The analysis of different contracts in
the individual defined contribution
schemes with variable benefits
Ren´
e van Pul
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: Ren´e van Pul
Student nr: 10681515
Email: rene.van.pul@aaa-riskfinance.nl
Date: August 18, 2017
Supervisor: Dr. Tim Boonen
This document is written by Student Ren´e van Pul who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Preferences in IDC with variable benefits — Ren´e van Pul iii
Abstract
This study examines the preferences of several reference persons between different contracts (or variants) of the individual defined contribution pension plans. The contracts include the possibility to take investment returns after retirement, that is we allow for variable benefits.
Stochastic economic scenario’s are used in combination with a projection model to generate the capital accrual and both the variable and fixed benefits in every scenario at every moment in time. In total 18 reference persons are analyzed. The reference person vary in state (active or retired), will have different salary levels and different levels of risk aversion. The contracts in turn vary in investment profile, premiums paid, smoothing option and implicit growth rate.
In order to be able to compare the different contracts we use a Constant Relative Risk Aversion (CRRA) utility function, with the consumption level equal to the discretionary income. By using this utility function we can obtain the certainty equivalents for the discretionary income both including and excluding the State old-age pension.
By analyzing the different reference persons and different con-tracts we first show that the variable benefit is preferred (by all reference persons) over the fixed benefit. Secondly we show that between contracts with variable benefits there is a trade-off between the premium paid and the benefits received. By decreasing the premium the discretionary income in the active phase may increase, however the available capital will decrease even further. Hence the pure trade-off between premium paid and benefit received will be in favor of the benefit received. However we show that by increasing the risk profile of the investments the preferences can change at least if we take into account the State old-age pension. By introducing a growth rate the expected cash flow of pension benefits will be less volatile, which is preferred by the reference persons. However smoothing the results will, although it decreases the volatility of the benefit as well, have a negative impact on the preferences for both the active and the retired reference person.
Keywords Pensions, Defined Contribution, Wet Variabele Pensioenuitkering, Variable pension benefit, Smoothing of results, Growth rate, Investments, Portfolio, Stochastic, Normal distribu-tion, Uncertainty, Discretionary income, Real Replacement Ratio, Year-to-Year Volatility, Utility function, Certainty equivelant, Netherlands, Money at risk
Preface vi
1 Introduction & overview 1
1.1 Pension systems in the Netherlands. . . 1
1.1.1 Classic defined contribution . . . 1
1.1.2 Defined contribution with variable benefit . . . 2
1.2 Objective of thesis . . . 3 2 Model set up 4 2.1 Active phase . . . 4 2.1.1 Premiums paid . . . 4 2.1.2 Investment returns . . . 5 2.1.3 Capital accrual . . . 5 2.2 Retirement phase . . . 6 2.2.1 Annuity pricing. . . 7
2.2.2 Classic defined contribution . . . 7
2.2.3 Variable benefit. . . 7
2.2.4 Smoothing of gains and losses . . . 8
3 Selection criteria 11 3.1 Main criterion . . . 11
3.2 Utility function . . . 11
3.3 Certainty equivalent . . . 12
3.4 Other relevant output . . . 13
3.4.1 Real Replacement Ratio . . . 13
3.4.2 Year-to-Year volatility . . . 13
4 Assumptions & Variants 14 4.1 Economic assumptions . . . 14 4.2 Parameters . . . 15 4.2.1 Contribution table . . . 15 4.2.2 Reference persons . . . 15 4.2.3 Lifecycles . . . 16 4.3 Variants . . . 16 5 Results 18 5.1 Analysis of variants. . . 18 5.1.1 Variant 1: basis . . . 18
5.1.2 Variant 2: Increasing risk . . . 20
5.1.3 Variant 3: Decreasing risk . . . 23
5.1.4 Variant 4: Downward slope in retirement phase . . . 26
5.1.5 Variant 5: Less premium . . . 30
5.1.6 Variant 6: More premium . . . 32
5.1.7 Variant 7: Smoothing period . . . 33 iv
Preferences in IDC with variable benefits — Ren´e van Pul v
5.1.8 Variant 8: Growth rate. . . 34
5.2 Sensitivity analysis . . . 36
5.2.1 Combination of variants . . . 36
5.2.2 Increase in equity volatility . . . 37
6 Conclusion 39
7 Discussion 41
After 3 long but fun and rewarding years my study finally (almost) comes to an end. Firstly, I would like to thank my supervisor Tim Boonen for his support while writing my thesis. Without his open mindedness and useful advice I would not have succeeded to write this thesis. Secondly, I would like to take my colleagues and bosses at Triple A - Risk Finance for giving me the time and sharing their knowledge to write this thesis.
Chapter 1
Introduction & overview
1.1
Pension systems in the Netherlands
In the Netherlands the pensions are regulated using a three pillar system: • the State pension system (AOW);
• occupational pension schemes; • individual pension products.
At this time there is a huge debate going on to decide the direction of the occupational pension system in the future. However as of now there are still, in general, two types of pension plans, the defined benefit (DB) pension plan and the defined contribution (DC) pension plan. A DB pension plan has a fixed accrual percentage which (more or less) guarantees the individual a certain pension benefit. The premium needed for the accrual is based on among others the interest and mortality rates, hence the pre-mium is uncertain. In comparison a DC pension plan has fixed prepre-miums. The level of premium is given by a certain contribution table and is therefore known beforehand. These premiums are invested in several asset classes, implying the capital accumulation is dependent on financial market returns and thus uncertain. Hence in a DC pension plan the ultimate pension benefit will depend on the level of premiums paid and the performance of the investments up to and including the retirement age. As of Septem-ber 2016 an extension to this classic DC pension plan has been introduced, the “Wet verbeterde premieregeling” (2016) (hereafter called the WVP.
In this paper we will focus on both the classic DC pension plan and its extension. The main focus will lie on the extension and the implications it has on, among others, the premiums paid, the investment profile and the resulting pension benefits. Below we will first describe the classic DC pension plan and afterwards discuss why it was necessary to extend this classic form of the DC scheme.
1.1.1 Classic defined contribution
As with any pension plan the goal is to obtain a sufficient consumption level after retirement. In the classic DC pension plan this level of consumption (the pension benefit) is uncertain. It depends on the premium paid, the investment returns and the price of the pension benefit (the annuity price). The premiums paid are defined beforehand and fixed during the investment horizon, as long as there are no changes in the pension plan. These premiums are invested in different asset classes. The capital accrued is held in an individual account. At retirement the available capital must be used to purchase an annuity. This annuity is a fixed guaranteed pension benefit which the individual will receive until the moment of death and is actuarially fair priced using mortality rates and interest rates.
Investment portfolio
The premiums are invested using the investment strategies that depend on the age of the participant. This lifecycle is used to ensure an appropriate asset mix which balances the risk- and return profile of the individual at any given age up to the retirement age. The main idea behind a lifecycle is that younger members, who have a longer time till retirement, tend to invest more in risky assets (for instance equities), while more mature individuals gradually transfer their assets to protection-seeking/low-risk investment classes (for instance bonds) (see for instance Merton (1969) and Merton (1971)).
Shortcomings
As with most pension plans this particular system has drawbacks or shortcomings. These can be summarized as follows:
• As the annuity has to be purchased at the retirement age, there is a strong depen-dency on the timing of retirement. If the interest rate happens to be low at that time an individual suffers from high annuity prices and thus low pension benefits; • The capital at retirement needs to be entirely converted into a fixed lifelong benefit. Since the benefit is not periodically increased with the inflation, the purchasing power of an individual will decrease;
• As the investments are based on the lifecycle investment strategies with a fixed moment of conversion, the investment risk needs to be gradually decreased, to match the movements of the annuity price, which lowers the investment returns. The above mentioned shortcomings are the main reasons for introducing the WVP, which is summarized below.
