The welfare implications of investment risks in the benefit phase of Individual Defined Contribution pension schemes

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The welfare implications of investment

risks in the benefit phase of Individual

Defined Contribution pension schemes

Jack Kroon

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: Jack Kroon

Student nr: 10001626

Email: jack kroon@hotmail.com Date: April 21, 2017

Supervisor: Dr. Servaas van Bilsen Second reader: . . .

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This document is written by Student Jack Kroon 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.

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Investment risks in the benefit phase of IDC — Jack Kroon iii

Abstract

This study examines the welfare implications of investment risks in the benefit phase of individual Defined Contribution pension schemes. With simple stochastic economic scenarios, the development of variable pension benefits are calculated and are compared with a fixed pension benefit. The equity returns are assumed to be normal distributed and non-correlated. A welfare comparison is made by using the weighted real replacement ratios with a constant relative risk aversion (CRRA) utility function. For the calculation of the real replacement ratios, all benefits in the retirement period, corrected with price inflation, are expressed as ratio of the final careers salary and are weighted with the relative survival rates. For nine different reference persons (three income levels and three risk-aversion levels), the welfare effects are compared for different variants of the variable pension benefit. The variants differ in investment policy, whether or not smoothing of investment results and the level of the discount rate. Taking investment risks in the benefit phase of an individual Defined Contribution pension scheme provides a welfare gain for all types of persons. For an average person the variable pension benefit gives a welfare gain between +3.0% and +9.0%.

Keywords Pensions, Variable pension benefit, Investments, Portfolio, Welfare, Stochastic, Normal distribution, Simple, Uncertainty, Utility function, Real replacement ratio, Certainty equivalent, Netherlands

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Preface vi

1 Introduction & overview 1

1.1 Defined Contribution in the Netherlands . . . 1

1.1.1 Old situation . . . 1

1.1.2 New situation . . . 1

1.2 Recent studies . . . 2

1.3 Objective of this thesis . . . 2

2 Data & Methodology 3 2.1 Data . . . 3

2.1.1 Economic scenarios. . . 3

2.1.2 Mortality rates . . . 4

2.2 Methodology in the benefit phase . . . 5

2.2.1 Determination of asset allocations in the benefit phase . . . 5

2.2.2 Determination of annuity prices. . . 5

2.2.3 Determination of starting capitals . . . 6

2.2.4 Determination of pension benefits . . . 7

2.2.5 Determination of real replacement ratios. . . 8

2.3 Measuring welfare . . . 8

2.3.1 Utility function . . . 8

2.3.2 Discount function. . . 9

2.3.3 Certainty equivalent of weighted real replacement ratios . . . 10

3 Results 11 3.1 Assumptions . . . 11

3.1.1 Parameters in basis scenario. . . 11

3.1.2 Reference persons . . . 11

3.1.3 Variants in the benefit phase . . . 12

3.2 Variant 1: Variable pension benefit - Basis . . . 13

3.3 Variant 2: Increase of initial allocation return portolio . . . 14

3.4 Variant 3: Decrease of initial allocation return portolio . . . 15

3.5 Variant 4: Reducing allocation of return portfolio . . . 16

3.6 Variant 5: Risk-free interest rate as discount rate . . . 17

3.7 Variant 6: Smoothing of investment results . . . 18

3.8 All combinations . . . 19

3.9 Sensitivities . . . 20

3.9.1 Change in risk premium on equity . . . 20

3.9.2 Change in volatility equity returns . . . 21

4 Conclusion 22

5 Discussion 23

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Investment risks in the benefit phase of IDC — Jack Kroon v

Appendix A: Smoothing of gains and losses 24

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While writing this thesis, I realized that my study nearly comes to an end. I am thankful to everyone who has supported me. First, I would like to thank Jack Tol, my supervisor at Triple A - Risk Finance, for sharing his knowledge, giving me useful instructions and for guiding me through my research. Secondly, I would like to thank Servaas van Bilsen, my supervisor at the University of Amsterdam, for his support and useful advice. I am also grateful to my parents for their support during my whole study. Finally, I would like to thank my girlfriend Elma for helping me to get my strength back to continue with this thesis.

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Chapter 1

Introduction & overview

1.1 Defined Contribution in the Netherlands

1.1.1 Old situation

Every month, participants in Defined Contribution (DC) schemes pay premiums which are invested in an asset mix. The allocation of these assets usually follows a lifecycle with more risky assets at a younger age and a reduced allocation of risky assets on the pensionable age. The Dutch Financial Markets Authority (AFM) prescribes that this risk reduction has to be ’timely’ and ’smoothly’. Specifically this means that a standard investment mix should be offered, which develops in such way that the uncertainty of the pension benefit is reduced when retirement approaches. At retirement, the participant has to convert his accumulated capital into a lifetime guaranteed old age pension. The amount of the pension benefit depends on the size of the accumulated capital and the purchase price of the lifetime benefit. This price is strongly influenced by the interest rates at the time of purchase. In the benefit phase, there is no possibility to take invest-ment risk and the annual pension benefits will not be adjusted for inflation. However, it is possible to buy-in a annnual percentage increase of the benefit in advance. The shortcomings of DC in the Netherlands can be summarized as follows:

• The level of pension benefits are strongly dependent on the timing of pensioning, due to the interest rates at the time of the purchase of a lifetime old age pension. • There is no possibility to take investment risk in the benefit phase. This leads to

a lack of returns on investments.

An increase of investment risks leads to higher expected returns, see for example Fama

& MacBeth(1973). The absence of investment returns and the dependence on the date

at which you retire were the main reason to introduce a new pension act. This will be explored in the next section.

