The impact of life-cycle investments on the pension benets of participants in a dened contribution plan

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the pension benefits of participants in

a defined contribution plan

Yvette de Koning

Thesis for the Master Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Y.W. de Koning BSc

Student no.: 10346120

Email: yvettedekoning@hotmail.com

Date: July 15, 2016

Supervisor: dhr. dr. S. van Bilsen

Supervisors EY: dhr. drs. R. van Daalen AAG dhr. B. Baggen MSc AAG dhr. J. Nolting MSc

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Statement of Originality

This document is written by Yvette de Koning BSc who declares to take full responsi-bility 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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Summary

In recent years there has been a transition from DB to DC pension plans. In a DC pension plan, investing the premiums and earned investment returns is often done according to a chosen life-cycle. Most pension administrators offer a default life-cycle, but these default life-cycles differ a lot. Most people choose suboptimal savings and investment strategies in DC plans by following the given defaults. So before an employer can make a choice for a specific DC plan, the life-cycles of different pension administrators should be analyzed. The purpose of this study is to investigate the impact of life-cycle investments on the pension benefits of participants in a DC plan. To investigate the impact of life-cycle investments on the pension benefits of participants in a DC plan, pension benefits for a predetermined asset mix are deter-mined. Long-term investment returns for different asset classes (the risk-free rate, a two-year zero-coupon bond, a ten-year zero-coupon bond and the AEX index) are simulated and some predetermined asset mixes are defined. The investment returns of the risk-free rate, a two-year zero-coupon bond and a ten-year zero-coupon bond are simulated with the CIR-model. The returns of the AEX index are simulated with the jump-diffusion model. With the simulated investment returns and the predetermined asset mixes, the accrued pension capital at the retirement age is calculated. This capital is converted into a lifelong old age pension by a conversion factor. In this study, 10,000 scenario’s for the accrued capital and the pension benefit at retirement are simulated for a certain asset mix. A comparison for the predetermined asset mixes is made based on summary statistics and other performance criteria. To gain more insight in the strategies, an optimal asset mix is determined by maximizing the CVaR for a confidence level of 2.5%. Maximizing the CVaR should minimize the downside risk. A backward optimization algorithm is used and it is allowed to vary the asset mix at any point in time. Subsequently, the 2.5%-CVaR of the optimal as-set mix is compared with the values of the 2.5%-CVaR of the predetermined asas-set mixes. Based on summary statistics and other performance criteria it can be concluded that a risk-averse participant should choose the asset mix from the risk-free life-cycle model in which 100% is invested in the risk-free rate. The asset mixes of the constant life-cycle model, the deterministic life-cycle model and Merton’s life-cycle model lead to suboptimal results. The 2.5%-CVaR for the asset mix of the risk-free life-cycle model is the highest if it is not allowed to vary the asset mix at any point in time and the predetermined asset mixes are taken into account. When the 2.5%-CVaR is maximized by using a recursive optimization algorithm where it is allowed to vary the asset mix at any point in time, there should be mainly invested in the risk-free rate and ten-year zero-coupon bonds. However, since the difference between the 2.5%-CVaR for this optimal asset mix and the 2.5%-CVaR for the asset mix where 100% is invested in the risk-free rate is negligible, it can be concluded that a risk-averse participant should invest only in the risk-free rate to minimize the downside risk.

Keywords defined contribution plan, optimal asset allocation, life-cycle investments, CIR-model, jump-diffusion model

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Preface

This is the thesis ‘The impact of life-cycle investments on the pension benefits of partic-ipants in a DC plan’. This thesis concludes my Master of Science education in Actuarial Science and Mathematical Finance at the University of Amsterdam (UvA) in Amster-dam. The thesis is performed in the period February 2016 to July 2016 in collaboration with EY Netherlands at the department European Actuarial Services in Amsterdam. Several persons have contributed academically and practically to this master thesis. I would therefore firstly like to thank my supervisor at the UvA, dhr. dr. S. van Bilsen. Secondly, I want to thank EY Actuarissen BV for offering me this internship. In addi-tion, I would like to thank my supervisors at EY, dhr. drs. R. van Daalen AAG, dhr. B. Baggen MSc AAG and dhr. J. Nolting MSc for their time and valuable input. Finally, I would like to thank my family and friends for being helpful and supportive during my time studying Actuarial Science and Mathematical Fincance at the UvA.

I wish you a lot of reading pleasure. Yvette de Koning BSc

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Chapter 1

Introduction

The exceptionally poor financial market performances seen in 2008 sent a huge shock wave across the pension world. Due to the subsequent economic and financial crisis defined benefit (DB) pension plans are gradually losing their dominance in the occupa-tional pension system of many countries and are replaced by defined contribution (DC) pension plans (Broadbent, Palumbo & Woodman,2006). The transition from DB to DC plans involves shifting risks from the employers to the employees. With an aging popu-lation, decreasing market interest rates, changing social views and shifting employment patterns, occupational pensions in the Netherlands have become a timely and contro-versial policy issue.

In a DB pension plan, an employer promises a specified monthly retirement benefit to the employee that is predetermined by a formula based on the employee’s earnings his-tory, tenure of service and age, rather than depending directly on individual investment returns. In a DC pension plan, participants accrue capital in individual accounts admin-istered by the pension administrator. The pension entitlements accumulated during the employee’s working career will depend on the contributions made during participation in the pension plan and the return on investments achieved with those contributions. Through the investment of the premiums and the earned investment returns a capital at retirement is accrued which is used to purchase a lifelong annuity. The value of the retirement benefits depend mainly on the interest rate and the life expectancy at re-tirement. Since these elements remain uncertain until the retirement date, the level of retirement benefits remain also uncertain until retirement.

In a DC pension plan, investing the premiums and earned investment returns is often done according to a chosen life-cycle. Each life-cycle fund has a final date. In practice, the risk of the investments is gradually reduced as the final date comes closer by con-verting stocks into bonds. Also within an asset class the risk is reduced as the final date approaches. Each pension administrator has developed its own life-cycle profiles. Most pension administrators offer three different life-cycle profiles: (1) a default life-cycle; (2) an offensive life-cycle; and (3) a defensive life-cycle. However, the default life-cycles of different pension administrators differ a lot, for example in strategic investment al-location by age cohort, the way of risk reduction as the age of participant increases and the investment products in which is actually invested. Most people choose subopti-mal saving and investment strategies in DC plans by following the given defaults (Choi, Laibson, Madrian & Metrick,2004;Beshears, Choi, Laibson & Madrian,2009;Benartzi, Peleg & Thaler, 2013). So, before an employer can make the choice for a specific DC plan, the life-cycles of different pension administrators should be analyzed. Therefore, the purpose of this study is to investigate the impact of different life-cycle investments on the pension benefits of participants in a DC plan.

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A thorough analysis of the offered life-cycles is performed by comparing the annual pension benefit for a particular participant when premiums and earned investment re-turns are invested in different predetermined life-cycles and the accrued pension capital at retirement is converted into a lifelong annuity. Since the earned investment returns are important for the value of the pension benefits, long-term investment returns for different asset classes are simulated based on real market data. In this study four asset classes in which can be invested are chosen: (1) the risk free-rate; (2) a two-year zero-coupon bond; (3) a ten-year zero-zero-coupon bond and (4) the AEX index. The returns of the risk-free rate and the zero-coupon bonds are simulated with the Cox-Ingersoll-Ross (CIR) model of Cox, Ingersoll & Ross (1985) and the return of the AEX index is simulated with the jump-diffusion model ofMerton (1976). These models are chosen since they are commonly used in practice and reproduce a realistic financial market. The simulated long-term investment returns are used to determine the pension benefit in each scenario for a predetermined asset mix for a particular participant. The simu-lated pension benefits for a predetermined asset mix are compared with the simusimu-lated pension benefits for another predetermined asset mix by using summary statistics and other performance criteria. By performing a sensitivity analysis taking into account dif-ferent asset mixes, the question about the impact of difdif-ferent life-cycle investments on the pension benefits of participants in a DC plan can be answered.