1.1.2 Defined contribution with variable benefit
With the introduction of the WVP the aforementioned shortcoming are solved. The new pension act extends the classic defined contribution pension plan with the possibility to receive a variable benefit (we will call this the DC with variable benefit pension plan). In comparison to the classic DC pension plan the capital is no longer entirely converted in an annuity at retirement. During retirement the capital is still invested. This causes the pension benefit to vary every year. The variable benefit fluctuates with the investment results and longevity results (as a consequence of both micro and macro risk). As such the level of pension benefit to be received is not known beforehand. Below we will discuss the aforementioned shortcomings, and the way the WVP resolved them. Time dependency
In the DC with variable benefit pension plan the capital will be periodically and partly converted into a benefit. The price of this benefit is equal to the price of a lifelong annuity at that moment in time but the benefit will only be fixed during that year. Since there is no single moment at which the individual needs to convert its capital, the time dependency issue is resolved, which is especially interesting in low interest rates situations (like we have today).
Decline in purchasing power
By allowing the individual to keep investing after retirement the benefit is expected to increase over time (as we expect a positive risk return ratio). If the investment return is at least equal to the inflation in that period there is no decline in purchasing power. However as the benefit is no longer guaranteed and negative returns are also possible
Preferences in IDC with variable benefits — Ren´e van Pul 3
the decline in purchasing power may also be steeper in respect to the fixed benefit. This is an inevitable drawback.
Fixed conversion moment
In the classic DC pension plan the lifecycle decreases its allocation to return generating asset classes and increases its allocation to protection-seeking and low-risk asset classes. This is done to match the movement of the annuity price. An individual is not willing to accept a large decrease in pension benefit because the interest rate suddenly declines. Therefore the lifecycle balances the investment budget more in favor of asset classes that mimic the behaviour of annuity prices. The downside is that the allocation to risky asset classes, which are expected to generate higher returns, are decreased. By allowing the individual to keep investing after retirement his investment horizon increases. In turn he can for a longer time hold a higher allocation to risky assets and thus generate higher returns.
Although the shortcomings of the classic DC scheme have been resolved, other draw-backs may arise. Since the pension benefit is no longer guaranteed, the individual has the risk of fluctuating and lower pension benefits. In order to limit these risks the DC with variable benefit pension plan has the possibility to smooth the results and purchase an annuity with a fixed decay rate. As an individual may be unwilling to have large fluctuations in their benefits the gains and losses can be smoothed over a period of time. In this way a more stable pension benefit is generated. Another appealing feature to this DC plan is the purchase of an annuity with fixed decay rate. As the price of such an annuity is cheaper, the starting benefit at retirement is higher. However if no returns above the risk-free rate are made the benefit will decline periodically with this fixed percentage. As the individual has the possibility to keep investing after retirement it can be made plausible that the returns will be expected to be at least as high as the decay rate. If the returns are equal to this decay rate a stable, but higher benefit is obtained.
1.2
Objective of thesis
The objective of this thesis is to analyze the preferences of the reference persons by considering several contracts or variants of the DC with variable benefit pension plan. In such we way we hope to find an optimal contract or variant.
This thesis is an extension of the study done bySteenkamp(2016) and the thesis writ-ten byKroon (2017). In these papers only the retirement phase is considered, whereas I will focus on the active phase as well. By using different reference persons and looking at both phases we will try to find an optimal contract (i.e. a combination of lifecycle, smoothing period and chosen decrease). Sensitivity analyses will also be done in order to show the dependence of the outcome to the assumptions used.
In chapter 2 we will first introduce the model and give a set up. In the next chapter the selection criteria are discussed. Based on these outcomes we will analyze which lifecycle gives optimal results. Chapter 4 gives an overview of the assumptions used and discusses the variants to be analyzed. The analysis itself is done in chapter 5. In chapter 6 the conclusion will be given, whereas finally in chapter 7 we will have a discussion.
Model set up
As the goal of this paper is to analyze and possible find an optimal contract for an individual considering both the active as well as the retirement phase, we need to have a suitable model, which will be discussed in this chapter.
2.1
Active phase
In the active phase the individual works and capital is accrued. The capital will depend on the premiums paid and the investment returns during the investment horizon. The investment horizon in the active phase is equal to the number of years between the current age of an individual minus the pensionable age. In this paper we assume the pensionable age to be equal to 67 years.
2.1.1 Premiums paid
The premiums paid Pt,xare dependent on the pensionable salary St, the offset Otand the contribution percentage cpx. The contribution percentage is given by the contribution table, as published by the Dutch tax authority (2017), and consist of percentages per age bucket. Hence the premium paid at time t for an individual aged x is given by:
Pt,x = (M in(St, StM ax) − Ot) · cpx. (2.1)
It is assumed that the contribution table is fixed during the investment horizon. (Maximum) salary
The development of the pensionable salary of an individual is given by the following relation:
St= St−1· (1 + Π + θx) ,t < tP A, (2.2)
with tP A being the time at which the retirement age is reached. Formula 2.2 implies that the salary in the next period depends on the inflation Π and merit increase θx. The merit increase is defined as the additional increase in the salary of an individual based on past performances, in other words a career path. It is assumed that the inflation is fixed over the entire investment horizon and the merit increase is dependent on the age of the individual (in chapter 4 we will come back to this in more detail). Also the pensionable salary is capped off at a maximum salary StM ax, which only increases with the inflation Π:
StM ax= St−1M ax· (1 + Π). (2.3)
Preferences in IDC with variable benefits — Ren´e van Pul 5
Offset
The offset is the part of the salary for which no pension is accrued and therefore no premiums are paid. It is used to finance the State old-age pension (AOW) and increases with the inflation over time:
Ot= Ot−1· (1 + Π). (2.4)
The State old-age pension equals 75% of the offset at any given moment in time.
2.1.2 Investment returns
During the investment horizon the premiums paid will be invested and depending on the stochastic returns the capital will increase of decrease. Depending on the allocation to risky assets the expected return (and volatility) may be higher or lower. A lifecycle is used to determine to allocation percentages at a certain age. As many as there are institutions that provide these defined contribution policies, as many lifecycles there are. All have different investment beliefs and therefore different visions of what kind of asset classes to hold and in which ratio to hold them. The asset classes can range from equity to commodities and bonds. This is done to limit the risk of overexposure to a certain asset and to obtain diversification effects. In general, however, these asset classes can be classified in two types of portfolios:
• Return portfolio; • Matching portfolio.
The return portfolio will mainly consist of equity and other high risk asset classes and is used to generate high returns. On the other hand, the matching portfolio is used to match the price of the annuity to be paid. As a decrease in interest rate constitutes in a higher annuity price, the matching portfolio will mimic this behavior. In this paper we assume that the return portfolio will consist of equities. We will generate equity returns for every scenario s at every moment in time t. The equity returns are given by rs,t, as will be defined by formula 2.6. The matching portfolio is assumed to consist of bonds that have a return equal to the risk-free rate rf, which in turn is assumed to be fixed for every scenario in the investment horizon.
2.1.3 Capital accrual
The capital at any moment in time can be generated using the recursive formula in2.5: Cs,t+1=
Cs,t+ Pt 1 − qx+t,t
· (1 + αx+t· rs,t+ (1 − αx+t· rf), (2.5)
with rs,tbeing the equity return at scenario s at time t, 1−qx+t1(z+t) the biometric return given the age x + t (see section2.1.3) and forecast year forecast t (see section2.2.1), rf the risk-free interest rate and αx+t the allocation to the return portfolio at age x + t. Financial returns
Just like Kroon(2017) we use a simple model to generate the stochastic equity returns rs,t at time t. However unlike Kroon (2017) we do not assume the autocorrelation to be equal to 0. The Dutch central bank (DNB) has analyzed the consistency between several equity returns over the period 1988-2006. In DNB (2006) it is shown that the autocorrelation over that period will lie between 1% and 19%. For simplicity we assume the autocorrelation ρ in the model to be equal to 15%.
