Pension risk for a single participant in Defined Benefit and Defined Contribution systems : risk modeling and understanding risk behaviour

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Pension risk for a single participant in

Defined Benefit and Defined

Contribution systems

Risk modeling and understanding risk behaviour

I.N. Santoe (6356303)

MSc Actuarial Science & Mathematical Finance

FEB, University of Amsterdam

Supervisor UvA:

Dr. T.J. Boonen

Second supervisor UvA:

Dr. S. van Bilsen

March 2017

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This document is written by Student Iremm Santoe 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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Chapter 1 Introduction

One of the main issues in the financial world nowadays is the low inflation and the low stock market return rates This is also a big concern for the Netherlands that effect many pension funds by reducing the accrued pension. Low market rates for instance have an immediately great effect on the funding level for the Defined Benefit systems funds. In the Netherlands a shift, from the old on Defined Benefit (DB) to the Defined Contribution (DC) pension plan, is happening. In the paper by Most and Wadia (2015) it is stated that employers are withdrawing as risk sponsors of occupational DB pension plans because of the cost and cost volatility of these pension system. In contrary to the Defined Contribution market in for example the UK, the Netherlands DC market is still immature. If not managed well, this shift will bring great risks for pension outcomes. In the DC pension system, participants have more freedom to make pension choices. But the main question is, does a participant has enough knowledge about the risks in the pension world? This thesis attempts to provide a risk analysis of the Dutch DB and the DC systems and helps to understand risk taking.

The Dutch pension sector is described in chapter 2, where the pension history is reviewed and the differences between the DB and the DC systems is summarized. The participants bear all the risks in the DC system. According to Bovenberg and Nijman (2015) paper risk management in the DC system is not focused on providing a stable pension income. The liability part of the pension balance, where the mentioned risks are stated, are looked deeper into. After mentioning the risks, the challenge is to predict the risks with a stochastic model. The relation between the nominal rate and expected inflation is explained in this chapter with the Fisher hypothesis like emphasized in the paper by Ahmad (2008). Understanding this relation is important for the purchasing power of deferred retirement paying, which in the pension world is no longer a guarantee.

A description of the used stochastic model, called Black-Scholes-Vasicek model, is given in chapter 3. With the Black-Scholes-Vasicek model the stochastic nominal and stock return rates are calculated, these outcomes will be used in the analysis of chapter 4. Further the analytical part of both the DB and the DC systems is explained and the used data is described in this chapter.

In chapter 4 a one participant sample portfolio is build, which gives the DB and the DC outcomes for a single participant. This is a crucial part of the thesis that emphasizes the key buttons on which pension is achieved. The model analysis of both the DB and the DC systems are given and discussed. In the DB system the risk of conditional indexation is assumed, so the DB system is not a guaranteed DB. According to Bovenberg and Nijman (2015) paper the guaranteed pension benefits have become more expensive and the employers are withdrawing

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as risk sponsors of the DB system. In the Netherlands this withdrawing is also seen. Analysis of the DB system shows that the average pension wealth amount is 57% of the participant end salary. The pension benefit analysis of the DC system when chosen for the most risky investment results between 37% and 85% of the participant end year salary. This big range difference in outcomes of 48% within one chosen Life Cycle is for sure a huge risk for the participant. Three different Life Cycles investments options are analysed here, with the results that taking bigger risks gives higher returns. This reasoning is stated in many academic papers and this thesis also emphasizes this. The main conclusion is that there is a fundamental risk difference in outcomes within the Life Cycle investment and between the three Life Cycles investments. For the participants choosing a Life Cycle is a challenging task.

Chapter 5 attempts to find an answer to, why for every participant given the choice, the trigger of risks taking is different. Understanding Prospect Theory might help a participant to achieve more satisfied pension results for the future. Pension is deferred income and is related to the income the participant has. Setting a clear reference point, explained here, is very important for risks taking. The paper by Tversky et al. (1997) concludes that in some levels of the financial markets loss aversion is present. The loss aversion characteristic will be explained in this chapter 5. Further is shown that the highest expectation according to the standard expected utility theory is not always the best theory to describe ones wealth. Main finding is that the most highest DC system LC outcome is not always preferred by the participant.

Last, in chapter 6 the conclusion of the theory in the above chapters is summarized. Some explicitly important issues in the pension world, that are left out in this thesis, are discussed. Here it is important and it is emphasized that every party, in this thesis the future retiree, has a crucial role and for sure benefits to achieve better pension outcomes.

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Chapter 2 The Dutch Pension Sector

If we think about pension we simply think about the following: ”A pension is a fund into which a sum of money is added during an employees employment years, and from which payments are drawn to support the persons retirement from work in the form of periodic payments”. This thinking was seen in form of a retirement support plan in the early 17th century by the invention of Lorenzo de Tonti called tontine. The first true tontine in the Netherlands was around 1670. A glimpse of further development in the Netherlands is as follows:

• in the second half of the 19th century came the employment pension.

• begin 20th century came retirement pension, which is a basic pension called ”AOW pen-sion”.

• the widows pension and orphans pension followed in 1919.

• in 1949 came the law ”Wet bpf for employer”, to compulsory participate to sectoral pension fund.

• in 1954 came the ”Pensioen- en Spaarfondsenwet(PSW)” till 2007. During 2007 came the ”Pensioenwet”, with the main aim to protect the employees retirement.

The central thought from a consumer perspective, looking at the development above, is con-sumption smoothing and insurance safety. So originally the market development such as de-mography, market value, interest rates, etc. from financial perspective were not an issue. In this chapter a review of the available literature of our pension scheme is given. In section 2.1 the Dutch pension sector is described. In section 2.2 an introduction to defined benefit and defined contribution system is given. These systems their (dis)advantages, changes nowadays, etc are scrutinized here. Section 2.3 gives an overview of the risks from Assets and Liabili-ties perspective side. Further the Fisher hypothesis, which emphasize the difference between nominal and inflation rates, is explained. At last the stochastic model is introduced, called the Black-Scholes-Vasicek model.

2.1 The Dutch Pension Landscape

The Netherlands pension system is divided into three pillars, shown in Figure 2.1 below. These pillars are for sure related to one another, with aim to achieve a safe retirement.

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Figure 2.1: The three pillar Dutch pension landscape (Boonen, T., UvA 2015-2016).

2.1.1 First Pillar: Public Retirement Pension

The retirement pension, as stated above, arose begin 20th century and is a basic pension called ”AOW pension”. This social security is covering up for most disabilities that prevent individuals from working. It secures all residents in the Netherlands with a minimum income after retirement. The AOW pension age is being gradually increased, so is linked to the life expectancy. The insurance is funded by the government, primarily through taxes, called PAYG. The meaning of the financing PAYG is that it is a pay-as-you-go system, which denotes that workers pay contribution in the form of tax, in order to pay the pension benefits of the retirees. This gives a hugest pressure on the working class, because of more retirees due to getting older than expected. For every year that you are insured, you build up rights to 2% of the full AOW pension. The AOW pension is included in the computation of chapter 4.

