Defined contribution : a redesigned benefits phase

(1)

Defined Contribution

A redesigned benefits phase

Erica Kuin

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: Erica Kuin Student nr: 6092071

Email: ericakuin@outlook.com Date: August 29, 2014

Supervisors: dr. T. Boonen (UvA - FEB - ASE)

drs. W.A.M. Tuyp AAG (Towers Watson Netherlands B.V.) Second reader: prof. dr. J.B. Kun´e

(2)

(3)

Defined Contribution — Erica Kuin 3

Abstract

This thesis is about a redesigned benefits phase of defined contribu-tion agreements. The last years the legislacontribu-tion for the benefits phase of defined contribution agreements is strongly criticized. The current legislation obligate to fully annuitize the pension wealth at retirement to insure the consumer has a steady income until death, but offering a nominal guaranteed income until death turns down any investment risk so that the pensions will become expensive or low and it leads to bearing inflation risk. In this thesis we will therefor redesign the benefits phase to investigate the possibility of bearing investment risk at retirement. We will also investigate the possibility to use a part of the second pillar pension wealth for medical expenses, since the medi-cal expenses increases with age. Due to aging, those medimedi-cal expenses will also increase each year. For both redesign models, we will look at the consumption and the utility of the consumption for the ex-pected scenario, the optimistic scenario and the pessimistic scenario. Comparing those with the current pension income, it is not optimal to use the given model for a redesign considering medical expenses. The income in the first year will be much lower and it takes at least 10 years to exceed the current pension income. However, the first redesign model gave some interesting results which could be further investi-gated. According to the utility results, it is optimal to invest a part of the second pillar pension wealth. But the consumption results would probably mean the opposite since the income in the first year will be a bit lower and it takes 5 years to exceed the current pension income. It is questionable if participants would choose for a lower pension in the first year, even when the pension income at later retirement is much higher. Also the spouses pension and the longevity bonus could cause a rejection of this redesign, since the spouses pension could make it much more complicated and the investment returns may be lower than the longevity bonus which demands for a negative risk premium on the participants pension. However, since this is not investigated yet, it could still be optimal to invest a part of the pension wealth, so further research is recommended.

Keywords Retirement, Pension funds, Defined Contribution, Benefits phase, Investment risk, Annuities, Inflation, Conversion risk, Longevity, Medical expenses

(4)

4 Erica Kuin — Defined Contribution

Acknowledgements

Apart from the efforts of myself, the success of any project depends largely on the encouragement and guidelines of many others. I take this opportunity to express my gratitude to the people who have sup-ported me in the completion of this project. I would like to express my gratitude to my supervisor Tim Boonen for the useful comments and remarks through the learning process of this master thesis. Fur-thermore, I would like to thank Edwin Schokker for introducing me to the topic. Also, I would like to thank Wouter Tuyp and Vincent Tol for their support on the way. I would like to thank my loved ones, who have supported me throughout the entire process, I will be grateful forever for your love.

(5)

Introduction

Recently Bakels et al. (2014) published an article about defined contribution agreements (DC) [2], in which they analyze desired developments in the second and third pillars and the pitfalls that may occur. One of those developments is changing the benefits phase of the DC agreement since the current legislation is strongly criticized. The current legislation makes pensions expensive and results in low pensions. We will investigate the current legislation and possible solutions to take the cons away.

1.1 Research goal and structure

One of the key issues that Bakels et al. mentioned in their article, is that bearing in-vestment risk in the benefits phase is not allowed. They claim that this makes, given the pension ambition and some risk tolerance, the pension unnecessarily expensive. Al-lowing variable annuities makes it possible to face investment risk (including interest rate risk and inflation risk) without being confronted with undesirable conversion risk. Brown and Nijman (2011) [5] also claim that retirees should have more flexibility in their choice of annuity products, including having access to nominal, variable, deferred and other annuity payouts.

In addition to the annuitization problem, Bakels et al. outline a number of issues for the longer term. One of those issues is introducing the possibility to use a part of the pension capital for certain purposes, like medical expenses, and the advantages and disadvantages of partly annuitization. So they claim we should move away from the requirement that all retirement wealth has to be annuitized, to a requirement that in-dividuals annuitize enough to have a real income floor that is sufficient to cover basic needs and use the remaining retirement wealth for other goals like medical expenses. The minimum floor for having a real income then should have some secure inflation protection.

Other research for optimizing the benefits phase is done by Horneff et al. (2006) [14], Peijnenburg et al. (2011) [20] and Finkelstein and Potterba (2004) [12]. Horneff et al. (2006) show the advantages and disadvantages of annuitization, like liquidity risk and bequest motives. They also show different ways to decumulate investment capital in the benefits phase. Peijnenburg et al show the optimal annuity demand when health costs in early retirement is included, the optimal annuity demand decreases when wealth increases. Finkelstein and Poterba (2004) show the influence of adverse selection on annuitization, they show that longevity has influence on the annuity payments. They

(7)

claim that annuities are not actuarially fair from the standpoint of the individual with a lower life expectancy, since individuals with a higher life expectancy will receive more benefits for the same price. They therefor claim that the demand for annuitization will increase with income. This can not be compared with the research done by Peijnenburg et al., since Peijnenburg et al. included health costs into their model. However, both studies are interesting for further research.

The main research goal of this thesis is allowing for investment risk in the benefits phase of defined contribution agreements in order to protect against inflation risk. For this goal, we will describe the current pension system with the current defined contribu-tion agreement in particular. We will investigate the compulsory annuitizacontribu-tion in the benefits phase and the advantages and disadvantages of the annuitization. The fiscal requirements are not part of this thesis and thus will not be investigated.

In this thesis a model for a redesigned benefits phase is build, where we will investigate different fractions of the second pillar pension income to be invested while the other part will be used for annuitization. We will try to find an optimal investment fraction. For the investments we consider three different scenario’s: the expected scenario, the optimistic scenario and the pessimistic scenario. The utilities of the consumptions of these different scenario’s are compared.

In addition to the main research goal, this thesis will also consider the possibility of partly annuitization, invest the other part of the pension wealth and using a part of this invested capital for medical expenses. The motivation for using retirement wealth for medical expenses and the level of medical expenses will be given, the advantages and disadvantages of partly annuitization will be investigated and different scenario’s are given and compared.

1.2 Further research and conclusions

In these thesis we considered investment risk in the benefits phase of defined contribu-tion agreements and the possibility of using a part of the pension wealth for medical expenses. Results concerning the pension consumption and the lifetime utility of the pension consumption are given. From the results of the second model we can conclude that it is probably not optimal to invest a part of the pension capital when the pen-sion capital also has to be used for medical expenses. This model can be optimized when more information about medical expenses is available, since a lot of assumptions are made for this model. On the other hand, we can conclude from the results of the first redesign model that it could be interesting to invest a part of the second pillar pension income and partly keep an annuity. We would expect that this eventually pro-vides higher benefits than the current pension income. However, in early retirement the benefits are lower and it is not secure that the investment returns are high enough to receive a higher pension income. Also, the part of the pension capital that is invested will be fully consumed at age 95, so the pension income will drop after 30 years which means the longevity risk is higher. Theoretically it may be optimal to invest a part of the pension wealth, but in practice participants may not be willing to choose for an insecure pension income. However, it is still interesting to investigate this. Both models could be improved by taking the correlation between income and life expectancy into account in order to cover more of the longevity risk.

