A Coherent Look at Pension, Mortgage and Long-Term Health Care: An Economic Scenario Generator Approach

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A Coherent Look at Pension, Mortgage and Long-Term Health

Care: An Economic Scenario Generator Approach

January 7, 2018

Author: Maurice Rodrigues s2522403

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Master Thesis Actuarial Studies

Supervisor University of Groningen: prof. dr. R. J. M. Alessie Supervisor EY Actuaries Amsterdam: drs. R. van Daalen AAG

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A Coherent Look at Pension, Mortgage and Long-Term Health Care: An Economic Scenario Generator Approach

Author: Maurice Rodrigues

Abstract

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1 Introduction

The retirement provision is more extensive than just pension. Pension is intended for a life-time settlement after retirement age. This settlement should at least be able to cover all costs. Housing and care costs are two important drivers to the expenses of elder people. Solid, acces-sible health care and a↵ordable mortgage expenses are main preferences of those people. Due to an increase of the life expectancy, health care costs will increase and pension products will be more expensive (Spoor (2008)). Consequently, personal savings and investments are playing an important role for an adequate financial situation in the future. In order to ensure the af-fordability of aging, there should be a stable collective basis reachable for everyone along with an individual saving strategy based on personal preferences and circumstances. Risk sharing between participants is one of the advantages of a collective system and an individual saving system helps with achieving a sufficient contribution to the collective part of pension. In this paper, we are trying to make this individual saving system dynamic in the distribution between pension savings, long-term care savings and mortgage expenses. The main focus is not only to maximize the pension savings, however we also want to maximize the proportion who can a↵ord their care costs and we want to minimize the relative di↵erence between the pension savings and the total capital.

The Dutch pension system is distributed over three pillars. In 2013, the pillars contain 54%, 40% and 6% of the total pension wealth, respectively (see Figure 15 of Schmitz et al. (2015)). Contributions in the second pillar are made by about 90% of the employees (excluding self-employed workers), which is internationally seen large. As time progresses, the share in the first pillar will slowly decrease as the second and third pillar accrual will increase. A lot of discussions took place the last years with respect to the pension system. The regulations are changing over the years, leading to an increase in the retirement age and lower annual pension accrual due to limited fiscal space. Consequently, financial risks of changes in life expectancy will be more and more individually responsible.

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The Dutch public insurance dealing with major health risks is a compulsory public insurance called the ‘Wet Landurige Zorg’ based on solidarity: Everyone with a taxable income is paying a premium through the tax for this insurance. Due to the rising life expectancy and increasing health care costs, this research assumes that the current health care system with respect to the elderly will be slightly di↵erent. We are using the suggestion of Soede et al. (2014) yielding a personal contribution of about 8% for elderly care on top of the mandatory collective premium and a full personal contribution for the preferred home care (which can be seen as a luxury good).

Concerning the housing market, low costs contribute to a pleasant pension. There are two main types of acquiring an accommodation: renting or taking a mortgage. According to Eu-rostat, in 2016, 69% of the Dutch population is living in a house with (redeemed) mortgage. Owning a house serves as capital and can be used as pension if there develops a market of making house capital liquid. One possible way to do this is to move in a lower priced house, which converts a part of your capital to liquid cash. A more interesting method is the reverse mortgage. This product was originally made for covering monthly living and care costs by using the owners accumulated wealth. The loan is called a reverse mortgage, since the provider makes monthly payments to the borrower.

This research uses a scenario approach for di↵erent investment strategies for the allocation to pension savings, care savings and mortgage. The scenarios are created by using an Economic Scenario Generator consisting of the following variables: salary, inflation, bonds and stocks. Each of the 10,000 scenarios consists of drawings from the variables for 43 years (a person of age 25 till retirement age 68), such that we consider multiple possibilities of future development of the economic world. The di↵erent investment strategies will be used to get an insight whether a dynamic view of pension, health care and mortgage can result in a more efficient allocation of the disposable income. The pension savings are used to a↵ord a yearly annuity at retirement. The care savings are used for the out-of-pocket payments for elderly care and the preferred home care. The mortgage component will be linked to pension and health care in a way Bovenberg (2012) suggested. His suggestion is that it should be possible to use the pension savings to pre-pay a part of the mortgage. This research will give insights whether it would be advantageous to use pension savings and/or health care savings for prepaying the mortgage.

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2 Dutch Systems

2.1 Pension System

The Dutch pension system is very unique in comparison with other pension systems. It consists of three separate pillars: the government provision (AOW), supplementary collective pensions and the individual arranged insurance. The combination of these three pillars are forming the amount of pension after retirement. The characteristics of the Dutch pension system are the collectivity, risk-sharing and the efficient implementation. The AOW part for the population older than the retirement age will be paid by the premiums of the population younger than the retirement age, while collective pension is build up during employment.

First pillar

The first pillar is the pension through the government, called the AOW. It was introduced in 1957 as a guaranteed minimum pension income, forming the basis of the pension system. The amount of the AOW is linked to the minimum wage of the Netherlands. The whole population who lived or worked in the Netherlands before reaching the retirement age, are entitled to AOW when reaching this age. The first pillar is financed through a system of solidarity. This means that the current labor force is paying the AOW of the pensioners. Everyone who pays income tax, and is younger than the retirement age, pays the AOW contribution. This contribution is used directly to pay out the AOW benefits. Due to the rising life expectancy, the retirement age will gradually raise to 66 in 2018 and 67 in 2021.

Second pillar

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Defined Benefit pension, the employer is dealing with the investment risk and investment profits. After the financial crisis, more and more companies are shifting to Defined Contribution plans with the result of moving the investment risk to the employees. Consequently, in this paper, we will assume that the underlying pension plan is a Defined Contribution plan.

Third pillar

The third pillar is formed by individual, voluntary based pension plans. In particular, self-employers or employees without pension regulation are using this third pillar. Everyone is free to build extra pension in the third pillar at their own discretion.

2.2 Health Care System

International principles are the main keys for the Dutch health care system: Accessible health care for everyone, compulsory and accessible health insurance leading to solidarity, and high quality care. In addition, the current situation is a result of historical developments and social circumstances.

The care system exists of four subsystems: the ‘Zorgverzekeringswet’ (Zvw), the ‘Wet langdurige zorg’ (Wlz), the ‘Wet maatschappelijke ondersteuning’ (Wmo) and the ‘Jeugdwet’. These four laws together are the main drivers for the care system. Most of the available budget is allo-cated to Zvw and Wlz. The Wlz is implemented nationally, while the Zvw is implemented by private health insurers involving regulated market forces under public conditions. Wmo and the ‘Jeugdwet’ are dealing with other forms of care and support. Approximately 400 Dutch municipalities are primarily responsible for the implementation of these latter two laws. In this study, we are mainly focusing on long-term health care or elderly care. These types of care are regulated by the Wlz. Consequently, this law will be explained in more detail.

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get care at home or in a health care institution. The Wlz is controlled on national level and regulated by Wlz-executives on behalf of the government. Those Wlz-executives have assigned the actual execution to region-designated offices that are closely linked to one health insurer. These offices arrange the delivery of the care needed. The Wlz is one of the compulsory public insurances, which is based on solidarity, meaning that everyone who pays income tax is paying a premium for the Wlz.

Care types from the Wlz

The heavy and intensive care available from the Wlz are broadly defined. For this reason, there is a lot of freedom in arranging an agreement between the recipient and the provider. The most important types are:

• Residence at an institution • Personal care

• Nursing

• Transport to and from daily activities and daily care Finance of the Wlz care

The Wlz is a compulsory public insurance, for which an income-dependent premium is paid through the income tax. The amount of this premium is based on a fixed percentage of the income tax, over a maximum amount of e33,589. This percentage is currently 9.65%. Besides this premium, the population using the Wlz have to pay a personal contribution. This personal contribution is income-dependent and several other characteristics are taken into account. It plays a role whether the care receiver lives at home or in an institution, whether he is under or above the retirement age and whether he is living alone or together.

