The impact of the initial financial position of a pension fund and the impact of the draw-down function in personal pension with risk sharing

The impact of the initial financial

position of a pension fund and the

impact of the draw-down function in

Personal Pension with Risk sharing

Siouar Hachana

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:

Siouar Hachana

Student number:

Email:

siouar@hotmail.com

Date:

October ,

Supervisor:

Dr. T.J. Boonen

This document is written by Siouar Hachana 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 ref-erences 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

The Dutch pension system is moving from defined benefit (DB) pension plans to defined contribution (DC) pension plans. However, DC plans are more focused on wealth accu-mulation than on providing a stable retirement income. In this paper, we study a newly proposed pension contract the Personal Pension with Risk sharing (PPR) by investigat-ing the impact of the initial financial position of a pension fund and the impact of the draw-down function. The expected draw-down amounts at retirement age 67 in PPR are compared to the amounts of the guaranteed DB and the DC pension plans. The expected amounts are calculated by two draw-down functions in PPR. In the first function the total surplus and deficit is shared among the participants. While only half of the surplus and the deficit is shared in the second draw-down function and the other half is held in reserve by the pension fund. When the initial funding ratio FRppr0 equals 100 in the Dutch

pen-sion fund ABP, the expected draw-down amounts at age 67 calculated by both draw-down functions are equal to the expected draw-down amounts at age 67 in the DC pension plan. When FRppr0 equals 120, the expected amounts at age 67 of individuals younger than age

49 calculated by the second function are higher than the expected amounts calculated by the first function. When FRppr0 equals 90, the expected amounts at age 67 of individuals

younger than age 49 calculated by the second function are lower than the expected amounts calculated by the first function. In this case, the pension fund has insufficient assets to pay the expected draw-down amount of the last age cohort 25. Therefore, when pension fund ABP starts with a surplus of 20 it should use the second draw-down function for individ-uals younger than age 49 and the first draw-down function for older individindivid-uals. When it starts with a deficit of 10, pension fund ABP should offer a DC pension plan to all participants.

Contents

Chapter 1 — Introduction 1

Chapter 2 —The Dutch pension climate 3

§ 1 Pension plans in second pillar . . . 4

1.1 DB plan . . . 4

1.2 DC plan . . . 4

§ 2 Personal Pensions with risk sharing (PPR) . . . 5

2.1 Investment function . . . 5

2.2 Draw-down function . . . 6

2.3 Risk sharing function . . . 6

2.4 Advantages of PPR versus DB pension plans . . . 6

2.5 Advantages of PPR versus DC pension plans . . . 7

2.6 Possible disadvantages of PPR . . . 7

Chapter 3 — Methodology 8 § 1 Financial market model . . . 8

§ 2 Draw-down function . . . 9

2.1 Total wealth . . . 10

2.2 The investment policy . . . 11

2.3 Risk sharing . . . 12

2.4 Draw-down function at retirement age . . . 13

Chapter 4 —Results 15 § 1 The impact of the first draw-down function in PPR . . . 16

1.1 The case of FRppr0 equals 100 . . . 16

1.2 The case of FRppr0 equals 120 . . . 17

1.3 The case of FRppr0 equals 90 . . . 20

§ 2 The impact of the second draw-down function in PPR . . . 22

2.1 The case of FRppr0 equals 120 . . . 22

2.2 The case of FRppr0 equals 90 . . . 24

Chapter 5 — Discussion 26 § 1 Implications of the results if the model is real . . . 26

Chapter 6 —Conclusion 28

Bibliography 30

chapter 1

introduction

The Dutch pension system is highly regarded because it prevents old-age poverty by provid-ing lifelong benefits and risk sharprovid-ing. However, due to the increase in life expectancy and the developments of the financial market, the pension system is not sustainable anymore. Employers are withdrawing as risk sponsors of occupational defined benefit (DB) pension plans because of the cost and cost volatility of these pension plans (Most & Wadia, 2015). In DB pension plans, guaranteed pension entitlements are offered to employees during retirement and employers bear the risks. The withdrawing caused a shift from DB plans to defined contribution (DC) pension plans. DC pension plans offer fixed contributions, while the pension benefits are only determined at retirement and the risks are borne by employees. Most DC pension plans provide little guidance on how to draw-down wealth during retirement. The optimal draw-down product requires that the amount of the retire-ment income plus risk exposures are tailored to the specific needs of the employee. Fur-thermore, risks during the accumulation phase should be customized to the aims for the draw-down phase. According to Bovenberg and Nijman (2015) risk management in DC pension plans in the accumulation phase is not focused on providing a stable retirement income. For example, longevity risk is not insured in several DC plans or is only insured jointly with specific financial risk exposures.

A new pension product that provides an adequate, affordable and stable retirement in-come is necessary. Therefore, Bovenberg and Nijman (2014) propose a new type of pension plan: the Personal Pension with Risk sharing (PPR). This new type of pension plan inte-grates income adequacy, affordability and security in transparent and flexible ways, while tailoring pensions to the preferences of employees. PPR combines the strengths of DB and DC pension plans by unbundling the following three main functions of these pension plans: the investment function, draw-down function and risk sharing function.

The investment function is a characteristic of the DC pension plan. Every employee has financial assets on a personal account. This allows tailoring of the investment policy to the specific needs of an employee. The draw-down function allows to customize the payout policy. The risk sharing function is a feature of the DB pension plan. This function pools the idiosyncratic longevity risk to protect employees from outliving their retirement income. Idiosyncratic longevity risk is the risk of an employee living longer than average. In the risk sharing function, systematic risks that are not traded on financial markets can

be exchanged through internal swap agreements. Examples of systematic risks are changes in wage inflation rates and in survival probabilities. Pension providers can tailor pensions to the needs of employees by combining these functions in a flexible and transparent way. In this thesis we study the impact of the initial financial position of a pension fund and the impact of the draw-down function in PPR. The expected draw-down amounts at retirement age 67 in PPR are compared to the expected draw-down amounts at age 67 in guaranteed DB and DC pension plans. The guaranteed DB plans are modeled without any investment risk. The expected amounts are calculated by two draw-down functions in PPR. In the first function the total surplus and deficit is shared among the participants. In the second function only half of the surplus and the deficit is shared among the participants and the other half is held in reserve by the pension fund. The Dutch pension fund ABP and two fictional pension funds named Young and Old with respectively young and old people are evaluated. Moreover, an economy with two risk factors given by the nominal short rate and the stock return is modeled. The Black-Scholes-Vasicek model is used as the financial market model. When pension fund ABP starts with a surplus of 20 in PPR, the expected draw-down amounts at age 67 of individuals younger than age 49 calculated by the second down function, are higher than the expected amounts calculated by the first draw-down function. Especially the expected amounts of young individuals are higher because of the increase of the extra assets over time. However, when pension fund ABP starts with a deficit of 10 in PPR, the expected amounts at age 67 of young individuals calculated by the second function are much lower than the expected amounts calculated by the first function. The pension fund has insufficient assets to pay the expected draw-down amount of the last age cohort 25 due to the deficit of 10.

chapter 2

the dutch pension climate

The Dutch pension system consists of the following three pillars: public pensions, occupa-tional pensions and private pensions (Bovenberg, Mehlkopf, & Nijman, 2014).