This thesis is structured as follows. Chapter 2 provides a general introduction to re-tirement plans in the Netherlands. In Section 2.1 the Dutch pension system is discussed and in Section 2.2 a general overview of life-cycles is given. Chapter 3 discusses how the input for the calculation of the pension benefits is determined. In Section 3.1 methods to simulate investment returns on the long term are discussed. The CIR-model and the jump-diffusion model are implemented in MATLAB and the simulated investment re-turns are used as input for the calculation of the pension benefits. Then, in Section 3.2 different life-cycle models are discussed and four predetermined asset mixes are chosen as input for the calculation of the pension benefits. To determine the asset mix of the life-cycle model ofMerton (1969) expected utility is maximized taking into account the risk-free rate and the AEX index. In Chapter 4 the results of this study are presented. In Section 4.1 it is discussed how pension benefits are determined based on the investment returns (simulated with the methods described in Section 3.1) and a certain life-cycle model (as described in Section 3.2). To compare the simulated pension benefits of one asset mix with the simulated pension benefits of another asset mix, summary statistics and other performance criteria are used. This is described in Section 4.2. To maximize the accrued pension capital the pension administrator could, for example, follow an expected utility approach and maximize concave utility functions. This is done in the life-cycle model of Merton (1969) in Section 3.2. In addition to the optimization of ex-pected utility, a possibility is the optimization of (tail) risk measures. In Chapter 5 the Conditional Value-at-Risk (CVaR) is optimized by using a backward recursive optimiza-tion algorithm. The closing chapter, Chapter 6, contains a conclusion, the limitaoptimiza-tions of this study and suggestions for future work.

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Chapter 2

Retirement plans

In this chapter the three pillars of the Dutch pension system are described. The focus is on the second pillar which can be broadly distinguished in DB, DC and collective defined contribution (CDC) plans. In recent years there has been a transition from DB to DC plans (Broadbent et al.,2006). Therefore, this study focuses on DC schemes and in particular on life-cycles which are discussed in the second section of this chapter.

2.1 The Dutch pension system

2.1.1 The three pillars and the pension triangle

As in many other European countries, the Dutch pension system consists of three pil-lars: the state pension, the occupational collective pensions and the private individual pension products (Bovenberg & Meijdam,2001).

The first pillar consists of payments made by the government to everybody who is older than the retirement age, conditional on living in the Netherlands since the age of fifteen, and is called AOW (which is an abbreviation for the Dutch term Algemene Ouderdomswet ). AOW is meant to prevent poverty amongst the elderly.

The second pillar consists of the occupational collective pension schemes. Al-though there is no statutory obligation for employers to offer a pension scheme to their employees, more than 90% of the Dutch employees accrue a supplementary pension in the second pillar on top of the AOW (De Nederlandsche Bank,2015a). By Dutch law the employer and the pension administrator are strictly separated. The pension schemes in the second pillar are administered by a pension fund, an insurance company or a Pre-mium Pension Institution (PPI). In 2016 a new pension administrator, the so-called APF (Algemeen Pensioenfonds) has entered the Dutch market.

The third pillar of the Dutch pension system is voluntary and relatively small. It includes individual pension provisions, either through annuity or endowment insurance, encouraged by tax reliefs up to certain limits. The third pillar is mainly used by the self-employed and employees in sectors without a occupational collective pension plan. The first pillar is financed on a pay-as-you-go (PAYG) basis, in such a way that the currently active part of the population pays the contributions that are used to finance the retirement benefits of the retirees. As can be seen in Table2.1the pension system in the larger continental European countries relies almost exclusively on PAYG financing (B¨orsch-Supan,2004). The second and third pillars are organized on a funded basis. In the second pillar, employees and employers pay contributions to a pension fund, insur-ance company or PPI that executes the occupational pension arrangement offered by the employer to the employee. These contributions are part of the employment contract and can be considered to be deferred wage income. Third pillar pensions consist of pen-sion savings to provide additional retirement income that individuals privately decide on.

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Table 2.1: Pension systems in various European countries (seeB¨orsch-Supan(2004)).

Percentage (%) of total retirement benefits NL DE FR IT ES CH GB US PAYG public pensions 50 85 79 74 92 42 65 45 Occupational pensions 40 5 6 1 4 32 25 13 Personal pensions 10 10 15 25 4 26 10 42

In case of second pillar pensions, the employer, the employee and the pension adminis-trator are in a triangular relationship as can be seen in Figure2.1. Three relations that are described in this triangle are: (1) the relationship between the employer and the employee, who together have entered into a pension agreement; (2) the relationship be-tween the employer and an external pension administrator, to whom the administration of the pension agreement is outsourced on the basis of a financing agreement; and (3) the relationship between the pension administrator and the employee, as a result of the outsourcing of the administration of the pension agreement. The pension administrator provides the employee with the pension scheme and starting letter.

The pension agreement is an agreement between the employer and the employee and is part of the labour agreement between employer and employee. Since the employer and the pension administrator should be strictly separated by law in the Netherlands, the pension agreement is administrated and executed with an external administrator. The financing agreement contains the funding and exit conditions between the employer and the pension administrator. The pension administrator will prepare the pension plan rules in which the details of the pension agreement are documented.

Figure 2.1: The triangular relationship between employer, employee and pension administrator.

As of January 2015 fiscal legislation with respect to second pillar pensions in the Nether-lands has changed (Rijksoverheid,ndc). The pensionable salary in second pillar pensions is capped ate100,000. As of January 2014, the fiscal normal retirement age for second pillar pension plans has been increased from 65 to 67 years. For second pillar pension plans, the maximum accrual rate for old age pension in career average DB plans is currently 1.875% per year. This used to be 2.25%. For final pay plans, the accrual rate has decreased from 2.0% to 1.657%. In the Netherlands, fiscal boundaries for DC plans are derived from the fiscal accrual rates of DB plans. Along with the decrease in fiscal maximum accrual rates for DB plans, the fiscal maximum contribution rates in DC plans have also been lowered in 2014 and 2015.

2.1.2 Types of pension agreements

In the Netherlands there are different types of pension agreements. Broad distinction can be made between DB plans, DC plans and collective defined contribution (CDC) plans.

A DB pension plan is a type of pension plan in which an employer promises a specified monthly retirement benefit to the employee that is predetermined by a formula based on the employee’s earnings history, tenure of service and age, rather than depending

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directly on individual investment returns. Employers can choose between several for-mulas when determining the final retirement benefits where the most commonly used formulas are career average formulas and final pay formulas. In career average formulas, the benefit at retirement equals a percentage of the career average wage, multiplied by the participant’s number of years of service. Final pay formulas base benefits on average earnings during a specified number of years at the end of a participant’s career; this is presumably the time when the salaries are the highest. The benefit equals a percentage of the participants final average wage, multiplied by the number of years of service. In the Netherlands, career average pay schemes are very common while final pay schemes continue to decline (De Nederlandsche Bank,2015b). The contribution is calculated by taking into account the life expectancy, the market interest rate, the retirement age, the age of entering the pension plan, the annual accrual rate and eventually indexation. For each participant the same uniform contribution in percentage of the wage should be paid. This percentage is based on the total contribution needed to cover the total annual accrual. Since individual differences such as age, gender, health and income are not taken into account when determining the amount of contribution to be paid, (in-tergenerational) solidarity is achieved. Since the contribution is regularly evaluated and adjusted to minimize the risk that the pension administrator cannot pay the benefits, the contribution payer is the one that bears the investment risk, longevity risk and interest rate risk. If the pension administrator earns a lower than expected yield, the employer will have to make additional contributions in order to provide the promised benefits.

