Longevity risk in the dutch pension : transfer of longevity risk from the pension fund to the individual participant.

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University of Amsterdam, Amsterdam

Faculty of Economics and Business Department Quantitative Economics

Section Actuarial Sciences

Longevity Risk in the Dutch pension

system

Transfer of Longevity Risk from the Pension Fund to the

Individual Participant

Master’s thesis of Kevin K. Keijzer

Thesis committee

Supervisor (UvA): Prof. dr. ir. M. Vellekoop Supervisor (Towers Watson): drs. W. Hoekert AAG

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Summary

The longevity risk is still taken by the pension fund as a whole and will probably be taken by individual participants in the upcoming new Dutch pension system. In this thesis the main question is how the value of pension entitlements will change if the mortality risk is transferred from the pension fund to the participant. The focus of the research in this thesis is on the Dutch population because of these upcoming changes.

As a basis for the research three mortality models have been used which are discussed and analyzed extensively. These are the Lee-Carter model, the Plat 2009 model and the AG Prognosetafel 2012-2062 and all three models are applied to the Dutch population. The last model is originally a deterministic model, but within this thesis a method is described to transform it into a stochastic model.

Pricing of longevity products has to be done using theorectical assumptions, since there is almost no real longevity market. Along with his mortality model Plat introduced a method to determine risk neutral prices for his mortality table. These prices are calibrated by using q-forwards. In this thesis this method is implemented for the AG Prognosetafel 2012-2062 and the Lee-Carter model as well.

The resulting risk neutral prices for all the models are used to value longevity risk by using a s-forward. In this way the value of the transfer of longevity risk from the pension

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Introduction

1.1 Motivation

In June 2011 the Dutch employers and the unions reached an agreement (Pension Accord) about the outlines of a new pension system. In May 2012 the government formulated their plans for the new pension system (Hoofdlijnennota), in September 2012 the government pub-lished some outlines for 2013 and in July 2013 the nFTK was formulated. At this moment the new regulatory framework (nFTK) has been canceled and the government’s plans are back to the writing table. One thing is certain: the Dutch (second pillar) pension system is about to change. However it isn’t certain how and when. In the current pension system the pension entitlement is an unconditional pension benefit which will only be reduced as a last resort. In the new plans there will probably be a more conditional variant of the entitlements.

One of the things that is about to change is the way of dealing with longevity risk. This is the risk that people live longer than expected. In the current pension system the costs for longevity risk are borne by the whole pension fund without any compensation (only as a last resort, reduction of entitlements). This weakens the financial position of the pension fund each time life expectancy rises. In all the plans for a new pension system a method was formulated which changes the pension benefits when the life expectancy will rise.

In the memorandum about the Pension Accord (Stichting van de Arbeid (2011)) it is suggested to update the pension entitlements in an actuarial neutral way when the mortality table will be updated. This is the adjustment mechanism for life expectancy changes, which states that the effect of an update of the pension benefits can be implemented over a period of a maximum of 10 years. In the other plans similar ideas were suggested.

1.2 Research objective

The new plans for increasing life expectancy results in an extra uncertainty for the pension benefits of all participants. The formerly unconditional entitlements become more uncertain.

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1.3. PAPER STRUCTURE

People essentially don’t like to have added uncertainty in their (future) income. From this point of view our research question will be:

How does the transfer of longevity risk from the pension fund to the individual participant effect the value of his/her pension entitlement?

The idea behind the research question is that under the new plans the pension funds probably stop bearing (a part of) the longevity risk, but we may wonder what happens if the pension fund will continue to bear the longevity risk. A pension entitlement of 1000 Euro each month has a certain value. That value is higher than the value of an expected pension entitlement of 1000 Euro each month in which longevity risk is not insured. We want to analyze the extra cost for having this longevity risk.

