Dutch legislation on variable annuities?
Jurijn Jongkees
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: Jurijn Jongkees
Student nr: 10375821
Email: jurijn.jongkees@hotmail.com
Date: August 15, 2017
Supervisor: dhr. dr. S. (Servaas) van Bilsen Second readers: dhr. dr. R.J. (Roel) Mehlkopf
dhr. prof. dr. R.J.A. (Roger) Laeven Second readers EY: dhr. H.J.F. (Bert) Baggen MSc AAG
mevr. drs. E. (Eva) Wierenga AAG dhr. drs. M.L.P (Michel) van den Berg
Statement of originality
This document is written by Jurijn Jongkees 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 is 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.
Abstract
This thesis investigates the welfare effects of the new Dutch legislation on variable annuities (‘Wet Verbeterde Premieregeling’). Based on the seminal work of Merton (1969), we derive a model for the optimal contract and the old contract. Furthermore, we thoroughly investigate the new legislation on variable annuities and we demonstrate how to model this new legislation (with and without smoothing of shocks). Contracts under this new legislation perform well in eliminating the welfare loss of the old contract (compared to the optimal contract) in an economy with solely equity risk. When shocks are smoothed out over time the welfare loss increases for a CRRA-participant and decreases for a CRRA-patricipant with habit formation. If interest rates and inflation rates are stochastic, the new legislation seems less successfull in reducing the welfare loss. We provide some socially and policy regarding issues with respect to this new research area.
Keywords Variable pensions, Optimal consumption choice, Optimal portfolio choice, Defined contribution, Smoothing shocks, Welfare loss, Habit formation, Stochastic inflation, Stochastic interest rate
Preface vii
1 Introduction 1
2 Optimal life cycle contract with equity risk 4
2.1 Assumptions and preferences . . . 4
2.2 Financial market . . . 5
2.3 Wealth accumulation . . . 5
2.4 Maximization problem . . . 6
2.5 Optimal life cycle contract . . . 6
2.6 Parameter input and results . . . 7
2.6.1 Parameter input . . . 7
2.6.2 Programming results . . . 8
3 New Dutch legislation on variable annuities 9 3.1 Modelling the old contract: fixed annuities . . . 9
3.1.1 Model . . . 9
3.1.2 Programming Results . . . 11
3.2 Qualitative analysis of the new legislation on variable annuities . . . 11
3.3 Modelling the new legislation on variable annuities . . . 14
3.3.1 Variable life cycle contract without smoothing financial shocks . 14 3.3.2 Variable life cycle contract with smoothing of financial shocks . 17 3.4 Comparing the different contracts . . . 22
4 Welfare losses 24 4.1 Welfare indicators . . . 24
4.1.1 Certainty equivalent . . . 24
4.1.2 Compensating variation . . . 25
4.1.3 Parallel shift . . . 25
4.2 Welfare losses for different contracts and parameters . . . 25
4.3 Minimum welfare losses . . . 27
4.4 An alternative utility function: habit formation . . . 31
4.5 Some concluding remarks, discussion and future research . . . 33
5 Stochastic interest rate and inflation 35 5.1 Modelling the optimal life cycle contract . . . 35
5.2 Modelling the new legislation on variable annuities . . . 41
5.3 Results . . . 42
5.3.1 Assumptions and parameter choices . . . 42
5.3.2 The fixed adjustment terms . . . 42
5.3.3 The annuity factors . . . 43
5.3.4 Programming results . . . 44
5.3.5 Interest and inflation shocks . . . 45
5.3.6 Welfare implications . . . 45 v
5.4 Some concluding remarks, discussion and future research . . . 47
6 Conclusion 49
References . . . 50 Appendix . . . 54
This thesis is the result of approximately 5 months of hard work and concludes my Master of Science in Actuarial Science & Mathematical Finance at the University of Amsterdam (UvA). The thesis is performed in the period March 2017 to August 2017 in collaboration with EY Netherlands at the department European Actuarial Services in Amsterdam. I really enjoyed the process of writing this thesis. I especially liked the combination of mathematics, programming, studying literature (and legislation) and discussing all this with my supervisor at the UvA. I used the mathematical and mod-elling skills I gained during my bachelor and master education to solve and elaborate on an interesting socialeconomic problem.
The subject of the new legislation on variable annuities was mathematically challenging, while at the same time being a hot and relevant research area. Many pension discussions nowadays concern the issue of investing in risky assets after retirement and individual life cycle optimalization. Studying this subject was a great opportunity for me to use my theoretical knowledge for a practical and socially important issue. Also, I am sure that the knowledge I gained on the law on variable annuities, programming with Matlab, mathematical modelling, pension literature and more, is very valuable for the rest of my career.
During the period I was writing my thesis, I have been supported and encouraged by a couple of extraordinary people. First and most of all, I would like to thank my supervisor at the UvA, dhr. dr. S. van Bilsen. It has been a priviledge to benefit from his guidance and supervision. Many thanks goes to him for his time, energy, ideas, enthusi-asm and intellectual and critical thinking. Without his help and frequent advice, I would not have been able to write my thesis as I have now. He always replied very quickly to my questions and if I needed some time to discuss my problems, he was always prepared to do this. I also liked the fact that he was very involved. He wasn’t just my supervisor, but also a great inspiration to me. He encouraged me to work hard and to get the most out writing my thesis, which I really appreciated. It has been a big learning process for me, which has certainly woken my interest for research on pensions. Moreover, the knowledge I gained during his course ‘Caput Financing of Pension’ proved to be very useful for writing this thesis. Finally, I am also grateful for the fact that he was willing to discuss my thesis at the research center Netspar.
There are some other people that I would like to thank. I am very thankful to dhr. dr. R.J. Mehlkopf for being the second reader of my thesis (while being employed by a different university) and discussing his view on my thesis. I thank dhr. prof. dr. R.J.A. Laeven for reading my thesis and for his involvement during my study. Moreover, I want to thank dhr. prof. dr. ir. M.H. Vellekoop for offering me a job at the UvA as a student assistent. For me it was a great experience to teach the introduction of actuarial science to first year students. I really enjoyed doing this. Lastly, I want to thank dhr. prof. dr. R. Kaas. He was the first to interest me for Actuarial Science. He has always been involved during the courses I took and also when I was a student assistent.
Secondly, I want to thank EY Actuarissen for offering me an internship. It has been a nice place to write my thesis and I am happy to start working there as an advisor soon. I am thankful for the nice contact I have with my colleagues. In particular, I would like to thank my mentor at EY dhr. H.J.F Baggen MSc AAG for his time, advice and for helping me find my way in the company. Also for the fact that he was willing to check and advice me on mainly the qualitative parts of my thesis. Moreover, I thank mevr. drs. E. Wierenga AAG and dhr. drs. M.L.P van den Berg for their willingness to provide me with valuable feedback on my thesis.
A special thanks goes to David Koetsier. He helped me a couple of times when I got stuck during the process of programming in Matlab. His intelligence, programming expe-rience and of course friendship was of great importance to me. Moreover, I thank Joris Plaatsman for his friendship, encouragement and support during this period. Also, I am grateful for the fact that he was willing to proofread the main parts of my thesis. Besides, I owe a lot to my girlfriend Noortje Dinkelman for her love, encouragement, enthusiasm and flexibility in this process. Also, my parents (Paul and Annemarie), my sister (Bente) and brother (Tijmen) have been very important for me during my time studying. Finally, I would like to thank the rest of my family and friends whose friend-ship and encouragement have turned out to be very important while I was studying Actuarial Science & Mathematical Finance at the UvA.
This research can be extended in many different directions. I wish I had more time to do this, but I only had 5 months. I have tried to investigate the most interesting and socially important aspects. Including a chapter on interest and inflation risk. I hope that this chapter provides a good kick-off for discussion and future research as it leaves many opportunities for extention.