First pillar pension

The first pillar pension is the state pension (AOW) and its main purpose is to prevent old-age poverty. It is a pay-as-you-go (PAYG) system which denotes a system that workers pay contribution in the form of tax, in order to pay the pension benefits of the retirees. The participation of workers is mandatory and the pension is paid out as an annuity which is a lifelong benefit. Everyone who has reached the AOW pension age and lives or has lived in the Netherlands is entitled to the AOW pension.

Second pillar pension

The second pillar consists of occupational (worker) pension, and aims at maintaining the standard of living during retirement. Often it is organized in a collective fashion to pool investment and longevity risks. This pillar is financed by capital funding. Capital funding is a method of financing of pension entitlements where the paid pension contributions are saved and invested in order to realize the pension benefits. So the pensions are financed from the total saved contributions in the collective pension scheme and from the invest-ment return of these contributions. Occupational pensions are part of labor agreeinvest-ments. Therefore, employees are obliged to save for their pension.

Third pillar pension

The third pillar is formed by private, personal pensions. Its main objective is to tailor pensions to specific individual needs in order to save extra pension and in the case of self-employed individuals and employees in sectors without a collective pension scheme, to somewhat save a pension. This pillar is voluntary and tax-favored up to a ceiling.

1

Pension plans in second pillar

Pension plans in the second pillar can be categorized into two types: defined benefit (DB) pension plans and defined contribution (DC) pension plans. The major difference between the two types of pension plans is which party, the employer or the employee, bears the risks involved with providing the pension benefit.

1.1 DB plan

A DB plan is an employer-sponsored retirement plan where the employee accrues a guaran-teed monthly pension benefit from the date of retirement until death (Broadbent, Palumbo, & Woodman, 2006). The pension benefit is predetermined by a formula based on factors such as duration of employment and wage history (Bodie, Marcus, & Merton, 1988). There are two forms of Dutch DB plans: the final salary plan where the employee’s final salary is considered, and the final average pay plan where the average salary over the final years of the employee’s career is taken into account.

An advantage of DB pension plans is that the employee has no risk of outliving the pension benefit because it is guaranteed for life. Another advantage is that the employer bears the investment and the financial risk by spreading the risks over the entire plan popu-lation. However, as a result of low interest rates and aging of the workforce, the guaranteed pension benefits have become more expensive and the employers are withdrawing as risk sponsors of DB pension plans (Bovenberg & Nijman, 2014). Due to the withdrawal, the pension benefits are no longer guaranteed but are variable (Broadbent et al., 2006). The employer is then forced to cut the pension benefits if the investment results lead to un-derfunded pension plans. Moreover, in DB plans the risks are shared equally among the participants through a nominal funding ratio. This leads to intergenerational conflicts be-cause the participants who are nearing retirement are affected most. Another disadvantage is that the value of the pension benefit entitlement has become unclear and non-transparent. This is not only because the pension benefit has become variable but also because the ac-crual pattern is nonlinear. The accrued pension amount depends on the salary and on the tenure, which both have no constant pattern. Therefore, the valuation of the pension ben-efit entitlement has become difficult and complex in a DB plan and does not facilitate its portability (Broadbent et al., 2006). This can result in accrual losses when an employee changes to an other employer.

1.2 DC plan

In a DC plan an employer pays contributions on the employee’s behalf to an individual pension account during the accumulation period. The contributions are then invested and the investment returns are added to the individual’s account. Finally, on or before retirement, this account is used to provide pension benefits by converting the capital in the individual account to a lump sum or to a stream of life annuities (de Haan, Lekniute, & Ponds, 2015). In a DC plan, only the employer contributions are fixed rather than the pension benefit. So the amount of pension benefit entitlements is unknown in advance.

An advantage of DC pension plans is that the contributions are fixed in contrast to DB pension plans. The pension benefit entitlement is individual which makes it clear and portable across employers (Segal, 2010). A disadvantage is that the employee bears the investment and longevity risk. So the employee gets all investment gains but has to bear all investment losses as well. Besides, the employee must calculate the amount of pension income needed to retire with a decent income and must also make complex investment decisions to realize the desired pension ambition (Broadbent et al., 2006). However, in general employees have limited investment experience and have difficulty in understanding which amount is necessary for a decent retirement income. Nevertheless, DC pension plans provide default options based on the life-cycle investment strategy. This strategy attempts to maximize the utility of the employees by balancing the investment risk and investment return profile based on the number of years the employees have until retirement, see Towers Watson (2011). Young employees take more investment risk and older employees nearing retirement take less investment risk. This provides some protection from extremely negative investment returns to employees nearing retirement. Another drawback is that participants in Dutch DC pension plans are forced to buy a guaranteed lifelong income at retirement (Bovenberg & Nijman, 2015) and to live only from this income during the decumulation phase. According to Bovenberg and Nijman (2014) DC pension plans are more focused on wealth accumulation rather than on providing a stable retirement income during the draw-down phase.

2

Personal Pensions with risk sharing (PPR)

Bovenberg and Nijman (2014) present a new pension plan the ‘Personal Pensions with risk sharing’ (PPR) where pension income in the accumulation phase as well as in the decumulation phase is defined in terms of an individual pension account with additional agreements about risk sharing. PPR unbundles the three main functions of DB and DC plans: the investment function, draw-down function and risk sharing function (Bovenberg & Nijman, 2015).

2.1 Investment function

Every employee has a personal investment account with financial assets. The employee pays contributions and receives investment returns on the assets. Each personal account contains a hedge and a return portfolio. A hedge portfolio covers the impact of tradable risk factors, such as interest-rate fluctuations and expected inflation on the cost of providing a particular pension ambition. The objective of a return portfolio is to optimize the trade offs between the mentioned risk factors and the risk premiums on these factors (Bovenberg & Nijman, 2015). A risk premium is the investment return minus the return that would be earned on a risk free investment. It can be seen as the amount that an investor is willing to receive for the risk involved in a particular investment.

