Optimal portfolio choice over the life-cycle in the upcoming Dutch pension system in a low interest rate environment
Master’s Thesis Actuarial Studies Supervisor: prof. dr. Laura Spierdijk
Supervisors Milliman: Sjoerd Brethouwer AAG & Steven Hooghwerff AAG Second Assessor: Agnieszka Postepska, PhD
Optimal portfolio choice over the life-cycle in the upcoming Dutch pension system in a low interest rate environment
Lucas van der Maas, S2765497 February 13, 2020
Abstract
Contents
1 Introduction 1 2 Literature review 3 3 Methodology 5 3.1 General formulation . . . 5 3.2 Preferences . . . 6 3.3 Assets . . . 7 3.4 Labor income . . . 8 3.5 Pension products . . . 93.6 Investment strategy of the pension fund . . . 11
3.7 Model . . . 13
3.8 Numerical optimization . . . 15
3.9 Model calibration . . . 17
3.10 Performance measures . . . 21
3.11 Monte Carlo simulation . . . 22
4 Results 23 4.1 Performance of investment strategies . . . 23
4.2 Pension contributions . . . 24
4.3 Base case results . . . 28
4.4 Variable Payout Annuity . . . 32
4.5 Variations of the risk free rate . . . 35
4.6 Bequest motive . . . 40
4.7 Withdrawal of part of pension wealth at retirement . . . 42
4.8 Sensitivity Analysis . . . 44
5 Conclusion 46 6 Further research and implications 48 References 50 A Appendices 52 A.1 Results sensitivity analysis . . . 52
A.2 Variables, parameters and abbreviations . . . 53
A.3 Dutch income tax . . . 54
1
Introduction
The Dutch pension system, although widely considered as one of the best systems worldwide, has several limitations that need repair. There is agreement between labor unions, employer organisations and the government about the main characteristics of a new system that will be further specified and introduced in the near future. However, in the last few years, while the new system was conceived, interest rates have dropped to historically low levels and even to negative rates.
As a consequence of the negative interest rates, banks find themselves in a difficult situation. As of November 2019, Rabobank, ING en ABN AMRO have lowered their interest rates to 0.01%. This implies that savings at a bank practically yield no returns, especially when taking inflation into account. While the yield of a 3-month Dutch government bond already turned negative five years ago, this year (2019) the yield of a 30-year Dutch government bond also dropped below zero. Although few individuals directly buy bonds, they are nevertheless affected by the low yields as pension funds invest in them.
Pension cuts are expected to be only a matter of time, particularly for members of pension funds that have hardly covered interest rate risk in the past. To mitigate this problem for new generations, the new ‘Pensioenakkoord’ proposes a large transition. Currently Defined Benefit (DB) schemes are far more popular than Defined Contribution (DC) schemes in The Netherlands, as DB plans constitute 94% of total pension plan wealth and DC schemes only 6% (Willis Towers Watson, 2019). In a Defined Benefit scheme, individuals pay variable pre-miums and expect to receive a fixed payout after retirement. In the newly proposed system individuals will pay a fixed contribution (Defined Contribution) and future pension payouts are based on performance of their pension savings in the stock and bond markets (SER, 2019). Although bank account savings deliver no returns, Dutch households barely invest in stocks apart from their accumulated wealth in pension funds nevertheless. The fraction of house-holds that invest in stocks peaks around age 45 (27%) and then slowly declines (Kooiman et al., 2019). Moreover, in the current pension system, it is expected that 43% of the Dutch households that include self-employed individuals (zzp’ers) will not reach a replacement rate (income after retirement divided by pre-retirement income) of 70%, whereas for households consisting of employees this is 31% (Zwinkels et al., 2017). The proposed Dutch pension system is intended to reduce the risk of pension cuts, but can leave individuals vulnerable to a low replacement rate if many negative stock shocks occur or interest rates remain ex-tremely low during the accumulation of pension wealth. To reduce the risk, or mitigate the effect of a low replacement rate, individuals may have to alter their asset allocation, labor and consumption behavior.
enhance welfare?
A life-cycle model for an individual is adopted in this thesis. Life-cycle models address the problem of choosing an appropriate asset allocation for the personal savings of an eco-nomic agent as he or she moves through the life-cycle, as well as determining the optimal leisure/work ratio and how much income to allocate to consumption. Assuming that pen-sion contributions are a fixed percentage of income, time spent on leisure/work will have a large influence on reaching a satisfactory pension income. Personal investments in stocks and bonds can mitigate the effect of a large drop in income due to low accumulated pension wealth. Determining optimal behaviour is addressed by means of dynamic programming (Samuelson, 1975). To determine the effects of the proposed pension system on welfare, we will compare lifetime utility between variable payout annuities and fixed payout annuities. Next to that, it is verified if the option to withdraw a maximum of 10% of accumulated pension wealth at retirement is valued by modeled individuals.
We will build upon existing literature by focusing on the persistent low interest rate environ-ment for Dutch individuals. The Dutch taxing system as well as the Dutch pension system proposed in the ‘Pensioenakkoord’ will be incorporated in this thesis. This is an extension of the research of V. Horneff et al. (2018) where an US context is considered and annuities are excluded. In Chai et al. (2011), annuities are included, but low interest rates and the Dutch pension system are not. Similar methods are used to obtain optimal behaviour. One of the main findings of this thesis is that Dutch households invest very little in stocks compared to outcomes of the model.
2
Literature review
Merton’s (1975) revolutionary paper on optimal consumption and portfolio allocation in continuous-time paved the way for a large body of further research. Samuelson (1975) was the first to review the discrete multiperiod consumption-investment problem, where an in-vestor wants to maximize utility by deciding on optimal consumption and what proportion to invest in stocks and bonds for a given number of periods. He made use of dynamic pro-gramming, and determined optimal policy functions by means of backward induction. Due to the relative simplicity of this problem, he was able to provide the optimal solution analyt-ically. In years to follow, many adaptations of this problem have been studied. In the next two paragraphs, studies that either cover the accumulation phase of assets or decumulation phases are described. Thereafter, studies that cover both phases are introduced. Finally, we focus on different utility functions used in the studies and we include studies that focus on empirical evidence.
due to access to Guaranteed Minimum Withdrawal Benefits variable annuities is researched by V. Horneff et al. (2015), also providing an overview of fees for variable annuities.
Gomes et al. (2008) included both the accumulation and decumulation phase and used dy-namic programming to determine optimal consumption, asset accumulation and portfolio decisions in a life-cycle model with flexible labor supply. This model was extended by Chai et al. (2011) by including freedom in choosing the age of retirement and the possibility to buy annuities (fixed payout annuities and variable payout annuities). One of the main findings of that paper is that access to fixed payout annuities compared to no access only gave a small increase in utility, whereas access to variable annuities provided a relatively large increase in utility compared to no access at all. We study if full annuitization of pension wealth at retirement shows similar results for increase in utility when comparing fixed payout annu-ities and variable payout annuannu-ities. Flexible labor supply was also included by Low (2005), whose results showed that individuals work longer and consume less when young under more uncertainty. V. Horneff et al. (2018) investigated the effect of a persistent low return en-vironment on saving, investing, and retirement behaviors, while not including annuities in their research. They assume a constant risk equity premium for different risk-free rates, whereas we assume that the equity returns do not change when the risk-free rate changes. A wide range of utility functions are adopted in different papers. Gomes et al. (2008) use a power utility function under the assumption of Epstein-Zin preferences, allowing them to separate risk aversion from the elasticity of intertemporal substitution. A disutility function is adopted by Gerrard et al. (2004), where deviations (also positive) from yearly targets are penalized. Cocco et al. (2005), V. Horneff et al. (2018) and Chai et al. (2011) all use a mod-ified Cobb-Douglas utility function that is also employed in this thesis. A slight adaptation of this function was made by Fan et al. (2015), as consumption and leisure are included as two separate terms.