All contributions will be paid into the fund long-term care, managed by the ‘Zorginstituut Nederland’. The financing will occur in two ways, depending on the personal choice:

• One way is that a part of the fund will be transfered to the ‘Centraal Administratie Kantoor’ (CAK), which subsequently pays the care providers.

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personal-based budgets. The bills from the care providers are sent to and handled by the SVB.

2.3 Mortgage System

The mortgage system in the Netherlands is based on some (new) underlying rules. Loan providers apply strict rules for the provision of mortgages. These rules are determined by the government and included in the law. Lately, new rules are adopted in the law to prevent payment and debt problems for the homeowners. The determination of the maximum mortgage provided by banks, insurers or other providers depends on three main factors: income, market property value and financial obligations. On average, the mortgage which can be received is around four times the yearly gross income. There is no strict rule for this multiplication factor since a lot of personal aspects play a role on determining the amount of the loan. This research will assume that the maximum loan is four times the yearly gross income.

There are several types of mortgages, or payment options. The two most popular are the following:

• Linear mortgage (see Appendix Figure 15) • Annuity mortgage (see Appendix Figure 16)

Figures 15 and 16 in the Appendix show the mortgage payments and the interest payments over the duration of the mortgage. The gray surface denotes the interest payments and the yellow surface denotes the mortgage payments. Linear mortgage pays the same amount of redemption every period such that the interest will be decreasing over time. An annuity mortgage has the same total payment the whole duration such that the redemption will increase and the interest will decrease over time. The total costs of a linear mortgage will be lower than the total costs of an annuity mortgage due to the interest payments.

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3 Literature Review

The main focus of this research is to combine savings for pension and long-term health care with the expenses to mortgage and investigate whether this can result in better living standards when reaching the retirement age. The meaning of better living standards is that we want to be able to pay the care we need after retiring, combined with a high enough income at retirement after dealing with the mortgage expenses. In this study, we are using a scenario-based method by using an Economic Scenario Generator (ESG) which will include the mapping of asset returns, bond returns, inflation and salary. There is limited econometric research about this topic in which an Economic Scenario Generator is used. However, there are other studies which compare di↵erent methods to combine health care, mortgage and pension.

3.1 Combination of Pension, Mortgage and Care

The papers Bovenberg and Koelewijn (2011) and Bovenberg (2012) argue that pensions build in the second pillar should be more dynamic, due to the fact that macro risks will be shifting more and more to the individual. More flexibility is needed in the working phase as well as in the retirement phase. A more dynamic pension also fits with the increased heterogeneity in the labor market and can contribute to the development, maintenance and utilization of human capital during the construction phase. Furthermore, a dynamic view of pension with health care and mortgage will contribute to the accessibility and a↵ordability of high quality care and living facilities at the retirement phase.

M. Amand did research on the possible macro-economic ways to combine living, care and pen-sions. According to the paper Amand (2012), a market for making housing equity liquid could be part of the solution. However, the financing of this method will introduce new risks to the gov-ernment balance sheet, where it is important who will be responsible for these risks. Due to the uncertainty in housing prices, this results in an uncertain value of the annuity by using house eq-uity. Consequently, it is likely that pensioners will receive a lower amount than the market value.

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4 Data Description

This study is using monthly data for inflation, stock market and the short rates. The data is from January 2005 up to and including December 2014, leading to 120 data points of each class. We will give an overview of the di↵erent classes in the subsections below.

4.1 Inflation

One of the most important risk factors is the inflation rate. A lot of pension products are yearly “indexed” based on the inflation rate to maintain the purchasing power. Not only pension prod-ucts are dealing with this phenomenon, also other insurance prodprod-ucts and income are growing with the inflation rate. Dutch inflation rates are calculated by the Dutch Central Bureau for Statistics (CBS) and is defined as the percentage change in the Consumer Price Index (CPI) of a certain period compared with the corresponding period in the previous year. The Consumer Price Index is a compiled and weighted index, consisting of a basket of services and consumer goods purchased by Dutch households based on EU regulations. The data for inflation is ob-tained from the CBS. Monthly inflation is defined as the monthly percentage change in the Consumer Price Index transformed by using the following formula:

It=

CP It CP It 1

CP It 1

,

where It is the inflation at time t and CP It is the Consumer Price Index at time t.

Figure 1: Monthly inflation rate in %

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Figure 1 shows the monthly inflation rate. It can be seen that the monthly inflation rate follows a mean reverting process with a high volatility. A mean reverting process means that the inflation rate will move back to the mean after positive or negative shocks.

Table 1: Monthly inflation summary statistics

Mean Standard Deviation Minimum Maximum

Monthly Inflation (%) 0.146 0.471 -1.100 1.200

The summary statistics of the monthly inflation are shown in Table 1. The mean is 0.146%, which will correspond to an annualized inflation mean of 1.766%. The standard deviation of 0.471% is high compared to the mean, with a minimum of -1.1% and maximum of 1.2%. Consequently, the monthly inflation rate moves around the mean with a respectively high volatility.

4.2 Bonds

This research uses the 1-month Euribor as a proxy for the nominal short rate. We will use this nominal short rate as the underlying process for calculating the 1 year bond returns. The 1-month Euribor process in our sample is given in Figure 2 below.

Figure 2: Annualized monthly Euribor rate in %

Data source: Eurostat

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consequently improve the economy. Since we are doing a scenario-based approach, where we are dealing with bond investments, it is reasonable to assume that these rates are strictly positive. The summary statistics of the 1-month Euribor are given in Table 2.

Table 2: 1-Month Euribor rate summary statistics

1-Month Euribor rate (%) 0.137 0.127 0.001 0.394

The mean is 0.137%, indicating an annualized mean of 1.656%. The mean is lower than the mean of the inflation, as a consequence of the decrease in the short rate after the financial crisis. The standard deviation is 0.127%, with a minimum and maximum monthly short rate of 0.001% and 0.394%, respectively. After modeling the short rate, it could be reasonable to increase the predicted long-term mean, since this will probably be unreliable due to the huge drop after the crisis.

4.3 Stocks

In this research, the AEX stock index is used as a proxy for the risky asset returns. The monthly stock returns are shown in Figure 3 below.

Figure 3: Monthly AEX returns in %

Data source: Yahoo Finance

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Table 3: Monthly AEX return summary statistics

Monthly AEX return (%) 0.306 5.217 -19.710 11.170

The mean is 0.306% with an annualized mean return of 3.734%, clearly the highest of the three asset classes. However, it also has by far the highest monthly standard deviation of 5.217%, indicating that the stock market can be used as the risky asset. In the financial crisis, the minimum monthly loss of 19.710% was realized. The maximum monthly return is 11.170%. Investing in the stock market can result in high returns, however it can be very risky.

4.4 Health Care Costs after Retirement

In this study, enough reserves for health care costs after retirement is one of the key requirements. How much are these costs and does it di↵er between the preferences of individuals? We are trying to divide the costs in five stochastic segments ECi for i = 1, 2, 3, 4, 5 and three personal-based

segments HCj for j = 1, 2, 3. To define these segments, we are using a Netspar design paper

Soede et al. (2014) to have an indication of how elderly care is distributed. The total costs for elderly care per person is on average e69,000 based on prices of 2008. In Figure 4 below, the distribution of elderly care is illustrated (replicated Figure 5.2 from Soede et al. (2014)).

Figure 4: Quantile distribution of care costs after retirement

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On top of this, we will assume that an additional payment for home care is paid individually from the personal health savings, based on the preferred home care. Figure 5 below shows the di↵erent classes for home care (replicated Figure 5.1 from Soede et al. (2014)).

