Risk tolerance and a suitable lifecycle

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Risk tolerance and a suitable lifecycle

Louana van Dijk

Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business 22 October 2014

Supervised by: Prof. Dr. J. Kuné

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Master's Thesis to obtain the degree in

Actuarial Science and Mathematical Finance

University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Louana van Dijk

Student nr: 5921465

Email: Louana.1988@gmail.com Date: 22 October 2014

Supervisor: Prof. Dr. J. Kuné Second reader: Prof. Dr. R. Kaas Supervisor: Dr. G.C.M. Siegelaer

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Acknowledgements

For a long time I was looking forward to finishing my study and this thesis was the last stage of it. Recently I was thinking that almost my entire life I have been in school or at the University and now that door will be closed and another one will be opened. One of my favorite aphorisms is “Remember yesterday, dream about tomorrow but live today.” I am very happy today that I have finished my study now, but I will always remember this time with a smile on my face. And the future will be full of surprises, whether it will be work or study I will see what will happen from here on. Of course there are also some people I would like to thank. At first I would like to thank my mother, one of the most important people in my life, she has always been such a great support, tried to help me as much as she could and always showed interest. Besides my mother my partner Remiek has always been one of the greatest supporters and I am very grateful for that. Of course I would also like to thank my supervisor Gaston Siegelaer, it was such a great time working on this thesis with him always being there to talk and discuss matters. Also I would like to thank my colleagues, especially Erwin Blom, for all the help, motivation and time.

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1. Introduction

Before the global crisis of 2008 pensions weren’t associated with risk, but now most people know better than that. In the last couple of years participants in pension funds have been confronted with the harsh reality of cuts and the fact that their pension rights aren’t as

guaranteed as they thought. Also employers don’t want to bear the risk anymore in case there is a deficit in the pension fund.

Because of the growing amount of retirees, the increasing life expectancy and the economic crisis the pension system as we knew it is drastically changing in all sorts of aspects. There has been a raise of the retirement age for the state pension, and we are moving progressively towards a Defined Contribution contract.

Banks and insurers are obliged to act in the best interest of their customers. They have to know the needs of their customer, what level of knowledge he has and which risks he can bear, i.e. the Know-Your-Customer rule. However this isn’t the case for pension funds, they just execute the pension management contract which they agreed upon with the employer. Nevertheless it is of great importance that a pension fund knows what the actual risk is that the participants are willing to bear. When there is a big difference between the risk that the participants are bearing and the risk they are actually willing to bear this might result in a confidence crisis.

But according to the Frijns committee in 2010, most pension funds pay too little attention to tailoring the risk policy and investment policy to the risk tolerance of pension plan members. Even though this is an important aspect of the pension funds especially in times like this where pension funds are having hard times. In the long run the ambition of most pension plans is an indexed pension but at the same time the pension fund has to guarantee a nominal accrued pension. So therefore pension funds are concentrating on finding the balance between the long term and the short term horizon. But they ususally do not look at the characteristics of the participants of the fund. Because of the great impact an investment policy can have on the pension capital, and the importance of this pension capital to a lot of people, pension funds have to consider the risk their participants can and want to bear. The committee recommends that the risk that the participants can and want to bear becomes the leading principle in determining the investment policy. The investment policy should not be based solely upon aiming at the highest investment return.

The Dutch Central Bank (DNB) has stressed out the importance of good risk management (Visie DNB toezicht 2014 – 2018). The trust in the current pension system has dropped because the promises made before the crisis, have been broken in case of cuts in pension rights. DNB wants this trust to be rebuilt in four years mainly by improving the communication about costs, expectations and risks. Pension funds can’t make promises any more about the pension

payments that they are not able to fulfill. This will hopefully lead to a recovery of the trust in the pension sector. DNB also wants that pension funds have a financial policy which is linked to the risk profile of the participants. The Financial Markets Authority (AFM) (7-okt 2013) also

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So altogether the risk profiles of the participants should play an important role in the

determination of an investment policy, and according to DNB all pension funds must adapt their investment policies to the risk profiles of their participants within the next four years.

In the research done by the CBS (Meningen over pensioenkwesties CBS) people are asked four questions and with each question they are given three options, Option A, Option B and the last option is that they have no opinion or don’t know. The first question regards to the fact that people are living longer and therefore the pension system will cost more money. They are asked if they think it is more sensible to get a lower pension or to work longer. 53 percent chose option B, work longer and 23 percent chose option A, get a lower pension. So it turns out that there are a lot of people willing to work longer in order to pay for their pension. The second question was about the situation in which there won’t be enough money to pay out all the pensions. People are asked if they think the contribution should be increased or if the pensions should be reduced. 48 percent chose option A, increase of the contribution and 28 percent chose option B. This

conclusion could also be drawn from the third question, so there is a large group of people who want to save extra or work longer in order to get a higher pension.

Towers Watson is one of the companies that set up a risk tolerance research. They have developed two thorough questionnaires, one for participants of a pension fund and one for retirees. It has taken a great amount of time to design these questionnaires and the questions in it are well considered. The answers that a person gives are being used to estimate the risk appetite and the risk capacity of a person. The risk appetite can be defined as the person’s attitude to risk and the risk capacity can be defined as the ability of a person to take risk. This is done by allocating certain weights to the questions, and by giving scores to the answers of the questions.

The risk appetite can be measured by the person’s preference between taking risks and

certainty; this is processed in a part of the questionnaire. For example in one of the questions the participant is asked to choose between four different options. The four options differ with respect to expected pension payment and variability around that expected pension payment. A similar question is being asked in terms of the pension age instead of the pension payment. Besides these questions, there are more questions that play a role in determining the risk appetite.

The risk capacity can be determined by four objective factors. The first factor is financial capital which consists of accumulated pension rights, savings et cetera. The second factor is human capital which depends on factors like salary, education and career opportunities. The third factor is personal characteristics like age, gender, healthiness et cetera. The last factor is governance capacity and this is influenced by for example the amount of time a person spends on sorting out his financial matters, the interest in pension matters and the level of expertise he has. Questions about these objective factors have been implemented in the questionnaire.

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With the questionnaire that Towers Watson has developed, the risk appetite and the risk capacity of a person can be estimated. The risk tolerance of a person is being determined by combining 25 percent of the risk appetite score and 75 percent of the risk capacity score of a person. For an overview of the risk tolerance see Figure 1 below.

