis

A portion of the salary remains after paying the premium for the pension plan. This portion can be used for consumption. After the retirement date, the policyholder receives the corresponding benefit of the DB and PPR pension plan for that year for consumption. Therefore, the consumption stream is defined as follows:

Cx = (

S_tx0 − πSx

t, for x < R;

bx,DB_t + bx,PPR_t , for R ≤ x < D. (3.22) The level of consumption depends, among other things, on the choice of γ. In this thesis we simulate the hybrid pension contract for a given value γ. This is done nsim times and nperiods number of periods. This results in nsim consumption streams for each policyholder. This process is repeated for different values of γ.

3.4.1 Utility function

In order to compare the consumption streams we use the expected lifetime utility. The expected lifetime utility of a pension plan is the expected value of the discounted sum of the utilities. Therefore, the expected lifetime utility of a consumption stream can be described by:

U = E h_XD t=0 e−δtu(Ct) i = D X t=0 e−δtu(ce). (3.23)

Here δ stands for the subjecive rate of time preference, ce is the certainty equivalent and u(·) is the constant relative risk aversion (CRRA) utility function described by:

u(Ct) = ( ₁ 1−ϑC 1−ϑ t , for ϑ ∈ (0, ∞) \ {1}; log(Ct), for ϑ = 1, (3.24)

where ϑ is the coefficient of relative risk aversion. We assume δ to be equal to rf and ϑ to be equal to 7 (O. Johansson-Stenman, 2010).

The properties of this utility function are:

u0(Ct) > 0; (3.25)

u00(Ct) < 0. (3.26)

The marginal utility u0(Ct) is positive which means that the utility increases with consumption (i.e. more consumtion means higher utility). Diminishing marginal util-ity u00(Ct) is negative, which implies that the utility function is concave. A concave utility function means that the marginal utility diminishes for each additional unit consumed.

The pension plans are compared by looking at the certainty equivalent. The cer-tainty equivalent of a stochastic consumption stream is defined as the amound ce such that the agent is indifferent between the stochastic consumption stream and receiving ce with certainty.

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Chapter 4

Welfare analysis

The consumption stream of the hybrid pension plan is investigated for different values of γ and β1. First, the current DB pension system is investigated. In this system γ equals 0 and the required ambition β1 is equal to 75% average wage. Then the DB pension plan that will be communicated to the policyholder in the hybrid pension contract will be analyzed. This is called the base of the hybrid pension plan. In this pension plan γ also equals 0, but the required ambition β1 is equal to 65% average wage. After that, the hybrid pension system is investigated in which the values of γ range from 0% to 100% in increments of 20%, where a γ equal to 0% corresponds to the current DB pension system.

In order to investigate these pension contracts, the financial market is modelled with the Black-Scholes-Merton model mentioned in Section 3.2. Furthermore, we assume there is one risk factor given by the stock price index according to equation 3.5. The values of the parameters in our Black-Scholes-Merton model are summarized in Table 4.1. A summary of the values of the remaining parameters can be found in Appendix B.

Parameters Symbol Value

Real interest rate rf 0.02 Wage inflation πwage 0.03 Price inflation πprice 0.02 Initial value bank account B0 1.00 Initial stock price S0 1.00

Mean stock return µ 0.07

Volatility of stock price σ 0.20

Table 4.1: Value of the parameters in the Black-Scholes-Merton model The structure of this chapter is as follows. In Section 4.1 the results of the current DB pension contract are given. The results of the base DB pension contract underlying the hybrid pension plan are given in Section 4.2. Section 4.3 gives the results of the hybrid pension plan. Section 4.4 will compare all the results by using the expected lifetime utility mentioned in 3.4.1. Finally, in Section 4.5 the results of the sensitivity analysis are discussed.