In a DC pension plan, participants accrue capital in individual accounts administered by the pension administrator. The pension benefit accumulated during the employee’s working career will depend on the contributions made during participation in the pen-sion plan and the return on investments achieved with those contributions. Contribu-tions may be made by the employee, the employer, or both. In DC plans the employee bears the investment risk, the longevity risk and the interest rate risk. The typical pay-off of a DC plan is a lump sum benefit at retirement. This capital should not be taken as a cash amount but should be used to finance an old age pension whether or not in combination with a spouse pension. During the years of participation, the spouse pension and the disability pension are often insured on risk basis. In addition to the contributions, the employer will have to pay the risk premiums and administration costs to the pension administrator separately.

In Table 2.2the advantages and disadvantages of DB and DC plans are shown.

Table 2.2: Advantages and disadvantages of DB and DC plans.

Pension plan Advantages Disadvantages DC-plan - Pension expenses more manageable

- No balance sheet obligation

- Pension benefit at retirement not known in advance

- Risks are transferred from employer to employee

DB-plan - Level of pension benefit is known in advance

- More easy to communicate and to understand

- Pension expenses less manageable - Possible balance sheet obligation - Could be expensive due to low interest

rates or high guarantee fees or back-service obligation

In the Netherlands, there are also CDC plans in addition to the DB and DC plans as discussed above. A CDC plan is a pension plan where the employer is only required to pay a fixed premium of which the level is agreed upfront. The employer is not responsi-ble for additional payments in case of underfunding. The difference with an individual DC plan is that the collective element is retained in CDC plans.

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2.1.3 Transition of DB to DC plans

Traditional DB plans are gradually losing their dominance in the occupational pension systems of many countries (Broadbent et al., 2006). The transition from DB to DC plans in pensions shifts various risks from the employers to the employees. Employ-ers are declining to continue to sponsor employer-sponsored DB pension plans due to the high risks these pension plans are now seem to impose on employers. As a result, employees are becoming increasingly exposed to financial risks, and retirement income may be subject to greater volatility than before. Some empirical evidence suggests that people do not like fluctuations in pension income (van Els, van den End & van Rooij,

2005). Hence, the shift away from DB plans is considered to be undesirable based on the current market conditions from the perspective of the employees. An advantage for employees is that the shift to DC plans creates the opportunity to design new pension plans that are better tailored to the specific needs of participants and the current market environment. From the perspective of the employers the shift from DB plans to individ-ual plans is suitable since employers do not longer bear pension risks and can strengthen their competitive position in a more and more complex dynamic world economy.

With an aging population, decreasing market interest rates, changing social views and shifting employment patterns, occupational pensions in the Netherlands have be-come a timely and controversial policy issue. Dutch pension plans have mainly preserved their DB character in recent years, although they have switched from final pay to average wage plans (Ponds & van Riel,2007;De Nederlandsche Bank,2015b).

2.2 Life-cycles in practice

DC pension plans offer each participant the freedom to choose and to implement the optimal consumption and investment strategy according to their own needs. Life-cycle theory has shown how to determine the optimal saving and investing strategies. In life-cycle investment strategies it is assumed that human capital, which is defined as the discounted value of future labor income, can be seen as a risk-free asset (Merton,1969;

Samuelson, 1969; Bodie, Merton & Samuelson, 1992; Campbell & Viceira, 2002). As a consequence, the optimal proportion of financial assets invested in equity should be decreased over the life-cycle and the optimal proportion invested in bonds should be increased. This is because the fraction of human capital is high early in life compared to the fraction of financial wealth, as can be seen in Figure2.2(Ibbotson, Milevsky, Chen & Zhu,2007). Young employees are less dependent on financial wealth for consumption since they have labor income as alternative income source. It is therefore affordable for them to take more risk with financial wealth than elderly employees who almost entirely depend on this type of wealth for their consumption.

Figure 2.2: Financial capital and human capital as share of total wealth over the life-cycle (seeIbbotson et al.(2007)).

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Pension administrators offer different life-cycle profiles. The concept of life-cycle profiles is that age-dependent equity allocation presents optimal life-cycle saving and investment models. Within the life-cycle supply of pension administrators, participants can often choose between different risk profiles. The pension administrators have different profiles ranging from highly offensive to very defensive. The asset mix of a default life-cycle has a lower risk than an offensive life-cycle and a higher risk than a defensive variant. All profiles contain the same concept of reducing risk as the retirement age approaches. The life-cycle profiles differ in strategic investment allocation by age cohort, the way of risk reduction as age of participant increases and the investment products in which is actually invested.

In Figure 2.3 (Centraal Beheer Achmea, 2015) three life-cycles offered by one of the biggest insurance company in the Netherlands, Centraal Beheer Achmea, are shown. Achmea invests in nine investment funds and offers three life-cycles profiles: (1) a de-fault life-cycle; (2) an offensive life-cycle; and (3) a defensive life-cycle.

In the offensive life-cycle a small percentage of the assets is invested in funds with a low risk exposure. At the start of the working career, 67 per cent of the assets is invested in shares which are volatile. This is almost equal to the percentage of assets invested in shares in the default life-cycle at the start of the working career. However, in the offensive life-cycle the reduction of risk is started five years later than in the default life-cycle. So, the offensive life-cycle yields a higher expected pension benefit, but the risk is also higher which may result in a lower expected pension benefit. In the defensive life-cycle only 40 per cent of the assets is invested in shares at the start of the working career. The reduction of risk is started five years earlier than in the default life-cycle. The defensive cycle yields a lower expected pension benefit than the default life-cycle, but less fluctuations are expected in the pension capital at retirement.

Figure 2.3: Achmea life-cycle investments: the default life-cycle, the offensive life-cycle and the defensive life-cycle (seeCentraal Beheer Achmea(2015)). The reduction of risk starts first for the defensive life-cycle followed by respectively the default life-life-cycle and the offensive life-life-cycle.

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Most people choose suboptimal saving and investment strategies in DC schemes by following the given defaults (Choi et al., 2004; Benartzi et al., 2013; Beshears et al.,

2009). The pension administrator is responsible for the construction of optimal life-cycle models, but the default life-life-cycle models of various pension providers are very different. So, before an employer chooses a certain life-cycle a thorough analysis is of importance.

2.3 Summary

The Dutch pension system consists of three pillars: the compulsory PAYG state pension, the occupational funding-based occupational pension and private individual pension products. In case of second pillar pensions, the employer, the employee and the pension administrator are in a triangular relationship. The relationship between the employer and the employee is captured in a pension agreement. DB, DC and CDC schemes are different types of pension agreements. In the Netherlands, the occupational pensions have preserved the DB character in recent years, although they have switched from final pay to average wage plans. There is also a shift from DB to DC plans. DC plans offer each participant the freedom to choose and to implement the optimal consumption and investment strategies according to their own needs, often according to a chosen life-cycle. Dutch pension administrators have designed their own life-cycle profiles which differ in strategic investment allocation by age cohort, the way of risk reduction as age of participant increases and the investment products in which is actually invested. Most pension administrators offer three different life-cycles: (1) a default life-cycle; (2) an offensive life-cycle; and (3) a defensive life-cycle. Since most people choose suboptimal saving and investment strategies by choosing the default life-cycle, a thorough analysis of the offered life-cycle is of importance. In Chapter 3 it is discussed how the input for the model used in this analysis is determined and in Chapter 4 the analysis is performed.