Figure 5: Quantile distribution of home care costs after retirement

By looking at the graph, 40% only need e1,000 or less for home care, while the following three 20% groups are paying e10,000, e31,000 and e59,000, respectively. We will divide the home care costs in three classes: ‘low’, ‘medium’ and ‘high’. These costs are based on personal pref-erences and could therefore be seen as a luxury good. Consequently, the total care costs exists of two parts: uncertainty costs and personal-based costs. This leads to the following stochastic and personal segments (with index year 2008):

Elderly Care EC (8% contribution)

• EC1 = 0

• EC2 = 1, 000

• EC3 = 14, 000

• EC4 = 58, 000

• EC5 = 273, 000

Home Care HC (100% contribution)

• HC1 = 0 (Low)

• HC2 = 20, 000 (Medium)

• HC3 = 45, 000 (High)

The total costs for these segments will grow with a growth rate rh which is probably higher

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5 Methodology

5.1 Framework

In this study, the focus lies on a dynamic way of distributing disposable income (Dt) to three

targets:

• Pension savings (Pt)

• Mortgage payments (Mt)

• Long-term health care savings (Ht)

By using several dynamic strategies and comparing them, we are trying to increase net income at retirement age. The main focus of this paper is not only to maximize the pension savings (or yearly pension annuity bought from total pension savings), however we also want to maximize the proportion who can a↵ord their care costs and we want to minimize the relative di↵erence between the pension savings and the total capital. An assumption is that the person will buy a house directly if the pension savings are minimal 10% of the house price and 4 times the pre tax salary is at least the house price. This age will be denoted as a and the mortgage will be paid in 30 years. Starting with a person of age 25 who has a modal income (named average Joe), assuming that he has no capital before entering the ‘system’. We will denote the gross income at time t by Et, the net income at time t by Yt and the interest for his mortgage at time t by

Rt. This means that the following equation has to hold:

pt(1 rtax)Et,i = ptYt= Dt,i = Pt,i+ Mt,i+ Rt,i+ Ht,i,

where pt is the ratio from the net income at time t which can be distributed to the three

components and is the same for all the scenarios i. The consumption rate is denoted by 1 pt. This consumption rate does include all expenses except savings and housing expenses (see

Table 9 and 10 in the Appendix). Due to lack of data from the Netherlands, we will use the consumption rates from the United States as a proxy. The tax rates rtax are adopted from the

new coalition agreement from the Netherlands (see Table 8 in the Appendix). The value of the three components at retirement age of 68 can be determined by creating all di↵erent scenarios which can be build by using a Economic Scenario Generator consisting of the following economic variables:

• Stock prices (St,i)

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• Inflation (It,i)

• Salary growth (Et,i)

We use various models for these economic variables and model the correlation between these models to prevent unrealistic scenarios. These scenarios are used to investigate whether dynamic strategies can result in a higher pension annuity at retirement age, while having enough health care reserves. We are also interested in whether it is wise to pay additional mortgage redemption (this additional redemption is denoted by M Et,i). Bovenberg and Koelewijn (2011) suggested

to be able to use the pension savings for prepayment of the mortgage, which will be denoted by = 1. If we set = 0.5 then the additional redemption will be equally paid from the pension savings and care savings. If we set = 0, the additional redemption will be paid from the care savings. The capital at retirement in pension and health is then written as follows:

F Pi = 67 X k=25 ↵k(Pk,i M Ek,i) 67 Y j=k ✓ 1 +Sj+1,i Sj,i Sj,i ◆ + 67 X k=25 (1 ↵k)(Pk,i M Ek,i) 67 Y j=k ✓ 1 +Bj+1,i Bj,i Bj,i ◆ (1) F Ci = 67 X k=25 k(Hk,i (1 )M Ek,i) 67 Y j=k ✓ 1 +Sj+1,i Sj,i Sj,i ◆ + 67 X k=25 (1 k)(Hk,i (1 )M Ek,i) 67 Y j=k ✓ 1 +Bj+1,i Bj,i Bj,i ◆ (2)

The stock and bond allocation for the two di↵erent funds will di↵er. The pension fund (investing ↵k in stocks) will be more aggressive than the care fund (investing k in stocks), meaning

↵k> k. The reasoning for this di↵erence is that a pension fund needs to provide an additional

amount on top of the risk-less AOW amount. Based on the investment strategies of di↵erent care funds and pension funds, this research uses an aggressive ↵ = 0.7 for the pension savings and an defensive k = 0.3 for the care savings. For example, pension fund ABP allocates 40%

in bonds and 60% in stocks and pension fund ‘Zorg en Welzijn’ allocates 30% in bonds. The costs at retirement, which can consist of outstanding mortgage debt (DMi) and care costs (Ci)

are written as follows (HPa denotes house price bought at age a and is assumed to be constant

over time): DMi= HPa 67 X j=a Mj,i+ a+29_X 68 Rj,i

Ci= d1,iEC1+ d2,iEC2+ d3,iEC3+ d4,iEC4+ d5,iEC5

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where d1,i, d2,i, d3,i, d4,i, d5,i, l1,i, l2,i, l3,i are dummy variables corresponding to one of

the five randomly assigned elderly care segments (EC1, EC2, EC3, EC4, EC5) and the three

personal chosen home care segments (HC1, HC2, HC3), respectively, as defined in Section 4.4

with d1,i+ d2,i+ d3,i+ d4,i+ d5,i = 1 and l1,i+ l2,i+ l3,i= 1. We denote the present value at age

68 for the costs by P V (DMi) and P V (Ci), where we use an discount rate of 3%. Now the net

income at age 68 can be written as:

Fi = F Pi+ (F Ci P V (Ci)) P V (DMi) (3)

= F Pi+ Zi P V (DMi),

where Zi = F Ci P V (Ci). We denote the number of created scenarios by N. The first two

measurements to compare the di↵erent strategies will be E(Fi) = _N1 PN_i=1Fi and E(IZi>0) =

1 N

PN

i=1IZi>0, where IZi>0is 1 if Zi> 0 and 0 otherwise. The first measurement E(Fi) shows the

expected total capital (transformed to a yearly annuity), which could be realized when perfect information about the care costs is known and E(IZi>0) shows the fraction of scenarios in which

the total care costs could be paid from the total care savings for a given investment strategy. The third measurement is the utility function which will be explained in Section 5.7. The choice of these first main measurements gives insights for the aim to have the highest possible net income at retirement (F ), while having a strategy resulting in sufficient health care savings. A sufficient health care savings is reached when Z is positive. A simple visualized summary is given below:

Figure 6: Visualized summary framework

5.1.1 Assumptions

In all our saving paths, we have the following assumptions:

• The total house costs will be e270,000, from which 10% will be directly paid by using the pension savings.

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house costs and four times the gross salary is greater or equal then the needed loan (Lt),

which is 90% of the total house costs.

• The yearly interest rate is set at rm = 3.5%.

• We split the strategy in two-parts: before and after age a . 5.1.2 Definition of Variables

First the definition of the variables are given below:

• Pt,i: Contribution to pension savings in period t for scenario i.

• Ht,i: Contribution to health care savings in period t for scenario i.

• Mt,i: Mortgage payments in period t for scenario i.

• MEt,i: Additional mortgage redemption in period t for scenario i.

• Rt,i: Rent payments for mortgage in period t for scenario i.

• Lt: The initial loan.

• Et,i: Gross salary in period t for scenario i.

• Dt,i: Disposable income in period t for scenario i.

• rs,t: Maximum premium rate for the pension savings at time t. (See the first column of

Table 18 in the Appendix)

• rm: Yearly interest rate for the loan.

• a: Age of buying a house.

• x1,t: The rate of the maximum contribution to pension savings at time t before age a.

• x2,t: The rate of additional mortgage redemption at time t.

• x3,t: The rate of the maximum contribution to pension savings at time t after age a.