3
Methodology
3.1
General formulation
To investigate the two research questions, we model a dynamic utility maximizing investor that can decide on leisure, consumption and personal investments in a risky asset (stock) and riskfree asset (bond) over the course of a lifetime. It is assumed that individuals base their decisions on maximizing utility, where utility consists of consumption and leisure. Figure 1 shows the cash flows of an individual. Prepension income is based on the amount of hours worked per week, income in the previous year, income shocks and growth. After pension contributions, taxes and housing costs have been deducted from income, the investor can decide on what proportion to spend on consumption and what proportion to invest in stocks and bonds. It is assumed that the investor can consume in the next time period from his personal investments in stocks and bonds. Hence, the investments in stocks and bonds should not be considered as traditional third pillar investments, but as personal capital ac-cumulation. A fixed proportion of the individual’s income will be put in a pension savings account. It is assumed that the investment strategy of pension wealth is determined by the pension fund and the individual can not influence this strategy. In section 3.6 it is explained why a Glide Path strategy is assumed to be adopted by the pension fund. When pension age is reached, accumulated pension wealth is converted into either a variable payout annuity or a fixed payout annuity. It is important to note that stocks and bonds in personal wealth are taxed, and that a fee is paid to the pension fund every year, but these were excluded from Figure 1 to maintain its clearness.
State variables are variables that individuals employ to determine optimal values for the decision variables. Personal wealth, pension wealth, income and age are the state variables that we use. Using these state variables, we can observe optimal reactions to either a high or low expected replacement rate for various interest rates.
Figure 1: Schematic view of the development of personal wealth and pension savings, in-cluding all decision variables
3.2
Preferences
In this section the utility function which the individual is assumed to follow is introduced. A dynamic utility maximizing agent is modeled in discrete time over t ∈ (1, 2, ..., T ), where T is equal to 78. The age of the individual is given by 22 + t, which implies that we model for the ages 23-100. Modeling starts at age 23 as it is inferred that individuals start accumulating pension wealth from this age. It is assumed that preferences for the agent in each period are represented by a time-separable and iso-elastic power utility function
Ut(Ct, lt) =
(Ctltα)1−γ
1 − γ (1)
based on current consumption Ctand leisure time ltnormalized as a fraction of total available
time (Gomes et al., 2008). This function is part of the Cobb-Douglas utility functions family. The parameter γ ∈ R>0\{1} denotes the coefficient of relative risk aversion for consumption,
than an individual with a high α. The recursive definition of the value function is as follows: Vt= (Ctlαt)1−γ 1 − γ + βEt(p s tVt+1) (2)
Parameter β < 1 is a discount factor that represents the time preferences of an individual. The probability of surviving until age t + 1 while having already reached age t is represented by pst.
In the base case a bequest motive will not be included in the analysis. However, in the sensitivity analysis a bequest motive is included. When the bequest motive is included, the value function is given by:
Vt= (Ctlαt)1−γ 1 − γ + βEt(p s tVt+1+ (1 − pst)b Mt+11−γ 1 − γ) (3)
with terminal utility
VT =
(CTlTα)1−γ
1 − γ + MT1−γ
1 − γ (4)
The strength of the bequest motive is characterized by b, Mtdenotes the wealth that remains
when the individual dies.
3.3
Assets
The agent has two options for investing his wealth: stocks and bonds. Returns on stocks and bonds determine the development of pension wealth and personal wealth over the years, next to pension contributions and the part of disposable income that is not consumed respectively. Stocks can be considered as a risky asset where returns are modeled as follows:
St+1= St
(1 + µS+ σSQt+1)(1 − τS)
The average yearly return and yearly standard deviation are denoted by µS and σS
respec-tively. Qt is a random variable that follows a standard normal distribution. As the focus is
put on real wealth (adjusted for inflation), the realized return on stocks is divided by 1 + inflation π. To keep computation times within reasonable limits, inflation π is assumed to be a constant throughout the life of an individual. The Dutch taxes on stocks are denoted by τS. While currently, the Dutch wealth taxing system focuses on total wealth, plans for 2022
are that assets will be taxed based on expected returns. We consider bonds (government bonds) as a riskfree asset and returns evolve accordingly:
Bt+1= Bt
(1 + µB)(1 − τB)
1 + π (6)
As bonds are assumed to be liquid, investments in bonds are equivalent to savings in a bank account. Taxes on bonds are given by τB. Once again, to lower computation time, the return
on bonds are kept constant over time, as stochastic interest rates would imply inclusion of another state variable and the addition of an extra source of stochasticity that has to be included in the Gauss-Hermite quadrature. Realized returns on stocks and bonds on time t are denoted by Rs,t and Rb respectively. While it is assumed that the individual wants
his investments in stocks and bonds to be liquid, this is not the case for pension savings. Durations of bonds that pension funds invest in are often high, to mitigate interest rate risk. Therefore, it is inferred that the pension fund obtains returns on bonds as follows: µB+p = µB+ p, where p is a premium on top of the riskfree rate, representing illiquidity.
3.4
Labor income
To create a realistic setting, stochastic labor income is included and the individual is given the option to decide on the number of hours worked per week. In this thesis, an individual starts working at the age of 23 and stops working at the age of 68, when the retirement age is reached. It is assumed that there are no years where the individual is unemployed. The labor dynamics work as follows:
Yt= 1 − lt 0.4 (1 − ht)(1 − τb)(1 − 0.5φ)w(t)Et (7) Et= Et−1Nt (8)
Yt is the annual real disposable labor income, where expenses for housing ht are already
deducted. However, inflation π is excluded from the equation as for simplicity it is assumed that income is always adjusted to inflation. w(t) is a deterministic function that allows us to calibrate wages according to age, as a strong growth is often observed in the first years of work. Permanent wage shocks are included by the variable Et, where the shocks are
dis-tributed as ln(Nt) ∼ N (−0.5σp2, σp2).
on work influence the yearly labor income. Assuming 100 waking hours per week, working 40 hours implies a normalized leisure of lt = 0.6. Working 40 hours a week is often considered
equivalent to having a full time job, 1−lt
0.4 = 1 in this case, which implies an individual earns
full wages. Hence, working less than 40 hours a week results in less income, and working more than 40 hours a week results in more income. We allow an individual to choose work hours between 22.5 and 42.5 hours per week, thus between a parttime- and a fulltime job with some flexibility. Leisure lt is set equal to 0.775 after retirement, which is equal to
the maximum leisure before retirement. Setting leisure equal to 1 after retirement creates erratic policy functions for the last years before retirement. The Dutch income tax burden before pension is denoted by τb. Although income taxes are progressive in The Netherlands,
in section 3.7 it is explained why income taxes are assumed to be a fixed proportion of income.