2.2 Draw-down function

A PPR aims at financing a certain pay-out ambition by the financial assets on the personal pension account. Bovenberg and Nijman (2015) define the draw-down rate as the fraction of the individual assets on the personal account that is drawn down when an individual retires. This fraction depends on how the surplus or the deficit of the pension fund is shared among the participants. Furthermore, the draw-down rate is the equivalent of the inverse of the actuarial annuity factor. The actuarial annuity factor converts the capital in the personal account into an annuity and depends on the assumed interest rate (AIR). The AIR is used as the discount rate. It is provided by the pension fund and is defined as the minimum interest rate that must be realized in order to meet the investment expectations and the pay-out ambition (Bovenberg & Nijman, 2015). It is important to choose the correct value of AIR as it will determine the draw-down policy. The larger AIR, the more capital can be draw-down at a given retirement age and vice versa.

2.3 Risk sharing function

Pension in the PPR is a combination of investment and insurance. When longevity insur-ance is included to the pension contract, the individual receives a so-called biometric return for staying alive. When the individual passes away, the personal assets will be distributed among the surviving participants and will not be left for the heirs. The biometric return is related to mortality probabilities. Because mortality probabilities rise with age, this re-turn will be high at advanced ages. The biometric rere-turn thus insures against idiosyncratic longevity risk. The risk sharing function allows also to exchange systematic risks, that are not (yet) traded on financial markets between participants. The systematic risks can be ex-changed through internal swap agreements. These risk-sharing agreements can be viewed as non-market investments and generate a non-market return.

2.4 Advantages of PPR versus DB pension plans

PPR offers the possibility to tailor risk exposures to the specific needs of an individual. Young individuals are prepared to take more investment risk because of the long investment horizon. Individuals nearing retirement are more interested in hedging inflation risk in stead of investment risk. In PPR this can be achieved by the life-cycle investment strategy as in DC pension plans. In this strategy an individual invests less in risky assets and more in risk-free assets as the individual nears retirement, see Towers Watson (2011). Regarding inflation risk, the same can be achieved by letting young individuals hedge inflation risk less than old individuals. Risks are no longer shared equally among the age cohorts causing intergenerational conflicts like in DB pension plans. Moreover, individual property rights are defined in terms of personal financial assets in PPR. As such individual property rights are easier to value, clear and transparent. The clarity facilitates the portability of pension rights.

2.5 Advantages of PPR versus DC pension plans

In contrast to DC pension plans, employees in PPR are not the risk bearers and the risk management is more focused on the decumulation phase. Retirees have the possibility to continue investing in risky assets during the decumulation phase. So retirees can continue to earn risk premiums and earn biometric returns for staying alive at the same time.

2.6 Possible disadvantages of PPR

PPR offers more individual freedom in choice than other pension contracts. This might increase the will to make individual decisions when participants have personal accounts (Bovenberg & Nijman, 2015). However, the general participant is not well-equipped enough to make important choices in for example which investment strategy. Therefore, it is nec-essary to create a proper choice architecture in PPR (Bovenberg & Nijman, 2014). Another drawback that might arise is that the transparency of the value of the pension rights leads to investing less in risky assets when the volatility of the financial market is observed. This can be compensated by communicating the expected value and the minimum value of the pension benefit that can be achieved when a financial shock occurs.

chapter 3

methodology

For the purpose of investigating the impact of the initial financial position of a pension fund and the impact of the draw-down function in PPR, the expected draw-down amounts at retirement age 67 in PPR are compared with the expected draw-down amounts at age 67 in DB and DC plans for the pension funds ABP, Young and Old. The first section describes the chosen financial market model. In the second section, the draw-down functions to de-termine the expected and the 95-confidence interval of the expected draw-down amounts at age 67 in each pension contract are defined.

1

Financial market model

In this paper t represents time in years; hence t ∈ Z_⩾0where time t = 0 represents the year 2016. We model an economy with two risk factors given by the nominal short-term interest rate rntand the stock price index St. Thousand scenario’s are generated with the

Tilburg Finance Tool (TFT) using the Black-Scholes-Vasicek model as the financial market model. The nominal short term interest rate rntis given by the Vasicek one-factor model:

drnt= a (b− rnt)dt + σrdWrt, (3.1)

where a, b and σrare positive constants. The speed of mean-reversion per year is given by

a. The drift term a (b− rnt)follows a mean-reverting process around its long-term mean

bof the short rate, in order to ensure that interest rates cannot rise infinitely. The second term is stochastic and leads to fluctuations around b in an unpredictable but continuous way (Vasicek, 1977). In equation (3.1), Wrtis a standard Wiener process and dWrtcan be

seen as a shock in the term structure.

The price index is given by dIt= ciItdt where cidenotes the constant inflation level.

The stock price Stis defined as

dSt= (rnt+ πrisk) Stdt + σSStdWSt. (3.2)

In equation (3.2), a stock risk premium πriskis added to the nominal short rate from

equa-tion (3.1). The volatility of stocks and the shock in stock prices are represented by respec-tively σS and dWSt. The risk premium cannot be zero, otherwise it is not interesting to

invest in a particular asset. The nominal short rate and the stock prices are volatile so σrand

σS cannot be zero as well. However, the stock prices are more volatile than the nominal

short rate. Therefore, σris smaller than σS. Given the fact that interest rates and the stock

market move in opposite directions (Alam & Uddin, 2009), the instantaneous correlation

ρ (Wrt, WSt)must be negative. The market price of interest risk λ is also negative, see

Hull (1993). Note that the price index, equation (3.1) and equation (3.2) depend on the initial values of respectively the price index, the nominal short rate and the stock price.

2

Draw-down function

We evaluate three pension funds in this paper: two fictional pension funds Young and Old, and the Dutch pension fund ABP. Pension fund Young consists of young employees until age 45 and pension fund Old consists of employees of age 46 until 66. Pension fund ABP is the largest pension fund in the Netherlands. The weights of the participated age cohorts of this pension fund are derived from the population of 2015 as reported by Stichting Pen-sioenfonds ABP (2015). The population in this report is given in age classes of several age cohorts τ together. Therefore, the weights are taken equally for each age cohort in one age class. Only the age cohorts from 25 until 66 are taken into account which results in a total number of participants equal to 865,953 in pension fund ABP. The pension funds Young and Old are constructed with the same total number of participants. Table 3.1 provides the weights wτ_{of the participants of each age cohort τ in these pension funds. The size of each}

age cohort τ in every pension fund is equal to

sτ= 865953 wτ. (3.3)

The employees join one of the evaluated pension funds in year 2016 at time t = 0 and stay there until retirement at age 67. The pension funds are designed in such a way that no new participants join the pension funds which means that the pension funds are closed. We also assume that the employees stay alive until retirement. Therefore, the size of each age cohort sτ _{in equation (3.3) does not change over time. Career development and the}