1.1.2 New situation

To solve above-mentioned problems, a new act was adopted in September 2016: “Wet

verbeterde premieregeling”(2016). This new regulation introduces an alternative choice

to the participant: a variable non-guaranteed old age pension. This variable pension benefit fluctuates periodically due to investment risk, changes in macro longevity and mortality result realized on the micro longevity (when taking part into a collective of participants). The level of the benefit will be determined every period in such a way that it can last a lifetime, give the actuarial assumptions. This means that the level of the pension benefit is based on the expected return on assets (EROA), the assumed mortality rates and the existing accumulated capital. The buy-in is not completely reliant on the economic situation at the pensionable age. The variable pension benefit

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extends the investment horizon of the participant, since it is possible to take investment risks in the benefit phase. Steenkamp (2004) says that, under certain assumptions, an extended investment horizon gives the participant opportunity to take more investment risks. This gives the participant the opportunity to invest in more risky assets and receive higher expected returns. With the variable pension benefit, the participant can choose for a discount rate equal to the EROA or equal to the risk-free interest rate. When choosing for a discount rate equal to the EROA, the level of the pension benefit starts at higher level than the fixed pension benefit, since the price of the annuity decreases due to a higher discount rate. With the EROA as discount rate, the variable benefit is expected to stay stable on average. On the other hand, when choosing for a discount rate equal to the risk-free interest rate, the level of the pension benefit starts at the same level as the fixed pension benefit and it is expected to increase annually. Finally, there is an option to reduce the volatility of the pension benefit. Gains and losses of investment and longevity risks can be smoothed over a period with a maximum of ten years.

1.2 Recent studies

Steenkamp (2016) discusses the effects of taking investment and longevity risks in the

benefit phase individually. The buy-in of a fixed guaranteed benefit is compared with a floating non-guaranteed benefit using a simple and evident model. The comparison is based on characteristics of the probability distributions of the pension benefit. Fur-thermore, a welfare comparison is made by using the weighted real pension result with a constant relative risk aversion (CRRA) utility function. For the calculation of the real pension result, all benefits in the retirement period, corrected with price inflation, are taken into account and are weighted with the relative survival rates. The conclu-sion of Steenkamp is that taking investment risk in the benefit phase of a individual defined contribution (IDC) scheme leads to an increase of welfare for the participant. This agrees with the study of Lane, Clark & Peacock (2014), which served as basis of the calculation model used by Steenkamp. Steenkamp based his conclusions on only one reference person: a 25-year old person with a starting salary of 28 000 EUR and CRRA utility function with risk aversion parameter 5. Furthermore, the effects of increased investment risk in the premium phase and investment risks in the benefit phase are not separated. The welfare gain as described, includes the effect of extra risk in the premium phase.

1.3 Objective of this thesis

This study will address the following question:

What are the welfare implications of investment risks in the benefit phase of an individ-ual defined contribution pension scheme?

The study of Steenkamp (2016) serves as the basis of this paper and will be extended with analyses of different reference persons. In this thesis the salary level, starting capital and risk aversion will vary to make up for the different reference persons. The objective of this thesis is to investigate the welfare implications of taking investment risk in the benefit phase at different levels of the aforementioned variables. This will be addressed in detail in Chapter 2.

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Chapter 2

Data & Methodology

2.1 Data

2.1.1 Economic scenarios

The capital can be invested in a return portfolio and a match portfolio. The return portfolio consists primarily of equity and the goal is to achieve the highest possible return. The matching portfolio is arranged so that the portfolio is a hedge against movements of the price of a fixed pension benefit. The interest rate risk and longevity risk are mitigated in this way. To concentrate on the investment risk, only the equity returns rs,t are stochastically simulated for scenarios s = 1, ..., 10000.

rs,t∼ N (µr, σ2_r) (2.1)

where: s = scenario

t = year in benefit phase µr = mean return

σr = standard deviation return

Just asSteenkamp (2016), we assume in the basis scenario that the equity returns rs,t are normally distributed with mean µr = 6.75% and standard deviation σr = 20.00%. These assumptions agree with the report ofCommissie parameters (2014). The equity returns in our model are supposed to be uncorrelated for simplicity. In reality, equity returns are auto-correlated. The risk-free interest rate (rf) and the price inflation (Π) are assumed to be fixed. The expected risk premium on equity πe is defined as:

πe= µr− rf (2.2)

The risk premium is a fee for investing in risky assets. Generally, it applies that the riskier the investment, the higher the risk premium. The assumptions in the basis sce-nario are summarized below:

Parameter Value Description

µr 6.75% Mean of equity returns

σr 20.00% Standard deviation of equity returns rf 1.00% Risk-free interest rate

Π 2.00% Price inflation

πe 5.75% Risk premium on equity

Table 2.1: List of used economic parameter values in basis scenario

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2.1.2 Mortality rates

The most recent mortality tables of The Royal Dutch Actuarial Association(2016) are used: AG Prognosetafel 2016. It is assumed that all reference persons have the age of 67 in 2017. For simplicity, we only used the male mortality rates. The survival probability kpx of a person with age x to live k more years is defined in2.3.

kpx= k Y i=1 (1 − qx+i−1) (2.3) where:

qx = the mortality rate of a person with the age of x k = the years of survival

By using the survival probabilities it is possible to calculate the remaining life ex-pectancy L67+t at time t: L67+t = 1 2+ 121−(67+t) X i=1 ip67+t (2.4)

It is assumed that a person will die on average at the half of the year. We will use the remaining life expectancy for the determination of the life cycles in the benefit phase with the reducing allocation of risky assets. Figure2.1shows the plot of the remaining life expectancy L67+t in years at age 67 + t for a male in the Netherlands with t = 0 at the beginning of 2017.

Figure 2.1: Remaining life expectancy L67+tin years at age 67 + t for a male in the Netherlands

with t = 0 at the beginning of 2017

0 2 4 6 8 10 12 14 16 18 20 67 72 77 82 87 92 97 102 107 112 117 Age

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Investment risks in the benefit phase of IDC — Jack Kroon 5

2.2 Methodology in the benefit phase

2.2.1 Determination of asset allocations in the benefit phase

Fixed allocation

If the variant in the benefit phase has a fixed allocation of the return portfolio, then this allocation αt at time t is equal to the intitial allocation α0 for every period in the benefit phase.