Figure 1 Risk tolerance

In this risk tolerance research persons are, inter alia, asked if they would accept a lower pension in case of adverse pension results. They are also asked if and how many years they are willing to work longer in case of an adverse pension result. The participants are asked how much money they could save extra in case of an adverse scenario. These three factors are all in different terms; accepting a percentage of which the pension will be lower, working an amount of years longer and saving an extra amount of money. This last can be done in euros but can also be expressed in terms of a percentage of the wage or percentage of the pension base. So these three factors are hard to compare to each other.

Because of the importance of good risk management and the connection to the risk profile of the participants, the main research question that will be central in this thesis is as follows; how can the risk tolerance of a person be translated into a suitable lifecycle? The term “risk” is used to indicate that there is a certain probability that a certain loss or damage will occur. In this thesis the term “risk” is used to indicate that there can be a probability that there will be a certain impact on the accumulated pension rights with the result that the desired pension will not be accomplished. To help answer the central question there are also two sub-questions. The first sub-question is; how to translate extra work years, extra savings and accept a lower pension into one term? And the second sub-question is; how to translate extra work years, extra savings and accept a lower pension into an appropriate lifecycle?

Risk tolerance

Risk capacity (ability to take risk)

Financial

capital

Human

capital

Individual

features

Governance

Risk appetite

(attitude to risk)

Preference between

taking risks and

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Figure 2 below pictures this balancing act between investment policy and risk tolerance of the pension plan membership. Risk tolerance is being measured by a single measure capturing ‘accepting a lower pension’, ‘working longer’ and ‘saving extra.’ The investment policy is

determined as the weights in return seeking assets and matching assets. This thesis will focus on tailoring the dynamic asset allocation, i.e. dynamic during pension accrual phase, to the risk tolerance of a pension plan member. Further explanation about both asset classes will be given in paragraph 2.3.1.

Figure 2 Risk tolerance and a suitable lifecycle

In summary, this thesis has the following scope. The Dutch Central Bank (DNB) and the Financial Markets Authority (AFM) have stressed out the importance of good risk management and a suitable risk profile of the participants for each pension fund. But according to the Frijns

committee, most pension funds pay too little attention to tailoring the risk policy and investment policy to the risk tolerance of pension plan members. Towers Watson is one of the companies that set up a risk tolerance research. In the research people are being asked if they would accept a lower pension, if and how much they could save extra and if and how many years they could work longer in case of adverse pension results. These three factors are all in different terms so they are hard to compare to each other. Because of the importance of good risk management and the connection to the risk profile of the participants, the main research question that will be central in this thesis is as follows: how can the risk tolerance of a person be translated into a suitable lifecycle? The emphasis will hereby lie on working longer, saving extra and accepting a lower pension.

The structure of this thesis is as follows. In the next chapter the methodology and a description of the models and methods that were used can be found. This chapter will be followed by a chapter with the results, in which we investigate the results for different options for saving extra and working longer. After that the analysis of the results will follow and at last the thesis will finish with the conclusion.

Investment

Risk

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2. Methodology

In order to answer the research question, the risk survey methodology applied by Towers Watson is being described. Furthermore the two existing models for computing the pension outcomes of different investment policies are being described. These models are the Global Asset model and the Life Cycle Model both developed by Towers Watson.

2.1 Risk survey

In the Risk tolerance survey people are being asked to choose between four options. Each option has been characterized by three possible pension incomes, a low pension income, an average pension income and a high pension income. The low income and the high income have an

exceedance probability of 2.5 percent towards the tail of the probability distribution each. In the first option there is a guaranteed pension income, so the low income is the same as the average and the high income. In the second option the high and low income deviate a little bit from the average income, in the third they deviate somewhat more and in the fourth they deviate the most. Table 1 below shows one example of the tables used in the survey. From the research paper it is concluded that the majority of the people choose option II, which is the option that contains some uncertainty but not too much.

Option I Option II Option II Option IV

Low income scenario (97.5 percent probability of

exceeding this outcome) € 1,600 € 1,400 € 1,200 € 975

Average scenario € 1,600 € 1,725 € 1,825 € 1,950

High income scenario (2.5 percent probability of

exceeding this outcome) € 1,600 € 2,025 € 2,400 € 2,750

Table 1 Scenarios and pension incomes

In case the investment strategy for a person exhibits a low income scenario that is beyond the acceptance level, this person can choose to work longer, accept a lower pension anyway or save extra1_{. The research paper that Towers Watson has published, analyses how many years people}

could work longer, according to the answers people give. On average a person is willing to work about 1.2 years longer. Conditional on the people who are willing to work longer the average is 3 years. Based on the answers given in the survey that Towers Watson performed it turns out that on average people are willing to save about 5 percent of their income. Because of the results found in the research paper the amount of years that people are willing to work extra is maximized at 6. The amount of extra savings that people are willing to save lies between 0 percent and 7 percent of the total income.

1_{In the model the assumption is made that people can save any amount of extra money within their pension plan. In practice this is}

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2.2 Global asset model

The Global Asset Model is a model which is being used by Towers Watson worldwide and represents the view of Towers Watson with regard to realistic asset return assumptions. This model produces a set of possible future investment returns, inflation rates, wage growth rates and interest rates. The underlying assumptions are determined by the global investment committee, and the following factors are taken into account: market yields, P/E ratios, other market data and the economic background; historical data on investment returns, correlations and volatilities; central banks' forecasts and objectives; their annual survey of investment managers' expectations; meetings with economic commentators and investment managers; feedback from clients and their advisers; forecasts provided to or by the US/UK Treasury and other Government bodies and the latest academic papers on the subject. The committee tries to use as many different sources of information as possible into the process. The model is being updated regularly.

The Global Asset model produces 3000 scenarios, thus approaching the continuous joint

probability distribution by 3000 discrete outcomes of future investment returns, inflation rates, wage growth rates and interest rates. The investment management costs are already taken into account in the investment returns. These scenarios are the input for the Life Cycle Model.

2.3 Life cycle model

The Life cycle model is a model which is being used to determine the replacement rate for a person. The replacement rate is calculated as the annual pension payment divided by the last pension base. The pension base is defined as the salary minus the pension offset. The

replacement rate measures the level of wealth a person is experiencing at retirement, relative to the income level he was used to; in sofar the second pillar pension is involved. The replacement rates are determined for the 3000 scenarios that are produced by the Global Asset model. In this thesis the 2.5 percent percentile replacement rate is chosen as the replacement rate in a ‘bad weather’ scenario and furthermore the average replacement rate is analyzed. But in order to determine the replacement rate some extra input is needed.