2.1.2 Third Pillar: Individual Pension Savings

Here individuals establish their own life insurance contract or DC-scheme with a financial institution. Mostly used by individuals not covered by the second pillar (e.g. self employed) or individuals wanting extra pension income. The insurance is fully funding, meaning that the contributions are saved and invested in order to buy an annuity at the retirement age. Individuals can deduct taxes from the invested amount up to a ceiling. In the next sections we will talk about the second pillar, with Defined Benefit and a Defined Contribution contract.

2.2 Second Pillar: The DB and DC Pension Contracts

2.2.1 The Defined Benefit (DB) system

Traditionally on in the Netherlands we had the DB system, which is still the biggest in number. But a shift to the new type of pension plan, namely DC system, is for sure visible. While most

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private firms turned to a defined contribution scheme. There are two types of Dutch DB plans: the final salary plan where the employees final salary is considered, and the final average pay plan where the average salary over the final years of the employees career is taken into account. Let us look at a summary of the following characteristics of the traditional DB scheme:

• promised benefit is linked to career final-wage or average-wage which we will discuss later; • guaranteed a certain standard of living, so benefits are certain;

• pension benefit is linked to the career with annual accrual rate (ar %) of 1.88% in this paper as will be seen in the next chapter;

• pension benefit is linked to the career part-time work factor in period (k, k + 1), which is 1 in this paper;

• unconditional indexation to inflation, for employees, DB offer security. Participants get a secured pension payment with inflation protection;

• DB schemes require no management decisions by employees;

• employer bears the funding risks here and guarantees are still a major cost for the em-ployer. Pension payments under this scheme vary from one year to another, causing an undesired uncertainty for the employer;

• participants prefer a stable contribution rate.

The biggest advantages of this DB system are the guarantee that it offers, it is simple to understand, and easy to communicate. The biggest disadvantage to the above is the that the DB schemes are much more expensive than one was expecting in the past. After the financial crisis of 2008 we all more than ever value the guarantees of our pension. More than ever we want to understand some of the impacts of financial developments for our self. Some of the traditional DB scheme items above are certainly not given nowadays. Certainty mostly depends on the funding ratio of the fund you are in, which mostly affected participants near retirement, this is seen as an inter generational issue. Like for example might not get the unconditional indexation, so benefits are variable. Now individual pension holders are exposed to falling returns in the financial markets so benefits not guaranteed anymore. The contributions in DB plans nowadays are increasing, because of longevity and low interest rates.

2.2.2 The Defined Contribution (DC) system

Individual DC scheme has the following characteristics:

• contributions of salary Sk are paid by both employer and employee, but the employee is

left with the risks;

• upon retirement, the accumulated capital (investment returns) will be converted to an-nuities to cover their retirement consumption, called an individual lump sum;

• so no risks exposure for the employer here;

• participants bear the risks and employees have to make complex financial decisions in-volving risks;

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• benefit is only determined at retirement.

Looking at the disadvantages of DB and the advantages of DC, a shift from DB to DC is understandable for sure. In short the most important drivers that causes the shift from DB to DC are:

• labor mobility;

• reduced relevance of long-term employment contract, hence the reduced added value of DB plan;

• stricter regulatory environment;

• changes in pension accounting. Historical cost accounting works for a defined contribution plan, where the future cash flows, arising from past transactions, are certain;

• improvements in financial sophistication;

• increasing costs of DB plans are not sustainable;

• biggest risk advantage of this DC system is that it tackle the ageing population problem automatically.

A shift from the DB to the DC pension system is seen now in the Netherlands, but what about the risks we still have even with the DC system? These DC system risks are:

• inflation risk;

• investment return risk;

• pension conversion risk: longevity and exposure to interest rates, which insurers use at pensionable age.

We will have a deeper look of the above risks in the next chapter, especially for the investment return risk.

2.3 Assets and Liabilities perspective

In Figure 2.2 we noticed on the balance sheet that a pension fund or insurance company is exposed to different risks. Let us here focus on the Liability side of the balance. Financial risk is caused by deviations from the expected future pension payments and corresponding expenses compared to the estimates included in the valuation of the liabilities. Low mortality rates, low interest rates or high expense due to inflation increases may result in higher liabilities. We will look deeper into the inflation and interest risk part, by trying to predict both risks. The other risks are also important but not part of this thesis. Should the inflation be hedge from Assets and Liabilities perspective? Hedging is not done in this thesis, because we assume that inflation will stay low.

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Figure 2.2: Assets and Liabilities balance sheet of a pension fund or insurance company.

Figure 2.3: Inflation movement year 2016 in the Netherlands. Source: http://www. tradingeconomics.com/netherlands/inflation-cpi/forecast. The inflation rate in 2016 was smaller than 2%.

Inflation means currency devaluation, i.e., the purchasing power per unit of currency is declin-ing. Indexation is a technique to adjust income payments by means of a price index, in order to maintain the purchasing power. We like to have pensions that are protected from inflation risk. According to current reality, the inflation rate and stock market return are both low. For sure high stock market return can to some extent match the risk of indexation. High inflation rate severely decreases workers annuities purchase power, which makes indexation more necessary. Low stock market return, which is now the case, abates the possibility and necessity of index-ation. In the Euro zone low inflation there is a low inflation, see Figure 2.3 for the Netherlands rates. A fixed inflation rate of 2% is assumed in this thesis and is reasonable if we look at Figure 2.3 this . In Figure 2.4 the Netherlands interest rate is given.

Figure 2.3 and 2.4 shows inflation and interest rates for the Netherlands till end 2016. We clearly see that both rates are very low in year 2016, what confirms our concern.

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Figure 2.4: The interest rate movement year 2016 in the Netherlands. Source : http://www. tradingeconomics.com/netherlands/interest-rate. Looking for 10 years we see low inter-est rates end year 2016.

One of the largest risk for pension entitlements is inflation risk, because the pension entitlements are nominal according to the paper by Blommestein et al. (2009). Unlike nominal interest rate, real interest rate considers purchasing power. Let us look at the relationship between nominal rates and expected inflation given by the Fisher hypothesis. The Fisher hypothesis suggests that there is a relationship between interest rates and expected inflation and roughly says that:

Real interest rates(rrt) = Nominal interest rate(rnt) − expected Inflation(ϕek).

Meaning if the inflation is positive, the real rates would be lower than the nominal rates. Also if there is deflation the nominal rates would be lower than the real rates. We emphasize this equation because in the pension world often the real rate is confused with nominal rate. We hope that the purchasing power in this equation, given by the real interest rate, outcome is positive. Like mentioned before the inflation is positive fixed at 2% in this thesis.

There are two types of interest rate models: the no-arbitrage models and the equilibrium model. In the no-arbitrage model, interest rates are obtained from the existing market bond price. In the equilibrium model, dynamics are simulated from the nominal rates, which is an applicable prediction model for interest rate. We will look at the equilibrium model by using the variant called Black-Scholes-Vasicek for the financial market model, which will be explained in the following chapter.

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Chapter 3 Models, pension formulas and data

description

In section 3.1 of this chapter we will explain the stochastic model called Black-Scholes-Vasicek and generate outcomes that we will use for our model in chapter 4. In section 3.2 we describe the calculation behind the DB and the DC systems needed to do the analytical part. Further in section 3.3 we give the data description of the data we used for our calculation.