Reducing allocation

In case of a reducing allocation of the return portfolio, we assume that the proportion αtto α0 is equal to the proportion of L67+t to L67 (2.5).

αt α0 = L67+t L67 (2.5) where:

α0 = the start allocation of the capital in the return portolio

For simplicity it is chosen to use the remaining life expectancy as a measure of the expected investment horizon. The results of the determination of the asset allocations in the benefit phase are shown the figure below:

Figure 2.2: Allocation of return portfolio as percentage of total allocation for different investment life cycles at age 67 + t

0% 5% 10% 15% 20% 25% 30% 35% 67 72 77 82 87 92 97 102 107 112 117 Age

Defensive - fixed allocation Neutral - fixed allocation Offensive - fixed allocation Defensive - reducing allocation Neutral - reducing allocation Offensive - reducing allocation

2.2.2 Determination of annuity prices

Since we assume that the risk-free interest rate rf is fixed, we can determine the annuity prices before looping through all economic scenarios. We assume these are the ’fair’ prices, since they are abstracted from capital surcharges and other fees. The annuity price is based on annual prenumerando benefits.

Annuity price for fixed benefit

At the age of 67 the accumulated capital is converted into a fixed lifetime annuity. Since the level of the benefit is fixed and risk-free, the annuity price at time t P Ftis calculated

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considering the risk-free interest rate rf as discount rate and the survival probabilities kpx. Formula2.6 shows this calculation method for t = 0, ..., 54.

P Ft= 1 + 121−(67+t) X i=1 ip67+t (1 + rf)i , (2.6) where:

kpx = the survival probability of a person with age x to live k more years

By summing the present value of annual benefits of 1, from the age of 67 + t to 121, the price for a fixed benefit P Ft at time t can be determined. The price for a fixed benefit at time t P Ft decreases, if the risk-free interest rate rf increases.

Annuity price for variable benefit

According to the New Act, it is possible to determine the annuity price for a variable benefit with the Expected Return On Assets (EROA) as discount rate. The new act states that this discount rate is maximized at the EROA of an asset mix which consists of 35% equity. We use formula 2.7for the calculation of the price of a variable annuity P Vt for t = 0, ..., 54. P Vt= 1 + 121−(67+t) X i=1 ip67+t Qi j=1 1 + rf + αt+j· πe , (2.7) where:

kpx = the survival probability of a person with age x to live k more years αt = the allocation of the capital in the return portolio at time t

πe = the expected risk premium on equity

We assume that the EROA will take the development of the asset allocation into ac-count. The EROA from t to t + 1 is assumed to be rf+ αt· πe. The variable benefit will fluctuate annually, but is expected to be stable on average with this method of pricing. The price for a variable benefit at time t P Vt decreases, if the EROA increases.

2.2.3 Determination of starting capitals

Next to the determination of the annuity prices, the starting capitals are also determined before looping through all scenarios. For every scenario s it is assumed that, when choosing for a fixed benefit, the starting capital Cs,0is equal to a level where the pension income including AOW (BF + AOWt) at t = 0 is equal to ρ0· S−1 (2.8). ρ0 is here a predetermined intial replacment ratio and S−1 is the final salary during the reference person’s career. Furthermore, we assume the AOWtat t = 0 is equal to 0.75 · F0, where F0 is the state pension offset (franchise, in Dutch) in the Netherlands at t = 0.

BF + AOW0 = ρ0· S−1 (2.8)

Cs,0= P F0· (ρ0· S−1− AOW0) (2.9)

After some algebra, the starting capital can be expressed as 2.9. The fixed pension benefit (excluding AOW) BF is here replaced by Cs,0_{P F0}.

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Table 2.2: Development of fixed and variable annuity prices for all different variants a. Risk-free interest rate rf is equal to 1.00%

P Ft P Vt α0 = 15% α0 = 25% α0 = 35% t Age α0= αt α0> αt α0= αt α0> αt α0 = αt α0 > αt 0 67 16.86 15.50 15.85 14.69 15.22 13.95 14.63 5 72 13.82 12.89 13.26 12.33 12.90 11.80 12.56 10 77 10.91 10.31 10.62 9.95 10.44 9.61 10.27 15 82 8.26 7.91 8.13 7.69 8.05 7.49 7.97 20 87 6.00 5.81 5.95 5.69 5.92 5.58 5.88 25 92 4.27 4.17 4.25 4.11 4.24 4.06 4.23

b. Risk-free interest rate rf is equal to 3.50%

P Ft P Vt α0 = 15% α0 = 25% α0 = 35% t Age α0= αt α0> αt α0= αt α0> αt α0 = αt α0 > αt 0 67 13.37 12.83 12.96 12.49 12.70 12.17 12.45 5 72 11.39 11.00 11.15 10.76 11.00 10.52 10.85 10 77 9.34 9.08 9.21 8.91 9.13 8.75 9.05 15 82 7.32 7.16 7.26 7.06 7.22 6.96 7.19 20 87 5.49 5.40 5.46 5.34 5.45 5.29 5.43 25 92 4.01 3.96 4.00 3.93 3.99 3.90 3.99

2.2.4 Determination of pension benefits

Fixed benefit

The fixed benefit BF has only to be determined at t = 0 and is not dependent on the stochastic equity returns in the future. At t = 0 the starting capital Cs,0 is converted into the fixed benefit BF by using the price for the fixed benefit P F0 (2.10).

BF = Cs,0

P F0 (2.10)

This benefit stays at the same level in the next years. Note that the starting capital Cs,0 is equal for each scenario s.

Variable benefit

The variable benefit BVs,t has to be determined each year and is dependent on the stochastic equity returns in scenario s. At time t the capital Cs,t is converted into a benefit by using the price for the variable benefit P Vt (2.11).

BVs,t= Cs,t P Vt

(2.11)

The capital in the next period Cs,t+1 is calculated by using formula 2.12. This is de-pendent on the return on equity and the allocation of the asset portfolio of the past year.