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4.1 The current DB pension plan

In this section the current DB pension plan is investigated. The current DB pension plan is an average wage scheme which results in a benefit of 75% average wage assumed that the policyholders work for 45 years. As a consequence, the accrual percentage is equal to 1.67%. The value of the parameters relating to this pension contract are summarized in Table 4.2.

Pensioncontract Symbol Value

Ambition β 0.75

Required ambition β1 0.75

PPR γ 0.00

Premium π 0.10

Table 4.2: Value of the parameters in the current DB pension plan

Figure 4.1a shows the expected consumption stream of a policyholder when he enters the pension fund up until the moment he dies and Figure 4.1b shows a close up of the retirement phase. The initial policyholders can influence the results because of the initial assumptions made. Therefore, we only took participants into account after time t > 60. This way the initial participants are out of the system. Furthermore, the results are expressed in percentage consumption of the average wage throughout the active and retirement phase.

Figure 4.1: Expected consumption stream in current DB pension plan

(a) Active and retirement phase (b) Retirement phase

The 5% percentile is the value below which 5% of the consumption may be found and can be seen as the least favorable outcome. The figures show the 5% percentile to be precisely the required ambition of the pension plan. Thus, in the worst scenario the benefits are never indexed for inflation resulting in a benefit exactly equal to the guaranteed benefit. The 95% percentile is the value below which 95% of the consumption may be found and can be seen as the most favorable outcome. The figures show the 95% percentile to be an increasing and almost linear line. This means that the pension fund is able to give indexation every year. This results in a higher percentage consumption of the average wage at the retirement age and increases even further after that time.

The expected consumption stream lays in between these two lines. The figures show the expected consumption to be 75.11% average wage at the retirement age and could be sometimes indexed after that. It is expected that in 12 of the 15 years of retirement indexation is given with a mean of 0.14% resulting in an expected consumption of 76.69% average wage at the age of 84 (i.e. one year before death). Figure 4.1b also shows the consumption stream if the expected consumption on

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the retirement age was fully indexed each year. From that, we can conclude that the policyholder can expect to experience a backlog of inflation. Table 4.3 shows the expected backlog of the ages 70, 75, 80 and 84. It can be seen the backlog increases by approximately 8% every five years. In the case of the upper 5% (i.e. 95% percentile) the policyholder does not experience any backlog of inflation. This means there is a probability the policyholder does not experience any backlog of inflation in the current DB pension plan. On the other hand, the policyholder experience full backlog on inflation in the lower 5% percentile which means that the value of the benefits decreases each year with the price inflation.

Age Backlog (%)

70 0.00

75 8.72

80 16.72 84 22.63

Table 4.3: Expected backlog in current DB pension plan

4.2 The base of the hybrid pension plan

A feature of the hybrid pension contract is having lower required ambition than the current DB pension contract. In our model we assume the required ambition β1 of the hybrid pension plan to be 65% average wage instead of the current 75%. As a result, the accrual percentage is equal to 1.44%. We call this the base DB pension contract underlying the hybrid pension plan. In this section we investigate this base DB pension contract and compare it with the current pension plan. The value of the parameters relating to this pension contract are summarized in Table 4.4.

Ambition β 0.65

PPR γ 0.00

Premium π 0.10

Table 4.4: Values of the parameters in the base DB pension plan

Figure 4.2a shows the expected consumption stream of a policyholder in the base DB pension contract when he enters the pension fund up until he dies and Figure 4.2b shows a close up of the retirement phase. These figures show the expected consumption to be 65.20% average wage at the retirement age and 67% average wage at the age of 84. Therefore, we can conclude that the expected increase of the benefit in the base DB is greater than in the current DB. This implies the probability of the benefit to be higher than the communicated benefit is larger in the hybrid pension plan than in the current DB pension plan.

The figures also show benefits are indexed each year with an average of 0.20%. This is the result of paying the same premium as in the current DB pension plan but getting a smaller percentage guaranteed benefit. Therefore, the pension fund has less liabilities but the same amount of assets. This results in a higher funding ratio, which makes it possible to index more frequently and give higher indexation. Table 4.5 gives a summary of the results of the base and current DB pension plan.