3.1 The Black-Scholes-Vasicek stochastic model

In this thesis we consider nominal short-term interest rate rnt and a stock price index St, which

gives us the stock return rates rSt, as the two risk factors. Where t ∈ Z represents the time in

years. We do this by applying the Black-Scholes-Vasicek model as the financial market model. The nominal short term interest rate rnt is given by the Vasicek one-factor model:

drnt = −a (rnt− b) dt + σrdWrt. (3.1)

Here we have chosen initial values of the parameters on the based of the academic papers we have read, like by Draper (2014) and also on our knowledge from our studies. Our market assumptions are realistic from our viewpoint and set as follow:

• a is the speed of mean reversion of the nominal short rate and we set a = 15% default; • b is the long-term mean of nominal short rate and we set b = 4% (is set same as the rn0;

• σr is the volatility of the interest rate and we set σr = 1%;

• Wr,t is a standard Wiener process that results in dWr,t which is a shock in the term

structure;

• term −a (rnt− b) is called the drift term and follows a mean-reverting process.

Furthermore in the Black-Scholes-Vasicek model, the price index is given by

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We do not predict inflation here like mentioned in previous chapter, the parameter ci defines

the constant inflation level and we choose to set ci = 2%. At last in the Black-Scholes-Vasicek

model the stock price St is formulated as:

dSt= (rnt+ πrisk) Stdt + σSStdWSt. (3.3)

Here we have also chosen initial values of the parameters in order to keep the expected stock return in each scenario so realistic as possible from our viewpoint (after trying lots of different default values). The assumptions we made are as follow:

• πrisk is the stock risk premium and we set πrisk = 4%;

• σS represent the volatility of stocks price and we set σS = 20%;

• dWSt represent the shock in stock prices.

The stock prices is more volatile and also additionally the higher the chosen values are, the higher the average stock return will be. We know since interest rates and the stock market move in opposite directions, there is an instantaneous correlation ρ (Wrt, WSt) which must be

negative and we set this to −5%. Besides the market price of interest risk, called λ, is also negative and we have set this equal to −0.15%. For the model above we also have set the start values, for the nominal short rate rn0 = 4%, the price index I0 = 1 and the stock price are

S0 = 1. Now that the Black-Scholes-Vasicek model terms are explained and valued we compute

the stochastic nominal short-term interest rate rnt and a stock return rates rSt. The stochastic

nominal short rate rnt is generated with the above drnt equation by taking thousand scenario’s

(s = 1, 2, ..., 1000), with each a horizon of fifty years (t = 1, 2, ...., 50). We will firstly look at the average generated outcomes for the nominal rates. So each year age t the nominal short rate rnt is computed by taking the average (dividing by 1000) of all the following simulated

scenario’s. This average with a horizon of t = 50 years is given in Figure 3.1, starting with value rn0 = 4% stated as above. We secondly look at the uncertainty of the generated outcomes for

the stochastic nominal rates. We order the given thousand scenario’s (s = 1, 2, ..., 1000) for the same year (and this is done for fifty years (t = 1, 2, ...., 50)). We than take 95% of the most probable realization, so we look at the confidence interval and create 97.5% and 2.5% quantiles. These stochastic rates are also given in Figure 3.1, with starting value rn0 = 4% stated as

above.

Looking at the outcomes in Figure 3.1 we see the uncertainty of the nominal rates. We will use the outcomes for computation in chapter 4.

The stock return rates rSt are calculated by the generated stock prices according to:

rSt =

St

St−1

− 1. (3.4)

With the initial stock price of S0 = 1 we stimulate also thousand scenario’s, with each a horizon

of fifty years. Since the future value of the stock return is uncertain, the second value at t = 1 was taken from the second scenario, the third value from the third scenario and so on, where stock prices Stand St−1 are modeled in above equation dSt. We will firstly look at the average

generated outcomes for the stock return rates. So the yearly age t stock return rate rSt is

computed by taking the average (dividing by 1000) of all the computed scenario’s for every t. The average return rates are given in Figure 3.2. A different story is it when we secondly look at the uncertainty of the generated stock return rates outcomes. We order the given thousand scenario’s (s = 1, 2, ..., 1000) for every same year (and this for fifty years (t = 1, 2, ...., 50)). We take 95% of the most probable realization, so look at the confidence interval and we create

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Figure 3.1: The 95% confidence interval for the generated nominal rates rnt, for the q quantile,

here q being 0.025 and 0.975.

Figure 3.2: The 95% confidence interval for the generated stock return rates rSt, for the q

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Looking at Figure 3.2 outcomes we see the uncertainty of the stock return rates. We will use the outcomes for computation in chapter 4.

3.2 The pension formulas giving lifelong benefits

In this thesis, the retirement income including the social security, is related to the average salary called Sk with k the age. Also the future Sk developments have great certainty like is

described in the next section. The pension basis also called the pensionable providing earnings Gk is the difference between the salary Sk and social security offset of age k. We assumed a

start social security offset of €12,953 at age 21 and this amount yearly grows with 2%.

3.2.1 DB system

So here for the Defined Benefit system career average formulas are used, that is the benefit at retirement equals a percentage of the career average wage. The formulas for the at age k for the accrued entitlements for retirement OPk,s is given by:

OPk+1,s = OPk,s+ 1.88% · Gk, (3.5)

with for OP0,s = 0 such that after k years

OPk+1,s = k−1

X

j=t0

1.88% · Gj, (3.6)

Meanings of above used terms are:

• Gk is pensionable income over year k and explained in next section 3.3 in equation 3.16;

• we generate thousand scenario’s s for each working age year k so s = 1, 2, ..., 1000; • accrual percentage is set at 1.88%, which is a % given by the Dutch Government on

http://wetten.overheid.nl/BWBR0002471/2016-09-01 (article 18a lid 2) for career average wage.

The formula for the at time k for fictional to-be reached pension (FOP) is given by:

FOPk+1,s = k−1

X

j=k0

1.88% · Gj+ 1.88% · Gk· (kop− k), (3.7)

meanings of above used terms are:

• Gk is pensionable income over year k and explained in next section 3.3 in equation 3.16;

• we generate thousand scenario’s s for each working age year k so s = 1, 2, ..., 1000; • accrual percentage is set at 1.88%, which is a % given by the Dutch Government on

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• FOPk+1,s is the fictional pension when reaching retirement, as provided for at time k and

kop is the time until the first year of retirement.

If we want the accrued pension rights keep their value, so the claims must also be indexed for inflation with symbol ϕk≥ 0. So the OPk+1,s term in above equation 3.6 becomes:

k−1 X j=k0 1.88% · Gj· k Y i=j (1 + ϕi). (3.8)

We assume indexed accrued pension rights, so set ϕk = 2% in this thesis. As mentioned in

chapter 2 even for the DB system there is no indexation guaranteed nowadays. If the coverage ratio is above a certain level (say 105%) the pension fund is allowed to index pensions, but this will also lower the coverage ratio because it increases liabilities.