First the start year of the prognoses has to be chosen and the number of different variants that will be run can be set. Next the data for the person has to be filled in, the start age, the gender, the start salary, and the part-time percentage. Next the retirement age has to be chosen. A career growth can also be chosen, and this has to be filled in per age. The contribution base for DC has to be chosen and also the pension base. At last the investment strategy has to be chosen. In order to answer the research question several adaptions on the existing life cycle model have been made so that the model is more flexible to work with and gives the output that is needed for the aim of this thesis.

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2.3.1 Assumptions

In this thesis the focus will be on the Defined Contribution contract. According to the Dutch Pension Act, there must be an explicit contract between employers and employees about the agreement that they make concerning pension. This is necessary so that it is clear who takes on which risks. Within a Defined Contribution scheme, the participant pays an established

contribution, with which a pension capital is being accrued. On the retirement date the

participant buys a lifelong monthly payment from an insurer with this accrued pension capital. So every person has its own pension capital and the total pension payment is not fixed in advance.

In order to work with the model several assumptions had to be made about the pension offset, the contribution et cetera. In order to determine the pension base, the pension offset has to be deducted from the pensionable income. This pension offset is derived from the state pension (in Dutch: AOW) and is actually the amount of the wage over which one doesn’t pay contribution. But the pension offset is not a fixed amount, for a Defined Contribution contract it depends on the contribution scale that is used. Until now the minimum pension offset was determined by taking 100/70 of the state pension for a married person. This is based on the assumption that in 35 years a pension of 70 percent of the final pay can be built up assuming an accrual of 2

percent. But as from 2015 this will change. For a Defined Contribution contract the new

assumption is that in 40 years a pension of 75 percent of the final pay can be built up assuming an accrual of 1.875 percent. This will lead to a factor of 100/75 instead of 100/70. So the minimum pension offset in 2015 will decrease to € 12,552 (based on the AOW payment today) for a Defined Contribution contract2_{. For the contribution scale the most recent one per}

1-1-2015 is used, where the scale is derived from a 1.875 percent accrual rate for a Defined Benefit contract. Furthermore there is chosen for contribution scales based on a 3 percent interest rate and elderly pension and built up and postponed partner pension, this means that the partner pension is on risk basis. This contribution scale is the most common in the Netherlands; see Table 2 for the used contribution scale. The start year of the prognoses is set on 2015 because the contribution base and pension offset of that year has been used.

Age percentage of pension base

15-19 7.2 20-24 8.0 25-29 9.3 30-34 10.8 35-39 12.5 40-44 14.6 45-49 17.0 50-54 19.8 55-59 23.3 60-64 27.7 65-66 31.5

Table 2 Contribution scale

2_{Because of the changes in de ‘Wet verlaging maximumopbouw- en premiepercentages pensioen en maximering pensioeninkomen’}

and the ‘Novelle Wet verlaging maximumopbouw- en premiepercentages pensioen en maximering pensioeninkomen en het Belastingplan 2014′ from 27 may 2014 there will be a difference between the minimum pension offset for a Final Pay contract and a Defined Contribution contract. First this was the same, and in 2014 the minimum pension offset was € 13,449 for both contracts.Per 1-1-2015 the method of determining the minimum pension offset has changed.

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Every working person experiences an individual salary growth in his or her career on top of the general wage growth. But this depends heavily on the person itself and the branch where the person is working. For this model an assumption on the career growth had to be made. In consultation with Towers Watson an average career growth is chosen, see Table 3 below for the assumptions on the career growth.

Age Career growth (percentage)

25-34 3

35-44 2

45-54 1

Older then 55 0

Table 3 Career growth

Almost all pension funds use the most recent actuarial table issued by the Dutch Actuarial Association (Actuarieel Genootschap, AG). Every two years a new AG prognosis table is being issued, based on more observations and improved methodology where possible. The most recent3_{AG table is the 2012-2062 table, so that one is being used in the model. Also the}

retirement target age is 67, and the maximum pensionable salary amounts to € 100,000 (indexed with general wage growth rate). At retirement date the accumulated capital is being converted into an annuity. Because of the focus on the level of wealth from an individual perspective, the real interest rates are being used to calculate inflation-linked annuities. These inflation linked-linked annuities give a more correct picture of the purchasing power in

comparison to nominal interest rates based annuities, because of the inflation during retirement. For the analysis 19 different available lifecycles have been analyzed. Every lifecycle has a

different allocation to two asset categories: return seeking assets and matching assets. The matching assets aim to replicate the cash flow pattern of an inflation-linked annuity as of retirement date. One year before retirement date in every lifecycle the weight of the matching assets is set on 100 percent. There are different start weights of return seeking assets in the lifecycles. Furthermore there are different derisking start dates for the return seeking assets, i.e. the date on which the weight of the return seeking assets starts decreasing. Combining these two factors leads to 19 different life cycles (actually 20, but the most aggressive life cycle is left out, this one maintains 100 percent in return seeking assets until 5 years for retirement date). In Table 4 the different characteristics of the lifecycles can be found. As a basis scenario Lifecycle M has been chosen, starting with 85 percent of return seeking assets and starting to derisk 10 years before retirement date. There are 19 different life cycles so there will be a total of 19 different variants that can be run. These percentages are filled in with the investment strategy. For a full overview of the asset weights in the return seeking portfolio per age see Appendix A. The remaining weight i.e. 100 percent minus return seeking assets is being invested in the matching assets.

Finally the personal data for the scheme participant have to be filled in. For this several assumptions have to be made. The model will be run for a 25 year old male person with a start salary of € 25,000, a part-time percentage of 100 percent and a retirement age of 67. This person is chosen because it best represents the average person in the working population at the

beginning of their career. This person will be the straw man in this thesis.

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9 Lifecycle Percentage return seeking assets

Start derisking

(Number of years before the end of the contract)

A 70 20 B 77.5 20 C 85 20 D 92.5 20 E 100 20 F 70 15 G 77.5 15 H 85 15 I 92.5 15 J 100 15 K 70 10 L 77.5 10 M 85 10 N 92.5 10 O 100 10 P 70 5 Q 77.5 5 R 85 5 S 92.5 5 Table 4 Lifecycles

The metrics that are used to measure a person’s risk tolerance are the number of years a person is willing to work longer, the percentage he is willing to save extra, and the minimum

replacement rate he is willing to accept in a ‘bad weather’ scenario. 19 different lifecycles have been investigated and for each lifecycle the 2.5 percent percentile replacement rate and the average replacement rate have been calculated using the life cycle model. This model is run for 3000 scenarios, and each scenario contains a set of future investment results, inflation rates, wage growth rates and interest rates. Furthermore the model applies to the straw man; a male person, age 25 and contributing for his pension according to a defined contribution contract. In the next chapter the results that were found with the lifecycle model for the straw man have been described.