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Figure 4.2: Expected consumption stream in base hybrid pension plan

(a) Active and retirement phase (b) Retirement phase

Consumption (%) Base Current Retirement age 65.20 75.11 5% 65.00 75.00 95% 66.30 76.50 D − 1 years old 67.00 76.68 5% 65.00 75.00 95% 87.48 99.12 Indexation Base Current number of times 15/15 12/15

mean value 0.20 0.14

Table 4.5: Summary of results of the base and current DB

Although indexation is given more often and at a higher level, it is not enough for participants to not experience a backlog of inflation. Table 4.6 shows the expected backlog of the ages 70, 75, 80 and 84. In the upper 5% consumption streams the policyholder does not experience any backlog on inflation. On the other hand, in the lower 5% consumption streams the policyholder experience full backlog on inflation. This is the same as in the current DB.

Age Backlog (%)

70 0.00

75 8.06

80 15.50 84 21.01

Table 4.6: Expected backlog in base DB pension plan

4.3 The hybrid pension plan

In this section the hybrid pension plan is investigated for different values of γ. It is assumed that the ambition of the contract is the same as in the current DB pen-sion contract. Therefore, the penpen-sion ambition β is equal to 75% average wage. The required ambition is assumed to be 65% average wage. This is also the percentage average wage that is communicated to the policyholders. As a result, the contribu-tion of 10% average wage remains and must be subdivided into the DB and PPR pension plan. In the model we assume that the values of γ are ranged from 0% to 100% with increments of 20%, where a γ equal to 0% corresponds to the current DB

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pension system. The values of the parameters relating to the pension contract are summarized in Table 4.7.

Pension contract Symbol Value

Ambition β 0.75

PPR γ 0 – 100

Premium π 0.10

Table 4.7: Values of the parameters in the hybrid pension plan

Figure 4.3 shows the expected consumption stream of a policyholder in the re-tirement phase for different values of γ and the expected consumption stream of the base DB pension contract underlying the hybrid pension plan. The figure shows the expected consumption stream is the largest with the lowest possible value for γ. This is because the more premium is put into the DB pension plan, the larger the guaranteed benefit will be. Furthermore, the pension fund beares more risk in the DB pension plan than the policyholder. In particular, if a loss is made on the risky asset (i.e. the stock return is smaller than the risk-free interest rate) the benefit in the DB pension plan remains the same. This is because, in case the loss causes the funding ratio to drop below 85%, the contribution of the working policyholders in-creases instead of the benefits of the retirees being cut. In the PPR system this loss can be seen directly in the benefit, because this loss is paid by policyholder itself. Therefore, we can conclude that the policyholder would rather go for certainty than taking advantage of the equity risk premium.

Figure 4.4 shows the 5% and 95% percentile of the consumption stream in the hybrid pension plan for different values of γ, respectively. These figures show the same result as the expected consumption stream. Therefore, we can conclude that the consumption stream is always the largest with the lowest possible value for γ.

Figure 4.3: Expected consumption stream hybrid pension plan

In addition, we look at the ratio between the expected consumption of the hybrid pension plan with γ equal to 100% and the communicated 65% average wage DB pension plan. This ratio is interesting because, from the point of view of the pension fund, they want to guarantee a benefit as low as possible. In order to still get a 75% average wage, the contribution of 10% average wage DB is put into the PPR. Figure 4.5 shows this ratio. This ratio declines when the policyholder gets older. As explained in Section 4.2, the premium that is put into the DB plan with 65% average wage is just as high as in the other pension plans, which results in a higher funding ratio making it possible to index the benefits more often and giving higher indexation. On the other hand, in the hybrid pension plan with a γ equal to 100%

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Figure 4.4: Consumption stream of hybrid pension plan

(a) 5% percentile (b) 95% percentile

the required ambition is also 65% average wage, but less premium is paid for this required ambition resulting in less indexation. Furthermore, in the PPR plan there are no investments being made in the risky stocks during the retirement phase. In addition, the biometric return is not high enough to give the same effect as the stock return. Therefore, the DB benefit increases over time while the PPR benefit does not. This is an explanation why the policyholder would rather put the premium in the DB pension plan than in the PPR.

