Rate of return guarantees in dutch defined contribution pension plans.

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Rate of Return Guarantees in Dutch

Defined Contribution Pension Plans

Patrick Herfst

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: drs. Patrick Herfst Student nr: 5008939 / 10458832 Email: pherfst@gmail.com Date: November 13, 2013

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List of Tables

1.1 Age-dependent maximum pace table for Dutch DC schemes . . . 10

1.2 Parameters used in mathematical desciptions . . . 14

2.1 Historical mortgage rates with Vasicek fit . . . 20

2.2 Comparing the cash flow model to literature . . . 25

2.3 Probabilities of resignation from employment . . . 26

3.1 Chances and frequencies of exercising guarantees . . . 29

3.2 Annual fees for standard guarantee structure; 18-year-old male . . . 30

3.3 Added value of guarantee for the standard scheme . . . 32

3.4 Survival probabilities starting employment in 2013 . . . 33

4.1 Depot fees adjusted after resignation . . . 35

4.2 Pace table for Dutch DC schemes with retirement at age 67 . . . 37

4.3 Annual fees when guarantee is lowered after resignation . . . 39

5.1 Probabilities for contract expiration . . . 42

5.2 Annual fees including Expiration . . . 43

5.3 Depot fees adjusted after resignation . . . 45

6.1 Differences in annual fees in second testrun . . . 48

6.2 Survival probabilities on different assumptions . . . 51

List of Figures

1.1 Development of the difference in capital build-up . . . 15

2.1 DNB Term Structure per 31 December 2012 . . . 22

2.2 Expected development of the future Mortality Rates . . . 24

3.1 Histogram of mortgage return rates . . . 29

4.1 Effect of mortality correction ES-P2 on male mortality chances . . . 36

4.2 Cumulative contributions for different retirement ages . . . 38

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Chapter 1

Introduction

1.1 Research questions

Most traditional second pillar pension schemes are based on the so-called Defined Benefit principle: an employer offers a predetermined amount of retirement wealth to its employees, for which it pays an uncertain actuarial premium to a pension fund or an insurance company. The management of the risks, liabilities and investments associated with this type of scheme are the sole responsibility of this institution. The contribution by the members can be unpredictable and volatile, easily adding up higher than was expected, especially for large or expanding companies. This is why there is a noticeable worldwide shift towards so-called Defined Contribution pension schemes in the last few decades. In this kind of scheme the employer pays a predetermined contribution to each employee’s individual ac-count. After deducting risk premiums and costs, the remaining amount must be invested to accumulate enough value to provide for an adequate income replacement upon the retirement date of the employee. In Defined Contribution (DC) schemes the economic and demographic risks are thus transferred to the employee only. The cash amount that is available on the retirement date depends highly on the value of the investment portfolio, interest rates and actuarial factors at that specific moment. The retirement income can therefore fluctuate significantly in a relatively brief lapse of time due to the volatility of the economic market place and developments in the actuarial field, creating a large amount of insecurity for the employee. On top of this, the average plan member has an insufficient understanding about the financial markets and pension plans to manage his portfolio carefully and efficiently.

That this insecurity is an immediate risk for the employee was made clear during the recent finan-cial crisis. Equity and other investment values as well as interest rates dropped rapidly and remained low for a long period of time. The impact of these events is still with us today. DC scheme members with a high exposure to these instruments have seen their retirement provision fund decrease dramat-ically in recent years, and also being unable to get as much value for their money as before because of higher annuity rates.

In light of these events, the request for higher security and reliance on pension income in DC schemes has become more prominent in recent years. Pension providers are therefore exploring op-tions on how to make DC schemes more stable and thus more attractive to its participants. One way to provide for this is for an insurance company to guarantee a certain level of return on the original contributions. Such a guarantee would protect the plan member against most risks involving the downward potential of their risky investments. In some countries, like Switzerland, Chile and Sin-gapore, different forms of guarantees are already mandatory within their respective retirement systems. In this research project, we will look at DC pension plans with a minimum rate of return. We want to know what the associated expected future payouts of different guarantee setups will be, what the involved cost levels are and what the best way to charge for these costs is. The possibility of an

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em-ployee leaving the pension plan before the retirement date through mortality, resignation or contract expiration is considered herein. We will make suggestions about which setup is the most balanced between risk and security for both parties. This entails an analysis containing the optimal value of the guaranteed return and a profit sharing mechanism, among others.

In the case that a pension scheme offers such a guaranteed rate of return and the actual return is lower than this guaranteed value, the provider has to make up for the resulting difference from its own reserves. We will determine the related liabilities of giving a guarantee for the provider using a cashflow model for different demographic groups. Since there is an expected future cost for the provider, this feature will naturally come at a price for either the participants or its funder. There are several ways to charge for these costs, the more practical of which we will treat in detail in terms of plan design feasibility, risk management of the provider and the build-up of capital. These scenarios should also display how the guarantee takes its role in the possible profits and losses of the provider, the development of the capital and the market-consistent replacement rate for the resulting income after retirement.

In this analysis, we will explicitly take in account several demographic parameters that are gen-erally ignored in the current literature. Two of these are the mortality rate and the yearly rate of resignation of the participants, both limiting the responsibilities of the pension provider with respect to the guarantee. We will evaluate the relative impact these additional parameters have on the over-all outcomes. Next to that, we will provide a risk analysis regarding the most important parameter assumptions that are necessarily put into our model. This includes the risks taken on mortality and resignation, as well as others such as wage inflation.

Within the Dutch pension system most employers are legally obliged to participate in an industry-wide pension fund based scheme. Companies that do not have this obligation, but have offered a pension plan to its employees, will have transferred their liabilities to an insurer. Contracts between these two parties usually have a fixed end date, typical terms running in the order of five years. We will look at what happens if the employer decides not to prolong its contract after this expira-tion date. The insurer still has obligaexpira-tions towards the employee that leaves his accumulated capital behind, including those involving the given guarantee. We will look at the implications of this expi-ration and towards possible alteexpi-rations in the scheme at this occurrence, including the guarantee setup. One of the main distinctive properties of this research is that it makes its modeling specific to the Dutch pension system’s regulations. The current literature on this topic is very general on this point. Most importantly, contribution levels are usually assumed to be a constant percentage of the annual wage throughout the lifetime of a participant. The Dutch version of the DC system works very differ-ently, using an age-dependent pace table of net contributions - percentages of the pensionable salary. Applying this in our model should yield significantly different outcomes than in previous researches, since the focal point of contributions is much closer to the retirement date.

The general outline of this thesis is as follows. In the rest of this chapter we will introduce the main background to our topic. This includes the role of DC products in the Dutch pension system as well as an overview of the possibilities and mathematical description of an implemented guarantee component. Next to that, we will give a summary of the findings of earlier research projects in this area as found in peer-reviewed literature. Ch. 2 described the steps we took and the assumptions we made to design a cash flow model to determine future payments and their financing costs. This includes a comparison of our output agains the literature. Ch.3 analyses the numerical results our model provided from different scenarios and plan setups, starting from a standard pension scheme. Ch. 5 discusses the changes that occur when incorporating the possibility for contract expiration into the model and Ch. 6 presents sensitivity analyses of the most important input parameters. Ch. 7 concludes.