Pension fund Young Pension fund Old Pension fund ABP

τ Weight τ Weight τ Weight τ Weight

25 20.00 46 0.02 25 0.20 46 3.20 26 15.00 47 0.04 26 0.20 47 3.20 27 10.00 48 0.06 27 0.20 48 3.20 28 9.00 49 0.08 28 0.20 49 3.20 29 8.00 50 1.00 29 0.20 50 3.71 30 7.00 51 1.10 30 0.26 51 3.71 31 6.00 52 1.30 31 0.26 52 3.71 32 5.00 53 1.40 32 0.26 53 3.71 33 4.50 54 1.50 33 0.26 54 3.71 34 4.00 55 2.00 34 0.26 55 3.87 35 3.00 56 3.00 35 2.73 56 3.87 36 2.00 57 4.00 36 2.73 57 3.87 37 1.50 58 4.50 37 2.73 58 3.87 38 1.40 59 5.00 38 2.73 59 3.87 39 1.30 60 6.00 39 2.73 60 2.28 40 1.10 61 7.00 40 2.84 61 2.28 41 1.0 62 8.00 41 2.84 62 2.28 42 0.08 63 9.00 42 2.84 63 2.28 43 0.06 64 10.00 43 2.84 64 2.28 44 0.04 65 15.00 44 2.84 65 2.28 45 0.02 66 20.00 45 3.20 66 2.28

Table 3.1: The weights of participants in the fictional pension funds Young and Old, and the

Dutch pension fund ABP. All pension funds have 865,953 participants in total.

2.1 Total wealth

Assume that the assets or total wealth Wτ

t+1that age cohort τ accrues per year at time t + 1

equals

W_t+1τ = sτ(W_tτ+ πτ) (1 + rnt+ ατ(rst− rnt)). (3.4)

The size of each age cohort in the pension fund sτ _{is defined in equation (3.3). Every}

year a premium πτ _{equal to 18 of the pensionable salary is paid in equation (3.4). The}

pensionable salary is the salary minus the franchise which is deducted from the amount of the state pension. The salary is linked to wage inflation ciequal to 3 and to individual

wage growth where the wage growth decreases as the individual ages. The salary pattern and thus the contribution pattern as well, are constant over time for each age cohort τ . The initial values of the total wealth Wτ

0 are chosen equal to the total paid contributions in the

past linked to the wage inflation cias is displayed in the following equation

W₀τ =(πτ−1+ W₀τ−1)(1 + ci). (3.5)

In equation (3.4) the nominal short rate rnt is generated according to equation (3.1) and

ατrepresents the proportion invested in stocks for each age cohort τ . The stock return rst

rst=

St+1

St − 1,

(3.6) where the stock prices Stand St+1are modeled in equation 3.2. The stock return rstis

only known at t + 1 in equation (3.6). As a result, Wτ

t and πτin equation (3.4) received at

time t will be invested when the stock return rstis known which is at time t+1. Therefore,

the total wealth Wτ

t+1is calculated on January 1st at time t + 1.

2.2 The investment policy

Guaranteed DB pension plan

A pension benefit is promised in a DB pension plan. To meet this promise, a guaranteed DB pension plan does not invest in risky assets such as stocks but invests only in risk-free assets such as bonds. Therefore, the DB plan is modeled without any investment risk. The total wealth of age cohort τ in DB is equal to

W_t+1τ,db= sτ ( W_tτ,db+ πτ ) (1 + rnt). (3.7) DC pension plan

In a DC pension plan individuals invest in risky assets in order to earn investment risk premiums. The individual assets of age cohort τ in the DC pension plan are invested according to the life-cycle investment strategy. The investment of the individual assets is modeled with a proportion invested in stocks ατ_{that declines as the individual ages. The}

invested proportion in stocks α25_{is equal to 70 and α}66_{is reduced linearly to 8.5.}

PPR

The individual assets on the personal account in PPR are invested with the same ατ _as

the DC pension plan. The total wealth in PPR and in the DC pension plan is calculated according equation (3.4) since the investment proportion in stocks ατ_{is the same in both}

pension contracts.

Subsequently, the individual assets of all participants are invested by the pension fund in PPR. In order to represent the investment interests of all participants in the pension fund, a different investment proportion than ατ _{is needed. The sum of the individual assets of}

all active participants in PPR is invested with:

αpprt = ∑66−t τ =25((W τ t + sτπτ) ατ) ∑66−t τ =25(W τ t + sτπτ) . (3.8) The term∑66τ =25−t(W τ

t + sτπτ)represents the available capital in cash in the pension

fund at time t. The proportion αpprt can be seen as the weighted average of the proportion

2.3 Risk sharing

Risk is shared in collective pension contracts through a collective funding ratio FRt. The

funding ratio is the market value of assets divided by the expected present value of future liabilities (Leibowitz, Kogelman, & Bader, 1994). It indicates the degree to which a pension fund is able to cover all future pension payments and should be equal to at least 100.

We consider only financial investment risk in this study. Since the guaranteed DB pension plan is modeled without any investment risk and with a constant contribution pattern, the liabilities and the assets are known and equal. Ergo, the funding ratio of the guaranteed DB pension plan is set to 100. In a DC plan individuals bear the investment risk so the risk sharing aspect is not included. Hence there is no funding ratio in a DC plan. PPR does include the risk sharing aspect through pooling of the investment risk among all participants. PPR is modeled with the following funding ratio

FRpprt =

At

Lt

, (3.9)

where the assets Atand the liabilities Ltof the pension fund are formulated as:

At+1= ( At+ 66_∑−t τ =25 sτπτ− PV(Dτt,ppr ))( 1 + rnt+ αpprt (rst− rnt) ) , (3.10) Lt= 67∑−t τ =25 W_tτ. (3.11)

The initial value A0is determined by the chosen initial funding ratio FRppr0 and the value

of L0which is determined by the sum of the initial values of W0τ calculated in equation

(3.5). The term PV(cWτ t

)

defines the present value of the pension benefit paid to age cohort τ that retires at time t and is equal to the present value of the draw-down amount. The formula of the present value pension benefit will be derived in the next subsection in equation (3.17). The term At+

∑66−t

τ =25s

τ_πτ_{− PV}(_Dτ,ppr

t

)

stands for the value of the available capital in cash in the pension fund after paying the draw-down amount Dtτ,ppr

to the retired age cohort at time t and will be invested at the beginning of time t + 1. The definition of the liabilities Ltin equation (3.11) is chosen as follows. The total wealth

Wτ

t that age cohort τ accrues per year at time t formulated in equation (3.4) can be seen

as the accrued pension entitlements. On the one hand, if the sum of the accrued pension entitlements Wτ

t is higher than the available assets Atin the pension fund, then the pension

fund has a deficit and can cut the draw-down amount by this deficit. On the other hand, if the sum of Wτ

t is lower than At, the pension fund can increase the draw-down amount by

the surplus. Hence, the accrued pension entitlements Wτ

t are actually virtual in PPR since

the value of the draw-down amounts depend on the funding ratio FRtand could turn out

to be lower or higher than the accrued pension entitlements. This is a consequence of the risk sharing aspect being included in a pension contract.