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3. Results

In this chapter the results of the different models that have been run will be given. All models are run for a male person with a salary of € 25.000 and age 25, with all the possible lifecycles and the 3000 scenarios as described before. By doing this the 2.5 percent percentile is determined, these percentages are the 2.5 percent lowest boundaries. The first paragraph will show the results of the replacement rates per lifecycle for the straw man assuming the straw man will not contribute extra on top of the given contribution table and will not work longer than the

assumed retirement age of 67. In the second paragraph the option of saving a certain percentage extra will be studied and in the third paragraph the option of working longer. In the last

paragraph the option of combining two options i.e. saving extra and working longer, will be examined.

3.1 Replacement rates for the base variant

At first the model was run for the straw man without saving extra or working longer. By doing this the 2.5 percent percentile and the average replacement rates were determined for every lifecycle, this will be referred to as the base variant. In Figure 3 the 2.5 percent replacement rate is set off against the average replacement rate of that lifecycle. It can be seen that the higher the average replacement rate, the lower the 2.5 percent percentile replacement rate is. To achieve a higher average replacement rate, more investment risk has to be taken on by choosing a lifecycle that exhibits a larger weight in return seeking assets, so in a bad weather scenario the 2.5

percent percentile will be a lot lower. See Appendix B for the detailed replacement rates for all the lifecycles.

Figure 3 Average and 2.5 percent percentile replacement rates per lifecycle

A B C D E F G H I J K L M N O P Q R S 50% 52% 54% 56% 58% 60% 62% 64% 66% 30% 31% 32% 33% 34% 35% 36% 37% 38% 39% 40% A ve ra ge re pl ac em en t ra te

2,5 percent percentile replacement rate

Replacement rate base variant for all lifecycles

A

B C D E F G H I J K L M N O P Q R S

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3.2 Replacement rates saving extra

The second step was running the model for different saving rates; the straw man is here assumed to choose a fixed percentage of his annual income that he will save extra every year starting at age 25. Figure 4 below shows the effect if the straw man decides to save a certain percentage extra. The bottom left point of every lifecycle is the basis scenario (i.e. not saving anything extra); every step to the right is saving 1 percent more.On the x axis the 2.5 percent percentile of the replacement rate is stated and the y axis shows the average replacement rate. In this case there are two possible methods to achieve a higher average replacement rate. The first method is choosing a lifecycle that is more risky, but in that case in a bad weather scenario the 2.5 percent percentile will be a lot lower. The second method is saving a certain percentage extra, the average replacement rate and the 2.5 percentage replacement rate will then increase relative to the basis scenario. It depends on the person’s preference for the 2.5 percent

percentile replacement rate and the average replacement rate which combination of lifecycle and percentage of saving extra is the best option for that person. See Appendix C for the replacement rates for all lifecycles.

Figure 4 Average and 2.5 percent percentile replacement per lifecycle for saving extra

50% 60% 70% 80% 90% 100% 110% 120% 30% 35% 40% 45% 50% 55% 60% 65% 70% A ve ra ge re pl ac em en t ra te

Replacement rates saving extra

A B C D E F G H I J K L M N O P Q R S

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3.2.1 Other options for saving

In paragraph 3.2 it is assumed that a person pays the percentage he wants to save extra immediately from the start until he will retire, in order to get a higher replacement rate. But there are also other options; two options will be examined in the following subsections. The first option explores various starting ages for extra saving. For the second option a percentage will be saved only if the replacement rate at that time is below a certain point.

Start saving at a later age

The option of starting at a later age with the extra saving will be shown here. The option is calculated for start ages 35, 45, and 55. In Figure 5 below the results can be found for lifecycle A, the other lifecycles have a similar shape. The bottom left point of every start age is the basis scenario (i.e. not saving anything extra); every step to the right is saving 1 percent more. On the x axis the 2.5 percent percentile of the replacement rate is stated and the y axis shows the average replacement rate. See Appendix D for the replacement rates for all lifecycles.

Figure 5 Average and 2.5 percent percentile replacement rates for different ages at which to start with saving extra for lifecycle A 50% 55% 60% 65% 70% 75% 80% 85% 90% 35% 40% 45% 50% 55% 60% 65% A ve ra ge re pl ac em en t ra te

2.5 percent percentile replacement rate

Replacement rates for different ages to start saving lifecycle A

start 25 start 35 start 45 start 55

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Dynamic saving

The concept of dynamic saving is that at the start of each year the straw man decides to save a fixed percentage if the replacement rate at the prior year is below a certain reference level. For this reference level the average replacement rate at every point in time is determined, so for every year the average replacement rate out of the 3000 scenarios is taken. The replacement rate for every point in time is calculated as the annual pension payment that can be bought with the DC capital at that point in time, starting on the pension date divided by the pension base at that point in time. So at the end of the year is checked whether or not the current replacement rate is below or above the reference level. If the replacement rate is below the reference level then the next year an extra percentage of the salary will be saved. At the end of every year this process will be repeated until the person will retire. The results of this can be found in Figure 6 below. The bottom left point of every lifecycle is the basis scenario (i.e. not saving anything extra); every step to the right is saving 1 percent extra. On the x axis the 2.5 percent percentile of the replacement rate is stated and the y axis shows the average replacement rate. In this case there are two possible methods to achieve a higher average replacement rate. The first method is choosing a lifecycle that is more risky, but in that case in a bad weather scenario the 2.5 percent percentile will be a lot lower. The second method is dynamically save a certain percentage extra: the average replacement rate and the 2.5 percentage replacement rate will increase relative to the basis scenario. It depends on the person’s preference for the 2.5 percent percentile replacement rate and the average replacement rate which combination of lifecycle and percentage of dynamic saving is the best option for that person. See Appendix E for the replacement rates for all lifecycles.