3.5
Pension products
In this section the Dutch pension system is described, as well as the assumptions that are made on pension income of an individual. The definitions of a fixed payout annuity and variable payout annuity are provided. The Dutch pension system consists of of three pillars: the first, second and third pillar respectively. In this thesis, only the first and second pillar are considered.
First pillar
The first pillar consists of payouts from the Dutch government, called AOW (Algemene Oud-erdomswet). From 50 years before reaching the retirement age, every year that someone lives or works in the Netherlands, 2% of the AOW is built up. This implies that an individual living in The Netherlands for his/her entire life will receive full AOW payments after retire-ment. For simplicity, we will assume that individuals live and work in The Netherlands for their entire lifespan. The annual AOW payout for a person with a partner equals e10, 124. It is assumed that the AOW payouts grow with inflation.
Second pillar
The second pillar consists of the work-related pension schemes. These pension schemes are administered by either an insurance company or a pension fund. Companies are affiliated with an insurance company or a pension fund and employees of this company are then also automatically affiliated. Most often, employees pay part of their income to accumulate pension, while employers also contribute to the pension savings of an employee. The total accumulated pension of an individual is based on the contributions over the years, and the return on the investments on these contributions.
the product (Chai et al., 2011): At= (1 + δ)Pt T X s=t+1 p(t, s) 1 + AIRs−t−1 (9)
At denotes the price of an annuity product with initial payout Pt. δ is an expense factor
set by the insurance company to cover its own costs. The cumulative conditional survival probability that an individual aged 22 + t will reach the age 22 + s is given by p(t, s) =
s−1
Q
t
pt.
The AIR is the Assumed Interest Rate that is set. For a fixed payout annuity the AIR is often set equal to the riskfree rate. In that case the individual receives a lifelong stream of payments that is constant. If the AIR is set higher, initial payments will be larger, but decrease every year. When setting a lower AIR, initial payments will be lower but increase in the following years.
A variable payout annuity (also called an investment-linked annuity) is an annuity that pays benefits based on the performance of the underlying investment portfolio. The recursive evolution of the payouts can be characterized as:
Pt+1 =
PtRt+1a
1 + AIR (10)
When the return of the underlying portfolio exceeds the AIR, the payout in the next period will increase and vice versa. Ra
t+1 = 1 + Rb+ ν(Rs,t+1− Rb) is the growth rate of the
under-lying portfolio, where ν is the proportion of the portfolio that is invested in the risky asset and 1 − ν the proportion that is invested in the riskfree asset. Rs,t+1 is the realized return
on the risky asset and Rb is the realized return on the riskfree asset. An investment linked
annuity is similar to the system that is proposed by the ‘Pensioenakkoord’, as positive and negative returns on the underlying portfolio will influence the payouts.
Third pillar
The Third Pillar consists of individual pension products. This pillar is mainly used by in-dividuals who are not part of a collective pension scheme. They are often self-employed people (zzp’er). It is assumed in this thesis that individuals will not invest in any third pillar pension products, but he/she does have the option to accumulate personal wealth next to pension wealth. As only 6% of Dutch pension wealth consists of Third Pillar wealth (Bruil et al., 2015), omitting the Third Pillar does not have large consequences for the usefulness of the model.
The total income of an individual after reaching the retirement age consists of AOW and annuity payouts Pt:
Parameter τp denotes the tax burden after retirement age, which differs in The Netherlands
compared to taxes before the retirement age. AOW and the annuity payouts are both a lifelong stream of income, which implies that all Dutch individuals that build up AOW and pension in the second pillar have some form of protection against longevity risk.
3.6
Investment strategy of the pension fund
There exists a wide range of possible investment strategies for pension wealth. While it is assumed that individuals can not influence the investment strategy of a pension fund, the chosen investment strategy does influence personal investments in stocks and bonds. A defensive strategy of the pension fund can lead to a more aggressive personal investment strategy.
The process of the pension build up Ft over the years runs as follows:
Ft+1 = 1 − cp 1 + π(ωt(Ft+ Φt)(1 + µS+ σSQt+1) + (1 − ωt)(Ft+ Φt)(1 + µB+p)), ∀t < K (12) Φt = 1 − lt 0.4 w(t)Et−1Ntφ (13)
ωt is the fraction of the pension savings that is invested in stocks and 1 − ωt the fraction
that is invested in bonds in year t. The age of retirement is denoted by K. The fixed contribution percentage that is deducted from pre-tax income is given by φ, where Φt is the
total contribution of an individual in year t. The fee the pension fund asks for managing the pension savings is given by cp. It is assumed that employer and employee each pay half of
the pension contributions.
The large computation times of the model do not allow us to include another decision vari-able, and individuals often can not influence investment strategies of pension funds and insurers. Therefore, a Glide Path strategy is assumed to be adopted by the pension fund, which is in line with V. Horneff et al. (2018).
In the Glide Path strategy ωt is determined by the following rule:
ωt= (100 − (t + 22))/100
This implies that at the age of 23, 77% is invested in stocks, whereas in the final year before pension at the age of 67, 33% is invested in stocks. This in line with the fact that pension funds and insurance companies lower the exposure to risky assets and invest more safely when an individual is coming closer to the retirement age.
eval-uated. These four investment strategies are: Constant Proportion, Buy and Hold, Time Invariant Portfolio Protection (TIPP) and Dynamic Life-cycle Investing.
The Constant Proportion strategy is characterized by a constant division of stocks and bonds as a percentage of total pension wealth. This implies that rebalancing has to be performed every year if the realized returns on stocks and bonds are not equal. A mix often chosen is a portfolio that consists of 60% stocks and 40% bonds (Ambachtsheer, 1987). Hence, we will also adopt this division.
As the name reveals, a Buy and Hold strategy implies that investments in stocks and bonds will be held and that no rebalancing is performed. In this thesis, it is assumed that 50% of initial pension wealth is invested in stocks and the other half in bonds. Likewise, 50% of the pension contributions that are added to pension wealth every year are invested in stocks and 50% in bonds. A potential risk of the strategy is that large stocks positions are built up when return on equity is much larger than return on bonds.
