Target-driven investing in deﬁned contribution plans: a comparative analysis

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Target-driven investing in defined contribution

plans: a comparative analysis

Author: M. Canter-Visscher Supervisors: prof. dr. T.K. Dijkstra drs. P. Platteel dr. C. Praagman Abstract:

We assess pension ratio driven static and dynamic investment strategies for defined contribution plans. Plan members set both interim targets and final targets. We distinguish between classical expected utility optimisers and plan members who adhere to

the maxims of behavioural finance, cumulative prospect theory in particular. These optimisations include an analysis of the impact of parameter uncertainty. Based on the model that ignores parameter uncertainty we find that optimal investment strategies are heavily tilted towards the asset class of equity. Moreover, pension ratio driven static and dynamic strategies generally imply a sizeable lower pension ratio in comparison to these

optimal investment strategies. The model which incorporates parameter uncertainty presents optimal investment strategies that are less tilted towards the asset class of equity.

We assess pension ratios by both coherent and incoherent risk measures, and it is found that deviation from coherence does not signify large diﬀerences in assessments of risk. We establish that parameter uncertainty impacts pension ratios by both a decreased downside

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Introduction

Originated by the global financial crisis, defined benefit (DB) pension providers were mostly unable to payout promised payments to their plan members. Poor performance endangered most of these pension providers, and sparked a debate on the tenability of DB pension funds. A defined contribution (DC) pension plan, which inherently transfers all financial risk from the pension provider to the plan member, is as a consequence becoming an increasingly popular alternative to a DB pension plan: with all risk transferred, a prospect of a sustainable pension system is in sight.

Yet, this development disfavours employees saving for retirementa. Under a DC regime income at retirement becomes uncertain, as plan members possess all risk, and plan providers have no legal or constructive obligation to pay additional contributions if the pension fund is unable to payout all plan members (cf. International Accounting Standards 19.8). Plan members hence lose the guarantee of a fixed level of income post-retirement due to the switch in pension schemes. DC plan providers therefore refrain from any form of guarantee, as it implies taking on risks which they are not required (and de facto are prohibited) to hold.

Instead of implementing DC plans which try to mimic comparable DB plans by means of well designed allocation strategies, DC plan providers oﬀer allocation profiles with a diﬀerent focus. To illustrate, the default option in pension plans resembles a strategy that is tilted to risky assets during the early working career of plans members, and as members age the allocation tilts to assets which are least risky (e.g, sovereign bonds). Depending on the risk appetite plan members may deviate from this default option, although the shift will merely be upwards or downwards (and thus will not alter the shape or the objective of the allocation strategy).

The rationale behind this strategy is that it reduces uncertainty of prospective retirement in-comeb_{, which is not necessarily in line with mimicking a DB plan. In particular, the default}

options do not take into account information about pension ratiosc or any progression thereof. Ideally, therefore, DC plan providers should try to focus their attention on obtaining a pension ratio of a comparable DB plan and incorporate this in their design of allocation strategies.

a

Even though it is an improvement in sustainability of the pension system, ironically, it might induce welfare losses for plan members: risk is transferred from institutions who are adequately equipped to manage it to plan members who are least equipped.

b

Although empirically it is found that this rationale may not always hold true (cf. Arnott et al. (2013)).

c_{The pension ratio is defined as the fraction of pension income received immediately after retirement to labour}

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To this end, we implement a target-driven investing strategy in a life-cycle model in this thesis. Plan members at the beginning of their working career set a desired pension ratio for retirement, and on an annual basis face corresponding interim and final targets for accumulated pension wealth. These targets help plan members on establishing the desired pension ratio at retire-ment, and include time-varying information on labour income and market expectations. Plan members adhere to target-driven objective functions and derive value from accumulated pension wealth above the contemporaneously set target. Similarly, plan members derive disvalue from accumulated pension wealth below the contemporaneously set target. Given these targets, plan members are assumed to annually decide on their choice of allocation.

As a baseline, we implement cumulative prospect theory for modelling the behaviour of plan members. We extend the baseline model by also incorporating expected utility models, and we provide a formal comparison of optimal policy functions. In addition, we provide a comparison between static and dynamic allocation strategies to infer the benefits of including time-varying information. In other words, we establish whether dynamic strategies, i.e., strategies which rely on conditioning sets, give rise to superior performance in establishing desired pension ratios. In eﬀect, we also analyse whether the option of choosing an own (optimal) allocation strategy adds sizeable potential. Furthermore, we infer the eﬀects of ignoring parameter uncertainty in our model by extending the model with a Bayesian approach. Lastly, we assess the cost of assuming a risk measure that is not coherent, by comparing it to a related coherent risk measure.

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Chapter 2

Theoretical framework

2.1 Life-cycle models

Extensive research has been done in the field of life-cycle models. These optimisation models hitherto are able to capture properties of financial markets such as a stochastic term structure of labour income (cf. Bodie et al. (1992)), mortality risk, time-varying investment opportunity sets (cf. Campbell et al. (2001)) and even behavioural traits as habit formation (cf. Gomes and Michaelides (2003)). Factors that plan members wish to optimise can be, e.g., consumption, leisure, wealth or allocation. Particularly asset allocation has been scrutinised in the literature. Arrow and Debreu (1954) pioneered the field of life-cycle models by introducing a stochastic asset allocation model in which individuals maximise their expected utility, and in addition Merton (1969) in his work introduced a life-cycle model which optimises both allocation and consumption.

Cairns (1997) and Owadally (1998) were among the first to introduce life-cycle models in the context of pension plans. Cairns (1997), using a dynamic programming approach, implements a continuous-time life-cycle model in a DB setting where optimal contribution rates and allocation strategies are derived. Owadally (1998) introduces a similar model in a discrete-time setting, and Thomson (1998) implements a multi-period setting for DC pension plans where optimal allocation strategies are derived using expected utility maximization. Yao et al. (2014) optimise asset allocation in the presence of salary and mortality risk using a multi-period mean-variance framework. Similarly, using again a dynamic programming approach, Haberman and Vigna (2002) in a multi-period setting derive optimal allocation strategies in DC plans where plan members have multiple assets at their disposal.

Blake et al. (2013), which this thesis is inspired by, implement a life-cycle model which optimises allocation during the accumulation phase, i.e., the working career of a plan membera. Among others, their life-cycle model accounts for salary risky and therefore possesses an important determinant of optimal allocation strategiesb. In the outlined studies of optimal asset allocation, although the main objective is similar, both framework and solution method varies: it is in fact common throughout the whole literature, as no consensus on it has been reached yet. A novelty in a_{Most literature focus on optimal decision making at only either the accumulation phase (cf. Battocchi and}

Menoncin (2004) and Le Courtois and Menoncin (2015)) or decumulation phase (cf. Gerrard et al. (2004) and Di Giacinto et al. (2014)).

b

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the framework of Blake et al. (2013) lies with the fact that they try to bridge this gap: instead of assuming rational life-cycle planners, they propose a framework in which plan members exhibit loss aversion and thus deviate from making assumptions about which rationality, i.e., which utility function plan members adhere to.

2.2 Decision theory

Decision theory mostly used to concern itself with the von Neumann-Morgenstern utility theorem which is the cornerstone of expected utility theory. However, literature has become increasingly sceptic of expected utility theory (cf. Starmer (2000)). Expected utility theory, although gener-ally describing agents their behaviour adequately, fall short on three key aspects which we will showcase.

