Generational eﬀects of a new term structure for Dutch pension funds

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Generational effects of a new term structure for Dutch pension

funds

by Lucas H.I. van Benthem s1703161

Econometrics, Operations Research and Actuarial Science (EORAS) Faculty of Economics and Business

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Abstract

As of October 2012 the Dutch central bank has introduced a new term structure for pension funds to discount their liabilities. The new interest rates converge with maturity to a constant that is called the Ultimate Forward Rate (UFR). In 2013 the UFR increases the interest rate of maturities higher than twenty years with respect to the previously used market interest rates. Therefore the present value of the fund liabilities has decreased and the funding ratio has increased. This improvement of the funding ratio has resulted in less benefit payment reductions for a substantial amount of pension funds. Other funds have been able to grant more indexation. The fund participants that are now retired immediately benefit from the introduction of the UFR. In case the UFR had not been used this money would have stayed in the fund for the younger age cohorts. A transfer of money seems to occur from the young to the old participants. Making use of value-based ALM and a generational accounts model, it is shown that in the current state of the economy indeed a value transfer from young to old participants takes place.

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Introduction

1.1 Goal of the paper

Pension funds in the Netherlands have seen their solvency position deteriorate since the financial crisis.1 _{Lower asset returns and losses had a negative impact on the equity side of the balance}

sheet. A decrease in the interest rate has lead to an increase of the fund liabilities. The together decreased the funding ratios of a substantial amount of pension funds. Because the government only has limited influence on the financial markets, a solution for the low interest rate has been an important topic of debate between Dutch politicians and other pension fund stakeholders. Among the proposals was for example the proposal to discount liabilities with expect returns. In December 2011 the Dutch Central Bank decided that the discount term structure would no longer be the most recent Euro swap curve but the average Euro swap curve over the past three months. This resulted in a reduction of the volatility of the discount rate. As of September 2012 a whole new term structure for discounting purposes was introduced by the Dutch Central Bank (DNB). The new term structure assumes that the interest rates on high maturities converges to a constant, which is derived from inflation and real interest rate long run expectations. In 2012 and 2013 this constant, which is called the Ultimate Forward Rate (UFR), lead to an increase in the interest rate for the high maturities. The higher interest rate resulted in lower liabilities and this gives the pension funds the opportunity to cut less on benefit payments or to grant extra indexation. While this is instantly profitable in the short run for retirees, it is not necessarily profitable for active participants. If the interest rates would not be altered, the money that is now spend on retirees would stay in the pension fund and would be used for future pension payments on the participants that have not reached the retirement age yet. The purpose of this paper is to investigate if all age cohorts take equally advantage of the new interest rate.

1_{See the AON-Hewitt pension index at http://www.pensioenthermometer.nl.}

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1.2 Dutch pension system

The Dutch pension system is composed out of three pillars.

1. The Dutch governement grants every Dutch citizen a state pension called the AOW.2

The state pension provides a minimal benefit payment that is of the same size as a social security payment. The state pension is independent of the wage of the participants and depends only on the number of years lived in the Netherlands.

2. Every Dutch employee has additional pension savings in the employers pension fund. This fund can be either exclusive to the firm, but there are also pension funds in which a complete industry participates. The pension savings of this type are usually wage dependent. In this paper we will focus exclusively on this type of pension arrangement. Later in this paper we will describe the Dutch second pilar in more detail.

3. Everybody is allowed to build up extra pension savings next to the state pension and the benefit payments of the pension fund. Some people save for extra income on their bank account. Others choose to buy a pension contract from an insurer. For self-employed, who are not required to participate in a pension fund, this could be the only type of additional pension savings next to the state pension.

1.3 Improving sustainability

Besides the new interest rate, other measures are being taken to improve the solvency and sustainability of pension funds. The funding ratio is the best known and most widely used measure of the solvency position of a pension fund, it is the ratio of the assets over the liabilities. Since 2008 a substantial number of pension funds have been struggling with a low funding ratio. Because this never happened before on such a scale, legislators, acedemics, employees, fund boards and other stakeholders propose several options to improve the pension system. Current topics that pension funds are dealing with are listed below.

• From a Defined Benefit (DB) contract to a Defined Contribution (DC) plan. As explained by Praagman (2012), in a defined benefit plan the employer guarantees that the employee will receive a fixed benefit payment upon retirement (average wage or final wage), regardless of the performance of the underlying investment pool. In a defined-contribution plan the employer makes predefined defined-contributions for the employee, but the final amount of benefit received by the employee depends on the investment performance. A scheme that is used more often is the Collective Defined Contribution scheme, where risk sharing occurs among members. The benefit payments still depend on the investment

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performance, but the contributions of the participants are pooled which means that risks can be shared between generations.

• Completing the pension contract. Nowadays for most pension funds it is up to the board of the pension funds to determine which measures will be taken if the funding ratio is very low or very high. In a complete contract all rules are laid down in advance and it is always clear how to respond to increasing or decreasing funding ratios.

• From a nominal to a real contract. In a nominal contract the fund usually guarantees non-indexed benefit payments and grants indexation only if the solvency position allows this. The real contract works differently. A non-indexed benefit payments is not guaran-teed but instead the fund strives to give the participant each year at least price-indexed benefit payments. If economic conditions are such that this goal cannot be met, the fund will cut the benefit payment in ten years.

• Increasing the retirement age. Since 1957 the Netherlands has a pay-as-you-go state pension system for all inhabitants. The retirement age was 65 years with the exception of certain physically demanding jobs. Since 1957 the retirement age has stayed the same, while life expectancy increased from 73 to over 81 years. This lead to an increase of the pension expenses for the government. Additional pensions provided by employers have also seen the consequences of the increased life expectancy. Over the years the increased life expectancy means that in the same number of active years the employees need to save for more retirement years. Until now this was solved by increasing the premium payments or by reducing the guaranteed benefit payments. A better solution seems to be the increase of the retirement age. This has been a topic of debate among politians for some years already. In Rutte & Samsom (2012) it has been announced that the retirement age will increase to 66 in 2018 and to 67 in 2021. After 2021 the retirement age will be linked to the life expectancy.

1.4 Balanced stakeholder value

The interest rate that pension funds use to discount liabilities has been the topic of conversation for some years. Until the Pension Act of 2007, a fixed discount rate of 4% was used, for all maturities.3 From 2007 on the market swap rate was used until the 3-month averaging was introduced in 2012. Because the low interest rate is not only a problem for pension funds and insurers in the Netherlands but also in other European countries, a new term structure under development on European level by EIOPA. In September 2012 the DNB finally introduced the UFR for Dutch pension funds.

The DNB stated that one of the main reasons to introduce the new term structure, is the low liquidity of interest rate swaps with a high maturity, on which the zero swap curve is based.

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Because of the low liquidity, the market interest rate is not reflecting the true interest rate. Another reason is that the government wants to help pension funds increase their funding ratio. This prevents large cuts in benefit payment which could harm the economy.