Figure 6 Average and 2.5 percent percentile replacement per lifecycle for dynamic saving

50% 60% 70% 80% 90% 100% 110% 120% 30% 35% 40% 45% 50% 55% A ve ra ge re pl ac em en t ra te

Replacement rates dynamic saving

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3.3 Replacement rates working longer

The decision of working some extra years can be made later on of course and will have no influence on the pension capital now. The assumption is made that if the straw man works longer, the same contribution rate applies at age 67 and no career growth will take place. Working longer impacts the replacement rate of a person in two ways namely (1) paying

contribution during a longer period will increase the total pension capital at the retirement date, so the attainable pension will increase and (2) the pension annuity that will be bought will be attained at a lower annuity rate, because the person is older. The results of this can be found in Figure 7 below. The bottom left point of every lifecycle is the basis scenario (i.e. working no extra years); every step to the right is working 1 year extra. On the x axis the 2.5 percent percentile of the replacement rate is stated and the y axis shows the average replacement rate. In this case there are two possible methods to achieve a higher average replacement rate. The first method is choosing a lifecycle that is more risky, but in that case in a bad weather scenario the 2.5 percent percentile will be a lot lower. The second method is working a certain amount of years longer: the average replacement rate and the 2.5 percentage replacement rate will

increase relative to the basis scenario. It depends on the person’s preference for the 2.5 percent percentile replacement rate and the average replacement rate which combination of lifecycle and extra work years is the best option for that person. See Appendix F for the replacement rates for all lifecycles.

Figure 7 Average and 2.5 percent percentile replacement per lifecycle for working longer

Replacement rates working longer

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3.3.1 Sensitivity working longer with no extra contribution

Because of the double effect, paying extra contribution and buying at a lower annuity rate, the replacement rate quickly improves. It is also interesting to investigate the separate influence of buying at a lower annuity rate, because the contribution effect can be seen as “saving extra”. In order to see the separate influence of the lower annuity rate the model was run again but now without the extra contribution payments after the age of 67. In Figure 8 the results of this can be found. The bottom left point of every lifecycle is the basis scenario (i.e. working no extra years); every step to the right is working 1 year extra. On the x axis the 2.5 percent percentile of the replacement rate is stated and the y axis shows the average replacement rate. Comparing this to Figure 5 it can be concluded that without the extra contribution during the time a person decides to work longer, the replacement rates increase less. See Appendix G for the replacement rates for the lifecycles.

Figure 8 Average and 2.5 percent percentile replacement per lifecycle for working longer

Replacement rates working longer (without contribution)

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3.4 Replacement rates combining save extra and work longer

In the previous paragraphs the effect of saving extra and working longer have been calculated in isolation. Now we will show the results when combining the two options. The bottom left point of every lifecycle is the scenario where 1 percent is saved and every step to the right is saving 1 percent more. In the first graph in every scenario the straw man will work one year longer, in the second graph he will work two years longer and so on. On the x axis the 2.5 percent percentile of the replacement rate is stated and the y axis shows the average replacement rate. In this case there are three possible methods to achieve a higher average replacement rate. The first method is choosing a lifecycle that is more risky, but in that case in a bad weather scenario the 2.5 percent percentile will be a lot lower. The second method is saving a certain percentage extra, the average replacement rate and the 2.5 percentage replacement rate will increase relative to the basis scenario. The third method is working a certain amount of years longer, the average replacement rate and the 2.5 percentage replacement rate will increase relative to the basis scenario. It depends on the person’s preference for the 2.5 percent percentile replacement rate and the average replacement rate which combination of lifecycle, percentage of saving extra and extra work years is the best option for that person. See Appendix H for the replacement rates for the lifecycles and also the combined graph of the four figures below.

Figure 9 Average and 2.5 percent percentile replacement per lifecycle for working 1 year longer and saving extra

60% 70% 80% 90% 100% 110% 120% 35% 40% 45% 50% 55% 60% 65% 70% 75% A ve ra ge r epl ac em en t r at e

Replacement rates working 1 year longer and saving extra

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Figure 11 Average and 2.5 percent percentile replacement lifecycle for working 3 year longer and saving extra

60% 70% 80% 90% 100% 110% 120% 35% 40% 45% 50% 55% 60% 65% 70% 75% A ve ra ge re pl ac em en t ra te

Replacement rates working 2 year longer and saving extra

A B C D E F G H I J K L M N O P Q R S 60% 70% 80% 90% 100% 110% 120% 35% 40% 45% 50% 55% 60% 65% 70% 75% A ve ra ge re pl ac em en t ra te

Replacement rates working 3 year longer and saving extra

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In summary, this chapter has shown how the 2.5 percent percentile and the average

replacement rates were found for the straw man where saving extra and working longer were not yet taking into account. By doing this the base variant was set up. The base variant is the starting point and after the outcome was found, the model was run for different options of saving extra and working longer. There are also a few assumptions made about the number of years a person is willing to work longer and the percentage he is willing to save extra. If a person is willing to work a few years longer it is assumed that in this period he will also pay an extra contribution and if a person decides to save extra he will save a fixed percentage of his yearly salary every year directly from the start. But running the models in order to calculate the

outcome of every possible combination of saving extra and working longer takes a lot of time. So in the next chapter the results are being analyzed in an attempt to find a method to estimate the replacement rates whithout running the model.

60% 70% 80% 90% 100% 110% 120% 35% 40% 45% 50% 55% 60% 65% 70% 75% A ve ra ge re pl ac em en t ra te

Replacement rates working 4 year longer and saving extra

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4. Analysis

In the previous chapter the 2.5 percent percentile replacement rates were calculated for all lifecycles. These were also calculated for different variants of saving extra and working longer. In this chapter the results will be analyzed with the help of trendlines and regression. The first paragraph will start with the analysis of the replacement rates per lifecycle and the percentage a person is willing to save extra. In the second paragraph the replacement rates per lifecycle and the number of years a person is willing to work longer are being analyzed. In the third

paragraph an analysis of the replacement rates with combining saving extra and working longer will be done. And in the last paragraph an application of the formulas that were found is looked at.

4.1 Analysis of the effect of saving extra

To investigate the relation between the percentage a person decides to save extra and the replacement rate, the effect on the replacement rate will be analyzed more extensively. In Figure 13 on the x axis the percentage that the straw man is willing to save extra is stated and on the y axis the effect on the 2.5 percent percentile replacement rate compared with the base variant can be found, this is the percentage with which the start replacement rate will increase. In Figure 14 the effect on the average replacement rate from the base variant is stated.

