Dynamic Life-Cycle Investing: The Holy Grail in a Changing Pension System? Gerben Berentsen S2375605

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Dynamic Life-Cycle Investing:

The Holy Grail in a Changing Pension System?

Gerben Berentsen S2375605

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Master’s Thesis Actuarial Science

Supervisor University of Groningen: Prof. Dr. T.K. Dijkstra

Supervisors PGGM: Agnes Joseph MSc AAG & Pascal Janssen MSc AAG Second Assessor University of Groningen: Dr. D. Ronchetti

Acknowledgement: I would like to thank all professionals at the AA&A department of PGGM for their

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Master’s thesis:

Dynamic Life-Cycle Investing:

The Holy Grail in a Changing Pension System?

Gerben Berentsen S2375605 Supervisors:

Prof. Dr. T.K. Dijkstra (University of Groningen) Agnes Joseph MSc AAG (PGGM/Achmea)

Pascal Janssen MSc AAG (PGGM) March 1 2018

Abstract

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1 Introduction

Life-cycle investment models (life-cycles in short) are concerned with the problem of choosing an asset allocation for the pension savings of an economic agent as he or she (henceforth I will use the term ‘he’) moves through the life-cycle from young adulthood, to middle age and retirement. The use of life-cycle models has greatly increased as there is currently a major shift in the occupational (second pillar) pension, where employers change the future pension system of their employees from defined benefit (DB) to defined contribution (DC) schemes. In a defined benefit pension scheme the pension of an employee is basically guaranteed. In a defined contribution scheme the employer and employee contribute to pension wealth. Unlike a defined benefit scheme, in a defined contribution scheme there is no guarantee on how much pension the employee will receive during retirement. When the employee wants to retire, his pension wealth is converted into an annuity. Hence, the size of the monthly pension payments greatly depends on past returns on pension wealth. As a result, pension providers look for ways to determine the optimal method to invest pension wealth for the employee. This is where life-cycle investment models come in play. Investing according to a life-cycle has several benefits: life-cycle models are versatile and can be adjusted to the specific needs and demands of an employee. Furthermore, they remove the necessity of making difficult financial decisions for the employee.

This thesis presents a new life-cycle model. In addition to the age of the agent, the life-cycle takes pension wealth and outstanding student loan into account. Each period, pension wealth and outstanding student loan are determined and based on this information the life-cycle gives advice on how much to contribute to pension wealth and how to allocate pension wealth. In contrast to previous literature on life-cycle models, the life-cycle presented in this thesis is a dynamic life-cycle. Where static life-cycles give advice on the allocation strategy beforehand and do not alter said advice, the presented life-cycle alters its advice based on past performance. Furthermore, the presented life-cycle takes Dutch regulations into account: pension contributions are limited to age-dependent tiers (Table 6) and pension payments during retirement are limited as well.

The life-cycle is optimised by means of a grid search, where all state and decision variables are discretized over a grid. Using dynamic programming the optimal decision is determined for each point on the grid. To solve expectations in the value functions, I resort to Gauss-Hermite quadrature. The life of the agent is partitioned into four phases. By dividing the agent’s life in phases and optimising for each phase separately, the required computation time is significantly reduced. For values in the state spaces that are not defined on the grid, the optimal decisions are estimated by means of interpolation.

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Wall Street crash of 1929, a simulation that involves unforeseen jumps in the stock returns is performed. Lastly, a simulation by means of a bootstrap analysis is performed. The bootstrap is based on yearly inflation adjusted returns of the S&P 500. All simulations show that on average the dynamic life-cycle performs much better than the alternatives. However, the dynamic life-cycle performs worse than the alternatives in the lower quantiles.

Dutch law restricts pension contributions to age-dependent tiers. The maximum allowed pension contribution as a percentage of income increases as the agent grows older. As employers pay a percentage of the total pension contributions, there is an inherent age discrimination in the labor market. Two agents earning the same wage and doing the same work, only having different ages, results in a larger contribution to pension wealth for the older agent that the employer has to pay. Hence, the Dutch government has proposed to change the way pension contributions are restricted.

This thesis explores the implications of changing the contribution methods for an agent. The life-cycle is optimised for three different methods of contribution. First, the life-cycle is optimised by limiting the contributions to the tiers. This results in a optimal life-cycle under the current Dutch regulation for pension contributions. Next, the pension contributions are unrestricted and agents can contribute freely to pension wealth. However, removing the limitations on contributions to pension wealth does not remove the limitations on received pension during retirement. Under current Dutch regulation, employees in a DC pension scheme are limited in how much pension they are allowed to receive. As pension payments and pension wealth are directly linked, this limits the maximum amount of pension wealth which an employee can convert into pension payments. This thesis explores the effect of removing the limitations on contributions, but still imposes the restriction on maximum pension during retirement. For both the limited and unlimited contribution methods, the average contribution rate is calculated. Using these two contribution rates two life-cycles with a fixed contribution rates are optimised. As a result, the thesis presented here compares the effect of the contribution method on the life-cycle and resulting pension and lifetime utility.

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2 Theoretical Framework

2.1 Life-Cycle Investing

Traditionally, life-cycle models only include age as a dependent variable and gave solely advice on asset allocation. Research in life-cycles has expanded by including more variables as input and output. As such life-cycle models can also give advice on the optimal consumption, contribution and labor supply. The first life-cycles were developed by Merton (1969) and Samuelson (1969). Merton (1969) uses a continuous time model and solves the model analytically for constant relative risk aversion (CRRA) and constant absolute risk aversion (CARA) utility functions. Samuelson (1969) uses a discrete time life-cycle, and presents a dynamic programming method for solving the model. In a follow up paper Merton (1971) investigated whether including more than one risky asset would alter the optimal solution of the life-cycle. This lead to the separation theorem which states that, under a set of conditions, a pair of funds can be constructed from a set of risky assets such that the agent does not prefer a linear construction of the original assets over a linear construction of the two funds.

The literature on life-cycle investing has expanded by increasing the complexity of life-cycle models. Richard (1975) includes uncertain time of death and the option for agents to buy life-insurances and finds that the separation theorem of Merton (1971) also holds under these conditions. Koo (1998) adds a liquidity constraint and restricts income to be uninsurable, which results in the agent taking less risk than he would if the market had been perfect. Housing and interest rate risk are incorporated in the life-cycle by Hemert et al. (2005), who find that agents start buying houses from the age of 25 and by 40 all agents own a house.

The impact of a correlation between labor income and returns of the financial market on the asset allocation in the life-cycle model has been widely investigated. Bodie et al. (1992) incorporates such a correlation and allows for flexible labor supply. They show that, due to the correlation, human capital can be treated as a combination of the risky and risk-free asset. As a result, this influences the asset allocation of wealth for the agent. Viceira (2001) shows that if income risk is idiosyncratic one should invest riskier than when income risk is systematic. Benzoni et al. (2007) assume a cointegration between labor income and dividends, which leads to the young holding a small fraction of risky assets, due to their human capital a having large exposure to the market.

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labor income shocks and finds that agents save much more and are more conservative with their asset allocation when there is a chance for disastrous income shocks. Gomes et al. (2008) builds on this model by including flexible labor supply. Chai et al. (2011) build even further on this model by allowing the agent to set his own retirement age. Both papers find that over the life-cycle agents first invest fully in the risky asset and slowly become more conservative as they age.