Figure 4.5: Ratio γ=100% and base DB

Furthermore, Figure 4.6 shows the ratio between the expected consumption of γ and γ+20%. This figure again shows that the consumption will be higher when more premium is put into the DB pension plan. This ratio becomes larger when the value of γ increases. We can conclude that the impact of choosing a γ of 20% instead of 40% is less than the impact of choosing a γ of 80% instead of 100%. This is because the indexation in absolute terms has a larger effect on higher benefits resulting in a higher ratio.

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Figure 4.6: Ratio

4.4 Utility analysis

In this section the consumption streams are compared by using the expected lifetime utility mentioned in Section 3.4.1. Table 4.8 shows the certainty equivalent, the 5% and the 95% percentile certainty equivalent as ratio of the initial salary for a given γ. The certainty equivalent can be seen as a measurement of preferences, therefore the policyholder prefers the model with the highest certainty equivalent. As expected, Table 4.8 shows this is the case when γ is equal to zero.

γ (%) Certainty equivalent 5% percentile 95% percentile

0 1.18 1.17 1.32 20 1.16 1.15 1.30 40 1.15 1.12 1.29 60 1.13 1.11 1.27 80 1.11 1.09 1.26 100 1.10 1.07 1.24 Base DB 1.03 1.01 1.15

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4.5 Sensitivity analysis

In this thesis we made some economic assumptions and assumptions about the value of the parameters in the Black-Scholes-Merton model. These assumptions may have an affect on the results. Therefore, we examine the extent to which the results are influenced by some assumptions. The assumptions made about the age of death D and the retirement age R are reconsidered and investigated with another value. Fur-thermore, the impact of the new legislation which allows risk taking in the retirement phase is also investigated.

4.5.1 Age of death

The current Dutch pension system is under pressure, partly because people are getting older. As mentioned in 3.1, the current life expectancy for someone who is born right now is 81.54 years. In our model we assumed the age of death to be 85 years. Due to the increasing life expectancy, this may increase even further. Therefore, we investigate the model with the age of death D equal to 95. This way a DB pension fund will get less premium relative to the duration of the guaranteed benefit, but this also applies to the PPR contract.

Figure 4.7: Consumption stream of DB and PPR

(a) Consumption stream of DB (b) Consumption stream of PPR

Figure 4.7 shows the DB and PPR consumption stream in percentage of the average wage, respectively. Figure 4.7a shows the benefit to be exactly the required benefit. This implies there is no indexation in the retirement phase. Figure 4.7b shows the percentage average wage is relatively low in the PPR contract. This is all due to the higher age of death, which means that the premium is not high enough to compensate for this increased age of death. Therefore, we can conclude that the increasing age of death would still be a problem in the hybrid pension contract.

4.5.2 Retirement age

Traditionally, the retirement age is 65 years. Over the coming years, this retirement age will increase to 67 years in 2021. In 2022, the retirement age is set at 67 years and 3 months. From 2022 and onwards, the retirement age depends on the life expectancy of people. An increase in the life expectancy automatically results in a rise of the retirement age by three months. This raises the question whether this would also be necessary in the hybrid pension contract. Therefore, we investigated the model with the retirement age R to be equal to 65, while the age of death D remains 85 years.

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(a) Consumption stream of DB (b) Consumption stream of PPR

Figure 4.8 shows the DB and PPR consumption stream as a percentage of the average wage. The benefit of the DB pension plan is equal to the required ambition. This means that the pension fund is not able to give indexation if the retirement age is equal to 65 years. In order to index the benefits, the premium should be increased or the required benefit reduced.