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Chapter 3

Life-cycle investments

Simulating the correct expected rates of return for stocks and bonds on the long-term is important for pension planning. Applying incorrect returns has severe consequences. When low expected returns are used, more contributions are required than in the case when high expected returns are assumed to buy the same annuity. In Section 3.1 the CIR-model ofCox et al.(1985) and the jump-diffusion model ofMerton (1976) are dis-cussed and implemented in MATLAB. In Section 3.2 different types of life-cycle models are discussed. The simulated long-term returns from Section 3.1 and four predetermined asset mixes chosen from different life-cycle models as discussed in Section 3.2 are the input for the analysis which is performed in Chapter 4.

3.1 Simulation of investment returns on the long term

3.1.1 CIR-model for risk-free returns and bond returns

One of the oldest approaches to simulate returns for bonds are the one-factor models of

Merton(1973) andVasicek(1977) where the short-rate determines the future evolution of all interest rates. The short-rate is assumed to be normally distributed, having the theoretical possibility to become negative. To overcome this drawback Dothan (1978) andRendleman Jr. & Bartter(1980) proposed a log-normal distribution for the instan-taneous short-rate andCox et al.(1985) proposed a non-central chi-square distribution. Nowadays, central banks give negative interest rates to the deposit as a monetary policy tool to discourage banks from depositing their excess cash with the central banks. For example on 10 March 2016 the European Central Bank (ECB) cut rates again, charging banks 0.4 percent to hold their cash overnight. At the same time, it offered a premium to banks borrowing in order to issue more loans (European Central Bank, 2016a). Al-though the interest rate of the ECB is negative, banks in general have been reluctant to go below zero on retail rates to prevent losing customers. As such, the assumption that interest rates can not become negative has become realistic. In the Vasicek model the assumption of mean reversion, a theory suggesting that rates of returns tend to revert back to their long-term average, is made. The mean reversion property in the Vasicek model is preserved in the CIR-model which is not the case for the models of

Dothan (1978) andRendleman Jr. & Bartter (1980). Since mean-reversion for risk-free returns and bond returns is well-accepted in the literature, the CIR-model is preferred over the model of Dothan (1978) and Rendleman Jr. & Bartter (1980). Based on this observations this study uses the CIR-model.

In the CIR model the instantaneous short-rate follows the stochastic differential equa-tion (SDE)

drt= θ(µ − rt)dt + σ

√

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where rt is the short-rate and a time dependent stochastic variable, µ > 0 is the mean

reversion level, θ > 0 is the speed of mean reversion, σ > 0 is the volatility and dWt

is a standard Brownian motion increment. Cox et al. (1985) showed that whenever 2θµ > σ2, the interest rate is strictly larger than zero, which is the key advantage of the CIR-model over the Vasicek model. Furthermore, there is empirical evidence for the volatility term in the CIR model because whenever interest rates are high, the volatility is likely to be high as well.

The CIR model provides explicit solutions for the bond prices. The price of a zero-coupon bond at time t with maturity T has the form

P (t, T ) = A(t, T ) exp (−B(t, T )rt) , (3.2) where A(t, T ) = 2γ exp ((θ + γ)(T − t)/2) 2γ + (θ + γ)(exp (γ(T − t)) − 1) 2µθ σ2 , (3.3) B(t, T ) = 2(exp((T − t)γ) − 1) 2γ + (θ + γ)(exp (γ(T − t)) − 1) (3.4) and γ =pθ2_{+ 2σ}2_. _(3.5)

P (t, T ) is the value of a zero-coupon bond at time t that pays one euro at maturity time T and R(t, T ) is the corresponding continuously compounding interest rate. In continuous time it holds that

P (t, T ) = exp(−R(t, T )(T − t)). (3.6)

Rewriting equation (3.6) to express the interest rates as a function of the value of a bond yields

R(t, T ) = − 1

T − tlog(P (t, T )). (3.7)

Rewriting equation (3.6) as in (3.7) and then substituting (3.2) in this equation gives a function to compute the term structure in the CIR model:

R(t, T ) = − 1

T − t[log(A(t, T )) − (B(t, T )rt)] , (3.8) with A(t, T ) as in (3.3), B(t, T ) as in (3.4) and γ as in (3.5).

A discrete version of the short-rate with time step ∆t = ti − ti−1 for i = 0, 1, 2... is

defined as

r∆t = θ(µ − rt)∆t + σ

√

∆t√rtZ,

with Z standard normally distributed. When the property of additivity of term structure models is taken into account, the model can be written as

rt+∆t= rt+ r∆t

= rt+ θ(µ − rt)∆t + σ

√

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The risk-free rate is the yield on high quality government bonds. The short-rate can be simulated with the discrete equation as defined in (3.9). Subsequently, the full term structure can be deduced from equation (3.8). The term structure of interest rates, which is also called the spot rate curve, at a certain time t defines the relation between the zero rates and the time to maturity T − t. By construction the continuous CIR model as defined in (3.8) does not produce negative short-rates, but the discrete model as defined in (3.9) does. Therefore, the negative yields must be replaced by values strictly larger, but close to zero, in the simulation of the discrete term structure. The estimates for θ, µ and σ are necessary in this simulation. To estimate θ, µ and σ the density of the CIR-process is required. Cox et al.(1985) defined the density of rti given rti−1 as

f rti|rti−1; θ, µ, σ, ∆t = c exp(−uti−1 − vti)

vti uti−1 q₂ Iq(2 √ uti−1vti) with c = 2θ σ2_{(1 − exp(−θ∆t))},

uti−1 = crti−1exp(−θ∆t),

vti = crti,

q = 2θµ σ2 − 1,

and Iq(2√uti−1vti) is a modified Bessel function of the first kind of order q. The

distri-bution function is the non-central chi-square distridistri-bution χ2[2crti, 2q + 2, 2uti−1] with

2q + 2 degrees of freedom and parameter of non-centrality 2uti−1 proportional to the

current spot rate. When the non-centrality parameter tends to zero, the probability density function simple reduces to a chi-squared probability density function.

The parameters θ, µ and σ can be estimated with maximum likelihood estimation (MLE). The MLE approach is often preferred because the estimators are unbiased, asymptotically normally distributed and asymptotically the best since the Cram´er-Rao lower bound is attained (Serlin,2007). The log-likelihood function is given by

ln(L(θ, µ, σ)) = N X i=0 ln f rti|rti−1; θ, µ, σ, ∆t = N X i=0 ln c exp(−u − v) v u q 2 Iq(2 √ uv) = N X i=0 ln(c) + N X i=0 −u − v +q 2ln v u + ln Iq(2 √ uv) = N ln(c) + N X i=0 −u − v + q 2ln v u + ln Iq(2 √ uv) , (3.10)

where N is the number of observations. Maximizing the log-likelihood function gives the following optimization problem

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Equation (3.11) can be solved with the function fminsearch in MATLAB, which finds the minimum of a scalar function of several variables, starting at an initial estimate. This is generally referred to as unconstrained nonlinear optimization. Initial points of optimization for convergence to the global optimum are crucial. Initial estimates can be found by applying Ordinary Least Squares (OLS). Equation (3.9) can be rewritten as

rt+∆t− rt= θ(µ − rt)∆t + σ √ rtt or also as rt+∆t− rt √ rt = θµ∆t√ rt − θ√rt∆t + σt, (3.12)

where t∼ N (0, ∆t). The initial estimates ˆθ and ˆµ can be found by minimizing the OLS

objective function (ˆθ, ˆµ) = arg min_θ,µ N X i=0 rti− rti−1 √ rti−1 −_√θµ∆t rti−1 − θ√rti−1∆t !2 . (3.13)

The initial estimate ˆσ is found as the standard deviation of the residuals.

The initial estimates are the starting points for maximizing the log-likelihood func-tion in equafunc-tion (3.10). Before applying maximum likelihood estimafunc-tion to estimate the optimal parameters ˆθ, ˆµ and ˆσ which are necessary for the simulation of the short term rate, the modified Bessel function of the first kind Iq(2√uti−1vti) must be evaluated.