I hope you will enjoy reading my thesis. Jurijn Jongkees
Introduction
In September 2016 a new law has been introduced in the Netherlands which gives par-ticipants of a defined contribution (DC) pension plan the possibility to convert their pension savings into a variable annuity. Before the introduction of this new legislation, it was compulsory to convert pension wealth at the retirement age into a life-long fixed annuity. Under the new legislation the pension wealth that is accrued during the work-ing years is not entirely used to purchase a fixed annuity, but some of it will be invested in risky assets during the retirement phase. This results in higher expected pension benefits. Risk and uncertainty, on the other hand, increase simultaneously. This new legislation on variable annuities (in the Netherlands known as: ‘Wet Verbeterde Pre-mieregeling’) opens up a whole new set of possibilities for DC-pension plan participants and for pension providers such as pension funds and insurance companies. The purpose of this thesis is to investigate the welfare effects of the new Dutch law on variable an-nuities.
We conclude that the new legislation on variable annuities is a great opportunity to get rid of the welfare loss that arises between the old contract and the optimal con-tract. This results, however, only holds in an economy with solely stock risk. If we consider interest rate risk and inflation risk as well, the new legislation is less successful in eliminating this welfare loss. Also, we find that smoothing of shocks can significantly decrease the year-on-year volatility of pension benefits. This is welfare improving for participants with a CRRA-utility function with habit formation, but welfare decreasing for a CRRA-participant without habit formation.
Since the well-known pioneering work of Merton (1969, 1971) and Samuelson (1969), a large variety of authors have studied optimal consumption and portfolio choice over the life cycle in many different ways. See, for example, Van Bilsen(2015) for a compre-hensive elaboration on optimal consumption and portfolio choice. Many different utility functions can be studied for this matter. Standard life cycles models, however, assume a constant relative risk aversion (CRRA) utility function (see e.g. Cocco, Gomes, and Maenhout,2005;Liu,2007;Gomes, Kotlikoff, and Viceira,2008). Under these standard CRRA-preferences optimal consumption is directly adjusted to financial shocks (i.e. a financial shock is immediately absorbed into the consumption of the pension plan par-ticipant). In this thesis we investigate the welfare effects of the new law on variable annuities using these standard life cycle models, but we also investigate an alternative utility function that includes habit formation (see Van Bilsen, Laeven, and Nijman, 2016). For this alternative utility function the pension plan participant derives utility from an endogenously determined reference (habit) level and therefore his optimal con-sumption gradually adjusts to financial shocks, instead of immediately.
We model the optimal life cycle contract using the work-horse model ofMerton (1969). 1
The assumptions underlying this model are also used to model the old contract in the Netherlands: pension wealth is converted into an annuity with fixed pension benefits. The modelling of the new legislation is based on the paper ofVan Bilsen and Bovenberg (2016). We consider the new contract with and without smoothing of financial shocks. This smoothing of financial shocks (see e.g. Van Bilsen and Linders, 2016; Balter and Werker,2016) is fairly new and the literature on this issue is relatively scarce. Moreover, smoothing of financial shocks is not very commonly applied in current pension contracts. Delta Lloyd offers, as the only Dutch pension provider, pension contracts with variable smoothed pension benefits (Boschman,2017). Smoothing of financial shocks brings some important advantages as it decreases year-on-year volatility of variable pension benefits. The advantages, disadvantages and the modelling of this smoothing mechanism under the new legislation are thoroughly discussed in this thesis.
Our analysis is extended in the spirit of the seminal work of Brennan and Xia (2002). Hence, we develop a framework to analyze the asset allocation problem for a participant in a economy with inflation and real interest rate risk. Our investigation aims to show the effect of stochastic inflation and interest rates on consumption and welfare. To the best of our knowledge, this analysis has not yet been applied to the new legislation on variable annuities in the literature before. As a consequence, we also provide some socially and policy regarding suggestions for future research.
It is very relevant to investigate the possibilities of this new legislation on variable annuities in the Netherlands for multiple reasons. First of all, many pension providers in the Netherlands are not sure what contracts to offer. There is uncertainty about the fraction of wealth that is to be investigated into risky assets. On the one hand, each pension provider wants their participants to be able to pick up the equity risk premium, on the other hand they want to avoid scenarios in which pension payment go down dramatically. The new possibilities and opportunities that the new legislation brings are yet to be explored and investigated.
Secondly, there have been many sounds that the pension system in the Netherlands needs a reform (Sociaal Economische Raad, 2015). The DB-orientated pension plans are no longer sustainable due to the low interest rate of the last periods. In this transi-tion phase, the possibilities and opportunities of the new law on variable annuities need to be thoroughly studied and communicated.
Thirdly, corporations are increasingly withdrawing as sponsors from defined benefit (DB) pension plans (De Nederlandsche Bank,2017a). DC-pension plans are gaining in popularity in the Netherlands at the cost of DB-pension plans. This means that pensions are becoming much more often individual savings with collective risk sharing (Sociaal Economische Raad,2016).1 This increasing emphasis on personal pensions immediately implies that the theory on optimal life cycle investment becomes much more important. The above issues, thus, highlight the growing demand for adequate individual savings, drawdown and investment decisions over the life cycle of a participant.
The main contributions of this thesis are the following. We discuss the law on variable annuities in a qualitative way. An extensive qualitative description of the key elements of the new legislation on variable annuities is given. We then proceed to set up a model for the three different main contracts: the optimal contract for a participant with con-stant relative risk aversion (CRRA), the old contract (i.e. a fixed annuity) and the new
1
For more on these personal pensions with risk sharing (PPR) see also:Bovenberg and Nijman(2016) andVan Bilsen(2015).
contract under the law on variable annuities. More specifically, we derive a model for the contract under the new law with and without smoothing of financial shocks. Sub-sequently, we calculate welfare losses (compared to the optimal contract) for different contracts and we calculate minimum welfare losses. Also, we investigate welfare losses under an alternative utility function with habit formation. Finally, we investigate the effect of the new legislation on consumption and welfare in an economy with stochastic interest and inflation. Furthermore, we provide some relevant discussion points regard-ing the new legislation in this economy.
In this thesis no model calibration is done. We choose the parameters in correspon-dence with the literature on life cycles. We leave the relevant parameter choice and exact calibration of the parameters for the Dutch population to policy makers2. In this thesis we, thus, purely aim at indicating how and why welfare losses arise. Moreover, we only consider the second pension pillar and we exclude the first pension pillar in our calculations.
The remainder of this thesis is organized as follows. Chapter 2 derives the optimal life cycle contract. Chapter 3 elaborates on the new legislation on variable annuities. Furthermore, we derive a model for the old contract and the contract under the new legislation. In Chapter 4 the welfare effects of the different contracts are presented. Chapter 5 extents the research of the previous chapters as it allows the interest and inflation rate to be stochastically determined as well. Finally, Chapter6 concludes this thesis. Some figures, tables and derivations of complicated calculations are relegated to the Appendix.
2
SeeBovenberg, Koijen, Nijman, and Teuling(2007) for the welfare losses that arise due to wrongly chosen parameters.
Optimal life cycle contract with
equity risk
In this chapter we formulate a model for optimal consumption and portfolio selection over the life cycle of a participant of a DC-pension plan. The work of Merton (1969) is the pioneering paper with regard to optimal consumption and portfolio selection. Accordingly, we use that paper as the foundations of the calculations done in this chapter. We start by considering a financial market with only stochastic stock return risk. This means that the interest rate is fixed and for now we abstract away from inflation (Chapter 5 elaborates on what happens when interest rate risk and inflation risk are present). In Section 2.1 we describe the assumptions and preferences we use. Section2.2 describes the available assets in the financial market we assume. In Section 2.3we elaborate on the accumulation of pension wealth over time. Section2.4describes the maximization problem and in Section 2.5we describe the optimal solution. Section 2.6, at last, shows the programming results for the parameter choices we assume and it shows the optimal life cycle contract graphically.
