Target based asset allocation in a defined contribution pension scheme

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Target based asset allocation in a

defined contribution pension scheme

Jelle Joosten

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Jelle Joosten Student nr: 11147407

Email: jehjoosten@gmail.com Date: August 27, 2017

Supervisor: dr. S. van Bilsen Second reader: dr. T. Boonen

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Statement of Originality

This document is written by Student Jelle Joosten who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

There has been a transition from DB to DC pension plans in recent years. In these plans, the participant’s capital is often invested according to a predetermined glide path. If the participant has a specific retirement target he would like to reach, the lack of a dynamic adjustment of the asset allocation makes this a suboptimal strategy. Under a set of fairly restrictive assumptions, an analytical solution for the based allocation problem can be derived. The target-based allocation strategy is introduced and, by means of simulation, compared to other allocation strategies, including the predetermined glide path.

The target-based strategy achieves a capital level within a 3% margin with respect to the target in over 70% of the simulated scenario’s.

Keywords Defined Contribution, Optimal asset allocation, Quadratic utility, Retirement, Target-based investing

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Introduction

1.1 Background

Pension schemes can generally be classified as either a benefit (DB) or a defined-contribution (DC) scheme. In a DB scheme, the participant gradually accrues the right to future pension benefit payments during his working life. In the basic situation, the premium level is periodically adjusted to cover the cost of these accrued rights. This way the participant knows in advance the exact amount of pension benefits he will re-ceive after he retires. For common accrual levels in the Netherlands, the total of accrued pension benefits over a full working life in combination with social security amounts to a retirement income of about 75% of the participant’s average pay during his working life. In a DC scheme on the other hand, only the level of the premium contributions is fixed. No pension rights are accrued by the participant during his working life. The contributions are invested in the financial markets where they generate a return. The contributions and returns accumulate until retirement. The participant then uses the accumulated capital to finance his consumption during retirement. This can be done through the purchase of a lifelong annuity or by periodically drawing down a fraction of the capital.

Historically, most of the funded pensions in the Netherlands have been accumu-lated in defined benefit schemes. Over the last two decades however, dropping interest rates, increasing life expectancy and multiple changes in regulation have led to a sig-nificant increase of the cost of guaranteed pension accruals. Also, these developments have increased the uncertainty with respect to the cost of future accruals. Companies - historically paying most of their employees premium contributions - have therefore started switching to defined contribution schemes. This protects the companies from large fluctuations in premium contributions.

From the point of view of the participant, this shift has increased the uncertainty, since the level of premium contributions is no longer adjusted to finance a specific level of retirement income. During his working life, when his accumulated capital is invested in the financial markets, he faces investment risk. This raises the question of how the accumulated capital can be optimally allocated across the available financial instruments (e.g. stocks, bonds), and how this allocation should evolve over the working life of the participant.

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2 Jelle Joosten — Target based asset allocation

1.2 Target date investing

Target date investing has become a widely adopted asset allocation strategy in defined contribution schemes. In a target date strategy, the allocation of the accumulated capital among different classes of financial assets is determined by the remaining time until the target (retirement) date. While the remaining time horizon is large, capital is heavily allocated to high risk assets. These assets have a higher expected return over the long term, but also have a higher volatility. As the target date comes closer, the participant has less time to recover from a possible downward financial shock. To decrease the pos-sibility of such a shock, capital allocation is then gradually shifted towards assets with lower volatility and lower expected return.

In the US, target date retirement funds have seen enormous growth over the past decade, from an estimated total invested capital of $70 billion late 2005 (Ibid), to $700 billion by the end of 2015 (Morningstar).

In the Netherlands, pension asset managers are legally required to invest the partici-pant’s capital in a prudent responsible fashion. A target date strategy is advocated (but not mandated) by the Dutch supervisor and has become the default strategy among pension providers.

1.3 Research question

A key feature of a target date strategy is the predetermined nature of the shift in asset allocation from high risk to low risk assets. Being determined only by the remaining time until the target date, the strategy does not utilize all available information. For instance, the asset allocation at any point in time during the investment horizon is in-different to realized returns - and thus the value of the accumulated capital - up to that point.

If the participant has a specific retirement capital target, the probability of achiev-ing this target might be increased by utilizachiev-ing information on both the remainachiev-ing time horizon and the remaining gap between accrued capital and target capital instead of only the former. Also, there may be other factors that could be relevant for the asset allocation decision.

The main objective of this thesis is to compare such a capital target based strategy with other asset allocation strategies, including the the target date strategy, and inves-tigate whether a capital target based strategy might be a viable alternative for asset allocation in the accumulation phase of a defined contribution pension scheme. This objective can be formulated in the following research question:

Can the asset allocation strategy in defined contribution pension schemes be improved by considering the participant’s retirement capital target and past returns as well as the remaining investment time horizon?

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1.4 Structure

The remainder of this thesis is structured as follows. Chapter 2 introduces the well known classical approach in academic literature for solving optimal asset allocation problems. Chapter 3 proposes and describes the alternative - target based - strategy which is the primary strategy of interest for this thesis. In Chapter 4, both of these strategies are compared to each other and to some suboptimal strategies that are often encountered in practice. Chapter 5 provides a conclusion and discussion of the findings.

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Chapter 2

The classical approach for

optimal asset allocation

The objective of the classical asset allocation problem can be stated as:

What is the optimal investment strategy at each point in time that will maximize the expected final utility of a risk averse investor?

Under certain assumptions on the investors preferences and on the dynamics of the financial market, an analytical solution for this problem has been derived in Merton (1969). A derivation based on the solution method introduced by Merton (1969) is given in section 2.1. This will be the starting point for the introduction of a target based strategy.

2.1 The stylized financial market

In the stylized financial market considered here, commonly called a Black-Scholes mar-ket, the investor can allocate his capital between two different assets: a risk-free asset (a bank account denoted by B ) and a risky asset (a stock, denoted by S ). The evolution of the price of the risk-free asset is given by the differential equation:

dBt= rBtdt (2.1)

where r is the continuously compounded risk-free interest rate. The evolution of the price of the risky asset is given by the stochastic differential equation:

dSt= µdt + σdZt (2.2)

where µ and σ are the expected return and volatility of the stock and Ztrepresents

a Brownian motion. It is assumed that an investor receives a premium for taking on the risk of investing in the stock, implying that the expected stock return µ is greater than the risk-free interest rate r.