2.4 Draw-down function at retirement age

At retirement age, an individual can convert the accrued capital to an annuity with the actuarial annuity factor ¨aτ

t. This factor is calculated as follows

¨ aτt = ∞ ∑ t=0 ∏t−1 s=0(1− qτ +s,i+s) (1 + rn)t . (3.12)

The annuity factor in equation (3.12) is defined at the beginning of year i when age cohort

τ reaches the retirement age 67 (Promislow, 2014). The term qτ +s,i+sis the probability

that someone aged exactly τ + s will die at year i + s. The actuarial interest rate rn is

chosen equal to 4 in the pension contracts which is the same as the initial value of the nominal short rate in the Black-Scholes-Vasicek model. Moreover, the annuity factor is gender neutral determined according to ¨aτ

t = βman¨aτt,x+ (1− βman)¨aτt,y. The variable

βman is the proportion of men participating in the pension funds and is chosen equal

to 0.49. The variables ¨aτ

t,x and ¨aτt,y are the actuarial annuity factors of men and women

respectively. Additionally, the annuity factors are calculated using mortality probabilities from The Royal Dutch Actuarial Association (2014). For the first time, this table is based on a stochastic model which considers the uncertainty of the projection in addition to future mortality. The uncertainty of the projection means that future mortality not only changes over age but also over time. Mortality probability q25,2016differs for example from

mortality probability q25,2026. This means that only longevity is taken into account and not

the longevity risk, because the mortality probabilities in the future are already determined. Moreover, the Projection Table AG2014 is partly based on historic mortality in a number of European countries of comparable wealth instead of only on historic mortality in the Netherlands.

The amount that eventually can be drawn down when an individual of age cohort

τ retires at time t, depends on the draw-down function in the pension contract. In the guaranteed DB pension plan, the total wealth Wτ,db

t defined in equation (3.7) is risk-free

and the funding ratio equals 100 because the pension benefit is guaranteed. The draw-down function in the guaranteed DB plan at time t when an individual of age cohort τ retires is equal to Dτ_t,db=W τ,db t sτ_a¨τ t , (3.13)

where the size of each age cohort is defined in equation (3.3) and the actuarial annuity factor is formulated in equation (3.12). The total wealth Wτ

t in the DC pension plan defined in

equation (3.4) is invested in stocks and therefore risky. The investment risks are not shared among the participants, hence there is no funding ratio in the DC plan. The draw-down function in the DC pension plan at retirement age at time t is equal to

Dτ_t,dc= W τ t sτ_¨aτ t . (3.14)

draw-down functions at retirement age at time t: Dτt,ppr1= WtτFR ppr t sτ_¨aτ t , (3.15) Dτ_t,ppr2=W τ t + 0.5Wtτ ( FRpprt − 1 ) sτ_¨aτ t , (3.16)

where the risky total wealth Wτ

t is defined in equation (3.4) and the funding ratio FR

ppr

t is

formulated in equation (3.9). In the first draw-down function in PPR in equation (3.15), the total surplus and deficit is shared among the participants while only half of the surplus and the deficit is shared among the participants in the second draw-down function in equation (3.16). The other half is held in reserve by the pension fund. The draw-down amount in PPR thus depends on the initial funding ratio. With an initial funding ratio of 100 in PPR, the second draw-down function is identical to the first draw-down function. The present value of the draw-down amount paid to age cohort τ that retires at time t in PPR is equal to the value of the accrued pension entitlements at pension age before converting the entitlements to an annuity:

PV ( Dτt,ppr1 ) = WtτFR ppr t , (3.17) PV ( Dτt,ppr2 ) = Wtτ+ 0.5W τ t ( FRpprt − 1 ) . (3.18)

Since the draw-down amounts in the guaranteed DB and DC pension plan do not depend on the funding ratio, the draw-down amounts from age cohort τ in these contracts will be the same in the pension funds ABP, Young and Old. For example, Dt36,dcwill have

the same value in the pension funds ABP and Young. PPR is the only pension contract where the investment risks are shared among the participants in the pension funds. So, the draw-down amounts in PPR will differ per pension fund as a result of a different FRpprt in

each pension fund.

Based on a thousand different economic scenario’s of the nominal short rate rntand

the stock price St, the draw-down amount at retirement of an individual from age cohort τ

in the three pension contracts is simulated in R for the pension funds. Finally the expected value and the 95-confidence interval of the draw-down amount are determined over all scenario’s. Participants will be interested in the expected value and more in the lower bound of the 95-confidence interval of the draw-down amount.

chapter 4

results

The impact of the initial financial position of a pension fund and the impact of the draw-down function is investigated in the pension contract PPR. Three pension funds are evalu-ated: the Dutch pension fund ABP, the fictional pension fund Young and the constructed pension fund Old. See Table 3.1 for the weights of the participants in these pension funds. The expected draw-down amounts at retirement age 67 in PPR are compared to the ex-pected draw-down amounts at 67 in the pension plans DB and DC in the pension funds. We model an economy with the Black-Scholes-Vasicek model as the financial market model. We use two risk factors given by the nominal short rate and the stock return according to the equations (3.1) and (3.2). Our choice of the parameters in the Black-Scholes-Vasicek model of our reference model is summarized in Table 4.1. These values of the parameters are chosen in order to keep the expected stock return in each scenario lower than 10.

Parameter Choice

Speed of mean reversion a 0.15

Long-term mean of the nominal short rate b 0.04 Initial nominal short rate rn0 0.04

Volatility of the nominal short rate σr 0.01

Price inflation level ci 0.05

Initial price index I0 1

Risk premium πrisk 0.02

Initial stock price S0 1

Volatility of the stock price σS 0.08

Correlation ρ (Wrt, WSt) -0.03

Market price of interest risk λ -0.05

Table 4.1: The choice of the parameters in the Black-Scholes-Vasicek model.

In the first section, the results of the expected draw-down amounts calculated with the first draw-down function with different initial funding ratio’s are given. The results of the expected draw-down amounts calculated with the second draw-down function with different initial funding ratio’s are given in the second paragraph.

1

The impact of the first draw-down function in PPR

The initial financial position of a pension fund determines the initial funding ratio FR0

and thereby determines the starting value of the assets A0through equation (3.9).