(8)

Fortunately, the State Secretary for Social Affairs and Employment is also convinced that considering investment risk in the benefits phase should be investigated. She send a letter on 25th of June 2014 to the House of Representatives of the Netherlands in which she states that making the benefits phase risk-bearing should be investigated and that she expects that she should be able to present some results in September 2014 [17]. I am convinced that they will come to the same conclusion.

(9)

Chapter 2

Current pension system

This chapter gives a description of the current pension system in the Netherlands, in particular the defined contribution agreement. It will give the current requirements of the accumulation phase and the benefits phase in defined contribution agreements.

2.1 Pillar-system

The Dutch pension system consists of three pillars, each of which provides annuities with only a few exceptions which can be found in the article of Brown and Nijman (2011) [5]. The first pillar is the state pension (AOW) and provides basic income above the level of the minimum wage as of the age of 65 until death1. Full AOW income is available for everyone who lived in the Netherlands for 50 years as of the age of 65 and is regulated by the government with a pay-as-you-go finance basis. In addition to the first pillar, the second pillar consists of (almost always) mandatory contributions to a pension fund or insurer. The selection of the pension fund and the level of payments are mandated. In the second pillar the pension income comes in the form of annuities and individuals are generally not allowed to convert their pension income to a lump-sum payment. Most second pillar entitlements are typically characterized as defined benefit (DB) plans, which can be seen as capital funding. While most second pillar schemes are DB, a limited number of pension funds offer DC schemes in the second pillar, where individuals pay a premium, but have no certainty about the level of capital on retirement age. The final pension capital will be depending on the loaded premium, on costs and on returns. These DC plans are mandatory and also required to be annuitized. The rules and legislation of these second pillar DC plans are close to those for third pillar products. The third pillar of the pension system consists of tax-free voluntarily pension products from banks or insurers. In particularly for the 5% of the working population that is self-employed and who do not have a second pillar coverage, this pillar is very important. This third pillar is also used by participants in pension funds who criticize that their first and second pillar coverage is insufficient. An important distinction between the DC plans in the second pillar and the DC plans in the third pillar is to what extent the individual decision power is delegated to the social partners or to the pension fund. This may lead to differences in the degree of choice freedom, desired product forms and appropriate regulation. The tax legislation concerning the tax deductibility of contributions require that the accrued pension is less than a certain percentage of income.

1_{In 2019 and 2023 the retirement age will be 66 years and 67 years respectively. After that, the}

retirement age will be adjusted for the life expectancy, see Rijksoverheid (2012) [22]).

(10)

2.2 DC: Accumulation phase

As mentioned in the previous section, participants of the defined benefit plan pay a premium but have no certainty about the level of capital on retirement. This will be depending on the loaded premium, on costs and on returns. The premium is subject to fiscal requirements. These requirements are made up to make sure the pension wealth at retirement age is equal for defined benefit plans and defined contribution plans. A de-crease of the fiscal maximum pension accrual rate in defined benefit plans has the result that the fiscal maximum premium rate in defined contribution plans will decrease as well. For the defined contribution plans the first possibility is discounting the maximum pension accrual rate with a fixed interest rate of 4%, which lead to a fiscal maximum premium ladder depending on the age of the participant. This means that for young participants the maximum premium rate is much lower than for older participants, but young participants have longer maturities. Several studies claim that this maximum premium rate is currently too low since the current interest rate is a lot lower than 4%. The second possibility is discounting with an interest rate of 3%. Since the critics of several studies, Bakels et al. (2014) [2] suggest that the discounting has to be done with the actual long-term interest rate structure and use this as the single possible ladder. In these DC schemes, the participants usually have plenty choice opportunities, in par-ticular with respect to the investment strategy. Unlike in DB schemes the participant faces investment, interest and longevity risks directly. The collectivity reduces the in-dividual risks and costs, which makes the pension affordable. Inin-dividuals choose their own life cycle and bear the investment and conversion risks (interest and longevity). The choice possibilities can be directed with a default life cycle, which is the “auto-matic choice”. Bakels et al. (2014) claim that 95% of the participants will choose for such a default option, since most of the participants don’t have the knowledge to make a good investment choice. For this reason, it is important to have a proper default option. Defaults can be designed for a whole population, or can be custom made and vary per pension plan (basis, excedent or additional on a voluntary basis). Another possibility is that participants draft and perform their own investment strategy by using opting-out. This means the participants will face choices and risk and need some guidance to do this, which is expected from the pension provider by the regulator.

2.3 DC: Benefits phase

The general rule for decumulation of pension wealth in the Netherlands is that the pension wealth has to be converted to an annuity. The exception to the rule is “bank saving”2 products in the third pillar. Second pillar pension income always comes in the form of annuities, individuals are not allowed to convert pension income to a lump-sum payment. This requirement to annuitize brings advantages, but also some disadvantages, which are more explored in subsection3.1. The main reason to annuitize is the protection against longevity risk. One of the most important disadvantages is the lack of inflation-protection. Essentially, true inflation-protection in the Netherlands is limited to the first

2_{These third pillar “bank saving” products are fixed term (minimum 20 years) rather than life long}

and therefore do not provide insurance against outliving your assets if one reaches a very high age. So these bank saving product do not offer any longevity risk, but these products do allow for bequest. In case of premature death of the insured, the money is transferred to the heirs rather than to the pool of annuitants.

(11)

pillar. In the second pillar DC plans, as wel as in the third pillar, annuity choices are typically limited to fixed nominal annuities. An important shortcoming of the focus on wealth accumulation is that one thinks about how to save, but an extensive retirement planning strategy also requires thinking about how to spend. It fails to consider how one’s assets will be converted into a stream of consumption in retirement.

(12)

Chapter 3

Redesign

In the benefits phase of a defined contribution agreement it is compulsory to annuitize all accrued pension wealth at retirement age. Since there are a lot of discussions on this subject, this chapter will consider the advantages and disadvantages of annuitization and give some basic properties for a redesigned benefits phase of a defined contribution agreement.