During the investment horizon starting at time 0 and ending at time T , the investor must continuously decide which fraction αtof his capital to invest in the stock at time t.

The remaining portion of his capital, 1 - αtis then invested in the risk-free cash account.

It is important to note that the investor’s total capital consists of both his financial capital (monetary wealth), and his human capital (non-monetary wealth). In the con-text of this thesis, human capital consists of the present value of future contributions to the pension scheme and is perceived to be risk-free. In other words, the participant already has an implicit investment in the risk-free asset through his human capital.

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This investment decreases during the investment horizon, since the amount of future contributions decreases as the retirement age comes closer.

The inclusion of human capital reflects the fact that, at the beginning of the partic-ipants working life, the impact of fluctuations in his financial capital is only marginal with respect to his total capital. During the investment horizon, this impact gradually increases. As will be seen in the following sections, this has an important effect on the optimal allocation of the financial capital throughout the investment period.

2.2 The power utility function

To determine the optimal asset allocation level at each point in time, it is essential to determine the participants definition of optimal. In this thesis, this means that every possible outcome of financial capital at the end of the participants working life needs to be given a score that reflects the desirability of this outcome. In academic literature, this is done by using a utility function. This utility function should meet some intuitive criteria. It should:

1. Assign a higher utility value to a higher wealth level

2. Decrease the marginal gain in utility as the wealth level increases

A popular utility function that meets these criteria is the power utility function:

u(xT) = ( 1 1−γx 1−γ T γ ∈ (0, inf) \ 1 ln xT γ = 1

This function has the property of constant relative risk aversion (CRRA), Meaning that decision making based in this function is unaffected by the scale of (in this case) the accumulated total capital. This utility function is used in the next section to derive the optimal asset allocation for the financial market described in the previous section.

2.3 The classical optimal allocation solution

The investors financial capital develops according to the stochastic differential equation dXt= {Xt[αt(µ − r) + r]} + XtαtσdZt (2.3)

Using the financial market and utility function from the previous sections, an optimal allocation rule for the investors total capital at each point in time t can be derived using either dynamic programmingMerton(1969) or by using Martingales (e.g.Brennan and Xia (2002)). There are many academic papers that elaborate on the solution method (see for instance), so in this thesis attention is restricted to the final result. This result is the famous Merton ratio which shows that the optimal fraction ρt of total capital to

be invested in the risky asset at any time t is a constant proportion given by: ρt=

µ − r

σγ (2.4)

Some intuitive conclusions regarding the allocation to the risky asset can be drawn from this ratio. First, it increases when the expected excess return of the risky asset over the risk-free asset, µ − r, increases. Second, it decreases when the volatility of the risky asset, σ, increases. Finally, it also decreases as the investor is more risk averse.

Since human capital is assumed to decrease during the investment horizon, the frac-tion of total capital invested in the risk-free asset also declines. To keep a constant

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fraction of total wealth invested in the risky asset, more financial capital is gradually allocated to the risk-less asset. Figure 2.1 gives an example of how the allocation can evolve during the investment horizon.

Figure 2.1: Evolution of Human Capital and Financial Capital over the investment horizon

2.4 Shortfall of the classical solution / review of the

as-sumptions

In the context of a pension scheme, the applicability of the CRRA utility function can be seen as questionable for multiple reasons. First of all, it requires that the participant is able to accurately state his risk aversion parameter γ. Studies show that most partici-pant are not capable of doing this in a consistent mannerHodegaard and Vigna(2007). Second, participants often have a specific retirement target in mind that they want to reach at the end of the investment horizon. This retirement target can be expressed as either a (nominal or real) financial capital value, or a (nominal or real) level of lifelong retirement income (Arnott et al. (2013)). In the Netherlands for instance, the rule of thumb target for a good pension is a final capital value that is sufficient to ensure a life-long retirement income of 70% of the participants final wage or 75% of his average wage. When viewed this way, a possibly more applicable utility function would be one that specifically aims for his target. The asset allocation could then be optimally adjusted at each point in time to ensure that, in expectation, the final capital level is as close to the target as possible.

A classic asset allocation method that meets this adapted view of the participants utility is through mean variance optimization. Pioneered byMarkowitz (1952) in a one period setting, mean variance portfolio optimization leads to an optimal trade-off be-tween return and risk. In this setting, risk is measured as the variance of the total portfolio return. The resulting asset allocation ensures that given a specific level of ex-pected return, the variance of the portfolio returns is minimized. Alternatively, given a specific level of portfolio variance, the optimal asset allocation maximizes the expected portfolio return.

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allocation in a defined contribution pension scheme, the single period framework has to be extended to a multi-period framework. Also, the framework should take account of the participants present and future premium contributions.

The next chapter describes the setup of such a framework and the solution method required to solve it in an optimal manner.

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Chapter 3

Target based asset allocation

This section introduces the target based asset allocation strategy. Section 3.1. starts with introducing the investment problem. Sections 3.2 and 3.3 follow with describing the solution method and the solution to this problem. Section 3.4. digresses on which target should be chosen by the investor. The last section provides an overview and examination of the optimal solution

3.1 Problem formulation

When using a mean variance or target based approach for optimizing the allocation over the investment horizon, the problem formulation is different from the one in the Merton model as formulated in section 2.1. It can be stated as:

What is the optimal investment strategy at each point in time that will in expectation reach the participant’s retirement target, and minimizes the variance of capital outcomes around this target?

In the situation studied in this thesis, the participant is assumed to enter the pension plan at the start of his working life at age 25. He leaves the plan when he retires at age 65. At the moment of entering, he has not yet accumulated any financial capital. Financial capital here is restricted to the sum of already contributed pension premiums and the realized returns on those premiums. Any financial capital the participant may have outside of the pension scheme is not taken into the optimization procedure. Hu-man capital is restricted to the present value of the sum of expected future premium contributions to the scheme, where the level of the contributions is assumed to be deter-ministic. Any human capital the participant has outside of the pension plan, for instance the expected value of future labor income to be used for immediate consumption, is also not taken into the optimization procedure.