1.1 The case of FR

equals 100

An initial funding ratio FRppr0 = 100% implies that A0=L0. Figure 4.1 shows the results

in pension fund ABP. It can be seen that the expected value and the 95-confidence interval of the draw-down amounts at age 67 in PPR are identical to the expected value and the 95-confidence interval of the amounts in the DC plan. The expected draw-down amounts of an individual from age cohort τ are therefore equal in the pension funds ABP, Young and Old because the funding ratio FRpprt remains equal to 100 over time in the three pension

funds. The guaranteed DB pension plan achieves the lowest expected draw-down amounts due to the lack of the stock investment results. The 95-confidence interval in the three contracts increases as τ decreases. This is due to the fact that young age cohorts have a longer investment horizon than old age cohorts and experience more uncertainty in the nominal short rate and the stock return.

Figure 4.1: The impact of the funding ratio FRppr0 = 100% in PPR on the draw-down amounts

of an individual from age cohort τ at retirement, calculated in equations (3.13), (3.14) and (3.15) in pension fund ABP. The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in the pension contracts guaranteed DB, DC and PPR.

When the volatility of the stock price σSis raised from 0.08 to 0.35, the 95-confidence

interval of the draw-down amounts in PPR and the DC pension plan increases, see Figure 4.2. The upper bound of the 95-confidence interval of the draw-down amounts increases more than the lower bound. The lower bound of a 25-year old decreases almost € 20,000 in PPR and the DC plan which is a big loss. The expected draw-down amounts increase slightly.

Figure 4.2: The impact of the increase of the uncertainty in the stock market on the draw-down

amounts of an individual from age cohort τ at retirement with FRppr0 = 100%, calculated

in equations (3.13), (3.14) and (3.15) in pension fund ABP. The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in the pension contracts guaranteed DB, DC and PPR.

In Figure 4.3, different values are chosen for the proportion invested in stocks ατ_in

PPR . This means that all participants in PPR invest the same amount in stocks. The results show that as ατ _{increases and as age cohort τ decreases, the expected value and the}

95-confidence interval of the draw-down amounts become larger over time . The difference between the expected draw-down amounts increases as age cohort τ decreases. This is also the case for the upper bound of the 95-confidence interval of the draw-down amounts in PPR. However, the lower bound decreases as age cohort τ decreases.

1.2 The case of FR

equals 120

An initial funding ratio FRppr0 = 120% implies that A0 = 1.2· L0and that the pension

funds start with a surplus of 20. Figure 4.4 shows the results in the three pension funds. The expected draw-down amounts increase more than the expected amounts and the 95-confidence interval of the amounts with FRppr0 = 100% in PPR and become higher than

the expected amounts in the DC pension plan due to the surplus at t = 0. The funding ratio in pension funds ABP and Young decreases over time but remains above 100. This results in a smaller difference between the expected draw-down amounts in PPR and the DC plan as τ decreases. The funding ratio in pension fund Old remains equal to 120 at all times and the expected draw-down amounts of all individuals in PPR are 20 higher than the expected amounts in the DC plan. This may be explained by the fact that the size

Figure 4.3: The impact of a flat proportion invested in stocks ατ _{on the draw-down amounts}

of an individual from age cohort τ at retirement with FRppr0 = 100%, calculated in equations

(3.13), (3.14) and (3.15) in pension fund ABP. The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in the pension contracts guaranteed DB, DC and PPR.

of age cohort τ = 66 is the largest in pension fund Old, see Table 3.1. Pension fund Old accrues a large amount of capital in the beginning which is apparently sufficient to keep the funding ratio at 120 after age cohort τ = 66 is retired and to pay the draw-down amounts to smaller age cohorts. The same results apply to the 95-confidence interval of the draw-down amounts in the pension funds ABP, Young and Old. Figure 4.4 also shows that the expected draw-down amounts of an individual of age cohort τ in PPR in pension fund ABP are lower than in the pension funds Young and Old. Pension fund Young is more able to uphold the interests of young age cohorts and pension fund Old is more able to uphold the interests of old age cohorts than pension fund ABP. Hence in case of a surplus in PPR as a starting position of the pension funds and in case of participants having the freedom to choose between the pension funds, then young individuals should choose pension fund Young and old individuals should choose pension fund Old.

Figure 4.4: The impact of the funding ratio FRppr0 = 120% in PPR on the draw-down amounts

of an individual from age cohort τ at retirement, calculated in equations (3.13), (3.14) and (3.15) in pension funds ABP, Young and Old. The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in the pension contracts guaranteed DB, DC and PPR.

1.3 The case of FR

equals 90

An initial funding ratio FRppr0 = 90% implies that A0= 0.9·L0and that the pension funds

start with a deficit of 10. Figure 4.5 shows that the expected draw-down amounts in PPR decrease more than the expected amounts with FRppr0 = 100% and become lower than the

expected amounts in the DC pension plan due to the deficit at t = 0. The funding ratio in pension funds ABP and Young increases over time but stays below 100. This results in a smaller difference between the expected draw-down amounts in PPR and the DC plan as τ decreases. The funding ratio in pension fund Old is equal to 90 at all times and the expected amounts of all individuals in PPR are 10 lower than the expected amounts in the DC plan. Pension fund Old accrues a large amount of capital in the beginning through age cohort τ = 66 when FRppr0 = 90%. However, because the size of the younger age cohorts

decreases in this pension fund, the accrued capitals in the years after are not high enough to diminish the deficit and the deficit stays equal to 10. The same results apply for the 95-confidence interval of the draw-down amounts in the pension funds ABP, Young and Old. Furthermore, the expected draw-down amounts of an individual from age cohort τ in PPR in pension fund ABP are higher than in the pension funds Young and Old. Pension fund ABP is more able to decrease the deficit over time. When the pension funds start with a deficit in PPR and participants are allowed to choose between the pension funds, then young and old individuals should choose pension fund ABP. The expected draw-down amounts of the DB plan are the lowest of the three pension contracts for the young age cohorts in the pension funds.

Figure 4.5: The impact of the funding ratio FRppr0 = 90% in PPR on the draw-down amounts

of an individual from age cohort τ at retirement, calculated in equations (3.13), (3.14) and (3.15) in pension funds ABP, Young and Old. The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in the pension contracts guaranteed DB, DC and PPR.

2

The impact of the second draw-down function in PPR

The draw-down amounts at retirement age 67 are sensitive for the choice of the draw-down function in the pension contract. A second draw-down function formulated in equation (3.16) is tested. Individuals from age cohort τ receive only half of the surplus or the deficit with this function. The other half is held in cash by the pension fund.

2.1 The case of FR

equals 120

When FRppr0 = 120%, the expected draw-amount of a 66-year old at time t = 1 calculated

by the second draw-down function is raised by 10 due to the surplus in PPR. The other 10 is kept in cash in the pension fund. As a result, the assets of the pension fund increase more than the assets calculated with the first draw-down function as τ decreases, especially for young individuals. The funding ratio becomes higher as τ decreases.