2.1
Assumptions and preferences
We consider the life cycle of a single pension plan participant who starts working at t = 0, retires at time t = TRand passes away at time t = TD. The date of death TD and
the date of retirement are known a priori and are therefore non-random. This means we do not take into account macro longevity risk. We assume that micro longevity risk is pooled away by the collectivity of the Dutch pension funds. Another assumption is that a pension plan participant will earn labor income Y for every t ∈ [0, TR]. This labor
income is risk-free, meaning that we do not take disability risk into account.
Unless stated otherwise, we assume that each pension plan participant has constant relative risk aversion (CRRA). This is modelled by the CRRA-utility function, which is defined as follows: u(ct) = ( 1 1−γc 1−γ t , if γ ∈ (0, ∞)\{1} log ct, if γ = 1 , (2.1)
where γ denotes the coefficient of relative risk aversion. This utility function has two nice properties:
1. Positive marginal utility. The first order derivative u0(ct) is positive. Which is
intu-itively obvious: more consumption means higher utility.
2. Diminishing marginal utility. The second order derivative u00(ct) is negative. Or,
equiv-alently, u(ct) is a concave function in ct. This is also a desirable property, because it
indicates that a participant prefers a fixed amount over a fair lottery: more risk with equal expectation leads to less utility.
2.2
Financial market
The asset menu in the financial market corresponds withMerton (1969). Two financial assets are available: a risk-free asset (bank account) and a risky asset (stock price index). A constant interest rate rt= r is assumed and the value Bt of the bank account is the
solution of the following stochastic differential equation: dBt
Bt
= rdt, B0 > 0 given. (2.2)
The stock price index St is described by a geometric Brownian motion (see e.g. Hull, 2012, p. 282):
dSt
St
= µdt + σdZt, S0 > 0 given. (2.3)
µ denotes the expected rate of return on the stock price index, σ is the volatility of the stock returns, and Ztrepresents a standard Brownian motion, i.e., dZt∼ N (0, dt).
The value of the bank account at a certain time t can easily be derived from (2.2):
Bt= B0exp Z t 0 rds . (2.4)
Using Itˆo’s lemma (Itˆo,1944) and (2.3) we can derive a similar expression for the value of the stock price index at a certain time t:
St= S0exp Z t 0 µ − 1 2σ 2 ds + σ Z t 0 dZs . (2.5)
Note that the evaluation of the stock price consist of a deterministic part and a stochastic part, while the evaluation of the value of the bank account is fully deterministic.
2.3
Wealth accumulation
We define total wealth at time t (Wt) as the sum of human capital Ht and financial
capital Ft:
Wt= Ht+ Ft. (2.6)
Human capital is equal to the discounted value of future premium contributions. Human capital is converted to financial capital over the years. Financial capital consists of all capital at a certain time that is build up from the past premium contributions (premium payments and the returns from investing these premia). Wealth at time 0 can be determined as the discounted value of future premium contributions (Gollier, 2005, Chapter 4). We assume these future premium contributions to be risk-free and therefore we abstract away from disability risk. Since we are considering DC-pension plans, we assume that the premium contribution is an age dependent fraction ptof a participant’s
labor income Yt1. This means that we can calculate W0 as:
W0 =
Z TR
0
ptYte−rtdt. (2.7)
pt is the premium contribution that is paid by employer and employee as a percentage
of labor income Yt. ptand yt are exogenously given in our maximization problem.
1
For more on optimal consumption and portfolio choice under flexible labor seeBodie, Merton, and Samuelson(1992).
We assume that consumption before retirement is equal to labor income minus pre-mium contributions. Therefore,
ct= (1 − pt)Yt for 0 < t < TR. (2.8)
Wealth can be shown to change over time according to the following dynamic budget constraint (see Part Bof the Appendix for a derivation):
dWt=
(
(r + ωt(µ − r))Wtdt + ωtσWtdZt for 0 < t < TR
(r + ωt(µ − r))Wtdt + ωtσWtdZt− ctdt for TR< t < TD
, (2.9)
where ωtis the fraction of total wealth invested into risky assets at time t.
2.4
Maximization problem
We need to find an optimal consumption strategy c∗and an optimal portfolio strategy ω∗ over time. A pension plan participant wants to maximize his/her expected lifetime utility over the admissible consumption-portfolio pairs (c, ω). We cannot expect consumption to be equally valued over time, therefore we introduce the subjective rate of time preference, which we denote by δ. Note that consumption before retirement is fixed (see (2.8)), therefore our maximization problem runs from TR to TD. Hence, each pension plan
participant faces the following dynamic maximization problem: max c,ω E Z TD TR e−δtu(ct)dt , (2.10)
subject to the dynamic budget constraint given in (2.9).
2.5
Optimal life cycle contract
We now reach the point where we have to find an optimal solution to (2.10) subject to (2.9). There are two methods of solving this problem. The first technique is the dynamic programming technique which is described by Merton (1969). Another technique is by using the Martingale method (Cox and Huang,1989;Hull,2012). We do not derive the optimal solution, as these results have already been derived by Merton(1969) and Cox and Huang (1989), but we do provide the optimal consumption and optimal portfolio choices that we need in order to run our simulations.
Optimal consumption over time can be expressed analytically as follows (Merton,1969): c∗t = l−1γ exp 1 γ(r + 1 2λ 2− δ)t +λ γZt for TR< t < TD, (2.11)
where λ = µ−rσ is the market price of risk and l is the value of the lagrangian. The optimal allocation to the risky asset can be found to be (Merton,1969):
ωt∗= λ
γσ for 0 < t < TD. (2.12)
In Figure 2.1 we plot the average development over time of human capital, financial capital and total wealth for certain standard input parameters, which we define in Section2.6.1. From (2.12) we observe that for the optimal life cycle contract the optimal portfolio choice is constant over time. However, this does not mean that the same amount of money is invested in the risky asset every year. Indeed, in terms of financial wealth Ft we observe a decreasing pattern. As we assume that human capital is risk-free, the
25 35 45 55 67 75 87 Age 0 2 4 6 8 10 12 14 Value of Capital
Mean development of total wealth Mean development of financial capital Mean development of human capital
Figure 2.1: Mean development of human capital, financial capital and total wealth over time. For this figure the input parameters of Section 2.6.1 are used.
fluctuations in total wealth should match the fluctuations in financial wealth. This means that:
ωt∗σWtdZt= ˜ω∗tσFtdZt. (2.13)
Here ˜ωt is the fraction of financial wealth invested in the risky asset at time t. From
(2.13) we can now derive: ˜ ωt∗ = ωt∗Wt Ft = λ γσ Wt Ft = λ γσ 1 +Ht Ft . (2.14)
As we can see ˜ω∗t decreases, on average, as the participant ages. It converges to ω∗, which is shown in Figure 1of the Appendix.
2.6
Parameter input and results
2.6.1 Parameter inputIn this section the choice for the parameters are explained. We will choose our parame-ters in line with the literature on life cycle models. µ, r and σ are taken in correspondence with theAdvies Commissie Parameters(2014): µ = 0.07, r = 0.025 and σ = 0.2. For all our calculations we assume that a pension plan participant starts working at the age of 25, retires at the age of 67 and passes away at the age of 87. So we have: TR= 42 and
TD = 62. We define the labor income to be equal to 1 (i.e. Y = 1 for all t).