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5.1.3 Saving Path Strategies

We first define a general strategy and by changing the parameters we define individual strategies. The general strategy is given by:

Before age a:

• Pt,i= x1,trs,tEt,i

• Ht,i = Dt,i Pt,i

• Mt,i = 0 • Rt,i= 0 After age a: • Mt,i= min ✓ Lt Pt 1_j=1(Mj,i+ M Ej,i),L₃₀t ◆ • MEt,i= x2,tLt

• Rt,i = (Lt Pt_j=0(Mj,i+ M Ej,i))rm

• Pt,i = min (x3,trs,tEt,i, Dt,i Mt,i Rt,i)

• Ht,i = Dt,i Mt,i Rt,i Pt,i

Before presenting the individual strategies, we will explain the general strategy in words.

Before age a:

The investment in the pension savings is given by the fraction x1,t (di↵erent among the

strate-gies) of the maximum pension premium rs,tEt,i (the same for all strategies in scenario i), where

rs,t is the maximum pension premium rate which can be found in Appendix Table 18. The

investment in the health care savings is the disposable income (Dt,i) which can be allocated to

pension savings, health care savings or mortgage subtracted by the pension investment at time t. Furthermore, there are no mortgage expenses before age a.

After age a:

The mortgage expenses at time t will be the minimum of the outstanding debt (Lt Pt 1_j=1(Mj,i+

M Ej,i)) and the payment of the mortgage contract (L₃₀t). If we allow for additional mortgage

redemption (M Et,i) at time t of 10% (x2,t = 0.1) of the initial loan, it will be paid from the

pension savings, the health care savings or equally from the pension and health care savings. After this, the interest (Rt,i) over the outstanding debt is determined. The investment in the

pension savings will be the minimum of the fraction x3,t of the maximum pension premium and

the remaining of the disposable income subtracted by the mortgage expenses. At last, the health care investment is given by the disposable income subtracted by the mortgage expenses and the pension investment.

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The following strategies are used as benchmark:

Strategy 1: This is the strategy with the standard parameters. The following strategies are changing some or all of the parameters of the benchmark strategy.

The standard parameters are: x1,t = 1 and x3,t = 1.

Strategy 2: Before buying a house, the person will only invest 70% of the maximum pen-sion contribution such that the other 30% is invested in the care savings.

Parameter values: x1,t= 0.7 and x3,t = 1.

Strategy 3: Before buying a house, the person will only invest 50% of the maximum pen-sion contribution such that the other 50% is invested in the care savings.

Parameter values: x1,t= 0.5 and x3,t = 1.

Strategy 4: After buying a house, the person will only invest 70% of the maximum pen-sion contribution such that the other 30% is invested in the care savings.

Parameter values: x1,t= 1 and x3,t = 0.7.

Strategy 5: After buying a house, the person will only invest 50% of the maximum pen-sion contribution such that the other 50% is invested in the care savings.

Parameter values: x1,t= 1 and x3,t = 0.5.

Strategy 6: The person will only invest 70% of the maximum pension contribution such that the other 30% is invested in the care savings.

Parameter values: x1,t= 0.7 and x3,t = 0.7.

We will reproduce the output of the benchmark strategies for the same 10,000 scenarios where we allow for additional mortgage redemption of 10% of the initial loan every 5 years. This leads to the parameter x2,t = 0.1 if the additional payment is made at time t and x2,t = 0 otherwise.

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5.2 Salary

One of the main assumptions from this study is that we are using a modal income pattern. In Dutch this is called ‘Jan Modaal’ and is used as a reference point when examining the impact of government measures on income. Modal income is not the same as the average income. The di↵erence is that average income is calculated as the weighted average of all incomes, while modal income is the income which occurs the most. It is calculated as 79% of the average income per working year. We obtained data of the modal income per year and modal income per age.1 We are modeling income by using a year dependent growth rate, an age dependent growth rate and the inflation rate. We are starting with a person with gross income e25,000 at age 25. Then the salary function is as follows:

E(a,t)= E(a 1,t 1)(1 + ga+ gt+ inft), (4)

where ga is the age dependent growth rate, gt is the time dependent growth rate and inft is the

inflation of year t. The age dependent growth rates and modal income for 2010 until 2014 is given below.

Figure 7: Age dependent modal income (left) and growth rates (right)

Data source: Statistics Netherlands CBS

This graph shows that the age dependent modal income pattern shifts up every year, however it follows the same pattern. Looking at the growth rates, they are having the same structure

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over time. The growth rate lines are following almost a linear trend but not completely linear. By looking at the graph, a model with age and a polynomial of age seems to fit the data. The following model is used for ga:

ga= ↵ + 1(a 25) + 2(a 25)2+ ",

where " _{⇠ N (0,} 2). We are fitting this model by using Ordinary Least Squares. Based on the age dependent growth rates of 2010 until 2014 we have ˆ↵ = 8.77· 10 2 _(8.16_{· 10} 3_{), ˆ}

1 =

6.16_{· 10} 3 (8.15_{· 10} 4) and ˆ2 = 9.39· 10 5 (1.78· 10 5), with the corresponding standard

errors between parentheses. We estimate 2 _{by computing the variance of the residuals. The}

estimate is given by ˆ = 8.84_{· 10} 3. Now we can draw age dependent growth rates from the following distribution: ga⇠ N ⇣ ˆ ↵ + ˆ1(a 25) + ˆ2(a 25)2, ˆ2 ⌘

The yearly modal income growth rate minus the corresponding inflation rate is shown in Figure 8 below.

Figure 8: Yearly modal income growth rate minus corresponding inflation

Data source: Statistics Netherlands CBS

There is no clear relation between the growth rates. The model to use for the yearly growth rate will be a random draw from the normal distribution with the mean and variance of the historical data shown above. This is written as:

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where ˆµ = 4.56· 10 3 _(3.95_{· 10} 3_{) and ˆ = 1.48}_{· 10} 2_{, with the corresponding standard error}

between parentheses.

The last term which impacts the salary function is the inflation rate. How this is modeled is explained in next section. By using the models for ga and gt we can create realistic scenarios

of how salary can grow over time.

5.3 Inflation

In this literature, the model used for inflation is the Vasicek (1977) model. The main reason for this choice is the property of mean reversion in the Vasicek model. Earlier research showed strong evidence of having mean reversion in the inflation rates, see Lee and Wu (2001). The Cox-Ingersoll-Ross model we are using for modeling bond prices is an extension of this Vasicek model, where an extra term is added to avoid negative short rates. The Vasicek model assumes that the dependent variable follows an Ornstein-Uhlenbeck process:

dIt= ↵(µ It)dt + dWt

Due to the mean reversion process, ↵, µ and are strictly positive and Wtis a Wiener process.

The solution of the Stochastic Di↵erence Equation is as follows: Let f (It) = Ite↵t, then:

df (It, t) = ↵Ite↵tdt + e↵tdIt

= ↵Ite↵tdt + e↵t(↵(µ It)dt + dWt)

= ↵µe↵tdt + e↵tdWt

Integrating both sides between any instants s and t, with 0 s < t, gives: Ite↵t Ise↵s= ↵µ Z t s e↵xdx + Z t s e↵xdWx Ite↵t Ise↵s= µ(e↵t e↵s) + Z t s e↵xdWx It= µ(1 e ↵(t s)) + Ise ↵(t s)+ e ↵t Z t s e↵xdWx

The discrete time version described in Brigo et al. (2009) for time-step t is given by:

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where "t⇠ N (0, 1) and

c = µ(1 e ↵ t) (6)

b = e ↵ t (7)

The volatility of the innovations can be computed using the Ito isometry:

E "✓ e ↵t Z t s e↵xdWx ◆2# = 2e 2↵tE Z t s e2↵xdx = 2e 2↵t ✓ 1 2↵(e 2↵t _e2↵s₎ ◆ = 2 2↵ ⇣ 1 e 2↵(t s)⌘, since we are having time steps of t, this leads to

= r

1 e 2↵ t

2↵ . (8)

The calibration process is the OLS regression of Iton its lagged value It 1, providing Maximum

Likelihood estimators for c, b and . Rewriting the system of Equations (6), (7), (8) and using the obtained Maximum Likelihood estimators gives estimators for ↵, µ and :

ˆ ↵ = log(ˆb) t (9) ˆ µ = ˆc 1 ˆb (10) ˆ = q ˆ (ˆb2 _{1) t/(2 log(ˆb))} (11)

Parameter Estimates Inflation

The regression results of (5) can be found in Table 11a of the Appendix. The initial estimates are ˆc = 0.0009 (0.0004), ˆb = 0.3453 (0.0869) and ˆ = 0.0013, with the corresponding standard errors between parentheses. By using the transformations (9), (10) and (11), we obtain the estimates for ↵, µ and . The estimates of our the model we will use for the simulation are

ˆ

↵ = 12.7612 (3.0192), ˆµ = 0.0014 (0.0006) and ˆ = 0.0069, with the corresponding standard errors between parentheses (can also be found in Table 11b). These latter standard errors are found by using the Delta Method explained in the Appendix.