In Section 105-2 of the Dutch Pension Act, it is stated that the board of the fund should equally represent all stakeholders. Each age cohort is a different stakeholder, and the age cohorts are the only stakeholders that we will consider. When policy changes occur, value transfers between age cohorts should be minimized. Policy changes should be fair in a sense that value transfers between stakeholders should not occur. The theory behind stakeholder value in a pension fund was first described by Chapman et al. (2001). Ponds (2003) proposed value-based generational accounting as a suitable method to evaluate a pension fund policy based on intergenerational risk-sharing. Where Ponds explains the idea in a 1 timestep setting, Hoevenaars (2008) comes up with a clear framework for a longer horizon. A Generational Account (GA) gives the market value of the pension contract for each cohort that develops over time. The model works by means of simulation, so we need to set up a framework to simulate a pension fund over time. Multiple applications of value-based generational accounting can be found in recent literature. Stienstra (2010) investigates the generational effects of changing the retirement age from 65 to 67. Lekniute (2011) investigates the transition from a DB plan to a CDC plan. She also investigates using the real interest, the expected real returns and a combination of the two as a discount rate. Recently Soer (2012) described the effects of changing to a new financial assessment framework and how a buffer called the equalization reserve can reduce the value transfers between generations.

In preparation of a number of policy changes, CPB (2012) wrote a memorandum on the associ-ated generational effects. The main focus is on the transition from a nominal to a real contract, but also the UFR is discussed. They investigate the value transfers between generations on a 80-year horizon, which seems unrealistic concerning the continuous development of the pension contract and the economy. The authors only highlight the benefits of the UFR for the older age cohorts, while effects for younger participants remain unclear. Also they consider only one type of pension fund. This all gives rise to further investigation of the generational effects of the UFR.

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1.5 Outline

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Ultimate Forward Rate

Recently a new term structure for Dutch pension funds has been introduced. Why this happened is explained in Section 2.1. How the term structure with UFR can be derived is explained in Section 2.2. Section 2.3 shows how the UFR affects the term structure and Section 2.4 shows the effects on the liabilities of a pension fund.

2.1 History

In May 2012 the Dutch Ministry of social affairs published a memorandum on a new financial assessment framework for pension funds. One of the topics was the introduction of the UFR. The Dutch central bank decided that as of June 2012 Dutch insurance companies should use the UFR to alter the interest rate term structure for liabilities. As of end September 2012 the UFR is also used by pension funds.

Until recently the swap rate term structure was used by pension funds to discount future cash-flows. Since the financial crisis in 2008, when stocks rapidly decreased in value, the bond market also significantly changed. Countries that face depressions and financial shortages, like Greece, Spain and Portugal see the interest rates on government bonds increasing. Countries with a healthy financial household see their interest rate decreasing. This means that these govern-ments can lend money at extremely low rates. The Euro swap curve has also decreased since 2008. While decreasing asset returns since 2007 deteriorated the equity side of the balance sheet, the decreasing term structure increased the liabilities of the funds.

Swap rates with a maturity until fifty years are used to determine the swap rate term structure. A problem of long maturity swaps is that the market liquidity is low. Therefore the high maturity swaps were volatile, which has a big impact on the pension fund obligations. Therefore from December 2011 onward the Dutch central bank applied three month averaging of the swap curve. Because of the historically low swap rates and the low liquidity at long maturity swaps, it is argued that the swap curve is not suitable to discount cash flows far into the future. In CEIOPS

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(2010) it is stated that the last liquid point for the Euro swap curve is 20 years. Therefore on European level one has thought of a new interest rate for higher maturities.

The technique named and described by Smith & Wilson (2011) is used to extrapolate the swap curve beyond twenty years. For high maturities it is assumed that the forward rate will converge to a long term average. This long term average is called the UFR. The central bank decided to set the UFR at 4.2%. This is the sum of a long term expected real interest rate of 2.2% and an inflation expectation of 2%. From 20 years-to-maturity until 60 years-to-maturity the forward will converge from the market interest rate to the UFR.

2.2 Methodology

The Dutch central bank has set up a guideline on how to determine the new term structure that converges to the UFR.1 Let us define the Euro swap rate with maturity t by Rt. Using the

swap rates we calculate the 1-year forward rate Ft−1,t for t = 1 until t = 60 as follows

Ft−1,t=

(1 + Rt)t

(1 + Rt−1)t−1

− 1.

Here we assume that the interest rate at time zero is R0 = 0. The adapted 1-year forward rate

which we denote by F_t−1,t∗ is defined as

F_t−1,t∗ =          Ft−1,t if 1 ≤ t ≤ 20, (1 − wt) · Ft−1,t+ wt· U F R if 21 ≤ t ≤ 60, U F R if t ≥ 61.

Here wtis the weight determined with the method of Smith & Wilson (2011) and U F R = 4.2%.

The weights can be found in Appendix A.1. Out of the adapted forward rate we can compute the adapter swap rate using the following relation

(1 + R∗_t)t=

t

Y

j=1

(1 + F_j−1,j∗ ).

So the adapted swap zero curve at maturity t is given by

R∗_t =   t Y j=1 (1 + F_j−1,j∗ )   1 t − 1. 1

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2.3 Effect on the term structure

The effect of the UFR is best described alongside a figure. In Figure 2.1 the original term structure, term structure with UFR and forward curves are depicted.

0 20 40 60

Term structures with UFR 1995

Time to maturity (in years)

Interest Rate (in percentages)

0 2 4 4.2 6 Interest Rate Interest Rate with UFR Forward Rate Forward Rate with UFR

0 20 40 60

Term structures with UFR 2012

Time to maturity (in years)

Interest Rate (in percentages)

0 2 4 4.2 6 Interest Rate Interest Rate with UFR Forward Rate Forward Rate with UFR

Figure 2.1: Interest rate and forward rate with and without use of the UFR in 1995 and 2012 using extrapolated values on German zero coupon bond yields.

Both graphs show that the forward yield with UFR converges to 4.2% from twenty years until maturity to sixty years until maturity. At twenty years until maturity there is a kink point in the forward curve with UFR. The interest rate with UFR at twenty years until maturity is more smooth. The interest rate with UFR in both figures is not converged after 75 years. In fact it will not converge to 4.2% in finite time at all.

In the figures we see that the term structure in 1995 was 4.2 to 2.6 percentage points higher than in 2012. The 1995 term structure is always above 4.2% while the 2012 term structure is always below. If the UFR would have been applied in 1995, it would lower the term structure above twenty years. If liabilities in that time would be discounted with this a term structure, the introduction of the UFR would increase the present value of the liabilities. In 2012 the UFR increases the term structure after twenty years. This leads to a higher discount factor and a decrease in the present value of the liabilities of a pension fund. This will have a positive effect on the funding ratio and indexations will be granted more often.

2.4 Effect on the liabilities

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structure. The level of the ordinary term structure is one of the components that determines the size of the effect of the UFR on the liabilities.

Another factor that will determine the impact of the UFR is the fund duration. If a fund has a low duration it means that the size of the cashflows is large in the near future and small at higher maturities. Usually this is caused by a high average age of the pension fund participants. Retired participants contribute to the benefit payments at early maturities but die in the years following. When this is accompanied with substantially less younger participants than older participants, this results in a low duration. When the fund has a high duration, this usually implies that the average age of the participants is low.

The UFR only changes the term structure with a time to maturity of 20 years and higher, also its effect is increasing with the time to maturity. When we discount the cashflows, the distribution of the cashflows over the maturities will thus affect the impact of the UFR. This is depicted in Figure 2.2. It is assumed that an old fund has duration of 10 years, an average fund 16 years an a young fund 25 years. In the figure it is assumed that in all situations the total liabilities, which is the the sum of the discounted cashflows, is equal under the term structure without UFR.