3.2.2 DC system

For the Defined Benefit system the following hold for the capital accrual:

PRk = GRk% ∗ Gk, (3.9)

with PRk explained in section 3.3 in equation 3.17

DCCAPk+1,s = (DCCAPk,s+ PRk) ∗ (1 + retk,s). (3.10)

Here DCCAPk,s is the aggregate pension capital and retk,s is the return rates for s is the given

thousand scenario’s (s = 1, 2, ..., 1000) on investments of age year k mentioned in previous section 3.1. And PRk is the premium, Gk is pensionable income over year k given in equation

3.16 and GRk explained in next section 3.3 in equation 3.17. We get:

DCCAPk+1,s = k X j=t0 P Rj · k Y i=j (1 + reti,s). (3.11)

For DC the total DCCAPk,s, so called DCCAPtop is converted with a conversion factor to an

actuarial equivalent old-age pension. The conversion factor is the factor that is used to convert the DCCAPtop capital to a constant income after the pensionable age until the individual

deceases. The conversion factors change annually by changes in mortality chances and of course the time of conversion is also important.

At time pensionable age t we have:

Utop,s=

DCCAPtop,s

¨ axop

, (3.12)

where ¨axop is the present value of a constant payoff of 1 from the pensionable age xop until the

participant deceases given by:

¨ axop = ∞ X i=0 Qt−1 s=0(1 − qxop) (1 + r)t , (3.13)

the term qxop is the probability that someone aged exactly xop years now will die at year the

retirement age 67. The r is the actuarial rate over the life of the annuity. The actuarial interest rate r is chosen equal to 4% in this thesis, which is the same as the initial value of the nominal

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short rate rn0= 4% in the Black-Scholes-Vasicek model. The annuity factor ¨axop is male gender

determined. During the career, we use the fictional pension amount OPk,s, purchased at time

k with a deferred annuity DCCAPk,s like:

OPk,s =

DCCAPk,s xop−xk | ¨axk

, (3.14)

In our thesis we generate thousand scenario’s s = 1, 2, ..., 1000 for the DCCAPk,s, so in the

end we get our:

\ DCCAPk = 1 1000 1000 X s=1 DCCAPk,s. (3.15)

3.3 Used data description

Each year Statistics Netherlands (CBS) publishes the life expectancy and mortality rates for Dutch males and females. The Actuarial Society (AG) turned these mortality rates into de-terministic mortality tables for a five years period. This table is used for valuating pension liabilities. With the given mortality rates called qx we compute the survival rates px = 1 − qx.

We used the latest mortality rates 2009-2014 for Dutch males and females which can be found on the following website: http://www.ag-ai.nl/view.php?action=view&Pagina_Id=612. With formula (1 + r)(−t), where t is the working time left years, we computed the purchase deferred pension factor for every age k. The results of the pension purchase entitlement factor per age are given in Figure 4.6 in the next chapter. For computation of salary Sk growth we take

the parameters as given by the Netherlands law on website: http://wetten.overheid.nl/ BWBR0002471/2016-09-01 (article 18a lid 3) see Table 3.1. The salary Sk growth is next to

individual growth also linked to inflation, so an increased of 2% for next year k in this pa-per. The year k contribution premium P Rk is made available for forming the pension capital

Age k class Salary Sk Growth %

21 till 34 3% 35 till 44 2% 45 till 54 1% 55 till .. 0%

Table 3.1: Individual salary Sk growth from age 21 till 67 as given by the Netherlands law.

and is expressed as a percentage of the pension basis according to the graduated rates given on the following Netherlands law website: https://www.belastingdienstpensioensite.nl/ Besluit%20BPS%2012-02-2013.pdf

The graduated rates GRk % depends on the participant age k the moment we determine the

premium. GRk are shown in Table 3.2. The pension basis also called the pensionable income

Gk is the difference between the salary Sk and social security offset of that year.

Gk= Sk− social security offset. (3.16)

We assumed a start social security offset of €12,953 at age 21 and this amount is inflation linked grows with 2% yearly. So the contribution premium P Rk is computed as follow:

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Age k class 4 in % Pension 15 till 19 4,6 20 till 24 5,3 25 till 29 6,4 30 till 34 7,8 35 till 39 9,5 40 till 44 11,6 45 till 49 14,2 50 till 54 17,4 55 till 59 21,5 60 till 64 26,8 65 till 69 31,5

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Chapter 4 Quantitative Analysis

In this chapter we simulate the pension for a single participant working from 21 years till pension age 67. The goal of this chapter is to evaluate the difference in the outcomes of the pension when having a Defined Benefit (DB) system or a Defined Contribution (DC) system. We simply look at both systems, through the viewpoint of a single participant, with some possible pension turn buttons. We give an inside in both systems analytically with the computed stochastic nominal rates and stock return rates from chapter 3. The Defined Benefit system has a bigger guarantee from the employers, but contribution plays the key role. In the Defined Contribution system we can achieve different pension outcomes due to differently Life Cycle investments choices. First we establish a pension model with couple of assumptions, more about these assumptions are stated in section 4.1. In section 4.2 the DB and DC systems outcomes are simulated and discussed. In section 4.3 we compare the difference in the outcomes of the pension when having a DB system or a DC system. Conclusions about the chapter are drawn in section 4.3.

4.1 Model assumptions

Main remark about the model is that it is designed to show the pension outcomes each year for just one participant. The important aim is to understand what happens to our pension contribution and what turn buttons are there in the pension world. Basically pension funds aggregate pension contributions and they allocate the pension benefits in the same way like our model, only for lots of people. So the one participant model has the same basic that pension funds use. For the model we make some assumptions or better called simplifications, these assumptions are as follow:

• one male participant, with marital status unmarried the whole time. So no pension for spouse;

• participant salary Sk pattern growth is constant over time for each age, as given in Table

3.1. In Figure 4.1 the salary Sk pattern growth is given;

• the chosen starting salary is €20,323, ending at a maximum of €98,616, which is not the current cap of €101,519;

• contribution is paid yearly at the end of the year; • birthday is on the end of the year (31 December);

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Figure 4.1: Participant pensionable income Gk (= Sk - social security offset) pattern over the

47 working years.

• we assume a fixed 2% inflation and indexation;

• participant starts his career (working) and saving for pension from age of 21 till 67 year. So in total horizon of 47 years;

• we ignore possible unemployment during the not yet pensionable age, or possible employ-ment after the pensionable age;

• we look at the total pension at the end, so the first pillar (AOW), is also added. So for computation we use the Gk, which is explained in the previous chapter;

• longevity is not the issue we tackle here, pension payments for the male participant is paid until death.

We emphasize that with some simple adjustments, we can change the above simplifications. Like for instance adding different sort of career paths, which will give other starting and end salary. Similar adjustments can be made for every point above, but our focus is not on that bigger complete model. We simply want to explain why we do get or don’t get wanted pension outcomes. Why is the expected pension so uncertain? We use, the in chapter 3, computed rates to stimulate the pension outcomes for a participant. In Figure 4.1 the pensionable income Gk,

which is the difference between the salary Sk and social security offset of that year, is given.