The theorem inherently prescribes a normative model, i.e, imposing that agents are rational and act in an optimising way. In contrast, descriptive models portray how agents actually behave in situations where risk is involved. Although this branch of decision theory is relatively new, it produced a widely accepted alternative to expected utility theorem: prospect theory. Prospect theory postulates that expected utility theory is not fully able to explain agents their behaviour in the presence of risk. To illustrate, where elsewhere consistent risk preferences are assumed, under prospect theory risk aversion in the domain of gains and risk seeking behaviour in the domain of losses can be accounted for. Thus, it also asserts that agents care about gains and losses instead of absolute levels of wealth which is proposed in expected utility theory. As we are speaking about gains and losses, it also inherently implies the presence of a reference point which agents use to determine the extent of the gain or loss. Furthermore, prospect theory is able to include loss aversion: the finding documented in behavioural finance (cf. Kahneman and Tversky (1984)) which illustrates that agents are more sensitive to losses than gains of equal sizec.

Prospect theory therefore introduces a model in which agents make decisions under risk where empirically observed behaviour is incorporated. It deviates from the assumption of optimising agents, and as such strictly speaking prospect theory does not rely on utility functions. Rather, prospect theory builds on value functions: instead of measuring in absolute levels of wealth, value is thus derived by gains or losses with respect to a point of reference. In eﬀect, the value function should hence also be able to capture the three outlined behavioural traits: a convex value function is assumed in the domain of losses, and a concave function is adhered to in the domain of gains. Moreover, the value function is assumed steeper in the domain of losses as agents are more sensitive in this domain.

The set of axioms a value function should encapsulate was first introduced in the work of Kahne-man and Tversky (1979), but they refrained from introducing any formal mathematical model. To bridge this gap, Kahneman and Tversky (1992) analysed prospect theory from a mathematical point of view. This more formal model, cumulative prospect theory, introduced a value function which hitherto is still the workhorse of prospect theoryd. In this thesis we will implement this model in DC pension plans, following the framework of Blake et al. (2013).

c

Loss aversion should not be confused with risk aversion. Risk aversion illustrates the reluctance to uncertainty, whether it is potential upside or downside is unimportant. Loss aversion only restricts itself to reluctance of uncertainty in the domain of losses.

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Given the different maxims of prospect theory and expected utility theory, it is of importance to assess how differences in underlying models of agent behaviour affect policy functions in life-cycle models. Yet, research is lacking in this regard. A notable exception is Blake et al. (2013) where both cumulative prospect theory and a traditional expected utility model are implemented. In particular, in their work effects on optimal allocation are assessed in a DC plan. Yet, the method of comparison raises questions: rather than using parametrisations of the models which are common in the literature, and supported in empirical studies, parameters of the expected utility model are derived from optimal allocation strategies of the cumulative prospect theory model.

In particular, given the optimal strategy under cumulative prospect theory, through the relation of optimal portfolio rules for a myopic investor in the presence of a risk-free asset and mean-variance preferences the coefficient of relative risk averison is constructed. This rather artificial alignment has two main disadvantages: firstly, a coefficient of risk aversion is found through the assumption of myopic investors which do not resemble long-term investors such as DC plan members saving for retirement. Secondly, by deriving the coefficient of risk aversion in such a way, an objective comparison of policy functions between the two models is troublesome as the model of expected utility implicitly depends on the model of cumulative prospect theory. A novelty of this thesis lies in establishing a formal comparison of optimal policy functions be-tween cumulative prospect theory and expected utility models, and assert the extent to which policy functions alter under the different specifications. More concretely, we assume plan mem-bers in addition to prospect theory adhere to Epstein-Zin utility and its special case power utility. Epstein-Zin utility includes two important characteristics of plan members: the coefficient of rel-ative risk aversion and the elasticity of intertemporal substitution. The coefficient of relrel-ative risk aversion quantifies a plan member his reluctance to risk within any given time period, while the elasticity of intertemporal substitution quantifies his reluctance to risk over time (cf. Campbell and Viceira (2002)).

Power utility assumes these two characteristics to be inverses of each other (cf. Campbell and Viceira (2002)). While a high degree of risk aversion is likely to be accompanied by a low value of elasticity of intertemporal substitution, there exist empirically no real evidence supporting an inverse relation (cf. Blackburn (2008)). Epstein-Zin utility, in contrast, does not impose a relation on these two concepts and as such is a more general specification of power utility. Another novelty in this thesis therefore lies in investigating, using again common parametrisations in the literature, whether the assumed inverse relation aﬀect optimal policy functions compared to Epstein-Zin utilitye. We in addition follow Blake et al. (2013) and infer the eﬀects of changing policy functions by comparing the distributions of pension ratios.

2.3 Allocation strategies

A novelty in thesis lies in comparing optimal allocation strategies of our life-cycle model to various investment strategies. Minimum and maximum risk strategies are implemented to serve as benchmarks to our optimal allocation strategies and proposed alternatives. We will in addition implement the traditional glidepath strategy used in DC pension plans. The theory of human capital serves as a prime justification for DC pension plans to implement these strategies. From the perspective of human capital, a glidepath serves traditional plan members well: plan members

e

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whose labour income is assumed riskless. The other side of this reasoning is equally valid as a sizeable number of plan members do not share this property of labour income. In fact, labour income for these plan membersf are assumed risky such that an inverse glidepath strategy is a more justified strategy to adhere tog. We therefore in addition consider an inverse glidepath strategy, following the set-up of Arnott et al. (2013).

We opt for a comparison between static and dynamic investment strategies to infer what the benefits of including time-varying information into investment decisions are. In particular, we infer whether dynamic strategies, i.e., strategies which rely on conditioning sets, give rise to supe-rior performance in establishing desired pension ratios. A threshold strategy which incorporates information of performance of (interim) pension ratios, as well as a constant proportion portfolio insurance strategy of Perold and Sharpe (1988) are therefore implemented. More importantly, in this setting we also propose a novelty of analysing whether the option of choosing an own (optimal) allocation strategy is worthwhile and in particular we determine if it adds sizeable potentialh to deviate from the frequently employed allocation strategies.

2.4 Estimation risk

An issue in implementing optimal allocation strategies is that parameters of the return distri-butions, e.g., moments of the distribution, are unknown and estimatedi_{. Data in most cases is}

limited, especially if time-varying investment opportunities sets are present. Estimation error is therefore present, and is likely to influence optimal policy functions and investment perfor-mance. Broadly speaking, two branches in the literature exist to derive parameter estimates: a plug-in approach and a decision-theoretic approach. The plug-in approach mostly concerns itself with frequentist methods to derive parameter estimates, e.g., maximum-likelihood estimation. Provided that these estimates are constructed, it is assumed these parameters are true (thereby ignoring parameter uncertainty). Hence, only in the implausible event of parameter estimates exactly corresponding to the true parameter values optimality is achieved. Especially in the case of long-term investors, such as plan members saving for retirement, estimation error may have profound eﬀects as estimation error stacks over time. Most notably, Jobson and Korkie (1980) and Michaud (1989) showcase the severe consequences the plug-in approach can have on investment performance.

As an alternative, the decision-theoretic approach discards the frequentist maxim and adopts a Bayesian approach. Rather than using fixed parameter estimates, a (posterior) distribution is constructed for the parameters. Optimality is therefore achieved for the predictive distribution of returns, and thus theoretically this approach is more appealing. Since estimation error is incorporated, the approach is praised in the literature as a more robust alternative to conventional techniques (cf. Brandt (2007)). A downside to this approach is that generally analytical results are challenging to obtain.