0 20 40 60

Discounted cashflows low term structure

Time to payment (in Years)

Cash flo

w

Young fund Average fund Old fund Ordinary term structure UFR term structure

0 20 40 60

Discounted cashflows high term structure

Time to payment (in Years)

Cash flo

w

Young fund Average fund Old fund Ordinary term structure UFR term structure

Figure 2.2: Impact of the UFR for different fund durations when the ordinary term structure is lower and higher than the UFR curve

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funds have a funding ratio of 100% when the spot rate is used for discounting. In Table 2.1 the change in total liabilities, which is the sum of the discounted cashflows and the funding ratio with the UFR term structure, are reported.

Change in total liabilities New funding ratio

Low TS High TS Low TS High TS

Young fund -12.17% 6.93% 113.9% 93.5%

Average fund - 4.63% 2.19% 104.9% 97.8%

Old fund - 1.86% 0.98% 101.9% 99.0%

Table 2.1: Change in total liabilities after introduction of the UFR for different term structures (TS) and durations

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Chapter 3

Pension Fund Setup

For our analysis we will use a pension fund and regulations that are representative for the Netherlands. In Section 3.1 we describe which risks are assigned to whom. A pension fund needs a population, which we describe in Section 3.2. The active participants of the pension fund accrue pension rights, the technical details of this accrual are explained in Section 3.3. These accrued rights will later result in benefit payments, this will also be explained. The inflow of money occurs with the premium payments, the details of this are explained in Section 3.4. Section 3.5 to 3.7 describes the setup of the funding ratio, liabilities and assets. How indexation and benefit cuts are granted is described in Section 3.8 and 3.9.

3.1 Type of fund

Until the beginning of the 21st century, Dutch pension funds usually had a defined benefit plan. The employer guaranteed a certain amount of pension for the employees upon retirement. Any risk, such as longevity risk or investment risk, was on behalf of the employer. If the fund was on risk of running into a deficit the employee would sponsor the fund with a financial injection. Since last decade pension funds have been making a transition to (collective) defined contribution plans. In a defined contribution plan all the risks are with the funds participants. The employee contributes a prespecified amount to the fund for each employer. In the case that funds assets decrease and run into a deficit, the fund has to consider lowering benefit payments or increasing the premium. While in a pure defined contribution plan all investments are made on individual basis, in a collective defined contribution plan investment risk is shared among other participants and also between generations. Self-employed can buy a pension contract from an insurance company. If they do so, they have a pure defined contribution plan. Most pension funds have been making a transition to collective contribution plans, therefore we will apply this type of fund. In practice this transition does not go without a hitch since the financial risks are transferred to the employees in a DC arrangement. A number of funds have solved this by giving the fund a final financial injection.

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3.2 Population

An important element of a pension fund is its population and their demographics. Our pension fund consists of three types of participants. The active participants are people working for the firm of which the pension fund originates. Inactive participants are people that used to for work the fund’s firm and attained pension rights in the pension fund but are no longer active for the firm. The last group are the retired, which are people that used to work for the firm but have past the retirement age.

We assume that the pension fund only pays old-age pensions. We do not consider disabled and widow arrangements. The assumptions will not be restrictive since the not included obligations are only a small part of the total liabilities. Also it is not expected that the introduction of the UFR will have a different effect on these types of pension payments then on the old-age pension payments.

The population of our fund will consist of Dutch inhabitants as registered in the population administration. These demographics approximate a pension fund where all Dutch inhabitants particpate in. The participants will start working at age 25 and die at age 99. The starting age is chosen because the accrued pension rights at age 25 are still very low, because of low labor participation before that age. Furthermore, participation in a pension fund is only compulsory from age 21 onwards. The final age is not restrictive because in 2012 only around 3250 people in the Netherlands are 99 or older. The retirement age will be 67, in line with the proposal by Rutte & Samsom (2012). Plans are in place to increase the retirement age even further in the near future.

Because we want to make inferences about the future, we need a population forecast. Based on demographic forecasts of the Dutch Central Bureau for statistics (CBS1) and mortality forecasts made by the Dutch Actuarial Association (AG2) in 2012, we will construct the population. The population forecasts of the CBS give a forecast for the total population from 2012 until 2060 at January first of the year. While the population corresponds best with the actual population development, it also means that we need to consider both inflow and outflow of participants. As a simplification we will only use the full population in 2013 as published by the CBS in the first year. In the years following we apply the AG mortality table to make populations for the future years. Every future year we add the CBS forecast for the number of 25-year old to the table. We will assume that all participants are born on the first of January and die the 31st of December. If we denote the population of age x in year t by popx

t and the probability of death at

age x in year t by q_txthe population of age x + 1 at time t + 1 is given by popx+1_t+1 = popx_t· (1 − qx t)

and q_t99= 1 for all t.

1

CBS stands for ”Centraal Bureau voor de Statistiek”, population forecasts can be found on http://www. statline.cbs.nl/statweb/?LA=en.

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Our fund will consist of both men and women because there are big differences in mortality, wage and benefit payments. If relevant for calculations, men and women will have subscript m and w respectively.

Not all participants are still active for the fund’s firm, therefore we also need to model inactive participants in our pension fund. Inactives are people that differ from actives because they no longer build up new pension rights. The number of inactive people in a pension fund tends to increase with age because people voluntary switch to other employers or are laid off. This holds particularly for our fund since we do not assume any outflow of participants. As a proxy for the ratio of active participants over the total population we will use the Dutch labor participation rate. The labor participation rate gives the percentage of active workers among the Dutch population. This makes it suitable to model the ratio active participants among total participants. The real Dutch labor participation rate in 2012 is given in Table 3.1.

Age

25-34 35-44 45-54 55-64

Men 86.3 89.9 87.9 63.1

Women 78.4 74.0 69.9 38.9

Table 3.1: Real labor participation rate in 2011 in the Netherlands in percentages

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40 60 80 100 0 20000 40000 60000 80000 100000 120000 140000

Total and active population Men

Age P opulation Total 2013 Total 2038 Active 2013 Active 2038 40 60 80 100 0 20000 40000 60000 80000 100000 120000 140000

Total and active population Women

Age P opulation Total 2013 Total 2038 Active 2013 Active 2038

Figure 3.1: Total and active Population in 2013 and 2038, men and women separately

The figure shows that at the potential working ages, the differences in population between men and women are very small. At the retired ages the population for women decreases at a lower rate than for men. The active participants are in all cases decreasing relative to the total population. For women however this occurs at a much higher rate than for men. In 2013 at age 60 more then 70,000 men are working while only 45,000 women are still active. Furthermore we see a change in demographics. From age 40 onwards the groups of active participants is larger in 2013 than in 2038. At the same time we see that the number of retired in 2038 is higher than the number of retired in 2013. So in 2038 the ratio of retirees over active workers is a lot higher than it is in 2013. Because the Dutch state pension is of a pay-as-you-go form, the actives in 2038 contribute relatively more to the state pension than the actives in 2013.

Wages differ per individual because of education, careers and work sectors. While these factors will not be taken into account, we can differentiate wages between age and gender. The average Dutch wages by age and gender are yearly published by the CBS. The average yearly wage of Dutch inhabitants that worked more than 12 hours a week in 2011, can be found in Table 3.2.