5.4 Bonds

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which is one of the main assumptions in this paper. The CIR-model is an equilibrium model in which interest rates are determined by agents having a logarithmic Neumann-Morgenstern utility function. The short term rate, rt, follows a single factor ‘mean-reverting square-root’

process:

drt= ↵(µ rt)dt + prtdWt,

where the parameters ↵ > 0, µ > 0 and > 0 correspond to the speed of the mean reversion, the mean and the volatility, respectively. Wt is a Wiener Process, which is used to model the

random market risk. The discrete-time version of this process can be written as:

rt+ t rt= ↵(µ rt) t + p rt t"t rt+ t rt p_r t = ↵µ_pt rt ↵ tprt+ p t"t, (12)

where t is the time-shift and "_{⇠ N (0, 1). The discrete-time version (12) will be needed to find} initial values for the Maximum Likelihood estimation discussed below. First we want to solve the Stochastic Di↵erence Equation:

Let f (rt, t) = rte↵t, then:

df (rt, t) = ↵rte↵tdt + e↵tdrt

= ↵rte↵tdt + e↵t↵(µ rt)dt + e↵t prtdWt

= e↵t↵µdt + e↵t prtdWt,

Integrating both sides from the interval [s, t] gives:

rte↵t rse↵s = Z t s e↵x↵µdx + Z t s e↵x prxdWx rte↵t rse↵s = µe↵t µe↵s+ Z t s e↵xprxdWx rt= rse ↵(t s)+ µ(1 e ↵(t s)) + e ↵t Z t s e↵xprxdWx.

To find the distribution of rt, we are following Feller (1951) where it has been shown that the

short term rate rt follows a non central chi-squared distribution with 2q + 2 degrees of freedom

and non central parameter 2u:

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where c = ₂ 2↵ (1 e ↵ t₎ q = 2↵µ₂ 1 ui = crie ↵ t vi = cri.

The distribution function is then given by:

f (rt|rt 1) = ce ut vt+1(

ut

vt+1

)q2I_q[2pu_tv_t+1].

To estimate the parameters ↵, µ and we will use Maximum Likelihood. Firstly, we will estimate initial values for ↵, µ and by using OLS on a rewritten version of (12):

yt= 1xt,1+ 2xt,2+ "t, (13) where yt= rt+1 rt p_r t xt,1 = t p_r t xt,2 = tprt.

After obtaining the OLS estimators ˆ1, ˆ2 and ˆ, the initial values are then given by:

ˆ ↵I = ˆ2 (14) ˆ µI = ˆ₁ ˆ₂ (15) ˆI = ˆ p t (16)

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Bond prices: By using the parameter estimates for the short term model, we can price a bond with di↵erent time to maturities by using the following formula:

P (t, T ) = A(t, T )e B(t,T )rt_, ₍₁₈₎ where A(t, T ) = 2he (a+h)(T t) 2 2h + (↵ + h)(e(T t)h ₁₎ !2↵µ 2 B(t, T ) = 2(e (T t)h ₁₎ 2h + (↵ + h)(e(T t)h ₁₎ h =p↵2_{+ 2} 2

Parameter Estimates Bonds

For the short rate model, we first needed initial values to use for our Maximum Likelihood estimation. The regression results of (13) are given in Table 12a of the Appendix. They are given by ˆ1 = 0.00002 (0.00006), ˆ2 = 0.1103 (0.1175) and ˆ = 0.0037, with the corresponding

standard errors between parentheses. We will transform these estimates to obtain initial values for ↵, µ and by using (14), (15) and (16). The initial values are given by ˆ↵I = 0.1103 (0.1175),

ˆ

µI = 0.0002 (0.0007) and ˆI = 0.0127, where the standard errors are found by using the

Delta method explained in the Appendix (can also be found in Table 12b of the Appendix). The final estimates are obtained by using the Maximum Likelihood method, i.e. Equation (17). The final estimates are ˆ↵M L= 0.1376 (0.1236), ˆµM L= 0.0001 (0.0004) and ˆM L = 0.0134 (can

also be found in Table 12c of the Appendix). As one can directly see, the estimated long-term mean is much lower than the mean of our historical data. This estimated long-term mean is a consequence of the low short rates in the last approximately 10 years, causing very unrealistic short rates in our scenario approach. We will set our long-term mean for the short rate model at the historical mean, which is 0.00137, preventing very unrealistic scenarios where the short rate will remain close to zero for the whole scenario building period (43 years).

5.5 Stocks

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random variable follows a Brownian Motion with drift. The GBM process is defined as follows:

dSt= µStdt + StdWt, (19)

where Wt is a Wiener Process, µ is the constant drift term and is the constant volatility.

Rewriting Equation (19) yield:

dSt

St

= µdt + dWt

The left hand side relates to the derivative of log(St) and since St is an Ito process, we can use

Ito’s Lemma on d log(St):

d log(St) = log(St)0dSt+ 1 2d log(St) 00 2_S2 tdt = 1 St dSt 1 2 2_dt = µdt + dWt 1 2 2_dt = (µ 1 2 2_{)dt + dW} t

Integrating both sides between any instants s and t, with 0 s < t, gives: Z t s d log(Sx) = Z t s (µ 1 2 2_{)dx +} Z t s dWx log(St) log(Ss) = (µ 1 2 2_)(t _{s) + (W} t Ws) St= Sse(µ 1 2 2)(t s)+ (Wt Ws)

If we define the time step t by t = t s and using the property of a Wiener process that Wt Ws⇠ N (0, t s), we have: St= Sse(µ 1 2 2) t+ p t"t

where "t⇠ N (0, 1). The calibration of this model is based on the OLS regression of the following

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After obtaining the OLS estimators ˆc and ˆ, we are able to transform these in estimators for µ and by: ˆ = pˆ t (21) ˆ µ = cˆ t+ 1 2ˆ 2 ₍₂₂₎

Parameter Estimates Stocks

For the estimation of the parameters for our stock model, we first estimate (20) and by using these estimates we can use the transformations for estimates of µ and . The estimates are given in Tables 13a and 13b, respectively. The estimates of (20) are ˆc = 0.0014 (0.0050) and ˆ = 0.0540, where ˆ corresponds to the monthly historical standard deviation. The transformations (21) and (22) yield the estimates needed for our stock model. The estimates of µ and are given by

ˆ

µ = 0.0340 (0.0594) and ˆ = 0.1871, which should be similar to the yearly historical return mean and standard deviation of the AEX. The standard errors are again given between parentheses and the transformed parameter standard errors are obtained by using the Delta Method explained in the Appendix. The yearly historical mean and standard deviation are given by 3.734% and 19.710%, respectively. The estimates of the parameters of our stock model are slightly lower than the historical values.

5.6 Scenario Approach

After estimation of the parameters for each individual class model, we are able to create scenarios based on these models. However, we have to use a discretization method and incorporate the correlation between the three classes to prevent unrealistic scenarios. The following subsections are explaining the methods for discretization and the correlation implementation.