Heaton and Lucas (1997) and Hindy et al. (1997) allow for habit forming in the utility function. In Heaton and Lucas (1997) the agent gets utility from past consumption. In Hindy et al. (1997) the utility function is compromised of two goods that decay over time. Polkovnichenko (2006) also includes a habit function and provides simulation results, she finds that agents become more conservative as savings are near the habit. These models still find that the young should invest quite aggressively in the risky asset.

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2.2 Optimisation

Early and relatively simple life-cycle models, such as Merton (1969), can be solved analytically. Most of these analytical solutions use the dynamic programming method described in Bellman (1954) and Samuelson (1969). Yet, for more complicated models most literature resorts to numerical optimisation. In such cases, it is impossible or too hard to find an analytical solution for the expected value function in each period. In a life-cycle model investment setting the value function gives the implicit expected future utility of an agent depending on the state variables, such as pension wealth. Most literature resorts to performing a grid search, where the state and decision variables are distributed over a grid and the optimal decision is determined for each state. Hence, for each state an optimal policy is derived. Next, given the optimal decision for each state, the policy function can be derived, which is a function of the state variables and gives the optimal decision. For observations in the state variables that do not lie on the grid, some sort of interpolation must be utilised. Furthermore, most literature use some form of numerical integration to compute the expected value function. This thesis uses Hermite spline and bilinear interpolation, for integrals in the value function Gauss-Hermite quadrature is used.

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2.3 Allocation Strategies

Besides life-cycle investing, there are many popular alternative investment strategies. One of those alternative strategies is a buy-and-hold strategy, where the agent initially decides on some asset allocation and never interferes afterwards. For each contribution to pension wealth, the agent invests this contribution according to the initially determined asset allocation. Hence, there is no re-balancing for this strategy. A constant mix investment strategy is also a popular alternative to life-cycle investing. As with a buy-and-hold strategy, with a constant mix, the agent initially decides on an asset allocation and does not alter this allocation. However, unlike a buy-and-hold strategy, the agent re-balances his portfolio such that his initial asset allocation holds again. The investor not only needs to decide on the allocation, but also on when to re-balance. Merton (1969) found that for his life-cycle model a constant mix over the life of the investor is optimal. Note that this constant mix strategy holds for all wealth of an agent. Hence, if human capital is included, the allocation of financial wealth to the risky asset decreases as age increases. However, more recent and complex life-cycle models do not favour the constant mix strategy1. Most life-cycle models result in a glide-path allocation, in which the agent invests heavily in stocks in his early life and this allocation decreases as the agent ages. For instance Cocco et al. (2005), suggest investing 100% in stocks until the age of 40, after which the agent should slowly decrease his exposure to the risky asset according to (200 − 2.5t)%, where t is the age of the agent. At the age of 60, the agent should not further decrease his exposure to the risky asset, and follow a constant mix with an allocation of 50%.

Arnott et al. (2013) perform an historical analysis of what would happen if one had invested in an inverse glide-path. An inverse glide-path is a glide-path in which the agent starts with a low exposure to the risky asset and increases as the agents ages. And finds that, overall, the inverse glide-path leads to a higher mean pension wealth and a higher minimum pension wealth, at the cost of increased volatility in the final pension wealth. Estrada (2014) performs a similar analysis from an international perspective and finds similar results. This thesis also incorporates an inverse life-cycle, an investment strategy that invests according to the opposite of what is prescribed by the life-cycle.

1_{Some models prescribe a constant mix between bonds and stocks. However, these models are dynamic on a deeper level}

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2.4 Risk Measures

To compare the life-cycle investment strategy with other widely used investment strategies: one can compare descriptive statistics, such as the mean and standard deviation of the resulting pension. However, risk measures also provide useful insight into how the life-cycle compares to the alternatives in the extreme cases.

One such risk measures is the value-at-risk (VaR). The VaR gives, for some confidence level ψ ∈ (0, 1), the smallest pension payment. The VaR of a pension investment strategy with pension payment X at the confidence level ψ is given by the smallest pension payment x such that the probability that the pension payments X are lower than x is no larger than ψ. Formally the ψ% VaR is defined by

VaRψ = inf{x ∈ R : FX(x) ≥ ψ}, (1)

where FX(·)is the cumulative distribution of pension payments.

The VaR gives a quantile of the pension payment. The VaR is a useful risk measure, yet it is not a coherent risk measure, since it does not satisfy the sub-additivity requirement2of coherent risk measures. Another widely used risk measure is the expected shortfall (ES). Unlike the VaR, the ES is a coherent risk measure. The ES gives the expected return in the q% worst cases. Hence, what return do we expect in the q% worst returns. The q% ES is defined by

ESq=

1 q

Z q

0

VaRu(P )du = E (P |P ≤ VaRq) . (2)

The VaR and ES will be reported in terms of lifetime utility and pension payments. The 1% VaR will be reported, as well as the 1% ES.

There has been much debate on which risk measure to use, the VaR or the ES. However, this debate is beyond the scope of this thesis. As a result both risk measures are reported as well as the conventional descriptive statistics.

2_{A risk measure satisfies the sub-additivity requirement if the risk measure of the combination of two funds is at most the}

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3 The Life-Cycle Model

3.1 Life-Cycle

At the basis of any life-cycle investment problem lies the utility function. Agents care about consumption, they want consumption to be as smooth as possible over the life-cycle. The agents do not directly care about their pension benefits. Unlike Blake et al. (2013), agents are not characterised by having a pension target. Instead, they want a smooth consumption pattern over their life-cycle. Following literature, the utility function is given by

U (Ct) =

C_t1−γ− 1

1 − γ for γ ≥ 0 and γ 6= 1, U (0) = −∞,

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where Ctis consumption at period t and γ is the relative risk aversion. The given utility function is of

the family of constant relative risk aversion (CRRA) utility functions. Furthermore, if γ = 1, the utility function is given by U (Ct) = ln(Ct).

Each period the agent makes a contribution to his pension wealth. The dynamics of the budget equation Wtare given by Wt+1= Rαtt Wt− Ct+ Y (t, Z) , (4) where Rαt

t is the rate of return on pension wealth between t and t + 1 depending on the asset allocation

αt, and wage is defined by Y (t, Z). Wage is dependent on age t and individual characteristics Z.

As stated above, the agent has to decide on his allocation to the risky and the risk-free asset. The dynamics of the risk-free asset Btare given by

Bt+1

Bt

= rf, (5)

that is the risk-free asset yields a constant return of rf3. For the risky asset St, I assume the following

dynamics

St+1

St

= rf + µ + σsνt≥ 0, (6)

with a risk premium µ for holding the risky asset, a standard deviation of shocks in the risky asset σs

and i.i.d. shocks νtin the risky asset, with νt∈ N (0, 1).