Age

21-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65 Men 21,000 30,700 38,700 45,900 50,200 51,600 51,500 50,900 50,500 Women 18,000 25,700 28,300 28,100 27,700 27,400 27,800 26,800 25,200

Table 3.2: Average Dutch wages in Euros in 2011 for men and women

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Just as with the participation rates we will linearly interpolate the wages after setting the wage for the median age of a group equal to the group wage. Wages on average increase with wage inflation over time. Therefore the wages will be increased yearly with wage inflation.

Different types of schemes are available to determine the yearly benefit payment of a retired. We use the one that is the most common in the Netherlands, this is the average wage plan. In an average wage plan, every year the employee saves a percentage of his pension base for his pension. The pension base is the wage minus the franchise. The franchise is an amount that is substracted from the wage because the state also provides an old age pension to every Dutch national. The amount of pension that the employee has already saved is called the accrued amount. Once retired the employee will receive this amount yearly until death. The accrual rate is the percentage of the pension base that is yearly saved and is maximized to 2.25% in the Netherlands. This is also the percentage that we will use.

To determine the pension base the franchise has to be set. The major Dutch pension funds use a franchise of e 10850 for full-time active participants. Part-time employees have a franchise equal to ratio of the number of hours they work relative to the hours a full time employee works. Using data from the CBS on working hours, the franchise per gender and age group can be determined. The working hours are reported in Table 3.3.

Age

21-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65 Men 26.50 37.10 38.90 39.70 39.90 40.00 39.40 38.30 34.00 Women 22.20 30.20 29.00 26.90 25.90 26.10 25.80 25.10 22.70

Table 3.3: Average working hours per week for Dutch inhabitants in 2012

The amount of working hours increases until age 45 and decreases afterwards. Women on average work less hours a week than men. This effect increases until age 45 and then decreases again. Because women work less their franchise will be lower, this results in a higher pension base compared to using the full franchise.

3.3 Accrual of pension rights and benefit payments

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In Table 3.4 a number of variables that are used to generate benefit payments cashflows are defined.

Variable name Description

accr Accrual rate

pbx Pension base for age x

partx Active participants ratio for age x

shx _{Ratio of participants that becomes inactive at age x.}

popx_t Total fund population of age x at time t

ARx_A,t Accrued rights of active participants of age x at time t ARx_IA,t Accrued rights of inactive participants of age x at time t T ARx_A,t Total accrued rights of active participants of age x at time t T ARx_IA,t Total accrued rights of inactive participants of age x at time t T ARx_t Total accrued rights of all participants of age x at time t N ARx_A,t New accrued rights of active participants of age x at time t N ARx_IA,t New accrued rights of inactive participants of age x at time t T N ARx_A,t Total new accrued rights of active participants of age x at time t T N ARx_IA,t Total new accrued rights of inactive participants of age x at time t

Table 3.4: Variable description

New accrued rights on an active participant of age x in year t are equal to the pension base at age x times the accrual rate

N ARx_A,t= pbx· accr.

The total new accrued rights of all participants of age x in year t are equal to the individual accrued rights times time number of active participants.

T N ARx_A,t= N ARx_A,t· partx· popx_t

If we exclude indexation and wage inflation, the accrued rights for a x year old active participant is the sum of the previously attained new accrued rights

ARx_A,t=

x

X

i=26

N ARi_A,t.

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of actives and inactives are given by

T ARx_A,t= ARx_A,t· partx· popx_t, T ARx_IA,t = x X i=26 ARi_A,t· shi ! · popx_t.

The total accrued pension rights for men and women in 2013 and 2038, excluding wage inflation and indexation, is given in Figure 3.2.

40 60 80 100 0 1 2 3 4

Total accrued rights Men

Age

Accr

ued r

ights (in billion Euro)

Active 2013 Active 2038 Inactive 2013 Inactive 2038 40 60 80 100 0 1 2 3 4

Total accrued rights Women

Age

Accr

ued r

ights (in billion Euro)

Active 2013 Active 2038 Inactive 2013 Inactive 2038

Figure 3.2: Total accrual rate in 2013 and 2038 of active and inactive participants, man and women seperate

The figure shows that there is dramatic difference in accrued pension rights of actives between men and women. In both years at age 66 the accrued rights of men are more than twice the accrued rights of women. Furthermore note that the accrued rights of the inactives are negligible until age fifty. For men the accrued rights of the inactives always remain significantly lower than those of the actives. However, for women the total accrued rights at age 66 of actives and inactives is almost equal. At age 67 everybody is inactive, therefore the total accrued rights of actives is 0 from this age onwards.

The total accrued rights did not include inflation on wages and indexation on the accrued rights of the participants. For the simulation we will only use the total accrued rights calculated in 2013. This because new accrual in future years is subject to wage inflation. Also indexation can be granted to the accrued rights.

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These accrued rights result in a series of benefit payments for each age group. For example the accrued rights of a 26-year old will result in a cashflow after 41 years until death. The benefit payments for an x-year old known in 2013 in k years from now is equal to

Bx,k₂₀₁₃= T ARx₂₀₁₃· 1_x+k≥67·

k

Y

j=1

(1 − q₂₀₁₃x+j−1).

The new accrual also equals a series of future payments. These are the extra benefit payments from the accrual in a year. They are defined as

N B_tx,k = T N ARx_t · 1_x+k≥67·

k

Y

j=0

(1 − qx+j_t ).

In every future year the benefit payments will be equal to the benefit payments in last year corrected for indexation or cuts, plus the new benefit payments.

B_tx,k = Bx−1,k+1_t−1 ∗ (1 + Index_t−1) + N B_t−1x−1,k, t > 2013

3.4 Premium payments

Every time the pension fund pays the retirement benefits an outflow of money occurs. The inflow of money in the pension is arranged with premium payments. Every period a part of the income of an employee is payed to the pension fund. Usually, the employer makes this payment directly to the pension fund. If the total premium in a year exactly covers the present value of the benefit payments that arise from new accrual, it is called the break-even premium. We multiply the break-even premium by the Solvency Capital Ratio (SCR) as a buffer. In Appendix A.1 the technical aspects of the evaluation of the premium payments can be found.

To determine the break-even premium, the present value of the new benefit payments cashflows (N B) is calculated. According to government regulations the discount rate used to determine the break-even premium should be the same discount rate that is used to determine the total liabilities of a fund.3 Hence the premium will change in case we introduce the UFR. This can make the comparison between a pension contract with and without UFR more difficult to interpret. Pension funds are obliged to set the premium as least as high as the break-even premium but are allowed to set a higher premium. We will therefore set the premium at 23.0%, which is under both term structures well above the mean dynamic break-even premium percentage of 15.4%.

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19

3.5 Funding ratio

The health of a pension fund is measured using the funding ratio. A funding ratio of least 100% means that a pension fund can be expected to pay all current and future obligations if the fund would not encounter any risks in the future. The funding ratio is the ratio of the total assets and total liabilities of the pension fund, that is F R = A/L.

We will consider the funding ratio at the beginning of the year (primo) and at the end of each year (ultimo). The primo funding ratio at time t will be denoted by F RP ri

t and at the ultimo

funding ratio by F RU lt_t . The primo funding ratio is the one we will report, the ultimo funding ratio will be used as a decision variable for indexation. Premium and benefit payments will occur at the start of the year. Investment returns are earned during the year. Therefore Primo and Ultimo values differ.