A novelty in this thesis is therefore that we scrutinise diﬀerences in optimal policy functions and distributions of pension ratios when both a plug-in approach and a decision-theoretic approach is followed. To the best of our knowledge, quantifying estimation risk in DC pension plans and

f

For example, employees in the field of investment banking or coal mining.

g_{Outpeformance of these inverse glidepath strategies are also documented in the literature (cf. Arnott etl al.}

(2013)).

h

Either a decreased downside risk or increased upside potential of the distribution of pension ratios.

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assessing the eﬀects on policy functions and distributions of pension ratios has hitherto not been applied in the literature.

2.5 Measures of performance

A question still unanswered is how to assess diﬀerences in the distributions of pension ratios. From these distributions Value-at-Risk (VaR) measures are derived. VaR has become the stan-dard of risk measures in financial management, where even major regulations (e.g., Solvency II and Basel II) rely on it. As such, in an attempt to provide risk measures which are communi-cable to the general reader, VaR serves as a baseline risk measure in this thesis. Yet, from an academic point of view, the VaR measure exhibits undesirable properties. Most prominently, VaR is criticised as not being able to satisfy all axioms of coherencej. The VaR measure has weak aggregation properties, and therefore does not meet the subadditivity requirement: when two portfolios are combined, the VaR measure of the merged portfolio is not necessarily bounded from above by the sum of the risk measures of the individual portfolios. Therefore, we in addi-tion propose a theoretically more appealing risk measure: Expected Shortfall. This risk measure satisfies all axioms of coherence, and is a more representative measure of risk for fat tailed dis-tributions. In this thesis we hence also propose a novelty by assessing the cost of assuming a risk measure that is not coherent, by comparing it to a related coherent risk measure.

In summary, this thesis contributes to the literature as following: firstly, we provide a comparative analysis of policy functions under different models of agent behaviour. Using parametrisations common in the literature, we assess the different distributions of pension ratios using both risk measures used by practitioners and risk measures which theoretically are more valid. Secondly, we provide an overview of performance between static and dynamic allocation strategies. In addition, we also analyse whether the option of choosing an own optimal allocation strategy is worthwhile for plan members compared to these frequently used strategies. Finally, we spearhead the effort by attempting to quantify significance of estimation risk in our dynamic DC context setting by both ignoring and allowing for parameter uncertainty.

j

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Chapter 3

The life-cycle model

3.1 Cumulative prospect theory

In this thesis we will implement cumulative prospect theory as a baseline, and adhere to the basic set-up of Blake et al. (2013). A functional form that shows consistency with behaviour of plan members follows from (cf. Kahneman and Tversky (1992))

U (W ) = { _(W_−f)_v1 v1 if W ≥ f −λ(f−W )v2 v2 if W < f where

• U(·) denotes the value a plan member derives, • W denotes the pension fund value.

• f denotes a prespecified target of the pension fund,

• 0 ≤ v1 ≤ 1 and 0 ≤ v2 ≤ 1 denote curvature parameters for the domain of gains, i.e.,

outperformance of the pension fund given the prespecified target and domain of losses, i.e., underperformance of the pension fund given the prespecified target, and

• λ > 1 is the loss aversion ratio.

Kahneman and Tversky (1992) define this value function as a two-part power function. Along with proposing this model, they empirically investigate which parameter values hold true. It is observed that the parametrisation v1 = 0.88, v2 = 0.88 and λ = 2.25 represent real world

behavioural best. We will use the same parametrisation in this thesis.

3.2 Asset classes

In this thesis the investment menu consists of a risky asset and a risk-free asset:

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We assume a constant annual real return rf of the bond fund. The annual return on the stock

fund during year of age x to x + 1 is given by:

rx = rf + µ + σZ1,x

where µ denotes the annual risk premium on the stock fund, σ the annual return volatility and

Z1,x denotes the realisation of a standard normal variable that is i.i.d. at year of age x. In

other words, we assume a return specification that contains both an expected and unexpected component.

3.3 Labour income

In this section we describe the labour income structure plan members follow. At the start of every year of x to x + 1, for x = 22, 23, . . . , 67, plan members receive their annual salary. We implement the following structure for modelling of salary promotions:

CSPx= h1  −(√3(x− 20) 45 )2 +4(x− 20) 45 − 1   + h2 ( (x− 20) 45 − 1 ) + 1

which is also used in Blake et al. (2007). In their work a least squares methodology is used in order to estimate the parameter values of h1 and h2, and it is found that h1 = 0.75 and

h2 =−0.19a.

Similarly, we define labour income growth as

Gx = gL+

CSPx− CSPx−1

CSPx−1

+ σ1Z1,x+ σ2Z2,x

which is the growth structure introduced in Cairns et al. (2006). gLdenotes a constant and equals

the long-term average yearly rate of growth. We also account for two stochastic components: Z1,x

denotes a stochastic factor which also drives the stock marketb_{, and thus σ}

1has the interpretation

of volatility to labour income from stock market performance. Similarly, Z2,xdenotes a stochastic

component which is inherent to labour income growth and thus σ2 resembles its volatility.

To construct labour income the following relation is assumed, where Ix denotes labour income

at x,

Ix= Ix−1exp(Gx) for x = 23, . . . , 67

To normalise results, we set I22= 1.

3.4 Pension fund accumulation and targets

3.4.1 Pension fund accumulation

When plan members enter their working career at year of age 22, pension wealth W is assumed to be zero. Therefore initially it follows that W22 = 0, and accumulation of pension wealth

a

We will also implement these parameter values in our thesis.

b

Since income growth shares a common stochastic factor with the return on the stock fund, we allow for correlation between market performance and income growth. In particular, this correlation is given by √σ1

σ2 1+σ22

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adheres to

Wx = (Wx−1+ πIx−1) [αx−1exp(rx) + (1− αx−1) exp(rf)] for x = 23, . . . , 67

• Wx denotes pension wealth at year of age x,

• Ix−1 denotes labour income received at year of age x− 1,

• π is the annual contribution rate, and

• αx−1 is the chosen allocation profile at year of age x, i.e., the proportion of pension wealth

invested in the stock fund.

3.4.2 Final target derivation

The annual amount of pension income received depends on the accumulated wealth and the price of a life annuityc. This price at age x is derived using

¨ a67= 53 ∑ s=0 sp67e−rfs

wherespx is the probability that a plan member of age x survives to age x + s. To obtain these

probabilities, the male mortality table of the Koninklijk Actuarieel Genootschap 2016 is used. In order to establish the final target, plan members need to form an expectation of labour income at year of age 67 immediately at the beginning of their working career. This expectation is given by E22[I67] = exp ( 45× ( gL+ σ₁2+ σ₂2 2 )) _∏67 x=23 exp ( CSPx− CSPx−1 CSPx−1 )

where a full derivation can be found in the Appendix.

If we assume that plan members desire a pension ratio of 2₃, the expected final target of pension wealth given year of age 22 follows from

f22(67) = ¨a67×

2

3 × E22[I67]

We, however, also need to establish final targets for every other year of age. For every year of age onwards, expected labour income at retirement, given labour income Ix at year of age x, is

given by

Ex[I67] =

Ix

E22[Ix]× E22

[I67]

Thus, the expectation is adjusted to reflect diﬀerences between realised and expected labour income at year of age xd.