Figure 4.6 shows that in PPR the expected value and the 95-confidence interval of the draw-down amounts of individuals from age cohorts τ = 49 until τ = 66 calculated by the second function, are lower than the expected value and the 95-confidence interval of the amounts calculated by the first function in pension fund ABP. The expected value and the 95-confidence interval of the draw-down amounts of the younger individuals calculated by the second function are higher. The expected draw-down amount of a 25-year old at time t = 42 calculated by the second draw-down function is € 270,000 higher. Figure 4.7 shows that the expected value and the 95-confidence interval of the draw-down

Figure 4.6: The impact of the draw-down function with FRppr0 = 120% on the draw-down

amounts of an individual from age cohort τ at retirement in pension fund ABP. The first draw-down function is formulated in equation (3.15) and the second draw-draw-down function is formulated in equation (3.16). The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in PPR.

amounts of individuals older than age 25 calculated by the second function are lower than the expected value and the 95-confidence interval of the amounts calculated by the first

function in pension fund Young. In pension fund Old, the expected value and the 95-confidence interval of the amounts of all individuals calculated by the second function are lower than the expected value and the 95-confidence interval of the amounts calculated by the first function. Participants in pension funds Young and Old consist of fewer age cohorts compared to pension fund ABP. Therefore, the extra assets calculated by the second draw-down function are not enough to exceed the assets calculated by the first draw-down function for individuals older than age 25 in pension fund Young and all individuals in pension fund Old.

Figure 4.7: The impact of the draw-down function with FRppr0 = 120% on the draw-down

amounts of an individual from age cohort τ at retirement in the pension fund Young and Old. The first draw-down function is formulated in equation (3.15) and the second draw-down func-tion is formulated in equafunc-tion (3.16). The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in PPR.

2.2 The case of FR

equals 90

When FRppr0 = 90%, the expected draw-amount of a 66-year old at t = 1 calculated by

the second draw-down function is lowered by 5 due to the deficit in PPR. The other 5 is kept in cash in the pension fund. As a result, the assets of the pension fund decrease more than the assets calculated with the first draw-down function as age cohort τ decreases. The funding ratio becomes much lower as age cohort τ decreases.

The results of PPR in pension fund ABP are shown in Figure 4.8. The expected value and the 95-confidence interval of the draw-down amounts of individuals of age cohorts 49 until 66 calculated by the second function are higher than the expected value and the 95-confidence interval of the amounts calculated by the first function. The expected draw-down amounts of the younger individuals calculated by the second function are much lower such that the assets the pension fund are insufficient to pay the draw-down amount of a 25-year old. Figure 4.9 shows that in PPR, the expected value and the 95-confidence

Figure 4.8: The impact of the draw-down function with FRppr0 = 90% on the draw-down

amounts of an individual from age cohort τ at retirement in pension fund ABP. The first draw-down function is formulated in equation (3.15) and the second draw-draw-down function is formulated in equation (3.16). The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in PPR.

interval of the draw-down amounts of individuals older than age 25 calculated by the second function are higher than the expected value and the 95-confidence interval of the amounts calculated by the first function in pension fund Young. The expected value and the 95-confidence interval of the amounts of all individuals calculated by the second function are higher than the expected value and the 95-confidence interval of the amounts calculated by the first function in pension fund Old. Fewer age cohorts participate in the pension funds Young and Old compared to pension fund ABP. As a result, pension funds Young and Old have sufficient assets to pay the draw-down amounts of all individuals.

Figure 4.9: The impact of the draw-down function with FRppr0 = 90% on the draw-down

amounts of an individual from age cohort τ at retirement in the pension fund Young and Old. The first draw-down function is formulated in equation (3.15) and the second draw-down func-tion is formulated in equafunc-tion (3.16). The dashed lines represent the 95-confidence interval of the amounts and the solid line represent the expected draw-down amounts in PPR.

chapter 5

discussion

1

Implications of the results if the model is real

The draw-down amounts in PPR depend on the funding ratio of the pension fund because of the risk sharing aspect in this pension contract. The expected draw-down amounts are calculated by two draw-down functions. In the first draw-down function in PPR formu-lated in equation (3.15), the total surplus and deficit of the pension fund is shared among the participants. In the second draw-down function formulated in equation (3.16) only half of the surplus and the deficit of the pension fund is shared among the participants. The other half is held in reserve by the pension fund.

When the initial funding ratio FRppr0 equals 100, the second draw-down function

was identical to the first draw-down function. The results showed that in this case the expected draw-down amounts at age 67 in PPR were equal to the expected draw-down amounts at age 67 in the DC pension plan and higher than the expected draw-down amounts at age 67 in the guaranteed DB pension plan in the pension funds ABP, Young and Old. When FRppr0 equals 120 the expected draw-down amounts at age 67 in PPR

calculated by the first draw-down function became higher than the expected draw-down amounts at 67 in the DC pension plan due to the surplus in the start. In pension fund ABP, the expected draw-down amounts at age 67 of individuals younger than age 49 calculated by the second draw-down function were higher than the expected draw-down amounts at age 67 calculated by the first draw-down function. This is due to the extra assets that are kept in cash calculated by the second draw-down function in the pension fund. Hence, when pension fund ABP starts with a surplus of 20 it should use the second draw-down function for individuals younger than age 49 and the first draw-down function for the older individuals. Especially the amounts of young individuals were higher due to the increase of the extra assets that are kept in cash in the pension fund over time. The results also showed that when FRppr0 equals 90, the expected draw-down amounts at age 67 in PPR calculated

by the first draw-down function became lower than the expected draw-down amounts at 67 in the DC pension plan due to the deficit in the start. In pension fund ABP, the expected draw-down amounts at age 67 of young individuals calculated by the second draw-down function were much lower than the expected amounts calculated by the first draw-down function. The assets in the pension fund are insufficient to pay the draw-down amount

of a 25-year old. Pension fund ABP was not able to recover from a deficit of 10. There-fore, pension fund ABP should offer a DC pension plan to the participants in the case of a deficit of 10. The impact of the second draw-down function on the expected draw-down amounts was lower in the pension funds Young and Old than in pension fund ABP. This is because fewer age cohorts participate in these pension funds. When pension fund Young starts with a surplus of 20, it should calculate the draw-down amounts of individuals older than age 25 with the first draw-down function and the draw-down amounts of age cohort 25 with the second function. If pension fund Old starts with a surplus of 20 then it should use the first draw-down function for all participants in PPR. When both pension funds start with a deficit of 10, they should offer a DC plan to the participants.