The initial funding ratio will be equal to F RI = 100%

3.6 Liabilities

We let the liabilities of our fund be denoted by Lt in year t. The liabilities in year t are defined

as the present value of all future benefit payments known in year t. We will make a distinction between beginning and end-of-year liabilities. The liabilities at the start of the year are denoted by LP ri _{and at the end of the year by L}U lt_.

The primo liabilities in year t are equal to the expected present value of future benefit payments, that is LP ri_t = 74 X i=0 B_ti· 1 1 + ri_t i .

The ultimo liabilities in year t are equal to the expected present value of future benefits minus the benefits payments made in year t. The payments are discounted using the interest rate at year t + 1 because evaluation takes place at the end of the year.

LU lt_t = 74 X i=1 B_ti· 1 1 + ri−1_t+1 !i−1 .

3.7 Assets

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assets will be equal to AIni = LP ri₂₀₁₃· F RI_{. In the first year the primo assets are equal to the}

initial assets. In the years following the primo assets in year t are equal to the ultimo assets in year t − 1. AP ri_t =    AIni if t = 0 AU lt t−1 if t > 0

The ultimo assets in year t are equal to the primo assets in year t plus that years premium payments minus the benefit payments and that multiplied with that years portfolio return r_tp.

AU lt_t = (AP ri_t + Pt− Bt) · (1 + rtp)

The major part of the assets is invested in stocks, bonds and property. In 2011 in the Netherlands on average 27% was invested in stocks, 58% in bonds, 14% in other investments like property and commodities and 1% was cash.4. The return on stocks will be denoted by rS, the return on 5-year bonds by rB and the return on property by rP. Then our portfolio return rp at time t is equal to

rp_t = w1· rtS+ w2· rBt + w3· rP.

3.8 Indexation

Usually a pension fund grants indexation if the funding ratio is above a certain threshold. Ideally indexation by a pension fund means that the accrued rights are increased with wage inflation. The Dutch Central Bank (DNB) is the supervisor for Dutch pension funds and they have declared that a pension fund cannot grant indexation if the funding ratio is below 105%. Inflation can be either price or wage inflation. Whether price or wage inflation is used, differs per pension fund and might also differ per actives and inactives. We assume that our pension fund indexes with price inflation, denoted by ip_t at time t. Depending on the level of the funding ratio, pension funds may also grant partial indexation. This occurs when the funding ratio is above the minimal capital ratio (M CR) but lower than the solvency capital ratio (SCR). We assume that the M CR = 105% and the SCR = 125%. In practice the SCR depends on the investment portfolio but 125% is a reasonable approximation. When the funding ratio is 105% indexation is equal to 0. When the funding ratio is equal to 125% full indexation is granted. We assume that between the M CR and the SCR indexation increases linearly and this is extended to funding ratios above 125%. If inflation is negative and the funding ratio is higher than 105% there will be no indexation. The indexation at time t is given by

4

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21 Indext=    0 if F R ≤ M CR, F RU lt t −M CR SCR−M CR · max(i p t, 0) if F R > M CR.

3.9 Benefit cut

In case the funding ratio falls below the minimal funding ratio, a pension fund is in danger of not meeting all future obligations. Therefore the Dutch Central Bank obliges the funds to propose an at most three year recovery plan in case the funding ratio is below the M CR. A Recovery plan can for example mean that premium payments are increased, future accrual rate is decreased or current accrued rights will be decreased. The latter will be our method of choise because it is most fair. It is most fair because with this method all participant will see a decrease in current and future benefit payments. The other methods will only have negative financial consequences for active participants. We assume that if the funding ratio in year t falls below the M CR, the fund sets a target funding ratio (F R∗_t) for the upcoming three years as follows

F R∗_t+1 = F RU lt_t +1 3(M CR − F R U lt t ), F R∗_t+2 = F RU lt_t +2 3(M CR − F R U lt t ), F R∗_t+3 = M CR.

To obtain the target funding ratio we apply negative indexation in each year of the recovery period. By decreasing the liabilities the the funding ratio will increase. The liabilities need to be changed to LU lt,T_t such that

AU lt_t

LU lt,∗_t = F R

∗ t.

If we set LU lt,∗_t = (1 + It) · LU ltt , it can be shown that

It=

F RU lt_t F R∗_t − 1.

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Modelling

The model used to assess generational effects of the new term stucture is explained in Section 4.1. As described in Kortleve & Ponds (2006) the model uses value-based ALM which requires economic scenarios for simulation. Different types of models to make economic scenarios are explained in Section 4.2.

4.1 Generational Accounts

First introduced by Ponds (2003) we will use a Generational Accounts (GA) model to get insight into value transfers between age cohorts. Particularly the value transfers resulting from the introduction of the term structure with UFR. We will determine the market value of future stochastic cashflows in the pension contract, for each age cohort. The resulting gains and losses incurred by an age cohort are than evaluated in the generational account. Let T denote the number of years over which we record the generational accounts and V gives he current market value of an uncertain future cashflow. The definition of a generational account at time t for age x in 2013 is given in equation (4.1). GAx_t = T X i=1 Vt(Bix) − T X i=1 Vt(Pix) + (Vt(AxT) − Ax0) (4.1)

The generational accounts record for each age cohort how much benefit payments are received and how much premium is payed. At time T the remainder in the fund is shared among living participants according to the liabilities of the age cohorts Lx_T. If the total assets in the fund at time T are equal to AT, each age cohort receives AxT =

Lx T

LTAT. The starting assets are

substracted because they are already known for each age cohort at time 0. This way we end up with the value transfers between cohorts between time 0 and T . Because the benefits of an age cohort should always be compensated with the losses of another, the sum of the generational accounts equals zero.

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23

The generational accounts also have the following respresentation.

GAx_t = (Vt(LxT) − Lx0) + T X i=1 Vt(Bix) − T X i=1 Vt(Pix) | {z }

Net Benefit Option

+ (Vt(RxT) − Rx0) | {z } Residue Option (4.2) Here Rx_t = (At− Lt)L x t

Lt is the claim on the residue, the residue is the money in the fund that

is used as a buffer and has not been assigned to specific participants. The first component is called the net benefit option and reflects the value transfers between cohorts by time T . The second term is called the residue option and give the change in the claim on residue.

The generational accounts on itself are interesting because they show the value transfers between cohorts within a given plan. We are more interested in value transfers due to a new plan. Let us denote the generational account of the old plan by GAx,old_t and of the new plan by GAx,new_t , where t is time and x is the age. Then value transfers due to a new plan are given by

∆GAx_t = GAx,new_t − GAx,old_t .

In general the old plan will imply using the zero swap curve and the new plan will use the UFR curve. We can also compute the value tranfers of the net benefit option ∆N BO and the residue option ∆RO.

To generate the future stochastic cashflows, we need to model the development of key economic variables. Furthermore we need to be able to determine the market value of the uncertain cashflows in 2013. This will be discussed in Section 4.2.

4.2 Economic Scenarios

In an ALM study one investigates the financial risk that a pension fund may encounter in the future. It is a tool that pension funds use to set premiums and determine benefit payments such that current and future obligations can be fullfilled. Also the impact of interest rate risk on the liability side can be investigated and interest rate hedges can be used to minimized interest rate risk.