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1.2 The Dutch pension system

The existing literature on guaranteed defined contribution pension plans make a general model frame-work for annual contributions, pension design, government oversight and other factors crucial to the expected pension wealth on retirement. One of the main additions of this research project is the inclu-sion of the explicit framework of the Dutch peninclu-sion system in terms of scheme setup and legislation into the model calculations.

In this section we will give a short summary of the workings of the Dutch pension system and the implications it has on the design of our model in further chapters. For brevity we will only consider the old age provisions, not those made for either mortality or invalidity. This section is largely based on Bannink & De Vroom (2007) and Broeders & Ponds (2012).

Three pillars

The Dutch pension system consists of three pillars. The first pillar is a government-run pay-as-you-go provision, the AOW. Every living inhabitant above the pensionable age (currently 65) has a right to the benefit payments of the AOW, which are related to the minimum wage and therefore automatically indexed annually. This ensures a certain minimal level of retirement income, even for those people without other provisions.

The second pillar consists of arrangements between the social partners; these consist of collec-tively arranged pension benefits obtained by each individual’s active working career path. Employers and employees together pay for the retirement income of plan participants during their employment. These assets and liabilities must be placed outside of the company, at a pension fund or insurer, and be funded in advance to minimize the chance of pension rights not being covered. The increase of capital in these schemes contains a high level of collective solidarity among the participant’s demography.

For some employment sectors, participation in an overall pension scheme at a pension fund is mandatory. Although most companies fall outside of this obligation, more than 90% of Dutch employ-ees build up retirement income during their employment. The majority of this is in a Defined Benefit framework, since larger businesses in particular have DB pension schemes, but smaller companies have more and more been tending towards DC pension plans in recent years.

The third pillar constitutes individual insurances and lifelong annuities. Everyone can invest capi-tal into these products for old-age benefits, enjoying tax advantages for the build-up of capicapi-tal within certain fiscal boundaries to provide for retirement wealth.

This research project will focus solely on second pillar old-age pension plans. Defined contribution

In line with the worldwide trend, there is a trending shift in pension system towards a Defined Contribution mindset, the pace of which the current financial crisis has further accelerated. Its main attractions for employers are the relative ease with which costs can be anticipated and the deferrence of any further risks.

The basis for calculating the periodic contributions is the pension base. This is the pensionable salary minus a suitable threshold, which is deducted from pension build-up because of the future income by the state pension AOW. Within a DC pension plan, the contribution consists of a certain prefixed percentage of this pension base, depending on the age of the participant. The fiscal maximum pace table of DC pension plans in effect as of 2013 (Staatscourant 2013) is shown in Table 1.1. It simulates the increasing contributions that would be made to a maximal pension plan in a DB envi-ronment, creating a time-proportional capital build-up. This characteristic of DC plans is a distinctive

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Table 1.1

Age-dependent maximum pace table for Dutch DC schemes; old age pension only; percentages of pension base.

Age Percentage Age Percentage

15 - 19 5.5% 40 - 44 14.0%

20 - 24 6.3% 45 - 49 17.0%

25 - 29 7.7% 50 - 54 20.9%

30 - 34 9.4% 55 - 59 25.8%

35 - 39 11.4% 60 - 64 32.1%

property of the Dutch pension system compared to DC plans worldwide. All pension rights and claims must be financed a priori.

Most pension plans use investment funds and life-cycle principles to generate sufficiently high but reasonably secure returns, usually with some freedom of choice for the participant. Upon reaching the retirement age, the total available capital in the pension plan depot at that precise moment must be used to purchase a lifelong monthly annuity, within fiscal limits. The income that a retiring participant receives will therefore strongly depend on the economic climate at that specific moment as well as developing actuarial insights. Surrender or transfer of capital is not allowed.

Future developments

The current financial crisis has put significant stress on the current pension system in the Nether-lands. In recent years, numerous suggestions to lower the levels of risk and uncertainty for providers and participants and to stabilize the costs for employers have been made. The high level of solidarity principles, characteristic for this system, came under duress. In respond to this, the legislator has made thorough alterations to the system, which will take into effect in a short time. The common denominator is that the accrual of pension wealth will be limited in coming years.

The biggest social issue today is the raising of the retirement age. Ever since its introduction in 1956, the pensionable age for the AOW and most pension arrangements has been set at 65. However, due to increasing life expectancy and shifting demographics the costs of these future benefits have grown accordingly. It will now be gradually raised by a few months each year until it reaches 67 by the year 2024, and adjusted with increasing life expectancy after that date. This leads to lower an-nual accrual rates, since the build-up phase is extended and the payment phase shortened. This also affects DC arrangements; the maximum pace table of contributions has been altered as of 2013. In this project, we will use the most recent pace table as our starting point. We will look what the effect of this increase in pension age is in a seperate section.

Several other, smaller adjustments have been implemented and proposed. For example, in an ear-lier stage it was decided that administrative costs should no longer come out of these contributions, but should be charged to the employer on-top. Plans for the restriction of pension accrual to a cer-tain general maximum salary are being looked at, as well as making the entire framework on a real basis instead of the current nominal one. These and other developments fall outside the scope of this research project.

Apart from these and other adjustments to the Dutch pension system, the new international directive of Solvency II will come into effect by 2014. This proscribes a new way of looking at risks and risk management in general and dictates another method of maintaining sufficient buffers for unforeseen scenarios. Next to that, the upcoming IFRS 4.2 accounting guidelines should provide more transparency in the annual reports for investors and other outsiders.

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1.3 Guarantees in DC schemes

A provider that offers a guaranteed rate of return has to make several important decisions about the setup of this product before it can be implemented and put onto the market. In this section we will present an overview about what kind of possibilities there are regarding the setup of a guaranteed rate of return in a DC pension scheme. First we will look at the possibilities regarding the actual design of the guaranteed rate. Then we will discuss the most common and practical ways to finance the implied costs related to such a product. Several outcomes from other research projects into this subject matter will be given.

Rate of return

In this section we will look at the most important options regarding the primary design of a rate of return guarantee in a pension scheme. There are much more factors that can be considered when constructing such a product besides the ones mentioned here, but we have restricted ourselves to the ones that are most relevant to this research project. The paragraphs below describe the basic design issues of a predetermined setup of a guaranteed DC scheme in terms of when payments will occur and how large they will be. For more extensive information on this subject, see for instance Turner (2001). For a detailed discussion on the optimal design of a guarantee in DC funds, see for instance Deelstra, Grasselli & Koehl (2002).

Height of guarantee

The most noticeable part of such a product design is the height of the guarantee. The most trans-parent and easiest to administer form would be a fixed return rate that never changes in time. With every application period, the present capital would increase with at least this minimal rate - if actual returns are lower, the provider must make up for the difference from its own reserves.