4.2 Sensitivity analysis

The starting position in both systems is from year 2016 till 2075, where the participant joins the company at age 21 and retires at age 67. Furthermore, it is assumed that pension benefits are payed until death. The starting salary is€20,323, first paid at age 22 end of year 2016. The pensionable pay Gk (= Sk - social security offset) starts at age 21 with €7,370 and ends at age

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Figure 4.2: The outcomes of the indexed accrued pension rights for the DB system, as given in equation 3.8, compared with the Salary Sk outcomes.

4.2.1 The DB system

In Chapter 2 we already discussed the advantages of the DB system, the participant here bears less risks and for sure no longevity risk. We model the DB system without any investment risks, so the returns of the liability-driven investment strategies of the fund is not a concern here. Although this is not complete true also, because we know that rate of return plays a crucial role for the annual contribution. If low returns are earned than the annual contributions should rise also. We will look at the average career formulas of the DB system and the participant bears no risk here. Nowadays the pension fund funding ratio, given by:

FRDB_t = Assets

DB t

LiabilitiesDB_t ,

mostly predicts if indexation is given or not. So the risk is that the pension entitlements can be cut, meaning we have a conditional indexation rather than a guarantee benefit for in-actives, if the fund is underfunded. Here we set the funding ratio of the DB system at 100%. For the determination of the pension liabilities provision we do not take into account the risk for conditional indexations during active employment. In chapter 3 we assume indexed accrued pension rights and set ϕk = 2% in equation 3.8 in this thesis. Also the contributions for

pensions, is regulated by the pension fund funding ratio. For the DB system we compute the yearly indexed accrued entitlements for retirement, OPk+1,s term, as given in equation3.5. The

outcomes of the indexed accrued pension rights in the DB system are given in Figure 4.2. Looking at Figure 4.2 we see an end OPk+1,s average wage amount of €42,88. This is 57% of

the participants end salary Sk.

For the DB system in this thesis, where we have one participant, we don’t look at the yearly contributions. In contrast to the DC system in the DB system the contributions are not fixed. Next we will look at the DC system outcomes, which should be more interesting in risks bearing

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4.2.2 The DC system

Like in the DB system we assume the same working age, salary, etc. But here at the retirement age of 67, we buy an individual lump sum with the total input. As stated in different finance papers the mean idea behind the Life Cycle (LC) is, that it should be related to taking risks according to the participant his age k. In principle, contributions may be invested in any asset class, although in practice most plans limit investment to bonds, shares and money market funds. Portfolio diversification is needed to have spread among many different investment vehi-cles such as cash, stocks, bonds, mutual funds and perhaps even some real estate. In this paper the DC system has three Life Cycle (LC) investments choices, which are a diversified by five elements, namely:

• (1) stocks which is seen as the most risky one, but also the one we need for higher returns. Returns are computed with the stochastic model, like stated in previous chapter 3; • (2) real estates here have a fixed rate of 7%;

• (3) investment grade credits here have a fixed rate of 4.8%; • (4) bonds here have a fixed rate of 4.5%;

• (5) Pension stabilizer (Ps ) is a retirement stabiliser factor used by most companies which gives more security for last years before retirement. It has an opposite direction than the daily market interest, it goes up when market interest goes down and vice versa. So with a Ps one is able to buy more pension when the market rate is low, which is the case nowadays.

We assumed that these elements appear in different % according to participant his age for each of the three fixed types of Life Cycles investments. The investments of the individual assets is modeled such that proportion invested in stocks (also real estates) declines as the individual ages. General financial advice is to shift the portfolio composition towards relatively safe assets, certainly away from risky stocks, as the participant grows older and reaches his retirement age. For every Life Cycle above we also stick to this advice, having lower stocks % at higher age. A strong argue of this financial advice is given by Bodie, Merton and Samuelson. They argue that younger participants can invest with higher risks in early years since they have the opportunity to work harder if faced with lower returns to cover up their losses. We notice that all the these different Life Cycle investments have management costs, going from high to low when one ages. But in this paper we don’t take these costs into account. The costs are mostly 0.5% and lower for a certain age and left out for our participant. We will highlight one of the three fixed types of Life Cycles investments, but the steps are the same for the other Life Cycles investments. The DC system options based on three Life Cycle (LC) investments strategy are called:

• Defensive Life Cycle, beginning at age 21 with 50% in stocks, that is the less risky cycle. • Neutral Life Cycle, beginning at age 21 with 65% in stocks, that is neutral risky cycle. • Offensive Life Cycle, beginning at age 21 with 85% in stocks, that is the most risky cycle. The Offensive Life Cycle investment is the one we will highlight in details, the other two are idem ditto analyse. Computation details for the Defensive LC and the Neutral LC are the same like the Offensive Life Cycles investment case. The outcomes of OPk,s as given in equation 3.14

of all three LC are generated and discussed. In Figure 4.3 the most risky cycle, called Offensive Life Cycle, is given for a working horizon of 47 years.

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Figure 4.3: The Offensive Life Cycle, given for working horizon of 47 working years, diversified by five elements. We emphasize that for the last working years a 100% Ps element.

Figure 4.4: The 95% confidence interval for the generated return predictions when the partici-pant invests in the Offensive Life Cycle, for the q quantile, here q being 0.025 and 0.975.

The outcomes of the rates computed in chapter 3 are used to predict the three Life Cycle investments outputs. We will highlight the Offensive Life Cycle out of the three investment, but the steps are the same for the other Life Cycles investments. If the participant invests in the Offensive Life Cycle, the return predictions are given in Figure 4.4.

If we zoom in on the outcomes for the last working years for the Offensive Life Cycle return, we see that the outcomes are the same in the end. This is because we have chosen the Offensive Life Cycle such that in the four ending years we invest 100% in the Ps element. Seen the low market interest nowadays this choice is also made for the Defensive LC and the Neutral LC. This choice gives a maximum protection for the participant, because in the end he wants to buy more pension when the market rate is low. We will calculate the DC system outcomes in several steps with the in chapter 3 given formulas.

Step 1 We compute the contribution premium P Rk that is given in equation 3.17 with the

rates outcomes of section 3.1. The outcomes are given in Figure 4.5, we see that in the last year the participant contribute 31.5%, which is an amount of €20,916.

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Figure 4.5: The DC system contribution premium P Rk that is given in equation 3.17.

Figure 4.6: The factor ¨axop given equation3.13 form age 21 till 67 years.

Step 2 We compute the in equation 3.13 given ¨axop annuity factor, this is given in Figure 4.6.

Here is defined that the participant survives the retirement age 67.

Step 3 Calculation of the pension benefit OPk,s as given in equation 3.14 outcomes, where

purchasing a deferred annuity with DCCAP\ k as in equation 3.15, is given in Figure 4.7.

We generate in Figure 4.8 Offensive Life Cycle the for the 47 working years, each thousand scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s, so in the end we get our 1000 pension benefit

OPk,s, where purchasing a deferred annuity withDCCAP\ kas in equation3.15. These thousand

scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s outcomes, with a peek around €56,700, are

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Figure 4.7: The 95% confidence interval outcomes for the pension benefit OPk,s as given in

equation 3.14 when the participant invests in the Offensive Life Cycle, for the q quantile, here q being 0.025 and 0.975.