The equity returns are stochastically simulated for 10,000 scenarios, over an investment horizon of 60 years, as follows:
rs,t= µr+ ρ ∗ σr∗ s,t−1+ (1 − ρ) ∗ σr∗ s,t. (2.6)
We assume the error term s,tto be independent and identically distributed, using a stan-dard normal distribution (N (0, 1)). Hence the equity returns are normally distributed as well. The assumptions regarding the mean µr and the standard deviation σrs are aligned with the assumptions used in the report ofCommissie parameters (2014), that is µr = 6.75% and σr = 20%.
The risk-free rf is be fixed over the entire investment horizon. Non-financial returns
Next to the financial returns we also have to take any non-financial returns into account. As we assume there is no insurance for the spouses after the individual dies (that is there is no shift of capital to the spouse) and the (longevity) risk is to be shared collectively, the capital that is released is to be shifted to the other participants in the collective. The return form this death-benefit is called the biometric return and is dependent on the mortality rates. With the above assumptions the biometric returns rbiocan be calculated using the method described by Bovenberg & Nijman(2015):
rbio = 1 1 − qx,t
. (2.7)
In the figure below the different types of returns are shown. It can be clearly seen that the biometric returns increase significantly with age, due to the higher mortality rates. Figure 2.1: Graphical representation of the cumulative wealth given the various return types
2.2
Retirement phase
At the pensionable age the capital is converted into an annuity, at an actuarial fair price. Whereas in the classic DC situation the process of converting capital into an annuity is only done once at the pensionable age, with the DC pension plan with variable benefit this process is repeated every year. Every year the actuarial fair price of the annuity is determent to obtain the benefit for that specific year. Hence the benefit level is also dependent on the capital at hand.
Preferences in IDC with variable benefits — Ren´e van Pul 7
2.2.1 Annuity pricing
The actuarial fair price of an annuity is based on the survival probabilities kpx,t, the interest rate rf and possibly the growth rate g. In which the growth rate is a fixed parameter indicating the growth or decline (in case g < 0) of the cash flow of the pension benefits. The price is given by the following formula:
¨ at= 1 + TM ax−(67+t) X i=1 ip67+i,t+i· (1 + g)i (1 + rf)i , (2.8)
where TM ax is the time we know (for sure) the individual has been deceased.
Survival probabilities
The mortality rates of the most recent projection table is used. As of September 2016 theThe Royal Dutch Actuarial Association(2016) has published a new projection table, the AG Prognosetafel 2016. The AG Prognosetafel 2016 is a projection table, that is a table in which the mortality rates depend on the age x as well as the forecast year t. This implies that as an individual ages, the year forecast is also incremented, thus the table is traversed diagonally. The starting year of the forecast is given by t. The survival probabilities can thus be calculated from the mortality rates as follows:
kpx(t) = k Y
i=1
(1 − qx+i−1,t+i−1), (2.9)
where qx,t is the mortality rate of an individual with the age x and being in the year t.
Growth rate
With the extension of the classic DC pension plan to allow for investments after retire-ment, it has also been allowed to incorporate a fixed incline or decline in the price of the annuity, which we call the growth rate. In the case of a fixed annuity, the cash flow is guaranteed for life and fixed at a certain level. By incorporating a growth rate the cash flows automatically increases or decreases by a fixed percentage over time. This percentage is given by the growth rate g, and can both be positive as well as negative. In practice a negative growth rate is used most of the time as it increases the initial benefit level.
2.2.2 Classic defined contribution
At the pensionable age the entire capital Cs,t (see equation 2.5) is converted into an annuity. This annuity is fixed from the pensionable age onwards and guarantees a lifelong benefit. We call this the fixed benefit BFs,x:
BFs,x=
Cs,67−x ¨ a67−x
. (2.10)
In this case the growth rate in the annuity price ¨a67−x is equal to g = 0.
2.2.3 Variable benefit
The DC with variable benefit pension plan allows for the possibility to keep investing the capital in a lifecycle during the retirement phase. The benefit at time t is therefore
also dependent on the available capital, which is again dependent on the stochastic returns. Hence the level of benefit at time t in scenario s, is given by:
BVs,t = Cs,t
¨ at
. (2.11)
In this case the growth rate in the annuity price ¨atmay be unequal to g = 0. Also note that the annuity price is variable over time, since the individual ages.
Capital accrual after retirement
As the capital is still invested in the retirement phase we can use a recursive formula to obtain the capital at any moment in time. The recursive formula in 2.12 is used to calculate the capital at time t + 1:
Cs,t+1 =
Cs,t− BVt 1 − qx+t,t
· (1 + αx+t· rs,t+ (1 − αx+t) · rf). (2.12) The above formula is similar to formula2.5, with the exception that a variable benefit is subtracted rather than a premium added.
The above formula immediately takes into account the total financial result (which could be a gain or a loss) at any given moment. This will result in very volatile benefits. Since this may not be desirable the possibility of smoothing has been introduced. This is discussed hereafter.
2.2.4 Smoothing of gains and losses
In case of the DC with variable benefit pension plan there is the option to smooth the financial result in excess of the risk-free rate. We call this the excess return, which can be both positive as well as negative. The idea behind smoothing the excess returns is to receive a less fluctuating pension benefit, that is the excess returns are not taken immediately (as is the case in formula2.12) but are smoothed over time. Even though the year to year fluctuations will be lower if the results are smoothed, the total result during the retirement phase is not altered (actuarially).
Step 1: Determining excess return
At any time t in every scenario the absolute excess return (hereafter just excess return) is given by:
Res,t= (Cs,t−1− BVs,t−1) 1 − qx,t
· αx+t· (rs,t− rf). (2.13) Hence the excess return Rs,te is given in euro’s. The excess returns will be equally dis-tributed over the smoothing period.
Step 2: Determining the smoothing period
The smoothing period is defined as the period over which the result (from equation
2.13) is equally distributed. In the WVP only two restrictions are given as to how the smoothing period can be chosen. The first restriction is that the smoothing period may not exceed ten years. Secondly the the smoothing period may not be higher than the remaining life expectancy ex,t:
τt= 1, if ex,t< 2, bex,tc, if 2 ≤ ex,t< τ0, τ0, if ex,t≥ τ0. (2.14)
Preferences in IDC with variable benefits — Ren´e van Pul 9
The above implies that as long as the (initial) smoothing period τ0 meets these require-ments it may be chosen freely. In equation2.14 it is shown that the smoothing period will automatically adjust if the remaining life expectancy becomes lower. The remaining life expectancy ex,t is defined using the definition of the cohortlevensverwachting (see The Royal Dutch Actuarial Association (2016)):
ex,t = 1 2+ ∞ X i=0 Πij=0(1 − qx+i,t+i). (2.15)
In equation 2.15 it is assumed that an individual will die on average at the half of the year.
Step 3: Determine recognized amortizations
We assume the excess return is smoothed equally over the smoothing period. However this cannot simply be done by dividing the result by the smoothing period. This must be done actuarially fair. Implying that we need to account for interest and survival probabilities. Similar to the price of annuity we can define the price of a temporary annuity ¨at:τt, as: ¨ at:τt = τt X i=0 ip67+t,i· (1 + g)i (1 + rf)i . (2.16)
Using this price we can define the amortization As,t. The amortization being the result that is recognized directly and is simply defined as the excess return divided by the temporary annuity price:
As,t= Rs,te ¨ at:τt
. (2.17)
Thus at the end of the first year t = 1, the amortization As,1 is determined using the excess return and the smoothing period τt. For every year in the smoothing period k ∈ (1, ..., τt) we now have an amortization. In the next year we again (may have) the amortization from the past year, As,1, but we also must take into account the realized excess return in that year As,2. If we call the result that is recognized in any year the cumulative amortization CAs,t,k, we get:
CAs,t,k = As,t, if t = 1 ∨ k = τt As,t+ CAs,t−1,k+1, if t > 1 ∧ k < τt 0, else. (2.18)
In figure 2.2the smoothing algorithm as described above is visualized. Step 4: Determine discounted unrecognized amortizations
In the previous step we have determined the amortization that is recognized. In the case that τt≥ 1 we also have amortizations that are unrecognized. These amortizations are thus not yet taken into account. In order to correctly determine the benefit an individual can afford, we need to correct the capital available with the discounted value of these unrecognized amortizations: U As,t= ( 0, if τt= 1 Pτt k=2 CFs,t,k (1+rf)k−1, else. (2.19)
Figure 2.2: Graphical representation of the smoothing of the excess returns over a smoothing period of 5 years. The first part of the figure shows the situation without smoothing, whereas a very volatile benefits are a result. In the second part the smoothing is shown. For simplicity it is assumed that the excess returns are divided by the smoothing period, hence no mortality or interest rate effect is included. In the bottom part the total smoothed effect is given, as the sum of the the smoothed results.Again, for simplicity no mortality or interest rate effect is included.