An extra reason for finding an answer to this question is that at this moment it isn’t sure whether or not it will be allowed to change the unconditional pension entitlements for longevity under FTK. If it won’t be allowed there should be another way to handle longevity and our analysis could help in finding a solution.

1.3 Paper structure

Chapter 2 gives a brief description of the Dutch pension system, primarily the second pillar and the most recent information about the suggested changes of the recent years. In Chapter 3 we discuss four mortality models. These are the mortality models described by Lee and Carter (1992), the model of Cairns et al. (2006a), the most recent mortality table of the AG, the Prognosetafel 2012-2062, and a mortality model described by Plat (2009). In chapter 4 a method for valuing longevity risk described by Plat (2009) is discussed and applied to the other mortality models as well. In chapter 5 the findings and results are presented and chapter 6 contains the conclusion.

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Chapter 2

Pension system in the Netherlands

In this chapter the pension system of the Netherlands will be discussed. First the current pen-sion system will be described, then the main points of the Penpen-sion Accord, Hoofdlijnennota, September Package, nFTK and some debate points will be discussed and finally a more de-tailed description is given about a possible way of dealing with the increasing life expectancy. This is the method described in the Pension Accord, but in the other plans similar methods are suggested.

2.1 Current pension system

The government obligates most working people to participate in the pension system, in this way the less well off will be guaranteed some income after the pensionable age. The risk sharing in the pension fund is for a great part based on solidarity. In the Netherlands a pension benefit after retirement age is based on three types of pension:

1. State pension

2. Occupational pension accrued by the employer and employee

3. Private pension

2.1.1 First pillar: State pension

The first pillar of the Dutch pension system is the state pension. The state pension is arranged in the Algemene Ouderdoms Wet. All residents older then 15 and living in the Netherlands are insured till the age of 65 and 1 month. Each year you live in the Netherlands between age 15 and 65 counts for 2% of the total benefit. The value of the benefit differs for single and married people. For a single person the payment is approximately 70% of the minimum wage. The state pension is a pay-as-you-go scheme. The contributions paid by the working population are immediately used to finance the benefits for the current beneficiaries. Because

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2.1. CURRENT PENSION SYSTEM

the population is getting older, more people retire every year and the working population is decreasing, the pay-as-you-go scheme becomes more expensive each year. Therefore there is a lot of discussion about the cost of the state pension. This has resulted in a raise of the pensionable age in increasing steps of 1, 2 and 3 months to 67 in 2023 (an agreement of the government states 2021 as the year to reach 67 years, however these plans have not been implemented yet) and thereafter the pensionable age will be adjusted in line with life expectancy increases.

2.1.2 Second pillar: Occupational pension

The second pillar consists of occupational pension supplied by the employer. The rules for occupational pension accrual are arranged in the Pension Act. If an employer facilitates a pension scheme employees are obliged to participate. Hence most of the people participate in a pension scheme. Pension schemes can be administered by a pension fund or by an insurer. There are three types of pension funds:

• Industry-wide pension funds (bpf): All companies in the industry are obliged to partic-ipate in the offered pension scheme. The government decides whether an industry gets a industry-wide pension fund. The only exception is that when a company offers its employees a better or equal pension scheme than the industry-wide pension fund does, that in that case the company can offers the better pension scheme.

• Professional pension funds: Some professions have their own pension plan, in that case the pension contract for that profession is mandatory.

• Company pension funds (opf): If a company isn’t obliged to join an industry-wide or professional pension fund it can start its own pension fund, it doesn’t have to however.

The accrual of pension is exempt from taxes, taxes are paid whenever a pension benefit is in payment. The pension funds can provide different types of pension. The Pension Act describes three different types of pension schemes:

• A defined benefit agreement (DB): This is a guaranteed pension benefit where the contribution varies. The final pay plan and the average pay plan are two types of DB schemes. The final pay plan guarantees a pension entitlement based on the career end salary whereas the average pay plan is guarantees a pension entitlement based on the salary earned on average in a career.