Direct implementation into MATLAB of the Bessel function Iq(2√uti−1vti) results in

an estimation failure because the Bessel function diverges to plus infinity and optimiza-tion routines are not able to handle this problem. Kladivko (2007) solves the problem of divergence by suggesting a scaled version of the Bessel function. The scaled Bessel function is given by Iq(2

√

uti−1vti) exp(−2

√

uti−1vti) and is available under the

com-mand besseli(q, 2*sqrt(u.*v),1). Equation (3.10) is adjusted for the scaled Bessel function into (N − 1) ln(c) + N X i=0 −u − v +q 2ln v u + ln I_qs(2√uv) + 2√uv, (3.14) where I_qs(2√uv) is the scaled Bessel function.

The MATLAB-implementation of estimating the initial parameters and estimating the parameters in the CIR-model with the log-likelihood function is given in Appendix A. After simulating the yield curve from equation (3.8) the short-rate is simulated over time. The return on the zero-coupon bonds is derived from the short-rate. These imple-mentations can also be found in Appendix A. In this study is the return of a two-year zero-coupon bond and a ten-year zero-coupon bond simulated in addition to the short-rate.

The 6-month Euro Interbank Offered Rate (Euribor) rate from the period 4 May 2006 until 4 May 2016 is chosen as input for the CIR-model. The 6 month Euribor is the in-terest rate at which a selection of European banks lend one another funds denominated in euros whereby the loans have a maturity of six months. Alongside the 6-month Eu-ribor interest rate there are another fourteen EuEu-ribor interest rates with different time to maturities. The Euribor interest rates are the most important European interbank offered interest rates. The LIBOR for interbank loans in currencies such as US dollars and pounds sterling on the London money market is similar to the Euribor for loans

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in euros on the European money market. When the Euribor interest rates rise or fall there is a high likelihood that the interest rates on banking products such as mortgages, savings accounts and loans will be adjusted. As the 6-month Euribor rate is the most liquid rate, the 6-month Euribor rate is chosen as input for the CIR-model.

In Figure 3.1 the daily 6-month Euribor rates are shown. The data is obtained from Bloomberg. The 6-month Euribor rate became negative for the first time on 11 Novem-ber 2015. Since the CIR-model only provides rates strictly larger than zero, all negative rates in the data set are substituted by 1 · 10−8. In Figure3.2the estimated yield curve is shown at t = 0. It can be seen that the yield curve is concave downward sloping, also called normal, which indicates that investors require a higher rate of return for taking the added risk of lending money for a longer period of time. When the starting point for simulating the short-rate is chosen as 4 May 2006 it can be shown that the simu-lated paths for the short-rate over time t (for the period 4 May 2006 until 4 May 2016) approximate the 6-month Euribor rate well. Therefore, the CIR-model can be used for forecasting over a period from the age of a particular participant until the pension age of this person. In Figure 3.3 the forecasting over 52 years is shown from 2016 until 2068. A forecasting period of 52 years is chosen since the AOW is accrued since the age of fifteen as described in Section 2.1.1. If a pensionable age of 67 is assumed and a particular participant starts to accrue pension in the second pillar since the age of fifteen, a forecasting period of exactly 52 years is needed.

Figure 3.1: The 6-month Euribor rate (daily) for the period 4 May 2006 until 4 May 2016, obtained from Bloomberg.

Figure 3.2: The yield curve estimated by the CIR-model ofCox et al.(1985). The yield is equal to the return on a zero-coupon bond of maturity T at t = 0.

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Figure 3.3: In the figure on the left the original time series, the 6-month Euribor rate (daily) for the period 4 May 2006 until 4 May 2016, is shown. In the figure on the right the simulated (20 simulations) paths for the short-rate for the period 2016-2068 are shown.

3.1.2 Jump-diffusion model for stock returns

A classical model to simulate stock returns is the log-normal diffusion process such that the log-return process has a normal distribution. However, this model differs from em-pirical observations in the real market. The market distribution for stocks has several properties which are not reflected in the log-normal diffusion process: (1) large random fluctuations such as crashes must be allowed by the model; (2) since large downward outliers are larger than upward outliers a skewed distribution is demanded; and (3) the distribution should have a kurtosis greater than three (leptokurtic) because the distribu-tion will have a higher peak and two heavier tails than those of the normal distribudistribu-tion. For modeling these extra properties, a diffusion process with log-uniform jump-amplitude Poisson process is usually used.Merton(1976) was the first to introduce the jump-diffusion model. Jump-diffusion models are particular types of exponential Levy models in which the frequency of jumps is finite.

In a jump-diffusion model, the stock price St follows the random process

dSt

St

= µdt + σdWt+ dJ. (3.15)

The first two terms are familiar with the Black-Scholes model and represent fluctuations in the stock price due to general economic factors such as supply and demand. These factors cause small movements in the price and are modelled by a geometric Brownian motion with a constant drift term, where µ is the drift rate, σ is the volatility and dWt

is the increment of a Wiener process. The last term models the arrival of important information into the market that will have an abnormal effect on the price. By its nature information only arrives at discrete points in time and will be modelled by a jump, where J = PNt

i=1Ji is a compound Poisson process where the jump sizes Ji are

non-negative independent and identically distributed (IID) with distribution F and the number of jumps Nt is a Poisson process with jump intensity λ:

P (Nj = k) =

(λt)j

j! exp(−λt) for j = 0, 1, 2, ... .

Ntis a Poisson process with intensity λ because the process has the following properties:

(1) Nt− Ns is independent of Ns; (2) N0 = 0, Nt ∈ N+; (3) Ns ≤ Nt if s < t; and (4)

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proportional to the length of ηt: Nt+ηt =     

Nt with probability 1 − ληt − o(ηt)

Nt+ 1 with probability ληt + o(ηt)

Nt+ k with probability o(ηt).

When market information causes an instantaneous jump in the asset price, moving the price from Stto JtStwhere Jtis the absolute magnitude of the jump, the relative price

change is given by dSt St = J St− St St = Jt− 1.

Merton (1976) considers the case where the jump variables Jt are modelled as

non-negative log-normal random variables in order to provide a realistic jump term such that the jump is less probable when the magnitude of the jump is larger. So the jump variable J is compound Poisson distributed and jt is distributed as

jt= ln(Jt) ∼ IID N (α, δ2).

The relative jump size is (Jt− 1) where Jt is log-normally distributed with mean α and

variance δ2 and therefore

Eh(Jt− 1) i = exp(α +1 2δ 2_{) − 1 ≡ K} _(3.16) and Varh(Jt− 1) i = exp(2α + δ2) exp(δ2) − 1 .

The jump diffusion model of Merton (1976) incorporating the above properties takes the SDE of the form

dSt

St

= µdt + σdWt+ (Jt− 1)dNt, (3.17)

where the Poisson process Nt is independent of the Wiener process Wt. The random

jump variable Jtis assumed to follow the log-normal distribution and it is assumed that

J is independent of Wt and Nt. The variable J is also independent through time so

Cov(Js, Jt) = 0 for s 6= t.

The model can be written as (

dSt

St = µdt + σdWt if no Poisson event occurs

dSt

St = µdt + σdWt+ (Jt− 1) if a Poisson event occurs.

To find an expression for ln(St), Itˆo’s lemma for jump-diffusion processes is used as

described inCont & Tankov(2003). Using this theory for f (·) = ln(·) and St described

by the stochastic differential equation (3.15) it follows that d ln(St) = ∂ ln St ∂t dt + µSt ∂ ln St ∂St dt +σ 2_S2 t 2 ∂2ln St ∂S2 t dt + σSt ∂ ln St ∂St dWt+ h ln JtSt− ln St i = µSt 1 St dt +σ 2_S2 t 2 − 1 S_t2 dt + σSt 1 St dWt+ h ln Jt+ ln St− ln St i = µ − 1 2σ 2 dt + σdWt+ ln Jt.