With CFs,t,kbeing the cumulative amortization accounted for survival probabilities and the growth rate:
CFs,t,k= CAs,t,k·k−1p67+(t−1)+(k−1),(t−1)+(k−1)· (1 + g)k−1. (2.20)
Step 5: Determine benefit next year
Equation 2.12 shows the way the capital is developed over time if the entire result is taken into account. However the smoothing algorithm as discussed above will split the result in two parts, the cumulative amortization CAs,t,k and the unrecognized amor-tizations U As,t. Thus in case the results are smoothed over time we need to adjust equation2.11. By subtracting the unrecognized amortizations from the capital available for buying an annuity we can define the benefit level the next year:
BVs,t =
Cs,t− U As,t ¨ at
Chapter 3
Selection criteria
In order to be able to properly analyze the different variants of contracts we need to define some output measures. We will compare these output measures in both the active phase as well as the retirement phase. These variants as well as the assumptions used are discussed in chapter4. In this chapter we will focus on the decision variables.
3.1
Main criterion
We need to define an output variable in which both the active phase and the retirement phase are taken into account. As such we have defined the discretionary income. Discretionary income
In this paper we assume there are no taxes, hence the net income equals the gross income. The discretionary income is defined as the gross income minus any fixed payments. In the active phase we assume the gross income to be equal to the salary (equation2.2) and the fixed payments to be the premiums paid (equation2.1). This salary is assumed to be after the payments for the State old-age pension. In the retirement phase no more fixed payments are present. We will consider two cases with respect to the gross income, the pension benefit including and excluding the State old-age pension. Equation 3.1shows the definition of the discretionary income when taking into account the State old-age pension DIs,tincl and equation 3.2shows the discretionary income without State old-age pension DIs,texcl: DIs,tincl= ( St− Pt, if t < tP A, Bs,t+ AOWt, if t ≥ tP A (3.1) DIs,texcl= ( St− Pt, if t < tP A, Bs,t, if t ≥ tP A (3.2)
In which tP A is the time at which the reference person reaches the pensionable age, and Bs,t being the benefit, either variable or fixed. If there is no need to differentiate between the discretionary income including or excluding the State old-age pension we will talk about discretionary income or use DIs,t.
3.2
Utility function
As the decision may be different for a risk averse individual in comparison to a risk seeking individual we need a way to account for the risk preferences of an individual. By using a utility function, with a risk aversion parameter, this can be achieved. Utility functions account the ’worst’ outcomes more heavily in respect to the ’best’ outcomes.
The risk aversion parameter tells us how heavily the ’worst’ outcomes are punished in comparison to the ’best’ outcomes. The higher the risk aversion parameter the more risk averse the individual is. At every time t and for every scenario s, the utility of the individual is calculated. As we consider a deterministic scenario setting the average utility U over the entire lifecycle is given by:
U = 10,000 X s=1 60 X t=0 f (t) · u(cs,t) ·tpx,t). (3.3)
Whereas u(cs,t is the commonly used Constant Relative Risk Aversion (CRRA) utility function (see for instance Gollier (2008)) as given by:
u(cs,t) = c1−γs,t
1 − γ, (3.4)
where γ represents the relative risk aversion parameter implicit in the utility function, with the restriction that γ > 0 and γ 6= 1. A higher risk aversion parameter implies a more risk averse individual. The parameter cs,t represents the consumption level at a certain time in a scenario. This consumption level is assumed to be equal to the discretionary income as defined in equations3.1 and3.2.
3.3
Certainty equivalent
As we have defined the utility function and the consumption level, we are able to calcu-late the utilities for every scenario over the entire investment horizon. Now all we need is a way to compare these outcomes with each other. Using the definition of certainty equivalent this is possible. The certainty equivalent is an objective measure which equals the guaranteed level of consumption (or utility) at which an individual is indifferent be-tween receiving this guaranteed amount or the uncertain level(s) of consumption (or utility) over time. In other words we can represent the uncertain cash flow of consump-tion over time in a single measure of consumpconsump-tion cash flow fixed over time, that is the certainty equivalent (CE). When taking into account a discount function f (t) and the chance an individual actually receives the consumption we have the following relation:
60 X t=0 f (t) · u(CE) ·tpx(z) = 10,000 X s=1 60 X t=0 f (t) · u(cs,t) ·tpx,t). (3.5)
Which can easily be rewritten to obtain the certainty equivalent of a contract as follows:
CE = U−1 1 10,000 P10,000 s=1 P60 t=1f (t) · u(cs,t) ·tpx,t P60 t=1f (t) ·tpx,t ! . (3.6) Discount function
In equations3.5 and 3.6 the discount function f (t) is included. This discount function is used to account for the ’time value of money’. In general it holds that an individual prefers to have an amount of cash (or consumption) now rather than tomorrow, hence we must have a decreasing function over time. In this paper we assume (like Kroon (2017)) a simple discount function:
Preferences in IDC with variable benefits — Ren´e van Pul 13
3.4
Other relevant output
As we do not want to focus solely on the discretionary income as the selection criteria we will discuss two more output variables that can provide more background on the optimal choice and which will mainly focus on the retirement phase. These output variables are the Real Replacement Ratio and the Year-to-Year volatility.
3.4.1 Real Replacement Ratio
The Real Replacement Ratio (RRRs,t), as defined by Kroon (2017)) and Steenkamp (2016), is a way to show the purchasing power effects including the State old-age pension (AOW). The Real Replacement Ratio is defined as the total discretionary income after retirement including the State old-age pension, as a percentage of the gross salary at retirement (corrected with the price inflation):
RRRs,t=
DIs,tincl StP A· (1 + Πp)t
. (3.8)
The discretionary income after retirement DIs,tincl is given by equation3.1in case t ≥ tP A and StP A being the gross salary at retirement.
3.4.2 Year-to-Year volatility
The Year-to-Year volatility Y Y Vs,tis a definition introduced by theVerbond van Verzek-eraars (2017). It is used as a risk measure for variable pension benefits (in comparison with fixed pension benefits). This risk measures shows the absolute volatility of year-to-year fluctuations in the benefits received:
Y Y Vs,t = 100% · | DIs,t DIs,t−1
− 1|. (3.9)
The above implies that the Year-to-Year volatility for a fixed benefit is always equal to 0%. In order to easily compare the outcomes we will look at the average (equation3.10) and the largest deviation (equation3.11):
Y Y VAvg = 1 10, 000 10,000 X s=1 1 T T X t=1 Y Y Vs,t, (3.10) Y Y VDev= 1 10, 000 10,000 X s=1 maxtDIs,t mintDIs,t . (3.11)
Assumptions & Variants
This chapter will give an overview of the assumptions used in the model. Also the different variants to be considered will be discussed. In chapter 5 the outcomes are discussed.
4.1
Economic assumptions
In the table 4.1the economic assumptions used in the basis variant are given.