Figure 13 Effect of saving extra on 2.5 percent percentile replacement rate per lifecycle

0% 10% 20% 30% 40% 50% 60% 70% 80% 0% 1% 2% 3% 4% 5% 6% 7% Eff ec t on 2 .5 pe rc en t pe rc en til e re pl ac em en t ra te Saving extra

Effect on replacement rate for saving extra

A

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Figure 14 Effect of saving extra on average replacement rate per lifecycle

As can been seen from these figures it looks like there is a linear relation between the amount of extra savings and the effect on the replacement rate. By applying a trendline to both graphs it is found that a linear trendline gives the best fit for every lifecycle. With this knowledge two regressions are done, one on the whole data set of Figure 13 and one on the whole data set of Figure 14. Because the linear trendline gives the best fit a linear regression with one

explanatory variable is set up, namely the percentage of saving. From this regression formulas 1 and 2 follow4_{, with Y}₁_{the effect on the 2.5 percent percentile replacement rate compared with}

the base variant, Y2 the effect on the average replacement rate compared with the base variant

and x1 the percentage a person decides to save. This formula can be applied for every lifecycle,

see appendix I for the regression output.

[1]

[2]

4_{Regression coefficients are rounded off to four decimals.}

0% 10% 20% 30% 40% 50% 60% 70% 80% 0% 1% 2% 3% 4% 5% 6% 7% Eff ec t on av era ge re pl ac em en t ra te Saving extra

Effect on replacement rate for saving extra

A

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4.1.1 Analysis of other options for saving

The effect of start saving at a later age

In paragraph 4.1 two regressions were executed to estimate the replacement rate given a certain percentage a person decides to save extra. So here a linear relation between the effect on the replacement rate and the percentage of saving is found. The underlying assumption in

paragraph 4.1 is that the straw man starts saving extra at age 25 and keeps on saving during 42 years. However, in paragraph 3.2.1 the results are based on saving extra when the straw man starts saving at a later age. To see if there is any relation between the percentage a person decides to save extra, the saving period and the effect on the replacement rate these three variables are plotted in a graph. In Figure 15 and 16 these graphs can be found for Lifecycle A, the other lifecycles have a similar shape. In Figure 15 on the x axis the percentage that the straw man is willing to save extra is stated, on the y axis the effect on the 2.5 percent percentile

replacement rate compared with the base variant can be found and on the z axis the saving period is set. In figure 16 the effect on the average replacement rate compared with the base variant can be seen.

Figure 15 Effect of saving extra and saving period on 2.5 percent percentile replacement rate for lifecycle A

0 22 42 0% 10% 20% 30% 40% 50% 60% 0% _1% 2% _3% 4% _5% 6% _7% Saving period Eff ec t on 2 .5 pe rc en t pe rc en til e re pl ac em en t ra te Saving extra

Effect on replacement rate save extra for different saving periods

lifecycle A

50%-60% 40%-50% 30%-40% 20%-30% 10%-20% 0%-10%

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Figure 16 Effect of saving extra and saving period on average replacement rate for lifecycle A

It can be concluded from these figures that it looks like there is a relation between the variables. With this knowledge of the preview chapters a new regression on the whole data set is set up. If the effects on the replacement ratios are plotted with the saving period on the x-axis it turns out that polynomial trendlines give the best fit for every percentage of saving, see Appendix J for the graphs. So a regression with two explanatory variables, namely the product of the saving

percentage and saving period; and the product of the saving percentage and the saving period squared, is set up. From this regression formulas 3 and 4 follow, with Y1 the effect on the 2.5

percent percentile replacement rate compared with the base variant, Y2 the effect on the average

replacement rate compared with the base variant, xt the percentage a person decides to save

during a certain period and T the saving period. This formula can be applied for every lifecycle, see appendix K for the regression output.

[3]

[4]

An effect of for example 10 percent on the replacement rate compared with the base variant can be achieved in certain different ways. Somebody can begin with saving directly at the start, or a person can decide to start later with saving but then at a higher rate. In graph 17 below this is shown for various saving periods.

0 22 42 0% 10% 20% 30% 40% 50% 60% 70% 0% _1% 2% _3% 4% _5% 6% _7% Saving period Eff ec t on av era ge re pl ac em en t ra te Saving extra

Effect on replacement rate save extra for different saving periods

lifecycle A

60%-70% 50%-60% 40%-50% 30%-40% 20%-30% 10%-20% 0%-10%

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Figure 17 Effect of saving extra and saving period on replacement rate

In Figure 17 it can be seen that an effect of 10 percent on the replacement rate compared with the base variant can be achieved by saving about 1 percent right from the start, this is 42 years before retirement date. However if a person decides to start saving 12 years before his

retirement date he has to save 4 percent in order to still get the effect of 10 percent on the 2.5 percent percentile replacement rate compared with the base variant. And even 6 percent to get the effect of 10 percent on the average replacement rate compared with the base variant. So to get a certain effect on the replacement rate compared with the base variant with a chosen saving period formulas 3 and 4 have to be solved for xt.

In paragraph 4.1 we have derived two formulas to calculate Y1 and Y2, the effects on the

replacement rates compared with the base variant, given the percentage the straw man wants to save directly from the start, so with a saving period of 42 year. So in formula 1 the assumption is made that the saving period is always 42 year and in formula 3 the saving period can be chosen. In Figure 17 it can be seen that if the scheme participant wants to achieve a 10 percent effect this can be done in various ways. Figure 17 only shows the effect of 10 percent. By setting formula 3 equal to formula 1 the various values for x1, xt, and T for all effects can be found. This can also be

done with formulas 4 and 2. Solving these equations leads to two new formulas, namely formula 5 and 6. In these two formulas the saving period T and the saving percentage during that period xt must be filled in so that the percentage x1 can be found. Thus it can be seen what the effect is

of saving during a shorter period. The formulas can also be rewritten so that xt will be solved for

a chosen saving period and percentage x1. With formulas 5 and 6 the effect of choosing to start

saving a later point can be seen and can be compared with saving right from the beginning. And percentages xt that will be saved during period T can be defined in terms of x1.

[5] [6] 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 0 5 10 15 20 25 30 35 40 45 Sav in g ext ra Saving period

Effect of 10 percent on replacement rate base variant

Y1 = 10 percent Y2 = 10 percent

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The effect of dynamic saving

To investigate the relation between the percentage a person decides to dynamically save and the replacement rate, the effect on the replacement rate will be analyzed more extensively. In Figure 18 on the x axis the percentage that the straw man is willing to save extra is stated and on the y axis the effect on the 2.5 percent percentile replacement rate compared with the base variant can be found, this is the percentage with which the start replacement rate will increase. In Figure 19 the effect on the average replacement rate compared with the base variant is stated.

