bond portfolio in a life-cycle fund
Joris Plaatsman
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: Joris Plaatsman
Student nr: 10375708
Email: jorisplaatsman@gmail.com
Date: January 15, 2018
Supervisor: dhr. dr. S. (Servaas) van Bilsen Second reader: dhr. dr. T.J. (Tim) Boonen Supervisor EY: dhr. ing. J. (Joost) Csik AAG
Statement of originality
This document is written by Joris Plaatsman who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this docu-ment are original and that no sources other than those docu-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. Besides, the views in this thesis do not necessarily represent the views of EY.
iv Joris Plaatsman — Interest rate risk hedging in life-cycles
Abstract
This thesis investigates the welfare effects of not optimally hedging against interest rate risk in a life-cycle fund. The research takes place in a setting under the new law on variable annuities, which was introduced in September 2016. We start by deriving the theoretically optimal life-cycle contract under interest rate risk and equity risk. Two key assumptions for this are a complete market setting and CRRA-preferences. Thereafter we collect both Dutch and American life-cycle data, which we compare to the derived theoretical life-cycle. We determine the welfare effects of investing suboptimally in the hedging part of the bond portfolio for the duration of one year. The main takeaway from the comparison between the data and the theoretical benchmark model is that the welfare losses take on a parabolic pattern over the life-cycle. The highest welfare losses occur during the years just before retirement. Both the setting where a variable premium is paid as well as the setting where a fixed premium is paid are taken into account. The mean welfare losses are approximately 28% to 36% lower in the variable premium setting than in the fixed premium setting. Several theories are presented and discussed to explain the parabolic pattern the welfare losses take on over the life-cycle. Finally, a sensitivity analysis of the results is performed, where some of the benchmark parameter values are altered. This causes the absolute welfare losses to shift. However, the earlier found parabolic pattern remains the same, meaning that the main result of this thesis is that optimally hedging against interest rate risk in a life-cycle is most important just before retirement.
Keywords Interest rate risk, Stochastic interest rate, Hedging, Life-cycle fund, Target date fund, Fixed premium, Variable premium, Variable pensions, Optimal portfolio choice, Optimal consumption choice, Welfare loss
Preface vi
1 Introduction 1
2 Optimal life-cycle model 4
2.1 Assumptions . . . 4
2.2 Financial Market . . . 5
2.3 Wealth accumulation . . . 7
2.4 Optimization problem . . . 8
2.5 Benchmark parameter values . . . 8
3 Optimal life-cycle contract with interest rate risk and equity risk 10 3.1 Optimal consumption path . . . 10
3.1.1 Wealth development . . . 11
3.2 Optimal life-cycle . . . 12
3.2.1 Optimal portfolio strategy in terms of total wealth . . . 12
3.2.2 Graphical results total wealth . . . 13
3.2.3 Optimal portfolio strategy in terms of financial wealth . . . 14
3.2.4 Graphical results financial wealth . . . 15
4 Data life-cycle funds 18 4.1 Target date funds . . . 18
4.1.1 Background . . . 18
4.1.2 Target date fund data . . . 20
4.2 Dutch life-cycle . . . 22
5 Welfare analysis 23 5.1 Procedure . . . 23
5.1.1 General idea and assumptions. . . 23
5.1.2 Certainty equivalent . . . 24
5.2 Welfare losses benchmark parameter set . . . 24
5.2.1 Results target date funds . . . 24
5.2.2 Results Dutch life-cycle . . . 26
5.2.3 Discussion. . . 27
5.3 Sensitivity analysis . . . 28
5.4 Concluding remarks and future research . . . 31
6 Conclusion 33
Bibliography 35
Appendix 38
Preface
Before you lies my Master’s Thesis ‘The welfare effects of a suboptimal bond portfolio in a life-cycle fund’. It is the result of 4.5 months of hard work and has been written to fulfill the graduation requirements of the Master of Science in Actuarial Science & Mathematical Finance at the University of Amsterdam (UvA). I was engaged in the investigation from September 2017 to January 2018, during which I did an internship at EY Netherlands in Amsterdam. This was at the department of European Actuarial Services. Both the recent introduction of the law on variable annuities in September 2016 and the fact that all Dutch pension providers were obligated by January 1, 2018, to offer an investment strategy that allows for a variable annuity, makes this research a relevant and interesting addition to the existing literature about life-cycle funds. Something I enjoyed these last months is that I was finally able to put the learned techniques, methods and obtained programming skills during my bachelor and master eduaction into practice. During the research period, there have been both difficult as well as more easy times. I particularly found the programming part difficult, and there have been a few moments where the help of my supervisor from the UvA, dhr. dr. S. van Bilsen, turned out to be really helpful. I am therefore very grateful to him. The way he helped yet challenged me was excellent. It has been a privilege to be supervised by him. His enthousiasm, creativity, willingness to help and think along with me have been very important to me during these last months. Furthermore, he had a visible desire to create something relevant and new, and therefore it often felt like we were a team while searching for solutions to the different problems I encountered. Also, he was willing to answer my questions using email or by video calling, even when he was not in the Netherlands. Without his help, this thesis would not have been the same. I would also like to thank dhr. dr. T.J. Boonen from the UvA for being the second reader of my thesis. Furthermore, I want to thank dhr. dr. R.J. Mehlkopf for answering some of the questions I had during my research. Moreover, I would also like to thank the actuarial department of the UvA in general. They introduced me to the area of actuarial sciences and made me enthousiastic to continue my study in this workfield. Without the different actuarial courses I followed, which I mostly found very interesting, I would not have been where I am today.
Something I am beyond grateful for is the internship I got offered by EY Actuaris-sen. My colleagues at the department I had the opportunity to work with have been really supportive during the creation of my thesis. A very special thanks goes to my supervisor from EY, dhr. ing. J. Csik AAG, for all his time and effort he put into me. His willingness to proofread my thesis was really helpful. The help and advice he gave were mainly aimed at increasing the practical use of this thesis. Also, he challenged me to think about the different decisions I made during the research. Furthermore, though it was difficult to find relevant Dutch data to include in the welfare analysis, he encour-aged and helped me to find it. Dhr. H.J.F Baggen MSc AAG also helped me with this, which I am grateful for as well.
A special thanks goes to Jurijn Jongkees, who has been very supportive throughout not only this period, but also during the rest of my study. His encouragement and friendship really helped me to focus on the end result whenever I was less optimistic. Also, he was willing to proofread my thesis, which turned out to be really helpful. Dur-ing my time of studyDur-ing at the UvA, several people other have been of great importance too. First of all, I would like to thank my parents (Dick and Saskia), sisters (Amber and Bente) and brother (Wout). They have always encouraged and supported me during this time. Finally, I would like to thank the rest of my family and friends for their optimism and faith in me, during both my bachelor and master study. Their confidence in me has really paid off.
Even though I enjoyed writing this thesis, I am also glad that I will soon be able to start my career at EY Actuarissen as advisor. I hope you enjoy reading my thesis. Joris Plaatsman
Introduction
The reform of the Dutch pension system has been the subject of many discussions in the past few years. Although this discussion is still ongoing, it recently resulted in new legislation. This new legislation allows participants of defined contribution (DC) plans to receive a variable annuity after retirement and was introduced in September 2016. This is realized by investing in so-called life-cycle funds (life-cycles for short). Although this investment type is not new in the Dutch pension landscape, the recently introduced legislation entails new challenges in the construction of life-cycles. A key aspect of in-vesting in life-cycles is that the amount of risk that is taken can be tailored to the life phase and risk preferences of the pension plan participant (Jurri¨ens,2014). Previously, Dutch life-cycle funds mainly aimed at maximizing the capital at retirement age, in order to purchase a fixed annuity. The focus has now turned to providing a stream of (variable) pension benefits during retirement, while maintaining an investment portfo-lio. This thesis investigates the welfare effects of not optimally hedging against interest rate risk in a life-cycle fund. This research takes place in a setting under the new law on variable pensions.