5.6.1 Euler Scheme Approach

An often used method for approximating a numerical solution of a Stochastic Di↵erential Equa-tion is the Euler-Maruyama method, also called the Euler Scheme. This method is used to simulate scenarios based on the stochastic models, since it is not possible to simulate continuous time paths. There are several methods for discretization, however the Euler-Maruyama method is feasible and straightforward for our chosen stochastic models.

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Di↵erence Equation:

dXt= µ(Xt)dt + (Xt)dWt,

where Wt is a Brownian motion and Xt can be one of our three classes inflation, short rate

and stocks. If we want to simulate monthly data for T years (m months), then we have to use this discretization method for 0 < t1 < t2 < ... < tm = T . The time-step is then given

by t = _mT = _12TT = ₁₂1 . To obtain the Euler-Maruyama method, the Stochastic Di↵erence Equation above has to be integrated between s and t, where t s = t:

Z t s dXu= Z t s µ(Xs)du + Z t s (Xs)dWu Xt Xs= µ(Xs)(t s) + Z t s (Xs)dWu Xt= Xs+ µ(Xs) t + Z t s (Xs)dWu,

where the Euler-Maruyama method is obtained by approximating the integralR_st (Xs)dWu by:

Z t s (Xs)dWu ⇡ (Xs)(Wt Ws) = (Xs) Wt = (Xs) p t"t,

where Wt = Wt Ws ⇠ N (0, t) and "t ⇠ N (0, 1). Consequently, the complete general

discretized version we are using for the simulation of our scenarios is the following:

Xt+1 = Xt+ µ(Xt) t + (Xt)

p t"t.

More specific, this leads to the three class-related discretized versions for inflation, short rate and stocks, respectively:

It+1= It+ ↵I(µI It) t + I p t"I,t rt+1= rt+ ↵r(µr rt) t + r p rt t"r,t St+1= St+ µSSt t + SSt p t"S,t. 5.6.2 Correlation Approach

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values of a 3-dimensional multivariate normal distribution for the discretized version of our models, i.e. draws from _N⇣0, ˆ⌃⌘. We can draw from this multivariate distribution by using the Cholesky-decomposition. The Cholesky-decomposition states that a positive-definite matrix can be written as a lower triangular matrix times the transpose of itself. Applying this to our estimated correlation matrix gives:

ˆ

⌃ = LLT,

where ˆ⌃ is our positive-definite correlation matrix and L is the lower triangular matrix satisfying the Cholesky-decomposition. Let Z = (z1, z2, z3) be a 3⇥1 vector with z1, z2and z3independent

draws from the standard normal distribution. Using the calculated lower triangular matrix L and the 3 _{⇥ 1 vector Z of independent standard normal draws, we can generate values from} N ⇣0, ˆ⌃⌘ by using LZ_{⇠ N} ⇣0, ˆ⌃⌘. The estimated correlation matrix between the residuals of our models, which we are using in this research, is given by:

ˆ ⌃ = 0 B B B @ 1 0.0399 0.0255 0.0399 1 0.0201 0.0255 0.0201 1 1 C C C A 5.7 Utility Function

One of the most important measurement instruments for investment studies is the expected utility. This paper is using the power utility function satisfying the Constant Relative Risk Aversion (CRRA). The expected utility is calculated based on the number of scenarios N and strategy j by: E(Uj) = 1 N N X i=1 U (Fi,j),

where Fi,j is the final capital at retirement age. The power utility function Up(·) is given by:

Up(x) = 8 > < > : x1 ₁ 1 , if 6= 1 log(x), if = 1

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Finances and found that it is realistic to take greater than 2. Pindyck (1986) uses a structural model of equity pricing to find an estimate of between 1.6 and 5.3. In this paper, we want the most standard and realistic coefficient. A combination of the results from the earlier research will result in a of approximately 3, which will be used in this study.

The main focus of this paper is not only to maximize the pension savings (or yearly pension annuity bought from total pension savings), however we also want to maximize the proportion who can a↵ord their care costs and we want to minimize the relative di↵erence between the pension savings and the total capital. In the utility function, we have to incorporate these latter two criteria. De Nardi et al. (2010) is using the power utility with an extra component of healthy status:

Un(x) = (1 + h)Up(x),

where h equals 1 if someone is healthy and 0 otherwise. In our context, giving a reward to the utility if the person is able to a↵ord the care costs and a penalty for the relative di↵erence between pension savings and total capital would be sufficient. The utility function used in this paper will be as follows:

Ui(xi) = ✓ 1 + 1IZi>0 2 F Pi Fi Fi ◆ x1_i 1 ,

where we will give the same weights to 1 and 2, resulting in 1 = 2 = 1. The input variable

xi will be the yearly pension annuity bought from the total pension savings F Pi in scenario i

(this will be explained in the following section). In this utility function, IZi>0 is the part giving

a reward for the sufficient care reserves for the needed care, while F Pi Fi

Fi gives a penalty for

having to much or less total capital than the total pension savings. When F Pi > Fi, it means

that the yearly pension annuity from the pension savings is needed for a part of the care costs and the outstanding mortgage. When F Pi < Fi, the total care savings are more than needed,

resulting in a loss of capital, since this part is not used to buy a yearly annuity. By adding these two terms, the utility will be adjusted to the objectives of this paper.

5.8 Transforming Total Capital to Yearly Pension

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the yearly pension available from a capital of e100,000 can be seen in the following graph:2

Figure 9: Annuity versus interest rate

Data source: https://www.bank-sparen.nu/banksparen/vergelijken/

By looking at the graph, there seems to be a upward linear trend between the interest rate and the yearly pension amount. To transform our accumulated pension (from the total pension savings FP and from the total final capital F), we will use the following equation:

Yearly pension = Capital_{· (c + · interest) ,}

where c is the intercept and is the coefficient for the interest rate. By OLS we have ˆc = 3.947· 10 2 _(2.021_{· 10} 5_{) and ˆ = 5.864}_{· 10} 1 _(1.217_{· 10} 3_{) with the corresponding standard}

errors between parentheses. This leads to the following equation for transforming capital into a 25 year annuity with yearly payments:

Yearly pension = Capital· (0.03947 + 0.5864 · interest) ,

where Capital is the total pension savings or the total final capital and the interest is set at a rate of 2%.

2_{The data is obtained by using 8 di↵erent providers from https://www.bank-sparen.nu/banksparen/}

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6 Simulation

After estimation of the models and correlations, we can start the scenario simulations. This research is simulating N = 10, 000 paths for all the di↵erent variables. The following subsections illustrates the paths corresponding to the standard quantiles (0.05, 0.25, 0.5, 0.75, 0.95) at retirement age. The light blue line is the 95%-quantile, the dark blue line the 75%-quantile, the green line illustrates the 50%-quantile, the red line corresponds to the 25%-quantile and the black line shows the 5%-quantile. The paths can cross each other since we are not taking the quantiles of each period, but only the quantile of the last period (retirement age) with their corresponding paths.

6.1 Salary

The quantiles of our 10,000 simulations are given in Figure 10 below.

Figure 10: Gross and net salary

(a) Yearly Gross Salary

Using Equation (4)

(b) Yearly Net Salary

After using the taxrates from Appendix Table 8

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while the bottom 5% has at most e83,435.40 (e50,757.90). The median of the total gross salary simulations are e101,308.45 (e59,783.79) at retirement age. Note that these salaries are not discounted, meaning that we cannot compare these salaries with the median salary of today.

6.2 Inflation

The left Figure 11a below shows the accumulation possibilities for a e1 investment in the Consumer Price Index. Figure 11b below illustrates the corresponding inflation paths.

Figure 11: Inflation graphs

(a) 1 Euro investment in the inflation (b) Yearly inflation

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6.3 Bonds

The simulation of the short rate can be seen in Figure 12 below.