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over time and whether premiums, benefit payments, indexation or investment strategies should be adapted.

For the generational accounts we perform a so called value-based ALM. This means that all future uncertain cashflows will be priced today at market values. Hence besides generating future paths for the pension funds cashflows, we need a method to determine their currents market values.

4.2.1 Generate economic scenarios using a VAR and affine term structure model

There are multiple models at hand to generate economies used for the Monte Carlo simulation. A review of different models to use for an ALM study can be found in Scherer (2003). Stienstra (2010) uses a Black-Scholes model to generate a future economy and the deflator method to determine market values. The model is simple to use and intuitive but it is too restrictive for our purpose, because it assumes a flat interest rate and it is not calibrated to current market values. However this model could be extended with a with a stochastic interest rate model like the one of Vasicek (1977). Also more advanced extensions including stochastic inflation exist as described in Van Haastrecht & Pelsser (2011).

A popular model in recent papers about generational effects is a VAR model for the stochastic variables and an arbitrage free affine term structure model to model the interest rate. In an introductory paper by Hoevenaars & Ponds (2008) on intergenerational value transfers the authors use this model to investigate value transfers that may arise from a plan redesign or from changes in funding policy and risk sharing rules. Soer (2012) applies the same model to investigate the impact of a new financial assessment framework for Dutch pension funds. Lekniute (2011) uses the model extended with stochastic volatility and jumps to investigate the generational transfers of a Defined Benefit to Defined Contribution and discounting with the expected investment return.

The VAR and affine term structure model seem suitable to generate our economic scenarios. In section 4.2.1.2 the variables of interest will be discussed.

4.2.1.1 Model description

To describe the return dynamics we will use a VAR model. In Cochrane & Piazessi (2005) an affine term structure for a VAR model is described. Moreover this model allows for market consistent valuation using deflators or risk neutral valuation. Let us define the VAR(1) model as follows

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Here zt= (r3t, rt120, rt360, xst, it)0 are the stochastic variables. c ∈ R5 and Φ ∈ R5×5 are the VAR

parameters. V ∈ R5×5 is the covariance matrix of the error term. If we let Σ be the lower Cholesky decomposition of V such that ΣΣ0 = V , the model can also be specified as

zt+1= c + Φzt+ Σt+1, t+1∼ N (0, I). (4.4)

The parameters of the VAR model can be consistently estimated using ordinary least squares. As described in Scherer (2003) a property of VAR models is that when simulating, on the long-run the mean over scenarios of every variable will converge to the historical mean. However when simulating, one does not always want variables to converge to their historical mean but to another value. Scherer describes a method to change the constant of the VAR model such that when simulating, scenarios on average will equal the chosen mean. Let µ∗ denote the desired mean and ˆΦ the estimated values of Φ. Then c is changed to c∗ as follows

c∗= (I − ˆΦ)µ∗. (4.5)

The VAR model let us simulate our key economic variables including the interest rate on three bonds with different time to maturities. Out of these three interest rates it is necessary to make a full term structure for discount purposes. For the calculations of the generational accounts the interest rate model should be arbitrage free. Hoevenaars (2008) uses the affine term structure model of Cochrane & Piazessi (2005). In this model the interest rate at time t with maturity n is given by

rn_t = −An

n −

B_n0

n zˆt. (4.6)

The parameters An and Bn are recursively calculated,

An = An−1+ Bn−10 (c − Σλ0) + 1 2B 0 n−1ΣΣ 0_B n−1 (4.7) Bn = −δ1+ (Φ − ΣΛ1)0Bn−1 (4.8)

with A0 = 0 and B0 = 0. Slagmolen (2010) gives a procedure how to estimate the parameters

of the affine term structure model. The parameters that need to be estimated are λ0 and Λ1.

Instead of estimating λ0 and Λ1 we estimate H0 = Σλ0 and H1 = ΣΛ1, which are the risk

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4.2.1.2 Variable selection

For the future state of a pension fund there are certain variables of which the development is of key importance.

Bond prices can be derived from interest rate term structures. Interest rate term structures also play an important role in evaluating the liabilities of a pension fund. We will simulate the 3-month, 10-year and 30-year interest rate. Then we use an affine term structure model to model the complete term structure. Our model will also model the excess return on stocks. We define the excess return as the return on stocks minus the return on a 3-month bond. Using both the return on stocks and bonds, the return on assets can be determined. Another important variable is the price inflation. In prosperous economic scenarios, the pension rights of the participants will be (partially) indexed with inflation. On the long run indexation has a significant impact on the size of benefit payments to the retirees. Table 4.1 gives that variables that we will simulate with the VAR model.

Notation Description

r3_t 3-month interest rate r120_t 10-year interest rate r360_t 30-year interest rate xst Stock return

it Inflation

Table 4.1: Stochastic variables used for simulation of future economy

4.2.1.3 Data description

The VAR parameters are estimated using historical time series. One has several options in choosing interest rate time series. Pension funds in the Netherlands currently discount their liabilities using the Euro swap curve. This swap curve is constructed by the European Central Bank (ECB) and this is the term structure for lending and borrowing by Eurozone national banks. This curve is assumed to have no risk premium. A disadvantage of using this curve is that currently the ECB does not issue bonds. Because our data will also be used to construct bond prices, the Euro swapcurve is not a proper candidate. Another candidate would be to use bonds issued by the Dutch central bank, the drawback is that in this term structure a risk premium is included and a long history cannot be obtained. German zero coupon bonds have a long history of trading and the term structure in recent years is close to, or even below the Euro swap curve. Therefore German bonds will be used to discount liabilities and to construct bond prices.1

1

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As a proxy for stock returns we use data of the MSCI world index.2 The MSCI world index is quoted in dollars and is converted to euros using the exchange rate.3 We will the Dutch consumer price index to model inflation.4

German zero coupon bonds with 30-year maturity are traded since 1989, we can not take an observation period that starts earlier than 1989. We will use the series January 1991 to December 2012. Summary statistics of the data are reported in Table 4.2.

3-month rate 10-year rate 30-year rate Excess stock return Inflation

Mean 0.034 0.049 0.053 0.029 0.022 Median 0.032 0.047 0.051 0.072 0.021 Standard Deviation 0.022 0.017 0.015 0.173 0.008 Kurtosis 0.243 -0.362 -0.763 0.557 0.493 Skewness 0.720 0.272 0.063 -0.633 0.545 2.5% Quantile 0.001 0.017 0.023 -0.403 0.008 97.5% Quantile 0.085 0.084 0.080 0.320 0.041

Table 4.2: Sample statistics of the the historical data

The table shows that interest rate mean increases with time-to-maturity. Furthermore the standard deviation seems to decrease with maturity. The kurtosis and the skewness decrease with maturity. While the 3-month interest rate is skewed to right with a skewness of 0.720. The excess stock return is on average 2.9% and the median is 7.2%. Since the skewness is −0.633, extreme negative stock returns occur more often than extreme positive returns. The inflation mean is 2.2% and the median is similar. Also note that inflation is somewhat skewed to the right.

A more detailed view can be given using graphs. In Figure 4.1 the data is plotted.

2

Source of the MSCI world index is Global Financial Data at http://globalfinancialdata.com.

3

Before the introduction of the Euro in 1999 the European Currency Unit (ECU) is used.