The fiscal maximum premium contribution in DC-pension plans in 2017 in the Nether-lands (Loonheffingen, inkomstenbelasting. Pensioenen; beschikbarepremieregelingen en premie- en kapitaalovereenkomsten en nettopensioenregelingen,2016) is given in Table 1 of the Appendix. This table includes only old age pension. In reality one also saves for spouse pension. If one does not have a spouse, the accrued spouse pension can usu-ally be converted into a higher old age pension. For simplicity we exclude spouse pension.
In most cases we use the following values for the risk aversion parameter γ and the time preference δ: γ = 5, δ = − log(0.97) ≈ 0.0305. However, in Section 2.6.2 we also illustrate the sensitivity of consumption to different values of γ and δ (see Figure2and 3 of the Appendix). From the above parameters, straightforward computation yields: λ = µ−rσ = 0.07−0.0250.2 = 0.225 and ωt∗= γσλ = 0.2255·0.2 = 0.375.
2.6.2 Programming results
In this section some simulation results are shown using the input parameters of Section 2.6.1. In Figure2.2we show 5 random simulations by 5 different colors for the optimal life cycle contract. We show consumption by solid lines (read on the left axis) and the
25 35 45 55 67 75 87 Age 0.2 0.4 0.6 0.8 1 1.2 1.4 Consumption 0 2 4 6 8 10 12 14 16 18 20 Total Wealth
Figure 2.2: 5 random consumption streams (solid line, read on the right axis) and cor-responding wealth development (dotted lines, read on left axis) for the optimal life cycle contract. All parameters are chosen as described in Section 2.6.1.
corresponding wealth development by dotted lines (read on the right axis). Note that the results vary greatly. The starting point of the variable annuity at age 67 depends on the wealth development during the working period of a participant, which again depends on the shocks.
In Figure 2 of the Appendix we see different mean consumption streams for different values of γ. Note that as the risk aversion parameter increases, the mean consumption decreases after retirement due to the fact that the fraction of wealth invested in risky assets decreases as a function of γ (see (2.11)). In Figure 3 of the Appendix we show mean consumption streams for different values of δ. Note that δ plays a big role in the way wealth is converted into consumption. Figure4 of the Appendix shows the average consumption path for the optimal life cycle contract given the parameter choices as de-scribed in Section2.6.1. Note that for these parameter choices the average consumption path for the optimal life cycle contract after retirement is not flat but increasing.
New Dutch legislation on variable
annuities
This chapter now moves away from the optimal life cycle contract as described by Merton (1969) and describes the new legislation on variable annuities. Equipped with the knowledge of the optimal life cycle contract, we first derive a model for the old contract (Section 3.1.1) and elaborate on the old and still most commonly used way of determining pension benefits in the Netherlands for a DC-participant: pension wealth at retirement is converted into a fixed annuity. Secondly, the new legislation in the Netherlands1 is described and explained in a qualitative way in Section 3.2. The key items and restrictions that are important to do our welfare analysis are extensively discussed. Section 3.3turns the qualitative discussion of Section3.2into a quantitative model. We create a new model for estimating and simulating the optimal life cycle contract in the Netherlands under the new legislation on variable annuities. Section3.3.1 models the new contract without smoothing of shocks, Section3.3.2derives a model for the new contract that includes smoothing of shocks. Finally, Section 3.4 compares all contracts we derived so far graphically.
3.1
Modelling the old contract: fixed annuities
3.1.1 ModelWe now look at the way in which pension benefits have been paid out by providers of DC-pension products before the new legislation on variable annuities was introduced in the Netherlands. In a DC-pension system a participant saves capital throughout his/her working period. The participant is in the old pension system forced to convert all ac-crued pension wealth at retirement into a fixed annuity.
To model this, we distinguish between two periods. First we study the period before retirement and then we consider the period after retirement. We assume the same as-sumptions, preferences, financial market and wealth development as described in Section 2.1- 2.3 respectively, however we now have a different maximization problem.
There is no maximization problem concerning consumption before retirement, because consumption is known a priori (similar to Chapter 2). A fixed contribution rate pt of
1
The new legislation can be found in the following two law adjustments:Besluit van 7 juli 2016 tot wijziging van het Besluit uitvoering Pensioenwet en Wet verplichte beroepspensioenregeling en enige andere besluiten in verband met de Wet verbeterde premieregeling(2016) andWet van 23 juni 2016 tot wijziging van de Pensioenwet, de Wet verplichte beroepspensioenregeling en de Wet op de loonbelasting 1964 in verband met verbetering van premieregelingen (Wet verbeterde premieregeling)(2016)
25 35 45 55 67 75 87 Age 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Consumption 0 2 4 6 8 10 12 14 16 18 Total Wealth
Figure 3.1: 5 random consumption streams (solid line, read on the left axis) and corre-sponding wealth development (dashed lines, read on the right axis) for the old contract. All parameters choices are as described in Section 2.6.1. Note that we use the same simulation set as in Figure 2.2.
wage ytis paid at time t. The remainder of wage yt is consumed:
ct= (1 − pt)yt for 0 < t < TR. (3.1)
The dynamic wealth accumulation before retirement follows from (2.9) and can be represented as
dWt= (r + ωt(µ − r))Wtdt + ωtσWtdZt for 0 < t < TR. (3.2)
We let the portfolio choice be equal to the optimal portfolio choice (2.12). So we get: ωt∗= λ
γσ for 0 < t < TR. (3.3)
At the retirement date total wealth (which then equals financial wealth WTR = FTR) is
converted into an annuity with known payments until death. So we can define:
ct= WTR/¯aTR for TR< t < TD, (3.4)
where ¯aTR is the annuity factor that holds at age TR, paying 1 (continuously) each year.
Since the date of death and the date of retirement are deterministic, we can derive: ¯
aTR =
Z TD−TR
0
e−rtdt. (3.5)
There is one restriction we have to take into account. The lowest pension benefit may in the Netherlands not be less than 75% of the highest pension benefit (in Dutch also
know as ‘Hoog-laag’ rule), which can be found inPensioenwet(2006), article 63. Since a participant is for the old contract described in this section only allowed to buy a fixed annuity, this restriction is not a problem. Note, however, that it may become a problem for certain values of δ under the optimal life cycle contract of Chapter2 (see Figure 3 in the Appendix).
3.1.2 Programming Results
In this section we again show some simulation results to show the difference between the contract in Chapter2and the old contract described in Section3.1.1. In Figure3.1 we show 5 random simulations (5 different colors) for the old contract. We again show consumption by solid lines (read on left axis) and the corresponding wealth development by dotted lines (read on right axis). We use the same financial shocks as in Figure2.2. Therefore, we see that the wealth development before retirement for the old contract equals the wealth development for the optimal life cycle contract as shown in Figure 2.2. However, for the old contract all wealth is withdrawn at the retirement date and a fixed annuity is bought. Again, we find that there is large variety in consumption after retirement dependent on the development of wealth before retirement.
3.2
Qualitative analysis of the new legislation on variable
annuities
In this section we will study the new legislation on variable annuities, which is known as the ‘Wet Verbeterde Premieregeling’ in Dutch and came into force the first of September 2016 in the Netherlands. This new legislation has made a couple of important changes in the Dutch Pension Law (Pensioenwet,2006). These changes are denoted in two law adjustments:Besluit van 7 juli 2016 tot wijziging van het Besluit uitvoering Pensioenwet en Wet verplichte beroepspensioenregeling en enige andere besluiten in verband met de Wet verbeterde premieregeling (2016) and Wet van 23 juni 2016 tot wijziging van de Pensioenwet, de Wet verplichte beroepspensioenregeling en de Wet op de loonbelasting 1964 in verband met verbetering van premieregelingen (Wet verbeterde premieregeling) (2016). We discuss the main and most important changes that are described in these law adjustments. The following documents contain a nice overview of the most important changes for pension participants and pension providers of the new legislation:Autoriteit Financi¨ele Markten (2016), Pensioenfederatie (2016), De Nederlandsche Bank (2017c) and PGGM (2016). See also Ortec Finance (2016) for a calculation of the benefits of variable pensions for current pension contracts.