3.1 Annuitization

In the current legislation it is compulsory to fully annuitize the pension wealth, to in-sure the consumer has a steady income until death. A payment in the form of income until death prevents participants of outliving their pension capital, and thus covers the longevity risk for the participant. This can be done by sharing the risk with a large group, so individuals don’t need to save reserves in case they live longer than expected. If the number of annuitants is sufficiently high, the mortality risk is hedged through diversification. The surviving participants receive the funds of the participants who die. Another advantage of full annuitization is that the participants are protected against the tendency to consume the whole pension wealth in an early state. The government wants to prevent that retirees need income support at higher ages. In general, annuities are economically efficient.

However, there are some disadvantages of full annuitization. One of the biggest disad-vantages is the lack of inflation protection. The pay-outs are nominal, sometimes with a predetermined fixed (indexation) increment percentage. This subjects individuals to uncertainty in the cost of living adjustment of their retirement income stream. Bakels et al. (2014) [2] and Brown and Nijman (2011) [5] claim that offering a nominal guar-anteed income until death is one of the biggest limitations of the current legislation. A guaranteed income until death turns down any investment risk so that the pensions will become expensive or low. Buyers lose liquidity because the assets usually cannot be recovered even to meet special needs. Besides, turning down interest rate risk leads to bearing inflation risk. Brown and Nijman (2011) claim there are two reasons to explain this lack of inflation protection in the Netherlands. The first reason is “money illusion” of individuals who are not willing to accept the lower initial payment in return for the long-term preservation of the cost of living adjustment. Most consumers are insufficient aware of the impact of inflation on the cost of living adjustment on longer term. The second reason is the lack of bonds indexed to Dutch inflation.

(13)

If shortly before the conversion date a lot is invested in equities or the duration of the portfolio is much lower than the duration of the annuity, a decrease in stock prices or a decrease in interest rates may lead to a much worse amount of the annuity benefits, which is not desired. This risk at retirement age is known as conversion risk. Some prod-ucts that are offered contain rate-of-return guarantee during the accumulation phase. Legislation forces insurers to offer only investment strategies that manage the conversion risk to annuities. These legislation require that the exposure to equities will be reduced close to the target date, and the duration of the fixed income portfolio is approximately the same as the duration of the annuity to be bought.

Another disadvantage of full annuitization is the lack of desire to annuitize pension wealth in case of bequest motives of the retiree. When a retiree has bequest motives, the desire to annuitize wealth will reduce. Horneff et al. (2006) [14] show that more than half of the retiree in the US expect leaving a bequest worth more than $10,000. Conditional on surviving, the rate of return on the annuity is greater than the rate of return on non-annuitized assets and thus individuals who do not have any bequest motives would annuitize all wealth. Brown and Nijman (2011) discuss several studies about the influence of bequest motives on wealth decumulation. Adverse selection could also be a reason against annuitization. Finkelstein and Poterba (2004) [12] show the influence of adverse selection on annuitization. Demand for annuities may be low if the expected annuity payments for a typical individual are low relative to the annuity’s premium. An example is longevity, if an individual has a lower life expectancy than the general individual in the population and annuities are priced to reflect the longevity of annuitant, then annuities will not be actuarially fair from the standpoint of the in-dividual with a lower life expectancy, since the annuity should be less expensive for lower life expectancy. Since individuals with higher income live longer, the demand for annuitization will increase with income, which makes the annuities more expensive due to the increased longevity.

Those disadvantages in combination with the annuity prices and the high insurance company loadings makes it appropriate to consider a redesigned benefits phase for de-fined contribution agreements. Retirees should have more flexibility in their choice of annuity products, including having access to nominal, variable, deferred and other an-nuity products. In this research we will therefor assume that partial annuitization is possible and the remaining second pillar pension wealth will be invested.

Assuming that neither zero nor complete annuitization is optimal, pinning down an optimal level of annuitization is difficult. The level of annuitization should be appropri-ate for any individual in a population. In the Netherlands, all retirement wealth in the first, second and third pillars is subject to mandatory annuitization. At the other end we have the United States. Brown and Nijman (2011) conclude that most workers in the United States have little, if any, annuitization in addition to the first pillar Social Security system, which itself provides an average replacement rate of approximately 40 percent of the average lifetime income. Most other countries fall at intermediate points along this spectrum.

In this research the annuity factors for male and female will be calculated with mortality rates using the generation table 2012 of the AG [1] and with interest rates using the term structure of December 13, 2013 published by DNB [10].

(14)

3.2 Utility and risk aversion

An important part of the redesign model is the utility function u(C), which transforms the value C (consumption) into opportune consumption benefits for the individual. An utility function u(C) should meetl the three requirements u0(C) > 0, u00(C) < 0 and u000(C) > 0, for every consumption variable C > 0. Peijnenburg et al. (2011) [20], Blake et al. (2008) [3] and Cipra (2014) [8] mention a utility function which is regular used in pension context for a risk coefficient γ 6= 1, Cipra also mention this function for the special case γ = 1: u(Ct) =          C_t1−γ 1 − γ for γ > 0, γ 6= 1, ln Ct for γ = 1, (3.1)

with γ > 0 the risk aversion and consumption Ct> 0. Since this is a regular used utility

function in pension context, we will also be using this utility function in this research. It can be shown that this utility function satisfies the given requirements. Cipra (2014) [8] denote the utility function (3.1) CRRA (constant relative risk aversion) since its relative risk aversion (Arrow-Pratt measure) RRA(Ct) is constant:

RRA(Ct) = −

Ct· u00(Ct)

u0_(C t)

= γ. (3.2)

If two individuals have different relative risk aversions, the one with the higher value of γ is more risk averse and the utility rises with the consumption. Next we have to find an appropriate risk aversion coefficient γ. Peijnenburg et al. (2011) claims that the coeffi-cient γ = 5 is consistent with previous studies. Kaplow (2003) [16] finds that empirical work indicates a coefficient of 2 or more, and that some empirical work indicates that an individuals coefficient may be above 10 and Horneff et al. (2006) [14] gives examples with an extreme risk aversion given by the coefficient γ = 9, a moderate risk aversion given by γ = 3, and a lower risk aversion coefficient of γ = 2. However, Chetty (2012) [7] claims that the central estimate of the relative risk aversion coefficient implied by labor is equal to 1 with an upper bound of 2. Kaplow (2003) concludes that there is no strong evidence that the relative risk aversion rises with income or wealth, but other studies like Stiglitz (1969) [23] and Paravisini et al. (2012) [19] argue that the relative risk aversion increases as wealth increases, and the absolute risk aversion decreases as wealth increases. Since the utility rises with respect to income for all risk aversion coef-ficients and without assuming a correlation between risk aversion and income, we will consider a risk aversion coefficient of 1, 2, 3 and 4 in this research with γ = 2 the average risk aversion coefficient.

Now let t = 1, . . . , t = T with the time horizon t = T at age 120, which we assume as the deterministic maximum age. Then for the lifetime consumption flow C1, . . . , CT we

find a lifetime utility given by

V = E " _T X t=1 βtu(Ct) # = T X t=1 ( βt t Y s=1 ps ! u(Ct) ) , (3.3)

with psthe survival rate at time s and β the time preference discount factor. We assume

a time preference discount factor of β = 0.96, as given by Peijnenburg et al. (2011) [20] and Horneff et al. (2006) [14] as regular used time preference discount factor.