4

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1995 2000 2005 2010 Interest rate Year Interest Rate 0 % 2 % 4 % 6 %

8 % _{3 month maturity observed} 10 year maturity observed 30 year maturity observed

1995 2000 2005 2010 Inflation Year Rate 0 % 1 % 2 % 3 % 4 % 5 % 1995 2000 2005 2010

Excess stock return

Year Retur n −40 % −20 % 0 % 20 % 40 %

Figure 4.1: Annualized historical data in the period January 1991 - December 2012 used to fit parameters of the VAR model

The interest rate series show evidence of unit roots behaviour. Inflation and excess stock return move around the mean and look stationary. Performing an Augmented Dicky Fuller (ADF) and Phillips-Perron (PP) we can find out whether the series are stationary. For both tests the alternative hypothesis is stationarity. In the ADF test the regression include a linear trend and a constant. Test results are given in 4.3.

ADF PP

statistic p-value statistic p-value

3-month bond -1.95 0.60 -2.11 0.53

10-year bond -2.90 0.20 -3.19 0.09

30-year bond -2.95 0.17 -3.32 0.07

inflation -13.73 <0.01 -12.55 <0.01 excess stock return -10.29 <0.01 -13.82 <0.01

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Under both tests only the inflation and excess stock return series reject the null and are sta-tionary. All the interest rate timeseries are non-stasta-tionary. Because the authors of the other papers using this model setup succeeded in estimating the model nonetheless, we will continue with our model estimation.

4.2.1.4 Model estimation

VAR model The VAR model is estimated using OLS per equation. Estimation results can be found in Table 4.4. r_t3 r_t120 r_t360 xst it R2 r_t+13 0.944 0.119 -0.153 0.001 0.003 0.991 (0.018) (0.072) (0.061) (0.000) (0.002) r_t+1120 0.041 0.794 0.163 0.001 0.006 0.985 (0.018) (0.070) (0.059) (0.000) (0.002) r360 t+1 0.011 -0.013 0.993 0.000 0.006 0.984 (0.017) (0.067) (0.056) (0.000) (0.002) xst+1 -2.345 -13.392 16.748 0.158 0.371 0.024 (4.927) (19.361) (16.293) (0.061) (0.641) it+1 0.299 -0.431 0.215 0.000 0.281 0.067 (0.461) (1.810) (1.523) (0.006) (0.060)

Table 4.4: Estimates and standard errors of Φ

The interest rates show a high R-squared values of at least 0.98 which could indicate a very good fit. Because of the high coefficient values while all variables are on the same scale, the interest rates are mostly explained by their own lag. The excess return and inflation have low R-squared values of 0.025 and 0.067 respectively. The large standard errors on the lagged interest rates of excess return and inflation are likely to indicate that these variables are not significant.

Because of the evidence that the interest rates are not stationary, the high R-squared values not necessarily indicate a good fit. Granger & Newbold (1974) describes that a misspecification of the model can lead to high R-squared values and inefficient estimates of the VAR model.

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H0= Σλ0 =         θ1 0 c3 θ2 θ3         , H1 = ΣΛ1 =         θ4 θ5 θ6 θ7 θ8 0 0 0 0 0 Φ3,1 Φ3,2 Φ3,3 Φ3,4 Φ3,5 θ9 θ10 θ11 θ12 θ13 θ14 θ15 θ16 θ17 θ18        

We estimate the parameters θ by minimizing the difference between the true historical interest rates and the fitted interest rates on the 3-month, 10-year, 30-year and 70-year maturities. We include the 70-year interest rate because otherwise the affine term structure decreases after the 30-year interest rate until almost value 0 at the 75-year interest rate. We assume that the 70-year interest rate is equal to the 30-70-year interest rate. Let T index our historical time period, then min θ X n=3,120, 360,840 X t∈T r(n)_t − ˆr(n)_t 2 .

The estimated risk premium parameters of the affine term structure model are as follows

ˆ H0 =         −0.00028 0.00000 −0.01115 0.00045 0.00003         ˆ H1=         0.05884 −0.03631 0.00395 −0.01491 0.04067 0.00000 0.00000 0.00000 0.00000 0.00000 −2.34491 0.37068 0.15834 −13.39207 16.74754 −0.01552 −0.04261 −0.00337 −0.04769 −0.02702 −0.00099 0.00932 0.00006 −0.00297 −0.00628         .

The estimated parameters of the VAR model are used to estimate the risk premium parameters. Because our VAR model fits the interest rates poorly, it is likely that this will have an impact on the risk premium parameters.

4.2.1.5 Failure of the model

In Section 4.2.1.3 it is described that the interest rates series have a unit root which according to Granger & Newbold (1974) results in inconsistent estimates for our model parameters. Our simulation results will show us whether the model is still usefull.

The uncertain future cashflows in a pension agreement can be seen as embedded options de-pendent on the state variables zt. To value these cashflows in a correct way we need to value

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A deflator is a stochastic discount factor. It can be used to calculate current market values of future cashflows generated under real world scenarios with measure P. The stochastic discount factor attaches greater value to economic scenarios with a lower than average risk premium. The same way it attaches less value to scenarios with a higher than average risk premium. In this way the risk-averseness of investors is incorporated is current value of a future cashflow. An appealing property of the deflator is that we can compute market consistent values of cashflows while generating real world scenarios. Let Mt denote the discount factor at time t. Hoevenaars

(2008) proves for our model setup, which is defined in Section 4.2.1.1, that the deflator is given by Mt+1= exp(−δ1zt− 1 2λ 0 tλt− λ0tt+1), (4.9)

where λt= λ0+Λ1zt. The value of a random cashflow CFt+1at time t in the deflator framework

is equal to Vt(CFt+1) = EPt(CFt+1·Mt+1). Hence if we generate N scenarios and let cft+1n denote

one cashflow realization and mn_t+1 the stochastic discount factor at time t + 1 in scenarios n, the market value of CFt+1 at time t is

Vt(CFt+1) = 1 N N X i=1 cf_t+1i mi_t+1 .

A cashflow at time t + k, k ∈ N has value Vt(CFt+k) = EPt(CFt+k·Qk_i=1Mt+i) at time t. Let us

define the cumulative deflator by M_t+k∗ =Qk

i=1Mt+i.

Let us denote the risk free one year interest rate at time t by r_tf. Telmer (2007) shows that the mean of the deflator is equal to the one year bond price EP

t(Mt+1) = 1

1+rf_t.

Theoretical aspects of the deflator seem elegant. Let us try to estimate the deflator in a three variable model. Our state variables are the three month interest rate, excess stock return and inflation, zt = (r3t, xst, it)0. Note that this model would not be suitable to investigate

implications of a new term structure. This is mainly due to the fact that the model is not able to realistically simulate term structures. We use monthly data from 1975 until end 2012, the long observation period should minimize stationarity problems in our short rate. The estimated model is similar to the one used by Soer (2012), who used it to value options in the pension contract.

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2015 2020 2025 2030 2035 0 1 2 3 4 5 Deflator Year V alue Median Mean 95% Confidence Interval 99% Confidence Interval 2015 2020 2025 2030 2035 0 1 2 3 4 5 Cumulative Deflator Year V alue Median Mean 95% Confidence Interval 99% Confidence Interval

Figure 4.2: Deflator and Cumulative deflator in the Soer model over 100,000 scenarios.