Figure 12: Short rate graphs

(a) 1 Euro investment in the annualized short rate (b) Annualized short rate

For the simulation of the short rate, we used the Cox-Ingersoll-Ross model (see Cox et al. (1985)) with the property of positive short rates (see Section 5.4 for an explanation). The left Figure 12a shows the quantile paths of the accumulation of an e1 investment in the short rate. The right Figure 12b illustrates the corresponding short rates for the di↵erent quantiles. Investing in the short rate does not have many risks, since all the investment paths yields positive returns with the bottom 5% of at most e1.47 and the top 5% of at least e3.21 with corresponding average annualized short rates of 0.892% and 2.746%, respectively. The median has a final value of e1.94 with an average annualized short rate of 1.553%, which is a fraction lower than the historical annualized average of 1.656% (see Section 4.2). These results and the additional 25%-and 75%-quantile are given in Table 15 of the Appendix.

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Figure 13: Bond return graphs

(a) 1 Euro investment in the 1 year bond (b) Yearly bond returns with maturity of 1 year

The left Figure 13a shows the quantile paths of e1 investment in the 1 year maturity bond and the right Figure 13b illustrates their corresponding yearly returns. The median is increased with respect to the investment in the short rate by only 0.051%, having a value of e1.98. The bottom 5% will have a value of at most e1.73 and the top 5% a value of at least e2.53, yielding an increase of 0.389% and decrease of 0.56%, respectively. This indicates it has on average slightly more return with less risk. The results can also be found in Table 16 of the Appendix.

6.4 Stocks

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Figure 14: Stock return graphs

(a) 1 Euro investment in stocks (b) Yearly stock returns

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7 Results

After the simulations and the explanation of our suggested strategies, the results can be pre-sented. The results are given in the Tables 4, 5, 6 and 7. Firstly, we will explain the output for the di↵erent strategies which are given in the tables. F P denotes the final pension savings transformed to a yearly annuity and inflation adjusted with index year 2018. The 25%, 50% and 75% quantiles and the standard deviation are given. FH denotes the total savings for health care after retirement. F denotes the total capital determined by Equation (3) in Section 5.1, which is also transformed to a yearly annuity and inflation adjusted with index year 2018 such that we can compare the annuity bought from the pension savings with the maximum annuity bought from the total capital. The annuity bought from the total capital is a representation of the annuity which could be realized in a perfect saving path. We determined the total capital F for the three care classes to see the di↵erence for the three choice variables as defined in 4.4. ‘Low’, ‘medium’ and ‘high’ care corresponds to the classes HC1, HC2 and HC3, respectively.

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Table 4: Results of benchmark strategies 1 2 3 4 5 6 F P? 25%-q 10820 10023 9366 9005 6842 8264 50%-q 14353 13216 12339 11867 9101 10754 75%-q 19532 17873 16569 15970 12369 14360 Std. dev. 8031 7101 6444 6567 5327 5561 F H 25%-q 55787 86028 111602 135885 235605 171081 50%-q 69366 107132 137938 173885 288476 211748 75%-q 86975 134124 171937 217769 348777 259480 Std. dev. 25263 38783 48538 63423 87820 69824 F? Low Care 15084 (50.26) 14809 (47.41) 14560 (45.59) 14914 (46.52) 14644 (42.94) 14628 (43.42) Medium Care 12991 (57.29) 12715 (54.27) 12466 (52.39) 12821 (53.24) 12551(49.36) 12534 (49.91) High Care 10374 (69.43) 10098 (66.26) 9849 (64.41) 10204 (64.98) 9934 (60.70) 9918 (61.40) E(Z > 0) Low Care 82.23 90.73 96.6 98.4 99.97 99.74 Medium Care 10.45 45.31 68.15 80.46 97.86 90.8 High Care 0.02 1.42 6.03 18.78 71.48 36.58 Utility Low Care 0.867 [81.85%] 0.890 [7.33%] 0.900 [5.52%] 0.885 [4.01%] 0.803 [0.40%] 0.862 [0.89%] Medium Care 0.470 [9.76%] 0.671 [30.15%] 0.786 [20.89%] 0.831 [21.26%] 0.842 [9.39%] 0.866 [8.55%] High Care 0.046 [0.21%] 0.165 [1.00%] 0.127 [4.48%] 0.280 [15.44%] 0.227 [62.31%] 0.446 [16.56%]

? denotes the transformed capital to a yearly annuity by using Section 5.8 and adjusted for inflation Standard deviations are given in % between parentheses

Proportion of scenarios in which the strategy results in highest utility is given between brackets

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and ‘high’ classes, respectively. By looking at the strategies with the highest chance of achieving the highest utility, strategy 2 (30.15%) and strategy 5 (62.31%) are performing the best for the ’medium’ and ’high’ classes, respectively. For the ‘medium’ class, this strategy has a low chance of being able to a↵ord the health care costs by using the care savings (45.31%). For the ‘high’ class, strategy 5 has a lot more chance of being able to pay the health care costs comparing with the strategy based on the highest expected utility, namely 71.48%.

Table 5: Results of strategies where redemption is fully paid from pension savings

1 2 3 4 5 6 F P? 25%-q 9187 8667 8210 6201 4077 5840 50%-q 11883 11140 10600 8030 5288 7434 75%-q 15595 14559 13889 10572 7052 9714 Std. dev. 5836 5239 4991 4131 3004 3499 F H 25%-q 146560 161245 176277 283220 379244 295706 50%-q 178001 195016 208455 339053 448190 350013 75%-q 212098 239022 248623 399911 525857 415816 Std. dev. 50445 59488 58992 89711 113021 92827 F? Low Care 15182 (41.08) 14907 (39.27) 14647 (38.68) 14888 (36.13) 14665 (32.59) 14666 (34.31) Medium Care 13089 (47.02) 12814 (45.11) 12553 (44.57) 12795 (41.59) 12571 (37.69) 12573 (39.64) High Care 10472 (57.39) 10197 (55.41) 9936.6 (55.04) 10178 (51.28) 9955 (46.86) 9956 (49.19) E(Z > 0) Low Care 99.38 99.87 99.94 100 100 100 Medium Care 83.9 88.82 91.18 99.28 99.97 99.88 High Care 16.12 27.39 33.12 86.71 97.85 90.64 Utility Low Care 0.883 [67.82%] 0.868 [28.63%] 0.855 [3.48%] 0.761 [0.05%] 0.659 [0.00%] 0.743 [0.02%] Medium Care 0.856 [56.14%] 0.866 [28.64%] 0.867 [7.59%] 0.804 [2.95%] 0.690 [0.12%] 0.787 [4.56%] High Care 0.443 [11.86%] 0.506 [12.47%] 0.538 [13.17%] 0.803 [25.37%] 0.738 [8.02%] 0.809 [29.11%]

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class, the maximum utility slightly increases from 0.866 to 0.867, moving from strategy 6 to 3. By looking at the chance of reaching the highest utility, strategy 1 will be the best strategy for the ‘medium’ class with a chance of 56.14% and expected utility of 0.856. This indicates that when one is using the additional redemption option, the optimal investment strategy will be strategies with more contribution in pension savings compared to the benchmark. For the ‘high’ classes, the additional redemption option is extremely useful by comparing the maximum expected utility, the probability of a↵ordability of the care costs and the total capital. The optimal strategy is strategy 6 where 70% of the maximum pension contribution is invested in pension, with an expected utility increase from 0.446 to 0.809. This is also the strategy with the most chance of reaching the highest utility, namely 29.11%. The probability of being able to a↵ord the care costs through the care savings is increased from 36.58% to 90.64%. The annuity from the total capital is increased from e9,918 to e9,956 with a lower standard deviation.