The total yield of a portfolio having an allocation of αtto the risky asset at time t is given by

Rαt t = (1 − αt) Bt+1 Bt + αt St+1 St = rf + αt(µ + σsνt). (7)

3_{The real risk-free rate ¯}_r

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Then the budget equation evolves according to Wt+1= rf + αt(µ + σsνt) Wt− Ct+ Y (t, Z) (8) = rf + αt(µ + σsνt) Wt+ φtY (t, Z), (9)

where φtis the pension contribution percentage at time t defined as φt = Y (t,Z)−C_{Y (t,Z)} t. In accordance with

Dutch law, I assume that the agent cannot withdraw from pension wealth before retirement, φt≥ 0. The

agent faces a borrowing and a short sale constraint, i.e. for all t the following must hold

0 ≤ αt≤ 1, (10)

Wt≥ 0. (11)

The agent has three phases during the active (non-retirement) part in his life: • a study phase,

• a working phase in which he also has to pay back his student loan,

• a working phase in which the agent does not have to pay back his student loan.

During his study phase the agent takes on a student loan, which he has to pay back later in his life. Income and consumption during the study phase are defined as

Y (t) = C(t) = βLfor 18 ≤ t < tw, (12)

where 0 ≤ β ≤ 1 is his loan rate, L the maximum student loan, and twthe pre-determined age at which

the agent starts working. I impose the restriction that the agent is not able to build up pension over his student loan.

The agent has to pay back the student loan, the agent pays back an amount of τ Y (t, Z) each year until the loan is paid back or until time tL. At time tL+ 1the loan is either paid back or forfeited and the agent

no longer pays back his student loan.

The student loan accumulates interest according to the risk-free rate. The total value of the loan at time t, Lt≥ 0, evolves according to L_t+1=      max 0, rf Lt+ 1t<twβL − 1t≥twτ Y (t, Z) if t + 1 ≤ tL 0 if t + 1 > tL . (13)

Income during the working phase, consist of three parts, namely a deterministic part (f (t, Z)) of wages on age and personal characteristics of the agent, a transitory shock (ηt∈ N (0, σ2Y)) with variance σY2 and

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has to pay back his student loan is given by

Y (t, Z) = (1 − τ )f (t, Z)eηt_. ₍₁₄₎

The income shocks in labor income are correlated with the returns of the risky asset with correlation ρ. These shocks could be interpreted as for instance a bonus or taking unpaid leave.

Permanent shocks in labor income are omitted, since this would result in another state space. As stated above, the debt is forfeit after several years, after which the agent enters a phase during his working life in which he does not have to pay back his student loan. This phase is characterised by the following income equation

Y (t, Z) = f (t, Z)eηt_. ₍₁₅₎

At retirement the agent uses all his pension savings to buy an annuity. It is obligatory by Dutch law to buy an annuity at pension age. While there are options to buy variable annuities, for which payments depend on market performance, I restrict my analysis to fixed annuities. These annuities pay a fixed income, regardless of market performance. The price of such an annuity bought at retirement age, tr,

paying 1 euro is given by

¨ atr = T −tr X t=0 tPtr rt f , (16)

where tPtr is the cumulative probability of surviving t years, given retirement age tr, and T is the

maximum age an agent can life (T − tr is the maximum amount of years an agent aged tr can enjoy

retirement before death).

I assume that after retirement an agent only has income from his annuity and a state pension and completely consumes both each year. These assumptions allow me to quantify the total utility of retirement with a corresponding pension wealth of fWtr. An agent with pension wealth fWtr consumes each year:

e C = Wftr

¨ atr

+AOW for t ≥ tr, (17)

where AOW is the state pension provided by the Dutch government, which all citizens receive unconditionally4. The total utility of having pension wealth fWtr at retirement, is given by

e Utr = T −tr X t=0 tPtrδ tCe1−γ− 1 1 − γ , (18)

where δ is the time preference of money.

Converting pension wealth at retirement to an annuity is in line with Dutch regulation and it saves on computation time, as I do not have to model the optimal consumption stream out of pension wealth during retirement.

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3.1.1 Maximum Pension

In the Dutch DC second pillar pension scheme pension premiums are limited. The maximum premium is based on an equivalent build up in a DB pension scheme. I use an equivalent build up rate of 1.875% per working year in an average wage DB scheme. The maximum contribution percentage of pension yielding wage is called a tier (in Dutch also known as ‘staffel’) and is dependent on the agent his age. The tiers are given in Table 6.

Pension yielding wage is the agent his wage (limited toe103,371 in 2017) minus a franchise. The goal of the Dutch second pillar pensions is to allow an agent to keep his standard of living after retirement. As the state pays a fixed state pension, pension build up in the second pillar does not need to compensate the entire standard of living during the working life. To reflect the state pension, a franchise is subtracted of the agent his wage to determine his pension yielding wage. The pension yielding wage of the agent is given by YP(t, Z) = max min Y (t, Z), Ymax − F, 0 , (19)

where F is the franchise, and Ymax_{the maximum wage (here}_e103,371).

The maximum pension benefit at retirement is defined as 1.875

100 twork ¯

YP, (20)

where tworkis the number of years the agent worked with an average pension yielding wage of ¯YP. If

the agent has large pension savings, such that his benefit would be larger than (20), the agent will only receive up to (20), any excess wealth is forfeit to the pension provider or employer. Hence, the agent has no incentive to save pension wealth beyond

Wmax = 1.875 100 tworkY¯ P ¨ atr. (21)

The agent does not want to save pension wealth beyond the level specified in (21). To determine this value he must form some expectations. The agent knows his past wages, he also knows the average future wage growth rates5 (Table 7). Then, the expected average pension giving wage over the total working life of the agent at time t is given by

Et Y¯P = 1 (tr− tw) t X s=tw YP(s, Z) + tr X s=t+1 Et YP(s, Z) !! . (22)

The maximum expected pension wealth for an agent at time t is given by Et(Wmax) =

1.875

100 (tr− tw)¨atrEt Y¯

P_. ₍₂₃₎

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3.1.2 Contribution Method

Currently contributions to pension wealth are limited to the tiers in a DC pension scheme in The Netherlands. Similar, the United States of America have limitations on pension benefits and contributions to pension plans6, however these limitations are much less restrictive than the Dutch regulations.

The recently elected Dutch coalition has announced it wants to change the way pension payments are limited7. Currently, the tiers increase over age. As most employers pay a relative part of the pension contributions, for any young and old worker whose only difference is age, the young worker will be cheaper. This is due to the tier for the young being lower than for the old. To remove this ‘age discrimination’ from the labor market, the coalition announced that they want to set one maximum percentage of pension yielding wage for pension contributions.

The model described previously allows the agent to freely choose his contributions, while contributions can also be limited to the tiers. Hence, the effect of the restrictive tiers can be explored. Even more, it allows to find the average wage-weighted pension contribution percentage ¯φ of both contribution methods ¯ φ = Ptr t=twφtY P_{(t, Z)} Ptr t=twY P_{(t, Z)} . (24)

The average wage-weighted pension contribution percentage gives a good substitute of what the fixed contribution percentage to pension wealth should be if the tiers were abolished and replaced by a fixed premium. Hence, the effect of the announced pension system changes can be explored.

6_{https://www.law.cornell.edu/uscode/text/26/415}

7_{https://www.kabinetsformatie2017.nl/binaries/kabinetsformatie/documenten/publicaties/2017/10/10/}

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3.2 Investment policy

The agent faces the following problem when selecting his consumption and asset allocation

max E0 T X t=1 δt t Y s=0 ps ! U (C(t)) ! , (25)

subject to the budget constraint.