• A defined contribution agreement (DC): In this case the benefit is not guaranteed, but the contribution is fixed. The pension benefit accrued at retirement will depend on interest, change of mortality rates and changing value of investments. This type of pension is more insecure for participants, all risks are transferred from the pension fund

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2.2. DEVELOPMENTS OF THE PENSION SYSTEMS

and the company to the employee. In recent years more and more companies have changed their DB agreement to a DC agreement to reduce their risk.

• An agreement to payment of a capital sum: Each year a part of the guaranteed capital is accrued, sometimes raised with profit sharing. The capital at retirement should be used to purchase a pension benefit at the pensionable age. During the accrual period the investment risk is for the pension provider. The longevity risk and interest rate risk are for the participant. This type of pension is not common in the Netherlands. An important part of the Pension Act is that a pension fund and the sponsoring company should be strictly separated. This is done to prevent that when the company goes bankrupt the pension fund will go bankrupt as well. Otherwise the employees will lose not only their jobs but their pension entitlements too.

2.1.3 Third pillar: Private pension

The third pillar is there to facilitate the opportunity to accrue private pension with the advantage of tax benefits as long as you stay within the fiscal boundaries. This makes it possible to accrue extra pension benefits for people who want a higher pension on top of their additional pension or people without a pension agreement or an employer who want to pay for the extra pension. As long as you stay within the fiscal boundaries this pension is handled with the same tax benefits as in the second pillar.

2.2 Developments of the pension systems

Due to the financial crisis and the ageing population the sustainability of the current pension system is at risk. In recent years many suggestions have been done and plans have been made to change the current pension system. In this paragraph an overview of the suggested changes is given.

2.2.1 Pension Accord

The social partners negotiated to solve some of the weaknesses of the current contract. This regards the first but mostly the second pillar pensions. All is stated in the memorandum of the Stichting van de Arbeid (2011). The main points agreed by the social partners will be discussed here.

The goal of the social partners was a new pension system that was more sustainable than the current pension system without losing the main features of the current contract. Important features to maintain are collectivity, solidarity and mandatory participation. The new contract is based on the idea that the following points should be improved:

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• Increasing life expectancy may not cause an extra risk for the pension fund.

• Pension contracts must be resistant to shocks.

• The pension contracts must be transparent and clear for participants, so that it can be easily communicated.

With these points in mind the social partners reached an agreement (Pension Accord). The current contracts are still allowed under the Pension Accord. The regulatory framework (FTK) is split up in two parts, FTK 1 for the current contracts and FTK 2 for the new contracts. The biggest difference for the current contracts between FTK and FTK 1 is that pension funds need to hold a bigger buffer under FTK 1. In the memorandum of the Stichting van de Arbeid (2011) the new contract is worked out. The most important changes to the pension contract are the following:

1. There must be a transparent link between the collective risk profile of the participants and the investment policy of the pension fund. Furthermore there must be a balanced allocation of risks. The risks must be divided equally over all participants: both the current and the future participants.

2. Pension funds should aim for a real indexation ambition i.e. aim to compensate for inflation. The pension funds should create an awareness among particpants that pension entitlements become more uncertain depending on the selected investments.

3. The nominal goal should be dropped, because the nominal unconditional entitlements prevents the funds to maintain a real ambition in a responsible way. This means that the entitlements are no longer unconditional. The social partners worked out an ad-justment mechanism for absorbing financial shocks. To make sure the pension benefits of participants aren’t moving too much the social partners stated that there should be an equalisation reserve (a buffer).

4. It must be possible to absorb an increasing life expectancy, without transferring risk from one generation to another. This is worked out in the adjustment mechanism for life expectancy changes (see section 3.4.2 for more details).

In the end the government makes the decisions about the pension system hence the sug-gested changes in the Pension Accord are guidelines at most.