The Time Invariant Portfolio Protection (TIPP) strategy is an extension of the Constant Proportion Portfolio Insurance strategy (Dichtl & Drobetz, 2011). The starting point is an investor’s risk capital at time t, called the cushion. The current cushion (ηt) is equal to the
difference between the current pension wealth at time t (Ft) and the value of the floor (κt):
ηt = Ft− κt
The exposure (et) to the risky asset at time t is calculated by multiplying this cushion with
the multiplier(m):
et = mηt
The multiplier can freely be chosen. However, it should be noted that choosing a large multiplier can result in large stock investments. The difference between the CPPI and TIPP strategy is that the floor is updated each year when following the TIPP strategy, while the floor is constant for the CPPI strategy. The floor is updated using the following recursion:
κt = max[κt−1, · Ft]
Bt. The state variables are pension wealth Ft and the permanent component of stochastic
labor income Et. Final utility is computed as: 78
X
t=46
βt−1p(68, 22 + t)Ut((Pt+ AOW )(1 − τp)(1 − ht), 0.775)
where pension wealth is converted into a fixed payout annuity with payouts Pt when the age
of retirement is reached. The policy controls are determined by backward induction. Risk measures
10,000 Monte Carlo simulations are performed, where assumptions on stock and bond returns and stochastic labor income are equal to the assumptions of the main model. To evaluate the investment strategies, first we look at the risk measures Value at Risk (VaR) and Expected Shortfall (ES). The Value at Risk is defined as:
V aRα(X) = Inf {x ∈ R : FX(x) ≥ α}
and the Expected Shortfall (sometimes also referred to as Conditional Value at Risk) as: CV aRα(X) = 1 α Z α 0 V aRγ(X)dγ
3.7
Model
A mathematical formulation of Figure 1 is presented in this section, where all constraints are defined. Furthermore, the option to withdraw a part of pension wealth at retirement is intro-duced, in order to investigate if this idea proposed in the ‘Pensioenakkoord’ enhances welfare. At each point in time the individual has to decide on how much to spend on consumption Ct, stocks St and bonds Bt and how much time to allocate to work and leisure.
Wt= Ct+ St+ Bt (14)
The wealth of an individual in the next time period is given by: Wt+1 = St
(1 + µS+ σSQt+1)(1 − τS)
1 + π + Bt
(1 + µB)(1 − τB)
1 + π + Yt+1 (15)
If the individual dies, the heirs will receive the bequest Mt= St−1 (1 + µS+ σSQt)(1 − τS) 1 + π + Bt−1 (1 + µB)(1 − τB) 1 + π (17)
For simplicity, it is assumed that only accumulated financial wealth contributes to supporting the potential bequest motives. This implies that pension for the surviving relatives (spouse pension) is excluded from the analysis. The individual faces the maximization problem:
max
{St,Bt,Ct,lt}t=Tt=0
Vt (18)
subject to the following constraints:
Wt= Ct+ St+ Bt, ∀t (19) Wt+1 = St (1 + µS+ σSQt+1)(1 − τS) 1 + π + Bt (1 + µB)(1 − τB) 1 + π (20) + 1 − lt+1 0.4 (1 − ht+1)(1 − τb)(1 − 0.5φ)w(t + 1)EtNt+1, ∀t < (K − 1) Ft+1 = (1 − cp) 1 + π (ωt(Ft+ Φt)(1 + µS+ σSQt+1) + (1 − ωt)(Ft+ Φt)(1 + µB+p)), ∀t < K (21) Wt+1 = St (1 + µS+ σSQt+1)(1 − τS) 1 + π + Bt (1 + µB)(1 − τB) 1 + π (22) + (1 − ht+1)(1 − τp)(Pt+1+ AOW ), ∀t ≥ (K − 1) Pt+1 = Pt 1 + π 1 + µB+ νt(µs+ σSQt+1− µB) 1 + AIR , ∀t ≥ K (23) St≥ 0, ∀t (24) Bt≥ 0, ∀t (25) Ct≥ 0, ∀t (26)
The last three constraints are shortselling constraints for stocks and bonds and the assump-tion that consumpassump-tion can never be negative. K denotes the age of retirement.
Withdrawal of part of pension wealth at retirement
The ‘Pensioenakkoord’ contains a proposal to give people the option to withdraw a maximum of 10% of pension wealth when retiring. While in the base case this option is excluded from our analysis, in the sensitivity analysis the option to withdraw is included. When investigating this option, decision variable ρ is added to the set of decision variables, where ρ is in the range of [0,0.1]. Only at time period t = 46 this decision variable is added. (1 − τp)ρF46is added to personal wealth, while (1 − ρ)F46is deducted from pension wealth. While
are not left with insufficient income after retirement, and also to ensure that administrative expenses do not increase too heavily, we assume that every individual can withdraw up to 10%.
Figure 2: Technical version of the schematic view of the development of personal wealth and pension savings, including all decision variables
3.8
Numerical optimization
Optimization problems of this type cannot be solved analytically due to the untradable labor income and the shortselling restrictions. Therefore, we resort to numerical optimization for solving the problem posed. The four decision variables of the model are labor, consumption, investment in stocks and investment in bonds. The state variables consist of age, income, personal wealth and pension wealth. The number of state variables and decision variables in the model leads to large computation times. However, the utility function that is used displays homothetic preferences. When we combine this with the assumption of proportional income- and wealth tax, a fixed proportion of income that is deducted for housing costs and pension contributions, the implication is that our model is scale free with respect to the permanent component of wages (Et). In other terms, if the permanent component of wages
double, all choice variables double (Gomes et al., 2008). Hence, to reduce computation times, the set of state variables is reduced by one variable as personal wealth, pension wealth and consumption are normalized by the permanent component of labor Et. Note that we only
that a modeled individual makes decisions before retirement based on normalized variables as this person focuses on a pension income relative to his prepension income. As income after retirement only consists of an annuity and AOW, previous labor income should not in-fluence decisions in these years anymore. A grid is created of the remaining state variables. While age is a discrete variable, pension wealth and personal wealth are continuous. Hence, we need to discretize these variables in order to create a grid. Personal wealth is split in 40 evenly spaced intervals, while pension wealth is split in 20 evenly spaced intervals. This results in a 40(W ) ∗ 20(P ) ∗ 78(t) grid.
For every point on the grid, optimal policy functions are determined. An optimal policy consists of decisions for labor, consumption and allocation of wealth to stocks and bonds, based on the previously defined value function
Vt(Wt, Ft) =
(Ctlαt)1−γ
1 − γ + βp
s
tEt(Vt+1(Wt+1, Ft+1)) (27)
To determine the non linear component Et(Vt+1(Ut+1, lt+1)), Gauss-Hermite quadrature is
used. Part of this method is that shocks are discretized. Labor income shock ln(Nt+1) and
equity shock Qt+1 are both discretized into 5 nodes. The non-linear component is calculated
as: Et(Vt+1(Wt+1, Ft+1)) = Z ∞ −∞ Z ∞ −∞ Vt+1ψ(ln(Nt+1), Qt+1)dln(Nt+1)dQt+1 (28) ≈ π−1 5 X n=1 5 X m=1 wln(Nt+1)wQt+1V (ln(Nt+1), Qt+1; Ft+1, Wt+1) (29)
The joint density of the random variables ln(Nt+1) and Qt+1 is given by ψ(ln(Nt+1), QT +1).
wln(Nt+1) and wQt+1 denote the Gauss-Hermite quadrature weights and π is the mathematical
constant. After retirement there are no labor shocks. The non-linear component is then calculated as Et(Vt+1(Wt+1, Ft+1)) = Z ∞ −∞ Vt+1ψ(Qt+1)dQt+1 (30) ≈ π−1 5 X m=1 wQt+1V (Qt+1; Wt+1, Ft+1) (31)
3.9
Model calibration
The parameters are calibrated in such a way, that the model created closely tracks real life. An individual is modeled from age 23-100, where it is assumed that he starts working immediately at the age of 23. Dutch mortality rates are obtained from the ’Prognosetafel AG2018’, which is created by the Koninklijk Actuarieel Genootschap from The Netherlands. Not only current mortality rates are provided, but also best estimates for future mortality rates. These rates can be used for the pricing of pension products and for the recursive definition of the value function. Agents are modeled until the age of 100 as best estimates predict a large increase of life expectancy. It is assumed that agents are 23 in 2019. The retirement age is increased to 68 as an increase in life expectancy will result in an increase in the retirement age as well.