Let φtY (t, Z)be the contribution of wage to the pension wealth with φt the contribution percentage of

wage to pension wealth, τ the pay back rate of the student loan. Then, consumption is given by

Ct=                βL for t < tw, Y (t, Z) − (φt+ τ )Y (t, Z) for tw ≤ t ≤ tL, Y (t, Z) − φtY (t, Z) for tL< t < tr, e C for t ≥ tr. (26)

First, the model is solved when φt is limited to the tiers. Next, the agent is allowed to freely choose

his pension contribution, hence 0 ≤ φ ≤ 1. Lastly, the average wage-weighted pension contribution is calculated for both methods and the model is solved for both fixed contribution methods.

There are 3 state variables, namely time t, pension wealth Wt, and the value of the student loan, Lt.

Using backward induction I can derive a policy function for each state. Note that there are four different phases for which the agent needs to select investment policies. The four phases are (in chronological order):

1. The ‘study’ phase, at the start of the study phase the agent picks β (the loan rate) and builds up a student loan which has to be paid back in the next phase.

2. The ‘work + pay back’ phase, during this phase, the agent pays back his student loan and earns income for his work.

3. The ‘work’ phase. Following Dutch regulation, if the student loan has not been paid back after several years, the remainder is forfeit. Hence, during this phase the agent does not have to pay back his student loan.

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equation is given by Vt(Xt) = max Ct,αt U (Ct) + δptEt Vt+1(Xt+1) (27) It is extremely hard (or even impossible) to find an analytical solution for Vt(·). However, in the last

year before retirement the value function can be determined, since this is the present value of all annuity payments corresponding to the final wealth.

Equation (18) gives the total utility of retirement dependent on the resulting pension wealth. Next, using Equations (18) and (27) the value function one year before retirement can be derived. Using backward induction the value function two years before retirement can be determined, using this value function the value function three years before retirement, etc. Hence, the value function can be determined for every t using the backward induction method.

Next, the policy functions for the ‘work + pay back’ phase. During this phase, income is defined as in Equation (14). The Bellman equation for this phase (except for when t = tL) of the problem is given by

Vt(Xt, Lt) = max Ct,αt U (Ct) + δptEt Vt+1(Xt+1, Lt+1) , (28)

where Vt(·, ·)is the value function corresponding to this phase.

At t = xL, the value function VxL+1is known, as it is the first value function in the ‘work phase’, which

we can derive using backward induction. For this particular period we have the following Bellman equation VxL(XxL, LxL) = max C_xL,α_xL U (CxL) + δptExL VxL+1(XxL+1) . (29)

Note that VxLdepends on the state spaces X and L, whereas VxL+1only depends on X.

As described above, I use backward induction to determine the optimal policy for each t. By dividing the problem into phases I greatly reduce the computational required power for the total life-cycle, as this keeps the number of state variables at a minimum in each period.

To solve the expectation in the Bellman equations, I resort to Gauss-Hermite quadrature. Gauss-Hermite quadrature is a method of numerical integration. It quickly solves integrals given some accuracy level A. Note that there are two sources of risk, namely shocks in the risky asset νt and in income ηt. Then,

Gauss-Hermite quadrature gives

E(Vt(Xt)) = Z ∞ −∞ Z ∞ −∞ Vt(Xt)Φ(ηt, νt)dηtdνt (30) ≈ 1 π A X i=1 A X j=1 wijV (Xij), (31)

where Xij are the realisations corresponding to weight wij and Φ(·, ·) is the bi-variate standard normal

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The state variables are distributed over a search grid. Since, the utility corresponding to each pension wealth on the grid is known, I can compute the expected value function one year before retirement and compute an optimal investment policy. During the ‘work’ phase I use monotone Hermite splines for wealth realisations that are not defined on the grid. For the ‘work + pay back’ phase, I resort to bilinear interpolation of the wealth and student loan values.

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3.3 Jumps in Stock price

Unforeseen extreme shocks are sometimes observed in the stock market. Examples of such shocks are the Wall Street crash in 1929 and the crisis of 2008.

In this section I introduce these extreme shocks. The investor is unaware that these extreme shocks can occur, and hence does not take them into account when optimising the life-cycle model. The investor follows either the policy functions determined in the previous section or one of the alternative investment strategies.

To simulate the extreme shocks (positive or negative), stock prices follow the following dynamics

St+1= (rf + µ + σsνt) St+ N∆t

X

t≤s≤t+1

(ξs− 1)Ss, (32)

where ξsthe size a shock, with log(ξs) ∈ N (µs, σ2s)and N∆tthe number of shocks that happen between t

and t + 1. I assume the total number of shocks at time t (Nt) to be a Poisson process with mean λT . This

model is based on Broadie and Kaya (2006) and Burg (2016). Unlike these papers I do not use a Heston model to describe normal stock dynamics, as a Heston model would result in an additional state space for the variance of the risky asset.

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3.4 Robustness

One of the main focuses of this thesis is to compare the life-cycle investing strategy with other alternative investment strategies. This gives insight in the performance and the robustness of the life-cycle. The alternative investment strategies that are considered are

• Buy-and-Hold strategy, • Constant mix,

• Inverse life-cycle, • Static life-cycle.

The first two investment strategies require the investor to decide on some initial asset allocation ¯α. I assume the investor takes the wealth weighted mean allocation that the life-cycle prescribes in a benchmark case, which is defined as

¯ α = Ptr t=twαtWt Ptr t=twWt . (33)

The third alternative strategy, the inverse life-cycle, does the opposite of what is prescribed by the policy functions of the life-cycle.

For these three alternative strategies the agent also needs to determine how much to contribute to his pension wealth. I assume that that for all alternative strategies, the contribution to pension wealth is equal to the prescribed contribution given by the policy functions determined for the optimal life-cycle. The resulting life-cycle model as described in Section 3.1 is a dynamic investment strategy, the prescribed actions by the policy functions depend on time, pension wealth and outstanding study loan. The static life-cycle strategy calculates the optimal pension contribution and allocation beforehand, hence the allocation and contribution are only dependent on time. The allocation will not change due to changes in pension wealth. However, if pension wealth becomes too large, the agent will be informed and he will make no more contributions. If pension wealth drops below the maximum expected pension wealth, contributions will resume. The asset allocation will still follow the pre-determined asset allocation. For all investment strategies a Monte-Carlo simulation is performed. The method of simulation is described in Algorithm 1. The results of the Monte-Carlo simulation show how the investment strategies behave in contrast to each other.

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is, for all investment strategies, the same simulated realisations of asset and wage returns are used.

Algorithm 1Simulation method without stress

1: procedureSIMULATION OF INVESTMENT STRATEGIES

2: For each simulation i and each t: sample labor risk (ηit) and non-stress asset shocks (νit) from

multivariate normal distribution.

3: Find the optimal contribution to pension wealth and optimal allocation given the starting states using policy functions.

4: Calculate pension wealth Wit+1 and student loan outstanding Lit+1 using the previously

sampled ηit, νit.

5: Find the optimal allocation and contribution for t + 1 given the previously determined state variables.

6: Repeat steps 4 and 5 until t = tr.