The primary result of this thesis is that the welfare losses of hedging suboptimally take on a parabolic pattern over the life-cycle. This result is found by comparing life-cycles that are used in reality with a theoretically modeled life-cycle. The latter is derived in an economy with interest rate risk and equity risk. The results indicate that a suboptimal investment in the interest rate risk hedging part of the bond portfolio somewhere dur-ing the 10 to 20 years prior to retirement results in the largest welfare loss. Optimally constructing the hedging part of the bond portfolio is thus most important during this period. The research is performed for two settings. We show that in a setting where a freely variable premium is paid during the working life, the losses are about 28% to 36% lower than in a setting where a fixed premium is paid. Furthermore, an economy with a higher bond risk premium, higher mean reversion of the interest rate or lower risk aversion results in relatively higher welfare losses. However, the general pattern of the welfare losses does not change when altering these parameters.
Life-cycle investing has been studied for a long time. With their studies about life-time portfolio selection, Merton (1969, 1971) and Samuelson (1969) mark the start of a wide range of research about life-cycles1. The literature on life-cycle investing can be roughly divided into two parts. On the one hand, there is research in a setting with borrowing constraints, meaning that over the life-cycle no negative investments can be made (see e.g. Vila and Zariphopoulou, 1997;Cocco et al., 2005;Gomes et al., 2008).
1
For example:Cocco and Gomes(2012) include longevity risk in a life-cycle model and investigate
how this can be hedged, Kraft and Munk (2011) incorporate the rental and ownership of residential
real estate in a model for lifetime consumption over the life-cycle, and Van Bilsen (2015) analyzes
optimal consumption and portfolio choice over the life-cycle in the presence of loss aversion, internal habit formation and probability weighting.
2 Joris Plaatsman — Interest rate risk hedging in life-cycles
On the other hand, there is research without these constraints (see e.g. Merton, 1969;
Bodie et al., 1992; Brennan and Xia, 2002). The benchmark model we derive is from
the perspective of an individual in a complete market setting. However, a collective as-pect is introduced to be able to abstract away from borrowing restrictions, putting this research in the second category as described above. Bovenberg et al. (2007) argue for collective DC pension schemes and provide an analysis of how they change welfare losses by eliminating borrowing constraints. This thesis does not further extend on this sub-ject. However, a small discussion is provided that explains how this collectivity would work.
In view of the above, it is important to note that the purpose of this thesis is not to calculate the exact absolute welfare losses that arise when not investing optimally in bonds over a life-cycle nor to promote a certain life-cycle. The aim is to identify the points in the life-cycle where (relatively) the largest welfare losses arise and where extra attention should thus be paid when constructing life-cycle funds in practice, which aim for a variable annuity.
When constructing a life-cycle, two different demands play a role. On the one hand, there is the demand for picking up the equity and bond risk premia in the stock and bond market, respectively; on the other hand, there is the demand for hedging the (vari-able) pension benefits against interest rate risk using a bond portfolio. Most life-cycles used in practice take more risk (higher equity exposure) relatively in the earlier years than in the later years (first described demand). The reasoning behind this is that young people have a longer time until retirement, meaning they have a longer period of time to recover from possible losses. A relatively high equity exposure in the earlier stages of life also compensates for the relatively large amount of human capital that is to be received in the form of income, which is (mostly) considered risk-free. When nearing the retirement date, the maintained portfolio usually becomes more conservative. This way of reducing the investment risk over the course of a life-cycle is consistent with most literature about life-cycles (see e.g.De Vries and Teulings,2006;Horneff et al.,2008). In this thesis we focus on the second described demand: the demand for hedging the (variable) pension benefits against interest rate risk. Viewing life-cycles in the context of the law on variable annuities has slightly changed the purpose of the bond portfo-lio. Before the introduction of the new legislation, the bond portfolio mainly aimed at hedging a single lump sum against interest rate risk, which was to be received at the retirement age. Now, it aims to hedge the (variable) consumption stream that starts at the retirement date. The literature on the welfare effects of holding a misspecified bond portfolio is scarce. This thesis adds to the existing literature about life-cycles by analyzing these welfare effects.
We start the analysis by modeling the theoretical optimal life-cycle contract, using the life-cycle model of Brennan and Xia (2002). In their model, they balance interest rate risk, equity risk and inflation risk over the life-cycle of an individual, in order to provide for him a (variable) stream of pension benefits. This is done in a setting where an investment portfolio is also held after the retirement age. The investment portfolio he holds consists of stocks, bonds and cash. For simplicity, we abstract away from in-flation risk in this thesis. The objective is to model the optimal consumption path and to ultimately find the investment strategy underlying this consumption path.
We want to compare the theoretical optimal life-cycle with life-cycles that are offered in the market. Because of the relatively scarce amount of Dutch data available on life-cycles that include investing after retirement, only one Dutch life-cycle fund is used in
the analysis. For this reason, we look at their American counterpart: the target date fund. A target date fund is an investment life-cycle fund, aimed at providing lifelong pension benefits starting at a specific target date. They have been around for a long time in America and are very popular pension investment products.Holt(2017) reports that by the end of 2016, the total value of all assets in target date funds was approxi-mately $880 billion. Since the introduction of the new law on variable annuities, Dutch life-cycle funds have started to become very similar to the American target date funds. The ongoing discussion about the Dutch pension system makes this research very rele-vant. Goudswaard et al. (2010) argue that the recent economic crisis of 2008-2009 has shown the vulnerability of the Dutch pension system. The confidence of people in the Dutch pension system has been damaged, leading to the fact that a new system must be able to regain this trust (Sociaal Economische Raad,2015). Over the years, several new ideas have been proposed, which were analyzed by the Social Economic Counsel. The new law on variable pensions is one of the results of the ongoing reform. The previous system created a great dependency on the interest rate at the retirement age, because all money in someones pension account was then converted into a fixed annuity. This great dependency is no longer present under the new law. On January 1, 2018, all Dutch pension providers were obligated to offer an investment strategy that allows for variable pension benefits (De Vos and Jonk, 2017; Van Zanden and Treur, 2017). This means that the life-cycle funds offered in the Netherlands might still be subject to change. This thesis adds to the investigation of finding an optimal and practically feasible life-cycle. Of course, the results in this thesis are strongly dependent on the chosen parameters of the theoretically optimal life-cycle. No parameter calibration is performed, meaning that the exact magnitudes of the resulting welfare losses may not be accurate. However, as mentioned before, the aim is to get an insight in the welfare effects of hedging sub-optimally.
This thesis is structured as follows. In Chapter2, the optimal life-cycle model is derived. Chapter 3 uses this model to determine the optimal life-cycle contract under interest rate risk and equity risk. In Chapter 4, both the Dutch and American data used in the research are presented and discussed. After this, the welfare analysis is performed in Chapter 5. This chapter contains an elaboration on the procedure followed, the results of the analysis, a sensitivity analysis and a discussion of the results. Finally, this chap-ter ends with some concluding remarks about the research and some suggestions for future research. Chapter 6 presents the conclusion and thus ends the thesis. The Ap-pendix contains some of the more complicated derivations together with some figures and tables.