A similar adjustment is necessary for the final target of pension wealth, viz.

fx(67) = ¨a67× 2 3× Ex[I67] = ¨a67× 2 3× Ix E22[Ix]× E22 [I67] c

We therefore do not allow for phased annuitisation of pension wealth. Plan members by construction are forced to annuitise all pension wealth when they enter retirement.

d_{Alternatively, it also resembles diﬀerences between the realised and expected path of labour income growth}

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20 25 30 35 40 45 50 55 60 65 70 25 30 35 40 45 50

Figure 3.1: Final fund target and interim targets at year of age 22. Plan members at this year of age have unity labour income, and a mean real risk-free rate of 0.52% is used to discount the targets. In addition, a contribution rate of 7.5% and ¨a67= 16.45 is assumed for plan members.

3.4.3 Interim target derivation

Provided that we have obtained a final target at a given year of age, we can derive interim targets, i.e., transform the final target to a target which has to be met in that corresponding year of age. To this end, we construct recursively

[fx(t) + πEx[It]] exp(−rf) = fx(t + 1) for t = 66, 65, . . . , x

where fx(x) denotes the interim target of pension wealth at year of age x.

In line with pension accounting standards for valuation of pension plan liabilities, e.g., Inter-national Accounting Standards 19, yield on investment grade corporate bonds should strictly speaking serve as the discount rate. We do not pursue this, however, and we will employ the risk-free rate as the discount rate throughout this thesise.

Figure 3.1 illustrates the final and interim targets for a plan member which starts his working ca-reer. Using parameter values derived from the dataset of Goyal and Welch (2008) (cf. Appendix), it is found that, given his labour income of unity, the final target equals f22(67) = 49.245.

Sim-ilarly, we find f22(22) = 28.708 and as it is assumed that W22 = 0 it is immediately clear that

plan members start below plan. Plan members can remedy it by altering their annual contribu-tion rate to a higher level. However, we do not opt for increasing this rate. By specifying this rate, it becomes more diﬃcult for plan members to obtain their desired pension ratio. A sizeable share of plan members therefore will end with a lower pension ratio, thereby providing a better assessment for the risk measures. In addition, plan members can use their allocation strategies to try to reach their final target.

e_{Among others, lack of recent data on investment grade corporate bonds and a time constraint to employ yield}

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3.4.4 The target-driven objective function derivation

Given a realisation of pension wealth Wx and an interim target fx(x) of pension wealth at year

of age x, we are in a position to construct a prospect theory value function for plan members. More formally, Ux(Wx) = { _(W x−fx(x))v1 v1 if Wx≥ fx(x) −λ(fx(x)−Wx)v2 v2 if Wx< fx(x) for x = 22, 23, . . . , 67.

It should be noted that if this value function is maximised, plan members are essentially myopic as they only concern themselves with optimising the current year of age. Plan members in our setting implement the total discounted prospect theory value function, however. As for these plan members the final target trumps the interim target, we will assume that interim targets ω have a lower weighting than the final target of pension wealthf. Mathematically, for year of age

x, we define the discounted prospect theory value function as

Vx = 67_∑−x−1 s=0 βsωUx+s(Wx+s) + β67−xU67(W67) = ωUx(Wx) + 67∑−x−1 s=1 βsωUx+s(Wx+s) + β67−xU67(W67) = ωUx(Wx) + β 67∑−x−1 s=1 βs−1ωUx+s(Wx+s) + β67−xU67(W67) = ωUx(Wx) + βVx+1

where β denotes the personal annual discount factor. Hence, plan members maximise total discounted utility where appropriate weight is given to interim targets and the final target.

3.5 Constraints

The following assumptions are established for the baseline model:

• Plan members are assumed to begin working at year of age 22, having zero accumulated

pension wealth, and retire at year of age 67.

• A fixed contribution rate of labour income (7.5%) is assumed for plan membersg_{, and}

contributions to the pension plan are made annually.

• Plan members desire a pension ratio of 2

3, implying that members desire to receive 2 3 of

last income as pension income.

• Members evaluate pension wealth annually, i.e., they assess performance of their investment

choices annually.

• A uniform weight profile is used for each interim target, and a higher weight to the final

target at retirement. f

We will apply a unit weight to the final target.

g_{Although the Dutch pension system uses age-dependent contribution rates (Witteveen premiestaﬀels), a}

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• Plan members are loss averse and adhere to cumulative prospect theory. Moreover, plan

members maximise the expected total discounted value function over their entire future working career.

• Taking either a long or short position in a financial asset is prohibited for plan members.

3.6 Model optimisation

During the accumulation phase of pension wealth a plan member faces the following optimisation problem max αx Ex [Vx] = max αx ωUx(Wx) + βEx[Vx+1] subject to Ix = Ix−1exp(Gx)

Wx= (Wx−1+ πIx−1) [αx−1exp(rx) + (1− αx−1) exp(rf)]

0≤ αx≤ 1

for x = 22, 23, . . . , 66.

The optimisation problem at hand does not allow of an analytical solution, as no explicit solution can be found for the expectation term. This is an often encountered problem in life-cycle models, and hence numerical approximation techniques have to be implemented. In particular, we will use dynamic programming to find optimal policy functions. As a first step, the Bellman equation has to be constructed

Vx(Wx, Ix) = max αx

ωUx(Wx) + βEx[Vx+1(Wx+1, Ix+1)]

Our model hence assumes two state variables which are denoted by pension wealth and labour income.

Dynamic programming relies on backward recursion, and since plan members face a finite horizon we can use the year when a plan member retires as a terminal condition (thereby vastly reducing the complexity of the problem). At the terminal condition, i.e., year of age 67, we assert that

V67(W67, I67) = U67(W67)

where as final target it follows that f67(67) = ¨a67×2₃ × I67. The state variable pension wealth

is discretised into 300 grid points with an equal distance between consecutive points, i.e.,{ ˆWi= 200

299 × (i − 1) : i ∈ {1, 2, . . . , 300}}. In addition, the state variable labour income is discretised

into 20 grid points: {ˆIj = 0.91 + 12.09₁₉ × (j − 1) : j ∈ {1, 2, . . . , 20}}. Given the grid points of

pension wealth and labour income, we can derive V_i,j,67∗ = V67( ˆWi, ˆIj) ∀ i, j.

In addition, our model has the presence of one control variable: stock fund allocation αx for

x = 22, 23, . . . , 66. We discretise this control variable into 50 grid points to exclude a local

maximum, and with the presence of a short-selling constraint it follows that{ˆαk = ₄₉1 × (k − 1) :

k∈ {1, 2, . . . , 50}}.

Suppose a plan member is one year away from retirement, and hence has year of age 66, then his optimisation problem follows from

max

α66

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given the constraints. If we define the set {Ωi,j,t = ( ˆWi, ˆIj) : t ∈ {23, . . . , 66}}, we can invoke

dynamic programming and construct optimal policy functions. To illustrate, assume a plan member has state variable realisations ˆW1and ˆI1, i.e., is located at node Ω1,1,66. A plan member

then has to form an expectation about future prospects, and in particular he has to derive E66[V67(W67, I67)]. Since all stochastic factors in our model are assumed standard normal, we

may implement Gauss-Hermite quadrature to approximate this expectation term.

More specifically, Gauss-Hermite quadrature enables us to discretise the standard normal inno-vations into a set of any given number of distribution nodes and weights. We opt for discretising the distributions into 18 nodes and weights, and since in our model we have two stochastic factors we define the sets {Z1,m : m = 1, 2, . . . , 18} and {Z2,n : n = 1, 2, . . . , 18} as the

Gauss-Hermite quadrature nodes. Similarly, we define the set {w1,m : m = 1, 2, . . . , 18} and

{w2,n : n = 1, 2, . . . , 18} as the Gauss-Hermite quadrature weights. It can then be shown (cf.