7: end procedure

8: Repeat steps 3 until 6 above for each investment strategy

The results of the Monte-Carlo simulation show how the investment strategies compare in a non-stress scenario. The previous section described alternative stock price dynamics which include extreme shocks. Allowing such jumps in the stock price allows me to compare the investment strategies in unforeseen stress situations. Hence, another Monte-Carlo simulation is performed using the stock price behaviour that allows for jumps. Algorithm 2 shows how the Monte-Carlo simulation with jumps is performed. To further investigate the differences between the investment strategies, a bootstrap analysis was performed. A bootstrap analysis has the benefit that it removes the necessity to make assumptions on the distribution of the underlying data. There has been much debate in the risk management literature (for instance in McNeil et al. (2015)), on which distribution to use when modelling financial returns. However, a simple bootstrap can fail with auto-correlation in the data.

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Algorithm 2Simulation method with jumps

1: procedureSIMULATION OF INVESTMENT STRATEGIES

2: For each simulation i and each t: sample labor risk (ηit) and non-stress asset shocks (νit) from

multivariate normal distribution.

3: Generate the number of jumps for each simulation Ni where Ni is Poisson distributed with

mean λ(tr− tw).

4: Generate the time at which a jump happens, Jij ∼ dU (0, (tr−tw))efor j = 1, ..., N and d·e denotes

a ceil operator and U the uniform distribution.

5: For each jump, sample the jump size log(ξJij)from N (µs, σ 2 s).

6: Find the optimal contribution to pension wealth and optimal allocation given the starting states using policy functions.

7: Calculate pension wealth Wit+1 and student loan outstanding Lit+1 using the previously

sampled ηit, νit, Ni, Jij, ξJij.

8: Find the optimal allocation and contribution for t + 1 given the previously determined state variables.

9: Repeat steps 7 and 8 until t = tr.

10: end procedure

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4 Calibration

4.1 Model Calibration

In this section I define the characteristics of the agent and the calibration of the life-cycle. Section 7 reports a sensitivity analysis by varying some of the assumptions made here.

The agent enters the life-cycle model at the age of 18, this is at t = 0. I assume that the agent is born at the start of the year. He studies for 7 years, until the age of 25. Hence, he earns his first wage at the age of 25 (tw = 7). Retirement starts at the age of 67 (tr = 49), hence the agent works for a total of (tr− tw =)

42 years. For risk aversion I assume γ = 5, as do Gomes et al. (2008). The time preference parameter δ is set to 0.975 (Chai et al. (2011) use δ = 0.97, here agents prefer current consumption a bit more). The effects of an agent his risk aversion are explored in Section 7.

Data on income is provided by Statistics Netherlands (CBS), who give average wages for persons that worked the full year in age intervals of 5 years. The average wage for ages 20-25 ise21,100 a year, for the ages 25-30 ise29,700. Hence, the starting wage is set to21,100+29,70₂ = 24, 100. Expected wage growth percentages are given in Table 7, these wage growth percentages are industry-standard and used by the government8_.

The Dutch Royal Actuarial Association provides sex dependent estimated yearly death probabilities9for ages 0 till 120 for the years 2016 until 2186. From these estimated yearly death probabilities the yearly survival probability can be calculated

Px = 1 − qx, (34)

where Px is the probability of surviving one more year for an agent aged x and qx is the estimated

probability of dying during the coming year for an agent aged x.

To calculate these yearly survival rates, I assume agents are male and aged 18 in the year 201710_.

The risk-free rate is set to the same value as in Gomes et al. (2008), namely 1% ⇐⇒ rf = 1.01. Note

that the student loan also grows at the rate of the risk-free rate. The rate of return on the risky asset is determined on the basis of the inflation adjusted returns of the Standard & Poor’s 500 index (S&P 500). Monthly data on closing values of the S&P 500 are obtained from Yahoo! Finance11_{. The Federal Reserve}

Bank of Saint Louis, provides data on yearly U.S. inflation12. The inflation adjusted return on stocks is

8_{See: https://zoek.officielebekendmakingen.nl/kst-26020-3.html} 9_{Prognosetafel AG2016: https://www.ag-ai.nl/view.php?Pagina_Id=731}

10_{The analysis can also easily be done for females, however survival probabilities are gender dependent.}

11_{See: https://finance.yahoo.com/quote/%5EGSPC,dividendsareincludedinS&P500index, Yahoo! Finance is part of}

Yahoo! Inc.

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calculated according to

Inflation adjusted return = 1 +Return S&P 500 1 +Inflation − 1.

Merging the two data sets results in 65 yearly observations, spanning from 01-01-1951 to 01-01-2016 (see Figure 1). The average inflation adjusted return is 5%, with a standard deviation of 15%. Dimson et al. (2009) reports a larger average inflation adjusted return on equity and a larger standard deviation, however their data ends at 2001, whereas this data set runs till 2016.

This yields the following parametrisation

rf + µ = 1.05,

σs= 0.15.

The labor income shocks as defined in Equation (14) have a mean of 0, and a standard deviation of σY = 0.07which is estimated by Cocco et al. (2005). Unlike Cocco et al. (2005) and Gomes et al. (2008),

who set ρ = 0, the correlation between income and returns of the risky asset is assumed to be ρ = 0.2. This because the transitory shocks in this model can be interpreted as a sort of bonus or temporarily set-back due to an economic downfall, which naturally correlates with movements of the risky asset. When the agent retires, he does not only receive his pension but also a state pension (AOW). The value of the franchise (F ) in Equation (19) depends on the AOW. AOW is dependent on marital status, however for determining the franchise, the AOW of a married person has to be used. For 2017 the AOW of a married person is set toe801.05 per month (= e9,612.60 per year). The franchise is set to the minimum franchise a DC pension scheme is allowed to use, namely

F = 100

75 · AOW. (35)

For the study loan I set the pay back rate to 4% (τ = 0.04) and the pay back time is 35 years. This is in line with how Dutch students have to pay back their student loans13. However, Dutch ex-students only have to pay back the 4% over a part of their income. Yet, for convenience, students pay back over their entire income.

In order to test the performance of the life-cycle and the alternative investment strategies unforeseen shocks as described in Equation (32) are allowed to happen. These shocks happen only rarely. I assume that on average such a shock happens every 200 years. Hence, the probability of such a shock is 0.5% (λ = 0.005). The shocks are meant to be a sort of a stress test for each of the investment strategies. If a shock happens, the average shock will lead to drop of 50% in the price of the risky asset. Since ln(ξ) ∼ N (µs, σs), I choose µs = −0.7, then a shock with ln(ξ) = µs = −0.7 will result in a drop of

(exp(−0.7) − 1) St≈ −0.5St, i.e. a loss of 50% on the initial investment in the risky asset.