2

Assumptions of the model

In this study we used the Black-Scholes-Vasicek model as the financial market model. This choice has influence on the results found and the conclusions. Four other popular financial market models are: the Black-Scholes model, the Black-Scholes-Black model, the model of Koijen, Nijman and Werker and the model of Commission Parameters 2014. The nominal short interest rate is constant in the Scholes model. The assumption in the Black-Scholes-Black model is that the nominal rate and the stock price are uncorrelated and that the nominal rate follows a random walk. Both assumptions are unrealistic. The models of Koijen, Nijman and Werker (Koijen, Nijman, & Werker, 2010) and of Commission Parameters 2014 (see Draper (2014)) model more variables and have more parameters than is necessary for this study. The more parameters, the more sensitive the results are. In order to be able to compare PPR with guaranteed DB and DC plans, it was necessary to make assumptions on the pension contracts. In a guaranteed DB pension contract, the pension benefit is guaranteed. However, nowadays pension funds invest in risky assets in order to realize the pension benefits. The risky investments ended up in a mismatch between the assets and the liabilities as a result of the developments in the financial market in the last years. The pension entitlements had to be cut. Thus, the pension benefit in a guaranteed DB contract is not guaranteed. Moreover, since the employer bears the risks in a DB contract, it is possible that the employer can jump in at any moment. These two phenomena are not considered in this study. The construction of the DC contract is the most realistic of the three pension contracts. The risk sharing aspect is included in PPR. Only the investment risk was shared. Longevity risk was not considered. When longevity risk is included in PPR, participants receive a biometric return for staying alive. This return is closely related to the mortality probability and is high at advanced ages. Hence, the biometric return insures against living longer than average (Bovenberg & Nijman, 2015).

chapter 6

conclusion

The aim of this paper was to investigate the impact of the initial financial position of a pension fund and the impact of the draw-down function in the newly proposed Personal Pension with Risk sharing (PPR). The expected draw-down amounts at retirement age 67 in PPR were compared to the expected draw-down amounts at 67 in guaranteed defined benefit (DB) and defined contribution (DC) pension plans in the Dutch pension fund ABP and the fictional pension funds Young and Old with respectively only young and old participants. The guaranteed DB pension plan was modeled without any investment risk. The Black-Scholes-Vasicek model was used as the financial market model with the nominal short rate and the stock price as risk factors. The expected draw-down amounts were calculated by two draw-down functions in PPR. The total surplus and deficit of the pension fund was shared among the participants in the first draw-down function. In the second draw-down function one half of the surplus and the deficit of the pension fund was shared among the participants and the other half was held in cash by the pension fund.

The results showed that when the initial funding ratio FRppr0 equals 100 in the three

pension funds, the expected draw-down amounts at age 67 calculated by both draw-down functions are equal to the expected draw-down amounts at age 67 in the DC pension plan and higher than the expected draw-down amounts at age 67 in the guaranteed DB pension plan. When FRppr0 equals 120, the expected draw-down amounts at age 67 calculated

by the first draw-down function in PPR achieved the highest draw-down amounts at age 67 due to the initial surplus. In pension fund ABP, the expected draw-down amounts at age 67 of individuals younger than age 49 calculated by the second draw-down function were higher than the expected down amounts at age 67 calculated by the first draw-down function. Especially the expected amounts of young individuals were higher due to the increase of the extra assets that were kept in cash in the pension fund over time. Hence, when pension fund ABP starts with a surplus of 20 it should use the second draw-down function for individuals younger than age 49 and the first draw-down function for the older individuals. When FRppr0 equals 90, the expected draw-down amounts at

age 67 calculated by the first down function, became lower than the expected draw-down amounts at age 67 in the DC pension plan and lower than the expected draw-draw-down amounts at age 67 in the DB plan of old participants in the pension funds. In pension fund ABP, the expected draw-down amounts at age 67 of young individuals calculated by

the second draw-down function were lower than the expected draw-down amounts at age 67 calculated by the first draw-down function. As a result, the assets in the pension fund were insufficient to pay the expected draw-down amount of the last age cohort 25. Pension fund ABP was not able to recover from a deficit of 10 in PPR and should therefore offer a DC pension plan to the participants in the case of a deficit of 10. The impact of the second draw-down function on the expected draw-down amounts in the pension funds Young and Old was lower than in pension fund ABP. This is because fewer age cohorts participate in the pension funds Young and Old.

In a DC pension contract the focus is more on wealth accumulation rather than on providing a stable retirement income during the decumulation phase. Further research is necessary to calculate the expected draw-down amounts not only during the accumulation phase but also during the decumulation phase. Eventually, an individual is more interested in the draw-down amount after retirement. In contrast to DC pension plans, in PPR retirees have the possibility to keep investing in risky assets during the decumulation phase. Retirees can keep earn risk premiums and earn biometric returns for staying alive during the decumulation phase. The annuity that an individual can buy with the accrued personal assets is in reality not guaranteed in PPR. Therefore, the study could be extended with modeling longevity risk besides investment risk. We also recommend for further research to model the guaranteed DB pension plan with more realistic features of the Dutch DB pension plans. Nowadays, pension funds are allowed to cut the pension benefits if the the pension fund is underfunded. So the pension benefit in a DB plan is not guaranteed. Furthermore, the DB pension plan could ideally be modeled with a stochastic salary pattern in order make the paid contributions stochastic over time.

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appendix a: r code

# clear

rm(list = ls()) # read in

#test1 <- read.delim("C:/Users/Gebruiker/Desktop/Dropbox/Thesis/R/test1.txt")

#sigmas35 <- read.csv("C:/Users/Gebruiker/Desktop/Dropbox/Thesis/R/TFT/sigmas35.csv", sep=";") #loonstijging <- read.csv("C:/Users/Gebruiker/Desktop/Dropbox/Thesis/R/loonstijging.csv") #weights <- read.csv("C:/Users/Gebruiker/Desktop/Dropbox/Thesis/R/weights.csv", sep=";", dec=",") # variables looninflatie <- 0.03 max_sal_op <- 101519 max_salstijging_op <- 0.01 stijging_franchise <- 0.02 begin_sal <- 38400 begin_franchise <- 12953 contrib_rate <- 0.18 # read from tables jaar <- loonstijging$jaar t <- loonstijging$t indiv_loonstijging <- loonstijging$Indiv.loonstijging ages <- loonstijging$leeftijd act_factors <- weights$Actuarial.factors alpha <- weights$Alpha.DC # Calculate salary

cumulative_indiv_loonstijging <- cumprod(1 + indiv_loonstijging) cumulative_looninflatie <- (1 + looninflatie)^t

salaris <- begin_sal * cumulative_indiv_loonstijging * cumulative_looninflatie

# calculate maximum salary

max_sal2 <- max_sal_op * (1+max_salstijging_op)^t feasible_salary_op <- pmin(salaris, max_sal2) # calculate franchise

franchise <- begin_franchise * (1+stijging_franchise)^t # calculate pensionable salary PG