Cs,t+1=

Cs,t− BVs,t

1p67+t−1 · (1 + αt· rs,t+ (1 − αt) · rf) (2.12) We assume that the capital does not lapse to the heirs when death occurs. To consider

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this transfer at death, a so-called biometric return is credited annually to the accumu-lated capital. To add the biometric return, the capital is divided with 1p67+t in 2.12. When the participant gets older, the mortality rates and thus the biometric returns increases. By adding the biometric returns to the capital, participants are insured to individual longevity. This method is derived from Bovenberg & Nijman (2015).

Variable benefit with smoothing of gains and losses

If the participant chooses for the option to smooth gains and losses, the process for the determination of the variable pension benefit is a bit more complicated. The exact methodology can be found in Appendix A. The idea is that investment results are not taken into account immediatly, but are smoothed over a longer period. This results in a less fluctuating pension benefit. Though, if subsequent losses are shifted to the future, the pension benefits can decrease drastly.

2.2.5 Determination of real replacement ratios

To assess the pension benefits, we will look at the development of the real replacement ratio of the total pension income (including AOW). The real replacement ratios give the purchasing power effects. The AOWt is assumed to increase each year with the price inflation (2.13).

AOWt= AOW0· (1 + Π)t (2.13)

The real replacement ratio RRRt at time t are defined as the total pension income (corrected with the cumulative inflation) divided by the gross salary at retirement (2.14).

RRRs,t = BVs,t+ AOWt S0· (1 + Π)t

(2.14)

In the next section, it will be explained how we use the real replacement ratios as input for the measure of welfare.

2.3 Measuring welfare

Our goal is to analyze the welfare effects of the transition from a fixed to a variable pension benefit for different persons with different risk attitudes.

2.3.1 Utility function

The risk attitude can be represented by using utility functions with risk-aversion param-eter γ. The higher γ, the more risk-averse an individual is. Due to the characteristics of utility functions, a lower pension income will be ’punished’ heavier than a proportionally higher pension result will be ’rewarded’. All persons are supposed to have a constant relative risk aversion (CRRA) utility function u(x) 2.15.

u(x) = x

1−γ_{− 1}

1 − γ , (2.15)

where:

x = input of the utility function γ = level of risk aversion

We use the real replament ratios RRRtas input for the utility functions to measure the welfare of the participant at time t. Figure 2.3shows the plots of utility function u(x)

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for γ = 3, 5, 7. It can be seen that lower values of the input will be punished heavier for higher values of γ.

Figure 2.3: Plots of utility function u(x) = x1−γ_1−γ−1 for γ = 3, 5, 7

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 u(x) x γ = 3 γ = 5 γ = 7 2.3.2 Discount function

The discount function describes the weights placed on rewards received at different time periods. A person receives an amount of cash preferably early than late. For simplicity, we assume that f (x)2.16 is the discount function with discount factor is fixed at 99%.

f (t) = δt, (2.16)

where:

t = the year in benefit phase δ = discount factor

The plot for f (x) with δ = 99% is shown in figure 2.4.

Figure 2.4: Plots of discount function f (t) = δt_{for δ = 99%}

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 25 30 35 40 45 50 f(t) t

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2.3.3 Certainty equivalent of weighted real replacement ratios

For each reference person and variant in the benefit phase, all annual pension benefits and real replacement ratios are calculated for 10,000 different economic scenarios. We define U as the total weighted replacement ratios.

U = P10,000 i=1 P54 t=0u(RRRi,t) · f (t) ·tp67·_10,0001 P54 t=0f (t) ·tp67 (2.17)

The welfare is calculated as the certainty equivalent of the total weighted real replace-ment ratios U. The certainty equivalent gives the value that an individual wants to receive with 100 percent certainty to be indifferent in the choice of an uncertain option. The certainty equivalent CE is defined in formula2.18.

CE = u−1(U ) (2.18)

By using the certainty equivalent (CE), a certain fixed pension income can be compared with a uncertain variable pension income. The CE varies with an individual’s attitude towards risk. This may give a different insight than just looking at the pension benefits.

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Chapter 3

Results

3.1 Assumptions

3.1.1 Parameters in basis scenario

Table 3.1: List of used parameter values in basis scenario Parameter Value Description

µr 6.75% Mean of equity returns

σr 20.00% Standard deviation of equity returns rf 1.00% Risk-free interest rate

Π 2.00% Price inflation

πe 5.75% Risk premium on equity

ρ 70.00% Initial replacement ratio

δ 99.00% Discount factor for time preference F0 12 953 EUR Offset in the Netherlands in 2016

AOW0 9 715 EUR State pension in the Netherlands in 2016

3.1.2 Reference persons

The input for a reference person is the final salary of the participants career S−1 and the accumulated capital at retirement. Three different salaries are taken into account:

• S−1 = 36 500 EUR (Average income)

• S−1 = 73 000 EUR (Double average income)

• S−1 = 101 519 EUR (Maximum fiscal pensionable salary)

In the base scenario we assume that the initial accumulated capitals are supposed to be at such level where the replacement ratio ρ of the fixed pension benefit is equal to 70%. The replacement ratio is defined as the pension income at t = 0, including AOW, expressed as percentage of the final salary in the participants career. By using formula

2.9, the starting capital Cs,0are calculated and presented below. Table 3.2: Starting capitals Cs,0 for each different final salary

Cs,0

S−1 rf = 1.00% rf = 3.50% 36 500 267 011 211 665 73 000 697 831 553 185 101 519 1 034 449 820 029

Furthermore, three different risk aversion levels γ can be distinguished: 11

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• γ = 3 (Aggressive) • γ = 5 (Moderate) • γ = 7 (Conservative)

The reference person used by Steenkamp (2016) had a γ equal to 5. This applies to a person with moderate risk-aversion. By adding γ = 3 for an agressive person and γ = 7 for a conservative person, we can eveluate the effecs of the variable pension benefit for persons with different types of risk tolerance. Figure 2.3 shows the plot of the utility functions for the three different γ’s.