2.2.2 Hoofdlijnennota

In May 2012 the government sent the Hoofdlijnennota to the House of Parliament. The Hoofdlijnennota states the plans of the government with the Dutch pension system. The pur-pose for the Hoofdlijnennota is the same as for the Pension Accord. These are the increasing

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life expectancy and the continuing financial crisis. The main points of the approach of the government are the following:

• Social partners and pension funds must agree on the distribution of risks in the pension contract before the start of the contract. This is done to create more awareness in society about risks and aims to restore the trust in the pension system.

• Pension funds should communicate pension related news or changes in a clear way.

• There will be a more stable FTK (again consisting of two parts, a nominal and real part), wherein the dependence on daily rates will be decreased giving the contract a more longterm character.

• The FTK will give more flexibility in providing indexation.

• In new contracts it will be possible to implement shocks in the financial market and increases in life expectancy gradually in the pension entitlements.

2.2.3 September Package

The government formed the September Package (Ministerie van Sociale Zaken en Werkgele-genheid (2012)) in consultation with DNB and the Pension Federation. The September Pack-age is formulated because the new FTK was delayed and hence the packPack-age focuses mainly on the year 2013. The plans contain the following suggested improvements of the current system:

• A new interest rate curve for all pension funds. The 3-month average curve is manipu-lated in the long end by using an Ultimate Forward Rate of 4.2%.

• There is an option to spread the possible benefit cuts. It is allowed to maximize to effectuate a benefit cut of 7% immediate and the rest will then be effectuated later. This is only possible when the following three constraints are met:

1. Increase the pensionable age to 67 as at 1 January 2013 instead of 1 January 2014. 2. Implement a type of adjustment mechanism for increasing life expectancy, in which

indexation becomes conditional to an increasing life expectancy.

3. From now on indexation is only granted if the funding ratio is 110% or higher.

These three constraints are required if a pension fund wants to spread the upcoming benefit cuts. The idea behind the constraints is to stimulate a more future-proof pension system.

• A temporary suspension of the rule that contribution must help the recovery of the funding ratios.

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2.3. ADJUSTMENT MECHANISM FOR LIFE EXPECTANCY CHANGES

2.2.4 nFTK

In July 2013 the government published a new version for a new FTK. The nFTK is mainly based on the Hoofdlijnennota.

• In line with the Pension Accord the nFTK is split up in two parts. A part wherein it is possible to accrue an unconditional nominal pension entitlement and a part wherein it is possible to accrue a more conditional pension entitlement but with volatile indexations.

• There will be an adjustment mechanism for financial shocks and an adjustment mech-anism for life expectancy changes.

• In the nFTK two interest rate curves are stated; one for each contract. The interest rate curve of the real contract is corrected for inflation.

• There will be an obligation to investigate effects between different generations.

2.2.5 At this moment

At this moment the designs for the new pension system have been held off by the Senate; they have been sent back to the House of Parliament. The government and the House of Parliament are still debating which changes they have to make compared to the nFTK ideas. There will probably be only one FTK just as in the current contract. Another important item that is currently discussed is the maximal fiscal accrual rate.

2.3 Adjustment mechanism for life expectancy changes

Social partners recognised increasing life expectancy as one of the weaknesses of the current contract. If the life expectancy rises nothing is done to pay for the extra liabilities. This will weaken the financial position of the pension fund and will bring risk and extra costs to future generations. To tackle this problem the social partners defined an adjustment mechanism for life expectancy changes. This mechanism is explained in detail in this section. The idea is to adjust the entitlements in an actuarial neutral way based on the pension fund-specific situation. Hence there will be no extra costs when life expectancy rises and the financial position of a pension fund doesn’t become weaker when mortality rates decrease. The social partners identified two elements1:

• An adjustment mechanism for new pension accrual.

• An adjustment mechanism for the existing entitlements under the new contract.