The discount factor β is set equal to 0.97 and a risk aversion parameter γ is set to 5 (Gomes et al., 2008). Leisure preference parameter α is taken as 1.9 −1+t/601 , assuming that leisure is more preferred when growing older (Ben´ıtez-Silva et al., 2003). After retirement α is equal to 1.9 −1+46/601 , as it is not sensible that the leisure preference changes when no decisions on hours worked can be taken. Whereas the bequest motive b is not included in the base case, later b is included and set to values of 5, 10, 100 and 1000. The standard deviation of wage shocks is set to 10.6% as in Cocco et al. (2005).
The return and standard deviation of the risky asset are based on returns of the SP500 between 1930-2018. This gives µS = 0.0543 as the expected return and σS = 0.1890 as the
Figure 3: The observed data points compared to the normal distribution
In the model, for all cases, it is assumed that the inflation π = 1% each year. This is based on the Consumer Price Index development over the past 6 years (2013-2018) in The Netherlands. Data from earlier periods are not included as this thesis focuses on the low interest rate environment and the inflation that is observed during the last few years. The expectation of the development of the interest rate could be based on the DNB yield curve. However, as no stochasticity is included it is not sensible to assume that the DNB yield curve perfectly predicts yield developments and including it in the analysis will more likely create unwanted noise in the results. Therefore, in the base case, return on bonds µB is set equal
to 0%, which approximately equals the return on savings at banks in 2019.
As the interest rates are assumed to be deterministic, it is implicitly assumed that the agent possesses full knowledge of the development of the interest rates in all years. In the sensi-tivity analysis parallel shocks are applied to the interest rate to observe the effects of a low interest rate environment. It is assumed that the pension fund can receive a premium on their bonds compared to individuals. The pension fund obtains returns on bonds as follows: µB+p = µB+ 0.003
As explained earlier, the state space was reduced by one variable. Tables 11 and 12 show that the Dutch taxing system is of a progressive kind. To conform to the assumption that only personal wealth, pension wealth and consumption relative to income are important, propor-tional taxes have to be applied. Based on 10,000 runs, the average tax burden throughout the working life of an individual was taken. Using this, τb = 0.3817 is taken as the taxing
follows Table 12 in the appendix.
While currently wealth tax makes no distinction between saving money and investing money, new plans presented by the Dutch government show that from 2022 stocks will be taxed much more heavily than savings (and bonds). As bonds deliver nearly no return at the moment, we set τB = 0 and τS = 1.38%. While pension contributions can be made from pre-tax income,
it is assumed that the pension fund charges costs for managing pension savings. These yearly costs cp are set equal to 0.5%, based on comparison of various third pillar products, as no
information of these costs is directly available from second pillar products. Pricing factor δ for pension products is set to 2.37% for both the VPA and FPA (Steinorth & Mitchell, 2015). It is assumed that each individual has a partner and hence AOW provides a yearly income stream of e10, 124 after retirement.
Housing costs are based on data of the CBS (CBS, 2019b). As these costs are given as a percentage of income, they can be used directly in the model. Since no state variable is included for housing wealth, it is implicitly assumed that an individual always rents (or never runs down his housing wealth), meaning that there is no housing wealth that can be converted into liquid savings for consumption. Labor income is calibrated by deterministic wages component w(t). Data of the CBS (CBS, 2019a) of average Dutch income is used for calibration, where it is assumed that the individual is an employed man. Yearly grow rates are determined by:
g(t) = pre-tax income year t + 1 pre-tax income year t
w(t + 1) = w(t)g(t), where pre-tax incomes are taken from the CBS data. w(1) equals e25,000.
Figure 4: 10% and 90% quantile of expected annuity payouts for νt is 0.25, 0.5 and 0.75
Figure 5: 10% and 90% quantile of expected variable annuity payouts where the AIR equals -0.65%, 0.35% and 1.35%, ν = 0.25
can be imagined that the agent would prefer a slightly higher AIR than the expected return on the underlying portfolio. However, there is no such freedom in the Second Pillar in The Netherlands. In order to create an expected constant stream of payments, for both annuity products, the AIR is set as:
AIR = 1 + E[R]
1 + π (32)
where E[R] is the expected return on the underlying portfolio of the pension product. Infla-tion π is included in the AIR in order to ensure that expected real payouts are constant.
3.10
Performance measures
To compare the VPA and FPA and determine the effect of different stock exposures, total lifetime utility is examined. Total lifetime utility can be calculated as follows:
V1 = K−1=45 X t=1 βt−1 t Y i=1 psi(Ctl α t)1−γ 1 − γ + T =78 X t=K=46 βt−1 t Y i=1 psi(0.775 αC t)1−γ 1 − γ (33)
As utility is a rather abstract concept, the Utility-Constant Equivalent Consumption Stream (CE) is used to determine utility differences (Chai et al., 2011). The CE is a constant con-sumption stream that provides equal lifetime utility as the leisure and concon-sumption stream that can be financed by the life-cycle strategy.
V1 = K−1=45 X t=1 βt−1 t Y i=1 psi(CEl α t)1−γ 1 − γ + T =78 X t=K=46 βt−1 t Y i=1 psi(0.775 αCE)1−γ 1 − γ (34) CE = V1(1 − γ) PK−1=45 t=1 βt−1 Qt i=1psi(ltα)1−γ + PT =78 t=K=46βt−1 Qt i=1psi(0.775α) !1−γ1 (35)
The measure for the change of utility that results from differences in stock allocation is defined as:
∆U = CE1− CE2 CE2
In other words, the percentage increase in CE is determined. Note that consumption before retirement is normalized consumption. When looking at change of utility due to conversion of pension wealth in different annuities, only utility after retirement is taken into account. The CE is then given by:
CE = V46(1 − γ) PT =78 t=K=46βt−1 Qt i=1p s i(0.775α) !1−γ1 (37)
The replacement rate is determined by dividing net average disposable post-retirement in-come by average disposable average labor between ages 40-60. This average is taken as income can heavily fluctuate during the course of a working life: r = Y¯p
P38 t=18Yt/21
3.11
Monte Carlo simulation
Monte Carlo analysis is performed to determine the effects of following the optimal policy controls, and to be able to give a clear view of optimal behavior in a realistic setting. For all scenarios that are analyzed, 10,000 simulations are carried out. Pre-tax income for a 23-year old is set to 25,000, which is derived from the CBS wage data (CBS, 2019a). Starting wealth is equal to disposable first year labor income where it is assumed that an individual worked full-time (40 hours) that year. Death is certain at the age of 100. For every year in the analysis, wage shocks and stock return shocks are drawn from their distribution, where no correlation with previous years is assumed. Based on their personal wealth Wt,
pension wealth Ft and age 22 + t decisions on labor, consumption and investments in stocks
4
Results
In this section, results from the Monte Carlo analyses are presented. We start by analyzing the performance of five investment strategies in a low interest environment. In the second subsection, the optimal fixed pension contribution rate for different risk-free rates are ana-lyzed. Thereafter, optimal behavior when pension wealth is converted into a fixed payout annuity is presented. This is compared to optimal behaviour when pension wealth is con-verted into a variable payout annuity. In the subsection that follows, changes in optimal behavior when the riskfree rate is either increased or decreased are discussed. Furthermore, individuals with a bequest motive are presented and the option to withdraw a part of pension wealth at retirement is studied. The section is concluded by a sensitivity analysis.