The reason to define financial capital and human capital this way is that, in practice, the amount of premium contributed by the participant and or his employer is a fixed and cannot be changed periodically at the will of the participant.

For convenience, the participant’s salary, denoted by Y , is assumed to stay constant over his working life. He contributes a continuous stream of premium payments to the pension scheme at a constant rate per year, denoted by c.

As in the Merton model, the participant must choose which fraction ρ of his financial capital is invested in the stock. His remaining financial capital is invested in the risk-free cash account.

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The evolution of the participants financial capital, denoted by X, during the invest-ment horizon is then given by the stochastic differential equation:

dXt= {Xt[ρt(µ − r) + r] + c} + XtρtσdZt

X0= 0

(3.1) The participant wishes to accumulate a financial capital of F at the time of retire-ment. For the optimization procedure (described in the next section) to be efficient, the target has to be determined when the participant enters the pension scheme, and remain unchanged during the entire investment horizon.

The utility function used for optimization is the quadratic loss function. The quadratic loss function yields a minimal cost penalty of 0 when the financial capital at the moment of retirement is exactly equal to the target F . The cost penalty increases as the final financial capital lies farther from the target.

The goal function for the allocation optimization can then be denoted by: minimize

ρ0,...,ρT

E(XT − F )2 _(3.2)

By formulating the goal function this way, it equally penalizes both final financial capital values lower than the target and values higher than the target. Intuitively this not a desirable property since most participants will always prefer a higher financial capital over a lower one.

The reason for penalizing values this way is that to reach - in expectation - a fi-nancial capital level higher than the target, the participant would have to make riskier investments than he would if he aimed for the target. Making riskier investments auto-matically increases the possibility of falling short of the target, which is perceived to be undesirable.

3.2 Stochastic dynamic programming

By combining equations (3.1) and (3.2), a stochastic multi period optimization problem is formulated: minimize ρt:t∈[0,T ] E[(XT − F )2] subject to dXt= {Xt[ρt(µ − r) + r] + c} + XtρtσdZt X0= 0. (3.3)

Stochastic multi-period optimization problems generally have to be solved using nu-merical techniques such as value iteration or grid search. These techniques are often very computationally intensive and not always converge to an optimal solution. How-ever, since the problem in (3.4) has been (intentionally) formulated based on stylized assumptions regarding asset returns and participant premium contribution, it can solved analytically. This is done using a well known mathematical technique called (stochastic) dynamic programming. It is beyond the scope of this thesis to give a complete math-ematical introduction into stochastic dynamic programming, so attention is restricted to the basic underlying idea. An rigorous mathematical introduction can be found for instance in Bertsekas(2005)

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The basic idea of dynamic programming is to break the difficult multi period op-timization problem down into many simpler problems. If returns over the entire in-vestment period are known, an optimal asset allocation for the entire period can be determined by making use of a backwards recursion. Starting at the end of the invest-ment horizon and working backwards one step at a time, the optimal asset allocation at time t can be determined by using the optimal at time t + 1, which has been determined in the previous step. Working backwards this way all the way to time t = 0 gives the optimal asset allocation for the entire investment horizon. The set of optimal investment fractions at times 0, ..., 39 is called the optimal policy.

When an analytical solution (which will be derived in the next section) exists, a much nicer result can be obtained. Instead of an optimal allocation policy given one specific set of asset returns, a general allocation rule can be derived. In the present case of a multi-period asset allocation problem, this rule is a formula that, given the remaining time horizon and accumulated capital, yields the optimal asset allocation for the next time step. This rule can be applied at any point during the investment horizon (0, T ) and for any real financial capital value.

When restrictions are imposed on the policy, like a limit on short-selling, an ana-lytical solution can no longer be derived. The same holds true for a more complicated financial market where stock market returns do not follow a Gaussian distribution but, for instance, a jump-diffusion process. Derivation and implementation of the required numerical techniques and procedures is beyond the scope of this thesis. A numerical framework that allows for restrictions on leveraging and bankruptcy and a more com-plicated financial market market can be found in Dang and Forsyth(2016)

The target based allocation problem studied in thesis is restricted to the stylized situation as described in previous section, using a predetermined nominal final wealth target

3.3 Solution to the problem

This section gives an (abbreviated) solution to the stochastic dynamic programming problem described in the previous section. Most of the formulas and derivations pre-sented in this section can also be found in Gerrard et al.(2004),Hodegaard and Vigna (2007) and Vigna (2011).

The goal is to find the asset allocation policy at time t given accumulated capital xt for which, given the capital target, the variance is minimal. To find this policy, first

Ito’s lemma is applied to derive the infinitesimal operator of the stochastic differential equation f (t, x) that describes the development process of financial capital throughout the investment horizon. For the sake of clarity, f (t, x) is formally stated as:

f (t, x) = dXt= {Xt[ρt(µ − r) + r] + c} + XtρtσdZt (3.4)

Applying Ito’s lemma to (3.5) gives infinitesimal operator: Aυf (t, x) := ∂f ∂t + {x[υ(µ − r) + r] + c} ∂f ∂x + 1 2x 2_υ2_σ2∂2f ∂x2 (3.5)

The optimal asset allocation policy can now be determined by apply the Hamilton-Jacobi-Bellman (HJB) equation. This means combining the infinitesimal operator and the cost function (3.2). The HJB equation can then be written as:

inf

υ∈R[A υ_(X

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filling in Aυ from (3.5) yields: inf α∈Rφ(α, t, x) = 0 (3.7) where φ(α, t, x) = [∂H ∂t + {x[α(µ − r) + r] + c} ∂H ∂x + 1 2α 2_x2_σ2∂2H ∂x2] (3.8)

With boundary condition at final time T the cost function (3.2):

H(T, x) = (XT − F )2 (3.9)

Under optimal control, denoted by α∗, equation (3.9) can be reformulated as: inf

α∈Rφ(α, t, x) = 0 ⇒ φ(α ∗

, t, x) = 0 (3.10)

Now, the optimal asset allocation policy is the policy for which φ(α∗, t, x) has a minimum. Assuming that φ is a convex function of α∗, this minimum is reached when two conditions are satisfied simultaneously. The first order derivative of φ with respect to α should be 0 (indicating an extreme value) and the second order derivative of φ with respect to α should be > 0 (indicating that the extreme value is a minimum). In mathematical notation:

φ0_α(α∗, t, x) = 0

φ00_αα(α∗, t, x) > 0 (3.11) The first condition in (3.11) yields:

x(µ − r)∂H ∂x + x 2_α∗ σ2∂ 2_H ∂x2 = 0 (3.12) rewriting gives: α∗= r − µ xσ2 H_x0 H_xx00 (3.13)

where H_x0 and H_xx00 are the first and second order derivatives of H with respect to x. Substituting (3.13) into (3.8) yields:

0 = ∂H ∂t + (rx + c) ∂H ∂x − 1 2 r − µ σ 2 (H_x0)2 H_xx00 (3.14)

It turns out that this solution also meets the second order condition of (3.11), indi-cating that the extreme value for the cost function is indeed a minimum. For proof of this second condition, the interested reader is referred to Gerrard et al.(2004).

Solving for the partial derivatives of H to t and x and plugging them back into (3.10) yields the optimal asset allocation α∗(t, x) for any t , x ∈ R. The solution method for solving the system of partial differential equations required to solve equation (3.14) is very cumbersome and does not lie within the scope of this thesis. For a complete deriva-tion of the partial derivatives,the interested reader is once again referred to Gerrard et al.(2004).

Skipping the solution method, the resulting optimal asset allocation can be derived by plugging the partial derivatives H_x0 and H_xx00 in equation (3.14), which can than be rewritten as: α∗(t, x) = µ − r σ2_x [F e −r(T −t)_{− x −} c r(1 − e −r(T −t)_)] _(3.15)

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3.4 Setting the target

In the optimal asset allocation rule (3.15), the target F serves as an upper bound for the distribution of the capital at final time T . The mechanics of the allocation rule (3.15) yields the optimal allocation to reach a final capital xT very close to F , but never

actually reaches F . While this may seem as an undesirable property at first, it actually ensures that no excess risk is taken during the investment horizon.

However, since the participant actually does want to reach his desired capital and -in practice - would not m-ind overshoot-ing his target by a small marg-in, he should aim for a target F that is higher than his desired capital in order to reach his desired capital. In Forsyth et al. (2017), an intuitive interpretation of this mechanic of the allocation rule is given by comparing the investor with a basketball player. In order for his shot to end up in the basket, he must aim a slightly higher than the basket. A more technical explanation based on the result of a mean-variance analysis can be found in Hodegaard and Vigna (2007). The latter also derives the formal relationship between the desired final capital level and the target F the investor should aim for during the investment horizon. This relationship is given by:

F = F∗+ F

∗

e(µ−rσ ) 2

T (3.16)

where F∗ denotes the desired final capital level.

3.5 Examination of the optimal solution

By reordering terms, equation (3.15) can be reformulated as: α∗(t, x) = [F e−r(T −t)− x − c r(1 − e −r(T −t) )] ·λ − r σ · 1 σx (3.17)

Formulating the solution this way makes it easier to break down its individual com-ponents and analyze their effect on the asset allocation. The effects of the individual components are summarized in table 3.1.

Table 3.1: Interpretation of optimal allocation formula

Component Interpretation Effect of increase on stock allocation F e−r(T −t) Present value of the target up

x Accumulated financial capital at time t down

c r(1 − e

−r(T −t)₎ _{Present value of future contributions} _down λ−r

σ Sharpe ratio of the stock up

1

σx ”Rest term” up

The first part of this formula can be interpreted as the current deficit of the partic-ipant’s total capital - consisting of accumulated capital and the present value of capital to be accumulated through future contributions - with respect to the present value of the retirement target. The change in allocation of the accumulated capital towards the risky asset moves in the same direction as the change of the deficit. The second part, the Sharpe ratio, acts as a multiplier that tilts the allocation more towards the risky asset when its return characteristics grow more favorable with respect to the risk-less asset. The final term (the rest term) lowers the risky asset allocation when it’s risk (measured by standard deviation) increases, and also when the accumulated financial capital increases.

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The optimal solution presented in (3.15) and (3.16) shows some clear similarities with the optimal ”Merton ratio” derived in chapter 2. It is important to note that the solution to the target based approach given in (3.15) yields the optimal fraction of financial capital invested in the stock market given any allowable pair (t, x), while the Merton ratio yields the fraction of total capital to be invested in the stock market. To compare both ratios, the target based allocation (3.15) is reformulated as a fraction of total capital. This fraction is then given by:

α∗_tb(t, x) = µ−r σ2_x{F e −r(T −t)_{− x −} c r(1 − e −r(T −t)_}x x +c_r(1 − e−r(T −t)₎ = µ − r σ F e−r(T −t)− x −c_r(1 − e−r(T −t) x +c_r(1 − e−r(T −t)) (3.18)

Recall the Merton ratio (with slightly altered use of symbols for the sake of compar-ison):

αM erton(t, x) =

µ − r

σγ (3.19)

Combining (3.17) and (3.18), it can be seen that both strategies are equal when the participant’s risk aversion is given by:

γ = x +

c r(1 − e

−r(T −t)₎

F e−r(T −t)− x − c_r(1 − e−r(T −t) (3.20) Equation (3.19) shows that, in the target based approached, the participant’s risk aversion is not constant. It varies with accumulated capital x and with the remaining investment horizon T − t (where T is fixed).

To give some intuition, the value of γ at time t = 20 is plotted against the present value of the financial capital. The financial capital deficit is equal to the target F minus the present value of future premium contributions.

Figure 3.1: Asset Allocation

The red line indicates the optimal financial capital level; the level at which the deficit is exactly equal to 0. The figure can be explained as follows. For financial capital values much lower than the optimal level, the participant is willing to take a lot of risk, since this extra risk increases the probability of reaching the target. For financial capital levels closer to the red line (approached from the left side), taking risk would only decrease the probability of (exactly) reaching the target. Therefore, the risk aversion increases to very high levels.