The purpose of this section is not to give a complete overview of all the qualitative details of the new legislation on variable annuities. Nevertheless, we touch upon the most important changes in Dutch pension plans. Every part of the new legislation that is needed to do our modelling and simulating of variable pensions is thoroughly dis-cussed. This section thus contains a qualitative and summarized analysis of the new legislation. Section 3.3 discusses a quantitative way of modelling contracts under the new legislation on variable annuities.
The first aspect that is important to mention is that the new legislation on variable annuities is only applicable to premium or capital pension contracts, not to fixed bene-fit contracts. This means that the new legislation is only interesting for participants of DC-pension plans, not for participants of Defined Benefit (DB) systems. The new legis-lation now allows for a participant of a DC-pension plan to choose at retirement either a fixed annuity or a variable annuity. If the participant chooses the variable pension plan, pension wealth is partly invested into risky assets after retirement. The expected
pension benefits will increase due to the risk premium that is contained in risky assets. On the downside, the risk of lower pension benefits increases as well.
Each pension provider is by law obligated to inform the participant about his possi-bility to convert a fixed pension benefit payout to a variable pension benefit payout. Hence, it is the responsibility of the pension provider to inform the participant about the possibility of a variable pension as soon as this choice becomes relevant for the life cycle investments (so when the asset allocation starts to differ for a fixed and a variable pension plan). Besides, the participant must have enough time to make the decision. This means that the pension participant must be informed at least six months before the life cycles of fixed and variable pension plans start to differ (or six months before retirement if they do not differ before retirement).
If the pension provider does not offer variable pension plans, then the pension provider must let the participant know that he has the possibility to transfer his pension wealth to a different pension provider. This is also known as the right to go shopping (De Nederlandsche Bank, 2017b). Shopping is again only possible for premium or capital contracts, not for fixed payment contracts. If the participant does not respond to the possibility of a variable pension, then the pension provider is allowed to carry out a fixed default pension contract without variable pension payments.
Another matter that should be mentioned is the duty of care (in the Netherlands also known as ‘zorgplicht’) the pension provider has. This means that the pension provider should invest the pension wealth of the participant with the best intentions for the participant, taking into account the ambition and expectations of the participant. This is in accordance with the prudent person principle which is described in article 135 of Pensioenwet(2006) (see also De Nederlandsche Bank,2015).
Moreover, the new legislation on variable annuities denotes that pension benefits for a participant that chooses the variable annuity can only vary due to changes in three risk factors. The first, and of course most important one, is the return on the investments. The second is longevity risk (changes in life expectancy) and the third is mortality risk (mortality result of the pension provider). Investment risk and longevity risk can either be calculated individually or collectively. Mortality risk is, however, always determined collectively.
The results on the three risk factors can be spread out over maximally 5 years (col-lectively as well as individually). However, recently a change in the pension law was approved by the Dutch parliament allowing pension providers to spread their financial shocks over maximally 10 years, see Verzamelwet pensioenen (2017). See e.g. Nijman, van Stalborch, van Toor, and Werker(2013) for more on smoothing of shocks in Dutch pension plans. For more on the modelling of smoothing of financial shocks, see Section 3.3.2.
To discount future pension benefit, the new legislation on variable annuities tells us to use the risk-free interest rate (in Netherlands this discount rate is often referred to as ‘projectierendement’). This is done in order to prevent redistribution across genera-tions (see e.g. Bovenberg, Nijman, and Werker, 2016;Bonekamp, Bovenberg, Nijman, and Werker,2016a). However, for contract under the new law on variable annuities this means that the expected pension benefits will increase over time. A pension provider can compensate this by a fixed adjustment term (which is called ‘Vaste daling’ in Dutch). This fixed adjustment, thus, compensates for the expected risk premium that is earned on the risky assets (De Nederlandsche Bank, 2016b,c). By using a risk-free discount
rate and a fixed adjustment equal to the expected risk premium earned on the entire portfolio, the participant should have flat median pension payments during his entire retirement period (Bonekamp, Bovenberg, Nijman, and Werker,2016b). Or, put differ-ently, if all financial shocks are equal to zero, we should have flat consumption after retirement.
The law on variable annuities imposes a restriction on this fixed adjustment term. It may not be larger than 35% of the difference between the parameter of stock return and the risk-free rate (Pensioenwet, 2006, article 63a, paragraph 3). The parameter of the risk-free interest rate can be determined by a duration approximation to convert the risk-free interest rate curve into one percentage (seeBesluit uitvoering Pensioenwet en Wet verplichte beroepspensioenregeling,2016, article 17a, paragraph 2). The param-eter of stock return can be found in article 23a, section 1, part b of Besluit financieel toetsingskader pensioenfondsen (2016). If the asset allocation of the pension provider changes or the parameter of stock return or the parameter of the risk-free rate changes, then the maximal fixed adjustment will change as well.
Besides, the fixed adjustment that is chosen by the pension provider must be consistent with the investments the pension provider holds in order to finance the pension pay-outs (Besluit uitvoering Pensioenwet en Wet verplichte beroepspensioenregeling, 2016, article 17a, paragraph 1). This means that the pension provider must choose its asset allocation with the best intentions for the participant.
Concerning the fixed adjustment term, the law on variable annuities only elaborates on the compensation for equity return. However, it does not say anything relating to the excess return that is earned on bonds. Therefore, the new legislation on variable annuities seems to imply that the fixed adjustment term is only used to compensate for equity excess return (and not in particular for bond excess return). In Section 5.2 we will discuss this in more detail.
Moreover, each pension provider is expected to create a risk profile of every individual participant whose pension wealth is managed by them (Besluit uitvoering Pensioenwet en Wet verplichte beroepspensioenregeling, 2016, article 14c). At least some informa-tion concerning financial status, (investment/pension) knowledge2, investment experi-ence, pension expectations/ambitions and the degree of risk aversion must be gained on a regular basis. This information can be obtained for example by doing a repetitive (online) survey, doing an interview before entering the contract, or doing some sort of big data analysis. If the risk profile changes over time than this must be communicated with the participant.
The last point of the new legislation on variable annuities we mention here is that the 75/100-rule (‘Hoog/Laag’-rule in Dutch) as was mentioned earlier in Section 3.1 is not a restriction for variable pensions. This rule must only hold when entering the contract (so in expected value it should hold). However, due to financial shocks, mor-tality results or longevity results pension benefits are allowed to differ more than the 75/100-boundary (Pensioenwet,2006, article 63).
We end this section by discussing some variable pension contracts that have recently become available in the Netherlands. Some pension providers have started to offer the first variable pension contracts (Boschman, 2017). Delta Lloyd, for example, offers 3 possible contracts: 15%, 30% or 45% of pension wealth is invested into risky assets after retirement. Aegon offers a contract with 67% invested in risky assets. At Allianz a
ticipant can choose between a range of 7.7% to 31% of pension wealth to be invested in risky assets. Finally, Nationale Nederlanden offers a contract with 35% to be invested in equity in the retirement phase. Note that in some cases, however, part of the contract has a defined benefit character and part has a defined contribution character. We can conclude that the amount of wealth that is invested into equity after retirement for the different contracts varies greatly for the different pension providers.