Table 6: Results of strategies where redemption is equally paid from pension savings and care savings

1 2 3 4 5 6 F P? 25%-q 10624 9840 9244 7625 5418 6869 50%-q 13804 12657 11851 9925 7126 8837 75%-q 18340 16681 15502 13269 9594 11586 Std. dev. 7072 6123 5515 5314 4098 4294 F H 25%-q 77742 114707 135389 216580 317005 253192 50%-q 99470 138480 163024 263409 378062 301570 75%-q 124612 167356 195948 312876 444062 354732 Std. dev. 36363 41641 48072 75092 98538 79507 F? Low Care 15423 (44.69) 15145.118 (41.72) 14907 (39.91) 15110 (39.80) 14897 (36.10) 14840 (36.53) Medium Care 13330 (50.92) 13051.683 (47.74) 12813.6 (45.84) 13017 (45.60) 12803 (41.56) 12747 (42.05) High Care 10713 (61.69) 10435 (58.26) 10196.8 (56.30) 10400 (55.77) 10186 (51.23) 10130 (51.85) E(Z > 0) Low Care 89.13 96.98 99.03 99.93 99.99 99.98 Medium Care 37.31 69.98 80.41 96.15 99.7 98.67 High Care 0.4 3.84 10.83 61.92 92.69 77.37 Utility Low Care 0.884 [89.07%] 0.898 [7.71%] 0.889 [2.18%] 0.823 [0.98%] 0.728 [0.02%] 0.789 [0.04%] Medium Care 0.639 [37.28%] 0.799 [31.35%] 0.844 [11.47%] 0.857 [16.23%] 0.768 [1.32%] 0.834 [2.35%] High Care 0.243 [0.41%] 0.331 [3.44%] 0.399 [6.82%] 0.691 [51.65%] 0.793 [22.50%] 0.779 [15.18%]

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an improvement in the maximum expected utility compared to the redemption option fully paid by the pension savings. The maximum expected utility increases from 0.883 to 0.898, which is still lower than the maximum expected utility of the benchmark strategies where no redemption is allowed. However, strategy 1 still has the best chance of reaching the highest utility, namely 89.07%. For the ‘medium’ and ‘high’ classes, the maximum expected utility did decrease, due to the decrease of the probability that the care costs can be fully paid by using the care savings. From this we can conclude that overall it does not have an advantage to pay the additional redemption equally by the pension savings and care savings compared to when the redemption is paid fully by the pension savings.

Table 7: Results of strategies where redemption is fully paid from care savings

1 2 3 4 5 6 F P? 25%-q 11567 11143 10634 9027 6847 8288 50%-q 15299 14531 13738 11886 9101 10781 75%-q 20667 19321 18228 15988 12369 14391 Std. dev. 8331 7414 6770 6576 5327 5568 F H 25%-q 22761 44737 65323 141595 246848 182657 50%-q 30498 59245 84892 182087 300165 223005 75%-q 42616 79765 109368 226040 357329 267096 Std. dev. 19685 28669 34667 65118 85940 66028 F? Low Care 15426 (49.37) 15308 (45.90) 15126 (43.82) 15311 (43.91) 15152 (39.94) 15093 (40.43) Medium Care 13333 (56.13) 13214 (52.33) 13033 (50.11) 13218 (50.11) 13058 (45.76) 13000 (46.31) High Care 10716 (67.72) 10598 (63.43) 10416 (61.07) 10601 (60.84) 10442 (55.93) 10383 (56.62) E(Z > 0) Low Care 75.41 81.88 86.06 98.13 99.91 99.64 Medium Care 1.1 8.9 25.02 80.63 97.79 91.95 High Care 0 0.03 0.32 22.14 75.15 41.95 Utility Low Care 0.844 [75.33%] 0.865 [6.42%] 0.874 [4.14%] 0.874 [12.35%] 0.794 [0.36%] 0.851 [1.40%] Medium Care 0.411 [1.10%] 0.475 [7.74%] 0.571 [15.68%] 0.832 [56.48%] 0.832 [7.99%] 0.865 [11.01%] High Care 0.075 [0.09%] 0.170 [0.03%] 0.193 [0.38%] 0.456 [21.82%] 0.749 [57.80%] 0.595 [19.88%]

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8 Conclusion

This research combined the three components pension, health care and mortgage by using an Economic Scenario Generator to build 10,000 scenarios. These scenarios are used to compare di↵erent investment strategies for allocating disposable income to one of the three components. The main objective of this research was to investigate whether it is advantageous to use pen-sion and/or health care savings for prepaying the mortgage. The health care costs consist of a random assigned elderly care segment additional with the personal-based choice for the type of home care. We defined three choice classes for the home care costs. The first class is for people who do not desire to get home care (‘low’), the second class is for people preferring normal home care (‘medium’) and the third is for people who want very luxury home care (‘high’). We compared the strategy outputs for these three choice classes to look whether the same strategies would be optimal and whether it is useful to use the prepayment option for people preferring the ‘low’, ‘medium’ or ‘high’ home care class.

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9 Further Research

In our analysis, we made the assumption that the house prices will remain constant. For further research, the Economic Scenario Generator could be extended by adding a model for predicting and forecasting the paths of house prices. An additional extension to the salary variable used in our Economic Scenario Generator is to take the correlation between income shocks into account and adding for example ‘good’ and ‘bad’ economic states.

In our scenario approach, we did assign one random elderly care (EC) segment to each sce-nario from which we calculated the utilities for all the considered strategies. Since the utility function is a non-linear function, it could be an option to assign multiple elderly care groups to the same scenario and examine the utility function properties.

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Appendix

Formulas

Delta Method for Standard Errors

The Delta Method can be used to compute standard errors of transformed parameters obtained from Ordinary Least Square estimation. In the three variable models, we are using transfor-mation of estimates to obtain the needed parameter estimates to use in our model or to use as initial values. By definition of a linear regression, a consistent estimator ˆ✓ will converge in probability to the true value ✓ and by applying the Central Limit Theorem we have the following asymptotic normality property

p

n⇣✓ˆ ✓⌘⇠ N (0 , ⌃) ,

where n is the number of observations, X is the matrix of exogenous variables and ⌃ is the covariance matrix given by ⌃ = 2⇣XT_X

n

⌘ 1

(this is the case in an OLS setting). The Delta Method states that by using a transformation function g(·) to the parameter ✓, the following property of the transformed parameter estimates holds:

p n⇣g(ˆ✓) g(✓)⌘_{⇠ N} ✓ 0 , [_rg(✓)]T ⌃ [_rg(✓)] ◆ ,

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Graphs

Figure 15: Linear Mortgage

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Tables

Table 8: Dutch tax rates after the coalition agreement 2017 (Source: Statistics Netherlands CBS)

Pretax income (Euro) tax rate (%)

0 - 68600 36.93

68600 and higher 49.50

Table 9: Income and expenditures in dollars, by age of reference person, 2013 (Source: U.S. Bureau of Labor Statistics)

Age Income Expenditures Percentage left for pension and care savings

25 until 34 59002 48087 18.5

35 until 44 78385 58784 25

45 until 54 78879 60524 23.27

55 until 64 74182 55892 24.66

Table 10: Income and expenditures (excluding housing) in dollars, by age of reference person, 2013 (Source: U.S. Bureau of Labor Statistics)

Age Income Expenditures (excluding housing) Percentage left for pension savings, care savings and mortgage

25 until 34 59002 19994 47.7

35 until 44 78385 38165 51.3

45 until 54 78879 41523 47.4

55 until 64 74182 37955 48.8

Table 11: Regression results of inflation model

(a) Regression results of Equation (5)

Estimate Standard Error t-value P-value

c 0.0009 0.0004 2.155 0.0332

b 0.3453 0.0869 3.975 0.0001

0.0013 - -

-(b) Transformation results of (9), (10) and (11)

Estimate Standard Error t-value P-value

↵ 12.7612 3.0192 4.2267 0.0000

µ 0.0014 0.0006 2.2601 0.0256

A Coherent Look at Pension, Mortgage and Long-Term Health Care: An Economic Scenario Generator Approach