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When the financial capital is already higher than the optimal capital, the participant becomes ”risk loving”: he is willing to pay capital to take risk. The reason for this behavior is that taking risk comes with the probability to lose capital. Since losing capital would increase the probability of exactly reaching the target which - in this model - increases utility. This latter behavior is very unrealistic from a practical point of view. However, the scenario’s in which this behavior occurs are purely hypothetical. The same reasoning as in the previous section applies here: the red line - which functions as an intermediate target for time t = 20 - can never be reached in practice, since the allocation rule would shift financial capital away from the risky asset before this level is reached.

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Comparison of allocation

strategies

This section provides a numerical comparison of the target-based asset allocation strat-egy with other strategies. Other strategies include the Merton stratstrat-egy introduced in the second chapter and some allocation strategies that are commonly applied in defined contribution pension schemes in practice. The strategies and simulation setup are first described in sections 4.1 and 4.2, followed by a detailed analysis of the results in sections 4.3 through 4.6. Section 4.7 provides an overview of the overall results, and section 4.8 concludes with a welfare analysis.

4.1 Description of the strategies

The optimal target based strategy will be compared to a number of other asset allocation strategies observed in theory and practice. The strategies reviewed are:

1. target based strategy

2. constant proportion of financial capital in stock

3. constant proportion of total capital in stock (Merton strategy) 4. deterministic glide path(s)

4.1.1 Target based strategy

The specifics and underlying motivation of the target based strategy have been specified in the previous sections. For practical reasons, the strategy used in this thesis does not allow for either leveraging or short selling the stock market. This means the fraction of financial capital invested in the stock market at any point in time lies somewhere be-tween 0% and 100%. The strategy as applied here therefore differs from the ”pure target based strategy”, which would short the risk free asset early in the investment horizon in order to leverage the stock market investment to a fraction larger than 100%. However Hodegaard and Vigna (2007) have shown that this restriction has only marginal effect on the distribution of final wealth levels. This restriction is imposed since individual investors in practice are generally not allowed to leverage their investment. Further, rebalancing frequency is restricted to once per year. This is another deviation from the ”pure target based strategy” introduced in the previous chapter. This restriction is imposed for practical reasons.

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4.1.2 Constant proportion of financial capital in stock

A popular asset allocation approach for both investors in general and investors saving for retirement is to keep a constant proportion of the total invested (financial) capital invested in the stock market and the remaining fraction invested in the risk free as-set. During the investment horizon, returns in both asset classes distort the fraction. Therefore, annual re-balancing is applied to maintain pull the stock fraction back to the desired constant level.

4.1.3 Constant proportion of total capital in stock (Merton strategy)

The ”Merton strategy” has already been introduced in chapter 2. It states that, in order to maximize the investors utility, the constant fraction invested in the stock market should be a fraction of total capital rather than financial capital only. As with the target based strategy, the fraction of financial capital invested in the stock market is capped to values between 0% and 100%, and rebalancing takes place annually.

4.1.4 Deterministic glide path

The deterministic glide path is equivalent to the target date funds described in the first section of this thesis. It is a very common asset allocation strategy for defined contribution schemes in many countries. In this strategy, the investor invests a constant fraction of his financial capital in the stock market during the first part of the investment horizon. 15 years from the end of the investment horizon, he starts to linearly reduce his stock market participation to 0% at the time of retirement.

4.2 Simulation setup

The characteristics of the financial market, the participant and the defined contribution pension scheme that will be used throughout the analysis are given in Table 4.1.. The parameters for expected stock return and volatility and for the risk free interest rate are taken in line with the findings of the Dutch Commsie Parameters(2014)

Table 4.1: Stylized assumptions

Salary 30.000 Expected stock return 8.5% Contribution rate 10% Volatility of stock return 20 % Desired financial capital 350.00 Risk-free interest rate 2.4% Target capital (resulting) 358.00

Stock returns for the 40 year investment period are simulated for 5000 scenarios. Where in order, re-balancing takes place on an annual basis at times 1, 2, ...39.

In order to compare the outcomes for each of the strategies described above, the variables for each strategy are chosen such that the expected final capital level is the same for each of the strategies. For the alternative strategies, this means a parameters choice of:

1. Model 2: Constant proportion of financial capital in stock: α = 35% 2. Model 3: Coeffiecient of risk aversion in stock (Merton strategy): γ = 1.36 3. Model 4a Deterministic glide path(s): starting proportion of financial capital in

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4.3 Model 1: Target based strategy

Since stock returns vary, the target based allocation method yields a different optimal stock allocation over the investment horizon for different scenario’s. Taking the average allocation of each year of the investment horizon, an ”average asset allocation” can be constructed. With this average asset allocation, (see figure 4.1.). all of the available financial capital is invested in the stock market for the first years of the investment horizon. After 8 years, financial capital is gradually shifted to the risk-free cash account during the remaining years of the investment horizon. At the time of retirement, the remaining portion of financial capital invested in the stock market is - on average - 10%. As mentioned, the stock allocation in individual scenario’s differs from the average. To illustrate this fact, the allocation for one single scenario has been added to Figure 4.1. It can be seen that, while the average stock asset allocation develops in a strictly non-increasing manner, the stock allocation in individual scenario’s may go up and down multiple times as result of the realized stock returns. A high stock market return in a given year could lead to a decrease in allocation much faster than in the average, since the target can (in expectation) be reached whilst taking less risk. If in a subsequent year heavy losses are incurred, financial capital might be shifted back into the stock market to increase the probability of making up for the incurred losses and reaching the target at the end of the horizon.

Figure 4.1: Model 1:Asset Allocation

The simulation gives a resulting level of financial capital (which by then is equal to total capital) for each scenario at the time of retirement. The probability density for all occurred levels of capital is plotted in Figure 4.2. As expected, the major part of the observations is located very close to the target, indicating that the target based allocation strategy works quite well. In about 85% of the scenario’s, the final capital lies with a margin +/- 10% of the target. There are a few observations that overshoot the target by a small margin, and almost none that overshoot the target in a major fashion. In about 3% of the scenarios however, the target based strategy yields extremely poor results. In these scenario’s the final capital of the participant is even much lower than the sum of the contributed premiums. These scenario’s have in common that multiple highly negative stock market returns occur in years 10-15 of the investment period. At this point, a sizable amount of financial capital has already been accumulated and on average this is the point where the stock market risk is reduced at the highest pace in order to reduce the probability of major losses at a later stage. The negative returns in years 10-15 in combination with a relatively long remaining investment horizon lead to an asset allocation with heavy stock market exposure for a longer amount of time. A few - consecutive or spread out - years of negative stock returns later on are then

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sufficient to yield very poor levels of final capital. As mentioned however, this occurs in only a very small number of scenario’s.