Figure 18 Effect of dynamic saving on 2.5 percent percentile replacement rate per lifecycle

Figure 19 Effect of dynamic saving on average replacement rate per lifecycle

0% 5% 10% 15% 20% 25% 30% 35% 40% 0% 1% 2% 3% 4% 5% 6% 7% Ef fe ct on 2 .5 pe rc en t pe rc en til e re pl ac em en t r at e Saving extra

Effect on replacement rate dynamic saving

A

B C D E F G H I J K L M N O P Q R S 0% 5% 10% 15% 20% 25% 30% 35% 40% 0% 1% 2% 3% 4% 5% 6% 7% Eff ec t on av era ge re pl ac em en t ra te Saving extra

Effect on replacement rate dynamic saving

A

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25

As can been seen from these figures it looks like there is some sort of relation between the amount of dynamic saving and the effect on the replacement rate compared with the base variant. By applying a trendline to both graphs it is found that a polynomial trendline with order three gives the best fit for every lifecycle. With this knowledge two regressions are executed, one on the whole data set of Figure 18 and one on the whole data set of Figure 19. Because the polynomial trendline with order three trendline gives the best fit a regression with three

explanatory variables is set up, namely the percentage that is saved dynamically, the percentage of dynamic saving squared and the third power of the percentage of dynamic saving. From this regression formulas 7 and 8 follow, with Y1 the effect on the 2.5 percent percentile replacement

rate compared with the base variant, Y2 the effect on the average replacement rate compared

with the base variant and xd the percentage a person decides to save dynamically. This formula

can be applied for every lifecycle, see appendix L for the regression output.

[7]

[8]

In paragraph 4.1 two formulas were found to calculate Y1 and Y2, the effects on the replacement

rates compared with the base variant, given the percentage the straw man wants to save directly from the start, so with a saving period of 42 year. So in formula 7 only if the replacement rate is below a certain reference level a percentage will be saved and in formula 1 the percentage will always be saved. By setting formula 7 equal to formula 1, for a given percentage xd the

corresponding x1 can be found, where x1 is the percentage the straw man is willing to save extra

directly from the start. This can also be done with formulas 8 and 2. Solving these equations leads to two new formulas, namely formula 9 and 10. With these two formulas the effect of saving dynamically or not can be seen and percentages that will be saved dynamically xd can be

defined in terms of x1.

[9]

[10]

So for example if a person decides to save 3 percent dynamically this means that, by filling in formula 9, this gives the same effect of saving 2 percent at all the time.

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4.2 Analysis of the effect of working longer

To investigate the relation between the percentage a person decides to save extra and the replacement rate, the effect on the replacement rate compared with the base variant will be analyzed more extensively. In Figure 20 on the x axis the number of years that the straw man is willing to work longer is stated and on the y axis the effect on the 2.5 percent percentile

replacement rate compared with the base variant can be found. In Figure 21 the effect on the average replacement rate compared with the base variant is stated. In these figures the

combined effect of (1) paying contribution during a longer period and (2) buying the pension at a lower annuity rate, is shown.

Figure 20 Effect of working longer on 2.5 percent percentile replacement rate per lifecycle

0% 10% 20% 30% 40% 50% 60% 70% 80% 0 1 2 3 4 5 6 Eff ec t on 2 .5 pe rc en t pe rc en til e re pl ac em en t ra te

Number of extra work years

Effet on replacement rate per lifecycle for working longer

_A

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Figure 21 Effect of working longer on average replacement rate per lifecycle

It can be concluded from these figures that it looks like there is a relation between the number of extra work years and the effect on the replacement rate compared with the base variant. By applying a trendline to both of the graphs it is found that a polynomial trendline with order two gives the best fit for every lifecycle. With this knowledge two regressions are done, one on the whole data set of Figure 20 and one on the whole data set of Figure 21. Because the polynomial trendline with order two gives the best fit, a regression with two explanatory variables is set up, namely the number of extra work years and the number of extra work years squared. From this regressions formulas 11 and 12 follow, with Y1 the effect on the 2.5 percent percentile

replacement rate compared with the base variant, Y2 the effect on the average replacement rate

compared with the base variant and x2 the number of years a person decides to work longer.

This formula can be applied for every lifecycle, see appendix M for the regression output.

[11] [12] 0% 10% 20% 30% 40% 50% 60% 70% 80% 0 1 2 3 4 5 6 Eff ec t on av era ge re pl ac em en t ra te

Effect on replacement rate per lifecycle for working longer

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4.2.1 Analysis of the effect of working longer with no extra contribution

In the previous chapter the results of the replacement rates for working longer without the contribution payments during the extra work years can be found. The replacement rates were of course lower than when the contribution will be paid. To detect what the effect is of this

contribution during the extra work years the replacement rates of working longer with and without contribution payments have been compared with each other. In Figure 22 the effect of buying at a lower annuity rate can be found and in Figure 23 the effect of the contribution payments during the extra work years can be found. Both have about the same influence on the replacement rate, so both around 50 percent of the total impact. It depends on the lifecycle which effect is larger.

Figure 22 Effect of buying at a lower annuity rate on replacement rates per lifecycle

Figure 23 Effect of contribution payments during the extra work years on replacement rates per lifecycle

0% 5% 10% 15% 20% 25% 30% 35% 40% 1 2 3 4 5 6 Eff ec t on 2 .5 pe rc en t pe rc en til e re pl ac em en t ra te

Effect of buying at a lower annuity rate

A B C D E F G H I J K L 0% 5% 10% 15% 20% 25% 30% 35% 40% 1 2 3 4 5 6 Eff ec t on 2 .5 pe rc en t pe rc en til e re pl ac em en t ra te

Number of years paying extra contribution

Effect of contribution payments

A B C D E F G H I J K L M

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4.3 Analysis of the combined effect of saving extra and working longer

To see if there is any relation between the percentage a person decides to save extra, the number of years he is willing to work longer and the combined effect on the replacement rate compared with the base variant these three variables are plotted in a graph. In Figure 24 and 25 these graphs can be found for Lifecycle A, the other lifecycles have a similar shape. In Figure 24 on the x axis the number of extra work years is stated, on the y axis the effect on the 2.5 percent percentile replacement rate compared with the base variant can be found and on the z axis the percentage that the straw man is willing to save extra is set. In figure 25 the effect on the average replacement rate compared with the base variant can be seen.