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The above mentioned S&P 500 dataset is used for the bootstrap analysis (as described in Section 3.4). To be able to correctly perform a bootstrap, it is necessary that the data contains no auto-correlation. Figure 2 gives the estimated (partial) auto-correlation for different lags. To formally test for auto-correlation, a Box-Pierce and Ljung-Box test are performed with a lag of 1. Table 1 gives the results of these tests, both tests find no indication for auto-correlation. For further analysis of the S&P 500 returns see Appendix A.2

Table 1:Auto-correlation test results

Test p-value Box-Pierce 0.766 Ljung-Box 0.761 1950 1960 1970 1980 1990 2000 2010 −0.4 −0.2 0.0 0.2 0.4

S&P 500 inflation adjusted return

Year

Inflation Adjusted Retur

n

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4.2 Optimisation

To find the optimal life-cycle investment strategy I resort to a grid search, i.e. for a grid of values the optimal policy functions are calculated. The life-cycle model has two decision variables, namely the allocation to the risky asset and how much to contribute to pension wealth. Furthermore, it has three state variables, time, pension wealth and outstanding student loan. All these variables are discretized over a grid. For values of the state variables that are not on the grid, interpolation techniques are used. Hence, the more points on the grid, the more precise (and smooth) the policy functions will be.

Table 8 gives the grid sizes of all variables for each contribution method. Note that choosing a finer grid will result in more precise results, but comes at the cost of computation time. This restricts the maximum number of grid points.

There are three contribution methods investigated: free contribution, a limited contribution (tiers) and a fixed contribution. The fixed contribution method forces the agents to pay a fixed percentage of income to their pension savings. The agent is not allowed to change his contribution rate, hence it is not a decision variable in this case. Removing one decision allows to select a much finer grid for the other state and decision variables.

0 5 10 15 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF Auto−correlation S&P 500

(a)Auto-correlation S&P 500 inflation adjusted returns

5 10 15 −0.2 −0.1 0.0 0.1 0.2 Lag P ar tial A CF

Partial auto−correlation S&P 500

(b) Partial auto-correlation S&P 500 inflation adjusted returns

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5 Results

5.1 Optimal Policy

The life-cycle model as described in Section 3.1 was optimised for three different types of contribution methods. At first, the model is solved for the agent that is allowed to freely choose his contribution to pension wealth. Second, the agent is limited in his pension contribution to the tiers. The tiers can be found in Table 6. Third, the agent pays a fixed premium based on the results of the previously determined optimal contribution method in the case the agent is allowed to freely choose his contribution. The resulting policy functions give the optimal decision for the agent based on the state variables. There are three state variables: time, pension wealth, and outstanding student loan. The two decision variables are: allocation to the risky asset, and pension contribution. Note that for the fixed contribution method, pension contribution is given and not a decision variable.

To investigate the resulting policy functions, two benchmark cases are calculated for each contribution method. In these benchmark cases all asset returns are as expected. The agent, however, does not know beforehand that everything will happen as expected. Furthermore, all wage increases happen as expected and there are no income shocks. For each contribution method there is one agent with no student loan and one with the maximum student loan. In the graphs presented later, the red line represents an agent with no student loan and the black line an agent with the maximum student loan. Since pay back to the student loan is independent of the contribution method to pension wealth, the dynamics of the student loan are the same for all contribution methods. Figure 3 shows how the maximum student loan behaves over time. Note that an agent never fully pays back his student loan. The agent pays back the student loan till the age of 59, after which the remainder of his loan is forfeit.

20 30 40 50 60 0 20 40 60 80

Student loan outstanding over age

Age

Student Loan Outstanding

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5.1.1 Free Contribution

In this section, the life-cycle model is optimised for an agent who is allowed to freely choose his pension contribution. Figure 4a shows the optimal contribution pattern for each of the benchmark agents. Both agents do not immediately start saving for their pension when they start working. The agent without a student loan starts saving at the age of 37, whereas the agent with a maximum student loan starts saving at the age of 39. Both agents try to smooth consumption. Due to (expected) increasing wages14, agents postpone saving for their retirement, as they can finance their pension with these wage increases.

Figure 4c shows the consumption pattern of both agents, one with no student loan and one with the maximum student loan. Note the evident consumption smoothing, from their 40’s agents try to keep consumption constant. This explains why agents do not save for retirement before their 40’s. This also explains as to why the agent with the maximal student loan delays saving for retirement longer than the agent with no student loan, as this agent has to pay back 4% of his income. Furthermore, it is noteworthy that for both agents the contribution to pension savings increases sharply. When agents do save for retirement, most of the time they save beyond the tiers.

Both agents invest fully in the risky asset when they start saving. After approximately 20 years they start to decrease their allocation to the risky asset. The agent without a student loan starts to reduce his exposure to the risky asset sooner than the agent who has a student loan. Both agents show a sharp decline in their exposure and have a final exposure of ≈ 30%.

This is in line with other literature, where the allocation to the risky asset starts at 100% and declines as agents age. However, agents keep their exposure at a 100% for a long time, where in Chai et al. (2011) agents reduce their exposure to the risky asset in the beginning of their 40’s. Unlike most other literature, agents start late in their working life with saving for retirement. The delay in pension saving is in line with Gourinchas and Parker (2002), who empirically show that employees delay saving for their pension.

Figure 4d shows the value of pension savings over time for both agents. The dotted black line shows the maximum amount of pension wealth an agent can convert into an annuity. As stated in Section 3.1.1, pensions are limited, hence so is pension wealth. For both agents their pension wealth nears the maximum pension wealth which they can convert into an annuity. This also explains why agents lower their exposure to the risky asset as they age. If they would have kept their exposure at 100%, it could lead to a wealth larger than the maximum level, in which case agents are no longer fully compensated for taking on risks.

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20 30 40 50 60 0.0 0.1 0.2 0.3 0.4 0.5

Pension Contribution as percentage of Income

Age

Contr

ib

ution

(a) Optimal pension contribution as fraction of income. The blue dotted shows the tiers.

20 30 40 50 60 0.0 0.2 0.4 0.6 0.8 1.0

Optimal Allocation to the risky asset

Age

Allocation to r

iksy asset

(b)Optimal allocation to the risky asset as fraction of pension wealth. 20 40 60 80 0 10 20 30 40

Optimal Consumption over the life−cycle

Age

Consumption

(c)Optimal consumption. The green line gives the wage of both agents. Black dotted vertical line is the retirement age. 20 30 40 50 60 0 100 200 300 400

Wealth over the life−cycle

Age

P

ension W

ealth

(d) Wealth over time. The black dotted line gives the maximum wealth for which an agent can buy an annuity.

Figure 4:Benchmark results for an agent who can freely choose his pension contribution. Black denotes an agent

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5.1.2 Limited Contribution (tiers)

In Section 5.1.1 the agents are allowed to freely choose their pension contribution. However, as pension wealth and pension contributions are tax exempt, policy makers have limited pension contributions to age-dependent tiers. Here the two benchmark agents can freely choose their pension contribution, yet they are limited to these tiers. The tiers are given in Table 6.

Figure 5a shows the pension contribution of both agents, the blue dotted line shows the tiers. Note that the tiers increase every 5 years as the agents age. Initially both agents do not save for their pension. At the age of 30 the agent with no student loan starts making his first pension contribution, while the agent with a student loan starts at the age of 31. At first the agents are not limited by the tiers. However, from the age of 35 both agents seem to be limited in their contribution by the tiers. For the rest of their working lives both agents seem to be limited by the tiers.

Figure 5b shows the allocation to the risky asset of both agents. As in the previous section, both agents start with an asset allocation of 100%. Even though the agent without a student loan starts saving sooner than the agent with one, they both start reducing their exposure to the risky asset at the same age, namely at the age of 55. The decline in exposure follows a similar path for both agents, yet the agent without a student loan reduces his exposure a little bit faster. Just before retirement both agents have an allocation to the risky asset close to 55%.