As explained before, the mean deflator should corresponds to an risk-free discount rate. The median however converges to 0 quickly. Also the mean value of the deflator is very unstable, even after generating 100,000 scenarios it is still not very stable. Simulating the pension fund 100,000 times is already no longer feasible because of memory capacity problems. While the mean deflator might correspond to a risk-free discount rate, the quickly converging median seems unattractive. It says that more than half of the cashflows that occur in 2020 are worth nothing today.

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33 2015 2020 2025 2030 2035 0 1 2 3 4 5 Deflator Year V alue Median Mean 95% Confidence Interval 99% Confidence Interval 2015 2020 2025 2030 2035 0 1 2 3 4 5 Cumulative Deflator Year V alue Median Mean 95% Confidence Interval 99% Confidence Interval

Figure 4.3: Deflator and Cumulative deflator in the 4-variable model value over 10,000 sce-narios

The figure shows that the cumulative deflator is very unstable. Furthermore the 97.5% quantile is close to zero after 2025. This deflator seems unsuitable to use for market consistent valuation of cashflows. Note that the median deflator below is below the the median deflator in the Soer model.

Five variable models were also fitted and a deflator was obtained. We tried to reproduce the model used in Hoevenaars (2008). The deflator results were worse than in the 4-variable model. This also holds for the deflator in our own model.

Clearly a trend is visible that if we add more variables to our model, the deflator become more unsuitable. While the 3-variable model already produced a disappointing deflator, for the 4-and 5-variable models this holds even more. Hoevenaars (2008) says in a footnote that the deflator needs a lot of scenarios to obtain a high degree of accuracy with respect to risk neutral simulation. He decided not to use that deflator to obtain option values but to apply risk neutral pricing. Considering our estimation results it is arguable whether the amount of scenarios used is the underlying problem of the inaccurate option prices. It does not seem realistic to assume that more scenarios (e.g. >1,000,000) will results in a better deflator. Furthermore it is computationally impossible to obtain so many scenarios when generating 25 year future on a monthly basis.

4.2.1.6 Possible solution

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model. A disadvantage of assuming unit root behaviour of the interest rate is that it will behave like a random walk when we want simulate future paths. According to Jardet et al. (2013) the historical interest rates do not have a unit root but instead have extremely slow mean reversion. They propose a so called near-cointegrated VAR model. This model is an interpolation between an ordinary VAR model and a VEC model. They also describe an arbitrage free affine term structure model. Whether these models are able to give satisfactory results could be a topic of further research.

4.2.2 Achmea Holding scenarios

Alternatively we can use the scenario generator described by Achmea Holding NV (2012). The variables used and how they are modelled is explained in Section 4.2.2.1. A description of the data is given in Section 4.2.2.2.

4.2.2.1 Variable selection and model description

The scenario generator allows for a extensive amount of economic variables for simulation. The variables that will be used an how they are modelled is shown in Table 4.5.

Variable Model

Nominal Euro Zero Swap Curve Displaced Diffusion Stochastic Volatility LIBOR market model Assets World Stochastic volatility jump diffusion model

Property Listed Europe Stochastic volatility jump diffusion model Price Inflation Netherlands 1-factor Vasicek model

Wage Inflation Netherlands 1-factor Vasicek model

Table 4.5: Economy variables and their models used for simulation

On every model, normally distributed shocks are yearly added to simulate future years. The shocks on the different models are normally distributed and correlated with a correlation matrix. This correlation matrix is calibrated using market data. The scenario generator also produces a deflator series for market valuation of cashflows.

4.2.2.2 Data description

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term structure and the up economy a 2% point higher term structure. The other variables have the same starting conditions over all economies. Every economy consists of 2,000 scenarios. In Figure 4.4 the term structure in 2013 and 2033 are visualized.

0 20 40 60

Interest rate term structure 2013

Time to maturity (Years)

Interest r ate 0 % 2 % 4 % 6 % 8 % Down Economy Basic Economy Up Economy 0 20 40 60

Interest rate term structure 2033

Time to maturity (Years)

Interest r ate 0 % 2 % 4 % 6 % 8 % Down Economy Basic Economy Up Economy

Figure 4.4: Mean interest rate term structure in 2013 and 2033

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2015 2020 2025 2030 2035

Stock and Property mean return

Year Retur n 0 % 1 % 2 % 3 % 4 % 5 % Mean Stocks Mean Property 2015 2020 2025 2030 2035

Stock and Property return distribution

Year Retur n −150 % −100 % −50 % 0 % 50 % 100 % Median Stocks Median Property

95% Confidence Bound Stocks 95% Confidence Bound Property

2015 2020 2025 2030 2035

Price and Wage mean inflation

Year Inflation 0 % 1 % 2 % 3 % 4 % 5 % Mean Price Mean Wage 2015 2020 2025 2030 2035

Price and Wage inflation distribution

Year Inflation −15 % −10 % −5 % 0 % 5 % 10 % 15 % Median Price Median Wage

95% Confidence Bound Price 95% Confidence Bound Wage

Figure 4.5: Distribution and mean series of the inflation and assets over time

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Chapter 5

Results

Using 2000 economic scenarios we can simulate a future for our pension fund. In each scenario the funds cashflows will develop differently. Using deflators we can determine the current market price of these cashflows. This will help us to calculate the generational account. The simulation occurs on a 25-year horizon. A small horizon will not give significant results. A horizon longer than 25-year is unrealistic since pension arrangements may have been drastically changed by then. In each case we simulate the pension fund twice. The first time we only use the normal term structure for the complete horizon. The second time we alter the the term structure with the UFR in the course of 2013, until the end of the simulation period. For both simulations we calculate the generational accounts and investigate the difference between the two for each age cohort.

5.1 Cause of value transfers

As described in Chapter 2 the UFR will currently result in a decrease in total fund liabilities, which should give extra possibilities to grant indexation and reduce the size of benefit cuts. By looking at each component of the generational account seperately, it should be possible to see this. We split up the generational accounts as

GAx_t = (Vt(LxT) − Lx0) | {z } Liabilities + T X i=1 Vt(Bix) | {z } Benefit payment − T X i=1 Vt(Pix) | {z } Premium payment + (Vt(RxT) − Rx0) | {z } Residue

The basic economy described in Section 4.2.2.2 includes a term structure that is representative for the one in 2013. Both with the this term structure and this term structue with UFR all components are then calculated. To see how the UFR affects the components, we take for each component the difference between the situation with the new and the old term structure. Figure 5.1 shows how the different components change.

Generational eﬀects of a new term structure for Dutch pension funds

Generational effects of a new term structure for Dutch pension

funds

Contents

Chapter 1

Introduction

1.1

Goal of the paper

1.2

Dutch pension system

1.3

Improving sustainability

1.4

Balanced stakeholder value

1.5

Outline

Ultimate Forward Rate

2.1

History

2.2

Methodology

2.3

Effect on the term structure

2.4

Effect on the liabilities

Chapter 3

Pension Fund Setup

3.1

Type of fund

3.2

Population

3.3

Accrual of pension rights and benefit payments

3.4

Premium payments

3.5

Funding ratio

3.6

Liabilities

3.7

Assets

3.8

Indexation

3.9

Benefit cut

Modelling

4.1

Generational Accounts

4.2

Economic Scenarios

Chapter 5

Results

5.1

Cause of value transfers