The PPR benefit in Figure 4.8b is smaller in comparison with Figure 4.7b, where the retirement age was equal to 70 years and the age of death equal to 95 years. Therefore, we can conclude that the impact of a lower retirement age is greater than a higher age of death. An explanation is that the labor income is the highest in the last five years before retirement. Therefore, if the retirement age is 65 years, the wealth of a participant is less than when the retirement age is 70 years. In addition, the wealth only grows with the risk-free interest rate for this difference of 5 years. This effect is stronger than when the life expectancy is 10 years more.

Another thing we investigated is the effect of a retirement age R equal to 75 years. In this case, the pension fund receives 50 years of premium of each policyholder and has to pay benefits for 10 years to each policyholder. Figure 4.9 shows the DB and PPR consumption stream as a percentage of the average wage. The figure shows full indexation is given in the base DB contract. The figure also shows the percentage of the indexation to be higher in comparison with the case where the retirement age is equal to 70 years. The same applies to the PPR benefit. This is due to receiving premium five years longer, resulting in a higher funding ratio.

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4.5.3 Investing after retirement

Since September 1, 2016 there is new legislation which allows risk taking in the retirement phase. Therefore, we investigate the impact of this new legislation. We do not want the retired person to take more risk than the pension fund. Therefore, we assume the participant to invest 20% of his wealth in the risky stock.

Figure 4.10: Expected consumption stream

Figure 4.10 shows that by allowing risk taking in the retirement phase the optimal choice for γ is 100%. The expected consumption is larger at the retirement age than in the original hybrid pension plan. In the original hybrid pension plan the fraction invested in equity is linear decreasing from 70% at the entering age to 0% at retirement age. In a hybrid pension plan, that allows risk taking in the retirement phase, the fraction invested in equity is linearly decreasing from 70% at the entering age to 20% at retirement age after which it remains 20%. As a result, the policyholder invests more in average in equity in the working phase. This results in a higher expected consumption at retirement age. Furthermore, the benefit after retirement increases more than in the original hybrid pension plan. In addition, the increase is larger as γ increases. This is because the absolute effect of a high stock return is larger when a policyholder has more wealth, resulting in a higher increase relative to a lower wealth. We also investigate the effect of the original hybrid pension plan where the fraction invested in equity is linearly decreasing from 70% at the entering age to 20% at retirement age. In this pension plan the expected consumption stream is still the highest by choosing a γ as low as possible. Therefore, we can conclude that the aspect of risk taking in the retirement phase has a lot of effect on the result.

Figure 4.11: Consumption stream

(a) 5% percentile consumption stream (b) 95% percentile consumption stream

Figure 4.11 shows the 5% and 95% percentile of the consumption streams, re-spectively. In the 5% percentile the consumption streams barely increase. On the other hand, the 95% percentile consumption stream increases with the highest

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in-The hybrid DB/PPR pension plan — D. Ruiter 25

crease for γ equal to 100% and can even result in a consumption stream larger than the average wage.

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Conclusion

In this thesis the welfare of a hybrid pension plan which combines defined benefit (DB) and personal pensions with risk sharing (PPR) and the impact of decision making by the policyholder is investigated. For this purpose, a model for the finan-cial market has been set up. We assumed the finanfinan-cial market to be described by the Black-Scholes-Merton model with one risk factor given by the stock price index. Furthermore, a DB and PPR pension plan has been modelled. The DB pension plan is a pension plan with conditional indexation depending on the funding ratio. If the funding ratio is above 105% indexation is given. Furthermore, the height of the guaranteed benefit of the DB pension plan depends on the decision made by the policyholder to put a part of the premium in the PPR.