With three different salaries and three different risk-averson paramters, we defined nine different reference persons in total.

3.1.3 Variants in the benefit phase

The following effects of the floating non-guaranteed pension benefit can be acknowl-edged:

• Exposure to investment risks in the benefit phase • Expected return on assets as discount rate • Smoothing of gains and losses

• Reduction of allocation in risky assets in the benefit phase as the persons grows older

To analyze each of the above mentioned effects, we have constructed six different variants for the variable pension benefit. Variant 1 is the basis variant where the allocation of the return portfolio is fixed at 25%. Furthermore, the EROA is used as discount rate. This variant agrees with the neutral variant of Dutch insurer Zwitserleven. To see what the effect is of a higher or lower intitial allocation of the return portfolio, we calculate variant 2 (offensive) and 3 (defensive). These portfolios agree with the asset allocation of portfolios ’defensive’ and ’ambitious’ at insurer Zwitserleven. With variant 4, we can find out what the effect is of a reducing allocation of the return portfolio. The use of the risk-free interest rate as discount rate is shown in variant 5. Here the intitial pension benefit starts at the same level as the fixed pension benefit, but it is expected to rise. In variant 6, the effect of smoothing gains and losses are shown. The characteristics of each variant are summarized in table 3.3.

Table 3.3: Summary of used variants in the benefit phase

Variant Benefit α0 αt Discount rate Smoothing

Benchmark Fixed α0 = 0% fixed rf No

1 - Basis Variable α0 = 25% fixed EROA No

2 - Increase α0 Variable α0 = 35% fixed EROA No

3 - Decrease α0 Variable α0 = 15% fixed EROA No

4 - Reducing αt Variable α0 = 25% reducing EROA No 5 - rf as discount rate Variable α0 = 25% fixed rf No

6 - Smoothing Variable α0 = 25% fixed EROA τ0 = 5

For each variant, the simulated pension benefits and real replacement ratio’s are summarized in a graph and compared with the benchmark. The welfare effects of the transition from a fixed to a variable pension benefit are shown for each reference person. At the end, the welfare effects of all combinations are summarized in a table. With those results it is possible to select a optimal combination for each type of person.

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3.2 Variant 1: Variable pension benefit - Basis

Simulation results

The results of the simulations for Variant 1 are shown below. The benefit starts at a higher level than the benchmark and is stable on average. The variable pension benefit in Variant 1 is on average higher than the fixed pension benefit, though it can reach lower values than the benchmark in certain scenarios.

Welfare effects

With the used assumptions in the base scenario, every reference person receives a welfare gain when choosing for Variant 1, compared to the benchmark. A reference person with a lower risk-aversion parameter γ receives a higher welfare gain. The differences between γ = 3 and γ = 7 gets larger when the final salary S−1 increases. This can be explained by the assumption that the part of the total pension income with AOW certainty is relatively smaller for a person with a higher final salary S−1.

Figure 3.1: Graphical representation of simulation results for 10,000 economic scenarios for the pension benefits Btand real replacement ratios RRRt. The results of this variant are shown for

a person with a final salary of 36 500 EUR. (a) Pension benefits Btat ages 67 + t

- 5 10 15 20 25 30 35 67 72 77 82 87 92 In 1,000 EUR Age 95% probability interval Mean Benchmark

(b) Real replacement ratios RRRtat ages 67+t

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 67 72 77 82 87 92 Age 95% probability interval Mean Benchmark

Table 3.4: Certainty equivalent of weighted real replacement ratios CEγ in per cents for each

person. The welfare effect is the percentage difference of the CEγcompared with the benchmark.

Benchmark Variable - variant 1

CEγ CEγ Welfare effects

S−1 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7

36 500 61.8 61.4 60.9 66.1 64.8 63.4 +6.9 +5.6 +4.2

73 000 59.0 58.1 57.3 64.0 61.5 58.8 +8.5 +5.8 +2.6

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3.3 Variant 2: Increase of initial allocation return portolio

Simulation results

The results of the simulations for Variant 2 are shown below. The benefit starts at a higher level than Variant 1 and is again stable on average. The variable pension benefit in Variant 2 is on average higher than for Variant 1, though the lower bound of the 95% probability interval is lower for Variant 2.

Compared with Variant 1, the probability interval has increased. This can be ex-plained by the extra amount invested in risky assets.

Welfare effects

With the used assumptions in the base scenario, not every reference person receives a welfare gain when choosing for Variant 2, compared to the benchmark. Reference persons with a risk-aversion parameter γ equal or lower than 5 receive a welfare gain. The increase of the allocation in the return portfolio is better for persons which are less risk-averse. If γ = 7, then there is a welfare loss for reference persons with a final salary S−1 of 73 000 EUR or higher.

S−1 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7

36 500 61.8 61.4 60.9 67.3 65.3 63.1 +8.8 +6.3 +3.6

73 000 59.0 58.1 57.3 65.0 61.0 56.7 +10.2 +5.0 -1.0

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3.4 Variant 3: Decrease of initial allocation return portolio

Simulation results

For Variant 3 the opposite occurs. The benefit starts at a lower level than Variant 1. The variable pension benefit in Variant 3 is on average lower than Variant 1, though the lower bound of the 95% probability interval is higher for Variant 3.

Compared with Variant 1, the probability interval has decreased. This can be ex-plained by the reduced amount invested in risky assets.

Welfare effects

With the used assumptions in the base scenario, every reference person receives a welfare gain when choosing for Variant 3, compared to the benchmark. Variant 3 is not better than Variant 1, except if γ = 7 and final salary S−1 is 73 000 EUR or higher.