Chapter 2
Optimal life-cycle model
This chapter describes the optimal life-cycle model under interest rate risk and equity risk. We use the model proposed by Brennan and Xia(2002). In their paper “Dynamic Asset Allocation under Inflation”, they develop a model that optimizes the utility of a finite-horizon investor. Chapter 3 uses this model to derive the optimal life-cycle con-tract, for certain benchmark parameter values we elaborate on in the final section of this chapter.
The motivation for the paper of Brennan and Xia (2002) is that, given a certain value for the risk aversion parameter γ, they explicitly solve for the optimal duration of the bond portfolio over the life-cycle. This is an important feature, since we want to com-pare this optimal duration with the durations of the bond portfolios held in life-cycles offered in the market. As the title of their paper already suggests, inflation risk is also taken into account in their research. This is not incorporated in our model. We solely look at interest rate risk and equity risk, and thus abstract away from inflation risk. In Section 2.1, we discuss the assumptions that underlie the model. Section 2.2 elabo-rates on the financial market in which the model opeelabo-rates. The financial market consists of all assets in which an individual can choose to invest. In Section 2.3, we derive the dynamic equation which is satisfied by the wealth over the life-cycle. Section2.4presents the optimization problem. Finally, in Section 2.5, we show and justify the parameter values we take for the benchmark portfolio.
2.1
Assumptions
Similar to Brennan and Xia (2002), we assume a finite-lived individual who takes part in a (collective) DC pension plan. Note that we mention the collectivity of the plan, in order to get rid of the borrowing restrictions as discussed in Chapter 1. The individual starts saving for his pension at time t = 0, retires at time t = TR, and dies at time
t = TD. This means that we look at a period of TD years. Because the date of death
is known a priori, we do not take macro longevity risk into account. Furthermore, we assume that micro longevity risk is diversified away from by of the collectivity of the pension fund the individual is participating in. In the first TR years, the individual
re-ceives an income of Ytfor every t ∈ [0, TR]. Every year he saves a certain fraction of his
income for his pension (the premium); the remaining part he uses for consumption (ct).
This premium he invests in a certain way in a stock, bond and cash.
We distinguish two different premium settings in our research. In the first setting, the premium the individual pays is variable, meaning he can decide for himself every period how much of his income he saves and how much consumes. This makes the premium part of the optimization problem. In the second setting, the premium is known a priori,
and set in a way that is in accordance with the Dutch fiscal rules for maximal contri-butions to a DC scheme. By making this distinction, we can observe what part of the resulting welfare losses between the benchmark model and what is done in practice is caused by not being able to freely alter the premium contribution during the working life. See Section 2.3 and 2.4 for more information on how the theoretical model differs between both premium settings.
The objective of our individual is to maximize his total lifetime utility by choosing a certain savings and consumption rate, and by balancing the ratio between the stock, bond and cash in which he invests (see Section 2.2). He can adjust this ratio continu-ously. We assume his preferences exhibit constant relative risk aversion. Because of this, we use the CRRA utility function to value a certain consumption stream. It is defined by: u(ct) = ( 1 1−γc 1−γ t , if γ ∈ (0, ∞)\{1} log ct, if γ = 1 , (2.1)
where γ is the risk aversion parameter. See Section 2.5 for the benchmark value we assume for γ.
2.2
Financial Market
As mentioned above, we assume that the financial market in which the individual can invest consists of three assets: a stock, a bond and cash. The latter is considered to be a risk-free asset. This means that the economy we consider has two risk factors: the short-term interest rate rt and the stock price index St. The formulas given below show
how the values of all assets in the economy evolve over time. They jointly determine the return the individual receives each period from his investments.
Following Brennan and Xia (2002), the short-term interest rate rt in our model
fol-lows an Ornstein-Uhlenbeck process, also known as the Vasicek model (Vasicek,1977): drt= κ(¯r − rt)dt + σrdZtr, r0 > 0 given. (2.2)
The above is a mean reverting process. In equation (2.2), ¯r is the long-term mean level, κ is the speed of mean reversion, σrdetermines the volatility of the interest rate, and Ztr
represents a standard Brownian motion. The latter means that dZtr ∼ N (0, dt). Using Itˆo’s lemma (Ito, 1944) to determine the dynamics of rteκt, we derive the following
equation for rt (see Appendix A):
rt= r0e−κt+ ¯r(1 − e−κt) + σr
Z t
0
e−κ(t−s)dZsr. (2.3) As can be seen in the above equation, both r0 and ¯r affect the future value of the
short-term interest rate at time t. The influence of ¯r increases as t increases, and the influence of r0 decreases simultaneously. The same kind of pattern can be seen in the final term
of (2.3): shocks that are further in the past (closer to t = 0 than to t = t) have a smaller influence on rt than shocks that have occurred more recently. This means that shocks
in the interest rate are fading out over time.
The stock price index follows a geometric Brownian motion (see e.g. Hull, 2012, p. 282), which is defined as:
dSt
St
6 Joris Plaatsman — Interest rate risk hedging in life-cycles
In the above, rtis the short-term interest rate as defined in (2.3), σs is the volatility of
the stock returns, λs is the equity premium per unit of risk, and again, Zts represents
a standard Brownian motion. λs is also known as the Sharpe ratio (Sharpe,1966). We
assume that dZtrand dZtshave a correlation that is equal to ρsr. Stsatisfies the following
equation (see Appendix B):
St= S0e Rt 0(rv+σsλs)dv+σs Rt 0dZ s v. (2.5)
Finally, we define Pth as the value at time t of a (zero-coupon) bond with a payoff of 1 at maturity date t + h. The latter means that h represents the (remaining) time to maturity. Pth is equal to:
Pth = eA(t,t+h)−D(t,t+h)rt, (2.6) with A(t, T ) = [D(t, T ) − (T − t)] ¯ r − λr σr κ − σ2r 4κ3 2κ (D(t, T ) − (T − t)) + κ 2D2(t, T ) ; (2.7) D(t, T ) =1 κ 1 − eκ(t−T ) . (2.8)
See Appendix A of Brennan and Xia(2002) for the derivation of the above three equa-tions. In equation (2.7), λr represents the market price of risk associated with the
stochastic innovations in the interest rate. Note that the value of D(t, T ) only depends on the (negative value of the) remaining time to maturity T − t. The dynamics of Pth
(see Brennan and Xia,2002) are described by the following equation:
dPth Ph
t
= (rt− D(t, t + h)σrλr) dt − D(t, t + h)σrdZtr. (2.9)
As can be seen easily, the bond risk premium is given by: Et dPh t Pthdt − rt= −D(t, t + h)σrλr> 0. (2.10)
Since estimates of λr are negative (see e.g. Stanton, 1997), and D(t, T ) and σr are
al-ways positive, bond risk premia are also positive. Also, D(t, t + h) increases (though at a declining rate) with the horizon h, which means that the bond risk premia of long-term bonds exceed those of short-long-term bonds. Furthermore, D(t, t + h) decreases with κ (speed of mean reversion of rt), so that bond risk premia are small if κ is large. These
facts result in the conclusion that D(t, t + h) can be viewed as the interest rate duration of the bond.
Because the cash that is put in the bank account can be viewed as a bond with duration h = 0, the dynamics of the bank account can easily be derived from dPt0
P0 t
. Defining Bt
to be the value of the bank account, we get: dBt Bt = dP 0 t P0 t = rtdt. (2.11)
Performing a similar derivation as the one for Stin AppendixAresults in the following
value of Bt:
Bt= B0e Rt
0rvdv. (2.12)
The focus in this thesis is on hedging against interest rate risk in life-cycles. Note that the bond price in equation (2.6) is directly related to the interest rate rtand thus also
to the unexpected shocks in it (dZtr, see equation (2.9)). Because of this, we use bonds in our model to match the interest rate sensitivity of the pension benefits (liabilities) to the interest rate sensitivity of the investment portfolio (assets).