The expected consumption stream is the highest when as much as possible of the premium is put into the DB pension plan. This is because a loss causes the wealth of the policyholder to decrease, resulting in a lower benefit. While a loss does not reduce the benefit of the DB pension plan. Therefore, we can conclude that the policyholder would rather go for certainty than taking advantage of the equity risk premium. Furthermore, in the PPR plan there are no investments being made in the risky stocks during the retirement phase. In addition, the biometric return is not high enough to give the same effect as the stock return. Therefore, the DB benefit increases over time while the PPR benefit does not. This is an explanation why the policyholder would rather put the premium in the DB pension plan than in the PPR. Furthermore, the negative effect on the expected consumption increases with an increase in the percentage put into the PPR contract. This is because the indexation in absolute terms has a larger effect on higher benefits resulting in a higher percentage of the average wage.

The current Dutch pension system is under pressure, partly because people are getting older. If the age of death increases the pension fund will get less premium relative to the duration of the guaranteed benefit. This also applies to the PPR con-tract. If the age of death increases by 10 years the premium paid during the working phase is not high enough to compensate for this increased age of death. Therefore, the increasing age of death would still be a problem in the hybrid pension contract. Furthermore, the retirement age in our model is 70 years. Traditionally, the retire-ment age was 65 years. If the retireretire-ment age would still be 65 years, the pension fund would not be able to give indexation. Also, the PPR benefit would be lower than when the age of death increases with 10 years. The impact of a lower retirement age is greater than a higher age of death. This implies that not receiving labor income has more effect than dying 10 years later.

Since September 1, 2016 there is new legislation which allows risk taking in the retirement phase. The aspect of risk taking in the retirement phase has a lot of effect on the result. By allowing risk taking in the retirement phase the expected consump-tion stream increases as the premium in the PPR contract increases. Therefore, we

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can conclude that not being able to invest in the retirement phase is the main reason to put the premium in the DB contract. This is because the DB benefit may increase in the retirement phase while the PPR benefit can only stay the same of decrease.

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Discussion

In this thesis we assumed the financial market to be described by the Black-Scholes-Merton model. This model simplifies the markets for financial assets into a set of mathematical rules. The main advantage of this model is that the formula is rela-tively simple. The main disadvantage is that it makes several assumptions that are not necessarily true in reality, which can be seen in Appendix C. This includes the fact that even risk-free assets still carry a slim possibility of default. Another factor is the model does not take transaction costs into account. The model also fails to take any dividends of the underlying stock into account.

Another limitation is the assumption that the stock return follows a geometric brownian motion. This implies the stock return can be modelled as a lognormal distribution. In the real world stocks have jumps and downward outliers are heavier than upwards. Therefore, the jump-diffusion model used by Merton (1976) would be a better model.

Furthermore, the risk-free rate is assumed to be deterministic. However, pension funds have to calculate their liabilities on the basis of current bond yields nowadays (van Rooij et al., 2004). This is called the fair value accounting method. The infla-tion is also assumed to be deterministic. The model can be adjusted by taking into account the correlation between interest rate and inflation. This correlation implies that the central bank is inclined to increase interest rates when the inflation is high and vice versa (Vlaar, 2006).

In this thesis we also assumed the mortality rate to be deterministic. Naturally, this is not true in the real world. The mortality rate is stochastic. The Royal Dutch Actuarial Association (AG) publishes a projection table of the life expectancy every two years. These projections are based on a stochastic model that takes into account the historic mortality of the Netherlands and European countries which are similar to the Netherlands. This way a random shock in a year in the Netherlands would not have a large impact on the projection table. In addition, it is also possible to adjust the model in such a way that there is a link between the retirement age and the life expectancy.

Another limitation is the assumptions there is one policyholder for every age and every policyholder is identical. In the real world this is not true. Therefore, it would be better to make use of the population development which is provided by the CBS. Also, the policyholders can be divided into four categories based on their salary; low and flat, low and increasing, high and flat, and high and increasing. Furthermore, we assumed the labor income to be risk-free and there is no partial retirement. An ex-ample of a model that takes this into account is introduced by Johansson-Stenman, O. (2010).