3.3
Modelling the new legislation on variable annuities
In this section we investigate a new way of modelling contracts. We consider contracts that have variable pension benefits as described by the law on variable annuities. In Section3.3.1we first investigate how to model a variable pension plan without smooth-ing of shocks. Secondly, in Section3.3.2we extent the model described in Section 3.3.1 to allow for smoothing of shocks. The models we consider in this section are based on the work of Van Bilsen and Bovenberg (2016).
3.3.1 Variable life cycle contract without smoothing financial shocks
In this section we investigate a new model that takes into account the new legislation on variable annuities. This means that we do not assume that a participant of a DC-pension plan buys a fixed annuity after retirement, but he/she now gets variable pension bene-fits depending on the investment results of the risky asset that are held by the pension provider.
Note that, again, the date of death is fixed and therefore we abstract away from mor-tality results and longevity results. It should, however, be kept in mind that in reality, results on these two categories will influence the variable pension benefits as well. All assumptions are equal to those we described in Section2.1and2.2, i.e we assume again that one starts working at time 0, retires at time TR and passes away at time TD. Also,
we use the same CRRA-utility function (see (2.1)) and we have the same asset menu as described in Section 2.2.
A Financial market
In this section we describe the financial market. The financial market is equal to the financial market we have defined in Section 2.2. This means that the dynamics of the bank account can be represented by (2.2) and the dynamics of stock price index is represented by (2.3).
B Budget condition
We now add to our model the budget condition: the value of the individual’s investment account (i.e Wt) must match the value of his/her pension liabilities for every time
t ∈ [0, TD]. Let Vt denote the value of pension liabilities. Now, budget balance implies
that for each t ∈ [0, TD]:
Wt= Vt (3.6)
must hold. It easily follows from (3.6) that:
d log Wt= d log Vt. (3.7)
We now move on to explore the dynamics of d log Wt and d log Vt in part C and D,
C Wealth dynamics
Since we have the same asset menu as in Chapter2, wealth Wt develops equally. Thus,
(2.9) holds here as well:
dWt= (r + ωt(µ − r)) Wtdt + ωtσWtdZt− ctdt (3.8)
where ωtand ctdenote, respectively, the share of wealth invested into the risky stock at
time t and the (annualized) consumption (from now on referred to as pension benefit) at time t. Using Itˆo’s lemma, we can derive that:
d log Wt= r + ωt(µ − r) − 1 2ω 2 tσ2 dt + ωtσdZt− ctWt−1dt. (3.9)
D Dynamics of the value of pension liabilities We start by factorizing the pension benefit ctas
ct= Vt Ct = Wt Ct , (3.10)
where we define Ct to be the conversion factor at time t. The withdrawal rate 1/Ct
represents the speed at which the wealth of a participant is depleted. In fact, it models how the pension wealth of a participant is allocated over future pension benefits. From (3.10) we now find that:
d log Vt= d log Ct+ d log ct. (3.11)
Since the budget condition (see (3.7)) must hold at every moment, a participant must absorb a portfolio shock in either the conversion factor Ct or in the current pension
benefit ct or in a combination of both. In order to derive the dynamics of log Vt we,
thus, first need to derive the dynamics of log Ct and log ct. The following two parts of
this section deal with these two subjects, respectively.
Dynamics of the conversion factor Following the new legislation on variable an-nuities, we allow the conversion factor to be dependent on the risk-free interest rate r and on the fixed adjustment term xt. We, thus, define the conversion factor as follows:
Ct=
Z TD−t
0
exp {−(r + xt)u} du. (3.12)
We can now determine (using Itˆo’s lemma) that the log conversion factor evolves ac-cording to the following equation:
d log Ct= (r + xt)dt − Ct−1. (3.13)
Dynamics of pension benefits The pension benefits can be specified as follows:
ct= c0exp Z t 0 γsds + Z t 0 σωsdZs . (3.14)
Here, we let γt denote the median growth of pension benefits at time t. In Part E we
show that this γt needs to be determined endogenously. Finally, we can determine the
dynamics of pension benefits. Using (3.14) we get:
E Solving the budget condition
Using (3.13) and (3.15) we can finally derive the dynamics of the value of pension liabilities from (3.11):
d log Vt= (r + xt)dt − Ct−1dt + γtdt + σωtdZt. (3.16)
We now investigate under what conditions the budget equation (3.7) holds. Solving this budget equation now gives us the following (see Section E of the Appendix for a derivation of this result):
(r + ωt(µ − r) −
1 2ω
2
tσ2)dt = (r + xt)dt + γtdt. (3.17)
Note that the fixed adjustment term xtis determined by the pension provider and
there-fore given exogenously. In order for (3.17) to hold, we need to choose γt endogenously
in this equation. 25 35 45 55 67 75 87 Age 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 Consumption 0 5 10 15 Total Wealth x=0 x=0.5% x=1.0% x=1.5% x=2.0% x=2.5% x=3.0%
Mean Wealth Development
Figure 3.2: Mean consumption stream (solid lines, read on the left axis) for the contract under the law on variable annuities for different values of xt (note that xt is constant
over time). All other parameters are as described in Section 2.6.1. Besides, the mean development of wealth is added as a dotted line (read on the right axis).
F Programming results
In this section we show some simulation results. Note that in reality the fixed adjustment term can be set by the pension provider. Throughout this thesis we assume that the pension provider chooses the fixed adjustment term such that expected pension payment are flat. This means we let the fixed adjustment term be equal to the excess portfolio return. We, thus, choose γt in (3.17) equal to 0 and:
xt= ωt(µ − r) −
1 2ω
2
If we, subsequently, choose all parameters again as described in Section 2.6.1and use the same simulation set as in Figure 2.2and Figure3.1, then we get a slightly different output compared to the output of Figure 2.2. This output is shown in Figure 5. This difference is caused by the fact that for the optimal life cycle contract the consumption path after retirement is not flat (see Figure4in the Appendix). The difference between Figure2.2 and Figure5is not clearly visible, however. See Figure3.4in Section3.4for a clearer graphical representation of this difference.
Note, however, that in reality the fixed adjustment term is not always chosen equal to ωt(µ − r) − 12ωt2σ2. We can also determine the fixed adjustment term such that the
contract under the law on variable annuities exactly replicates the optimal life cycle contract (if we choose the portfolio choice equal to (2.12) and exclude smoothing of shocks). If the portfolio choice is not chosen optimally (as in (2.12)), this causes a wel-fare loss. Welwel-fare losses for different contracts under the new law on variable annuities are calculated in Chapter4.
Finally, to illustrate the effect of the fixed adjustment term we look at the mean con-sumption streams for different values of xtin Figure3.2. We add the mean development
of wealth as a dotted line. Note that the mean development of wealth does not depend on the value of xt, see (3.8). We again use the same parameters as described in Section 2.6.1.
3.3.2 Variable life cycle contract with smoothing of financial shocks
This section describes a model for consumption under the new legislation on variable annuities that includes smoothing of financial shocks (Balter and Werker, 2016). The purpose of smoothing financial shocks is to obtain a less volatile consumption stream by smoothing shocks over multiple periods. This gradual absorption of shocks into pension benefits is consistent with internal habit formation (see e.g. Fuhrer,2000; Gomes and Michaelides,2003). The underlying idea is that participants think in terms of reference levels, and not in terms of absolute levels of consumption (Van Bilsen, Laeven, and Nijman,2016) and do therefore benefit from less volatile pension benefits. Section 4.4 investigates this by calculating welfare losses using a new utility function which includes habit formation.
In this section we implement a buffering mechanism in which future (rather than cur-rent) pension benefits bear most of the current investment risk. As a consequence, the volatility per period of current payments is smaller than the volatility per period of wealth. For more about buffering mechanisms and modelling smoothing of shocks see: Van Bilsen and Linders (2016) or Van Bilsen (2015, Chapter 6).