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20 30 40 50 60 0.0 0.1 0.2 0.3 0.4 0.5

Age

Contr

ib

ution

(a) Optimal pension contribution. The blue dotted shows the tiers.

20 30 40 50 60 0.0 0.2 0.4 0.6 0.8 1.0

Age

Allocation to r

iksy asset

(b)Optimal allocation to the risky asset.

20 40 60 80 0 10 20 30 40

Age

Consumption

Age

P

ension W

ealth

Figure 5:Benchmark results for an agent who is limited in his pension contribution. Black denotes an agent with

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5.1.3 Fixed premium

The Dutch government wants to abolish the tiers and replace them by a fixed premium. At the moment of writing this thesis there is little known about the level of this fixed premium. Hence, I resort to calculating the average wage-weighted pension contribution (henceforth abbreviated as average pension contribution) of both previously calculated benchmarks. The average pension contribution is calculated for the agent with a maximum student loan. Table 9 gives the average pension contribution for both methods. For the free contribution method, the agent contributes 15.69% on average, while the agent who is limited by the tiers contributes on average 15.10%.

These pension contributions are relatively low compared to pension contributions for DB contracts, which also have a fixed premium for all cohorts. This can be explained by the fact that the model does not take administration and transaction costs into account. For comparison, the pension contribution for Pensioenfonds Zorg en Welzijn (PFZW), the pension fund for employees in medical care, has a pension premium of 23.5% (2017). However, excluding additional premiums due to the coverage ratio being too low, disability insurance and spouse pension the premium would be around 19%. Please note that comparing pension contributions in a DB and DC is relatively hard, as pension payments are ‘guaranteed’ in a DB contract, furthermore a higher premium in a DC scheme likely results in a higher pension, while this is not necessarily the case in a DB contract.

For the remainder of this thesis for the fixed contribution method, a fixed contribution of 15.69% is used. This is the average pension contribution that an agent with the maximum student loan contributes under the assumption that he is not restricted in his pension contribution. Note that the difference in average pension contribution for the agent that is limited to the tiers is only 0.59%. Tables 9 and 10 give the results of both contribution rates, 15.69% and 15.10%, for the interested reader.

Figure 6b gives the risky asset allocation of both agents. Please note that for these figures, if there is only one black line visible, the red line is obstructed by the black line. Hence, in this figure, the agents with and without a student loan have the exact same behaviour with regards to the asset allocation. This is easily explained by the fact they contribute the same amount of euros to their pension wealth, hence the dynamics of pension wealth are the same for both agents. Compared to the other two contribution methods, agents with a fixed contributions lower their asset allocation much sooner. By construction, the agents immediately start saving for their pension when they start working. Hence, their wealth starts growing much sooner. Agents with a fixed contribution start saving 14 years earlier than the agents that can freely choose any contribution, while they start decreasing their exposure at the age of 43 (15 years before the agents with unlimited contribution).

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The agents that are limited by the tiers have a much lower final wealth than the agents with a fixed contribution. For example for agents with a student loan, there is almost a difference of ≈ e3000 in annual pension payments.

The consumption paths of both agents with fixed pension contributions is given in Figure 6c. As expected the only difference in consumption is the pay back of the student loan, as an agent cannot lower/raise his consumption by any other means than by taking a larger/smaller student loan.

A fixed premium forces the agents to start saving early for their pension. This results in agents being more conservative later in life with their pension savings. This might be explained by the fact that, unlike the other contribution methods, agents are not allowed to contribute additional funds to their pension after low (or negative) returns of the risky asset.

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20 30 40 50 60 0.0 0.1 0.2 0.3 0.4 0.5

Age

Contr

ib

ution

(a) Pension contribution. The blue dotted shows the tiers. 20 30 40 50 60 0.0 0.2 0.4 0.6 0.8 1.0

Age

Allocation to r

iksy asset

(b)Optimal allocation to the risky asset.

20 40 60 80 0 10 20 30 40

Age

Consumption

Age

P

ension W

ealth

Figure 6: Benchmark results for an agent who fixes his pension contribution to 15.69%. Black denotes an agent

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5.2 Optimal student loan

One of the advantages of dynamic programming is that the optimal student loan can be calculated by only looking at the years of studying: the estimated life-cycle model already gives the optimal actions the agent has to follow after the study, furthermore it also gives the expected utility of all future states given the initial states.

The student starts working at tw, at this time he has not yet starting to save for his pension, Wtw = 0. The

student loan outstanding depends on the student his consumption during the study phase. The student is restricted to consuming the same amount each year during his study. The agent chooses β, which is kept fixed for all years of study, and the agent consumes his entire loan each year.

The method of backward induction and dynamic programming already yielded policy functions and corresponding expected utility values for all t ≥ tw. Hence, finding the optimal student loan leads to the

following maximisation problem

max β tw−1 X t=0 t Y s=0 ps ! δt(βL) 1−γ_{− 1} 1 − γ + Etw−1 Vtw(0, Ltw), (36) L_t_w = tw X s=1 βLrs_f. (37)

To find the optimal β, a grid search was utilised. The total student loan and expected lifetime utility at t = twwere calculated for each β ∈ (0, 0.01, 0.02, ..., 1). Furthermore, the total utility of consumption during

the study phase was calculated. Hence, for each β the total expected lifetime utility was calculated. Table 2 shows per contribution method the optimal student loan. For all contribution methods, the optimal student loan is the maximal student loan. Furthermore, Figure 3 shows that the benchmark agent with a maximum student loan, does not fully pay back his student loan. Hence, the agent has an opportunity to borrow money which he does not (fully) have to pay back. This makes the student loan attractive for the agent.

Table 2:Optmimal student loan per contribution method

Contribution method Optimal β Resulting Student loan (e)

Free 1 90,474.91

Limited 1 90,474.91

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6 Simulation

To compare the investment strategies listed in Section 3.4 with the dynamic life-cycle, several methods of simulations are performed, starting with a Carlo simulation. This is followed by another Monte-Carlo simulation in which jumps as described in Section 3.3 are allowed. Furthermore, a bootstrap analysis is performed.

The simulations are performed for each of the three contribution methods and five investment strategies. For all simulations, it holds that the simulated paths of stocks and wage are simulated a priori. The simulated paths are used as input for each investment strategy.

6.1 Monte-Carlo simulation

For each investment strategy and each contribution method a Monte-Carlo simulation with 100,000 agents is performed. For each agent his enitire working life is simulated. The asset returns and wage shocks are the same for each investment strategy. That is, the asset returns and wage shocks are drawn beforehand, after which they are plugged-in in each investment strategy.

When the agent is allowed to freely choose his pension contribution or is limited to the tiers the agent contributes as is optimal for the life-cycle investment strategy. In the case of a fixed pension contribution, the agent always contributes a fixed percentage of his income, except when his pension wealth growths larger than the maximum expected pension wealth Equation (22). Any addition to pension wealth beyond this point is likely to be ‘wasted’, since it cannot be used to buy an annuity at retirement or withdrawn later on. If this happens the agent stops making contributions to pension wealth until retirement or until his pension wealth drops below his maximum expected pension wealth.