Parameter Value Description
µr 6.75% The mean of the equity returns
σr 20.00% The standard deviation of the equity returns
rf 1.00% The risk-free rate
g 0.00% The growth rate of the benefit
Π 2.00% The inflation
δ 99.00% The factor indicating the ’time value of money’
Starting year 2017 The starting (forecast) year in the projection table AG2016
O0 e13,123 The current offset (2017) in The Netherlands
S0M ax e103,317 The current maximum salary (2017) in The Netherlands
AOW e9,842 The current State old-age pension (2017) in The Netherlands
Table 4.1: List of used economic assumptions in the basis variant
As we include the possibility of career opportunities we have the following, industry standard, merit increase (the ’3-2-1-0’ merit increase). A merit increase θx is an extra salary increase. It can be seen as a reward for past performances. Every individual will at any moment in time have an increase in salary by the wage inflation. This is an increase which holds in general. However to account for the fact that an individual who just starting working will on average have higher salary increases in respect to an individual who is close to retirement, we use the merit increase.
Age class Merit increase
15 - 30 3.0%
31 - 40 2.0%
41 - 50 1.0%
51 - 66 0.0%
Table 4.2: Table of merit increase as defined by θxin 2.2
Preferences in IDC with variable benefits — Ren´e van Pul 15
4.2
Parameters
Next to the economic assumptions different parameters need to be set for the basis vari-ant. Parameters include the contribution table, the reference persons and the lifecycles used.
4.2.1 Contribution table
The premiums paid will depend on the contribution table used, as published by Dutch tax authority(2017). The contribution tables are based on different interest rates. The Dutch tax authority publishes two tables, the 3% and the 4% table. These tables are based on the maximum accrual percentage in the average pay system (that is 1.875%) and the interest rates used to discount the cash flows. Using these assumptions the tables are aimed to provide for a pension benefit after 40 service years that does not exceed 75% of the average salary. In the basis variant we look at the 3% contribution table, as shown in the table below.
Age class Contribution percentage
15 - 19 7.3% 20 - 24 8.1% 25 - 29 9.4% 30 - 34 10.9% 35 - 39 12.7% 40 - 44 14.8% 45 - 49 17.2% 50 - 54 20.1% 55 - 59 23.6% 60 - 64 28.0% 65 - 66 31.8%
Table 4.3: The contribution percentages cpxused in2.1of the 3% contribution table as provided
by the Dutch tax authority.
4.2.2 Reference persons
We need to have different reference persons in order to represent the various states. We will define two reference persons:
• The ‘newly active’, aged 27; • The ‘retiree’, aged 67.
For both references persons we will distinguish between a risk averse reference person (with γ = 7), a conservative reference person (with γ = 5) and a risk seeking reference person (with γ = 3).
Just like Kroon(2017) we will define the following starting salaries: • S0 =e37,000 (Average income);
• S0 =e74,000 (Double average income);
• S0 =e103,317 (Maximum fiscal pensionable salary). Starting capital
As the ‘newly active’ is assumed to be at the start of its career and will have a total of 40 service years, he will not have any available capital yet. However as the ‘retiree’ is already retired we have to define a starting capital for this reference person, since no more premiums are paid. Just likeKroon (2017) we assume that the ‘retiree’ has a
capital such that the Real Replacement Ratio (equation3.8) is equal to 70%. From this the following relation holds:
Cs,0= ¨a0· (70% · S0− AOW ). (4.1)
The price of annuity ¨a0 (see equation 2.8), given the assumptions in table 4.1, for an individual aged 67 with starting year 2017 is equal to 16.97. With this the starting capitals can be calculated, which are shown in the table below.
Starting salary Starting capital
e37,000 e272,544
e74,000 e712,137
e103,147 e1,060,450
Table 4.4: Starting capitals for different starting salaries for ‘retiree’
4.2.3 Lifecycles
The goal of a defined contribution pension system is to generate enough income af-ter retirement. By simply setting aside the premiums the capital will not be sufficient to reach this goal. Therefore the premiums are invested, which implies that the par-ticipant suffers from investment risk. As the The Dutch Authority for the Financial Markets (AFM) mandates the investment profile to be in line with the risk aversion of the individual several lifecycles are used, ranging from very defensive to very offensive. In general the following rule of thump is used ”the more risk one takes the higher it’s expected return”. However the downfall of taking higher risk is that the returns can be significantly negative as well. As a default lifecycle in the basis variant we will use a neutral lifecycle, in this way it is possible to show both the effects of increasing and decreasing the investment risk profile.
As was discussed in chapter 2 we only consider an allocation in equity and (risk-free) bonds. In the figure below the default lifecycle is shown for an investment horizon of 60 years. In this (basis) variant the allocation to equity remains fixed at 30% after retirement.
Figure 4.1: Graphical representation of the default lifecycle used
4.3
Variants
The discretionary income is (mainly) dependent on the following assumptions or deci-sions:
Preferences in IDC with variable benefits — Ren´e van Pul 17
• Contribution table; • Economic scenario; • Lifecycle;
• Smoothing period; • Fixed growth rate.
We want to analyze the different effects of the aforementioned assumptions. Therefore we need to define several variants other than the basis variant. In the table below we have defined the different variants.
Table 4.5: Overview of different variants
Variant description Contribution table Lifecycle Smoothing period growht rate
1 - Basis 3% Neutral No 0%
2 - Increasing risk 3% Offensive No 0% 3 - Decreasing risk 3% Defensive No 0% 4 - Downwards slope 3% Decreasing at retirement No 0% 5 - Less premium 80% of 3% Neutral No 0% 6 - More premium 120% of 3% Neutral No 0% 7 - Smoothing period 3% Neutral 5 years 0%
8 - Growth rate 3% Neutral No -1%
We will introduce these variants in length in the next chapter while discussing the re-sults. Also some sensitivity analyses are performed with respect to the economic scenario and combinations of the variants above.
Results
In this chapter the results of the different variants will be discussed. All results are based on 10,000 stochastic scenario’s. First we will look at the basis variant with respect to the fixed benefit, that is we compare the classic DC pension plan with a variant of the DC with variable benefit pension plan. For the other variants the benchmark will not be the fixed benefit variant, but the basis variant. Every variant will firstly be described whereas afterwards the outcome is analyzed. Finally we will analyze some sensitivity analyses.
5.1
Analysis of variants
5.1.1 Variant 1: basis
Description
In the basis variant the neutral lifecycle is used, as shown in figure 4.1. The starting equity allocation is equal to 80%, which will decrease from the age of 50 to 30%. The decrease in equity allocation is called the glidepath. By increasing the allocation to the matching portfolio the duration between assets and the price of the annuity is better matched. In such a way the capital will show less fluctuations. In practice it is also used to diminish the impact on the annuity prices by fluctuations in interest rates. As we assume a fixed interest rate we do not cover this in this thesis. After retirement the equity allocation will remain fixed at 30%. The premiums paid for the ‘newly active’ are based on the 3% contribution table (see table 4.3), whereas the ‘retiree’, as he is retired, has no more contributions. The variable benefit does not take into account a fixed decrease over time nor a smoothing period.
Results
In the figures below the results for both the ‘newly active’ as well as the ‘retiree’ are shown. As can be seen the variable benefit is expected to be higher than then fixed benefit. Also the expected Real Replacement Ratio’s are higher in case of the variable benefit. This is in line with the conclusions by Kroon (2017), that is there are welfare effects by allowing to invest after retirement. Which is, at least for the ‘retiree’, not surprising since in Kroon (2017) only the retirement phase was analyzed. Considering the ‘newly active’ the conclusion does not change. However due to the uncertainty in available capital at the retirement age, the results are more volatile. That is an individual may face greater downsides, but greater upsides are possible as well. Moreover we can see that by investing in the retirement phase the benefits can vary year to year by as much as 12%, whereas for a fixed benefit there is, by definition, is no variation in the pension benefit.