A guaranteed return rate of 0% would construct a purely nominal guarantee of the contributions made in the associated timeframe. Higher percentages would warrant minimal investment returns for every application period. These can serve as inflation protection for the built up total capital.

The guaranteed return rate in each period can also be constructed to depend on certain external circumstances. For instance, it can be related to either the period’s price or wage inflation. It may also be dependent on some other freely observable capital market index, such as a benchmark investment portfolio or index. This makes the estimation and management of the financial risks involved much more difficult.

Actual return or benchmark return

The guaranteed rate of return may be the actual rate of return received on each individual’s pen-sion capital depot or it may instead be related to a predetermined benchmark portfolio. This latter choice would lay at least some responsibility for investment choices with the participant, if the pension plan offers such involvement. Since only a limited amount of underperformance would be compensated by the given guarantee, the risks created by moral hazard issues would be somewhat limited. In this research project, we will only focus on the realized return of an actual investment portfolio.

Application period

The guarantee application period is a predetermined and fixed period in time, at the end of which the given guarantee might be exercised in case of underperformance. Two varieties are particularly common in the insurance field: the maturity guarantee and the multi-period guarantee.

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the termination date is considered. In pension contracts, this would be the participant’s retirement date and will be the only point in time that the guarantee can be exercised. This guarantee covers the cumulative compounded return on the periodic contributions, since at maturity the actual available capital is compared to a fictitious depot which increased with the guaranteed return rate.

The multi-period guarantee has regular points in time at which the guarantee might be exersized, in practice typically ones per (calendar) year. After every application period the corresponding guar-antee can be called upon independently, after which the next application period with its corresponding guarantee steps in. This would in effect generate a series of successive guarantees until the retirement date of the participant.

The maturity guarantee is a weaker kind of guarantee than the multi-period guarantee. It is much safer for the guarantee provider, since incidental underperformance in one period can be compensated by better returns in other periods. With a multi-period guarantee, payments must be made more of-ten. Throughout this research project, we will consider both of these types of guarantee simultaneously. Profit sharing

Some guaranteed return plans offer only a so-called point guarantee: a fixed percentage with which the capital depot will grow. When the actual return exceeds the guarantee, the surplus goes to the provider or the employer (or both). The investment portfolio in such a scheme basically has the struc-ture of a fixed rate savings account.

In contrast, most providers offer some sort of risk sharing mechanism. In this kind of scheme, the surplus of the return is shared with its participants. Often, a part of the excess return is withheld to help finance the guarantee. The provider can also choose to place a cap on the return rate or take a percentage of the surplus. In this way, both the participant and the provider will benefit from higher yielding investment strategies.

Gross or net rate of return

The rate of return can either be guaranteed before or after several types of expenses. Some ex-amples of expenses that a provider can charge are purchasing and selling costs, management fees and administration costs. In this report we will neglect cost levels, since they are added on top of the required defined contribution premium and will be paid for by the employer separately. All provided guarantees will therefore be net return values. The only exceptions are the premiums that finance the guarantee, which will be discussed later on.

Financing of the return

A given guarantee will pay out a certain amount of cash in events of underperformance, but there is no possibility of contrary cashflows. The chance of coming “in the money” per period may be small depending on the exact design of the guarantee, but nevertheless it will have a strictly negative fair value for the issuer. A provider will usually charge an additional premium for these associated expected future costs. These additional payments must be financed by one or more of the parties concerned. There are several ways of doing this, the most relevant of which are listed below.

Annual risk premium

At the beginning of every new period, the provider can charge an additional risk premium to the employer on top of the DC premium for the possibility of the return rates falling short. The employer can then choose whether or not to defer these costs to its employees. The provider must make sure that these additional premiums will be sufficient to cover the liabilities should the guarantee be exercised.

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Instead of charging an additional premium, the required amount may also be taken from the annual contribution. In this case, the employee automatically pays for the return rate protection himself. This comes at the direct expense of the available capital and thus of the resulting wealth after retirement. The premiums must be made explicit to all parties and must therefore be determined by the provider a priori. They must be independent of the prevailing economic climate at that moment, but rather be sufficient for all scenarios. They can for instance be based on the height of the existing capital depot value and vary according to the given demographics of the employees.

Percentage of capital depot

Instead of charging extra premiums for the possible return rate risks, the provider can deduct the necessary amount from the already present capital depots of existing employees. This keeps the premiums for both employer and employees in a purer DC mindset, but as a consequence lowers the build-up of wealth available after retirement.

Another problem arises with this option. Because of Dutch legislation regarding costs in DC schemes, this type of deduction may not vary for different demographic groups. Therefore the to-tal cost of the guarantee must be turned over as a percentage of the built-up capito-tal depot value, which creates a numerical challenge for the provider.

Risk sharing

Instead of charging either the employer or the participants, the provider can make reservations about the risk sharing mechanism in the pension scheme. One possible way of doing this is by not sharing the potential surplus return completely, but by detaining a fixed percentage. This way a re-serve can be made to cover future expenses in case of insufficient returns.

Another possibility is setting a cap on the surplus return. Above this value, no risk sharing is done with the participant. The guarantee would function as a collar strategy: a long put option and a short call option. The money reserved in case of especially high returns provides a buffer against the liability of losses in case of low returns.

1.4 Mathematical desciption

In the previous sections, we have given an overview of the most important possibilities in the design of a DC pension plan containing a return rate guarantee. In this section we will provide a mathematical description of two different types of such schemes that can be implemented by Dutch pension insurers. We will regard only the “savings account” based on annual payments and the investment returns in this research project. Optional risk insurances like the spouse and orphan pensions, a waiver of pre-mium in case of disability (PVI) and the Netherlands specific “ANW-hiatus” insurance can be offered, but not looked into in this thesis.

As stated above, the annual contributions (usually also payable per month or per quarter) are based on an age-dependent pace table of net contributions - percentages of the pensionable salary (minding an offset) according to the Dutch governmental regulations. Contract-specific costs are added on-top. We assume here that all contributions are payed at the beginning of every year and ignore further cost levels.

Throughout this research project, we will focus on two different guarantee setups that may be used in Dutch insured pension contracts. The first is a maturity guarantee which, as described in Section 1.3, will only compensate for underperformance at the retirement date. The member receives

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Table 1.2

Parameters used in the mathematical desciptions of both the maturity and annual guarantees. At Built-up capital depot at time t

ci Pension contribution at time i

f Percentage of profit sharing of extra return gt Guaranteed annual return on time t

Gt Cost of the guarantee at time t

k Cost level of guarantee within profit sharing rt Realized return rate of past year at time t

r_t∗ Profit sharing return rate at time t e

rt Return rate in scheme without guarantee

x Current age of the participant ω Retirement age of pension plan

his capital depot to fund a life-long pension at this point in time, while the maturity guarantee states that this depot will not be lower than the paid contributions, compounded with an annual return of gt.