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Taking integrals on both sides of the equation gives ln St= ln S0+ µ − 1 2σ 2 t + σWt+ Nt X i=1 ln Ji. (3.18)

Adding a compound Poisson process to a geometric Brownian motion has affected the drift term. To compensate the increasing drift term, equations (3.17) and (3.18) become respectively dSt St = (µ − λK) dt + σdWt+ (Jt− 1)dNt (3.19) and ln St= ln S0+ µ − λK −1 2σ 2 t + σWt+ Nt X i=1 ln Ji. (3.20)

This last equation can be written as

St= S0exp µ − λK −1 2σ 2 t + σWt Nt Y i=1 Ji. (3.21)

As described in Section 3.1.1 MLE is applied to estimate the parameters in the CIR-model of Cox et al. (1985). In order to use the MLE approach for estimating the pa-rameters µ, σ2, α, δ2 and λ in the jump-diffusion model, the transition density of the jump-diffusion process has to be determined. Since the density of the log-return under the jump-diffusion model of Merton (1976) is given by an infinite mixture of Gaus-sian distributions, estimation by maximum likelihood is invalid. This is supported by

Press (1967) and Beckers (1981). Because the mixture of distributions may imply an unbounded likelihood function it is not possible to find a maximum likelihood estimator for the parameters in the jump-diffusion model.

Press(1967) andBeckers (1981) used the method of moments, matching sample and population cumulants to fit restricted forms of the model. Parameter estimation by cumulant matching is known to yield consistent estimators, but these estimators are not always efficient.Press(1967) frequently obtained negative estimates of the variance parameters σ2 and δ2 when he used the method of cumulants to a sample of monthly returns over the period 1926 through 1960 of ten NYSE listed common stocks. In con-trast to Press(1967),Beckers (1981) did not believe that a larger sample size will solve the inherent estimation problems. Therefore Beckers (1981) modified the Press proce-dure and set the mean logarithmic jump size α equal to zero rather than µ whichPress

(1967) assumed. This reduces the dimensionality of the estimation problem and ensures that the transition density is symmetrical. Due to symmetry of the distribution, the odd cumulants except the first one all vanish.Beckers(1981) shows that the population cumulants for the jump diffusion model, with α set to zero, in equation (3.15) are

K1 = µ∆t, K2 = σ2∆t + λ∆tδ2, K3 = 0, K4 = 3δ4λ∆t, K5 = 0, K6 = 15δ6λ∆t,

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where Ki represents the ith population cumulant. The sample cumulant functions ¯Ki

for i ≤ 1 are then ¯ K1 = m1, ¯ K2 = m2− m21, ¯ K4 = m4− 3m22− 4m1m3+ 12m21m2− 6m41, ¯ K6 = m6− 6m5m1− 15m4m2+ 30m4m21− 10m23+ 120m3m2m1 − 120m₃m3₁+ 30m3₂− 270m2 2+ 360m2m41− 120m61,

where m1 = _n1 Pni=1xi and mj = _n1Pni=1(xi − m1)j for j = 2, ..., 6 and xi the ith

logarithmic return. Matching the sample cumulants with the population cumulants and solving the system of equations yields the parameter estimates

ˆ µ = ¯K1∆t, ˆ λ = 25 ¯K 3 4 3 ¯K₆2 , ˆ δ2 ₌ K¯6 5 ¯K4 , ˆ σ2 ₌ _K¯₂₋5 ¯K42 3 ¯K6 ! /∆t.

Beckers(1981) applied these estimators to forty-seven NYSE listed common stocks, each with five hundred daily return observations over 15 September 1975 to 7 September 1977 and obtained negative estimates of the variance parameters σ2 and δ2. Only for stocks with high sample kurtosis, this estimation method generates sensible parameter values. The average kurtosis for the thirty stocks with negative variance estimates is 1.06 and the average kurtosis for the seventeen positive variance stocks is 3.17. Kurtosis is a mea-sure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions that are more outlier-prone than the normal distribution have a kurtosis larger than 3 and distributions that are less outlier-prone have a kurtosis less than 3. The method of Beckers(1981) is implemented in MATLAB and no negative estimates for the variance parameters σ2 and δ2 are obtained in this study. As input is chosen for the AEX index (daily) for the period 4 May 2006 until 4 May 2016 which is obtained from Bloomberg. In Figure3.4 this data is shown. The logarithmic return of this data has a kurtosis of 10.487. Because of a high sample kurtosis the result that no negative estimates for the variance parameters are obtained is what is expected and therefore the method ofBeckers (1981) can be applied.

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When the parameters are estimated, stock returns can be simulated by the jump diffu-sion process of Merton (1976). Monte Carlo simulation of the process is done in MAT-LAB by first simulating the number of jumps Nt and then simulating geometric

Brow-nian motion on intervals between jumps. The implementation of the simulation can be found in Appendix A. In Figure 3.5 the log-returns of the stocks simulated with the jump-diffusion model of Merton(1976) are shown. In Figure3.5athe initial stock price is equal to 30, the drift rate is 0.04, the maturity is 0.0833 and the volatility is 0.3. The jump frequency is equal to 3, the jump mean parameter is 0.05, the jump volatility parameter is 0.5 and the number of time steps is equal to 1000. It can be seen in Figure

3.5athat no jump occurred in the simulated path. When the jump frequency is increased to 20, jumps are visible. This is shown in Figure3.5bwhere the jump frequency is equal to 20 and all other parameters are similar to those of Figure3.5a. In the simulated path that is shown in Figure3.5bthree jumps are visible. In Figure3.6the logarithmic stock returns are simulated for the period 2016 until 2068 with the parameters estimated with the method ofBeckers (1981).

Figure 3.5: One simulation run of the stock log-return by the jump-diffusion model ofMerton(1976) for initial stock price of 30, drift rate of 0.04, maturity of 0.0833, volatility of 0.3, jump mean parameter of 0.05, jump volatility parameter of 0.5 and 1000 timesteps. The jump frequency is equal to 3 in subfigure (a) and equal to 20 in subfigure (b).

(a) Jump frequency equal to 3. (b) Jump frequency equal to 20.

Figure 3.6: In the figure on the left the original time series, the logarithmic return of the AEX index (daily) for the period 4 May 2006 until 4 May 2016, is shown. In the figure on the right the simulated (20 simulations) paths for the logarithmic stock return for the period 2016-2068 are shown. The parameters are estimated with the method ofBeckers(1981).

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3.2 Determination of asset mixes

3.2.1 Life-cycle models

Individual DC pension plans offer each participant the freedom to choose and to imple-ment the optimal consumption and investimple-ment strategies according to their own needs. Life-cycle theory has discussed how to determine the optimal saving and investing strate-gies. Through the saving decision, participants decide how to smooth consumption over time by setting the pension premium and the pension benefits. Through the investment decision, individuals decide how to invest the premium in the various financial assets so as to smooth consumption across various future contingencies that may arise in the future.

There are different life-cycle models: (1) the risk-free life-cycle model; (2) the con-stant life-cycle model; (3) the deterministic life-cycle model; and (4) Merton’s life-cycle model. In the risk-free life-cycle model only in risk-free assets is invested. In the constant life-cycle model in both risk-free assets as in risky assets is invested. The ratio between the different assets is constant for the whole life-cycle and therefore not dependent of age. In the deterministic life-cycle model the ratio between different assets is predeter-mined for the whole life-cycle and thus dependent of age. Although the deterministic life-cycle model is often used in practice, this model is not really realistic since the port-folio choice is not dependent of the state of the economy.Merton(1969) provides a more realistic model where the portfolio choice is dependent of the age of the participant and the state of the economy.