S−1 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7

36 500 61.8 61.4 60.9 64.6 63.8 63.0 +4.5 +4.0 +3.5

73 000 59.0 58.1 57.3 62.4 60.9 59.4 +5.7 +4.8 +3.7

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3.5 Variant 4: Reducing allocation of return portfolio

Simulation results

When the allocation of the return portfolio is reducing, the expected return on assets decrease. This gives a lower starting benefit compared to Variant 1. Furthermore, the 95% probability interval doens’t get much larger at higher ages, since the allocation of the return portfolio is then much smaller.

Welfare effects

With the same intitial allocation of the retun portfolio τ0, a reducing allocation of the return portfolio τt is only better for the conservative reference persons with higher incomes.

S−1 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7

36 500 61.8 61.4 60.9 65.1 64.2 63.3 +5.3 +4.6 +3.9

73 000 59.0 58.1 57.3 63.0 61.3 59.6 +6.7 +5.5 +4.1

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3.6 Variant 5: Risk-free interest rate as discount rate

Simulation results

When the risk-free interest rate is used as discount rate, the variable pension benefit starts at the same level as the fixed pension benefit. Since the expected return on assets are higher than the risk-free interest rate, the variable pension benefit is expected to increase. It can be seen that the real replacement ratios are almost stable on average.

Compared with Variant 1, the upper and the lower bounds are higher than those for Variant 1. This can be explained by the fact that there is more capital left over, due to the lower starting benefits.

Welfare effects

With the used assumptions in the base scenario, Variant 5 has the best results for every reference person. This can be explained by stable development of the average real replacement ratios. A lower risk premium on assets would probably lead to other conclusions. In the sensitivites we will explore the effects of a lower risk premium on assets.

S−1 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7

36 500 61.8 61.4 60.9 66.6 66.0 65.2 +7.8 +7.4 +7.1

73 000 59.0 58.1 57.3 65.2 63.9 62.4 +10.4 +9.8 +8.9

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3.7 Variant 6: Smoothing of investment results

Simulation results

Smoothing of investment results has no influence on the mean of the pension benefits. The mean of the pension benefits is equal as Variant for all ages. Next, the 95% prob-ability intervals of the pension benefits and the real replacement ratios are small in the first years, but they increase significantly after a few years. This can be explained as follows: when there are investment losses (gains), the pension benefit remains still at a higher (lower) level. This causes the capital to reduce even more (less). Therefore, smoothing of gains and losses can lead to very low (very high) pension benefits at higher ages in certain scenarios.

Welfare effects

Smoothing of investment results leads to a lower welfare level, compared to Variant 1. This is because the very low pension benefits in certain scenarios are punished heavily.

S−1 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7 γ = 3 γ = 5 γ = 7

36 500 61.8 61.4 60.9 66.0 64.5 62.8 +6.7 +5.0 +3.1

73 000 59.0 58.1 57.3 63.7 60.6 56.9 +7.9 +4.2 -0.6

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3.8 All combinations

Table 3.10: Welfare effects of all combinations of the variable pension benefit for the nine different reference persons

a. EROA as discount rate

Welfare effects

No smoothing Smoothing

Fixed Reducing Fixed Reducing

S−1 γ 15% 25% 35% 15% 25% 35% 15% 25% 35% 15% 25% 35% 36 500 3 +4.5 +6.9 +8.8 +3.3 +5.3 +7.0 +4.4 +6.7 +8.3 +3.3 +5.2 +6.8 5 +4.0 +5.6 +6.3 +3.1 +4.6 +5.7 +3.8 +5.0 +5.0 +3.0 +4.4 +5.2 7 +3.5 +4.2 +3.6 +2.8 +3.9 +4.4 +3.1 +3.1 +1.4 +2.7 +3.5 +3.4 73 000 3 +5.7 +8.5 +10.2 +4.4 +6.7 +8.6 +5.5 +7.9 +8.8 +4.3 +6.5 +8.1 5 +4.8 +5.8 +5.0 +3.9 +5.5 +6.2 +4.3 +4.2 +1.4 +3.7 +4.9 +4.8 7 +3.7 +2.6 -1.0 +3.4 +4.1 +3.5 +2.6 -0.6 -7.5 +3.0 +2.9 +1.1 101 519 3 +6.1 +8.9 +10.4 +4.7 +7.2 +9.1 +5.9 +8.1 +8.7 +4.6 +6.8 +8.4 5 +5.0 +5.7 +4.0 +4.2 +5.7 +6.1 +4.4 +3.5 -0.9 +3.9 +4.9 +4.4 7 +3.5 +1.6 -3.7 +3.5 +4.0 +2.8 +2.2 -2.8 -12.4 +3.0 +2.5 -0.3

b. Risk-free interest rate rf as discount rate

Welfare effects

S−1 γ 15% 25% 35% 15% 25% 35% 15% 25% 35% 15% 25% 35% 36 500 3 +4.9 +7.8 +10.2 +3.6 +5.8 +7.7 +4.9 +7.7 +9.9 +3.6 +5.7 +7.6 5 +5.0 +7.4 +9.0 +3.7 +5.6 +7.1 +5.1 +7.3 +8.6 +3.7 +5.6 +7.0 7 +5.2 +7.1 +7.8 +3.8 +5.5 +6.6 +5.2 +6.8 +7.2 +3.9 +5.5 +6.4 73 000 3 +6.8 +10.4 +13.1 +4.9 +7.8 +10.1 +6.8 +10.2 +12.6 +5.0 +7.8 +10.0 5 +7.1 +9.8 +11.0 +5.2 +7.6 +9.2 +7.1 +9.5 +10.0 +5.3 +7.6 +8.9 7 +7.2 +8.9 +8.3 +5.3 +7.3 +8.0 +7.3 +8.3 +6.5 +5.5 +7.3 +7.5 101 519 3 +7.5 +11.2 +14.0 +5.4 +8.4 +10.9 +7.5 +11.0 +13.3 +5.5 +8.4 +10.7 5 +7.8 +10.5 +11.4 +5.7 +8.2 +9.8 +7.8 +10.1 +10.2 +5.9 +8.3 +9.4 7 +7.8 +9.3 +7.9 +5.8 +7.8 +8.2 +7.8 +8.3 +5.3 +6.0 +7.7 +7.5