4.1
Performance of investment strategies
We evaluate the five investment strategies when the fixed contribution rate φ equals 18% and the risk-free rate equals 0%.
From Table 1 it can be seen that the Dynamic Life-cycle performs best in terms of the 10% VaR. However, Table 1 also shows that the Dynamic Life-cycle strategy performs worst when looking at the 5% CVaR. The Glide Path strategy performs well when looking at the 5% and 10% CVaR.
Table 1: 10% VaR and CVaR for the assessed investment strategies
Glide Path TPPI Buy and Hold Constant Proportion Dynamic Life-cycle
V aR0.05 187,313 172,806 171,129 180,118 175,495
V aR0.1 219,695 208,704 207,403 221,556 231,354
CV aR0.05 147,542 129,479 128,537 129,916 117,192
CV aR0.1 170,024 158,292 157,446 163,599 158,295
Although Table 1 shows that the Glide Path strategy performed well in the lowest decile of the distribution, Table 2 shows that in most quantiles it performs worst of all strategies. This is due to the fact that in the last years before retirement, pension wealth is mostly in-vested in bonds, which yields no returns. The Constant Proportion strategy shows positive results, ranking consistently 3rd or 4th of the investment strategies. As was expected, the Buy and Hold strategy only performs well in favourable market conditions. The parameters of the TIPP strategy were set rather defensively. In the assumed low interest environment this leads to a performance somewhat similar to the Glide Path strategy.
in stocks. This often leads to good results, but in case of serious stock market crashes, a pension fund that fully adopted a Dynamic Life-cycle strategy will face problems.
Table 2: Rank of the evaluated investment strategies for various quantiles where rank 5 indicates the investment strategy that performs best in terms of accumulated pension wealth in this quantile
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Glide Path 5 3 3 2 1 1 1 1 1 1 1
TPPI 4 2 1 1 2 2 2 2 2 2 2
Buy and Hold 3 1 2 3 3 3 3 4 4 4 5
Constant Proportion 2 4 4 4 4 4 4 3 3 3 3
Dynamic Life-cycle 1 5 5 5 5 5 5 5 5 5 4
4.2
Pension contributions
To determine the optimal fixed contribution percentage when an individual’s pension wealth is converted into a VPA (as proposed for the Dutch system) at retirement, maximum lifetime utility and replacement rate are investigated. Fixed contribution percentages between 14-24% are examined, for risk-free rates of -1%, 0% and 1%. Although in this thesis employer and employee each pay half of the pension contributions, when determining the optimal fixed contribution rate it is assumed that pension contributions are paid by the employee. A dis-torted view would be created when assuming that employers’ contributions also increase if the individuals wants his personal contributions to increase. In Figure 6 the average lifetime utility for different fixed contribution rates for each of the risk-free rates can be found. Contrary to what might be expected, based on lifetime utility for the risk-free rates that are investigated, an individual prefers the fixed contribution to be as low as possible. As pension savings provide fiscal advantages as no taxes on these savings are assumed, one expects that higher contribution rates would be optimal. However, lifetime utility is mostly influenced by consumption in the early years of working life for two reasons. In the first place, utility is not too heavily discounted by the discount parameter. Second, low labor income decreases even further if a high contribution percentage is set. As the individual is risk averse, total utility heavily decreases if the consumption is relatively low when being younger. Instead of illiquid pension savings, an individual prefers personal liquid wealth that can be consumed when facing negative income shocks. In addition, the investment strategy analysis showed that an investor that follows a dynamic life-cycle strategy wants to invest nearly all of pension wealth in stocks when being young. As the Glide Path strategy invests more defensively, an individual wants to manage his own wealth and invest more in stocks.
observed than in Figure 6. Although the utility differences are small, it is shown that from this age and on, a higher risk-free rate results in a lower optimal fixed contribution rate. This can be explained by the fact that lower pension contributions are needed in order to obtain a sufficient replacement rate if the risk-free rate is higher.
Figure 6: Average lifetime utility for various fixed contribution percentages. Utility is mul-tiplied by 1016
Although lifetime utility shows that modeled individuals prefer a low fixed contribution to pension wealth, the fixed contribution rate is set at 18% to calibrate a realistic scenario. Moreover, this rate is shown to be optimal from age 27 and on for a risk-free rate of 0%. As can be seen in Table 3, a fixed contribution of 18% results in a median replacement rate of 83% when the risk-free rate is 0%. In the other scenarios, the same rate is kept to observe changes in behavior when less or more accumulated pension wealth can be expected. It should be noted that using different investment strategies could result in different optimal fixed contribution rates.
Fixed contribution rate Replacement rate
14% 0.72 15% 0.75 16% 0.78 17% 0.81 18% 0.84 19% 0.88 20% 0.91 21% 0.94 22% 0.97 23% 1.01 24% 1.05
Table 3: Median replacement rate for various fixed contribution rates when converting pen-sion wealth into a variable payout annuity when the interest rate equals 0%
For a fixed contribution of 18% at a risk-free rate of 0%, Figure 8 shows that there is a heavy spread in replacement rates when the relatively risk-averse Glide Path strategy is used. Although many individuals reach a replacement rate that is close to or higher than 1, there are also many individuals who have a replacement rate close to 0.6. As a dynamic life-cycle model is employed, individuals with lower replacement rates consumed less in the years before retirement and saved more. In Figure 9 the replacement rate is shown if 1/33 of personal wealth (331(W46− C46)) is added to average retirement income ( ¯Yp). The group of
and “unlucky” generations are created under the application of a Defined Contribution plan.
Figure 8: Replacement rate for individuals when the fixed contribution equals 18% and the risk-free rate equals 0%
4.3
Base case results
In the base case of the model several interesting observations can be made. The low interest rate affects the individual in multiple ways. First of all, the growth of the pension wealth is relatively low. In the first years, the fraction of the pension wealth that is invested in stocks is still quite high, but this fraction drops steadily, making it increasingly hard to generate large returns on the portfolio. Figure 10 shows the growth of pension wealth in quartiles. The large differences between pension savings can be explained by the fact that income is very volatile when the standard deviation is set at 10.6% and returns on stocks vary heavily for different simulations. However, even when excluding any variations in income (wages are deterministic), large differences in accumulated pension wealth can still be found for different generations.