(15)

Chapter 4

Redesign considering investment

risk

This chapter will investigate the possibility of allowing investment risk in the benefits phase for defined contribution agreements. Currently, a lot of research is done to this subject to improve the pensions for a defined contribution plan in the Dutch pension system. This research is also inspired on a recent research of Bakels et al. [2] (2014) concerning several discussions about defined contribution agreements.

4.1 Inflation risk

Inflation is the annual percentage increase in the general level of prices for goods and services. When inflation rises, you can buy less goods with the money you own. The inflation rate is given in Figure4.1.

Figure 4.1: The inflation rate for the last years since 1988, given by CBS (2014) [6].

One of the main disadvantages of the mandatory annuitization is the lack of inflation protection. Given the pension ambition and risk tolerance, not allowing for investment risk in the benefits phase makes the pension unnecessarily expensive. Since the total pension consists of the social pension AOW from the first pillar and the additional pen-sion from the second pillar, the total penpen-sion will partly follow the inflation since the AOW is yearly adjusted to the inflation rate. Figure4.2shows the effect of the realized inflation on the pension income.

(16)

Figure 4.2: Pension income as a percentage of the inflation-protected pension income (cost of living adjustment) for a male participant with initial income of AC 35,000 at retirement in year 1988. The price of the inflation protected annuity is not included. We can see that a pension income without inflation protection is significant lower than a pension income with inflation protection. Since the participants are mandated to pur-chase an annuity at retirement age, they are not protected against inflation risk. To protect a participant against inflation risk, one could invest the second pillar pension wealth so that investment returns increase the total pension income. However, that does not protect against longevity, so it could be important to partly keep the annuitization. It also may be optimal to delay annuitization if returns on investment in the future might exceed current returns or if annuities purchased later in life are priced more fa-vorably than those purchased earlier, but this won’t be part of the research here. An interesting question is now which part of the pension wealth has to be annuitized and which part can be used for investments.

Participants have to choose between a guaranteed fixed pension and a chance for a higher pension, but there is also a chance that the pension will be lower than the guaranteed fixed pension. Participants who are healthier are more inclined to purchase annuities than retirees who are less healthy because they do not expect to live that long. Since participants with higher income are assumed to live longer, they will prefer annuitization against the longevity risk.

4.2 Investment capital and returns

Each year the invested capital will be adjusted with investment returns and each year a part of this capital is used as an additional income above the consumption from the annuity. The part of the invested capital that is used for consumption increases with age to raise the consumption in order to maintain a higher pension income and to avoid having a high investment capital at moment of death. Off course, a high investment capital could be preferred in case of bequest motives. Another possibility is a higher additional consumption when a participant prefers a higher income at early retirement and a lower at later retirement. Those cases will not be considered in this research. There are different ways to decumulate the investment capital, Horneff et al. (2006) [14] mention a few possibilities like using a fixed percentage. Another possibility is using the 1/T rule according to the life expectancy at retirement age. The

(17)

investment-Defined Contribution — Erica Kuin 17

consumption ratio at year t for t = 1, . . . , T is then given by _{T −(t−1)}1 . This decumulation fraction framework is not constant but rises with age. Another framework where the de-cumulation fraction rises with age, is taking the remaining life expectancy of the retiree for each age into account. Let T (t) be the remaining life time at year t, the investment-consumption ratio at year t is then given by 1/E[T (t)] ≤ 100% according to the life expectancy at time t = 1, . . . , T .

For investments return we will consider the Dutch AEX returns in the period 1992 -2013 adjusted to factors like dividends, stock splits and new stock offerings. The adjusted closing price represents a more accurate reflection of a stock’s value, since distributions and new offerings can alter the closing price. Since returns are assumed to be log-normal distributed, we can make simulations with the mean and the variance of the Dutch AEX returns. Since we only calculate yearly income, we will calculate the yearly AEX returns. For AEXtthe AEX index at time t, we define Xt+1= AEX_AEXt+1_t . The parameters µ and σ

of the log-normal distribution ∼ ln N (µ, σ) can be calculated with E[X] = exp(µ +1₂σ2) and Var[X] = exp(σ2_{) − 1}

E[X]2, which yields

σ = s ln Var[X] E[X] + 1 , µ = ln(E[X]) − 1 2σ 2_. _(4.1)

Based on AEX data, we get a mean of E[X] = 108.80% and variance of Var[X] = 6.13%. Using those values and the equations in (4.1), we find the following mean and standard deviation:

Parameter µ σ

Value for AEX data 5.70% 23.41%

Table 4.1: The parameters of the log-normal distribution based on AEX data. With the parameters from Table4.1we can make 10,000 simulations for the log-normal distribution for each age of the retiree, this gives us a total average return of 8.8% (denoted as the expected scenario or 50%), which is consistent with the average return of the AEX index. In this research we will also consider the optimistic scenario (90% quantile) and the pessimistic scenario (10% quantile), the average return for all the scenario’s are given in Table4.2.

Scenario Average return Expected (50%) 8.8% Optimistic (90%) 15.5% Pessimistic (10%) 2.4%

Table 4.2: The average investment returns, based on AEX data, for the expected scenario (50%), the optimistic scenario (90%) and the pessimistic scenario (10%).

4.3 Model

In the research of Bakels et al. (2014) they show the cost of living adjustments of three different annuities: the nominal guaranteed annuity, the real guaranteed annuity and the expected real flat variable annuity (variable 50%). For the variable annuities they also showed the optimistic scenario (variable 90%, which mean the 90% quantile) and the pessimistic scenario (variable 10%, which means the 10% quantile). For the variable

(18)

annuity the cost of living adjustments turned out to be highly insecure, but the chance that this pension income will be higher than for a nominal or real guaranteed annuity is high. The nominal annuity will probably give higher benefits in the first years, but after those first years the variable annuity will probably give a higher income.

This section describes such a model for calculations for the redesign with investment risk. We will consider different intitial pension incomes, starting at the AOW-level of an yearly salary of AC 9,850 and ending at AC 200,000 per year. The income after redesign will consist of three parts: the AOW-income and the second pillar pension divided in annuity income and income from investments. The invested pension wealth is computed with the initial income at retirement (minus AOW) and with the annuity factors, this causes differences in investment capital for male participants and female participants since the annuity factors of female participants are higher than for male participants due to the higher survival rates. The AOW income is different for married retirees or single retirees, married retirees will receiveAC 9,850 per year in 2014 and single retirees will receiveAC 14,256 per year in 2014. In this research we will assume that the absolute minimum income is AC 9,850 per year and we will consider the case that the remaining pension wealth is used for annuitization and the case where we have to find the optimal investment/ annuitization fraction. Since the inflation rate is approximately 2% (see figure 4.1), we will assume a fixed inflation of 2%.