1_{The two elements are directly quoted from the report of Stichting van de Arbeid (2011)}

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2.3. ADJUSTMENT MECHANISM FOR LIFE EXPECTANCY CHANGES

An adjustment mechanism for new pension accrual The adjustment is separated in three parts:

1. The pensionable age should be adjusted for new entitlements after the announcement of an increase in the pensionable age for state pensions. Hence if the pensionable age for state pensions is announced to increase, participants will accrue pension benefits based on the new pensionable age from that moment onwards. The agreed rise of the pensionable age for state pension in April 2011 will result in an increase of the pensionable age for new entitlements in the second pillar.

2. The liabilities should be adjusted to a new mortality table. The costs that occur through implementing the new mortality table will be balanced by the reduction of costs due to the higher pensionable age. The social partners suggest to adjust the mortality tables every five years.

3. The cost for each of the two above adjustments shouldn’t effect the contribution. The cost of the changes should be implemented in a way which is actuarially neutral:

• If the net cost increases, the contribution will stay the same, but there will be a negative ’adjustment factor for life expectancy’. This can result in a reduction of accrual or a negative indexation.

• If the net cost decreases the extra funds will be used for a positive ’adjustment factor for life expectancy’. This can be done by extra accrual or indexation for the participants. Extra funds will not be given back to the employer.

An adjustment mechanism for the existing entitlements under the new contract. Not only the new entitlements but also the existing entitlements should be adjusted when the mortality table is updated. This should be done in every situation, not just if the pension fund is underfunded. A sudden decrease of the pension benefits is unwanted. Therefore the social partners have decided to distribute the effect of the mortality table update over ten years.

In the next chapter the mortality models of Lee-Carter, Cairns-Blake-Dowd (CBD), AG Prognosemodel 2012-2062 and Plat are discussed. This will be the first step towards valuing longevity risk.

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Chapter 3

Mortality Modelling

One of the goals of our analysis is to value the longevity risk in a pension portfolio. For the valuation of the longevity risk a stochastic mortality model must be used. The mortality models should give a good fit for the Dutch population. A lot of different mortality models were defined in the last decades. In this chapter four mortality models will be discussed. The first two models that will be discussed are the ones by Lee and Carter (1992) and Cairns et al. (2006a). These two models are very popular and widely used. The problem with these models is that they were not designed for the Dutch population, therefore the fit for the Dutch population is not optimal without any adjustments. The other two mortality models that will be discussed were originally fitted on the Dutch population. The third model is the AG Prognosetafel 2012-2062, following Actuarieel Genootschap (2012). The main problem of the AG Prognosetafel 2012-2062 is that this is a deterministic table without any uncertainty. Since the AG Prognosetafel 2012-2062 is the most used table by pension funds in the Netherlands it would be preferable to work with this table. Therefore the AG Prognosetafel 2012-2062 will be described and then transformed into a stochastic model. The fourth model is the stochastic mortality model of Plat (2009). This model is an adjusted version of the Lee-Carter model and the Cairns-Blake-Dowd-model and gives a good fit on the Dutch population. But first the data that is used and the assumptions about the theoretical framework that are made will be described.

3.1 Data and Assumptions

The data used in this thesis consists of mortality rates published by the Dutch Central Bureau of Statistics (CBS)1 _{and numbers of deaths and exposures to risk published in the Human}

Mortality Database (HMD)2. The CBS data consists of 1-year mortality rates qx,t for ages

x = 0, 1, 2, ..., 99 and years t = 1950, ..., 2011. The data published by the CBS is of high quality

1

http://statline.cbs.nl

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3.2. LEE-CARTER MODEL

and therefore suitable for this thesis. The data used to determine the central mortality rates mx,t, the number of deaths dx,t and the exposure to risk ex,t, are published in the Human

Mortality Database (HMD). The dx,t and ex,t are used for ages x = 0, 1, ..., 109 and years

t = 1950, ..., 2009. The data published by HMD is based on information given by the CBS and is therefore assumed to be of high quality.