Figure 4.8: Histogram: The Offensive Life Cycle investment for 47 working years, each year thousand scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s, so in the end we get our 1000

pension benefit OPk,s, where purchasing a deferred annuity with DCCAP\ k as in equation 3.15.

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equation 3.14when the participant invests in the Defensive Life Cycle, for the q quantile, here q being 0.025 and 0.975.

In Table 4.1 we find the outcomes of the pension benefits for different quantiles for the Offensive LC investment. We see the highest OPk,s outcome of€83,430, which is an amount that we find

realistic high for the taken risk. Likewise the lowest OPk,s outcome of €36,840 might be low

for the participant, but this happens only for 2.5th percentile. If we look at the 50th percentile outcome of €63,924 and compare this to the participant his end year 67 salary Sk of €98,616,

we get a result of 65%. In computation we get for the participant, if we look at the 2.5th and 75th percentile results between 37% and 85%. The highest result for the most risky Life Cycle investment might sure be satisfying for the participant we think.

The Offensive LC investment The pension benefit OPk,s in€

Lowest quantile (2.5th percentile) 36,840 First quantile (25th percentile) 50,738 Median (50th percentile) 63,924 Third quantile (75th percentile) 83,430

Table 4.1: The Offensive Life Cycle investment: The participants old-age pension benefit OPk,s

as given in equation 3.14 outcomes, where purchasing a deferred annuity with DCCAP\ k as in

equation 3.15.

Step 1+ 2 for the Defensive LC and the Neutral LC investments are the same as in the Offen-sive LC investment. We compute step 3 for the DefenOffen-sive LC and the Neutral LC investments. The calculation of the pension benefit OPk,s as given in equation 3.14 outcomes, where

pur-chasing a deferred annuity with DCCAP\ k as in equation 3.15, for the Defensive LC and the

Neutral LC investments are given in Figure 4.9 and 4.10.

We generate in Figure 4.11 the Defensive LC for the 47 working years, each thousand scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s, so in the end we get our 1000 pension benefit OPk,s,

where purchasing a deferred annuity with DCCAP\ k as in equation 3.15. These thousand

sce-nario’s (s = 1, 2, ..., 1000) for the DCCAPk,s outcomes, with a peek around€50,100, are given

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equation 3.14 when the participant invests in the Neutral Life Cycle, for the q quantile, here q being 0.025 and 0.975.

Figure 4.11: Histogram: The Defensive Life Cycle investment for 47 working years, each year thousand scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s, so in the end we get our 1000

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Figure 4.12: Histogram: The Neutral Life Cycle investment for 47 working years, each year thousand scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s, so in the end we get our 1000

Here given the frequent of the realized OPk,s, we will analyse the outcomes in Table 4.3.

We also generate in Figure 4.12 the Neutral Life Cycle for the 47 working years, each thousand scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s, so in the end we get our 1000 pension benefit

OPk,s, where purchasing a deferred annuity withDCCAP\ kas in equation3.15. These thousand

scenario’s (s = 1, 2, ..., 1000) for the DCCAPk,s outcomes, with a peek around €52,350, are

given in Figure 4.8. We will analyse these outcomes in percentiles further in Table 4.3.

In Table 4.2 we find the outcomes of the pension benefits for different quantiles of the Defensive Life Cycle investment. We see the highest OPk,s outcome of €56,472, which is an amount that

we find small but realistic high for the taken risk. Likewise the lowest OPk,soutcome of€39,757

might be to low for the participant, but this happens only for 2.5th percentile. If we look at the 50th percentile outcome of €51,200 and compare this to the participant his end year 67 salary Sk of €98,616, we get a result of 52%. In computation we get for the participant, if we

look at the 2.5th and 75th percentile results between 40% and 57%. The highest result for the less risky Life Cycle investment might sure not be satisfying for the participant we think. So the participant who has chosen this should also think to save pension in the third pillar as seen in Figure 2.1. We emphasize here is that the best result of 57% here is the same as in the DB system case for the participant.

The Defensive LC investment The pension benefit OPk,s in€

Table 4.2: The Defensive Life Cycle investment: The participants old-age pension benefit OPk,s

equation 3.15.

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The Neutral LC investment The pension benefit OPk,s in€

Table 4.3: The Neutral Life Cycle investment: The participants old-age pension benefit OPk,s

equation 3.15.

LC investment. We see the highest OPk,s outcome of€73,648, which is an amount that we find

realistic high for the taken risk. Likewise the lowest OPk,s outcome of €40,662 might be low

for the participant, but this happens only for 2.5th percentile. If we look at the 50th percentile outcome of €61,693 and compare this to the participant his end year 67 salary Sk of €98,616,

we get a result of 63%. In computation we get for the participant, if we look at the 2.5th and 75th percentile results between 41% and 75%. The highest result for the most risky Life Cycle investment might sure be satisfying for the participant we think.

The DC systems Tables outcomes compared to each other will be discussed further in the chapter 5.

4.3 Comparing the DB to the DC system

In the above section 4.2 we have seen the outcomes for the DB and the DC system for one participant. Both systems are totally different systems, so it is very hard to compare both systems to each other. Still we will try in this thesis to compare both systems to one another. Having all the information about the DB system we know that to give a certain guarantee the fund must invest in less risky assets, because less risks taking gives more guarantee. So in order to compare the both systems we assumed a DB Life Cycle (LC) investments choice for the DB system. DB Life Cycle, beginning at age 21 with only 20% in stocks, that is diversified by also five elements, namely stocks (starting with 20% that decreases when participant ages), real estates (7%), investment grade credits (4.8%), bonds (4.5%) and Ps. Further we assume the financing is a PAYG contract and we take a fixed yearly GRk of 18% in equation 3.17.

If the fund find it necessary in the DB system the contributions will be adjust. The DB Life Cycle (LC) investment choice is adjusted for the participant very close to the Defensive Life Cycle (the less risky of the DC one). In Figure 4.13 we find the contribution premium Pk and

in Figure 4.14 we find the DB Life Cycle.

In Table 4.4 we find the outcomes of the pension benefits for different quantiles of the DB Life Cycle investment. We see the highest OPk,s outcome of €72,477. In computation we get for

the participant, if we look at the 2.5th and 75th percentile results between 59% and 74%. We emphasize here is that the lowest result of 59% here is bigger than the DB system outcome for the participant. Let us now compare Table 4.4 when taken the same DB P Rkfor the Defensive

Life Cycle investment in the DC system case, given in Table 4.5. In Table 4.5 we find the outcomes of the pension benefits for different quantiles of the Defensive Life Cycle investment with DB premium. We see the highest OPk,s outcome of €76,836. In computation we get for

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Figure 4.13: The DB system contribution premium P Rk that is given in equation3.17for every

year a fixed GRk of 18%.

Figure 4.14: The DB Life Cycle, given for working horizon of 47 working years, diversified by five elements. We emphasize that for the last working years a 100% Ps element.