Figure 11: The average disposable income, average wealth (excluding pension savings) and average consumption over the lifespan of an individual when pension wealth at retirement is converted into a fixed payout annuity
(2011) describe that the ’consumption puzzle’ might not be a puzzle after all when looking at optimal behaviour. Although it is an interesting observation, the drop is more likely to be explained by a decrease in disposable income in our situation. A rational individual with a particularly high replacement rate does not consume less after retirement. Empirical evidence shows that bank deposits of households with median or high replacement rates are relatively stable around retirement. However, households that experience a sharp decline in gross income around retirement run down a large part their bank deposits in the first year of retirement, probably to adjust consumption gradually (Kooiman et al., 2019). The pattern that emerges from the model is slightly different. In any case (low, medium, high replace-ment rate), consumption decreases gradually and wealth is diminished at a rather steady rate. This can be explained by the fact that when using backward induction to derive optimal policies, an individual has no knowledge of previous income, and hence this can not directly influence consumption behaviour after retirement. However, still quite some wealth is accu-mulated as large differences in consumption before and after retirement have a heavy impact on utility. At the age of 75 a small drop in disposable income can be noted. This is due to the fact that the housing data consist of the population grouped in buckets of 10 years and hence a sudden drop can be noticed as average housing costs increase for people of these ages.
Figure 12: The average proportion of personal wealth that is invested in stocks (the remaining wealth is invested in bonds) when pension wealth at retirement is converted into a fixed payout annuity
re-tirement age. Although fully investing in stocks is optimal from a utility-maximizing point of view, the model does not incorporate the fact that individuals might have to cover large (unexpected) expenses in early working life. Therefore, in general it is not advisable to fully invest personal wealth in stocks. From age 50 onwards it can be noted that an individual smoothly reduces the fraction of wealth that is invested in stocks, reaching its lowest point when reaching the retirement age. Interestingly, stock exposure rises again after retirement when an individual receives a fixed payout annuity. This can be explained by the fact that before reaching retirement, an individual’s pension income is still subject to uncertainty, as large shocks in the stock market greatly influence accumulated pension wealth. When retire-ment is reached, this uncertainty vanishes, as the accumulated pension wealth is converted to a fixed payout annuity. Having a fixed income, where no labor shocks are possible, an individual is willing to take more risk in the stock market with the possibility of increasing wealth in bullish markets.
Figure 13: The average numbers of hours worked per week
ex-plained by the fact that an individual wants to keep enough wealth to make up for his lower pension income. As well as not reducing work hours, it can also be observed that consumption relative to total wealth decreases in the last years before reaching retirement if the accumulated pension wealth compared to income is low. This can either be explained by the fact that people who are aware of their low accumulated pension wealth already in-creased their personal accumulated wealth or that even in the last years before retirement they try to boost their personal wealth by consuming less. This implies that a rational risk averse investor takes into account that it would be wise for him to compensate for a low income after retirement by saving more before retirement. When looking at empirical evidence (Statline, 2019), no drops in numbers of hours worked occur in the early working life. Apart from that, our results show a similar pattern as the Statline data, where working hours are approximately constant until age 55 and then slightly decrease. In contrast to empirical evidence, where average number of working hours reach 34 as a maximum, average working hours that result from our model have 42 hours as a maximum. In a low interest rate environment, this might be optimal, to ensure that more pension wealth is accumulated.
4.4
Variable Payout Annuity
Figure 14: The average disposable income, average wealth (excluding pension savings) and average consumption over the lifespan of an individual when pension wealth is converted into a variable payout annuity
Figure 15: The average proportion of personal wealth that is invested in stocks (the remaining wealth is invested in bonds) when pension wealth at retirement is converted into a variable payout annuity
Whereas results from the base case for both the VPA and the FPA advise to invest a large fraction of liquid wealth in stocks, empirical evidence shows contrary results. Only 17% of the households in the Netherlands where the oldest member of the household is aged 40 hold stocks, which is in sharp contrast with our optimal results. There are two possible explana-tions. Firstly, it is possible that households possess no knowledge of stock markets and are afraid or do not know how to invest. For these households, welfare gains can be obtained by following a dynamic life-cycle investment strategy. Secondly, households with low income and wealth might need to save carefully and can get into trouble when they are confronted with large unexpected expenses. For these households it is sensible to keep a large part of their liquid wealth in savings or bonds.
4.5
Variations of the risk free rate
In the base case a riskfree rate of 0% was assumed. However, interest rates can fluctuate, and therefore, optimal behaviour is researched for parallel shocks to the riskfree rate. Parallel shocks of +1% and -1% are taken. First, a parallel shock of -1% is studied. In Figure 16 the accumulated personal wealth is shown. To determine differences in optimal behaviour, all other parameters were kept equal. This implies that pension contributions were also set at 18%, even if this may not be optimal.
Figure 16: The average disposable income, average wealth (excluding pension savings) and average consumption over the lifespan of an individual when the riskfree rate is -1%
An interesting difference between Figure 16 and Figure 11 is that a riskfree rate of -1% results in a large increase in average accumulated personal wealth. This implies that individuals ac-cumulate more personal wealth in order to compensate for a lack of income after retirement. This can also be seen from the fact that there is a larger difference between consumption and income after retirement in Figure 16 compared to Figure 11. It can also be noted that a lower riskfree rate has a drastic influence on pension income. Pension income is negatively affected in two ways: pension savings accumulate slower as the return on bonds is very low, and the Assumed Interest Rate has to be set lower to ensure a constant expected stream of annuity payouts.
onwards, in Figure 17 it can be observed that optimal stock exposure decreases from a later age. At the same time, the through that is reached just before retirement is not as deep as in Figure 12. After retirement, relative stock exposure is also higher for a lower risk-free rate.
Figure 17: The average proportion of personal wealth that is invested in stocks (the remaining wealth is invested in bonds) when the riskfree rate equals -1%
Figure 19: Average disposable income, average wealth (excluding pension savings) and av-erage consumption over the lifespan of an individual when the riskfree rate is +1%
Figure 20: The average proportion of personal wealth that is invested in stocks (the remaining wealth is invested in bonds) when the riskfree rate equals +1%
Optimal labor behaviour is shown in Figure 21. In the early years of working life we once again notice somewhat erratic optimal behaviour. Looking at the years before retirement, working hours decrease earlier and to a lower level compared to a riskfree rate -1%. As a higher replacement rate is more easily obtained, individuals can afford to work less during the last years before retirement. Although the differences are small, compared to a risk-free rate of 0%, a larger decrease in working hours before retirement is also observed.
respectively. On average, stock exposure is higher and wealth accumulation increases when the riskfree rate decreases and vice versa. Less hours are worked when the riskfree rate increases, more hours are worked when it decreases.
4.6
Bequest motive
Kooiman et al. (2019) show that in practice households barely run down their wealth after retirement. Interestingly, the trajectories of wealth and bank deposits at the end of life show hardly any difference between households with and without children. This raises the ques-tion whether or not there is a bequest motive present or that people focus on precauques-tionary savings for possible large unexpected expenses. In previous research, the bequest motive b is often set at a value of b = 2 (W. Horneff et al. (2010), Chai et al. (2011), Cocco et al. (2005)). Since the utility function in Chai et al. (2011) is similar to ours, it is surprising that they chose a value of b = 2. For this value we find that it is optimal to consume 46.39% of total wealth at time period T when death in the next time period is certain, for any level of wealth. This is counterintuitive when someone tries to save money for his heirs.