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4.4 Model 2: Constant financial capital proportion

The first alternative strategy to be compared to the target based strategy is the simplest strategy of keeping a constant proportion of financial capital invested in the stock market throughout the entire investment horizon. The fraction is enforced on an annual level by rebalancing, which is necessary in the light of new premium contributions and realized returns. As stated earlier, the constant fraction of financial capital invested in the stock market is chosen such that the expected value of the final capital is equal in all strategies. For the constant proportion strategy, this gives a constant stock exposure of 35% of financial capital during the entire investment horizon (see Figure 4.3). In contradiction to the target-based strategy, the distribution of final capital is skewed to the right (see Figure 4.4). As a result of the absence of a steering mechanism - as in the target based model - the final capital levels are distribute over a much larger spectrum of outcomes. Compared to the target based strategy, there is a significant probability of overshooting the target by up to 50%. This naturally means that there also is a significantly increased probability of significant shortfall with respect tot the target.

Figure 4.3: Model 2: Asset Allocation

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4.5 Model 3: Constant total capital proportion

Here, the (adjusted) ”Merton strategy” is applied. In order to arrive at the same ex-pected wealth at the end of the investment horizon as in the previous strategies, the participant’s risk aversion parameter is set equal to 1.36, which - according to general understanding - means that the participant has a relatively strong risk aversion. This level of risk aversion is chosen to increase the comparability of outcomes of the different models. The resulting average asset allocation shows similarities with the average target based allocation (see Figure 4.5). The shift from investments in the stock market to the risk-free asset starts a little earlier in the Merton strategy than in the target based strategy. The pace of the average shift is also faster than for the target based strategy in the first year, but slower in the later years. At the moment of retirement, the average stock market exposure is still almost 20%, compared to a little over 10% for the target based strategy.

Although the average asset allocation is very similar to the target based strategy, the distribution of final capital outcomes is not. As in the constant allocation strategy dis-cussed above, the distribution of final capital is distributed over a much larger spectrum of outcomes than the target based strategy. In fact, the distribution (see Figure 4.6) is almost identical to the distribution of the constant asset allocation. A slight difference can be observed at the endpoints of the distribution, where the probability mass of the Merton strategy is lower than in the constant allocation strategy.

Figure 4.5: Model 3: Asset Allocation

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4.6 Model 4: Deterministic glide path

Finally, the target based strategy is compared to the deterministic glide path strategy which is common in real world defined contribution pension schemes. In this strategy, a constant proportion of the financial capital is invested in the stock market during the first 25 years of the investment period. The asset allocation is then linearly shifted to towards the risk-free asset at such a rate that the stock market exposure is reduced to 0 at the exact moment of retirement. Within this construction, various different glide path strategies can be constructed by varying the stock exposure during the first 25 years of the investment period. Two glide paths are considered in this section. The first glide path considered is the one that, in expectation, achieves the same level of final capital as the previous strategies. This is achieved by setting the stock exposure during the first 25 years of the investment horizon equal to 50% (see Figure 4.7).

Glide paths in practice have a much higher stock market exposure during the first part of the investment horizon. A second and more realistic glide path, starting with a stock market exposure of 85%, is therefore is therefore added to the comparison (see figure 4.9). The final capital distributions of both glide path strategies are shown in figures 4.8 and 4.10. The shape of the distributions is similar to that of models 2 and 3. The second glide path strategy, with the higher stock market exposure, naturally has a much larger spectrum of outcomes than the strategy with less market exposure.

Figure 4.7: Model 4a: Asset Allocation

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Figure 4.9: Model 4b: Asset Allocation

Figure 4.10: Model 4b: Wealth Distribution

4.7 Overview

The results of the different strategy outcomes is summarized in Table 4.2. It summarizes the distribution of final capital levels as a percentage of the desired capital level. Table 4.2. again shows the accuracy with which the target based strategy manages to ’steer’ the capital level towards the target. In over 70% of the simulations, the final capital level lies within three percentage-points of the target.

Table 4.2: Distribution of final capital outcomes for the different models (as % of desired capital) p5 Q1 p50 Q3 p95 mean Model 1 63% 99% 102% 102% 102% 97% Model 2 57% 76% 92% 113% 151% 97% Model 3 60% 78% 93% 112% 143% 97% Model 4a 56% 74% 91% 113% 156% 97% Model 4b 53% 83% 120% 178% 323% 146%

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4.8 Welfare analysis

As pointed out in the previous sections, the target based allocation strategy manages to produce final capital levels (very) close to the target in most of the scenario’s. In order to measure the desirability of the capital outcomes for the discussed strategies, a welfare analysis based on the simulations of the previous sections is presented here.

So far, two different (dis)utility functions have been introduced and applied to find an optimal asset allocation. The Merton strategy uses the power utility function and the target based strategy uses the quadratic (dis)utility function. In the target based context, both strategies have a clear disadvantage when used to value the resulting cap-ital levels for the different strategies.

The power utility function equally penalizes outcomes higher than the target and outcomes lower than the target. In reality, participants will probably not penalize out-comes higher than the target at all. As discussed in the previous chapter , this is not a significant problem in the target based strategy since the asset allocation policy ensures that there is only a very small probability of achieving a capital level higher than the target. However, when valuing the outcomes of other strategies that do have a signif-icant probability of overshooting the target, the quadratic utility function is unfit to compare the results of the different strategies.

The power utility function does not have this particular problem, since it does not aim for a specific target. However, since it does not aim for a target it also does not adequately penalize final capital values below the target in the (target-based) context of this thesis.