Let W be the accrued second pillar pension income, then for the investment fraction α with 0% ≤ α ≤ 100%, there will be α W used for investments and (1 − α)W will be used to purchase an annuity. The invested capital is calculated with

Z0 = ¨a65α W, (4.2)

with ¨a65 the annuity factor at retirement age. Each year a fraction λt ∈ [0, 1] of the

investment capital will be used for consumption. Now let Zt be the investment capital

at time t for t = 0, . . . , T − 1 and let θtbe the yearly stochastic return from investments.

The additional consumption ˆctfor t = 0, . . . , T − 1 can then be calculated with

Zt+1= (1 + θt) (Zt− ˆct), (4.3)

ˆ

ct= λtZt, (4.4)

such that with the first pillar income of AOW =AC 9,850, adjusted to the inflation rate of 2%, the total pension income at time t will be

Ct= AOW(1 + 2%)t+ (1 − α)W + ˆct. (4.5)

We can find the utility with the formula (3.1) and with the formula (3.3) we can find the total lifetime utility. We will calculate the lifetime annuity for α = 0% till α = 100% with small steps. We can compare the lifetime utility for several fractions α and determine which fraction for investment leads to an optimal lifetime utility.

4.4 Results

This section will present the results of the first redesign model. First we will look at the pension consumption for several income levels, gender and investment fractions.

(19)

Figure 4.3: Pension consumption as a percentage of the inflation-based pension repre-senting a male participant with an initial income of AC 35,000 per year who invested 20% of his second pillar pension income, for the current pension consumption and the pension consumption of the three investment scenarios mentioned in subsection 4.2.

Figure 4.3 shows the pension consumption for a male participant with initial income who invested 50%. Other results concerning the pension consumption are presented in Appendix A. The figures display the pension consumption of a male participant with an initial income of AC 35,000. Different scenario’s relative to the inflation-based pen-sion consumption are given, where 50% is the expected scenario, 90% is the optimistic scenario and 10% is the pessimistic scenario. We can see that if the investment fraction increases, the expected income is lower for early retirement and higher for later years but will drop significant after 20 years.

Income AC 15,000 AC 35,000 AC 60,000 AC 100,000 Investment fraction 100% 92% 83% 81% 79% 75% 94% 88% 85% 84% 50% 96% 92% 90% 90% 30% 98% 95% 94% 94% 15% 99% 98% 97% 97%

Table 4.3: The expected early retirement income relative to the current pension income without inflation protection for different investment fractions and initial incomes. For a male participant it will take approximately 5 years to exceed the current pension income without inflation protection. Table 4.3 presents the expected early retirement income relative to the current pension without inflation protection. We can see that this percentage decreases when the investment fraction increases. For instance, a male par-ticipant with initial incomeAC 35,000 can receive a fixed pension ofAC 25,150 in addition to the AOW income when he doesn’t invest at all, which isAC 35,000 in the first year of retirement. If this participant invests 75% of his second pillar pension income, he will receive a pension ofAC 20,950 in addition to the AOW income, which isAC 30,800 in the first year of retirement. We expect that this will exceed the current pension income, but in the pessimistic scenario this will never be the case.

(20)

Figure 4.4: The lifetime utility for a male participant with initial income AC 35,000 and risk aversion 2, for different investment fractions (of the second pillar pension capital). The expected scenario is represented by “50%”, “90%” represent the optimistic scenario, “10%” represent the pessimistic scenario and “Inflation” represents the lifetime utility for an inflation based pension consuption.

Gender Male Female

Risk aversion 2 3 4 2 3 4 Income AC 15,000 100% 100% 100% 100% 100% 100% AC 25,000 100% 96% 68% 100% 100% 84% AC 35,000 100% 80% 58% 100% 90% 68% AC 50,000 100% 72% 52% 100% 76% 58% AC 60,000 98% 68% 50% 100% 72% 56% AC 80,000 92% 64% 48% 92% 66% 52% AC 100,000 88% 62% 46% 88% 64% 48% AC 130,000 84% 60% 44% 84% 60% 46% AC 160,000 82% 58% 44% 82% 60% 46% AC 200,000 82% 58% 42% 80% 58% 44%

Table 4.4: Optimal investment fraction for gender, income and relative risk aversion coefficient at retirement age, based on the lifetime utility.

Since it is uncertain if participants will still be alive in later years, consumption over 1 year has more value than consumption over 10 years. Because of this, we will use utility instead of consumption. Lifetime utility aggregates the utilities of each year and lowers utility for later years with the survival rates and time preference discount factor. Figure

4.4 shows the lifetime utility for different investment fractions of a male participant with an initial pension income of AC 35,000 and a risk aversion coefficient of 2, we see that in that case we expect that it is optimal to invest all second pillar pension wealth. More results for different initial pension incomes, different risk aversion coefficients and different genders are given in AppendixB. We get that the expected utility raises with income, the higher the income, the less attractive it is to invest a larger fraction of the second pillar pension capital. If the risk aversion coefficient γ > 1, investing the second pillar pension income becomes less attractive. We can also see that the lifetime utility is higher for male participants then for female participants, but this is due to the mortality

(21)

rates. Table 4.4presents the expected optimal investment fraction of the second pillar pension capital, based on the lifetime utility. Moreover, if the risk aversion coefficient equals 1, it always seems to be optimal to invest all the second pillar pension capital.

(22)

Chapter 5

Redesign considering investment

risk and medical expenses

This chapter considers the issue of moving away from the requirement that all retirement wealth has to be annuitized to a requirement that individuals only annuitize a part of the retirement wealth. The remaining retirement wealth will be invested and can be used for several purposes, but we will only consider using the remaining retirement wealth for medical expenses and additional consumption.

5.1 Medical expenses

A research done by the RIVM (2013) [21] to medical expenses shows that the total health care costs will increase significant for higher ages, as shown in Figure5.1.

Figure 5.1: The average total yearly health care costs (AC) per age for males and females (RIVM, 2013).

Van der Veen (2011) [24] claims that the need for long-term health care will increase in the next few years due to the aging of the population. Also Peijnenburg et al. (2011) [20] claim that the health care costs have increased substantially over the last decades, in all Western countries. At the same time the variation in health care costs (health risk) increased as well. They claim that health risk should be taken into account for pension

(23)

policies and social securities in general, since it provide financial security to individuals and health risk is one of the major financial risks for the elderly.

Due to the accumulated costs for health care, requiring partial annuitization and us-ing the remainus-ing retirement wealth for medical expenses could be interestus-ing. We are becoming more responsible for our own health care. In order to use a part of the retire-ment capital for health care, we will be using ’partial’ annuitization. One possibility is using the remaining retirement wealth for purchasing health care. Another possibility is saving a buffer with the remaining retirement wealth and use it for medical expenses until the savings in the buffer is completely consumed.