Preferences in IDC with variable benefits — Ren´e van Pul 19
Figure 5.1: Graphical representation variant 1 for both reference persons with a starting salary ofe37,000 of simulation results for 10,000 economic scenarios for the pension benefits Bt, Real
Replacement ratios RRRtand Year-to-Year volatitlity Y Y Vt. The figures on the left represent
the outcomes for the newly active. On the right the outcomes for the retiree are presented. (a) Pension benefits for the ‘newly active’ (b) Pension benefits for the ‘retiree’
(c) Real replacement ratios for the ‘newly
ac-tive’ (d) Real replacement ratios for the ‘retiree’
(e) Year-to-Year volatility for the ‘newly active’ (f) Year-to-Year volatility for the ‘retiree’
Comparison
In line with the conclusion by Kroon (2017) every reference person in variant 1 would prefer a variable benefit over the fixed benefit. In table 5.1 this can be seen since the certainty equivalent is higher in case of a variable benefit. The impact on the ‘newly active’ is not as strong as the impact on the ‘retiree’, because an extra element of uncertainty is introduced by not fixing the level of capital at retirement. Also when taking into account the State old-age pension (corrected with inflation) the impact will diminish. In this case the pension benefit will consist of an uncertain part and a certain (guaranteed) part. The uncertain part is the (variable) benefit which can purchased with the capital available and the guaranteed part is the State old-age pension is not scenario dependent (see also equation3.1). The guaranteed part will have a minor effect in the scenario’s in which high returns are received as the State old-age pension will be a relatively small part of the discretionary income. However in scenario’s with low returns and therefore a low capital at retirement age (and benefit) the State old-age pension will become a significant part of the discretionary income. Hence the results in these scenario’s will be ’shifted upwards’, which will decrease the impact on the certainty
equivalent gain.
Table 5.1: Certainty equivalent CEγ results of discretionary income including the State old-age
pension for variant 1 in comparison with the fixed benefit. The impact being defined relatively to the fixed benefit.
Reference Fixed benefit Variable benefit - variant 1 Impact person S0 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 ‘newly active’ 37,000 58,100 53,588 50,067 58,783 53,879 50,184 1.2% 0.5% 0.2% 74,000 110,331 102,599 96,485 112,193 103,665 97,113 1.7% 1.0% 0.7% 103,317 122,435 117,710 113,445 124,785 119,569 115,109 1.9% 1.6% 1.5% ‘retiree’ 37,000 28,064 27,955 27,855 30,204 29,568 29,063 7.6% 5.8% 4.3% 74,000 54,046 53,985 53,927 59,476 57,950 56,732 10.0% 7.3% 5.2% 103,317 74,594 74,548 74,504 82,607 80,331 78,487 10.7% 7.8% 5.3%
This effect can be seen in table 5.2. In this table the impact for the ‘newly active’ is higher. This also holds for the ‘retiree’. Note that the certainty equivalent of the fixed benefit (excluding the State old-age pension) is the same independent of the risk aversion parameter γ. This is because we only consider the retirement phase in case of the ‘retiree’and its benefit is fixed over time. Hence there is no time or utility preference. Table 5.2: Certainty equivalent CEγ results of discretionary income excluding the State old-age
pension for variant 1 in comparison with the fixed benefit. The impact being defined relatively to the fixed benefit.
Reference Fixed benefit Variable benefit - variant 1 Impact person S0 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 ‘newly active’ 37,000 49,735 43,624 38,277 51,748 45,609 39,916 4.0% 4.6% 4.3% 74,000 102,139 92,320 83,303 105,458 95,423 86,087 3.2% 3.4% 3.3% 103,317 112,758 102,360 90,227 116,774 106,801 94,278 3.6% 4.3% 4.5% ‘retiree’ 37,000 16,058 16,058 16,058 18,067 17,447 16,903 12.5% 8.7% 5.3% 74,000 41,958 41,958 41,958 47,207 45,587 44,166 12.5% 8.7% 5.3% 103,317 62,480 62,480 62,480 70,296 67,885 65,768 12.5% 8.7% 5.3%
5.1.2 Variant 2: Increasing risk
Description
As with the basis variant this variant will consist of a variable benefit with no smoothing of results or a fixed growth rate and the 3% contribution table. Only the lifecycle used will be different. We will consider two ways in which the investment risk profile of the lifecyle can be increased, namely:
• a: an increase in risk, only in the active phase;
• b: an increase in risk, both in the active phase as well as in the retirement phase. By increasing the starting allocation of risky assets by 20%-points we will increase the risk taking in the active phase. The fixed allocation to risky assets at the retirement phase will remain at the level of variant 1, that is 30%. In the very offensive lifecycle (variant 2b), the allocation to risky assets at the retirement phase is also increased by 20%. In both lifecycles the glidepath will start at age 50, the same age as in variant 1 (see also figure5.2).
Results variant 2a
As a rule of thumb one can say that an increase in allocation to risky assets will lead to higher expected returns and thus higher benefits. However on the downside there is a higher probability of a sudden and sharp decrease in benefit. These effect can be shown using the key figures in the table below. We show the following key figures:
Preferences in IDC with variable benefits — Ren´e van Pul 21
Figure 5.2: Graphical comparison of the lifecycles of variant 1, variant 2a and variant 2b. Only the allocation to equity is given.
• RRR2.5%: the 2.5% percentile of the Real Replacement Ratio;
• RRR50.0%: the 50.0% percentile (or median) of the Real Replacement Ratio ; • RRR97.5%: the 97.5% percentile of the Real Replacement Ratio;
• Y Y VAvg: the scenario average of the absolute volatility of the year-to-year fluctu-ations;
• Y Y VDev: the scenario average of the largest deviation of the absolute volatility of the year-to-year fluctuations.
These key figures are defined in equations 3.8, 3.10 and 3.11, and will only cover the retirement phase.
Table 5.3: Key figures Y Y VAvg, Y Y VDev and the 2.5%, 50.0% and the 97.5% of the Real
Re-placement Ratio’s RRR of variant 2a in comparison with variant 1. The impact being defined relatively to variant 1.
Reference Variable benefit - variant 1 Variable benefit - variant 2a Impact
person Criteria S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 ‘newly active’ RRR2.5% 34,7% 27,7% 20,4% 34,3% 27,3% 20,0% -1,0% -1,6% -1,9% RRR50.0% 53,7% 49,7% 37,6% 57,3% 54,0% 41,1% 6,7% 8,6% 9,2% RRR97.5% 106,7% 112,5% 89,1% 141,4% 154,5% 125,4% 32,6% 37,4% 40,7% Y Y VAvg 4,3% 4,3% 4,3% 4,3% 4,3% 4,3% 0,0% 0,0% 0,0% Y Y VDev 157,5% 157,5% 157,5% 157,5% 157,5% 157,5% 0,0% 0,0% 0,0% ‘retiree’ RRR2.5% 45,1% 37,4% 35,2% 45,1% 37,4% 35,2% 0,0% 0,0% 0,0% RRR50.0% 66,4% 65,3% 65,0% 66,4% 65,3% 65,0% 0,0% 0,0% 0,0% RRR97.5% 97,5% 105,9% 108,3% 97,5% 105,9% 108,3% 0,0% 0,0% 0,0% Y Y VAvg 4,3% 4,3% 4,3% 4,3% 4,3% 4,3% 0,0% 0,0% 0,0% Y Y VDev 319,1% 319,1% 319,1% 319,1% 319,1% 319,1% 0,0% 0,0% 0,0%
As can be seen from the table5.3by increasing the investment risk profile in the active phase the 2.5% percentile Real Replacement Ratio is decreased for the ‘newly active’. On the other hand the best results, that is the 97.5% percentile, are increased more. Looking at the year-to-year fluctuation there is no difference. As we only consider the retirement phase, at which the investment risk profile is the same, it is logically that this is the case. The same holds for the ‘retiree’.