2.3
Wealth accumulation
We distinguish two different kinds of wealth: total wealth (Wt) and financial wealth
(Ft). These two are related as follows:
Wt= Ht+ Ft, (2.13)
where Htequalizes the human capital at time t. As mentioned above, we consider both
the setting where the pension premium is variable and where it is not. In the variable premium setting, human capital is equal to the discounted value of future income. For 0 < t < TR, the individual decides for himself how much of his income he invests in
his financial portfolio, while the remaining part is used for consumption. When the in-dividual faces a fixed premium, human capital is equal to the discounted value of all future premium contributions (Gollier,2005, Chapter 4). This means that in this case, consumption before retirement is fixed as well (which equals income minus premium) and is thus not a part of our optimization problem (see Section 2.4 for more on this). In both cases, we have that Ht= 0 for t > TR.
Financial wealth Ft is equal to the sum of all premiums paid up to time t, plus all
returns and losses gained from investing these premiums each year in the financial mar-ket. In both premium settings, Ft starts at a value of 0, which means that W0 = H0.
When the individual faces a variable premium, the latter is equal to: H0 = E0 Z TR 0 MvYvdv , (2.14)
while in the case of a fixed premium, it is equal to: H0 = E0 Z TR 0 MvpvYvdv . (2.15)
In the above two equations, Mvis equal to the pricing kernel (also known as the
stochas-tic discount factor) of the financial market in which the individual invests. pv is equal
to the premium contribution both the individual and his employer contribute to the DC pension plan, expressed as a percentage of income Yv. See Section2.5for the values we
assume for pv and Yv.
As time passes by, human capital is converted into financial wealth according to the investment and consumption decisions the individual makes. We have:
Wt= ηStSt+ ηtPPt+ ηtBBt, (2.16)
where ηSt, ηtP, and ηBt represent the amount of stocks, bonds, and bank accounts the individual possesses at time t (in terms of total wealth), respectively. Another way to write Wt is as follows: Wt= W0+ Z t 0 ηSvdSv+ Z t 0 ηPvdPv+ Z t 0 ηvBdBv− Z t 0 cvdv. (2.17)
Equation (2.17) states that total wealth at time t is equal to the starting capital W0,
increased by the gains from investing in the stock, bond, and bank account, minus the cumulative consumption, all up to time t. Combining equation (2.17) with equation
8 Joris Plaatsman — Interest rate risk hedging in life-cycles
(2.4), (2.9), and (2.11), results in the following dynamic budget constraint (see Appendix
C):
dWt= rt+ ωtSσsλs− ωtPB(t, t + h)σrλr Wtdt+
ωtSWtσsdZts− ωtPWtB(t, t + h)σrdZtr− ctdt.
(2.18)
In the above equation, ωtS and ωtP are the fractions of total wealth invested in the stock and bond at time t, respectively. The fraction of total wealth invested in the bank account at time t is ωBt = 1−ωSt−ωP
t . As mentioned before, in the fixed premium setting
both the premium and consumption before retirement are not part of the optimization problem. This translates to ct = 0 for 0 < t < TR. Note that this does not mean that
the individual does not consume before retirement.
2.4
Optimization problem
Since we make a distinction between fixed and variable premiums, we have two different optimization problems. In both cases, equation (2.18) serves as the dynamic budget constraint (where ct = 0 for 0 < t < TR in case of fixed premiums). As mentioned
in Section 2.1, our individual aims to optimize his expected total lifetime utility, by choosing a certain consumption and investment strategy, ct and ωt = (ωSt, ωPt , ωtB),
respectively. Because people tempt to evaluate consumption at different points in time differently, we include a subjective rate of time preference in the model, denoted by δ.
Brennan and Xia (2002) exclude this parameter.
In the setting with a variable premium, the individual maximizes:
maximize cv E0 = " Z TD 0 e−δv c 1−γ v 1 − γdv # , (2.19)
such that both the dynamic budget constraint given in equation (2.18) is satisfied and that ct is financed by a feasible trading strategy with initial investment W0. In the
setting with a fixed premium, the individual maximizes:
maximize cv E0 = " Z TD TR e−δv c 1−γ v 1 − γdv # , (2.20)
such that again both the dynamic budget constraint given in equation (2.18) is satisfied and that ct is financed by a feasible trading strategy with initial investment W0.
2.5
Benchmark parameter values
This section elaborates on the parameter values we use in the above life-cycle model, in order to obtain the benchmark model. We assume that our individual starts saving at the age of 25, retires at the age of 68, and dies at the age of 88. This means that TR and TD are equal to 43 and 63, respectively. We choose to use a retirement age of
68, because the retirement target age has been changed to 68 since January 1, 20181. Although not every employer’s pension adjusts to this change immediately, sooner or later it probably will.
σs and λs are set to σs = 0.2 and λs = 0.2. This is in line with the advice of the
1
See Besluit van 21 december 2016 tot wijziging van enige wetten en uitvoeringsbesluiten op het
Advies Commissie Parameters(2014). For σr, we choose a value of σr = 0.015, and we
take λr in a way that the resulting fraction of total wealth invested in the speculative
part of the bond portfolio (aimed at picking up the bond risk premium) is around 20%. This results in a value of λr= −0.1. For ρsr, we assume a value of ρsr= 0.
The speed of mean reversion of the interest rate, κ, is directly related to the amount of time it takes until the current interest rate determines the future interest rate for less then 50%. We call this the half time (‘halfwaardetijd’ in Dutch), and attach the symbol η to it. κ and η are related by:
1 2 = e
−κη (2.21)
For the half time, we take a value of η = 5, resulting in κ = 0.1386.
Because similar literature about life-cycles usually takes a value for γ around 5 (see
e.g.Brennan and Xia,2002;Munk and Sørensen,2010;Chai et al.,2011), we take γ = 5
in our benchmark model. For δ, we take a value of δ = − log(0.97) and, finally, for h (the maturity of the bond in which the individual invests), we take a value of h = 20. This last value is not really relevant, though. Other values would yield the same results. Only the duration of the bond portfolio is important, which is the result of a combination of both the maturity of the bond and exposure to the bond. Adjusting the maturity leads at the same time to a change in the exposure, resulting in the same optimal duration. As starting values for the short-term interest rate and stock price index we take r0= 0.01
and S0 = 1, respectively, and we set the labor income Ytequal to Yt= 1 for 0 < t < TR.
The latter means that we assume a constant, non-stochastic value for the income. As can be seen in Chapter3 (in equation (3.1) and (3.3)), the optimal consumption path does not change relative to income when the income Yt would vary over time. Moreover, the
optimal life-cycle pattern also does not change. For this reason, we assume a constant value for Yt. For pt(the premium contribution), we use the values stated in Voorlopige
aangepaste premiestaffels bij pensioenrichtleeftijd van 68 jaar (Vraag & Antwoord
17-002 d.d. 290917)(2017) for the fiscal maximum contribution in DC-pension plans in the
Netherlands. See Table D.1 of Appendix D. While in reality, someone saves for spouse pension as well as old age pension, we exclude it for simplicity. This means that Table
D.1 only covers old age pension. We assume that the total pension contribution of the individual and his employer for a given age (pt) is equal to its maximum stated in Table D.1.