Therefore, an adapted disutility function is applied to compare the welfare outcomes of the different strategies. That is, the quadratic disutility function is applied to penalize capital levels lower than the target, but a disutility value of 0 is applied when the capital level is higher than the target. In mathematical notation:

du(x(T )) = ([F − X(T )]+)2 (4.1)

The cumulative distribution of resulting disutility values for each strategy are plotted in figure 4.11. Since lower disutility values symbolize a higher welfare score, the best scoring strategy is one with a high probability mass at low disutlility values. In figure 4.11, this is translated to the line starting at a higher probability level for a disutility value of 0. The target-based allocation scores best, with almost 80% of the scenario’s resulting in 0 disutility.

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The mean disutility values for each strategy are presented in table 4.3. It can be seen that the mean disutility of the target based allocation strategy is around a factor 2 lower than the other mean disutility of the other strategies.

Table 4.3: My caption Model Mean disutility (x 109 )

1 2.5

2 5.0

3 4.3

4a 5.4

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(a) Target based (b) Constant proportion FC

(c) Merton rule (d) Glidepath 1

(e) Glidepath 2

Figure 4.11: The empirical cumulative distribution function of the disutility of final capital levels for the different strategies

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Chapter 5

Conclusion

The main question in this thesis has been whether a target based asset allocation proach could be a viable option for defined contribution pension schemes. In this ap-proach, financial capital is dynamically allocated in order to reach a predetermined financial capital target level at the moment of retirement. This is achieved by using quadratic utility function which penalizes both capital levels higher and lower than the target. An analytical solution for this problem in a Black-Scholes financial market has been presented based on previous academic work (e.g. Hodegaard and Vigna(2007)).

Through simulations, the target based allocation policy has been compared to both the classical allocation solution derived by Merton and to deterministic strategies used in practice in defined contribution pension schemes.

In the stylized setting in which the simulation have been carried out, the target based asset allocation managed to reach a final capital level within a 3% margin with respect to the target in over 70% of the simulations. This is achieved by reducing unnecessary risks through strictly limiting the upside potential of capital outcomes.

In this setting, the target based approach can be seen as a major improvement compared to current practices, since most of the significant downside risk is eliminated. However, the assumptions required to obtain the analytical solution used in the simulations do not represented the real preferences and characteristics of the participant in an adequate manner. Also, the assumption of a Black-Scholes financial market does not hold in practice. A more realistic model could contain:

• A stochastic term structure of interest rates • A more realistic model for stock returns

• A model for stochastic salary of the participant • A model for inflation

The most important shortcoming of this model is the nature of the target itself. Instead of setting a predetermined financial capital, a realistic framework should allow the participant to set a predetermined income goal. A framework that captures all of these shortcomings could be the subject of future research.

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Model 1: Target based model

# Define financial market parameters rf <- 0.024 # risk free interest rate

mu <- 0.085 # expected stock return (arithmetic average) sigma <- 0.2 # standard deviation of stock returns

# Define Participant and plan parameters Y <- 30000 # Yearly income

c_perc <- 0.1 # contribution to pension plan (% of income) c <- Y * c_perc # Yearly premium contribution

Tar <- 350000

adj <- exp(((mu-rf)/sigma)^2 * nYears) / (2 * Tar) Tar_adj <- Tar + 1 / (2 * adj)

age_start <- 25 age_ret <- 65

nYears <- age_ret - age_start # investment horizon nScen <- 5000 # number of scenarios # Create matrices for data storage

X_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1) alpha_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1)

# Generate stock returns

set.seed(1309) # for comparable results

dZ <- matrix(rnorm(nYears * nScen), nrow = nScen, ncol = nYears) dS <- mu + sigma * dZ

# Simulation

for(i in 1:(nYears+1)){

if(i == 1){X_mat[, i] <- 1} # start with 1 instead of 0 to avoid error else{

X_mat[, i] <- (X_mat[, (i-1)] + c)* (1 + alpha_mat[, (i-1)] * dS[, (i-1)] + (1 - alpha_mat[, (i-1)]) * rf)

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}

alpha_mat[, i] <- pmax(0, pmin(1,

- (mu - rf)/(sigma^2 * X_mat[, i]) * (X_mat[, i] + c/rf * (1 - exp(-rf * (nYears-i+1))) - Tar_adj

* exp(-rf * (nYears-i+1))))) }

# RESULTS

# mean and median final wealth level

mean(X_mat[, (nYears+1)]) # mean = 338724.4 median(X_mat[, (nYears+1)]) # median = 356101.5 # percentiles of final wealth level

round(quantile(X_mat[, (nYears+1)], probs = c(0.05, 0.25, 0.5, 0.75, 0.95)))

# 5% 25% 50% 75% 95%

# 220019 347948 356102 357764 358237 scenex <- 13

# plot of average asset allocation and scenario 13

plot(colMeans(alpha_mat), type = "l", main = "Asset allocation", ylim = c(0,1), ylab = "alpha", xlab = "year")

lines(alpha_mat[scenex, ], lty = 2, col = "blue") # plot of final wealth distribution and scenario 13 plot(density(X_mat[, (nYears+1)]), type = "l",

main = "Distribution of final capital level", ylab = "Probability", xlab = "Final wealth")

points(x = X_mat[scenex, (nYears+1)], y = 0, pch = 4, cex = 1.2, col = "blue")

Model 2: Constant proportion of financial capital

Tar <- 350000

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X_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1) alpha_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1)

# Simulation

for(i in 1:(nYears+1)){

}

alpha_mat[, i] <- 0.35 }

# RESULTS

mean(X_mat[, (nYears+1)]) # mean = 339093.8 median(X_mat[, (nYears+1)]) # median = 322504 # percentiles of final wealth level

round(quantile(X_mat[, (nYears+1)],

probs = c(0.05, 0.25, 0.5, 0.75, 0.95)))

# 5% 25% 50% 75% 95%

# 198777 265151 322504 396347 530457 scenex <- 13

# plot of average asset allocation

plot(colMeans(alpha_mat), type = "l", ylim = c(0,1), main = "Asset allocation", ylab = "alpha", xlab = "year") #lines(alpha_mat[scenex, ], lty = 2, col = "blue")

# plot of final wealth distribution

plot(density(X_mat[, (nYears+1)]), type = "l", main = "Distribution of final capital level", ylab = "Probability", xlab = "Final wealth") points(x = X_mat[scenex, (nYears+1)],

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y = 0, pch = 4, cex = 1.2, col = "blue")