Comparison variant 2a
As the lifecycle after retirement in variant 2a is equal to the lifecycle in variant 1, we saw that the results of the ‘retiree’ are the same as the results from the previous section. Therefore we will only focus on the ‘newly active’ when comparing the two variants. As can be seen in table 5.4, the increase in risky assets in the active phase has a different effect depending on the risk aversion parameter. A risk seeking (γ = 3) individual will want to increase it’s exposure to risky assets. The risk averse ‘newly active’ is more or less indifferent. In the scenario’s where the realized returns do not exceed the inflation
level, the State old-age pension will be increased more (as it is incremented with the inflation). As the certainty of the State old-age pension becomes a relatively large part of the discretionary income the willingness to invest in a more risky lifecycle increases as well. If we look at the results in absence of the State old-age pension, we see a similar effect as in variant 1. The absence of a constant adjusted for inflation (the State old-age pension) shifts the outcome levels downwards and put more weight (by the utility function) at the scenario’s in the lower percentiles. Hence a (very) risk averse individual will be unwilling to take more risk in the active phase.
Table 5.4: Certainty equivalent CEγ results for the ‘newly active’ of discretionary income both
including and excluding the State old-age pension for variant 2a in comparison with variant 1. The impact being defined relatively to variant 1.
Variable benefit - variant 1 Variable benefit - variant 2a Impact CE S0 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 DIIncl 37,000 58,783 53,879 50,184 59,203 53,996 50,209 0.7% 0.2% 0.1% 74,000 112,193 103,665 97,113 113,141 103,913 97,082 0.8% 0.2% 0.0% 103,317 124,785 119,569 115,109 126,002 119,965 114,938 1.0% 0.3% -0.1% DIExcl 37,000 51,748 45,609 39,916 52,516 45,609 39,038 1.5% 0.0% -2.2% 74,000 105,458 95,423 86,087 106,696 95,354 84,491 1.2% -0.1% -1.9% 103,317 116,774 106,801 94,278 118,281 106,536 91,627 1.3% -0.2% -2.8% Results variant 2b
Compared to variant 2a there is a higher allocation to risky assets at the retirement phase. As a consequence the investment profile in the active phase is slightly more riskier as the glidepath (that is the decreasing trajectory of the equity allocation) is less steep. At retirement the investment profile remains more offensive as the allocation to risky assets has moved up by 20%-points. As we saw with variant 2a the more riskier the investment profile the higher the expected benefit and real replacement ratio’s. As a downside the probability interval is increased, thus increasing the possibility of bad results. This can also be seen by the Year-to-Year volatility, which on average is increase from 4.3% to 7.2%, see also table5.5
Table 5.5: Key figures Y Y VAvg, Y Y VDev and the 2.5%, 50.0% and the 97.5% of the Real
Re-placement Ratio’s RRR of variant 2b in comparison with variant 1. The impact being defined relatively to variant 1.
Reference Variable benefit - variant 1 Variable benefit - variant 2b Impact
person Criteria S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 ‘newly active’ RRR2.5% 34.7% 27.7% 20.4% 33.4% 26.3% 19.4% -3.7% -5.2% -5.2% RRR50.0% 53.7% 49.7% 37.6% 63.7% 61.4% 46.9% 18.7% 23.5% 24.6% RRR97.5% 106.7% 112.5% 89.1% 188.1% 208.9% 168.7% 76.4% 85.7% 89.3% Y Y VAvg 4.3% 4.3% 4.3% 7.2% 7.2% 7.2% 66.7% 66.7% 66.7% Y Y VDev 157.5% 157.5% 157.5% 212.8% 212.8% 212.8% 35.1% 35.1% 35.1% ‘retiree’ RRR2.5% 45.1% 37.4% 35.2% 44.0% 36.1% 33.8% -2.3% -3.6% -4.0% RRR50.0% 66.4% 65.3% 65.0% 74.7% 76.1% 76.5% 12.4% 16.5% 17.7% RRR97.5% 97.5% 105.9% 108.3% 187.2% 223.1% 233.3% 92.0% 110.6% 115.4% Y Y VAvg 4.3% 4.3% 4.3% 7.2% 7.2% 7.2% 66.7% 66.7% 66.7% Y Y VDev 319.1% 319.1% 319.1% 695.0% 695.0% 695.0% 117.8% 117.8% 117.8% Comparison variant 2b
As there is a difference in the lifecycle for both the active phase and the retirement phase, we will now look at both reference persons. As we saw in variant 2a the risk seekers are willing to have an even higher risk investment profile. However the risk averse individual with γ = 7 is unwilling to take more risk. The exception being the individual with average income. This individual is indifferent between choosing variant 1 and variant 2b. As the State old-age pension is a great part of his discretionary in-come after retirement he is guaranteed a relatively large sum of money. This increases
Preferences in IDC with variable benefits — Ren´e van Pul 23
his willingness to accept a more risky lifecycle.
Table 5.6: Certainty equivalent CEγ results of discretionary income including the State old-age
pension for variant 2b in comparison with variant 1. The impact being defined relatively to variant 1.
Reference Variable benefit - variant 1 Variable benefit - variant 2b Impact person S0 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 ‘newly active’ 37,000 58,783 53,879 50,184 59,637 54,083 50,204 1.5% 0.4% 0.0% 74,000 112,193 103,665 97,113 114,057 103,956 96,710 1.7% 0.3% -0.4% 103,317 124,785 119,569 115,109 127,143 120,011 113,889 1.9% 0.4% -1.1% ‘retiree’ 37,000 30,204 29,568 29,063 31,206 29,969 29,071 3.3% 1.4% 0.0% 74,000 59,476 57,950 56,732 61,463 58,103 55,498 3.3% 0.3% -2.2% 103,317 82,607 80,331 78,487 85,281 80,085 75,850 3.2% -0.3% -3.4%
In table 5.7 the certainty equivalent levels, when considering the discretionary income without the State old-age pension, is shown. The impact in this case are higher. As the risk averse individual does not have the guarantee of this State old-age pension he/she becomes more unwilling to take risk. The chance that the benefit will be too low will weigh more heavily and thus is punished more heavily if you compare it with the addition of the State old-age pension. Thus even a conservative individual will not choose a lifecycle in which the investment risk profile is too high and no guarantee is available. This holds true for all reference persons considered with the exception of the risk seeking individual.
Table 5.7: Certainty equivalent CEγ results of discretionary income excluding the State old-age
pension for variant 2b in comparison with variant 1. The impact being defined relatively to variant 1.
Reference Variable benefit - variant 1 Variable benefit - variant 2b Impact person S0 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 ‘newly active’ 37,000 51,748 45,609 39,916 53,108 44,299 35,019 2.6% -2.9% -12.3% 74,000 105,458 95,423 86,087 107,639 93,301 77,161 2.1% -2.2% -10.4% 103,317 116,774 106,801 94,278 119,414 103,536 81,888 2.3% -3.1% -13.1% ‘retiree’ 37,000 18,067 17,447 16,903 18,544 16,936 15,172 2.6% -2.9% -10.2% 74,000 47,207 45,587 44,166 48,455 44,252 39,643 2.6% -2.9% -10.2% 103,317 70,296 67,885 65,768 72,155 65,896 59,033 2.6% -2.9% -10.2%
5.1.3 Variant 3: Decreasing risk
Description
In the previous variant we focused on increasing the risk profile of the lifecycle by increasing the allocation to equities. Now we will focus on limiting the risk profile of the lifecycle by decreasing the allocation to equities. All other assumptions will remain the same as in variant 1. In line with variant 2 we will consider two ways in which the investment risk profile of the lifecyle can be limited, namely:
• a: a decrease in risk, only in the active phase;
• b: a decrease in risk, both in the active phase as well as in the retirement phase. By decreasing the starting allocation of the risky assets by 20%-points we will limit the risk taking in the active phase. The fixed allocation to risky assets at the retirement phase will remain at the level of variant 1, namely 30%. In the very defensive lifecycle (variant 3b), the allocation to risky assets at the retirement phase will also be decreased by 20%-points. In both lifecycles the glidepath will start at age 50, the same age as in variant 1.
Figure 5.3: Graphical comparison of the lifecycles of variant 1, variant 3a and variant 3b. Only the allocation to equity is given.