Chapter 5 compares the resulting life-cycle contract of Chapter 3, where the above parameters are used, with the life-cycles observed in practice, which are discussed in Chapter 4. The welfare losses resulting from this comparison will undergo a sensitivity analyses, where we will change the above parameter values. This way, we can see what the impact is of, for example, a different value of the risk aversion parameter γ.
Chapter 3
Optimal life-cycle contract with
interest rate risk and equity risk
This chapter gives the resulting optimal solutions of the maximization problems, stated in (2.19) and (2.20). We also present figures supporting these solutions, in which the benchmark parameter values of Section2.5are used (e.g. resulting optimal consumption paths and life-cycles). For this, we thus build upon the model presented in Chapter 2. We are in particular interested in the optimal exposure to the bond (for the benchmark maturity time), to ultimately compare this with the exposure to bonds in the life-cycles we discuss in Chapter 4.
Section3.1 presents the optimal consumption path over the life of an individual facing interest rate risk and equity risk. This section also discusses the mean wealth develop-ment of the individual. Section 3.2gives the optimal life-cycle, underlying this optimal consumption path.
3.1
Optimal consumption path
The derivation of the solution to (2.19) (variable premium) is given in Brennan and Xia (2002), so we refer to their paper for the proof. The optimal consumption c∗t for 0 < t < TD, when the individual is facing a variable premium, is given by:
c∗t = Q(0, TD)−1(Mt)−1/γW0e−δt/γ, (3.1) where Q(t, T ) = Z T t exp 1 − γ
γ (D(t, u)rt+ a(t, u))
e−δu/γdu. (3.2) Note that D(t, T ) has been defined in equation (2.8). We refer to equation (E.12) in AppendixEfor the definition of a(t, T ). Furthermore, in equation (3.1), W0 is equal to
the present value of all future income, as stated in equation (2.14).
Since Brennan and Xia (2002) do not discuss the maximization problem in (2.20), the proof of equation (3.3) is included in Appendix E. When having fixed premiums, the optimal consumption c∗t is:
c∗t =
(1 − pt)Yt for 0 < t < TR
Q(TR, TD)−1(Mt)−1/γW0e−δt/γ for TR< t < TD
, (3.3)
where Q(TR, TD) is as in equation (3.2), with D(t, u), rt, and a(t, u) replaced by D(0, u),
r0, and a(0, u), respectively. Furthermore, W0 is equal to the present value of all future
premium contribution, as stated in equation (2.15). 10
30 40 50 60 70 80 Age 0.6 0.7 0.8 0.9 1 1.1 1.2 Consumption level
(a) Variable premium
30 40 50 60 70 80 Age 0.6 0.7 0.8 0.9 1 1.1 1.2 Consumption level (b) Fixed premium
Figure 3.1: Mean optimal consumption path when facing a variable premium (figure a) or a fixed premium (figure b). All parameter values are chosen in accordance with Section
2.5.
Figure 3.1a and 3.1bshow the mean optimal consumption paths of equation (3.1) and (3.3), respectively. The benchmark parameter values defined in Section 2.5 have been used to create these figures. Remember that by a ‘fixed premium’ in Figure 3.1b, we do not mean that it stays at the same level. Rather, we mean that its value is known a priori.
Looking at the above figures, we see that the mean optimal consumption path in both premium settings increases on average after retirement. This is due to the term (Mt)−1/γ
in both equation (3.1) and (3.3), where Mtrepresents the stochastic discount factor (see
AppendixEfor more on Mt). The higher the value of t, the lower the value of Mt,
mean-ing that (Mt)−1/γ increases over time. The increasing pattern after retirement in Figure 3.1 is thus the result of positive results (on average) of the investment portfolio in the financial market.
In contrast, the consumption before retirement differs significantly between both fig-ures. This is due to the fact that in the setting with a fixed premium, the consumption before retirement is fixed to (1 − pt)Yt, which slowly decreases over time. In the
set-ting with a variable premium, the consumption pattern is influenced by the value of Mt in the same way as explained above. See AppendixFfor plots of several individual
consumption paths for both types of premium schemes.
3.1.1 Wealth development
As was described in Section 2.3, we distinguish three different kinds of wealth. Figure
3.2a and 3.2bdepict the mean wealth development for both the variable premium and fixed premium setting, respectively. It can be seen in both figures that human capital declines over the period t ∈ [0, TR] (= age ∈ [25, 68]) from W0 to 0. The reason behind
this is that the amount of future labor income (variable premium) and future premium (fixed premium) declines over the life of the individual.
In the variable premium setting, both total wealth and financial wealth are affected by an upward and a downward force. Every year, the wealth is increased by the re-turn from the investment portfolio (upward force on average), but the individual also consumes a part of his wealth (downward force). The absence of this downward force
12 Joris Plaatsman — Interest rate risk hedging in life-cycles 30 40 50 60 70 80 Age 0 5 10 15 20 25 30 35 Value of Capital Total Wealth Human Capital Financial Wealth
(a) Variable premium
30 40 50 60 70 80 Age 0 2 4 6 8 10 12 14 Value of Capital Total Wealth Human Capital Financial Wealth (b) Fixed premium
Figure 3.2: Mean total wealth, financial wealth, and human capital development when facing a variable premium (figure a) or a fixed premium (figure b). All parameter values are chosen in accordance with Section2.5.
before retirement in the fixed premium setting, explains the different patterns in the developments of total wealth and financial wealth observed in Figure 3.2a and Figure
3.2b. Moreover, note the different scaling on the vertical axes, resulting from the differ-ent ways human capital is defined and the fact that in the second figure, consumption only takes place after retirement. Connecting Figure 3.2 with Figure 3.1 also explains why the wealth in the fixed premium setting increases less than in the variable premium setting: a smaller consumption (after retirement) needs to be financed, so it needs to increase less.
3.2
Optimal life-cycle
The objective now is to find the optimal portfolio strategies, which finance the optimal consumption paths as shown in equation (3.1) and (3.3).Brennan and Xia(2002) provide the proof for the variable premium setting in terms of total wealth, but they do not discuss the optimal portfolio strategy in terms of financial wealth. For this reason, we split up the solution in two parts. First, we discuss the optimal portfolio strategies (variable and fixed premium) in terms of total wealth, and after that, we derive the solutions in terms of financial wealth.
3.2.1 Optimal portfolio strategy in terms of total wealth
As noted above, the proof of the optimal portfolio strategy for the variable premium setting is provided in the paper by Brennan and Xia (2002), so we just present the solution here. Define ω∗t = (ωtS∗, ωP ∗t , ωB∗t ) as the optimal fractions in terms of total wealth invested in the stock, bond and bank account (cash) at time t, respectively. The optimal portfolio strategy ω∗t for the variable premium setting is given by:
ωt∗= −1 γ φs σs , 1 −D(t, t + h) 1 − γ γ Dˆvar(t, TD) − φr σrγ , 1 − ωtS∗− ωtP ∗ , (3.4)
where φs and φr denote the constant loadings on the stochastic innovations in the
economy (dZtsand dZtr, see AppendixE). Note that, since D(t, T ) only depends on the value of t − T , and since we take a constant value for h (the chosen maturity of the bond
in which we invest), −D(t, t + h) is a constant. Furthermore, ˆDvar(t, TD) is defined as:
ˆ
Dvar(t, TD) =
Z TD
t
e−δu/γexph1−γγ (D(t, u)rt+ a(t, u))
i RTD t e−δv/γexp h 1−γ γ (D(t, v)rt+ a(t, v)) i dv D(t, u)du, (3.5)
which is a weighted average of D(t, u) (with u ∈ [t, TD]). a(t, T ) is defined in Appendix E in equation (E.12).