The CBS (Centraal Bureau voor de Statistiek) has done some research to basic health care costs with respect to different income groups. From this research we can conclude that the basic health care costs increases with the age and decreases with income. Van der Horst and ter Rele (2013) [15] show that individuals in the Netherlands with higher income, have lower basic health care costs but have to pay more premium than individ-uals with lower income. In their research to additional health care insurance, Van Dijk et al. (2013) [11] show who needs this insurance and who can or can’t afford it. They concluded that females do more often take an additional health care insurance than males, and that “healthier” individuals do not take insurance because they don’t need it. There is also a disturbing trend that individuals with lower income don’t take an additional health care insurance, because they can’t afford the premium. So individuals with higher income consume more additional health care than individuals with lower income.

For the distribution of health care costs between different incomes, we will approximately assume the same distribution as Van der Horst and ter Rele (2013) show and keep this as a variable to be able to investigate other distributions. The total average health care costs for male isAC 16,185 per year and for femaleAC 21,191 per year. Participants with a low income of AC 20,000 have to pay approximately AC 2,000 while participant with a high income of AC 80,000 have to pay approximately AC 4,500. For the dispersion of medical expenses between different ages we use the medical expenses given by CBS and RIVM. This leads to the dispersion given in Figure5.1.

Age 65 70 75 80 85 90 95 100

Male 18% 25% 37% 62% 105% 170% 226% 282% Female 15% 21% 35% 65% 118% 182% 223% 264%

Table 5.1: The medical expenses for male and female per age, relative to the average medical expenses for male and female.

With the dispersion of medical expenses given in Figure 5.1, we get the medical ex-penses for male or female per income and per age. We will assume that 40% of the additional health costs have to be paid individual and the remaining part will stay an age-independent insurance. Note that we assume average medical expenses for cohorts by gender, age and income to be able to do some calculations. In practice, the medical expenses can be much higher or much lower.

(24)

5.2 Model

A model for optimal retirement consumption with health cost risk is given by Peijnen-burg et al. (2011) [20], which considers out-of-pocket medical expenses in early retire-ment. We will adjust this model for medical expenses in the whole retirement phase and we will invest the buffer which is saved for those expenses. This buffer will be invested as we did in the model in the previous chapter. So we expect (50% scenario) an average return of 8.8%, the optimistic scenario (90% scenario) will give an average return of 15.5% and the pessimistic scenario (10% scenario) will give an average return of 2.4%. From this buffer, a part will be used for additional consumption and a part will be used for medical expenses. As described in the previous section, the medical expenses depend on gender, income and age.

The model looks like the model in the previous chapter, but with medical expenses inserted. Let W be the accrued second pillar pension income. The part α with 0 ≤ α ≤ 1 of the pension income W is invested and the part (1 − α)W is used for the annuity. With the first pillar income of AOW = AC 9,850, adjusted to the inflation rate of 2%, the total pension income Ct for year t = 0, . . . , T − 1 is calculated with

Ct= AOW (1 + 2%)t+ (1 − α)W + ˆct. (5.1)

with ˆct the additional consumption. The invested capital is calculated with

Z0 = ¨a65α W, (5.2)

with ¨a65 the annuity factor at retirement age. Let ht be the medical expenses at time

t and λt the fraction of the investment at time t which will be used for additional

consumption. If θt is the investment return at time t, the additional consumption can

be found with

ˆ

ct= λt(Zt− ht), (5.3)

Zt+1= (1 + θt)(Zt− ˆct− ht), (5.4)

= (1 + θt)(1 − λt)(Zt− ht). (5.5)

If the buffer is fully consumed, the medical expenses will have to be paid with the remaining second pillar pension that is annuitized. So the total consumption of the second pillar pension income is now

Ct= AOW(1 + 2%)t+ (1 − α)W + ˆct+ min{Zt− ht; 0}. (5.6)

The consumption Ct depends on the fraction λt of the investment capital that is used

for consumption. Like in the previous chapter, this fraction can depend on the remain-ing lifetime T (t) at year t, such that λt = _{E[T (t)]}1 with λt ≤ 100%. Alternatively, this

can be determined with an fixed percentage or a consumption ratio in year t given by λt = _{T −(t−1)}1 with λt ≤ 100%. Assuming the life expectancy at age 65 is 40 years,

then the fraction λt for year t equals _41−t1 . Since medical expenses are much higher at

later age, it could be more preferable to save more investment capital for this at later age. We can find the utility for both scenario’s with the formula (3.1) and with the formula (3.3) we can find the total lifetime utility. We will calculate the lifetime annuity for α = 0% till α = 100% with small steps. We can compare the lifetime utility for several fractions α and determine which fraction for investment leads to an optimal lifetime utility.

(25)

5.3 Results

Since it is hard to pin down the level of medical expenses that should be paid for individ-ually, these results are just an indication of a possible redesign for defined contribution agreements. Again, we will first look at the pension consumption for several income levels, gender and investment fractions.

Figure 5.2: Consumption of different scenarios relative to an inflation based pension, for a male participant with an initial income of AC 35,000 and risk aversion 2, in case 50% is invested. The current pension income without medical expenses and with medical expenses are given.

Figure 5.2 shows the consumption of a male participant with an initial income of AC 35,000 and risk aversion 2 in case 50% of the second pillar pension income is invested. Other results with different investment fractions for a male participant with an initial income of AC 35,000 and risk aversion 2 are given in Appendix C. In all the figures the current pension income without medical expenses and with medical expenses, the expected scenario, the optimistic scenario and the pessimistic scenario are given. If the investment fraction increases, we expect a lower income at early retirement and higher income at later retirement. However, the pension income can also be much higher (optimistic scenario) or much lower (pessimistic scenario).

Income AC 15,000 AC 35,000 AC 60,000 AC 100,000 Investment fraction 15% 97% 93% 92% 92% 30% 94% 87% 85% 84% 50% 89% 78% 75% 73% 75% 84% 67% 62% 59% 100% 79% 56% 49% 45%

Table 5.2: The expected early retirement income relative to the current pension income without inflation protection for different investment fraction and initial income, for a male participant with risk aversion 2.

From Table 5.2 we can see that in case a participant invests a fraction of its pension capital, the pension income that we expect in early retirement can be lower than the current pension without medical expenses. If the investment fraction increases, the ex-pected pension income in early retirement decreases. Also if the income increases, the

(26)

expected pension income in early retirement relative to the current pension income de-creases. In Table5.3we can see how long it takes to exceed the current pension income. If the initial pension income is AC 15,000 and 15% is invested, the expected pension will be higher than the current pension income for the first 15 years, after 15 years the expected pension will be lower than the current pension income. For other invest-ment fractions and income groups, the number of years we expect it will take to exceed the current pension income increases when the investment fraction increases, and also increases when the initial pension income increases.