Blake et al. (2013)) that the expectation term can be numerically approximated by E66[U67(W67)] = ∫ _∞ −∞ ∫ _∞ −∞U67(W67)Φ(Z1,m, Z2,n)dZ1,mdZ2,n ≈ π−1 18 ∑ m=1 18 ∑ n=1 w1,mU67 (√ 2× Z1,m, √ 2× Z2,n ) w2,n

where in addition Φ(Z1,m, Z2,n) denotes a bivariate standard normal distribution, and π is the

mathematical constant.

In other words, given ˆW1and ˆI1 at year of age 66, we simulate pension wealth and labour income

one period ahead according to specific discretised innovation shocks and assuming, say, ˆα1, we

find W₆₇(m) = ( ˆ W1+ π ˆI1 ) [ ˆ α1exp(rf+ µ + σ √ 2Z1,m) + (1− ˆα1) exp(rf) ] and I₆₇(m,n)= ˆI1exp ( gL+ CSPx− CSPx−1 CSPx−1 + σ1 √ 2Z1,m+ σ2 √ 2Z2,n ) for m = 1, 2, . . . , 18 and n = 1, 2, . . . , 18.

For each simulated set (

W₆₇(m), I₆₇(m,n)

)

we find by two-dimensional linear interpolation V₆₇(m,n)∗ from the already constructed values V_i,j,67∗ , and sum each V₆₇(m,n)∗ (weighted and divided appro-priately) to obtain the expectation term. Note therefore that the expectation term depends on allocation choice, and thus has to be constructed for every possible choice of ˆα separately.

Provided that we have derived the expectation term for every choice of allocation, we can find optimal policy functions of Ω1,1,66 by assessing

ωU66( ˆW1) + βE66[V67(W67, I67)]

A grid search is then implemented such that the optimal choice of allocation can be found. Consequently, for the respective values of the state variables, we obtain Ω1,1,66 → α∗1,1,66. A

similar argument provides us with Ωi,j,66 → α∗i,j,66∀ i, j. We can iterate this procedure backwards

to derive optimal policy functions Ωi,j,t→ α∗i,j,t for each node at each specified year of age. As a

last step, we construct Ω22= (0, 1) since plan members all face identical initial conditions (i.e.,

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Diﬀerent plan members are constructed with simulated paths of labour income and stock fund returns. For each plan member, given his specific labour income and stock fund return (and thus pension wealth), we find his optimal choice of allocation by finding the appropriate nodes Ωi,j,t. We employ a two-dimensional (linear) interpolation to approximate the optimal choice of

allocation given the two nodes that are least distant to the plan member his labour income and pension wealth realisation.

As can be noted subjectivity is present in determining the size of the intervals and number of grid points. A larger interval implies more accurate policy functions of pension wealth and labour income outliers. Furthermore, increasing the number of grid points makes interpolation more accurate as it is more likely that the nodes used for interpolation are closer to the realisations of state variables. Similarly, increasing the Gauss-Hermite quadrature nodes and weight will result in more accurate results as innovation shocks can be captured more appropriately. However, increasing these aspects indefinitely is not desirable. A problem in life-cycle models, and in particular dynamic programming, is that increasing these properties more than linearly increases the time to perform all necessary calculations. Our specification consists of the maximum number of nodes possible, given the limited time for this thesis. A linear interpolation is implemented in our setting. However, there is no justified reason to assume the Bellman equation is linear in either state or control variables. A spline interpolation is therefore theoretically a more valid methodology. Assuming a spline interpolation, however, is seen to result in heavily oscillating policy functions and consequently linear interpolation is more appealing.

3.7 Extensions

3.7.1 Expected utility models

In addition to the prospect theory value function, we extend this thesis with normative utility specifications. To this end, we introduce utility functions that are frequently used in analysing life-cycle models: Epstein-Zin utility and its special case power utility.

Mathematically speaking, the utility specifications boil down to

Ux(Wx) = { (Wx− fx(x)) 1 1−γ _{if W}_x≥ f_x_(x) −(fx(x)− Wx) 1 1−γ _{if W}_x_{< f}_x_(x) for power utility where γ denotes the relative risk aversion, and

Ux(Wx) =              ( (Wx− fx(x)) 1− 1_ψ 1−γ ) 1 1− 1_ψ if Wx≥ fx(x) − ( (fx(x)− Wx) 1− 1_ψ 1−γ ) 1 1− 1_ψ if Wx< fx(x)

for Epstein-Zin utility where ψ denotes the elasticity of intertemporal substitution. Note that Epstein-Zin utility reduces to power utility when γ = _ψ1.

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risky asset and risk-free asset (cf. Appendix), corresponds to our thesis. Using this dataset, it is derived that the coeﬃcient of relative risk aversion has a value of 5.65 and the elasticity of intertemporal substution has a value of 0.226. A similar optimisation problem as outlined in Section 3.6 is followed.

3.7.2 Estimation risk

An extension of allowing for parameter uncertainty will be incorporated (thereby invoking a decision-theoretic approach). We will attempt to quantify significance of estimation risk in our dynamic context setting, using the ideas of Barberis (2000). Incorporating plan members their beliefs about distribution parameters greatly increases our state space such that the dynamic programming problem becomes diﬃcult to solve. Hence, simplifying assumptions are used to make the optimisation problem feasible.

In particular, we will assume that plan members acknowledge that they are uncertain about parameters of the return distribution. Yet, plan members discard the impact on current optimal allocation of potentially altering beliefs about the parameters over time. He therefore solves the optimisation problem assuming that his beliefs of parameters of the return distribution remain unchanged throughout the accumulation period. These fixed beliefs of parameters are encapsulated within the posterior distribution which is calculated on data up until the start of his working career.

More specifically, we will generate a sample from the posterior distribution for the parameters

p(µ, σ2|r) at nodes Ωi,j,t ∀ i (and Ω22)h where r is the financial (real) return time series data of

the stock fund outlined in the Appendix. For each pair (µ, σ2), we draw p(r|µ, σ2, r) which is a

normal distribution thus providing us with the predictive distribution.

Mathematically, we assume a flat prior as we want to minimise influence the posterior distribution

p(µ, σ2)∝ 1

σ2

such that the posteriors are given by

σ2|r ∼ IG ( T− 1 2 , (T − 1)ˆσ2 2 ) µ|σ2_{, r}_{∼ t} ( T− 1, ˆµ,σ 2 T )

where ˆµ and ˆσ2 denote the sample mean and variance (cf. Zellner (1971)). For sampling of these distributions the following algorithms are used:

3.7.3 Alternative investment strategies and measures of performance

For the baseline model with cumulative prospect theory alternative investment strategies are evaluated. In particular, in addition to the optimal investment strategy, we will consider

h

In words, we sample a new draw from the predictive distribution at each diﬀerent grid point of pension wealth. To illustrate, the same draw is used (and necessary) for nodes Ω1,1,23 and Ω1,2,23: labour income and pension

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Algorithm 1 Sampling algorithm of the inverted Gamma distribution 1: procedure 2: Sample X∼ Gamma ( T−1 2 , 2 (T−1)ˆσ2 ) . 3: Set Y = _X1 ∼ IG ( T−1 2 , (T−1)ˆσ2 2 ) .