Figure 7 gives the simulated empirical cumulative density of resulting pension payments (ine1, 000) of each investment strategy and each contribution method. Note that these pension payments at retirement include AOW and are limited as described in Section 3.1.1. The sharp increase for the higher quantiles of pension, is where pension payments are cut off, due to being to high. The maximum pension depends on past income shocks, hence the maximum pension is not the same for each simulation. Each colored line corresponds to an investment strategy, black to the life-cycle, green the constant mix, blue the buy-and-hold, red the inverse life-cycle and purple the static life-cycle.

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limited or restricted in the other contribution methods.

For all contribution methods, the inverse life-cycle gives on average the lowest pension wealth. Yet, in the lower quantiles, the inverse life-cycle outperforms the alternative investment strategies. Tables 11, 12 and 13 reveal that for all contribution methods the 1% VaR and 1% ES are the largest for the inverse life-cycle. The inverse life-cycle seems to result in a much lower expected pension than the resulting pension of the other investment strategies, at the benefit of outperforming the alternatives at the lower quantiles. As a result the inverse life-cycle yields a stable, yet lowered mean pension.

On average, for all contribution methods, the life-cycle seems to result in a higher pension compared to the other alternative investment strategies. However, the alternative strategies seem to perform better in the lower and upper quantiles (with the exception of the inverse life-cycle in the upper quantile of pension payments). This phenomenon is easily explained as to when stock markets do not yield satisfying returns, the dynamic cycle increases its exposure to assets. This way the dynamic life-cycle tries to use the risk premium to increase pension wealth. However, if asset returns remain low (or even negative), the other strategies will yield a higher pension as their exposure to the stock market is lower. A similar result holds for the case when pension wealth is high, the dynamic life-cycle will reduce its asset allocation, whereas the other strategies will not.

In most cases, the buy-and-hold and constant mix strategies seem to yield a lower pension than the life-cycle, yet higher than the inverse life-cycle. Only in the extreme cases these strategies surpass the life-cycle.

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Pension payments do not necessarily reflect the performance of the investment strategy, as with the limited and free contribution methods, it might be the case that agents contribute more when following a specific investment strategy and as a result get a higher pension. Looking at the differences in lifetime utility provides a better comparison between the different investment strategies. Large contributions lead to a lower utility at the time of the contribution, but could yield an increased pension which in turn increases the utility at retirement. The empirical cumulative density lifetime utility functions are given in Figure 8.

Figure 8a shows the empirical cumulative lifetime utility density when the agent is free in his contributions and shows that there is not much difference between the investment strategies. This shows that if the agent is free to choose his contribution method, lifetime utility is not much dependent on the investment strategy. However, most countries enforce regulation that restricts the amount of contribution to pension wealth, hence extrapolating the differences between the investment strategies in these cases is of importance. It seems that following the life-cycle strategy leads to the highest utility on average. Yet, in the lower quantiles other investment strategies give a higher lifetime utility. The reasoning for this is similar as previously discussed in the case of pension payments.

For the inverse life-cycle, total lifetime utility seems to be much lower in most cases than lifetime utility of the other investment strategies. Yet, as with the pension payments, the inverse life-cycle outperforms all other investment strategies in the lower quantiles. Compared to the other strategies, the inverse life-cycle gives a good downside protection, at the cost of lower average lifetime utility.

The static life-cycle seems to perform better than the constant mix and buy-and-hold most of the time. Noteworthy is the fact that the difference in lifetime utility between the life-cycle and other investment strategies is lower when premiums are more restricted. The difference is the largest in the case of fixed premiums and lowest in the case of free premiums. Furthermore, the quantile for which the inverse life-cycle performs better than the alternative investment strategies is lower for a fixed premium. Simulations reveal that in most cases the dynamic life-cycle performs better than the alternatives. Only in the extremely good and bad cases the dynamic life-cycle performs worse than some of the alternatives. The underperformance in the lower quantiles is due to low returns of the risky asset, which causes the dynamic life-cycle to increass the asset allocation to the risky asset. If the next draw of returns is also low, pension wealth slips further down, resulting in lowered pension payments. The alternative strategies do not alter their asset allocation based on pension wealth, hence they are not susceptible to this phenomenon.

In the upper quantiles, the dynamic life-cycle lowers the asset allocation as pension wealth nears its maximum, hence the upside potential is lost.

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6.2 Stress

Besides a standard Monte-Carlo simulation, a simulation that includes jumps as described in Section 3.3 was also performed. This simulation is meant to be a stress test for the investment strategies. The agent has a chance to experience unforeseen disastrous shocks. These shocks are meant to represent market crashes. The agent does not expect that these crashes are possible, hence he optimises his life-cycle model with the assumption that stocks behave as in Equation (6).

Figure 9 shows the resulting pension of 100,000 simulations that include unforeseen stress jumps. Comparing Figures 7 and 9, shows that the inclusion of stress jumps does not result in big shift in pension results. Yet, the tail of low pension payments seems to be longer in the case of stress. The inverse life-cycle seems to be less affected by the jumps, this is as expected as the inverse life-cycle has little to none exposure to the risky asset when pension wealth is low. This reduces the probability of an agent being unlucky twice, first by having bad draws resulting in a lower pension and then having a large exposure when a shock occurs. Note that the dynamic life-cycle increases exposure when pension wealth is low. Hence, when pension wealth is low and a shock occurs, the agent with a dynamic life-cycle will be hit severely by the shock.

Figure 10 gives the resulting lifetime utility when jumps are included. As with the pension payments, for all contribution methods the tail of low lifetime utility is larger in the case of stress jumps. The inverse life-cycle seems to be affected the least by this, following the same reasoning as for the pension payments. Tables 11, 12 and 13 give descriptive statistics as well as the 1% VaR and 1% ES risk measures for resulting pension payments and lifetime utility when stress jumps are included in the simulation. These results are indicated in the tables by ‘(Stress)’.

Comparing these results with the results of the Monte-Carlo simulation shows that the risk measures are all lower when stress jumps are included. This is as expected since normal stock behaviour is the same as in the Monte-Carlo simulation. Note that there is a small possibility that the jumps are positive. However, analysis of the results shows that there was never a positive jump in the 100,000 simulations. The most extreme jump was ≈ −73%, while the least extreme jump was ≈ −7%.

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life-cycle experiences bad draws from the market and hence has a low pension wealth, he will have a 0% allocation to the risky asset, then if a shock occurs, the agent loses no money.

The main result of including stress jumps seems to be that, for all investment strategies, it increases the probability of a lower pension and lower lifetime utility. The inverse life-cycle is the least sensitive to the jumps. However for the most part, the empirical cumulative density functions are affected by the stress jumps on a minimal level. This is easily explained by the fact that the probability of experiencing a jump in a given year is only 0.05%. Hence, each simulation has a probability of 0.5(tr− tw) = 21%to

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Dynamic Life-Cycle Investing: The Holy Grail in a Changing Pension System? Gerben Berentsen S2375605

Dynamic Life-Cycle Investing:

Master’s thesis:

Dynamic Life-Cycle Investing:

Contents

1

Introduction

2

Theoretical Framework

3

The Life-Cycle Model

4

Calibration

5

Results

6

Simulation