Typically, the insurer offers some degree of freedom for the individual to choose in which insurer-based investment funds his savings premium will be invested.

Should the member leave service with the contracted company, or if the whole contract has ex-pired before the retirement age (with standard contract lengths in the order of five years) then the guarantee on return upon retirement can be lowered for the entire duration of the participation. In an alternative view, this can be seen as a contract until the retirement age, but with the possibility to cease payments after every five years.

Some formulas for the cash flow development can be deduced from the description of the guarantee setup above. We define the relevant parameters we need for this formulation in Table 1.2 shown below. This also shows the parameters we will need later in this paragraph. Using these parameters, the capital At that will be available in this scheme on the retirement date ω can be written as follows:

Aω−x = max (_ω−x−1 X i=0 ci ω−x Y j=i+1 (1 + rj); ω−x−1 X i=0 ci ω−x Y j=i+1 (1 + gω−x) ) (1.1) = max (_ω−x−1 X i=0 ci ω−x Y j=i+1 (1 + rj); ω−x−1 X i=0 ci· (1 + gω−x)ω−x−i )

which is the maximum value of the actual built-up capital and the minimal guaranteed amount. The costs to the insurer upon retirement that result from cumulative underperformance is the difference between these two values, which can be expressed as follows:

Gω−x = max (_ω−x−1 X i=0 ci· n (1 + gω−x)ω−x−i− ω−x Y j=i+1 (1 + rj) o ; 0 ) (1.2) The second type we will discuss here is an annual guarantee. These kind of schemes usually offer far less freedom of choice for the individual participants than the maturity guarantee plans. This is a way to keep more control over the investments and constrain moral hazard issues of the participant. The savings premium is set aside and invested by the insurer, the participant’s capital gaining periodical additions of the fixed point guaranteed return on premium gt. A profit sharing scheme is in place

ultimo each year; the actual surplus return above the guarantee is shared into the capital depot. For returns below the guarantee the insurer has to make up the difference for capital buildup immediately.

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Not the whole of the surplus might be shared with the participants. Later on we will look at the impact of two possible ways to reserve part of the excess return to see the resulting costs of the guarantee setup (partly) financed. One of those ways is to set an offset margin k above the guarantee value: in case the annual return is greater than the guarantee value, this part will go directly to the insurer’s reserves. Another way to limit the profit sharing is to only share a fixed portion f of the excess return with the participant.

Using the description above, we can formulate a recursive relation for the capital that is available within the depot after every application period as follows:

At= n At−1+ ct−1 on 1 + gt+ f · max(rt− gt− k; 0) o (1.3) =nAt−1+ ct−1 o ·n1 + r_t∗o

where r_t∗is the return rate that is actually shared with the participant. In the case that the full excess return is shared (f = 1 and k = 0) this definition reduces to the simple form:

r_t∗= max(rt; gt) (1.4)

The resulting capital that will be available within the depot upon the retirement date is the sum of the contributions compounded with these actual return rates:

Aω−x = ω−x−1

X

i=0

ci· (1 + r∗i)ω−x−i (1.5)

The cost of the guarantee is the payout that results from the guarantee setup relative to the most similar pension plan without a guaranteed return rate. In our specific case, the difference lies within the definition of the annual return rate. We compare our pension scheme with a guarantee to a scheme without the return offset k, since its only use is to finance the guarantee. However, the comparison scheme includes the profit sharing factor f , since this is also commonly used in other pension schemes to increases profitability. The actual returnr_et in such a scheme would equal:

e rt=

(

rt if rt≤ gt

gt+ f · (rt− gt) = f · rt+ (1 − f ) · gt if rt> gt

The annual cost resulting from such a scheme would therefore be the difference in capital build-up between the two plans as caused by the difference in the two return rates r∗_t −_ert, the typical

development of which can be seen in Figure 1.1.

Figure 1.1

The development of the difference in capital build-up with increasing return rate.

-1,0% -0,5% 0,0% 0,5% 1,0% 1,5% 2,0% 2,5% 3,0% 3,5% g g+k ~ ↑ r*-r r →

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Gt= n At−1+ ct−1 o ·nr_t∗−_ert o (1.6) with r∗_t −_ert=      gt− rt if rt< gt −f · (r_t− g_t) if gt≤ rt< gt+ k −f · k if rt≥ gt+ k

We derived general formulas to calculate the capital after each application period and upon the retirement date and the costs for the insurer due to the obligations resulting from the guarantee setup. To evaluate the expected future costs at the current time, we will need to determine the so-called present value of these cash flows by actuarial discounting:

P V (G) = ∞ X i=0 v(t) ·tpx· Gt= ∞ X i=0 tpx (1 + rt)t · G_t (1.7)

where the variable rt stands for the current term structure zero rate at maturity t. With the formulas

derived in this section, we can design and build a cash flow model that will price the future benefit payments. We will look into the construction of this model in detail in the next chapter.

1.5 Results of earlier research

One of the first theoretical analyses of annual or multi-period guarantees within insurance policies is made by Hipp (1996). He considers an equity-linked life insurance product containing a minimal guar-anteed amount with profit sharing that can either be the annual return from the index or the return at expiration, minding the total accumulated premiums. These payoffs are identical to a European call option that can be priced under a Black-Scholes type economy. Assuming a deterministic and constant interest rate he is able to obtain closed-form and intuitive solutions for the market consistent valuation of these two types of guarantee schemes. Also minding the size of the delta- and vega-risks, he finds that a forward cliquet option can be used to hedge the interest rate risk for this product.

Persson and Aase (1997) extend these results by taking stochastically changing interest rates ac-cording to the Vasicek model, and converting to risk adjusted probability measures. Miltersen and Persson (1999) use a Heath-Jarrow-Morton framework to price various types of guarantees, both on stock market return and interest rate return, and extend their formulas to maturity guarantees for two application periods. Some numerical results from the resulting theoretical closed-form integrals of both papers show that the market values of such guarantees can be quite large, and increase with expiration horizon. Logically, the maturity guarantees have much less value than the annual guarantees. Finally, Lindset (2003) is able to obtain closed-form solutions under stochastic interest rates for an arbitrary number of periods, which have the downside of demanding a high degree of calculations.

With these newly found formulas, various studies added one or more adjustments to the most gen-eral description of the insurance contract. For instance, the guarantee of Yang, Yueh & Tang (2007) is dependent on the n-year spot rate instead of investment returns in a Magrabe option pricing model. Numerical results for their formulation shows that the volatility of the stock market is replaced with interest risk, and the guarantee is therefore more stable in price. Nielsen, Sandmann & Schl¨ogl (2011) analyse the implied costs of several types of guarantee schemes and what additional contributions are necessary for the financing thereof. They find that an investment guarantee is more expensive than a contribution guarantee, which in turn is more expensive than surplus participation. They also explicitly allow for early termination of the contract, but find that this has very little impact on the guaranteed benefits. In addition, several studies focussed on the investment startegies for pension funds that offer minimum return guarantees.