Results variant 3a
Table 5.8: Key figures Y Y VAvg, Y Y VDev and the 2.5%, 50.0% and the 97.5% of the Real
Re-placement Ratio’s RRR of variant 3a in comparison with variant 1. The impact being defined relatively to variant 1.
Reference Variable benefit - variant 1 Variable benefit - variant 3a Impact
person Criteria S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 ‘newly active’ RRR2.5% 34,7% 27,7% 20,4% 34,8% 27,9% 20,6% 0,3% 0,6% 0,7% RRR50.0% 53,7% 49,7% 37,6% 48,7% 43,9% 32,9% -9,2% -11,7% -12,5% RRR97.5% 106,7% 112,5% 89,1% 75,7% 75,3% 57,7% -29,0% -33,1% -35,3% Y Y VAvg 4,3% 4,3% 4,3% 4,3% 4,3% 4,3% 0,0% 0,0% 0,0% Y Y VDev 157,5% 157,5% 157,5% 157,5% 157,5% 157,5% 0,0% 0,0% 0,0% ‘retiree’ RRR2.5% 45,1% 37,4% 35,2% 45,1% 37,4% 35,2% 0,0% 0,0% 0,0% RRR50.0% 66,4% 65,3% 65,0% 66,4% 65,3% 65,0% 0,0% 0,0% 0,0% RRR97.5% 97,5% 105,9% 108,3% 97,5% 105,9% 108,3% 0,0% 0,0% 0,0% Y Y VAvg 4,3% 4,3% 4,3% 4,3% 4,3% 4,3% 0,0% 0,0% 0,0% Y Y VDev 319,1% 319,1% 319,1% 319,1% 319,1% 319,1% 0,0% 0,0% 0,0%
An opposite effect as opposed to the variant 2a can be seen. The ‘worst’ scenario’s, that is the 2.5% percentile, will slightly increase the Real Replacement Ratio’s due to the risk limitation. The ‘best’ outcome, that is the 97.5% percentile, will be decreased more. As a result the probability interval in which the benefits will lie becomes more narrow. Again, as in variant 2a, the year-to-year fluctuation and the key figures for the ‘retiree’ do not change since in the retirement phase the risk profile is the same.
Comparison variant 3a
Again we will only focus on the certainty equivalents of the ‘newly active’, since the lifecycle at retirement is the same for the ‘retiree’.
Table 5.9: Certainty equivalent CEγ results for the ‘newly active’ of discretionary income both
including and excluding the State old-age pension for variant 3a in comparison with variant 1. The impact being defined relatively to variant 1.
Variable benefit - variant 1 Variable benefit - variant 3a Impact CE S0 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 DIIncl 37,000 58,783 53,879 50,184 57,996 53,583 50,087 -1.3% -0.5% -0.2% 74,000 112,193 103,665 97,113 110,237 102,837 96,839 -1.7% -0.8% -0.3% 103,317 124,785 119,569 115,109 122,203 118,115 114,445 -2.1% -1.2% -0.6% DIExcl 37,000 51,748 45,609 39,916 50,024 44,729 40,159 -3.3% -1.9% 0.6% 74,000 105,458 95,423 86,087 102,577 94,122 86,623 -2.7% -1.4% 0.6% 103,317 116,774 106,801 94,278 113,148 105,001 95,417 -3.1% -1.7% 1.2%
In contrast to the previous variant every individual will prefer to take more risk if we look at the discretionary income including State old-age pension. As the capital
Preferences in IDC with variable benefits — Ren´e van Pul 25
accrued will be expected to be less in comparison to variant 1, the State-old age pension becomes a more significant part of the discretionary income as the lifecycle becomes more defensive. Since the individual is guaranteed a relatively larger sum of money he is willing to accept more risk. If the State old-age pensions is not included this effect will diminish. A risk averse ‘retiree’ will prefer the lifecycle which is a bit safer.
Results variant 3b
As can be seen in figure 5.3 the lifecycle in variant 3b will decrease its allocation to equities in the retirement phase by 20%-points. As a result the glidepath will be steeper in comparison to the previous variant, that is the lifecycle will become even more de-fensive. Logically the Year-to-Year volatility will decrease as less risk is taken. Less risk implies that the expected excess return (that is the return above the risk-free rate) will be lower. In case of variant 3b it will be as low as 0.575%, as we only have a 10% allocation to equity with an expected excess return of 5.75% (from table4.1the surplus of the expected return µr over the risk-free rate rf). As the inflation is assumed to be fixed at 2%, the benefit is not expected to keep up with the inflation. Hence the Real Replacement Ratio’s will be lower as well. Only in the 97.5% percentile the ‘retiree’ has a Real Replacement Ratio equalling the starting ratio.
Table 5.10: Key figures Y Y VAvg, Y Y VDev and the 2.5%, 50.0% and the 97.5% of the Real
Replacement Ratio’s RRR of variant 3b in comparison with variant 1. The impact being defined relatively to variant 1.
Reference Variable benefit - variant 1 Variable benefit - variant 3b Impact
person Criteria S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 S0= 37, 000 S0= 74, 000 S0= 103, 317 ‘newly active’ RRR2.5% 34,7% 27,7% 20,4% 35.0% 28.1% 20.7% 1.0% 1.3% 1.2% RRR50.0% 53,7% 49,7% 37,6% 44.0% 38.5% 28.8% -18.1% -22.7% -23.6% RRR97.5% 106,7% 112,5% 89,1% 59.6% 56.9% 43.6% -44.1% -48.4% -51.1% Y Y VAvg 4,3% 4,3% 4,3% 1.4% 1.4% 1.4% -66.7% -66.7% -66.7% Y Y VDev 157,5% 157,5% 157,5% 116.4% 116.4% 116.4% -26.1% -26.1% -26.1% ‘retiree’ RRR2.5% 45,1% 37,4% 35,2% 43.2% 35.0% 32.7% -4.0% -6.3% -7.2% RRR50.0% 66,4% 65,3% 65,0% 55.3% 50.8% 49.5% -16.8% -22.3% -23.9% RRR97.5% 97,5% 105,9% 108,3% 70.0% 70.0% 70.0% -28.2% -33.9% -35.4% Y Y VAvg 4,3% 4,3% 4,3% 1.4% 1.4% 1.4% -66.7% -66.7% -66.7% Y Y VDev 319,1% 319,1% 319,1% 147.1% 147.1% 147.1% -53.9% -53.9% -53.9% Comparison variant 3b
Every individual considered in this paper prefers the neutral (default) lifecycle from variant 1 over the lifecycle from variant 3b. Even the most risk averse individual prefers the lifecycle with more risk. This is however not so surprising as the contribution of the lifecycle to the benefit received will become lower as the risk profile will decrease. The individual will therefore not perceive the reward from taking the risk. Hence in order to perceive a positive effect from taking risk he is willing to take more risk.
Table 5.11: Certainty equivalent CEγ results of discretionary income including the State old-age
pension for variant 3b in comparison with variant 1. The impact being defined relatively to variant 1.
Reference Variable benefit - variant 1 Variable benefit - variant 2b Impact person S0 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 ‘newly active’ 37,000 58,783 53,879 50,184 57,153 53,213 49,951 -2.8% -1.2% -0.5% 74,000 112,193 103,665 97,113 108,014 101,683 96,356 -3.7% -1.9% -0.8% 103,317 124,785 119,569 115,109 119,402 116,161 113,234 -4.3% -2.9% -1.6% ‘retiree’ 37,000 30,204 29,568 29,063 28,850 28,632 28,439 -4.5% -3.2% -2.1% 74,000 59,476 57,950 56,732 56,171 55,856 55,573 -5.6% -3.6% -2.0% 103,317 82,607 80,331 78,487 77,792 77,383 77,014 -5.8% -3.7% -1.9%
However if there is no guarantee to which the individual can fall back on, a risk averse ‘newly active’may be willing to accept a low risk profile. This is shown in table 5.12. The most risk averse ‘newly active’ considered now prefers the very defensive lifecycle.