As described in Section 3.1, in this study the four assets in which one can invest are the risk-free rate, a two-year zero-coupon bond, a ten-year zero-coupon bond and the AEX index. In addition to the rates of returns, some predetermined asset mixes are determined as input for the model. Four predetermined asset mixes are chosen as in-put: an asset mix from the risk-free life-cycle model, an asset mix from the constant life-cycle model, an asset mix from the deterministic life-cycle model and an asset mix from Merton’s life-cycle model.

In the asset mix of the risk-free life-cycle model 100% is invested in the risk-free rate and nothing is invested in the two-year zero-coupon bond, the ten-year zero-coupon bond and the AEX index. For the constant life-cycle model is chosen for a distribution where 25% is invested in each asset class. The deterministic life-cycle is often used in practice. For the determination of the distribution of the asset classes for the determin-istic life-cycle model the life-cycles of different Dutch pension providers are investigated including the life-cycles of Centraal Beheer Achmea (see also Figure 2.3). All variants contain the same concept of reduced risk as the retirement age approaches and a combi-nation of all variants is chosen as asset mix for the deterministic life-cycle in this study. For the determination of the distribution of the asset classes of the model of Merton

(1969), the model must be implemented in MATLAB. How this model is implemented is discussed below.

3.2.2 Optimization of expected utility

In the model of Merton (1969) a Black-Scholes market consisting of two assets is as-sumed. The financial market consists of one risk-free asset (i.e. bank account) and one risky asset (i.e. stock price index). The value of the bank account is given by the solution of the stochastic differential equation

dBt

Bt

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where r is the rate of return on the bank account and Bt the value of the bank account

at time t. The stock price index St is described by a geometric Brownian motion

dSt

St

= µdt + σdZt, (3.23)

where µ denotes the expected rate of return on the stock price index, σ is the volatility of the stock returns, and Ztrepresents a standard Brownian motion, i.e. dZt∼ N (0, dt)

with dZt a systematic shock that cannot be pooled away.

As already mentioned, in this study a financial market is assumed which consists of four assets: the risk-free rate, a two-year zero-coupon bond, a ten-year zero-coupon bond and the AEX index. In Section 3.1.1 the returns of the risk-free rate, two-year zero-coupon bond and ten-year zero coupon bond are simulated with the CIR-model and in Section 3.1.2 the AEX index is simulated with the jump-diffusion model. The CIR-model is a more realistic model than the stochastic differential equation given in equation (3.22) because the CIR-model assumes mean reversion, positive interest rates and there is empirical evidence for the volatility term in the CIR-model because whenever interest rates are high, the volatility is likely to be high as well. All these as-sumptions are not made in equation (3.22). The jump-diffusion is a more realistic model than the geometric Brownian motion in equation (3.23) because the market distribution for stocks has several properties which are not reflected in equation (3.23), for example large random fluctuations such as crashes. Since the focus in this study is on the CIR-model to simulate risk-free returns and bond-returns and the jump-diffusion CIR-model to simulate stock returns, the financial market in the model of Merton (1969) is not only smaller, but also less realistic than the financial market that is assumed in this study. However, since the model ofMerton(1969) is a well-known problem in continuous-time finance and in particular intertemporal portfolio choice, the closed-form solution of the workhorse model ofMerton (1969) is also chosen as a scenario for an asset mix in this study, but instead of simulating the returns of the risk-free asset and the AEX index with equation (3.22) and (3.23), the CIR-model and the jump-diffusion model are used. So, for the determination of the distribution of the asset classes in the predetermined asset mix of the model of Merton (1969) there is only invested in the AEX index and the risk-free rate where the risk-free returns are simulated with the CIR model instead of the stochastic differential equation in (3.22) and the returns of the AEX index are simulated with the jump-diffusion model instead of the geometric Brownian motion de-scribed in equation (3.23).

For the determination of the optimal consumption and portfolio choice over the life-cycle Merton (1969) analyzed combined consumption-investment decision problems in continuous time in a multi-period framework. An investor must choose how much to consume and how to allocate his wealth between a stock and a risk-free asset so as to maximize expected lifetime utility. To solve this maximization problem Merton (1969) introduced the method of dynamic programming and was able to derive closed-form solutions for optimal consumption and investment under the assumption of constant relative risk aversion (CRRA) utility functions and log-normally distributed risky as-sets. The work of Merton (1969) showed that all investors should hold the same risky portfolio. Furthermore, the work of Merton (1969) implied that all individuals should consume a fixed fraction of their wealth. The fraction of wealth invested in the risky stock is equal to

w_t∗= λ

γσ (3.24)

where λ denotes the Sharpe ratio, γ the coefficient of relative risk aversion and σ the volatility of stock returns. In Appendix B1 the workhorse model of Merton (1969) is derived. In Appendix B2 some extra derivations of the model ofMerton(1969) are given.

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Total wealth Wtis the sum of human capital Htand financial wealth Ft. This is already

explained in Section 2.2. Since there can only invested in financial wealth and not in hu-man capital, the fraction of financial wealth that is invested in risky stocks fω∗_t should be determined instead of the fraction of total wealth invested in risky stocks ω∗_t. If human capital is risk-free, then the fluctuations in total wealth must match the fluctuations in financial wealth. That is,

ω_t∗σWtdZt= fω∗tσFtdZt,

where fω∗_t is the fraction of financial wealth invested in the risky stock. It follows that f ω∗_t = ω_t∗Wt Ft = λ γσ Wt Ft = λ γσ(1 + Ht Ft ).

On average, as the agent ages, the fraction of financial wealth invested in risky stocks f

ω_t∗ decreases. This is because human capital is less risky than equity and the value of human capital usually declines as a proportion of an individual’s total wealth as one ages which can be also seen in Figure2.2. Since younger workers have more opportunities to alter their labor supply than older workers, for example the opportunity to work longer hours and take on extra jobs, the share of assets held as risky equity should decline with age. Furthermore, older participants have less time to recover from potential losses. In addition the fraction of financial wealth invested in risky stocks depends on financial wealth Ftwhich depends on the state of the economy. To determine fωt∗, human capital

Ht and financial wealth Ft should be determined for every t. In order to determine Ft

and Ht the gross pension premium must be determined.

When it is said that the year t is the period (t, t + 1], the gross pension premium that is (prenumerando) paid in the year t, πt, is equal to

πt= pr%t· Gt,

where the premium percentage pr%t can be a fixed, predetermined percentage, but is

often age dependent and where Gtare the pensionable earnings over the year t.

In this study the premium percentage pr% is assumed to be equal to the maximum contribution scale and therefore age dependent, but deterministic. In the Netherlands it is common to apply an age-dependent premium ladder with age categories of maximum five years and a contribution expressed as a percentage of the pensionable earnings. The Wages Tax Act in the Netherlands dictates the rules for the maximum contribution scale. As described in Section 2.1.1, in the Netherlands fiscal boundaries for DC plans are derived from the cost structure of DB plans. Along with the decrease in fiscal maximum accrual rates for DB plans, the fiscal maximum contribution rates in DC plans are also lowered. In the Netherlands there are four age dependent premium ladders. Since the second age-dependent premium ladder is usually used in practice, these maximized contribution rates are used in this study. The age-dependent premium ladders can be found at the website of Rijksoverheid (ndb), where also distinction is made between different accrual rates for average wage schemes and between discount rates of 3% and 4%. In this study, the maximal accrual rate for average wage schemes is assumed to be equal to 1.875%. In addition, since interest rates are currently very low a discount rate of 3% is assumed. The third column in Table 3.1 presents the second age-dependent premium ladder as per 1 January 2015 based on the maximum net contribution rate and a discount rate of 3%.