Also, assumptions have been made about the risk aversion in the utility func-tion. We assumed the relative risk aversion to be equal to 7 (O. Johansson-Stenman, 2010). This parameter is based on a small sample of people. However, pension funds

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are not interested in the preferences of people, but in the preferences of the policy-holders and these do not match (Van Rooij, et al., 2007).

Additionally, our model can be adjusted in such a way the policyholder has more moments to choose the percentage they wish to put into the PPR contract. In our model the policyholders choose this percentage at the entering age and are not able to change this. Therefore, the model can be adjusted in such a way that a young policyholder chooses a higher γ than an older policyholder. We also assume every policyholder to choose the same percentage γ which is not realistic. The model can be adjusted such that γ is stochastic.

Another adjustment is the ability to buffer portfolio shocks (van Bilsen et al., 2016). If you take investment risk during the retirement phase into account, the PPR benefit is going to fluctuate over time. To dampen the effect of portfolio shocks on PPR benefit you might want to buffer portfolio shocks. This is a feature to get a stable pension payment and is based on the prospect theory which states that people value reference point. The motivation behind this is that people think in terms of reference levels and not in terms of absolute levels of consumption. This feature can make the PPR contract more attractive.

As mentioned in Section 2.1.2 the pension fund has several policy instruments, such as the indexation policy, premium policy and investment policy. In this thesis we assumed these policies to be constant. In the real world the pension fund is, under certain conditions, able to adjust these policies. For example, a pension fund can adjust their investment policy if the funding ratio is low. Also, in our model the pension fund is not able to cut the benefits. In reality, a pension fund can reduce the benefits with a maximum of 7% per year (Rijksoverheid, 2012).

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Appendix A: Symbols and

abbreviations

Symbols

Simulation Symbol

Number of simulations nsim

Number of periods nperiod

Pensioncontract Symbol Ambition β Required ambition β1 PPR percentage γ Premium π Accrual percentage ap Demography Symbol Age x Time t Retirement age R Time of death D Economy Symbol Initial salary S₀0 Salary S_tx0 Pensionable salary S_tx Career path cx Consumption C_tx

Real interest rate rf

Wage inflation πwage

Price inflation πprice

Pensionfund Symbol Benefit DB bx,DB_t Benefit PPR bx,PPR_t Contribution DB cx,DB_t Contribution PPR cx,PPR_t Surplus SPt Funding ratio Ft

Financial parameters Symbol

Initial stockprice S0

Mean stock return µ

Volatility of stock return σ Fraction invested in equity α Fraction invested in equity at entering age αx0

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Abbreviations

AG Royal Dutch Actuarial Association AIR Assumed Interest Rate

AOW Algemene Ouderdomswet

CRRA Constant Relative Risk Aversion DB Defined Benefit

DNB De Nederlandsche Bank PAYG pay-as-you-go

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Appendix B: Parameters

Simulation Symbol Value

Number of simulations nsim 1000

Number of periods nperiod 150

Ambition β 0.75

PPR γ 0.05

Premium π 0.10

Initial values Symbol Value

Entering age x0 25

Retirement age R 70

Time of death D 85

Initial salary S0 20.000

Offset in year 0 Offset0 8.000

Minimum required funding ratio Fmin 1.05 Funding ratio for maximum indexation Fmax 1.30

Careerpath x0 – x0+20 cx 0.01

Careerpath x0+20 – R cx 0.005

Maximal increase of contribution Imax 1.00

Financial parameters Symbol Value

Real interest rate rf 0.02

Wage inflation πwage 0.03

Price inflation πprice 0.02

Initial stockprice S0 1.00

Mean stock return µ 0.07

Volatility of stock return σ 0.20

Fraction invested in equity α 0.30 Fraction invested in equity at entering age αx0 _0.70

Biometric return r_bx 0.01

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underlying the financial market

Assumptions of the Black-Scholes-Merton model (Merton, 1976):

1. Frictionless markets: there are no transactions costs or differential taxes.

2. The short-term interest rate is known and constant through time.

3. The stock pays no dividends or other distributions during the life of the option.

4. The option is European in that it can only be exercised at the expiration date.

5. The stock price follows a geometric Brownian motion through time which produces a log-normal distribution for stock price between any two points in time.