As can be seen in equation (3.4), the optimal fraction invested in stocks remains constant over the life-cycle. Furthermore, ωP ∗t consists of two parts: a constant part
φr
D(t,t+h)σrγ
and a part that changes over the life-cycle D(t,Tˆ D)
−D(t,t+h)
. The constant part represents the speculative bond portfolio, that aims to pick up the bond risk premium (see equation (2.10)). The second part of the bond portfolio is the part that has the largest correla-tion with the interest rate rt, meaning it tries to hedge the lifetime consumption stream
against interest rate risk (Brennan and Xia, 2002). We call this the hedging part of the bond portfolio. Note that an infinite risk-averse individual (γ → ∞) would invest all of his (total) wealth in this hedging part of the bond portfolio, trying to hedge as much as possible of his consumption stream against interest rate risk, without taking any additional risks.
The optimal portfolio strategy when having a fixed premium is given by: ω∗t = −1 γ φs σs , 1 −D(t, t + h) 1 − γ γ Dˆfix(t, TD) − φr σrγ , 1 − ωtS∗− ωP ∗t , (3.6)
where ˆDfix(t, TD) is defined as:
ˆ Dfix(t, TD) = Z TD max(TR,t) e−δu/γexp h 1−γ
γ (D(t, u)rt+ a(t, u))
i RTD max(TR,t)e −δv/γexph1−γ γ (D(t, v)rt+ a(t, v)) i dv D(t, u)du. (3.7) The derivation of equation (3.6) and (3.7) is provided in Appendix G. Note that the structure of the portfolio strategy in equation (3.6) is very similar to the optimal port-folio in equation (3.4). Looking at the bond portfolio, we again recognize a speculative and a hedging part. Important to mention is that the hedging part of the bond portfolio only tries to hedge the pension benefits against interest rate risk, not the entire lifetime consumption. The latter is the case in the variable premium setting.
3.2.2 Graphical results total wealth
Figure 3.3a and 3.3b show the optimal portfolio strategies of equation (3.4) (variable premium) and (3.6) (fixed premium), respectively. These figures display the mean opti-mal fraction of total wealth to invest in the different asset classes over the life-cycle of the individual.
In the setting where the individual faces a variable premium, the fraction invested in the bank account is negative for about 32 years, while in the fixed premium setting, it is negative for approximately 40 years. The cause for this difference is found in the fact that the hedging part of the bond portfolio is bigger in the fixed premium setting. Looking at equation (3.4) and (3.6), we see that the hedging part is directly influenced by ˆDvar(t, T ) and ˆDfix(t, T ). As discussed before, these terms can be interpreted as the
weighted averages of the duration of the (remaining) consumption stream. In the fixed premium setting, the (total) wealth is only used to finance the pension benefits starting
14 Joris Plaatsman — Interest rate risk hedging in life-cycles 30 40 50 60 70 80 Age -40% -20% 0% 20% 40% 60% 80% 100% 120% 140%
Proportion of Total Wealth
Bonds Hedge (maturity 20) Bonds Speculative (maturity 20) Stocks
Cash (bank account)
(a) Variable premium
30 40 50 60 70 80 Age -40% -20% 0% 20% 40% 60% 80% 100% 120% 140%
Proportion of Total Wealth
Bonds Hedge (maturity 20) Bonds Speculative (maturity 20) Stocks
Cash (bank account)
(b) Fixed premium
Figure 3.3: Mean optimal life-cycle over the life of the individual in terms of the fraction of total wealth, when facing a variable premium (figure a), or a fixed premium (figure b). The investment portfolio consists of a stock, bond (maturity 20) and cash. All parameter values are chosen in accordance with Section 2.5.
at age 68, while in the setting with a variable premium, the (total) wealth is used to finance the consumption over the entire life of the individual. This causes the weighted average duration of the consumption stream to be bigger in the fixed premium setting, explaining why the hedging part of the bond portfolio is bigger in this setting.
Note the kink in Figure3.3bat the retirement age of 68. Starting from this point, both figures are the same. This is because in the fixed premium setting, the consumption stream (pension) starts at the age of 68. However, despite the fact that the relative investment portfolios are the same from this moment on, they do differ in absolute terms. Furthermore, because at the end of the life of the individual there are no more payments to be made, the optimal fraction of total wealth to invest in the hedging part of the bond portfolio goes to 0% in both figures (no more payments need to be hedged at the end of the life-cycle). Lastly, because both figures assume that the total discounted value of lifetime income (variable premium) or lifetime premium (fixed premium) is received at the age of 25, a constant amount of risk is taken during the whole life-cycle. This amount of risk taken equals the sum of the amount invested in the stock and the speculative part of the bond portfolio. As we will see later on, this pattern changes when expressing the life-cycle in terms of financial wealth.
3.2.3 Optimal portfolio strategy in terms of financial wealth
In this section, we make the step from the optimal portfolio strategy in terms of total wealth to the optimal portfolio strategy in terms of financial wealth. Define
e ω∗t = (ωe S∗ t ,ωe P ∗ t ,ωe B∗
t ) as the optimal fractions in terms of financial wealth invested
in the stock, bond and bank account (cash) at time t, respectively. The basic idea of the derivation we perform to find ωe∗t is to find the dynamics of human capital. We use the latter to derive the dynamics of financial wealth, using the following relation:
Wt= Ht+ Ft ⇒ dWt= dHt+ dFt. (3.8)
Using the dynamics of financial wealth, we can eventually determineωe
∗
t. See Appendix H for the complete derivation. We find the following value for ωet∗, when the individual faces a variable premium:
e ωtS∗=ωtS∗ 1 +Ht Ft ; (3.9) e ωtP ∗=ω P ∗ t WtD(t, t + h) − R43 t D(t, s)e µ+12σ2 Ytds FtD(t, t + h) ; (3.10) e ωB∗t =1 −ωetS∗−ωetP ∗. (3.11)
In the above equations, ωS∗t and ωP ∗t refer to the optimal portfolio strategy in terms of total wealth, as defined in equation (3.4). Moreover, µ and σ2 in equation (3.10) are the mean and variance of a normally distributed stochastic variable. See Appendix H
for more details.
The optimal fractions invested in the stock and cash, equation (3.9) and (3.11), re-spectively, remain the same in the fixed premium setting. However, the optimal fraction invested in the bond does change a little bit. The premium rate ptis added to the model,
resulting in: e ωP ∗t = ω P ∗ t WtD(t, t + h) − R43 t D(t, s)e µ+12σ2 Ytptds FtD(t, t + h) . (3.12)
Note that we can no longer make a distinction between the hedging and speculative part of the bond portfolio in both equation (3.10) and (3.12).
3.2.4 Graphical results financial wealth
Relative life-cycles
Figure 3.4a and 3.4b present the optimal fractions of financial wealth to invest in the different assets over the life of the individual, in order to obtain the optimal consump-tion streams of equaconsump-tion (3.1) and (3.3), respectively.
(a) Variable premium (b) Fixed premium
Figure 3.4: Mean optimal life-cycle over the life of the individual in terms of the frac-tion of financial wealth, when facing a variable premium (figure a) or a fixed premium (figure b). The investment portfolio consists of a stock, bond (maturity 20) and cash. All parameter values are chosen in accordance with Section 2.5.