We use the same financial market, budget condition and wealth dynamics as was de-scribed in Section3.3.1in paragraphA,BandCrespectively. In this section we, however, adapt and extent the model we described to allow for smoothing of shocks.
A Dynamics of the value of pension liabilities We again factorize the pension benefit ct as
ct= Vt Ct = Wt Ct , (3.19)
where Ct again denotes the conversion factor at time t. From (3.19) we now find that:
Again, to derive the dynamics of log Vt, we first need to derive the dynamics of log Ctand
log ct. The following two paragraphs of this section deal with these two parts respectively.
Dynamics of the conversion factor We define Vt,h to be the value of a future
pension benefit ct+h. We now write the conversion factor Ct as:
Ct= Vt ct = Z TD−t 0 Vt,h ct dh = Z TD−t 0 Ct,hdh, (3.21) where Ct,h = Vt,h
ct . As a next step we will treat Ct,h as the present value of an annuity
payout at time t + h. We now let δt,h denote the discount rate at time t for an annuity
payment occurring at time t + h. This gives us the following relation:
Ct,h= exp{−δt,hh}. (3.22)
The discount rate models the speed at which the participant withdraws his/her pension wealth. A higher (lower) discount rate corresponds to a lower (higher) conversion factor which leads to more (less) pension benefits being payed out right now. In this way we can spread out financial shocks over multiple periods.
Subsequently, we allow our discount rate to depend on the risk-free interest rate r, a fixed adjustment term xt and past portfolio shocks. See Section3.2for more
informa-tion about the fixed adjustment. We now write δt,h as:
δt,h = r + xt+ δSt,h. (3.23)
The term δt,hS indicates how the discount rate depends on past portfolio shocks. By increasing (decreasing) δt,hS following a negative (positive) portfolio shock, the partici-pant (partially) absorbs a shock into future pension benefits. Note that the past portfolio shocks must depend on ωt, because a higher share in risky assets will increase the impact
of stock price index shocks on the total portfolio. To dampen the impact of a portfolio shock on current annuity units, a participant adjusts his conversion factor after each portfolio shock. We now use (3.22) and (3.23) to write Ct,h as:
Ct,h = At,hFt,h, (3.24)
where
At,h = exp{−(r + xt)h}, (3.25)
Ft,h = exp{−δt,hS h}. (3.26)
In the following, we refer to At,h and Ft,h as the horizon-dependent annuity factor and
the horizon-dependent funding ratio, respectively. The horizon-dependent annuity factor is the deterministic part of the conversion factor, where the horizon-dependent funding ratio contains a stochastic part (as it depends on past portfolio shocks). Note that the model we considered in Section 3.3.1is a special case from the model we define in this section (i.e. the horizon-dependent funding ratio is equal to 1 all the time). From (3.24) we can now easily derive that:
d log Ct,h = d log At,h+ d log Ft,h. (3.27)
Note that we define d log Ct,h here as d log Ct,h = log Ct+dt,h−dt − log Ct,h for dt → 0
(i.e. t + h is fixed, but t changes). The same holds for d log At,h and d log Ft,h. We can
now derive the dynamics for d log Ct(see PartFof the Appendix). In the following two
Dynamics of the horizon-dependent annuity factor The log horizon-dependent annuity factor log At,h = −(r + xt)h satisfies the following dynamic equation (by Itˆo’s
lemma):
d log At,h = (r + xt)dt. (3.28)
(3.28) shows us that future pension benefits are discounted one period less, as is ex-pected.
Dynamics of the horizon-dependent funding ratio The portfolio shock at time t is given by ωtσdZt(see (3.8)). We now want to define a buffering function qt,h (for more
on different possibilities concerning buffering mechanisms see Van Bilsen and Linders, 2016). We let a participant translate a fraction qt,h of a current portfolio shock into
ct+h. This means that the exposure of log Vt,h to a current portfolio shock equals qt,h.
By picking a buffering mechanism that increases with the horizon h, we make sure that a current portfolio shock yield a larger impact on pension benefits in the distant future than on pension benefit in the near future. To make sure that the entire portfolio shock is absorbed into current and future pension benefits, the following restriction must hold:
Z TD−t
0
αt,hqt,hdh = 1 ∀t ∈ [0, TD]. (3.29)
The buffering mechanism that is chosen is not always scaled in such a way that (3.29) holds. Therefore, we need a scaling parameter dependent on time, which we call βt. So
we split the buffering function into a buffering part qt,h and a scaling part βt. In order
for equation (3.29) to hold we now solve for βtgiven a certain buffering mechanism qt,h
in the equation below:
Z TD−t
0
αt,hβtqt,hdh = 1 ∀t ∈ [0, TD]. (3.30)
From (3.30) we arrive at: βt=
1 RTD−t
0 αt,hqt,hdh
∀t ∈ [0, TD]. (3.31)
The exposure of the log horizon-dependent conversion factor log Ct,h = log Vt,h− log Vt,0
to a current portfolio shock is equal to qt,h− qt,0. Thus, the horizon-dependent funding
ratio Ft,h can be defined as:
Ft,h = exp nZ t 0 βs(qs,t+h−s− qs,t−s)ωsσdZs o . (3.32)
Ft,h models the dependence of the conversion factor Ct,h on past portfolio shocks. From
(3.32) we can now show that the horizon-dependent funding ratio satisfies the following dynamic equation:
d log Ft,h = βt(qt,h− qt,0)ωtσdZt−
Z t
0
dqs,t−sβsωsσdZs. (3.33)
To make the above dynamics and notation clear, we also write (3.33) in terms of dt for dt → 0:
d log Ft,h= log Ft+dt,h−dt− log Ft,h
= βt(qt+dt,h−dt− qt+dt,0)ωtσdZt−
Z t
0
(qs.t+dt−s− qs,t−s)βsωsσdZs. (3.34)
Since we are working in a continuous world, (3.33) and (3.34) are equal. However, intuitively (3.34) might be more clear. The first term on the right-hand side of both
(3.33) and (3.34) represents the impact of a current portfolio shock on the horizon-dependent funding ratio. The second term corresponds to past portfolio shocks that are absorbed into current pension benefits, so that they should no longer be included in the horizon-dependent funding ratio. We now compare (3.26) with (3.32) and find that:
δSt,hh = − Z t 0 βs(qs,t+h−s− qs,t−s)ωsσdZs = − log Ft,h. (3.35)
(3.35) shows the relationship between the discount rate of the conversion factor and past portfolio shocks. The term δS
t,h thus arises from the gradual adjustment (depending on
the smoothing mechanism qt,h) of pension benefits to past portfolio shocks. If we exclude
smoothing of shocks (as described in Section 3.3.1), past portfolio shocks should not affect the conversion factor and are therefore equal to 1 for every horizon h. Indeed, for qt,h equal to unity for each horizon h, portfolio shocks are fully translated into current
pension benefits.
Dynamics of pension benefits The pension benefits are specified as follows: ct+h= c0exp nZ t+h 0 γsds + Z t+h 0 βsqs,t+h−sσωsdZs o . (3.36)
Here, we again let γt denote the average growth of pension benefits at time t. We
have defined the horizon-dependent funding ratio Ft,h to model the dependence of the
conversion factor on past portfolio shocks, so using (3.37) it follows that: ct+h= ctFt,hexp nZ t+h t γsds + Z t+h t βsqs,t+h−sσωsdZs o . (3.37)
We are now equipped to determine the dynamics of pension benefits. Using (3.37) we get:
d log ct= γtdt +
Z t
0
dqs,t−sβsσωsdZs+ βtqt,0σωtdZt= γtadt + βtqt,0σωtdZt. (3.38)
γat denotes the actual median growth of pension benefits:
γta= γt+ γtS, (3.39) where γtSdt = Z t 0 dqs,t−sβsωsσdZs (3.40)
represents the effect of past portfolio shocks on current median growth of pension ben-efits. For clarity, we write (3.38) again also in terms of dt for dt → 0:
d log ct= log ct+dt− log ct= γtdt +
Z t
0
βs(qs,t+dt−s− qs,t−s)ωsσdZs+ βtqt+dt,0σωtdZt
(3.41) Again, in continuous time (3.38) and (3.41) are equal.