Deelstra, Grasselli & Koehl (2003) analyse the effect of the minimal realized return on the optimal asset mix, finding that less risky investments are necessary and therefore the expected return will be

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lowered. Di Giacinto, Gozzi & Federico (2008) try to maximize the expected utility from the fund wealth and analyse the effect of solvency constrains.

Bacinello & Ortu (1996) price a minimum guarantee provision for unit-linked life insurance policies in which the benefits are linked to a benchmark portfolio. Payments can be chosen to be reinvested or set apart. They model the term structure in a Cox-Ingersoll-Ross framework. For the scheme without reinvestment they obtain closed form solutions for the single premium, while a simulation approach is given for the reinvestment case. Although Bacinello especially remained active in the field of guar-anteed minimal returns, later works focus mainly on the structure and impact of surrender options in unit-linked pension plans, which is not permitted in Dutch pension plans.

More recent studies of guarantee schemes in insurance products are using new insights in mathe-matical finance and increased computer power to make more complicated models where no closed-form solutions are possible. This has the added advantage that the results are more generally applicable, as fewer assumptions have to be made. Therefore the attention has shifted from theoretical formulations to more computational results in recent years. One example is Bakken, Lindset and Olson (2006), who make a Monte-Carlo approach to the model of Lindset (2003) and can thereby reduce growth in computational efforts.

Scheuenstuhl et al (2011) performed an analysis on the effect of introducing an investment guar-antee to DC savings plans, focusing mainly on the guarguar-antee exercized only at the retirement age. They model a standard DC pension plan with a contribution that is a constant percentage of the salary, neglecting administrative costs. The participant will remain in the plan for forty years, without lapse or mortality. They assume an average salary development consisting of both inflation and career dynamics. Investment is possible in two asset classes: a diversified equity portfolio, which follows an MSCI world equity index, and government bonds with different maturities; returns are simulated using the risklab ”Economic Scenario Generator” and averaged over 10,000 scenarios. A life-cycle strategy is implemented by linearly reducing the equity position over the last ten years.

They analyse several different types of guarantee: the nominal sum of contributions, a real sum including the inflation rate, fixed percentages at 2% and 4% annually, and a floating rate dependent on the current interest rate. These are all valid at the retirement age only; besides these, an ongoing nominal guarantee is also considered. The fair value of these guarantees is calculated based on a risk neutral framework, where the interest rate and inflation are modeled by a Hull-White model and the equity returns by an extended Black-Scholes model.

They find that a nominal guarantee on contributions, only exercised on the retirement date, can be quite inexpensive to maintain and can be simply implemented by a small deduction of the capital depot or the annual premium. The ongoing version has a significantly higher average cost level and is therefore discouraged. When the guaranteed return is set higher, the annual cost level rises sharply as well, as should be expected. A connection to inflation or the floating rate is more volatile than a fixed percentage, which drives up the cost level.

Antol´ın et al (2011) analyze the costs and benefits of different types of minimum return guaran-tees. They valuate the guarantee claims as a financial derivative (put option) in a stochastic financial market model and focus on several ways of financing these expected claims. The term structure is fixed and assumed from historical averaging, with fixed equity volatility. For forty years, the contributions are 10% of the salary with a career and inflation linked wage growth. The same life-cycle investment strategy as Scheuenstuhl et al (2011) is implemented.

They look at the same types of guarantee as Scheuenstuhl et al (2011) at the retirement date and investigate different ways of financing the associated costs: as a percentage of net asset value, of annual contributions or of the possible investment surplus. They find that nominal guarantees may cost as

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little as 0.1% of the net asset value per year or a relatively small addition to the annual contribution. Prices rise as guarantees become either higher or more volatile, since they will be exercised more often. They add that the cost of the guarantee also gets higher the shorter the contribution period and the riskier the investment strategy. They find that the best option to pay for the given guarantee might be to deduct from the built-up capital depot: besides the low annual payment, it is indicated that the participant has a higher level of wealth upon retirement.

Both these papers proceed upon the work of Pennacchi (1999) and Pennacchi (2002), which based itself on research into equity-linked life insurance products like Hipp (1996). A contingent claim anal-ysis and martingale pricing techniques are used to valuate two types of minimum return guarantee: one fixed in time, one set relative to a performance index. It values the guarantee at each point in time as a standard put option in a Black-Scholes framework, with a constant real interest rate. It is proven that, if the contribution rate is constant, the value of the total guarantee grows exponentially in time.

Some numerical results are given, assuming the interest rate follows a Vasicek model curve and the asset return and wage increase stochastically including positively correlated Wiener processes. Mortality was taken into account in this calculation. It is shown that the value of the guarantee rises significantly at lower wage levels and drops at increasing income due to the nature of the put option comparison.

Turner & Rajnes (2009) analyse pension funds in stock market crashes and find that the guarantee schemes that seem best able to withstand extreme conditions, such as the situation of the last few years, are those with low minimum returns. Cost levels in schemes with guaranteed returns above the risk-free rate tend to grow very high in extreme scenarios. Grande & Visco (2011) show that the re-sulting losses in these circumstances can best be absorbed by governmental bodies instead of insurers and recouped by employers and employees over longer time intervals.

A noticeable trend within this current literature is a shift from (increasingly refined) mathemati-cal environments like the Black-Scholes or Heath-Jarrow-Morton frameworks to more general models which require more computing power. In this thesis, we will continu this trend by designing a cash flow model of expected payments and thereby needing fewer assumptions. We do expect to retrieve some of the striking patterns within the most common conclusions of these earlier projects. These include the relatively large market value of these guarantees, especially the multi-period version, and the sharp increases in cost level with rising guarantee height and shorter contribution period.

All of the papers above which provide numerical results of their approaches have assumed a stan-dard asset mix containing both stock and bonds, mostly generating return rates on a fixed index. We will offer a very different way of determining average return rates, which gives rise to far more stable annual growth. As was occasionally done before, we will determine the height of several types of financing methods for benefit payments.

This project also adds to the literature by focussing on the customs and regulations of the Dutch pension system and implementing this in our modelling, where previous research has remained very general on this point. The most prominent deviation is the height of the annual contributions, which are assumed to come from a pension base and increase with age, instead of a fixed percentage of the salary. Another distinguishing feature of this thesis is the possibility of the participant of exiting the pension plan by either mortality or resignation from his current employer. Inactivity may also follow from contract expiration, where the employer shifts his future pension liabilities to another insurer. We will look into these options in detail later on.

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Chapter 2

Designing a Cash Flow Model

This research project performs a systematic analysis on the effects of several return rate guarantees to DC savings plans and their effect on the retirement income situation. This analysis focuses on a standard DC pension plan according to Dutch regulations. The specific characteristics of such a pen-sion plan lie mostly in the level of contributions and several prohibitions regarding possible usages of the pension capital. For instance, the options of lapse or surrender of available capital have not been taken into account, since they are explicitly forbidden in the Dutch pension system.