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The DB LC investment The pension benefit OPk,s in€

Table 4.4: The DB Life Cycle investment.

emphasize here is that the lowest result of 45% here is must lower than the DB Life Cycle investment lowest result of 59% system outcome for the participant. When comparing both systems in above way, our main finding is that the DB gives more guarantee than the DC system. The DB Life Cycle, beginning at age 21 with only 20% in stocks, is a save choice. But

The Defensive LC investment The pension benefit OPk,s in€

Table 4.5: The DB Life Cycle investment.

we know that a fund must take risk in order to guarantee benefits in the future. In the PAYG financing system a fund will take more risks, because of long horizon perspective for a 21 year participant. So we think the DB system outcomes compared to the DC system are even more guaranteed than Tables 4.4 and 4.5 show.

4.4 Summary of comparison of the DB and the DC

sys-tem

In this chapter we modelled a simple portfolio in which a participant enters a pension plan in year 2016. With the 95% confidence interval outcomes calculated in above Tables we see how very sensitive pension outcomes are to changes. Also changes in for instance to the required % annual contribution GRkor Salary Sk can be crucial. This effect is not calculated in this thesis,

but with simple adjustments can be calculated with this model. We tried to the highlight the key buttons of the DB and the DC systems. We see that if we compared both systems the DB system has the biggest guarantee with the highest of the lowest result of 59% of all lower result of the DC system. The amount of the DC system contracts is still not the biggest in the Netherlands, but we see a shift towards these contracts nowadays like stated in the paper by Most and Wadia (2015). The understanding of this chapter is a main part, because with this participants may get more insight in the complex systems. In the pension world the understanding of risks, needed for example to choose a Life Cycle investment, is crucial. We will try to understand what drives a participant to take risks with a theory called Prospect Theory in the next chapter.

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Chapter 5 Prospect Theory (PT)

We introduce a concept of behavioural finance, called Prospect Theory, which is an essential theory in the finance. Ultimately in the DC system the participant decides how much risks, by making the choice of a Life Cycle investment, he is willing to take for his pension. We will try to understand on what base a participant decides his choice. We will study the participant risk (aversion) behaviour, that is highly relevant for pension saving. The finding is that individuals tend to feel losses harder than gains. Therefore prospect participants, explained later in this chapter, would prefer their retirement income to increase during their working life rather than vary. The main finding is that participants don’t act rational like stated in the standard finance. There are different key concepts within behavioural finance, like: anchoring, mental account-ing, confirmation and hindsight bias, overconfidence, Prospect Theory, etc. In this chapter we highlight the key concept Prospect Theory and we explain why understanding this might be important by analysing the outcomes of chapter 4 with a PT viewpoint.

5.1 Prospect Theory description

Daniel Kahneman and Amos Tversky are the founding fathers of the Prospect Theory. There is empirical evidence showing that the choices made by individuals in decision situations are often inconsistent with standard Expected Utility Theory (EUT). In the EUT, which is driven by expectation, the following assumptions holds:

• there is an expected utility model;

• a risk exchange is given by a random vector (Y1, ..., Yn) satisfying the full allocation

requirement given as follow:

Y1+ ... + Yn = X1+ ... + Xn;

• let Xj, with j = 1, 2, ... be a sequence of independent risks defined on a common

proba-bility space (Ω, F , P) with a common probaproba-bility distribution and expectation µ. A risk is a real-valued random variable, that is, a function X that maps Ω to the real line R and satisfies {ω ∈ Ω|X(ω) ≤ x} ∈ F for any x ∈ R;

• for each given j = 1, ..., n, the utility function Uj is non-decreasing, concave and twice

differentiable;

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• the outcome (or state) space Ω is finite.

Taken the above utility function assumptions optimal is maximizing the Expected Utility (EUT):

w1E [U1Y1] + ... + wnE [UnYn] ,

with positive weights (constants) w1, ..., wn and allocations (Y1, ..., Yn).

Figure 5.1 gives the standard utility function Uj, which is non-decreasing concave for gains

and also losses. Counter examples in the paper by Kahneman and Tversky (1979) shows that inequalities arises when individuals choose. So in simple words we may say that the descriptive power of the EUT is falling. So we will look at an alternative descriptive theory called the Prospect Theory.

Figure 5.1: The Expected Utility Theory (EUT) function, which is non-decreasing concave for gains and also for losses, says that individuals are risk averse even for losses. But counter examples shows that inequalities arises when individuals choose.

In the Prospect Theory the value of a regular prospect is determined by a: (a) value function of the form:

V(x, p; y, q) = π(p)v(x) + π(q)v(y),

where p and q are (objective) probabilities, and x and y represent gains (or losses) rather than absolute values of wealth.

And also a (b) weight function of the form:

W(x, p; y, q) = π(p)v(x) + (1 − π(p))v(y),

where above π associates a decision weight to a given objective probability with following properties:

• the function π is increasing with π(0) = 0 and π(1) = 1;

• we have π(p) + π(1 - p) < 1 (gives subcertainty, which will be explained further); • we have π(p) > p for small p (gives overweighting of small probabilities, which will be

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In Figure 5.2 we see the value function operating for losses (negative x) and gains (positive x) given the reference point (explained further). This Figure 5.2 states a convex form for negative x, meaning risk seeking behaviour. And a concave form for positive x which is the standard risk averse behaviour. The weight function is plotted at the reference point and explains the term loss aversion, which will be explained further. First we analyse the nature of the function incorporating three characteristics, which we will later explain, namely:

• (1) reference point (framing): The function v operates on gains/losses with respect to a given reference level;

• (2) s-shape function (diminishing sensitivity): The function is typically concave for posi-tive x and convex for negaposi-tive x;

• (3) loss aversion: The function needs to be steeper for negative x than it is for positive x.

Figure 5.2: Prospect Theory: we see the value function operating for losses (negative x) and gains (positive x) given the reference point. This leads to a convex form for negative x, meaning risk seeking behaviour and concave for positive x which is the standard risk averse behaviour.

We describe the above three characteristics starting with (1) called the reference point. Pension participants are expected to be different in terms of income, like inherited wealth and so on, although they earn the same salary like in our model. This difference in begin wealth gives every participant his own reference point. This reference point, which is different for everyone, is essential to understand participants their preferences toward risks. The standard utility theory considered utility by the reference point of the final state, rather than on the current state. We will explain this reference point with an example for our model in chapter 4. Say that the expected pension is lower due to for example low AOW rights, because the participant did not live from say age 18 to 21 in the Netherlands (3 years shortage for full AOW rights). Than it seems hard to believe that this participant will be choosing the most