Figure 22: The average wealth (excluding pension savings) over the lifespan of an individual for various bequest motives
Figure 23: The average proportion of personal wealth that is invested in stocks (the remaining wealth is invested in bonds) for various bequest motives
4.7
Withdrawal of part of pension wealth at retirement
Figure 24: The average disposable income, average wealth (excluding pension savings) and average consumption over the lifespan of an individual when there is the option to withdraw from pension wealth at retirement and housing costs are included in the model
Figure 25: The percentage of pension wealth that is withdrawn for pre-tax second pillar income of 10000, 30000 and 50000 when housing costs are ignored
4.8
Sensitivity Analysis
Besides varying the risk-free rate and including a bequest motive, other sensitivity analyses are also performed. It is assumed that pension wealth is converted into a variable payout annuity at retirement. The impact of changing the leisure preference, risk aversion, stock return, and standard deviation of income are presented in Tables 4, 5, 6 and 7 in the ap-pendix. As can be expected, decreasing α results in more labor, and hardly any decline in work hours can be observed, even shortly before retirement. A more interesting pattern can be found for increasing α. On average, less hours per week are worked and especially in the first years of an individuals career, a through can be seen. This is mostly due to the fact that income in these years is relatively low, and hence leisure is more appreciated. Comparison of the wealth trajectories of α = 0.9 and α = 1.8, for α = 0.9 there is less wealth accumulated in the early years of working life. As leisure and income are low in these years, the agent increases consumption to smooth his utility over the lifetime.
with a low risk aversion as modeled in this thesis invest all of their personal wealth in stocks throughout their life. This is in sharp contrast with empirically observed behaviour in the Dutch population. When increasing the risk aversion, a modeled individual tends to accu-mulate a relatively large amount of personal wealth. Furthermore, our results show that already in the early forties of an individual, the relative exposure to stock declines. This is more in line with empirical behaviour. Nonetheless, consumption of the modeled individual shows an unusual pattern, as consumption in retirement increases steadily as a person grows older.
A decrease in the return on equity, when keeping the risk-free rate constant has a somewhat similar effect as increasing the risk-free rate on the asset portfolio: a switch from stocks to bonds is made earlier. Personal wealth accumulation is approximately similar when the return on equity is not decreased. This is due to the fact that an agent sets more of his income aside to mitigate the effect of a lower annuity payout after retirement. It can also be noted that work hours hardly decrease during the years before retirement.
A smaller standard deviation in income σp = 0.05 leads to minor differences in optimal
5
Conclusion
This thesis addresses two research questions concerning the new pension system that will be implemented in The Netherlands in the near future. The first question is: what is the optimal portfolio choice over the life-cycle of an individual in the upcoming Dutch pension system in a low interest rate environment? The second question is: how do the features of the proposed pension system enhance welfare?
An individual who has to decide on his portfolio consisting of consumption, labor and in-vestments in stocks and bonds in the proposed new Dutch pension system is modeled. In a realistically calibrated setting, we determined optimal behaviour for a Dutch risk averse individual.
Results for the first question show that in a low interest rate environment it is optimal to hold a large fraction of personal wealth in stocks, especially in the early years of working life. This is in sharp contrast to the empirically observed behaviour in the Dutch population. Shortly before retirement, a considerable decrease in stock exposure is optimal. Whereas agents with a fixed payout annuity keep their personal wealth highly exposed to stocks after retirement, agents with a variable payout annuity greatly lower their investments in stocks and opt for a larger proportion of their wealth in risk-free bonds. Although some caution is understand-able, Dutch individuals can improve utility by holding more stocks.
In any case, it is optimal for an individual to accumulate personal wealth next to pension wealth, as the Defined Contribution scheme that is modeled in this thesis shows that ac-cumulated pension wealth greatly varies. Moreover, a drop in consumption due to adverse labor shocks can be mitigated when more personal wealth is available. Although modeled in-dividuals work more than the average Dutch person, the small decline in work hours before retirement matches empirical behaviour. When accumulated pension wealth is low com-pared to income, it is shown to be optimal to work more and consume less in the years before retirement, as might be expected. A drop in income after retirement is present for most individuals in the low interest rate environment. To smooth consumption, personal wealth is accumulated before retirement and slowly decumulated after retirement.
The interest rate greatly affects optimal portfolio choices for an individual. Assuming equal fixed contribution rates, when exposed to a low interest rate, individuals consume less before retirement and save more. Furthermore, a larger fraction of personal wealth is exposed to stocks, and work hours are hardly decreased during the last years before retirement. When assuming higher interest rates, opposite changes are observed.
accu-mulated pension wealth is converted into a variable payout annuity. Optimal stock fractions after retirement for personal wealth display large differences between having a variable pay-out annuity versus a fixed paypay-out annuity. When given the option, individuals often choose for withdrawing a part of pension wealth at retirement. Determining an optimal fixed con-tribution rate showed that individuals want to contribute as little as possible, even though pension savings provide fiscal advantages compared to accumulating personal wealth. This is mostly due to the fact that when being younger individuals prefer more personal wealth to be able to cover labor shocks. As consumption later in life is heavily discounted, early years have the largest effect on lifetime utility. From age 27 and on a fixed contribution of 18% is optimal when pension wealth is converted into a variable payout annuity and the risk-free rate is 0%.
6
Further research and implications
The Dynamic Life-cycle model that is presented in this thesis is a good starting point for further research. Although the number of state- and decision variables were limited in the model, the results provided interesting insights in optimal behaviour in the proposed pension system. Nevertheless, several limitations should be mentioned. Many interesting additions can be made, but they would require much more computation power.
First of all, stochastic interest rates and stochastic inflation could be included, which would be more realistic than deterministic interest rates and inflation as assumed in our model. Mortality numbers are taken as deterministic, where in reality, they are stochastic. Including stochastic mortality rates and interest rates could provide insights as they influence annuity payouts and hence provide another source of uncertainty for an individual. However, it is not expected that they would drastically change behavior. It is assumed that an individ-ual pays for housing costs, but there is no state variable for housing wealth, implying that someone either rents or never decumulates housing wealth. Including a state variable for housing wealth gives the option to run down housing wealth. While there are few products available on the Dutch market to run down housing wealth, Kooiman et al. (2019) show that a considerable amount of households move from a bought house to a rented house after retirement, hence making it possible to run down housing wealth.
Before retirement, all variables were normalized by the permanent component of wages to reduce computation times. However, due to normalization, maximum pension accumulation of the Dutch system could not be included, nor could progressive taxes and a franchise for pension contributions be included.
As shown in section 4, especially for low personal wealth in the early career, optimal stock investments are very high. As people with low personal wealth might have trouble paying for large unexpected expenses, it is counter-intuitive to invest all of that wealth in risky assets. To get rid of this problem, another state variable could be introduced that accounts for a savings account for large unexpected expenses.
In this thesis, we model based on an average Dutch male. For further research it would be interesting to vary the gender, educational level, total household income and wealth and whether or not an individual has children. Changes in gender and educational level influence income and mortality rate, whereas having children can influence personal savings, labor and a bequest motive. Where a retirement age of 68 was assumed, the expected increase in life expectancy could lead to a higher retirement age.
of payments. This results in high annuity payouts in the years after retirement and much lower payouts later on. This product is especially interesting for individuals with a low per-ceived life expectancy. It would be interesting to investigate if this product enhances welfare. Implications
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