The next step is to define the theoretical framework that will be used. The random variable Tx,t is defined as the remaining random lifetime for a person aged x in year t. With

this random variable the qx,t en px,t are defined:

spx,t = P (Tx,t > s) (3.1)

Hence spx,t is the probability that a person who is alive at age x in year t will survive to

age x + s. Furthermore1px,t ≡ px,t. We also define:

sqx,t = P (Tx,t≤ s) (3.2)

Hence sqx,t is the probability that a person who is alive at age x in year t will die before

reaching age x + s. Furthermore1qx,t ≡ qx,t.

In this case the mortality and survival rates are discrete rates at integers only. The mortality rates show the percentage of people which died in a period, but not the distribution during the period. The force of mortality represents the instantaneous rate of mortality and is given by:

µx,t = lim s&0

P (Tx,t ≤ s)

s (3.3)

There are several approaches which can be chosen to link the force of mortality to the mortality rates. The one chosen here is a piecewise constant force of mortality following Brouhns et al. (2002):

µx+s,t+s= µx,t ∀ s ∈ [0, 1) (3.4)

Since some of the models are defined using the initial mortality rate and others using the central mortality rate or force of mortality the following approximation is used to compare the different models:

qx,t≈ 1 − e−µx,t (3.5)

3.2 Lee-Carter model

One of the most frequently used models is the Lee-Carter model described in Lee and Carter (1992).

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3.2.1 Model

The model is given by the following formula:

ln(µx,t) = αx+ βxκt+ σxεt with εt∼ N (0, 1)i.i.d (3.6)

where the αx, βxand κtare the parameters that need to be estimated. The αx parameters

represent the average form of the mortality curve for each age, the βx parameters represent

the pace of mortality improvement for each age and the change of mortality over time is given by the κt parameters.

Because the model haves an identifiability problem (there are too many degrees of freedom) therefor the parameters needs to be restricted. The following equations will lead to the same solutions for the model:

αx+ βxκt= αx+ (βxc)

κt

c (3.7)

αx+ βxκt= (αx− βxc) + βx(κt+ c) (3.8)

To solve this problem the following constraints will be set:

X x βx = 1 (3.9) X t κt= 0 (3.10) 3.2.2 Estimation

Brillinger (1986) states that the Poisson distribution is an appropriate distribution for mor-tality analysis. The number of deaths at age x in calendar year t is assumed to follow a Poisson distribution (see Brouhns et al. (2002) for a motivation):

Dx,t ∼ Poisson(Ex,tµx,t) with µx,t= exp(αx+ βxκt) (3.11)

The αx, βx and κtwill be estimated by maximum likelihood methods. The log-likelihood

function L(φ; D, E), where the parameter set φ contains (αx, βx, κt), is given by:

L(φ; D, E) =X

x,t

(Dx,tln[Ex,tµx,t(φ)] − Ex,tµx,t(φ) − ln(Dx,t!)) (3.12)

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3.2.3 Forecasting

The next step is to determine the ARIMA(p,d,q)-model for ˆκtusing the Box-Jenkins

method-ology described in Box and Jenkins (1976), to make a projection to forecast future mortality rates. Lee and Carter (1992) used a random walk with drift as the model for ˆκt, because this

model gave the time series the best fit:

ˆ

κt= ˆκt−1+ θ + εt with εt∼ N (0, σ2) (3.13)

in which σ is a constant standard deviation and θ is a constant drift. This model was fitted on the population of the United States and not developed for the Dutch population. The random walk with drift is possibly not the best model for ˆκt for the Dutch male population,

due to the steepening trend for males in recent history as follows from Keijzer (2011).

3.2.4 Bootstrapping the model

Normally when using maximum likelihood the standard errors of ˆαx, ˆβx and ˆκt are equal

to the diagonal elements of the inverse of the Fisher matrix. In this case the estimates are modified to meet the constraints so the standard errors can’t be derived that way. Hence the uncertainty of the forecasts as well as the confidence intervals are determined using a bootstrap method as described by Koissi et al. (2006). This method combines the uncertainty of the estimators with the prediction errors. This methodology is chosen because in this way there is no need to know the distribution of the residuals (see Efron and Tibshirani (1998)).