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3.9 Sensitivities

3.9.1 Change in risk premium on equity

Welfare effects

S−1 γ 15% 25% 35% 15% 25% 35% 15% 25% 35% 15% 25% 35% 36 500 3 +1.9 +2.7 +3.1 +1.5 +2.3 +2.8 +1.8 +2.4 +2.5 +1.5 +2.1 +2.5 5 +1.5 +1.6 +1.0 +1.3 +1.7 +1.7 +1.3 +1.0 -0.2 +1.2 +1.4 +1.1 7 +1.1 +0.4 -1.2 +1.1 +1.1 +0.6 +0.7 -0.7 -3.3 +0.9 +0.6 -0.5 73 000 3 +2.3 +2.8 +2.5 +1.9 +2.6 +2.9 +2.1 +2.1 +1.1 +1.8 +2.3 +2.3 5 +1.5 +0.6 -1.9 +1.5 +1.6 +0.8 +0.9 -1.1 -5.3 +1.3 +0.8 -0.7 7 +0.5 -2.2 -6.9 +1.1 +0.3 -1.5 -0.6 -5.4 -13.0 +0.6 -1.0 -4.1 101 519 3 +2.4 +2.8 +2.2 +2.0 +2.8 +2.9 +2.1 +1.9 +0.4 +1.9 +2.3 +2.1 5 +1.4 +0.0 -3.3 +1.6 +1.5 +0.4 +0.7 -2.3 -7.9 +1.3 +0.5 -1.5 7 +0.1 -3.6 -9.8 +1.0 -0.1 -2.5 -1.4 -7.9 -18.0 +0.4 -1.7 -5.8

Welfare effects

S−1 γ 15% 25% 35% 15% 25% 35% 15% 25% 35% 15% 25% 35% 36 500 3 +2.5 +3.7 +4.5 +1.9 +2.8 +3.6 +2.6 +3.7 +4.3 +1.9 +2.9 +3.6 5 +2.4 +3.2 +3.2 +1.8 +2.6 +2.9 +2.4 +3.0 +2.7 +1.9 +2.5 +2.7 7 +2.3 +2.6 +1.8 +1.8 +2.3 +2.2 +2.3 +2.2 +0.9 +1.9 +2.2 +1.8 73 000 3 +3.3 +4.6 +5.1 +2.5 +3.6 +4.4 +3.4 +4.5 +4.6 +2.6 +3.7 +4.2 5 +3.2 +3.5 +2.4 +2.5 +3.2 +3.1 +3.2 +3.0 +1.0 +2.6 +3.0 +2.5 7 +2.9 +2.0 -0.9 +2.4 +2.5 +1.5 +2.7 +0.7 -4.1 +2.4 +2.1 +0.4 101 519 3 +3.6 +4.9 +5.3 +2.7 +3.9 +4.6 +3.7 +4.7 +4.6 +2.8 +3.9 +4.4 5 +3.5 +3.6 +1.9 +2.7 +3.3 +3.0 +3.4 +2.7 -0.1 +2.8 +3.1 +2.3 7 +3.0 +1.5 -2.5 +2.5 +2.5 +1.1 +2.6 -0.4 -7.1 +2.5 +1.9 -0.5

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3.9.2 Change in volatility equity returns

Welfare effects

S−1 γ 15% 25% 35% 15% 25% 35% 15% 25% 35% 15% 25% 35% 36 500 3 +4.1 +6.0 +7.0 +3.1 +4.7 +5.9 +4.0 +5.6 +6.2 +3.1 +4.6 +5.6 5 +3.4 +4.0 +3.4 +2.7 +3.7 +4.0 +3.1 +3.1 +1.5 +2.6 +3.3 +3.2 7 +2.6 +1.9 -0.4 +2.3 +2.7 +2.1 +2.0 +0.3 -3.5 +2.1 +2.0 +0.6 73 000 3 +5.0 +6.7 +6.8 +3.9 +5.7 +6.7 +4.7 +5.7 +4.6 +3.8 +5.3 +5.8 5 +3.6 +2.6 -1.0 +3.3 +3.8 +2.9 +2.8 +0.0 -6.2 +2.9 +2.8 +0.8 7 +1.8 -2.1 -9.2 +2.5 +1.7 -1.0 +0.2 -7.0 -17.6 +1.9 -0.1 -4.6 101 519 3 +5.3 +6.7 +6.3 +4.2 +6.0 +6.8 +4.9 +5.5 +3.5 +4.1 +5.4 +5.6 5 +3.6 +1.7 -3.3 +3.4 +3.7 +2.3 +2.5 -1.8 -10.5 +3.0 +2.4 -0.5 7 +1.3 -4.3 -13.6 +2.4 +1.1 -2.5 -0.9 -11.0 -24.9 +1.6 -1.3 -7.2

Welfare effects

S−1 γ 15% 25% 35% 15% 25% 35% 15% 25% 35% 15% 25% 35% 36 500 3 +4.6 +6.8 +8.4 +3.4 +5.2 +6.6 +4.5 +6.6 +7.9 +3.4 +5.1 +6.4 5 +4.4 +5.9 +6.2 +3.3 +4.7 +5.5 +4.4 +5.6 +5.5 +3.3 +4.6 +5.2 7 +4.3 +4.9 +3.9 +3.3 +4.3 +4.3 +4.2 +4.4 +2.8 +3.3 +4.1 +3.8 73 000 3 +6.1 +8.7 +9.8 +4.5 +6.7 +8.2 +6.1 +8.3 +9.0 +4.6 +6.6 +7.9 5 +6.0 +6.8 +5.4 +4.5 +5.9 +6.0 +5.8 +6.0 +3.6 +4.6 +5.7 +5.3 7 +5.5 +4.5 +0.2 +4.4 +4.9 +3.5 +5.2 +2.8 -3.5 +4.4 +4.4 +2.1 101 519 3 +6.7 +9.2 +10.1 +4.9 +7.2 +8.6 +6.6 +8.7 +9.1 +5.0 +7.1 +8.2 5 +6.4 +7.0 +4.7 +4.9 +6.3 +6.0 +6.2 +5.9 +2.2 +5.0 +6.0 +5.1 7 +5.8 +3.8 -2.1 +4.7 +4.9 +2.9 +5.3 +1.5 -7.5 +4.7 +4.2 +1.0

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Conclusion

This study addressed the following question:

What are the welfare implications of investment risks in the benefit phase of an individ-ual defined contribution pension scheme?