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Appendix D: Closed form

discount factors

Sum of a geometric progression: n X k=1 aqk−1 = a(1 − q n₎ 1 − q (6.1)

For x0 ≤ x < R the discount factor is defined as:

R−x|¨axt = 1 (1 + r)R−x + 1 (1 + r)R−x+1 + . . . + 1 (1 + r)R−x+(D−R) = D−R X k=0 1 1 + r R−x+k = D−R X k=0 1 1 + r R−x · 1 1 + r k = D−R+1 X k=1 1 1 + r R−x · 1 1 + r k−1

Now we can use the closed form of the sum of a geometric progression with a =_1+r1 R−x and q = _1+r1 which results in:

R−x|¨axt = 1 1+r D−x ·1 −_1+r1 D−R+1 1 − 1 1+r = 1 1+r D−x ·1 −_1+r1 D−R+1 r

This is the closed form of the discount factor for x0 ≤ x < R. For R ≤ x < D the discount factor is defined as:

¨ ax_t = 1 + 1 (1 + r)1 + 1 (1 + r)2 + . . . + 1 (1 + r)D−x = D−x X k=0 1 1 + r k = D−x+1 X k=1 1 1 + r k−1 35

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Now we can use the closed form of the sum of a geometric progression with a = 1 and q = _1+r1 which results in:

¨ ax_t = 1 · 1 − 1 1+r D−x+1 1 − 1 1+r = 1 −_1+r1 D−x+1 r

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Appendix E: Solution of the

differential equations

The risk-free asset

We assumed that Btsatisfies the following differential equation: dBt

Bt

= rfdt,

where rf denotes the risk-free interest rate and B0 the value of the bank account at time 0 is given. We assume B0 to be equal to 1. This is an ordinary differential equation, because we assumed rf to be deterministic. Therefore, the solution can be found as follows: Z t 0 dBs Bs = Z t 0 rfds logBt B0 = rft Bt= B0erft

This is the solution of the differential equation.

The risky asset

We assumed that the stock price index satisfies the following differential equation: dSt

St

= µdt + σdZt,

where µ denotes the expected rate of return on the stock price index and σ the volatility of the stock returns, Z = {Zt|t ≥ 0} represents a standard Brownian motion and S0 > 0, the value of the stock at time 0, is given.

In order to solve this we need Itˆo’s Lemma. Assume Xt is an Itˆo drift-diffusion process that satisfies the stochastic differential equation:

dXt= µtdt + σtdBt,

where Bt is a Wiener process. Any twice differentiable scalar function f (t, x) of two real variables t and x one can be written as:

df (t, x) = ∂f ∂t + µt ∂f ∂Xt +1 2σ 2 t ∂2f ∂X2 t dt + σt ∂f ∂Xt dZt

To solve the differential equation we apply Itˆo’s Lemma on the function log(St) 37

(42)

which results in: d(log(St)) = ∂log(S_t) ∂t + µSt ∂log(St) ∂St + 1 2σ 2_S2 t ∂2log(St) ∂S2 t dt + σSt ∂log(St) ∂St dZt = 0 + µSt 1 St −1 2σ 2_S2 t 1 S_t2 dt + σSt 1 St dZt = µ − 1 2σ 2_{dt + σdZ} t

This equation is an integral formula and can be written as: log(St) − log(S0) = µ − 1 2σ 2_{t + σZ} t logSt S0 = µ − 1 2σ 2_{t + σZ} t St = S0e(µ− 1 2σ 2_)t+σZ t

Welfare of a hybrid DB/PPR pension plan