Figure 24 Effect of working longer and saving extra on 2.5 percent percentile replacement rate for lifecycle A

0% 1% 2% 3% 4% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 0 1 2 3 4 Saving extra Eff ec t on 2 .5 pe rc en t pe rc en til e re pl ac em en t ra te

Effect replacement rate save extra and work longer lifecycle A

80%-90% 70%-80% 60%-70% 50%-60% 40%-50% 30%-40% 20%-30% 10%-20% 0%-10%

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Figure 25 Effect of working longer and saving extra on average replacement rate for lifecycle A

It can be concluded from these figures that it looks like there is a relation between the variables. With this knowledge of the previous chapters a new regression on the whole data set is set up. A regression with four explanatory variables is set up, namely the percentage of saving, the number of extra work years, the number of extra work years squared, and the product of saving and working extra. From this regression formulas 13 and 14 follow, with Y1 the effect in the 2.5

percent percentile replacement rate compared with the base variant, Y2 the effect on the average

replacement rate compared with the base variant, x1 the percentage a person decides to save

and x2 the number of years a person decides to work longer. This formula can be applied for

every lifecycle, see appendix N for the regression output.

[13]

[14]

To get an impression of the impact of each variable in the equations above, the separate effects for saving three percent and working two years longer can be found in Table 5. The effect of the individual effects are a lot larger than the effects of the cross terms.

0.47 0.25 0.17 0.02 0.04

Table 5 Separate effects (in percentages)

Now for every lifecycle the 2.5 percent percentile replacement rate and the average replacement rate can be estimated for a certain amount a person is willing to save and work longer. It

depends on the person’s preference for the 2.5 percent percentile replacement rate and the average replacement rate which lifecycle he will choose.

0% 1% 2% 3% 4% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 0 1 2 3 4 Saving extra Eff ec t on av era ge re pl ac em en t ra te

Effect replacement rate save extra and work longer lifecycle A

80%-90% 70%-80% 60%-70% 50%-60% 40%-50% 30%-40% 20%-30% 10%-20% 0%-10%

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4.4 Linear approximation

With the formulas derived in the previous paragraphs a very good approximation of the value of the replacement rates can be found given the amount of years a person wants to work longer and the percentage he wants to save extra. But because of the cross terms x22 and x1x2 these

formulas cannot be solved analytically for given values of Y1 and Y2, they can only be solved by

trial and error. So in order to circumvent this non-linear feature, a linear approximation is used to solve for the linear equations. If a person has a certain 2.5 percent percentile replacement rate and a certain average replacement rate he want to achieve when he will retire these

replacement rates can be found by setting Y to the value of Min RR/Start RR-1. Here with Min RR the replacement rate the person wants to achieve is indicated and with start RR the replacement rate from the base variant is indicated. This is because the new replacement rate is the

replacement rate from the base variant for that lifecycle multiplied by (1+Y). A new, simplified, regression is now done with only saving extra and working longer as explanatory variables, so that these formulas can be solved for a given Y1 and Y2. See appendix O for the regression output.

[15]

[16]

At first formulas 15 and 16 will be put in matrices form for all the lifecycles.

( ) ( ) ( ) [17]

Now Y1 is set to the value of min RR1 /start RR1 -1 and Y2 is set to the value of min RR2 /start RR2

-1. Then equation 17 is being solved. In order to do this the inverse of the matrix with the regression values is taken. On the next page the new matrices can be found.

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32 ( ) ₍ ) ( ) [18]

Now by filling in the values for the 2.5 percent percentile replacement rate and the average replacement rate that a person wants to achieve, for every lifecycle the percentage of salary the person needs to save extra and the amount of years the person needs to work longer will be given. In Table 6 an example of this is given, in this example the value of the 2.5 percent

percentile replacement rate is set to 0.7 and the value of the average replacement rate is set to 1.

Lifecycle Percentage of saving extra; x1 Number of extra work years; x2 A 0.076 1.388 B 0.056 3.112 C 0.037 4.845 D 0.018 6.523 E -0.002 8.267 F 0.041 4.537 G 0.017 6.676 H -0.006 8.754 I -0.031 11.019 J -0.054 13.120 K 0.000 8.376 L -0.030 11.038 M -0.059 13.745 N -0.088 16.421 O -0.118 19.194 P -0.053 13.352 Q -0.090 16.777 R -0.124 19.872 S -0.160 23.329

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But there are also other possible solutions, namely replacement rates that are above the minimal replacement rate that someone wants to achieve. By filling in equation 18 only one angular point of the permitted solution will be found, namely the point where the replacement rates are exactly equal to the minimal replacement rates that someone wants to achieve. For example the 2.5 percent percentile replacement rate must be at least 0.7 and the average replacement rate must be at least 1. By filling in these numbers in equation 18, for every lifecycle a combination of working longer and saving extra is found where the 2.5 percent percentile replacement is exactly 0.7 and the average replacement rate is exactly 1. For lifecycle A the percentage a person needs to save extra to achieve these replacement rates is 7.6 percent and the number of years he needs to work longer is 1.4 years. So this is the first angular point of the permitted solution. To find the other solutions recall that Y1 is set to the value of min RR1 /start RR1 -1 and Y2 is set

to the value of min RR2 /start RR2 -1. Now equations 19 and 20 are found by filling this into

equation 15 and 16 and rewriting them. So now filling in x2 gives the value of x1 for a certain

replacement rate that someone wants to achieve. These formulas can also be rewritten so that x1 gives the value of x2 for a certain replacement rates that someone wants to achieve. If one of

the angular points of the permitted solution is negative then that point will be excluded from the permissible solution.

[19]

[20]

In figure 26 equations 19 and 20 are plotted for a 2.5 percent percentile replacement rate of 0.7 and an average replacement rate of 1 for lifecycle A. The point where the lines cross is the point that was found earlier by filling in equation 18. In this figure it can be seen that working 0 years longer and saving 9.4 percent extra is also one of the angular point of the permitted solution and saving 0 percent extra and working 9.7 years longer is the other angular point of the permitted solution. The green shaded area is the area of all the permitted solutions.

Figure 26 Permissible solutions for lifecycle A

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 0 1 2 3 4 5 6 7 8 9 10 11 Sav in g ext ra

Permissible solutions for lifecycle A

Min RR1 = 0.7 Min RR2 = 1

B A

Risk tolerance and a suitable lifecycle