As stated earlier, the expected future cost of the guarantee payments in a given pension plan is strictly positive. The issuer will usually defer these costs to the plan participant or the partaking company. It is therefore important that we can estimate a fair value for these costs, so that the issuer will be sufficiently protected against the downside potential of this setup, but maintains its competitiveness relative to other pension providers. A large part of our analysis will therefore focus on the fair pricing of the given guarantees.

2.1 Cash flow model

In general, this fair pricing entails finding the appropriate cost levels at which the present value of the expected future guarantee fees equal the present value of the expected future guarantee payoffs in a risk neutral framework. For this purpose, a model has been implemented that generates these expected future cash flows, given a set of input parameters.

To keep this model manageable, a number of assumptions have been made that simplify the practical circumstances of insurance portfolios. We assume that the timing of all payments is on an annual basis. This means that the contract between employer and pension provider starts at the beginning of the year (which in The Netherlands is the case for about 70% of all insured pension contracts) and that all contributions for each year are paid for a priori, as is proscribed. All participants start and may possibly end their employment on this same date, which is also their birthday, and work full-time throughout their career.

2.2 Wage development and contribution

For simplicity, we have assumed the same average wage process for all participants with a typical initial salary ofe 20,000 for an 18-year-old employee in 2013. This develops in time with a fixed wage inflation of 3% per annum. For the individual career development we use the so-called “3/2/1/0” approximation. This implies additional wage increases of 3% per year until age 35, 2% until age 45, 1% until age 55, and 0% until retirement. This is the same approximation of career dynamics that was used to construct the pace table for age-dependent contribution levels. Both factors are assumed to be independent of the financial market situation.

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From this wage process, the future employer’s contributions to the pension plan can be deduced. To allow for the AOW pillar I payments after retirement and to make sure the pillar II payments are congruous with the active period, an offset is applied. The minimal offset is e 13,227 in 2013 and increases every year with the inflation rate, which is fixed here at 2%. The annual contributions follow from applying the appropriate fraction from the pace table of Table 1.1. This shows the maximal allowed contribution percentages of Dutch DC plans for the purpose of acquiring an old-age pension only; we will not take any provisions for surviving partners or children into account. We use the minimal offset and the maximal pace table in all following calculations. The true height of a pension scheme’s pace table must be proportional to this maximum pace table. Since we will only look at relative ratios in this project, we can generalise to all tables.

2.3 Mortgage return rates

The resulting annual contributions are put into a separate savings account, ignoring secondary cost levels, where it will increase due to asset return. As stated before, the return rate with which the account will grow is the average interest rate for the outstanding mortgage portfolio of that year.

To model the future mortgage return rates, we first construct a fictitious pre-existing portfolio. This consists of unitary quantities of mortgages with a 30-year horizon without amortization pay-ments, but with full repayment upon the termination date. This is the most popular type of mortgage in the Netherlands at the moment. There is new legislation underway that limits these kinds of loans fiscally, favoring annuity mortgages instead, but we will ignore this for the moment.

Dating back for eleven years, these mortgage loans carry interest rates equal to that year’s average mortgage loan interest rate, which are displayed in Table 2.1. The difference between 15-year and 20-year fixed periods is on average 0.2%, which has been implemented in the model. The interest rate does not remain fixed for the entire maturity; every past year’s production consists of equal parts of policies with a 15-year and a 20-year horizon, after which the interest rate becomes variable in nature. Based on this portfolio, the expected return for the first year equals circa 4.6%.

For mortgages that will be closed in the coming years, we maintain the structure of 30-year horizons with equal parts of 15-year and 20-year fixed interest rate periods. To model future average mortgages return rates, we separate the mortgage rate into two parts: the general term structure for the given maturity and a spread on top of this:

rmortgage= sT + rspread (2.1)

where sT is the one-year spot rate at maturity T resulting from the term structure. The modeling of

future term structures will be discussed further below. To generate future spreads, we assume a Vasicek model underlying this process. This model was first described in Vasicek (1977) and constitutes a

one-Table 2.1

Historical mortgage rates of the past eleven years, with Vasicek parameter fit. Year r15 r20 Year r15 r20 Year r15 r20

2002 5.27% 5.47% 2006 4.83% 5.05% 2010 4.91% 5.12% 2003 5.06% 5.24% 2007 4.55% 4.78% 2011 4.94% 5.14% 2004 4.65% 4.88% 2008 4.78% 4.96% 2012 4.81% 5.00% 2005 4.32% 4.50% 2009 5.15% 5.33% Vasicek parameters: 15 year: a15= 0.55 Θ15= 0.8% σ15= 0.6% 20 year: a20= 0.55 Θ20= 1.0% σ20= 0.7%

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factor mean-reverting short rate model, as it describes interest rate movements around a fixed mean and driven by only one source of market risk. The model specifies that the instantaneous interest rate follows the stochastic differential equation:

drt= a · (ΘM − rt)dt + σ · dWt (2.2)

where rt represents the interest rate spread, ΘM is the long-term mean of the spread with maturity

M , a is the speed of mean-reverting and σ is the volatility. The factor dWt is a Wiener process under

a risk neutral framework, which allows randomness into the system.

To estimate the parameters mentioned above, we make a regression analysis using the ANOVA procedure onto the discreet version of Eq. 2.2:

r_t+1spread= (1 − a)rspread_t + aΘM + σt (2.3)

The resulting parameter values are also shown in Table 2.1. We see that the mean spread level ΘM

is around 0.8% for the 15-year fixed interest rate and about 1.0% for the 20-year horizon. However, there is a high degree of uncertainty in these parameters because of the variation in input values and the small number of data points. The standard deviations are also quite large relative to the intrinsic values. In a later section we will look at the influence deviations in these parameters have on the overall outcomes.

With the stochastic mortgage rate spread on top of the stochastic term structure, we generate fu-ture interest rates against which new loans are closed for fufu-ture periods. Together with the pre-existing portfolio these give rise to incoming interest payments, which are all summed and then divided by the still remaining total mortgage debt to produce that period’s return rate. Since this return rate is an average yield of thirty years’ portfolio, and one year’s results are smeared out over longer timeframes, a short period of low interest rates will not have a large enough effect to trigger guarantee payments. However, should there be several years of underperformance, it will also take a relatively long time to recover to sufficient levels.

We now have the expected future return rates, which are fed back back into our cash flow model. For every year the return drops below the guaranteed value, the annual guarantee immediately pays out the difference in the savings account in accordance to Eq. 1.3. For the maturity guarantee this happens only at the retirement date if the cumulative amount was less than the guaranteed value, as in Eq. 1.1. The cashflows of the pension liabilities are assumed to be fully covered by the mortgage portfolio, meaning that there is always at least as much incoming payments from outstanding mortgage loans as there are outgoing payments towards retired pension scheme participants. Otherwise, there would be additional risks involved due to the inherent uncertainty in the other asset classes. By this assumption, we limit ourselves to the dynamics of the mortgage market.