Adding terms In the previous parts we have determined the dynamics of log At,h,
log Ft,h and log ct, respectively. This part combines these results and derives the
dynam-ics of log Ct,h, log Ctand finally the dynamics of log Vt. Using (3.33), (3.28) and (3.40) we
can first calculate the dynamics of the log horizon-dependent conversion factor log Ct,h:
The dynamic equation of the log conversion factor log Ct is given by (this follows from
(3.21) and (3.42)):
d log Ct= µCtdt + (1 − βtqt,0)ωtσdZt− Ct−1dt, (3.43)
where µCt can be interpreted as the expected rate of return on the conversion factor defined as: µCt dt = (r + xt)dt − γtSdt + 1 2 Z TD−t 0 αt,hd log Ct,hd log Ct,hdh − 1 2 Z TD−t 0 Z TD−t 0
αt,uαt,vd log Ct,ud log Ct,vdudv. (3.44)
Here, the quadratic covariation d log Ct,ud log Ct,v is given by
d log Ct,ud log Ct,v = (qt,u− qt,0)(qt,v− qt,0)βt2ωt2σ2dt. (3.45)
Now, using (3.20), (3.38) and (3.43) we can derive the dynamics of d log Vt:
d log Vt= µCtdt + (1 − qt,0βt)ωtσdZt− Ct−1dt
+ γtdt +
Z t
0
dqs,t−sβsσωsdZs+ βtqt,0σωtdZt. (3.46)
B Solving the budget equation
In this section we set wealth equal to the pension liabilities in correspondence with the budget condition in (3.6) in order to derive the relationship between parameters. Working out d log Wt= d log Vt we get (see Part Hof the Appendix) :
(r + ωt(µ − r) − 1 2ω 2 tσ2)dt = (r + xt)dt + 1 2 Z TD−t 0 αt,hd log Ct,hd log Ct,hdh −1 2 Z TD−t 0 Z TD−t 0
αt,uαt,vd log Ct,ud log Ct,vdudv + γtdt (3.47)
We want the budget condition in (3.6) to hold. Note that the fixed adjustment term xt is determined by the pension provider and therefore given exogenously. In order for
(3.47) to hold, we thus need to choose γt endogenously in this equation.
C Programming results
We repeat the simulations we performed in Figure 2.2, Figure3.1 and Figure 5 in the Appendix including smoothing of shocks. To illustrate the effect of smoothing shocks we plot the results for no smoothing, a smoothing period of 5 year and a smoothing period of 10 year in Figure6in the Appendix. The consumption without smoothing is given by the solid line (which is equal to the line in Figure5 in the Appendix), the consumption with 5 year of smoothing is given by the dashed line and the consumption with 10 year of smoothing is given by the dotted line. We make sure (3.47) holds by choosing γt
endoge-nously, we take xt equal to (3.18) (as described in PartF of Section 3.3.1). Note that
we use the exact same simulation set as in Figure2.2,3.1and Figure5in the Appendix. Smoothing of shocks can be useful when the economy is relatively constant and calm since it reduces volatility for a participant. On the other hand, it can be useful in stress situations as well. A sudden decline of the stock index, for example, causes a sudden decline in consumption as well for the contract of Section3.3.1. Smoothing this decrease can help a participant adapt to his/her lower consumption level slowly instead of imme-diately. This is illustrated in Figure3.3. A sudden decline in consumption is observed at
67 70 75 80 85 Age 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Consumption No smoothing 5 year smoothing 10 year smoothing
Figure 3.3: Random consumption path with a sudden decline in stock prices at the age of 73 (an economic crisis). We look at three different consumption paths: the blue line is without smoothing of shocks, the red line is with 5 year smoothing and the yellow line is with 10 year of smoothing.
the age of 73. Note that when shocks are smoothed, consumption decreases gradually to this shock. Section 4.4 extents this analysis on smoothing of shocks by calculating the welfare implications for participants that derive utility from a certain reference level of consumption which changes over time (habit formation).
3.4
Comparing the different contracts
In this section we show simulation results for the different contracts we have considered so far. Figure3.4shows a random simulation for the optimal contract, the old contract, the contract under the law on variable annuities without smoothing of shocks and a contract under the law on variable annuities including smoothing of shocks. For the last two contracts we chose the fixed adjustment term equal to the excess return on the portfolio ωt(µ − r) −12ωt2σ2. For the contract that includes smoothing of shocks we
choose γt endogenously from (3.47).
Note that the contract under the law on variable annuities without smoothing of shocks first lies a little above the optimal life cycle contract. However, as time passes the con-sumption under the optimal life cycle contract starts to transcend the contract under the law on variable annuities without smoothing of shocks. Note that this is caused by the fact that the fixed adjustment term is chosen such that the mean consumption under the law on variable annuities is flat. Mean consumption under the optimal life cycle contract is, however, not flat (see Figure 4of the Appendix).
67 70 80 87 Age 0.8 0.85 0.9 0.95 1 1.05 Consumption Old contract
Optimal life cycle contract
Contract under law on variable annuities Contract under law on variable annuities including 5 years of smoothing shocks
Figure 3.4: Random simulation for the 4 different contracts we consider: the optimal contract, the old contract, the contract under the new legislation on variable annuities without smoothing of shocks and the contract under the new legislation on variable an-nuities including smoothing of shocks.
Welfare losses
In this chapter we calculate welfare losses (relative to the optimal life cycle contract discussed in Chapter 2) that result for the different contracts/models we discussed in Chapter 3. Besides, we investigate minimum welfare losses. The welfare losses are all estimated by simulation. First of all, Section 4.1elaborates on three important welfare indicators that can be used to measure the utility of a certain consumption stream by converting it to one value. We consider the certainty equivalent, compensating variation and parallel shifting. Section4.2shows the results of the welfare calculations done for a couple of contracts under the new legislation on variable annuities. Section4.3shows the results on the minimum welfare losses (for different values of γ) that result between the different contracts. Moreover, we discuss sensitivity to the risk aversion parameter γ and the subject rate of time preference δ. Finally, Section 4.4discusses welfare implications for the CRRA-utility function (see (2.1)) including habit formation. This utility function is, hence, dependent on a consumption reference level.
4.1
Welfare indicators
In this section we discuss three different welfare indicators that can be used to calcu-late welfare losses: certainty equivalents, compensation variation and parallel shifting of consumption streams. The three welfare indicators are a useful way of assessing the impact of different contracts/models on a participant’s well-being as they value a cer-tain consumption stream by one value. We start by evaluating cercer-tainty equivalents, subsequently we discuss the compensating variation and last of all we look at parallel shifting.
4.1.1 Certainty equivalent
The certainty equivalent of a stochastic consumption stream c is defined as the amount ce such that the agent is indifferent between a certain consumption stream c and receiving ce with certainty. A certainty equivalent maps a certain consumption stream to one value: U0= E Z TD 0 exp{−δt}u(ct)dt = Z TD 0 exp{−δt}u(ce)dt, (4.1)
where U0 is equal to the expected discounted value of utility at time 0. Welfare losses
are calculated as the relative decline in certainty equivalent of a certain consumption stream. Hence, the welfare loss can be computed as
ce∗− ce
ce∗ , (4.2)
where ce∗ is the certainty equivalent consumption level corresponding to the optimal consumption strategy.