By using a simple model, the welfare effects are calculated for different variants of the variable pension benefit for different types of persons. The less risk avers a person is, the better it is to invest with a higher allocation in risky assets after retirement. Additionally, the welfare effects for people with higher income are more sensitive to risk aversion than those with lower income. The option for a variable pension benefit which is expected to increase, gives a higher welfare than the variable pension benefit which starts at a higher level and stays stable. Further, smoothing unexpected investment results over a longer period leads to a welfare loss.

The variable pension benefit can provide a welfare gain for all types of persons. Even for highly risk averse persons, there are variants of the variable pension benefit which give a higher welfare than the fixed guaranteed pension benefit. Finally, for an average person the variable pension benefit gives a welfare gain between +3.0% and +9.0%.

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Chapter 5

Discussion

With our used utility function, the option for smoothing unexpected investment gains and losses leads to a welfare loss. This option can lead to very low pension benefits in certain scenarios, which are punished heavily. On the other hand, smoothing diminishes the year-on-year fluctuations and thus the year-on-year losses. Since people are loss averse, see Kahneman & Tversky (1984), smoothing could be better in reality. With other assumptions, such as loss punishments, the conclusion about smoothing of invest-ment results could be different.

We used a discount rate of 1.00% for time preference in our model. It can be expected that this should be higher for most people in reality. Namely, it is preferable to have a higher income at a younger age when you are physically able to do more with it. With a higher discount rate, the conclusion about the increasing variable pension benefit (see Variant 5) could be different.

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and losses

If the participant chooses for the option to smooth gains and losses, the process for the determination of the variable pension benefit is a bit more complicated. The pension benefit for the first year BVs,0is the same is in2.11. For t = 1, ..., 54 the following steps must be taken:

Step 1: Determine excess return in previous year Rs,t−1

First, the gain (or loss) has to be decided. This is defined as the excess return over the expected return on assets (5.1).

Rs,t−1=

Cs,t−1− BVs,t−1 1p67+t−1

αt−1(rs,t−1− (rf + πe)) (5.1) This Rs,t−1 can be seen as the unexpected change in the capital. To prevent that this unexpected change has a large impact on the benefit for the next year, it can be smoothed over a longer period. In the next step this smoothing period τt is defined.

Step 2: Determine smoothing period τt at time t

In this study we only look at the effects of a initial smoothing period of five years. Though it can have a maximum of ten years. By the new act “Wet verbeterde

pre-mieregeling”(2016), the smoothing period for an individual is maximized at its

remain-ing life expectancy. To ensure that the smoothremain-ing of gains and losses will not be shifted too far to the future as the life expectancy drops.

τt=      1, if Lt< 2 bL_tc, if 2 ≥ Lt< τ0 τ0, if Lt≥ τ0 (5.2)

The smoothing period τtat time t is defined in formula5.2and the method is shown in figure 5.1.

Step 3: Calculate smoothing annuity price P mt at time t for τt terms

In this study we assume that the unexpected change in the capital will be smoothed equally over the smoothing period, so the actuarial annuity price for τtterms has to be calculated. P mt=    1, if τt= 1 1 +Pτt_i=1−1_Qi 1p67+t+i j=1 1+rf+αt+j∗πe , if τt> 1 (5.3) 24

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Figure 5.1: Determination of smoothing periods at ages 67 + t with initial smoothing period τt= 5 0 1 2 3 4 5 6 7 8 9 10 67 72 77 82 87 92 97 102 107 112 117 Years Age

Remaining life expectancy Smoothing period Minimum Initial smoothing period

Step 4: Calculate amortization as,t at time t

We assume that the gains and losses are smoothed equally for each term of the smooth-ing period.

as,t= Rs,t−1

P mt (5.4)

This can be achieved by dividing the unexpected gain (or loss) by the smoothing annu-ity price P mt at time t.

Step 5: Calculate accumulated amortizations of gains and losses As,t,k at time t for terms k = 1, ...τt

For the first year, only as,1 is amortized. The next year, as,1 + as,2 and so on. To remember what the accumulated amortizations are, we calculate As,t,k at time t for terms k = 1, ...τt: As,t,k = ( as,t, if t = 1 or k = α0 as,t+ As,t−1,k+1, if t > 1 and k < α0 (5.5)

Term As,t,1 is amortized each year and k = 2, ...τt will be shifted to the next years. So, the accumulated amortizations at time t for term k is equal to as,t plus the accu-mulated amortizations at time t − 1 at the term k + 1.

Step 6: Calculate present value of unrecognized amortizations Us,t at time t

Only As,t,1 has to be recognized for the determination of the benefit at time t. The present value of the unrecognized accumulated amortizations Us,t is calculated as fol-lows: Us,t=    0, if τt= 1 Pτt k=2As,t,k s67+t+1,k−1 Qi j=2 1+rf+αt+j−1∗πe , if τt> 1 (5.6)

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Step 7: Determine benefit for coming year

The present value of the unrecognized accumulated amortizations Us,t has to be sub-tracted from the capital when determining the level of the variable benefit with smooth-ing:

BVs,t=

Cs,t− Us,t P Vt

(5.7)

If there are some years of unexpected losses, the level of the pension benefit is artificially higher due to a negative Us,t.

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Projection Table AG2016 . Retrieved fromhttp://www.ag-ai.nl/view.php?Pagina Id=731

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