Algorithm 2 Sampling algorithm of the non-standardized t distribution

1: procedure

2: Sample X∼ t(T − 1) from a standard t distribution with T − 1 degrees of freedom.

3: Set Y = ˆµ +√σ TX.

• A minimum risk strategy, i.e., fully invested in the bond fund. • A maximum risk strategy, i.e., fully invested in the stock fund.

• A balanced static strategy: plan members rebalance annually to a 50% stock fund and 50%

bond fund.

• A glidepath strategy: plan members are 80% invested in the stock fund at year of age 22,

and linearly decrease the holding in the stock fund each year of age onwards such that at year of age 66 the holding in the stock fund is 20%.

• An inverse glidepath: plan members are 20% invested in the stock fund at year of age 22,

and linearly increase the holding in the stock fund each year of age onwards such that at year of age 66 the holding in the stock fund is 80%.

• A threshold strategy which is 100% invested in the stock fund if the current pension ratio

is below a lower threshold TL, 100% invested in the bond fund if the current ratio is above

an upper threshold TU and linearly increasing in the bond fund as the current ratio rises

from the lower threshold to the upper threshold. In this thesis we will use TL = 0.4 and

TU = 0.8.

• A constant proportion portfolio insurance introduced by Perold and Sharpe (1988) where

αx= CM ( 1− CF ( Floor Fund )) = CM ( 1− CF ( Liabilities Fund ))

where CF denotes the weight plan members attach to the pension wealth exceeding a floor

level and CM denotes the significance attached to the surplus ratio. Note that in our setting

the ratio of the liabilities to the fund is equal to the inverse of the pension ratio. We will follow Blake et al. (2001) and use CM = 2 and CF = 0.5 as our parameter set. We set

an upper and lower bound on allocation so that plan members cannot take a long or short position. In other words, we in addition set a short-selling such that α ∈ [0, 1] implying that plan members cannot invest more than their entire pension wealth.

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We will analyse diﬀerences in pension ratios under the specified investment strategies. Given that we have obtained a distribution of pension ratios under a specific allocation strategy, we can derive Value-at-Risk (VaR) and Expected Shortfall (ES) estimates for any chosen confidence level. More specifically, Value-at-Risk denotes for some level of significance in the unit interval, i.e., α∈ (0, 1), the smallest number p such that the probability that the pension ratio P is less than p is larger than α:

VaRα = inf{p ∈ R : P(P > p) ≤ 1 − α} = inf{p ∈ R : FP(p)≥ α}

Expected Shortfall is an extension of the Value-at-Risk in the sense that it looks further into the tail, which mathematically is captured by

ESα =E [P |P ≤ VaRα]

for any confidence level α ∈ (0, 1). This risk measure thus averages Value-at-Risks over all confidence levels below α, and has the interpretation of describing how extreme events become conditional on VaRα being violated.

If either VaR or ES estimates are unsatisfactory, i.e., diﬀerent from the optimal strategy, the contribution rate is increased to an appropriate level which is done by referring to the relevant percentiles of the empirical distribution of pension ratios. Furthermore, we derive the critical VaR-confidence level. This confidence level denotes the percentile where plan members would just accept the DC plan (a pension ratio of 2₃) given their followed allocation strategy.

3.8 Model discussion

We have assumed the presence of two assets in which plan members may invest their pension contributions and accumulated pension wealth: a risky-asset in the form of a stochastic stock fund and a risk-free asset as proxied by a constant bond fund. We therefore do not account for other potential asset classes. However, this is not necessarily a shortcoming as DC pension plan providers usually solely oﬀer these two asset classes. In addition a stochastic term structure for income is assumed, thus acknowledging that plan members face uncertainty in their future income (cf. Figure 3.2).

Although it is of importance in life-cycle models, there is no clear consensus in the literature regarding this aspect: both deterministic labour income profiles and diﬀerent stochastic income profiles are seen in the literature. We opt for using the same labour income profile as Blake et al. (2013) since empirical data is used to calibrate labour income parameters. In addition, it implicitly assumes a logical positive correlation between stock fund performance and labour income growth. Of equal importance, through the use of salary promotions and innovation shocks both permanent transitions as well as temporarily transitions in labour income are accounted for: it supports the notion that if plan members make promotion and receive higher labour income, this eﬀect will persist.

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20 25 30 35 40 45 50 55 60 65 70 1 2 3 4 5 6 7 8 1st Decile 1st Quantile 4th Quantile 10th Decile Mean

Figure 3.2: Labour income profiles for simulated plan members (15,000 simulations). Labour income growth depends on stock fund market performance and contains stochastic components. Corresponding deciles and quantiles as well as mean income for plan members are given. As normalisation unity labour income is imposed at the start of the working career (year of age 22), and labour income is expressed in real terms.

potential sources from which plan members can derive wealth (e.g., housing wealth). The in-clusion of such sources are likely to alter allocation strategies, as a minimum level of wealth is already achieved in the form of his real estate holding.

We also ignore often documented statistical properties in stock prices. Among others, finan-cial return series are leptokurtic and skewed implying that, among others, small returns occur more often and returns exhibit fat tails. Assuming normality with its imposed symmetry and underestimation of extreme events is thus a deviation from reality. If a leptokurtic distribution is followed, it is expected that pension wealth accumulation will be mitigated, thereby making it more diﬃcult for plan members to reach their interim targets (and hence plan members will follow allocation strategies that are relatively higher exposed to risk).

Weak but significant relations of returns and, e.g., financial variablesi are commonly found in academic literature, and hence in this regard our models falls short by not incorporating them. Also worth mentioning is the oversimplified assumption of a constant bond fund return. We do not account for stochastic term structures of interest rates, although it enhances economic relevance of the model. Consequently, we also do not account for a possible dependence structure between the stock fund and bond fund returnsj. If a stochastic term structure of interest rates is assumed, it will induce extra uncertainty and therefore it is expected the distribution of pension ratios will exhibit more variability.

A drawback of our model is the exclusion of the decumulation phase after plan members retire. Plan members upon retirement are required to trade in their entire pension wealth for life annu-ities. We therefore exclude the option for plan members to keep their pension wealth partially

i

Dividend yield is a prominent example of exhibiting predictive power on financial return series.

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invested in the stock fund after retirement. As annuities become less costly over time, there is a higher probability of obtaining the pension wealth target and thus the desired pension ratio. In addition, pension wealth can accumulate a longer period so that targets are more likely to be achieved. However, it may also introduce extra downside risk when the stock fund is per-forming poorly after the moment of retirement. In addition, the requirement of fully investing accumulated pension wealth in annuities destroys a possible bequest motive plan members may exhibit.

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Chapter 4

Results

4.1 Cumulative prospect theory

In this section we will present the results of plan members adhering to cumulative prospect theory. As a baseline, we introduce the parameter values for the life-cycle model where both the labour income parameters and the personal discount factor β are similar to Blake et al. (2013) (cf. Table 4.1).

It is worth assessing how a more realistic setting in both return specification and loss aversion parameters aﬀect optimal policy functions one year before retirement. To this end, Figure 4.1 illustrates optimal policy functions for a plan member at year of age 66, given an income of 5.08 units. If plan members are below target, plan members follow a maximum risk strategy, i.e., opt-ing for a full allocation in the stock fund of their accumulated pension wealth and contributions. In the domain of losses, therefore, a full stock fund weighting is followed: properties of prospect theory with its risk seeking behaviour in the domain of losses hence carry over to optimal policy functions.