2.4 Term structure of interest rates

After estimating the expected future incoming cash flows due to the wage process and economic devel-opment and the consequential outgoing cash flows resulting from the guarantee, we need to calculate their present value at the start of the insurance contract to determine a fair value. For this, we need the “future value of money”, which can be represented by a term structure of interest rates.

There are several different ways to construct such a term structure, but the one we will take as a starting point is the yield curve that is published by the DNB and which is used by (almost all) Dutch insurers and pension funds to valuate their long-term liabilities. It is constructed by listing the weighted prices of interbank interest rate swaps that exchange fixed legs for the six-month Euribor in AAA-rated Eurozone countries and converting to zero-coupon spot rates. This yield curve is assumed

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Figure 2.1

The DNB term structure as per 31 December 2012, with and without the UFR.

0,0% 0,5% 1,0% 1,5% 2,0% 2,5% 3,0% 3,5% 4,0% 0 5 10 15 20 25 30 35 40 45 50

DNB Curve with UFR DNB Curve without UFR

to be both risk-free and arbitrage-free.

On the grounds that for longer maturities there is no liquid market for these commodities, DNB uses the Ultimate Forward Rate (UFR). The actual zero rates are converted to forward rates and for maturities between 21 and 60 years are weighted with the set UFR value of 4.2%. Above maturities of 60 years, the forward rates are set equal to the UFR. The resulting forwards are then converted back to zero rates. For illustration of this effect, Figure 2.1 shows the DNB yield curve per 31 December 2012 with and without the addition of the UFR.

For pension funds, this obtained yield curve is then averaged over the last three months. Since we are more interested in insured contracts rather than pension funds, we will not take this averaging along in our modeling.

To estimate future term structures, we implement a Hull-White model for the underlying process. This is a generalized version of the Vasicek model we used for the estimation of future mortgage rates, where now the long-term mean is no longer fixed but has time dependence:

drt= a · (Θ(t) − rt)dt + σ · dWt (2.4)

This model, first described in Hull & White (1990), also constitutes a one-factor mean-reverting short rate model.

We use the DNB yield curve published 31 December 2012 as our starting value, where we first take out the weighted values of the UFR for longer maturities and calibrate the associated values of Θ(t). Next we generate new short rates from the input curve using the Hull-White model with parameter values α=5% and σ=10% and monthly interpolation of Eq. 2.4. These we convert to a term structure form using the affine structure bond pricing formula:

P (t, T ) = exp(A(t, T ) − B(t, T ) · r(t, T )) (2.5) where B(t, T ) = 1 − e −a(T −t) a (2.6) and

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A(t, T ) = ln v0(T ) v0(t) + σ 2 4a3 h

1 + e−2aT − e−2at− e−2a(T −t)− 41 + e−aT − e−at− e−a(T −t)i (2.7) In this expression vs(t) is the discount factor at time s with maturity t. This results in pure yield

curves, on which we then add the (weighted) UFR to obtain the term structure variety used by DNB to valuate long-term liabilities. In this manner we generate term structures with sixty years maturity for sixty years into the future. This is enough given the maximal horizon which we will consider in further calculations.

As stated before, we also use these yield curves in part to determine future mortgage interest rates. The one-year forward rates with maturities of 15 and 20 years can be seen as an approximation of the future spot rates at those maturities, and are used as the basis for the mortgage rate - this would be the term sT of Eq. 2.1.

2.5 Mortality rates

One of the key demographic attributes within an insurance portfolio is the mortality rate. The pos-sibility of an employee deceasing while still participating in the DC pension plan is explicitly taken along in our analysis, where the current literature most often ignores this. In such a case, the entire savings account will default back to the insurer only, in accordance with Dutch legislation.

The mortality rates that we use are given by the AG Prognosetafel 2012-2062, which was designed by the Dutch Actuarial Society and Institute (AG/AI). This is a prognostic table rather than a static table that takes into account the evolvement of future mortality rates. It is based on a mathematical model that uses historical data as well as expected coming developments. Since the average lifespan is generally increasing, the logical trend in these mortality rates is downward.

Some illustrations of this are given here to get a better feel for this. In Figure 2.2a the mortality rates for a 25-year old male and female are plotted from 2013 to 2062. Both show a definite downward slope, although it does flatten out. Figure 2.2b shows the expected male mortality rates in 2013 and 2062. This also shows a downward evolution, which is most obvious in the ages between 50 and 90. In this manner, the life expectancy of males increases from 79.3 years in 2013 to 86.4 years in 2062, and that of females from 82.8 years to 86.9 years.

Along with the stochastic determination of the term structure described above, the mortality rates are used to actuarially discount the cash flows resulting from the guarantee to obtain a fair value estimation of the guarantee price.

2.6 Guarantee fees

With the generated interest curves and the mortality rates, we can determine the present value of the cash flows that ensue from a given guarantee setup. This is the amount that would have to be reserved on the insurer’s balance sheet to ensure being able to meet with the expected future liabilities. An insurer would ideally charge for these costs by lump sum before entering into the contract with the employer. This is not common practice, since it would be a large initial charge and would undermine the market position of the insurer. Therefore the costs must be retrieved in the form of annual fees which are reckoned along with the annual settlements.

There are several possibilities for doing so, and in this project we will look at three distinct options. The first is an additional annual risk premium, a fixed percentage on top of the annual contributions to be made by the employee. The second option is to take this risk premium out of the contribution

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Figure 2.2

The expected development of the future mortality rates.

(a) 25-year old male and female from 2013 to 2062. (b) Male mortality rates in 2013 and 2062.

0,0000 0,0001 0,0002 0,0003 0,0004 2013 2018 2023 2028 2033 2038 2043 2048 2053 2058 0,00 0,05 0,10 0,15 0,20 50 55 60 65 70 75 80 85 90

itself instead of on top of it. The final option is to charge a fixed percentage of the capital depot at the end of every year. The height of these three percentages logically will rise with increasing levels of the guarantee.

The additional premium in this form would probably be the best option for the employee, since the fee would be charged outside his capital build-up. It would on the other hand also be the most costly for the employer, so we presume that in schemes that use this mechanism the two parties will share the costs. In the other cases the participant pays for the protection the guarantee setup offers himself, which keeps the pension scheme in a purer DC mindset but reduces the capital available for investment and lowers the prospective retirement wealth.

2.7 Replacement rates

The main output of our model consists of two matters. It calculates the fair value of the cash flows resulting from the guarantee setup and the three levels of annual fees described above to finance this. It also calculates the average replication rate of participants upon retirement, where this rate is given by the ratio between the total income after retirement (including the AOW payments) and the last salary earned. For this quantity, we determine the monthly annuity R that can be bought by the resulting total capital upon retirement K:

R = K ¨ ax (2.8) where ¨ ax= ∞ X t=0 v(t) ·tpx= ∞ X t=0 tpx (1 + zt)t (2.9) and where v(t) is given by the term structure applicable at the time of retirement andtpx is the

cumu-lative survival rate. These quantities are calculated on a monthly basis by interpolating. Both these types of output depend not only on the guarantee parameters themselves, but also on the age (because of investment horizons) and gender (because of mortality rates) of the participant upon entering the pension scheme.