The above life-cycles look much more similar than the ones in terms of total wealth do (see Figure3.3). A pattern that both of the above life-cycles exhibit is that in the first few years, a lot of risk is taken through a large investment in the stock. This invest-ment gets smaller over the life-cycle. The bond portfolio has the opposite structure: it
16 Joris Plaatsman — Interest rate risk hedging in life-cycles
starts small, after which it increases to its maximum. After this point, it decreases again. In terms of total wealth, the amount of risk taken through an investment in the stock is constant over the entire life-cycle, but, as we can see, in terms of financial wealth it is not. This large stock portfolio is created to compensate for human capital, which is to be received in the form of income or premium. Human capital is considered a risk-free bond, and the (relatively) large stock portfolio compensates for this in order to obtain the optimal amount of risk. Of course, in reality human capital is not completely risk-free, but we abstract away from the possibility of a change in income Yt. Since human
capital declines over the life of the individual (see Figure 3.2), the optimal fraction of financial wealth invested in the stock also decreases over the life-cycle, until it eventually coincides with the optimal stock portfolio of the life-cycle in terms of total wealth. In fact, after the pension age of 68, both life-cycles in terms of financial wealth are entirely equal to their counterparts in Figure3.3aand3.3b, because human capital is equal to 0 from this moment, meaning that Wt= Ft for t > TR. In both figures we recognize this
moment by the kink at retirement age.
A noticeable difference between Figure 3.4a and 3.4b is the following. For most of the time, the bond portfolio is much bigger compared to the stock portfolio in Figure 3.4b
than in Figure3.4a. This pattern originates from the same fact which we explained ear-lier for the life-cycle in terms of total wealth. Because the weighted mean duration of the consumption stream is bigger before retirement in the setting with fixed premiums, the bond portfolio, which tries to match this duration, is also bigger in this case. Remember that, despite the fact that Figure3.4a and3.4b look very similar, the resulting (mean) optimal consumption paths and wealth developments do differ significantly.
Nominal life-cycles
Something that may disturb when looking at the above life-cycles is that, during the first 40 years, a substantial fraction of financial wealth needs to be borrowed in order to finance the optimal stock and bond portfolio. This could lead to the conclusion that this life-cycle is infeasible in reality, because not all pension funds are in the position to borrow large amounts of money. However, the presented life-cycles may be misleading regarding this negative investment in cash. Because the financial wealth of the indi-vidual is much smaller during the earlier years of his life than later on, the amount of money that actually needs to be borrowed is less dramatic than it may look like. Furthermore, as mentioned before we assume that our individual is participating in a collective DC pension plan. When we also assume that there is an equal amount of participants of each specific age, the older generations could (partially) invest their money in the younger generations. The interest that the younger people would otherwise pay to the bank, they would now pay to the older people.
To illustrate the above concepts, we present two figures that show the life-cycles of Figure3.4aand3.4bin nominal terms. By looking at the figures below, the possibilities of the collectivity described above become more insightful. Figure 3.5ais for a variable premium, and Figure3.5bis for a fixed premium. Note the different scaling on the ver-tical axes.
Looking at these life-cycles, it becomes clear what the effect of collectivity would be. The people older than (approximately) 65 can ‘invest’ their money in the investment portfolios of the younger people, leaving a smaller amount of money that would actually need to be borrowed by the younger people. On top of this, a (payer) swap derivative also provides an additional solution to the problem of borrowing money. Since a payer
swap is actually equal to a bond (long) and cash (short), this could replace the actual investment in bonds and cash. This idea is originated in the fact that, in the presented life-cycles in this chapter, the individual is not aiming at a particular investment in a specific asset. Rather, he is searching for a particular return and interest rate sensitivity. Since a payer swap has the same return and interest rate sensitivity as an investment in both a bond and cash, this derivative could replace the actual investment in these two assets. This thesis does not extend further on the subject of the feasibility of the presented life-cycles.
(a) Variable premium (b) Fixed premium
Figure 3.5: Mean optimal life-cycle over the life of the individual in nominal terms, when facing a variable premium (figure a) or fixed premium (figure b). The dashed blue line represents the value of financial capital. The investment portfolio consists of a stock, bond (maturity 20) and cash. All parameter values are chosen in accordance with Section
Chapter 4
Data life-cycle funds
This chapter elaborates on several life-cycle funds that are used in practice. We eventu-ally use these life-cycles in Chapter 5 for the welfare analysis. The financial market of the optimal model we developed in Chapter 2 consists of a stock, a bond and cash. In order to get reliable results when comparing real life-cycles with the theoretical model, we need data of life-cycles that invest in these same three assets. Furthermore, the aim of these life-cycles must be to provide a variable annuity instead of a fixed annuity. Un-fortunately, the data for the various pension plans in the Netherlands satisfying these restrictions is not easily accessible. On top of this, since the law on variable annuities is relatively new, it is likely that Dutch pension products that comply with the new legislation are still subject to change1. However, we did manage to obtain the data for one Dutch life-cycle, which is offered by a subsidiary company of a Dutch bank. For the rest of the data, we focus on the American life-cycle market, where it has been possible for a long time to receive a variable annuity after retirement.
In America, there is a very alive market in life-cycle funds, which are also known as target date funds. A target date fund is a life-cycle investment product, which aims at providing a variable pension benefit, starting at the target date. The target date is usually equal or close to the retirement date of the holder, and is specified in the name of the target date fund. Section4.1further elaborates on the American target date fund market and also provides a description of the specific target date funds that are used in the welfare analysis. Section4.2discusses the Dutch life-cycle fund.
4.1
Target date funds
Target date funds have a very commonplace in the American retirement landscape. To see this, we first need a little insight into the American pension system. This will be given in Section4.1.1, together with some general information about target date funds. Section4.1.2provides an overview of the several target date funds that are used in the welfare analysis of Chapter5.
4.1.1 Background
Where pension systems usually consist out of three pillars (the state pension, the sup-plementary employer-sponsored pension plans and the private individual savings), the
Investment Company Institute (2017) states that the American pension system is best
thought of as a five-layer pyramid. Arranged from most important to least important, these layers are: social security, home ownership, employer-sponsored retirement plans (DB and DC plans), individual retirement accounts (IRAs), and other assets. The third
1
On January 1, 2018, all Dutch pension providers were obligated to offer an investment strategy that
presorts for a variable pension benefit (De Vos and Jonk,2017;Van Zanden and Treur,2017).
and fourth layer have become more important over the years: approximately eight out of ten households that were close to their retirement date in 2013, had money invested in either DB or DC plans, or IRAs. Concerning the ratio between DB and DC plans, there is a shift visible in the American pension system from DB to DC contracts (Investment
Company Institute,2017), just as in the Netherlands (De Nederlandsche Bank,2017).
Focusing on the third layer, employers in America can choose to sponsor a 401(k) plan. A 401(k) plan is a tax-qualified DC pension account, and is defined in subsection 401(k) of theInternal Revenue Code(2017). When employers choose to sponsor a 401(k) plan, it means that the retirement savings contributions that go to the employee’s DC ac-count are deducted from his or her paycheck before taxation2. The employer has the freedom to decide about several features which affect the structure of the contributions to the 401(k) plan, one of which being the feature whether employees are automatically enrolled or not. Another feature employers can decide on, is which investment options to make available to the employees. For example, equity funds and bond funds are pop-ular in large 401(k) plans. Next to these, target date funds are also common investment choices: approximately three-quarters of big 401(k) plans offer target date funds. These plans offer on average nine target date funds (Investment Company Institute,2017). As explained above, a target date fund is a life-cycle investment product where a prede-termined investment allocation over time is followed. The life-cycle is based on a specific target date. After the target date, the fund provides a variable pension benefit to its holder. Usually, target date funds offered by the same insurer or pension fund are five years apart in target date. This means that people who invest in a target date fund, search for a fund with a target date that is as close as possible to their (expected) retirement date. As we will see in the next section, different target date funds have dif-ferent objectives. For example, some target date funds aim at capital growth and a high total return, while other target date funds also include capital preservation (see Sec-tion4.1.2for the definition) in their goal, both before and after the specified target date. The first target date funds entered the market in 1994 (Faassen,2017), and have shown a rapid growth in recent years. At the end of 2014, about 18% of the total assets in 401(k) plans was invested in target date funds. In contrast, this percentage was only 5% at the end of 2006 (Investment Company Institute, 2017). Something that made this growth possible, among other things, is the fact that the target date fund was included in the set of acceptable default options within 401(k) plans (Elton et al.,2015).