Income AC 15,000 AC 35,000 AC 60,000 AC 100,000 Investment fraction 15% - 9 10 11 30% 6 11 11 11 50% 8 11 12 12 75% 10 11 12 12 100% 10 12 12 12

Table 5.3: The number of years we expect it takes for the expected early retirement income to exceed the current pension income, for a male participant with a risk aversion coefficient of 2. When the initial income isAC 35,000 and the investment fraction is 15%, the expected pension will be higher than the current pension income for the first 15 years and lower after 15 years.

For instance, a male participant with initial income AC 35,000 can receive a pension income of AC 24,271 in his first year of retirement (second pillar pension income minus medical expenses) in addition to the AOW income when he doesn’t invest at all. The first 20 years of retirement he will receive at least this amount of pension income. If this participant invests 30% of his second pillar pension capital, we expect that he will receive an pension income of AC 20,542 in addition of the AOW income in the first year of retirement. So we expect that he will receiveAC 30,392 instead ofAC 34,121 and that it will take 11 years to exceed the current pension income when 30% is invested. However, in the optimistic scenario it will take 7 years to exceed the current pension income, but in the pessimistic scenario it will never exceed the current pension income.

As mentioned in the previous chapter, it is uncertain if participants will still be alive in later years, consumption over 1 year has more value than consumption over 10 years. Because of this, we will use utility instead of consumption. Lifetime utility sums the utility of each year and lowers utility for later years with the survival rates and time preference discount factor. Figure5.3 shows the lifetime utility for different investment fractions of a male participant with an initial pension income of AC 35,000 and a risk aversion coefficient of 2. More results for different initial pension income, different risk aversion coefficient and different gender are given in Appendix D. We get that the expected utility increases with income, however the expected utility drops for higher investment fractions when the income increases. The higher the income, the lower the investment fraction where the utility drops. Since the invested capital is fully consumed after 40 years, a high investment fraction results in a low consumption since only the lower annuity income is left. If the investment fraction is lower, the annuity income is higher and the medical expenses can be paid for. We also see that the utility increases when the risk aversion coefficient γ > 1 increases, but the utility drops more when

(27)

the investment fraction is higher. The utility is higher for male participants than for female participants, but the utility increases when the investment fraction increases for female participants while the utility decreases for male participants when the investment fraction decreases. However, also for female participants the utility drops for very high investment fractions.

Figure 5.3: The lifetime utility for a male participant with initial income AC 35,000 and risk aversion 2, for different investment fractions (of the second pillar pension capital). The expected scenario is given by 50%, 90% represent the optimistic scenario and 10% represent the pessimistic scenario.

(28)

Chapter 6

Conclusion and further research

In this chapter the conclusions are made for both models and compared to the conclu-sions of other researchers. Also, a few possibilities to optimize and improve the models are given.

6.1 Conclusion considering investment risk

This section will provide a conclusion for the redesign model considering investment risk. The main reason for the redesign was the lack of inflation protection for the cur-rent nominal pension income. Curcur-rently, it is compulsory to annuitize all second pillar pension wealth in order to protect against outliving the pension wealth and to protect against the tendency to consume the whole pension wealth in an early state. This re-design model considered the possibility of bearing investment risk in the benefits phase by comparing the consumption and utility for several investment fractions.

Brown and Nijman (2011) [5] claim there are two reasons to explain the lack of in-flation protection in the Netherlands. The first argument is the lack of bonds indexed to the Dutch inflation. The second reason is “money illusion” of individuals who are not willing to accept the lower initial payment in return for the long-term preservation of the costs of living adjustment, which is also mentioned by Bakels et al. (2014) [2]. There is an important argument for this “money illusion” of individuals, according to the results of the redesign model considering investment risk, the pension income in case of investments is expected to be lower than the current pension income in early re-tirement. However, after 5 years the pension income in case of investments is expected to exceed the current pension income. The higher the investment fraction, the lower we expect the pension income to be in early retirement, but the higher we expect the pension income to be in later retirement. If we consider the utilities of the different investment fractions, we would expect that participants with lower income would prefer a higher investment fraction, while participants with higher income would prefer a lower investment fraction, due to the low expected consumption in early retirement. If a par-ticipant is given the option between an assured fixed pension income (sometimes with an indexation increment percentage), or an variable pension income what is expected to exceed the assured fixed income within 5 years, the participant would probably choose for the assured pension income since the variable pension income is insecure and may also be way lower than the assured pension income. The volatility of the investment re-turn can also be very high, which gives a volatile pension income when this is not hedged.

(29)

Another issue is the longevity bonus. Normally, each participant receives a longevity bonus matching his age. But in case of investing a part of the pension capital and the participant dies, the remaining capital could accrue to the participants heirs. This means that the pension provider can not add this remaining capital as a longevity bonus to the pension capital of the other surviving participants. But even if the remaining cap-ital after death can be used as a longevity bonus for the surviving participants, the investment returns may be lower than the longevity bonus you need. So a negative risk premium on the participants pension could be necessary.

So the disadvantage of this redesigned benefits phase is the insecure pension income. Another disadvantage is the lack of protection against outliving assets. The part that is used for investments will be fully consumed at age 95, so the pension income will drop after 30 years. In case of bequest motives bearing investment risk could be interesting for participants with a lower life expectancy, since their investment capital is not fully consumed. Another advantage is the expected higher benefits in case of bearing invest-ment risk in the benefits phase. However, theoretically it may be optimal to invest a part of the pension wealth, but in practice participants may not be willing to choose for an insecure pension income.

6.2 Conclusion considering investment risk and medical

expenses

This section will provide a conclusion for the redesign model considering medical ex-penses. We expect that the need for long-term health care will increase in the next few years due to the longevity of the population. The medical expenses already have increased substantially over the last decades. Since it is not clear which part of the additional health care has to be paid individually, assumptions on medical expenses are made to give an indication of a possible redesign for defined contribution agreements concerning medical expenses.

Peijnenburg et al. (2011) [20] has done a similar research, but they only considered health costs shocks for a fixed number of years in early retirement. They claim that whether full annuitization remains optimal depends mainly on the amount of health cost risk early in retirement. But, research on medical expenses show that the need for health care is higher in later retirement than in early retirement. Also, Peijnenburg et al. (2011) did not investigate the possibility to combine bearing investment risk and medical expenses. However, from the results of this thesis we conclude that it is not optimal to invest at all in case medical expenses are considered, since the early retirement income is expected to be lower and it will take approximately 10 years to receive a pension income equal to the current pension income or higher than the current pension income. Saving a buffer for medical expenses can be more optimal, but only saving this for medical expenses in early retirement leads to a lower income at later retirement while the medical expenses in later retirement are higher.