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risky Life Cycle investment. He would rather sit tight on the pension he can have and not take the most risky Life Cycle investment. On the other hand, if the same participant is already wealthy now, like for instance owns already at age 20 a big house, he may be willing to take on higher risk because he can afford the losses. The reference point gives another dimension to the (long) horizon investment, which stated that when one is young one take more risk and vice versa. In short a reference point is a benchmark level that is unique for every pension participant. Knowing this we must take into account the fact that participants obtain utility by making losses or gains relative to their reference point, when analyzing the pension income obtained at retirement age. So it is crucial for a participant to set his proper reference point. It is the key as it determines the domain in which the outcome is assessed, that is, it determines whether the participant is considering losses or gains. Also in the DC system the decisions vary considerably in the different manners of describing the investment opportunities. We called the Offensive LC investment in chapter 4 the most risky cycle. We think by for example calling it the High Growth LC investment, participants would be more triggered to invest in this. This is also important for the reference point and is called framing, what must be well understood. Let us look at (2) the s-shape function in Figure 5.2, which gets the s-shape due to the reference point. In PT we separate between risk averse, risk neutral and risk seeking individuals which means the following:

• risk averse individual has an inherent resistance to risk and will always prefer the safest (for sure) option;

• risk neutral individual will be indifferent between the two options, he mostly looked at expected value;

• risk seeking individual not chooses the best outcome above the certain alternative(s); The Defined Contribution system is about making choices at this moment for the future. PT states that an individual is more risk averse towards gains and more risk seeking towards losses. This risk seeking towards losses seems less intuitive in terms of pension saving than we think. Let us get a better understanding of the above Prospect Theory Figure 5.2. The risk seeking effect for negative x outcomes for example can be explained with the following example:

• Problem I: choose between (a): 4,000 with probability 0.8 or nothing with probability 0.2, which gives an outcome of 3,200 and (b): 3,000 for sure.

• Problem II: choose between (a’): −4,000 with probability 0.8 or nothing with probability 0.2, which gives an outcome of −3,200 and (b’): −3,000 for sure.

Most people seems to prefer (b) to (a) and (a’) to (b’) in above problems. The outcomes in both above problems can not be describe with standard expected utility theory. According to EUT the outcomes should be: Most people seem to prefer (a) to (b) and (b’) to (a’) in above problems. In problem I we see the risk averse behaviour. But we see that when there might be losses people choose different, which is explained with PT and is called risk seeking. Having noticed this we may understand more about the risk behaviour of a participant. So when a participant of the DC system knows(/thinks) that there might be a small probability of loss, he will overestimate this probability. This might result in less good decisions to choose proper types of Life Cycles investments.

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to help them plan for their financial future. Looking at the last years we think that the word pension triggers negative emotions by the most participants such as fear, pain, dead, etc. By pointing out the risks points to pension participants the feeling will only be more triggered. So this might lead to no action at all for the participants side for the Defined Contribution system. The decisions also vary in the frequency of the feedback on their returns they received according to paper by Tversky et al. (1997). We think that it is necessarily to give updated information about the returns. Another thing is that pension is something were the so called delay discounting hold. In delay discounting hold we don’t value future outcomes so much as values now, which results in low interest for the pension now.

5.2 Analysis of the DC system outcomes from a Prospect

Theory viewpoint

In the former chapter we described a DC system, which has three Life Cycle investments choices. The reason we looked at only three choices is because we think that participants get discouraged if they have to many LC choices. The benchmark we will use for describing the PT consists of the DC system LC investments outcomes of chapter 4. First let us explain why we use the outcomes in % of the end salary. We think that many participants associate the end pension with the DB system average-wage FOPk+1,s fictional pension in equation 3.7. In this

FOPk+1,s the ambition of 70% pension percentage is set by many participants. For our model

participant who works in total 47 years, this FOPk+1,s is more than 70% of his salary. Lots of

pension participants participating in the pension system still expect this percentage. But for this expectation there is no room now, especially in the DC system case. Most participants we think still, do like to see the pension outcomes in terms of % of their salary (this is called editing phase). We will use this pension ambition to understand the participant behaviour with a PT viewpoint. In Figure 5.3 we find the three DC LC investments outcomes in % of salary Sk for

the lowest of 2.5% to highest of 75% quantiles computed in chapter 4. Looking at all the three 2.5% quantiles in Figure 4.15 we see not much difference in the outcomes (4%). Characteristic (1) of the PT is setting a proper reference point, that is the key as it determines the domain in which the outcome is assessed. This reference point determines whether we are considering losses or gains. Looking at the three LC 75% quantiles outcomes let us assume a certain pension of at least 70% and take this as the reference point. This start reference point is likely to be the standard choice for the participant so his staring choice is the Offensive or the Neutral LC. For sure his ambition is 70% and PT states that the participant cares much more about losses relative to his reference point than about gains. We assume a participant has an overview of his pension status say every five years. If the outcome at a certain point in those five years is below the reference point of 70%, this is than a loss from the participant his viewpoint. This outcome will lead to overestimation behaviour of the risk seeking participant, this is characteristics is the (2) s-shape function. This participant could be in any of the two LC, but because of the risk seeking behaviour the probably is the biggest that he will choose for the Offensive LC now. So at a certain time step of five years the participant ends with the most risky choice. We want to emphasize here that seen from PT viewpoint the participant, who overviews his pension, with an ambition of 70% pension ends soon (with probability higher than 50%) in the most risky LC. In addition to the previous section, the reference point can also be affected by other relevant factors such as expectations. We now take the reference point as expectations outcomes so in the three 75% quantiles outcomes a participant can expect the highest of 85%. This also gives the Offensive LC investment as choice, where the risk element is the biggest. With both assumptions of reference point we end up with the most risky LC. It is crucial that

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Figure 5.3: The three DC LC investments outcomes in % of Salary Sk for the lowest of 2.5%

to highest of 75% quantiles as seen in chapter 4. In section 5.2 we analyse these outcomes with the viewpoint of Prospect Theory.

participants are provided with adequate information on assets accumulated and concomitant downside risks to help them plan for their financial future.

Further for a risk averse or a neutral participant for both assumptions of the reference point his feeling will not be more triggered with the most risky case. So from the two LC investments choices named the Neutral LC and the Defensive LC, the participant ends up with the Neutral LC investment now. A much happening effect explained by characteristics (3) loss aversion is the staus quo bias. This means participants tend to maintain the choice they had previously, even if that plan is no longer the optimal choice. Participants seem to be unwilling to make adjustments to their LC investments as they age because they love status quo bias. We advice to let participants automatically roll into the Neutral LC investment choice. By making the Neutral LC the first choice with automatically enroll for new participants, but allow them to opt out, we think that this might be the best and safest advice. We also advise seen the in Figure 5.3 three DC LC investments outcomes in % of salary Sk for the lowest of 2.5% quantile

that a participant should think of Figure 2.1 third pillar saving for more pension. We think that participants should be stimulated for more pension awareness and in the next section we give some suggestions.

5.3 Suggestions to obtain more pension awareness

Like we say the main issues in the financial world nowadays is the low inflation and stock market return rates. We hope that the low interest nowadays must not give the participants ’low-interest’ in their pension. We should give the pension world more marketing transforma-tion, before it might be to late to prepare for the future we want. Looking at the Prospect Theory above we find that the average participant of the pension system is only partly ra-tional. Participants need more knowledge, than for example the Figure 2.1 three pillar basic information, about the pension we think. Like for instance information about how many in %

Pension risk for a single participant in Defined Benefit and Defined Contribution systems : risk modeling and understanding risk behaviour