Since their introduction, several papers and documents have been published to give insight into target date funds and their structure. For example, Morningstar, an in-vestment research and inin-vestment management firm, publishes a document each year, that describes the target date fund landscape (see e.g.Holt and Yang,2015,2016;Holt,
2017). A research that is somewhat in line with this thesis, is the research by Spitzer
and Singh(2008). In their study, they take a closer look at the underlying life-cycles, by
comparing several fixed allocation portfolios with different target date fund classifica-tions. They find that three of these classifications have a higher probability of running out of money than the fixed allocation portfolios, leading them to conclude that a revi-sion of the asset allocation strategy is necessary.
The reason why the target date fund is so popular in America and less well-known in the Netherlands, is twofold. On the one hand, the American pension system is less generous than in the Netherlands, meaning that people in America are more
depen-2
This went to a maximum of $18000 in 2017, and $18500 in 2018 (Retirement Topics - 401(k) and
20 Joris Plaatsman — Interest rate risk hedging in life-cycles
dent on individual pension accrual. On the other hand, target date funds are relatively easy investment vehicles: people only need to choose an investment horizon, and the fund does the rest (Faassen, 2017). With the new legislation on variable annuities in the Netherlands, we see that more pension funds and insurers create pension products that are similar to the American target date funds. This is in accordance withJurri¨ens
(2014), who expects that life-cycle investment products are on the rise.
4.1.2 Target date fund data
The American target date fund market has existed for over two decades. Over the years, a wide variety of pension funds, insurers, and financial institutions have started to offer an even wider variety of target date funds, each with their own specific objective. Ac-cording to the latest overview of the target date fund market, three firms collectively hold more than 70% of the target date fund market (Holt,2017). These companies are Vanguard, Fidelity Investments, and T. Rowe Price3. Because the target date funds these three firms offer represent such a large proportion of the entire target date fund market, we use the funds they offer in our welfare analysis.
The three mentioned firms each offer different target date funds, and the specific funds we use can be subdivided in either one of the two following categories. The first cate-gory contains the target date funds that seek to provide high total return and capital appreciation4. The target date funds in the second category seek high total return, with
a secondary objective of capital preservation5. This second category is considered to be less risky than the first.
We retrieve five-yearly data on the target date funds from Morningstar, which was last updated on September 30, 2017 when the data collecting took place. Since the life-cycle data is only available for every five years, linear interpolation is used to construct the complete life-cycle. Besides investing in stocks, bonds, and cash, the target date funds we look at also make an investment in a class named other. Since it is not clear what kind of investments this class holds, and since the investments in this class are relatively small, we make the assumption that this class consists of an extra investment in stocks, bonds, and cash. This means that we divide the investment in this category over the other categories, such that the ratios between the relative investments in stocks, bonds, and cash remain equal6. Moreover, we also collect the duration of the bond portfolio at the different moments in time, because these are important in Chapter5for the welfare analysis.
In the three subsections below, the target date funds that will be used in the wel-fare analysis are shortly discussed, their underlying life-cycle is shown, and they are subdivided into one of the two previously discussed categories. The final subsection presents and discusses the average life-cycle, based on the discussed target date funds.
3
Vanguard is an American investment adviser and one of the biggest asset management firms in the world, Fidelity Investments is a multinational financial services corporation, and T. Rowe Price is a publicly owned international asset management firm.
4In the case of target date funds, capital appreciation is a rise in the value of the fund. It is considered to be a more risky approach than capital preservation, which is defined below.
5In the case of target date funds, having capital preservation as a (secondary) objective means that
the goal is to preserve capital and to prevent a loss in the fund. 6
For example: assume that the portfolio consists of the following relative portfolio weights: 50% in stocks, 30% in bonds, 17% in cash, and 3% in other investments. Dividing the 3% in other investments
proportionally over the remaining classes, results in an investment of 0.5+0.3+0.170.5 · 0.03 + 0.5 ≈ 0.515 in
stocks, 0.3
0.5+0.3+0.17· 0.03 + 0.3 ≈ 0.309 in bonds, and
0.17
Target date fund Vanguard
Vanguard offers two different target date funds. The difference between these two funds, is that they both invest in different share classes. However, since their life-cycle paths are virtually the same, and since we are not interested in the stock portfolio, we use the data from the target date fund which has the largest total asset value. This means that we use the fund that goes by the name of Vanguard Target Retirement X Inv., where X represents the specific target date.
The prospectus of this target date fund states that the objective of the fund is to ‘pro-vide capital appreciation and current income consistent with its current asset allocation’
(Vanguard,2017). This means that this fund belongs in the first (more risky) category.
Furthermore, the prospectus also states that within seven years after the target date, the investment allocation becomes equal to that of the Target Retirement Income Fund. The life-cycle is displayed in I.1a.
Target date funds Fidelity Investments
Fidelity Investments offers several different target date funds. Almost every target date fund they offer has many different variants that each invest in a different share classes, while the underlying life-cycles remain the same. Since we are not interested in the stock portfolio, we do not take the different variants of a certain target date fund into consideration.
The first target date fund we look at is the Fidelity Freedom X, where X represents the specific target date. Its prospectus states that the fund ‘seeks high total return until its target retirement date. Thereafter, the fund’s objective will be to seek high cur-rent income and, as a secondary objective, capital appreciation’ (Fidelity Investments,
2017b). This places the fund in the first and more risk full category. Furthermore, the
fund reaches an allocation similar to that of the Fidelity Freedom Income Fund about 10 to 19 years after the retirement date. The life-cycle is displayed in Figure I.1b. The second target date fund offered by Fidelity Investments that we include in the analysis, is a fund with the name Fidelity Advisor Freedom X, were X again represents the specific target date. The objective of this fund, stated in the fund’s prospectus, is to ‘seek high total return with a secondary objective of principal preservation as the fund approaches the target date and beyond’ (Fidelity Investments, 2017a). The fund thus belongs in the second (less risky) category. Just like the previous fund, the allocation of this fund becomes similar to that of the Fidelity Freedom Income Fund about 10 to 19 years after the retirement date. The life-cycle is displayed in Figure I.1c.
Target date funds T. Rowe Price
T. Rowe Price offers two different target date funds, just as Vanguard does. But un-like the Vanguard funds, these funds do differ in their strategy. The first one prioritizes growth, while the other one aims to provide protection against volatility. In other words, one of the target date funds T. Rowe Price offers belongs in the first and more risky category, while the other one belongs in the less risky category. We include both these funds in our research.
The fund belonging in the first category goes by the name of T. Rowe Price Retire-ment Fund X, where X represents the specific target date. The prospectus of the fund states that its objective is to ‘seek the highest total return over time consistent with an emphasis on both capital growth and income’ (T. Rowe Price,2017a). Furthermore, the fund aims for a portfolio allocation of 20% stocks and 80% bonds 30 years after the