6.3 Further research

The models may give some clear results, but both can be optimized. It is important to investigate the risk tolerance of participants, since theoretically it may seems optimal

(30)

to invest a part of the pension capital, but in practice participant may not be willing to choose for an insecure pension income. Also, almost all researchers only consider the standard pension for the participant, while there is also a pension for the spouses. In-dications for costs and risks for the spouses pension are not investigated or mentioned. Including the spouses pension would make it technical, administrative and communica-tive much more complicated.

For the redesign considering medical expenses the model can be optimized when more information about variation in medical expenses and shocks is available. Both models could also be further improved by taking the correlation between income and life ex-pectancy into account. Since income and life exex-pectancy are positively correlated, higher annuity factors for participants with a higher income would be appropriate in order to cover the longevity risk more.

(31)

Bibliography

[1] Actuarieel Genootschap (AG) (2012): Prognosetafel AG2012-2062, http://www.ag-ai.nl/view.php?action=view&Pagina Id=480

[2] Bakels, S., B.J. Bosboom, G. Dietvorst, A. Joseph, K. Kamminga, M. Meniar, T. Nijman, T. Steenkamp and B. Werker. (2014): “Een toekomstperspectief voor de premieovereenkomsten”, Netspar, Occasional paper.

[3] Blake, D., D. Wright and Y. Zhang (2008): “Optimal funding and investment strate-gies in defined contribution pension plans under Epstein-Zin utility”, Pension In-stitute, Discussion Paper PI-0808.

[4] Brown, J.R. (2007): “Rational and behavioral perspectives on the role of annuities in retirement planning”, National Bureau of Economic Research, Working Paper 13537.

[5] Brown, J. and T. Nijman (2011): “Opportunities for improving pension wealth decumulation in the Netherlands”, Netspar, Discussion Paper 01-008.

[6] Centraal Bureau voor de Statistiek (CBS) (2014): Consumentenprijzen; inflatie vanaf 1963,http://statline.cbs.nl

[7] Chetty, R. (2002): “A new method of estimating risk aversion”,

http://scholar.harvard.edu/files/chetty/files/curvature aer.pdf

[8] Cipra, T. (2014): “Pension demand and utility: the life annuity puzzle”, Finance a ´

uvˇer-Czech Journal of Economics and Finance, 64(3), 213 – 232.

[9] Davidoff, T., J.R. Brown and P.A. Diamond (2005): “Annuities and individual welfare”, The American Economic Review, 95(5), 1573 – 1590.

[10] De Nederlandsche Bank (DNB) (2014): Nominale rentetermijnstructuur (zero-coupon) d.d. 31-12-2013, http://www.statistics.dnb.nl/financiele-markten/rentes/index.jsp

[11] Dijk, M., A. Brabers, M. Reitsma and J. de Jong (2013): “Is een aanvullende verzekering nog wel voor iedereen weggelegd?”, Nivel,

http://www.nivel.nl/sites/default/files/bestanden/Factsheet-aanvullende-verzekering-2012.pdf.

[12] Finkelstein, A. and J. Poterba (2004): “Adverse selection in insurance markets: pol-icyholder evidence from the U.K. annuity market”, Journal of Political Economy, 112(1), 183 – 208.

(32)

[13] Horneff, W.J., R. Maurer, M.Z. Stamos (2006): “Life-Cycle asset allocation with annuity markets: Is longevity insurance a good deal?”, University of Michigan Re-tirement Research Center, Working paper 146.

[14] Horneff, W.J., R Maurer, O.S. Mitchell, I. Dus (2006): “Optimizing the retirement portfolio: Asset allocation, annuitization, and risk aversion”, National Bureau of Economic Research, Working Paper 12392.

[15] Horst, A. van der and H. ter Rele (2013): “De prijs van gelijke zorg”, Centraal Planbureau, Policy Brief 01.

[16] Kaplow, L. (2003): “The value of a statistical life and the coefficient of relative risk aversions”, National Bureau of Economic Research, Working Paper 9852.

[17] Klijnsma, J. (2014): “Risicodragende pensioenuitkeringen in beschikbare premieregelingen”, Ministerie van Sociale Zaken en Werkgelegenheid, http://www.rijksoverheid.nl/documenten-en- publicaties/kamerstukken/2014/06/25/risicodragende-pensioenuitkeringen-in-beschikbare-premieregelingen.html

[18] Ling, E. van and J. Koopmans (2014): “Onderzoek optimalisering over-gang van opbouw- naar uitkeringsfase en de inrichting daarvan in premie-en kapitaalovereenkomsten”, LCP Netherlands, Onderzoeksrapport Min-sterie SZW, http://www.rijksoverheid.nl/onderwerpen/pensioen/documenten-

en-publicaties/rapporten/2014/07/10/onderzoek-optimalisering-overgang- van-opbouw-naar-uitkeringsfase-en-de-inrichting-daarvan-in-premie-en-kapitaalovereenkomsten.html

[19] Paravisini, D., V. Rappoport and E. Ravina (2012): “Risk aver-sion and wealth: Evidence from person-to-person lending portfolios”,

http://www0.gsb.columbia.edu/faculty/eravina/ RRA Wealth.pdf.

[20] Peijnenburg, K., T. Nijman and B. Werker (2011): “Health cost risk and opti-mal retirement provision”, Netspar, Discussion Paper 05/2010-018 (revised version February 2011).

[21] Rijksinstituut voor Volksgezondheid en Milieu (RIVM) (2013): “Kosten van Ziekten in Nederland 2011”, versie 1.3,

http://www.kostenvanziekten.nl/systeem/kosten-van-ziekten-tool/

[22] Rijksoverheid (2012): “AOW-leeftijd stapsgewijs omhoog naar 66 jaar in 2019 en 67 jaar in 2023”, http://www.rijksoverheid.nl/documenten-en-publicaties/persberichten/2012/05/25/verhoging-aow.html

[23] Stiglitz, J.E. (1969): “The efffects of income, wealth, and capital gains taxation on risk-taking”, Quaterly Journal of Economics, 83(2), 263 – 283.

[24] Veen, R. van der (2011): “De toekomst van de langdurige zorg”, CVZ,

http://www.zorginstituutnederland.nl/binaries/content/documents/zinl- www/documenten/publicaties/overige-publicaties/1109-de-toekomst-van-de-langdurige-zorg/De+toekomst+van+de+langdurige+zorg.pdf

(33)

Appendix A

Results redesign investment risk

-consumption

Figure A.1: Invested 30% of the initial income.

Figure A.2: Invested 50% of the initial income 33

(34)

(35)

Appendix B

Results redesign investment risk

-lifetime utility

(36)

(37)

Appendix C

Results redesign investment risk

and medical expenses

-consumption

Figure C.1: Invested 15% of the initial income.

Figure C.2: Invested 30% of the initial income. 37

(38)

(39)

Appendix D

Results redesign investment risk

and medical expenses - lifetime

utility

(40)

Defined contribution : a redesigned benefits phase