For the pensionable earnings it holds that Gt= max (St− Ft, 0) ,

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where St is the pensionable salary and Ft is the offset in year t. The pensionable salary

is maximized at e100,000 as of January 2015. As of January 2016 the pensionable salary is maximized at e101,519 (Art. 18ga of Wet LB 1964). Therefore, in this study a maximized pensionable salary ofe101,519 is assumed.

Table 3.1: Maximum net contribution rates based on an accrual rate of 1.875% per year for average wage plans and a discount rate of 3%. OP is the abbreviation of old age pension and PP is the abbreviation of spouse pension (seeRijksoverheid(ndb)).

Age Percentage of pensionable earnings OP OP and deferred accrued PP OP and immediately effective accrued PP OP and immediately effective obtainable PP 15-19 5.9 7.2 8.1 8.4 20-24 6.6 8.0 9.0 9.5 25-29 7.6 9.3 10.4 11.0 30-34 8.8 10.8 11.9 12.5 35-39 10.3 12.5 13.8 14.4 40-44 11.9 14.6 15.9 16.6 45-49 13.9 17.0 18.4 19.1 50-54 16.2 19.8 21.2 22.1 55-59 19.1 23.3 24.6 25.4 60-64 22.6 27.7 28.5 29.1 65-66 25.6 31.5 31.8 31.9

When the gross premium is determined, human capital Ht and financial wealth Ft can

be determined. Since it is assumed in this study that not the whole future labour income is used to invest in the life-cycle, but only the gross premium, adjusted human capital is calculated by discounting the gross premium by the stochastic short-rate for each scenario,

H_ti = Z TR

t

πt exp −rit t dt,

where H_ti is human capital at time t for scenario i, πt is the gross premium at time

t, TR is the pensionable age and rit is the short-rate at time t for scenario i (which is

determined in Section 3.1.1). What remains is the T × I matrix      H₁1 H₁2 · · · H₁I H₂1 H₂2 · · · H₂I .. . ... . .. ... H_T1 H_T2 · · · H_TI     

where I is the number of scenarios and T is the number of years.

When human capital is defined, financial wealth F_tiand the optimal fraction of financial wealth invested in the AEX index fωi∗

t can be determined for each scenario i = 1, ..., I

for every t by the following recursive formula F₁i = π1, f ωi∗ j = min max λ γσ 1 + H_ji F_ji ! , 0 ! , 1 ! , (3.25) Fj+1= (Fj+ πj+1) 1 + fωi∗ j Rij+ 1 − fωi∗ j r_ji,

for j = 1, ..., T , where Ri_tis the return of the AEX index at time t for scenario i (which is determined in Section 3.1.2). Since it must hold that 0 ≤ fωi∗

j ≤ 1, the minimum of

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What remains are two T × I matrices      F₁1 F₁2 · · · F₁I F₂1 F₂2 · · · F₂I .. . ... . .. ... F_T1 F_T2 · · · F_TI      and       g ω₁1∗ ωg2∗ 1 · · · ωg1I∗ g ω1∗ 2 ωg22∗ · · · ωg2I∗ .. . ... . .. ... g ω_T1∗ ωg2∗ T · · · ωgTI∗       . (3.26)

For the implementation of the recursive formula in equation (3.25) the fraction _γσλ must be determined. In Advies Commissie Parameters, a report from the Dutch govern-ment, parameters are recommended which set maximum limits on the expected returns that pension administrators may rely for their investments and minimum limits on the expected values of wage inflation and price inflation that determine the level of pension benefits. In Advies Commissie Parameters the standard deviation of stock returns σ is equal to 20% per year (Rijksoverheid,nda), therefore this value for σ is assumed in this study. The coefficient of relative risk aversion γ is assumed to be equal to 5 since this is common in the literature (Bovenberg, Koijen, Nijman & Teulings,2007;Gomes, Kotlikoff & Viceira,2008). The Sharpe ratio λ is defined as µ−r_σ where µ − r is called the mean equity premium.Gomes et al.(2008) assumed that the mean equity premium is equal to 4% (and that σ is equal to 20%) which implies that the Sharpe ratio is 20%.

Bovenberg et al. (2007) also assumed that the Sharpe ratio is equal to 20%. Since it is common in the literature to assume that λ is equal to 20%, this assumption is also made in this study. The fraction _γσλ is therefore equal to 20% and this is exactly the constant fraction of total wealth which must invested in risky assets according to the model ofMerton (1969).

The optimal fraction of financial wealth which must invested in the AEX-index f

ω_t∗ according to the model of Merton (1969) is determined by taking the mean of fωi∗_t over all scenarios, thus over the columns of the matrix in equation (3.26). The fraction of financial wealth invested in the two-year zero-coupon bond and ten-year zero-coupon bond is 0% for all timesteps since the financial market in the model of Merton (1969) consists only of the AEX index and the risk-free rate. The optimal fraction of financial wealth invested in the risk-free rate is therefore equal to 1 − fω_t∗.

In Figure3.7a human capital for a 25-years-old with a pensionable age of 67 is shown. Full-time salary is set ate36,500 which is the expected Dutch gross modal income for 2016 (Centraal Planbureau, 2016) and it is assumed that the general salary increase is 1.5% per year. The offset is set at e12,953 which is the minimum AOW-offset as of January 2016 (Belastingdienst, 2016). This amount is calculated based on the AOW benefits as listed in Annex II.1 of the Rekenregels published by the Ministry of Social Affairs and Employment at 30 November 2015 (Rijksoverheid, 2015a). Furthermore, it is assumed that the yearly offset increase is 1% per year, the person works full-time and the number of simulations is 10, 000. In Figure3.7bthe financial wealth for this person is shown. In Figure3.7bthe same assumptions are made as in Figure3.7a. Both human capital as financial wealth are dependent of age. It can be seen that for this 25-years-old human capital is zero and financial wealth is equal to total wealth after 42 years which is exactly when the person reaches the pensionable age.

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Figure 3.7: In subfigure (a) human capital is shown for a 25-age-old person with pensionable age 67. The salary is set ate36,500, the yearly general salary increase at 1.5%, the offset at e12,953 and the yearly offset increase at 1.0%. In subfigure (b) financial wealth for this person is shown and the same assumptions are made as in subfigure (a). The number of simulations in both figures is equal to 10.

(a) Human capital (b) Financial wealth

In Figure3.8the fraction of financial wealth invested in the AEX index is shown where the same assumptions are made as in Figure 3.7. It can be seen that the fraction ω_et

converges to 20% after 42 years which is exactly at the pensionable age. Furthermore, the value 20% is exactly the fraction of total wealth which is invested in risky stocks. This is because at the pensionable age human capital is zero and when human capital is zero the fraction of financial wealth invested in risky stocks is equal to the fraction of total wealth invested in risky stocks.

Figure 3.8: The fraction of financial wealth which must invested in the AEX index for a 25-years-old person with pensionable age 67. The salary is set ate36,500, the yearly general salary increase at 1.5%, the offset ate12,953 and the yearly offset increase at 1.0%.

To perform the analysis in Chapter 4, thus to compare different predetermined asset mixes for a particular participant, a 25-years-old with pensionable age 67 is assumed. In this study, it is assumed that it is not possible to invest after pensionable age since the investment of the available premiums and earned investment returns are used to purchase a lifelong annuity at retirement age. Therefore, the asset mixes are determined for 42 years. In Figure 3.9 the four chosen predetermined asset mixes are shown. The MATLAB implementation of the calculations which are done to determine the asset mix of the model of Merton (1969) can be found in Appendix A. In Appendix C the four predetermined asset mixes are shown in tables.