Initially, when plan members are slightly above target risk-averse behaviour is noticeable: plan members decrease their risk exposure by tilting their allocation slightly to the bond fund. We

Loss aversion parameters Labour income

λ 2.25 gL 2%

v1 0.88 σ1 5%

v2 0.88 σ2 2%

h1 0.75

h2 -0.19

Asset returns Other parameters

rf 0.52% β 96%

µ 6.01% π 7.5%

σ 19.65% ¨a67 16.45

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0 20 40 60 80 100 120 140 160 180 200 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 54.19

Figure 4.1: Optimal asset allocation given varying levels of pension wealth for a plan member at year of age 66 and an income of 5.08 units. Plan members in this setting are assumed to adhere to cumulative prospect theory. The vertical line with corresponding arrow denotes the pension wealth target of this corresponding plan member.

therefore document a locking-in eﬀect of minor size when pension wealth is slightly above its target. However, when realised pension wealth increases further from this point on plan members, even though they are risk averse in the domain of gains, become comfortable with increasing their risk exposure to the limit.

In addition, we illustrate the eﬀects of pension wealth on optimal policy functions for earlier years of age (cf. Figure 4.2). We note that at year of age 65 plan members have maximum risk exposure throughout all levels of wealth. Regardless of being below or above target, i.e., being in the domain of gains or losses, plan members maintain a full stock fund weighting. While this is not necessarily surprising in the domain of losses, as plan members are assumed risk seeking, it is remarkable for the domain of gains where plan members exhibit risk averse behaviour. We find similar risk exposures for earlier years of age, given the specified income. Plan members thus do not lock-in wealth when pension wealth is in line with the target during earlier years of age. We hence contradict findings of Blake et al. (2013) where (a diminishing) locking in of pension wealth is found at the previous years of age from retirement.

The absence of locking in pension wealth has its eﬀect on realised policy functions of plan members, which are found in Figure 4.3. Given the earlier findings, it is not surprising to see that on average plan members opt for maximum risk exposure. In very few cases a slight tilt towards the bond fund is present, although this tilt is only minor in size. We therefore do not document a gradual tilt towards the bond fund over time as found in Blake et al. (2013)a. However, with respect to the baseline model of Blake et al. (2013), there are three pronounced diﬀerences. Firstly, using the parametrisations found in the literature, we impose a higher curvature parameter for the domain of gains. Ceteris paribus, the level of risk aversion in the domain of gains is therefore decreased and hence plan members are less inclined to lock in

a

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0 20 40 60 80 100 120 140 160 180 200 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 Age 66 Age 65 54.19 52.91

Figure 4.2: Optimal asset allocation given varying levels of pension wealth for a plan member at years of age 66 and 65 with an income of 5.08 units. Plan members in this setting are assumed to adhere to cumulative prospect theory. The vertical lines with corresponding arrows denote the pension wealth target of these corresponding plan member.

20 25 30 35 40 45 50 55 60 65 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1st percentile 99th percentile mean

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their pension wealth. Secondly, we impose a lower loss aversion ratio: plan members are thus assumed less sensitive to losses. The decrease in loss aversion implies that plan members are more inclined to take on additional risk in the domain of losses, and invest their pension wealth more aggressively, i.e, opting for a higher allocation in the stock fund. Lastly, we specify diﬀerent parameter values for the return specification: instead of assuming parameter values without a

priori justification, we estimate return parameters based on historical financial data.

Especially the effects of the last alteration are profound. To illustrate, by estimating the real risk-free return from empirical data we find a return which is almost four times as low as the risk-risk-free return assumed in the baseline model of Blake et al. (2013). Similarly, our premium on the stock fund, i.e., the equity premium is almost three times as large while the volatility is not significantly different. By estimating return parameters from financial return series, the return on the bond fund pales in comparison to the stock fund. It affects optimal policy functions: while plan members are assumed risk averse in the domain of gains, the risk premium is disproportionally high with respect to their risk aversion. Put differently, plan members in the domain of gains do not want to lock in their wealth: the protection against downside risk does not outweigh the upside potential.

The positive risk premium therefore is seen to be an important determinant of allocating in the stock fund. This risk premium more than oﬀsets plan members’ aversion for taking on risk. It is worth mentioning that in fact our results are (partially) driven by the equity premium puzzleb. Although by construction stocks expose plan members to more risk, this property alone cannot account for the full expected diﬀerence between the two, i.e., the equity premium. Hence, even though the curvature parameters are of a realistic level, they are not of a high enough level to fully explain the risk premium. Plan members acknowledge this and consequently choose for a full allocation in the stock fund. These findings are in line with Gomes et al. (2008) in which they document that a historical equity premium and volatility give rise to allocation strategies that have unrealistically high exposure to risk.

4.2 Expected utility models

In this section we will investigate to what extent optimal policy functions and pension ratios will differ between different models of plan member behaviour, i.e, cumulative prospect theory and expected utility models. We introduce Epstein-Zin utility and its special case power utility and we will contrast it to policy functions and pension ratios under cumulative prospect theory. Using the set of parameters in Table 4.2, we will optimise the model as outlined in Section 3.6. Figure 4.5 illustrates policy functions under the assumption of Epstein-Zin utility and power utility. Notably differences in optimal policy functions between these expected utility models are minor, and hence imposing an inverse relation of the relative risk aversion and elasticity of intertemporal substitution does not alter policy functions significantly. In other words, deviating from the value of the elasticity of intertemporal substitution found in the literature to the inverse of the relative risk aversion found in the literature (0.226 and 0.1770, respectively) has negligible implications.

The minor implications of assuming power utility are also noticeable in Table 4.3. The summary statistics and corresponding risk measures show no real misalignments. Diﬀerences between

cu-b

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Loss aversion parameters Labour income γ 5.65 gL 2% ψ 0.226 σ1 5% σ2 2% h1 0.75 h2 -0.19

Asset returns Other parameters

rf 0.52% β 96%

µ 6.01% π 7.5%

σ 19.65% ¨a67 16.45

Table 4.2: Parameter values for the life-cycle model when plan members adhere to Epstein-Zin utility and power utility.

20 25 30 35 40 45 50 55 60 65 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5th percentile 95th percentile mean

(a) Epstein-Zin utility.

20 25 30 35 40 45 50 55 60 65 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5th percentile 95th percentile mean (b) Power utility.

Figure 4.4: Optimal stock fund allocation quantiles for 15,000 simulated plan members who adhere to Epstein-Zin utility and power utility. The 5th percentile and 95th percentile are given, as well as the mean optimal asset allocation.

Pension ratios

Mean Median Standard deviation 20 % VaR 10% VaR 1% VaR Critical VaR 1% ES

Cumulative prospect theory 0.8099 0.7187 0.4348 0.4292 0.3245 0.1797 44.9111 0.1549

Epstein-Zin utility 0.6958 0.7701 0.2634 0.4011 0.2614 0.1498 33.4738 0.1337

Power utility 0.6936 0.7693 0.2635 0.4027 0.2596 0.1489 34.0471 0.1319

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-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 Prospect theory Epstein-Zin Power 2/3

(a) Probability density estimates of pension ratios. Estimates are based on a normal kernel function.

0 0.5 1 1.5 2 2.5 3 3.5 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F(x) Empirical CDF Prospect theory Epstein-Zin Power 2/3

(b) Empirical cumulative distribution function of pension ratios.