Model 3: Constant proportion of total capital

Tar <- 350000

X_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1) HC_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1) alpha_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1)

# Simulation

rho <- 0.225 # constant proportion of total capital invested in stock for(i in 1:(nYears+1)){

}

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alpha_mat[, i] <- pmax(0,

pmin(1, (rho * (HC_mat[, i] + X_mat[, i])) / X_mat[, i])) }

# RESULTS

# 5% 25% 50% 75% 95%

# 210217 273053 327096 392692 500998 scenex <- 13

# plot of average asset allocation and scenario 13

lines(alpha_mat[scenex, ], lty = 2, col = "blue") # plot of final wealth distribution and scenario 13 plot(density(X_mat[, (nYears+1)]), type = "l", main = "Distribution of final capital level",

ylab = "Probability", xlab = "Final wealth") points(x = X_mat[scenex, (nYears+1)], y = 0, pch = 4, cex = 1.2, col = "blue")

Model 4: Deterministic glide paths

# constant proportion of financial capital invested in stock until 15 years before retirement. # from 15 years before retirement, linearly decrease stock investment until 0 at retirement setwd("C:\\Users\\joostenj\\OneDrive - Korn Ferry\\MSc thesis\\R")

Tar <- 350000

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nYears <- age_ret - age_start # investment horizon nScen <- 5000 # number of scenarios LC_start <- 0.493

LC <- c(rep(LC_start, 25), c(LC_start - (1:15)/15*LC_start)) LC2 <- c(rep(0.85, 25), c(0.85 - (1:15)/15*0.85))

# Create matrices for data storage

X_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1) X_mat2 <- matrix(NA, nrow = nScen, ncol = nYears + 1) HC_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1) alpha_mat <- matrix(NA, nrow = nScen, ncol = nYears + 1) alpha_mat2 <- matrix(NA, nrow = nScen, ncol = nYears + 1) # Generate stock returns

# Simulation

for(i in 1:(nYears+1)){ if(i == 1){

X_mat[, i] <- 1 # start with 1 instead of 0 to avoid error X_mat2[, i] <- 1} # start with 1 instead of 0 to avoid error else{

X_mat2[, i] <- (X_mat2[, (i-1)] + c)* (1 + alpha_mat2[, (i-1)] * dS[, (i-1)] + (1 - alpha_mat2[, (i-1)]) * rf) } alpha_mat[, i] <- LC[i] alpha_mat2[, i] <- LC2[i] } # RESULTS

# results a) deterministic strategy with same expected value as other strategies # mean and median final wealth level

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# 197352 260032 317564 395370 545939 scenex <- 13

# plot of average asset allocation

plot(density(X_mat[, (nYears+1)]), type = "l", main = "Distribution of final capital level", ylab = "Probability", xlab = "Final wealth")

points(x = X_mat[scenex, (nYears+1)], y = 0, pch = 4, cex = 1.2, col = "blue") # results b) deterministic strategy as common in practice

mean(X_mat2[, (nYears+1)]) # mean = 512309.3 median(X_mat2[, (nYears+1)]) # median = 418726.6 # percentiles of final wealth level

round(quantile(X_mat2[, (nYears+1)], probs = c(0.05, 0.25, 0.5, 0.75, 0.95)))

# 5% 25% 50% 75% 95%

# 181796 291964 418727 623311 1132211 # plot of average asset allocation

plot(colMeans(alpha_mat2), type = "l", ylim = c(0,1), main = "Asset allocation", ylab = "alpha", xlab = "year")

plot(density(X_mat2[, (nYears+1)]), type = "l", main = "Distribution of final capital level", ylab = "Probability", xlab = "Final wealth")

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Bibliography

Arnott, R.D., K.F. Sherrerd, and L. Wu (2013), “The glidepath illusion... and potential solutions.” The Journal of Retirement.

Bertsekas, D. P. (2005), Dynamic Programming and Optimal Control, third edition. Athena Scientific, Nashua.

Brennan, M.J. and Y Xia (2002), “Dynamic asset allocation under inflation.” The Jour-nal of Finance.

Commsie Parameters (2014), “Advies commissie parameters.”

Dang, D.M. and P.A. Forsyth (2016), “Better than pre-commitment mean-variance portfolio strategies: A semi-self-financing hamilton-jacobi-bellman equa-tion approach.” European Journal of Operaequa-tional Research. Available at SSRN: https://ssrn.com/abstract=2243204.

Forsyth, P.A., K.R. Vetzal, and G. Westmacott (2017), “Target wealth: The evolution of target date funds.”

Gerrard, R., Haberman S., and E. Vigna (2004), “Optimal investment choices post-retirement in a defined contribution pension scheme.” Insurance: Mathematics and Economics.

Hodegaard, B. and E. Vigna (2007), “Mean-variance portfolio selection and efficient frontier for defined contribution pension schemes.” Department of Mathematical Sci-ences.

Markowitz, H. (1952), “Portfolio selection.” The Journal of Finance.

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Vigna, E. (2011), “On efficiency of mean-variance based portfolio selection in dc pension schemes.”

Target based asset allocation in a defined contribution pension scheme

Target based asset allocation in a

defined contribution pension scheme

Jelle Joosten

Statement of Originality

Contents

Introduction

1.1

Background

1.2

Target date investing

1.3

Research question

1.4

Structure

Chapter 2

The classical approach for

optimal asset allocation

2.1

The stylized financial market

2.2

The power utility function

2.3

The classical optimal allocation solution

2.4

Shortfall of the classical solution / review of the

as-sumptions

Chapter 3

Target based asset allocation

3.1

Problem formulation

3.2

Stochastic dynamic programming

3.3

Solution to the problem

3.4

Setting the target

3.5

Examination of the optimal solution

Comparison of allocation

strategies

4.1

Description of the strategies

4.2

Simulation setup

4.3

Model 1: Target based strategy

4.4

Model 2: Constant financial capital proportion

4.5

Model 3: Constant total capital proportion

4.6

Model 4: Deterministic glide path

4.7

Overview

4.8

Welfare analysis

Chapter 5

Conclusion

Model 1: Target based model

Model 2: Constant proportion of financial capital

Model 3: Constant proportion of total capital

Model 4: Deterministic glide paths

Bibliography