The preparation of future term structures and mortgage interest rate spreads is not a determin-istic affair. We introduce a rather large degree of variation into both data series by random motions

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(the Wiener processes in Eqs. 2.2 and 2.4) and thereby making it a stochastic process. To establish statistically significant results we make an average over 10,000 simulations of every guarantee setup we want to calculate in a Monte Carlo-like approach.

2.8 Comparing the model to literature

To summarize, our model has been designed to estimate future term structures and mortgage interest rates, starting from historical data and well-known mathematical modeling. Together with the styl-ized cash flow patrons of the annual contributions for different participants into DC pension plan, this results into expected payments of the guarantee setup in cases of underperformance. We calculate a fair value for these payments and derive three types of annual fee to finance this. From the ultimate capital depot, we determine replication rates for participating employees.

To test whether the model we have described in detail above gives reasonable results, we try to reconstruct the results given by Scheuenstuhl et al (2011) as mentioned before in our overview of the current literature. We chose this particular paper because, although the input parameters and the investment model differ significantly, the overall setup of the cash flow model is the most similar to ours. We alter our valuation model to match the design of Scheuenstuhl as much as possible. The most profound adjustment is the setup of the investment plan, where the asset universe now consists of a mix of broadly diversified portfolios of equity (µ=7.5%, σ=20%) and government bonds (µ=4.8%, σ=3.0%, duration 5.6 years). They use a life-cycle strategy where an initial 80% stock allocation is reduced to 20% in the final ten years before retirement.

They rely on the risklab Economic Scenario Generator (ESG), a hybrid of econometrics-based and pricing-based modeling, for their conceptual modeling framework of interest rates and other macroeco-nomic factors. They combine this with a Black-Scholes type formula for vanilla options. Since we lack access to this ESG, we continue to use our own constructed interest rate model using the Hull-White framework and the fair value calculation resulting from it. Other adjusted input parameters include the wage process according to the Panel Study of Income Dynamics as described in Cocco, Gomes & Maen-hout (2005) and the absence of mortality during employment. The annual wage inflation is fixed at 2%, the price inflation is stochastic around 2%, and the annual contribution is set at 10% of the salary. With these assumptions in mind, the average level of the annual fees out of the contributions or out of the capital depot to finance both a maturity and an annual guarantee are determined. This is done for several values of the guarantee height on a 40-year horizon, which corresponds to a 25-years old employee entering the DC pension scheme and remaining until retirement. The results from these calculations are given in Table 2.2 alongside those from our own adjusted model.

This shows that the annual fee structures that are the result of our model’s calculations correspond well with those of Scheuenstuhl in their order of magnitude. We see that our model produces values that overall lie somewhat higher than their comparisons, where the percentages of the contribution fee

Table 2.2

Comparative values of our cash flow model to Scheuenstuhl et al (2011).

Scheuenstuhl Our model

Type Height Contribution Depot Contribution Depot

Maturity 0% 1.244% 0.055% 1.487% 0.069%

2% 4.936% 0.218% 5.109% 0.259%

4% 18.709% 0.887% 20.092% 1.004%

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correspond more accurately than the capital fee. This can most likely be attributed to the remaining differences in the model setup. Our term structure model has no correlations with other economic vari-ables, so that it can probably vary more freely than for Scheuenstuhl, which is done in a risk-neutral Black-Scholes framework.

So from these calculations, we can conclude that our model produces plausible output values. The fair pricing that results from our model might be somewhat overestimated in comparison with Scheuenstuhl, but this can be explained from remaining model dissimilarities.

2.9 Resignation from employment

The previous calculations all assumed the participant to stay in the pension plan until the retirement date. In reality the development of the economy during the last few decades has seen a more liquid job market emerge, with employees changing employer or even entire career paths more and more frequently. As a consequence most participants in pension plans will eventually become passive, which means that they no longer receive contribution payments. This will happen more often at early stages in their career. Previous research projects have not implemented the possibility of resignation into their modelling, which makes it one of the most important additions of our specific design.

To meet with this more realistic outset, we add to the existing model a module in which the par-ticipant might leave employment before his retirement date. For this we use a dataset that we derived from real historical administrative policies dating back to 1996 from pension schemes of a Dutch in-surer. These specify the relative amounts (which can therefore be seen as conditional probabilities) of pension scheme participants resigning at each age. These probabilities are smoothed out using a cubic polynomial to reduce statistical noise. The obtained probabilities are shown in Table 2.3. We see that the trend of these probabilities is ever downward, indicating that employees will on average be less inclined to leave service the closer they are to their retirement.

Table 2.3

Age-dependent probabilities of resignation from employment.

Age Prob. Age Prob. Age Prob. Age Prob. Age Prob. Age Prob.

18 24.3% 26 16.1% 34 11.0% 42 8.0% 50 5.8% 58 3.3% 19 23.1% 27 15.3% 35 10.5% 43 7.7% 51 5.5% 59 2.9% 20 21.9% 28 14.5% 36 10.1% 44 7.4% 52 5.3% 60 2.5% 21 20.8% 29 13.9% 37 9.7% 45 7.1% 53 5.0% 61 2.1% 22 19.7% 30 13.2% 38 9.3% 46 6.9% 54 4.7% 62 1.6% 23 18.7% 31 12.6% 39 9.0% 47 6.6% 55 4.4% 63 1.1% 24 17.8% 32 12.0% 40 8.6% 48 6.3% 56 4.1% 64 0.6% 25 16.9% 33 11.5% 41 8.3% 49 6.1% 57 3.7% 65 0.0%

At the end of every year, the model makes a randomized decision in every scenario whether or not the participant leaves employment based on these chances. The annual contributions will stop, but the capital present in the depot will still grow according to the return rates determined earlier. Therefore the guarantees may still pay out (annually or at retirement) and therefore different cost levels may be determined. These calculations are being done parallel to the earlier ones, so the cash flows of the guarantees use the same interest and mortgage rate paths in every simulation whether or not resig-nation is taken into account. The rest of the simulation proceeds as described earlier. Unfortunately, this process does add to the statistical noise manifest within the results.

Because contributions will stop after resignation, the capital available between this point and retirement will naturally be lower than when remaining in employment. As a result, the payments

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that the insurer has to make in case of underperformance, and thereby the risk the insurer takes, will be reduced. However, the amount of time in which the total risk must be financed will be limited, counteracting this effect. The influence on the annual fees to finance the total risk will therefore be highly dependent on the input parameters, chiefly among them the horizon until retirement and the chances of resignation. We will look at this effect more closely later on.

Rate of return guarantees in dutch defined contribution pension plans.