Figure 4.5: Probability density estimates and empirical cumulative distribution functions of simulated plan members adhering to cumulative prospect theory and expected utility models (15,000 simulations).

mulative prospect theory and expected utility models, however, are profound. To illustrate, under cumulative prospect theory plan members achieve a higher mean pension ratio due to the higher risk exposure (cf. Figure 4.3 and Figure 4.4). The higher risk exposure enables plan members under cumulative prospect theory to receive more of the equity premium, thereby establishing a higher mean. Plan members under expected utility models (partially) lock in their pension wealth nearing retirement when pension wealth is in accordance with the contemporaneously targets, and thus forsake potential upside. Consequently, a lower mean and standard deviation of pension ratios is found under expected utility models.

It is therefore not surprising that cumulative prospect theory implies a higher upside potential of pension ratiosc, but it is surprising to see that downside risk (i.e., until the 20th percentile) is slightly lower under an almost full risk exposure strategy of cumulative prospect theory (cf. Figure 4.5). It again illustrates the eﬀects of the high risk premium in the sense that return of the bond fund is small in comparison and is likely to be outperformed by the stock fund. The (partial) locking in of pension wealth present in expected utility models has its benefits, however. For the middle region of simulated plan members (i.e., the 20th percentile to the 60th percentile) a higher pension ratio is achieved. Furthermore, the realised pension ratios are considerably more centered at the desired pension ratio of 2₃ in Figure 4.5.

In the Appendix (cf. Table 5.2) we also provide implications of assuming diﬀerent behavioural models for (ending) pension wealth and pensionable income from annuities. Several summary statistics on pension wealth, annuity income and final 10-year change in pension income are outlined in this section. These tables are inspired by the work of Arnott et al. (2013), and serve as an addendum for practitioners in the field. These statistics provide practitioners with a more natural interpretation of the life-cycle model and its policy functions, but are essentially generalisations of the earlier findings on the distribution of pension ratios. Therefore we will not consider them separately. A notable exception is Panel C which considers growth in pensionable income for the final 10-years: as a higher risk exposure is present under cumulative prospect theory in the final years, on average a higher change in annuity income is present compared to

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Pension ratios

Mean Median Standard deviation 20 % VaR 10% VaR 1% VaR Critical VaR 1% ES Optimal strategy 0.8099 0.7187 0.4348 0.4292 0.3245 0.1797 44.9111 0.1549 Minimum risk 0.1876 0.1828 0.0322 0.1607 0.1509 0.1313 ∅ 0.1269 Maximum risk 0.7916 0.7054 0.4223 0.4200 0.3164 0.1753 46.3087 0.1504 Balanced 0.3790 0.3742 0.0754 0.3130 0.2834 0.2316 99.9772 0.2172 Glidepath 0.3280 0.3236 0.0496 0.2855 0.2676 0.2329 ∅ 0.2244 Inverse glidepath 0.4370 0.4207 0.1381 0.3165 0.2736 0.1977 93.6655 0.1768 Threshold 0.5808 0.5626 0.1934 0.4575 0.3426 0.1755 76.5532 0.1505

Table 4.4: Summary statistics on the distribution of pension ratios (15,000 simulations). Plan members are assumed to follow various allocation strategies. Value-at-Risk (VaR) values are provided for several percentiles, as well as the Expected Shortfall (ES) for the first percentile. The Critical VaR illustrates the percentile of plan members which just accept the pension scheme (a pension ratio of 2₃).

the expected utility models. It also implies a higher variability in the change of pensionable income. However, by comparing the ratio of the last and first deciles, it follows that this higher standard deviation is mostly attributed to upside potential instead of downside risk.

4.3 Allocation strategies

In this section we will use our baseline model of cumulative prospect theory, and consider both static and dynamic allocation strategies to assess the benefits of including time-varying infor-mation into allocation decisions. In addition, we assess whether an own optimal strategy adds sizeable potential compared to frequently employed allocation strategies by comparing the dif-ferent risk measures.

To this end, we firstly consider summary statistics on the distribution of pension ratios. Table 4.4 outlines characteristics of the distributions when plan members both may choose their own optimal allocation and are forced to follow either static or dynamic allocation strategies. We leave out the constant proportion portfolio insurance strategy in conjunction with a short-selling constraint, as in almost all cases the short-selling constraint holds with equality.

It is not surprising that a maximum risk strategy approximates the optimal strategy adequately, as plan members under cumulative prospect theory opt for allocation strategies that exhibit high exposure to risk. By construction the optimal strategy outperforms the maximum risk strategy, but diﬀerences between the two are minor. With respect to the other allocation strategies, there are pronounced diﬀerences. The minimum risk strategy performs suboptimal: with a mean pension ratio of 18.76% and a standard deviation of 3.22%, plan members are far from their desired pension ratio of 2₃. VaR and ES estimates signify equally suboptimal prospects for these plan members. The critical VaR is non-existent: no simulated plan member manages to obtain the desired pension ratio.

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-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 12 14 Optimal Glidepath Inverse Min Max Balanced Threshold 2/3

(a) Probability density estimates of pension ratios. Estimates are based on a normal kernel function.

0 0.5 1 1.5 2 2.5 3 3.5 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F(x) Empirical CDF Optimal Glidepath Inverse Min Max Balanced Threshold 2/3

(b) Empirical cumulative distribution function of pension ratios.

Figure 4.6: Probability density estimates and empirical cumulative distribution functions of simulated plan members using diﬀerent allocation strategies (15,000 simulations).

ratio. Although a lower standard deviation is present compared to the optimal strategy, this lower variability mostly explains downside risk whereas the higher standard deviation in the optimal strategy is mostly attributable to upside potential (cf. Table 5.3 in the Appendix). The table also outlines the suboptimal performance of the (inverse) glidepath strategy and the balanced strategy. Out of the three alternatives the inverse glidepath strategy performs best in terms of mean and median, albeit inferior to the dynamic and maximum risk strategies. Since all three alternatives on average stay equally long in the stock and bond fund, we infer that the timing of investments is an important aspect in establishing the pension ratios. We therefore also support the findings of Arnott et al. (2013) by establishing that an inverse glidepath on average achieves a higher ending wealth. Moreover, although standard deviation is higher for the inverse glidepath, the excess standard deviation is mostly explained by potential upside instead of downside risk (cf. Table 5.3 in the Appendix).

Figure 4.6 illustrates the suboptimal performance of the (inverse) glidepath strategy, the balanced strategy and the minimum risk strategy. Where 44.91% of simulated plan members are below the desired pension ratio using an optimal strategy, at least 93.67% of these plan members under the outlined strategies achieve a similar result. The figure also illustrates the increase in upside potential when an optimal strategy is adhered to, and the inability of the threshold strategy to also realise a similar upside.

Target-driven investing in deﬁned contribution plans: a comparative analysis

Target-driven investing in defined contribution

plans: a comparative analysis

Contents

Chapter 1

Introduction

Chapter 2

Theoretical framework

2.1

Life-cycle models

2.2

Decision theory

2.3

Allocation strategies

2.4

Estimation risk

2.5

Measures of performance

Chapter 3

The life-cycle model

3.1

Cumulative prospect theory

3.2

Asset classes

3.3

Labour income

3.4

Pension fund accumulation and targets

3.5

Constraints

3.6

Model optimisation

3.7

Extensions

3.8

Model discussion

Chapter 4

Results

4.1

Cumulative prospect theory

4.2

Expected utility models

4.3

Allocation strategies