In this method draws will be done with replacement from the residuals. The algorithm is as follows:

1. Estimate the Lee-Carter model and derive the Pearson residuals rx,t using ˆαx, ˆβx and

ˆ

κt. Where rx,t is defined as:

rx,t = Dx,t− ˆDx,t q ˆ Dx,t (3.14)

in which ˆDx,t is the estimated number of deaths following from the model.

2. Reduce the residuals with their own averages (˜rx,t = rx,t− ¯r). This is done to ensure

that the expectation of ˜rx,t is equal to zero. Then draw with replacement from the

group of residuals.

3. The next step is to create new realisations Db_x,t (see England and Verrall (1999)) using:

D_x,tb = ˆDx,t+

q ˆ

Dx,tr˜x,t (3.15)

in which ˆDx,t is the estimated number of Deaths following from the model.

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3.3. CAIRNS, BLAKE, DOWD MODEL

4. Reestimate the model using the Bootstrap sample Db_x,t. In this way new ˆαb_x, ˆβb_x and ˆκb_t will be derived and then determine the residuals again.

5. Estimate the time series for ˆκb_t. Compared to the original forecast the model will not change for ˆκb_t. Only the estimated parameters will differ.

6. Repeat step 2 till 5 for b = 1, ..., B.

The B sets of estimators for ˆαb_x, ˆβ_xb and ˆκb_t (given by ˆφb for each b = 1, ..., B) are used to determine the confidence intervals with percentiles. The (1 − 2α)-percentile interval is given by [ ˆφ(α), ˆφ(1−α)], in which ˆφ(α) is the (α)th percentile and equal to the (B times αth) element of the sorted list of replica’s ˆφb.

3.2.5 Remarks

An advantage of this model is that it is relatively simple to produce a forecast of the future mortality rates and determining the uncertainty using bootstrap. At the same time the fact that it is a rather simple model is a disadvantage as well. For example in the original Lee Carter model it’s not possible to incorporate cohort effects as discussed in Haberman and Renshaw (2006). Furthermore the model is not developed for the Dutch population. In it’s current form the model is insufficient to model the Dutch population due to the steepening trend for males as stated in Keijzer (2011). This model is still implemented for comparison reasons.

3.3 Cairns, Blake, Dowd model

3.3.1 Model

The CBD model described in Cairns et al. (2006a) is given by the following model:

qx,t =

exp(κ0

t+ κ1tx)

1 + exp(κ0_t+ κ1_tx) (3.16) The model of Cairns, Blake and Dowd estimates the mortality rates qx,t instead of the

force of mortality µx,tin the Lee-Carter model. Compared to the Lee-Carter model this model

consists of two trend parameters κ0 and κ1. The parameter κ0 has equal impact on mortality at all ages and the parameter κ1 has more impact at the older ages. The two parameters give the model more flexibility over time compared to the model of Lee and Carter, however the model has less flexibility over age since βx = x.

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3.4. AG PROGNOSETAFEL 2012-2062

3.3.2 Remarks

A disadvantage of the CBD model is that it is not developed for the Dutch population and only fits well for ages above 60. The mortality rates for the lowers ages can not be modelled by a straight line. For the purpose of this thesis this model is therefore not implemented.

3.4 AG Prognosetafel 2012-2062

3.4.1 Model

The latest generation table published by the Actuarieel Genootschap (2012) is described in this section. The method consists of a number of steps. The first step is preparing the data. For all the historic mortality rates published by the CBS the two year average is taken:

qx,t/t+1 = 0.5[qx,t+ qx,t+1] (3.17)

After taking the average the mortality rates will be smoothed with the Van Broekhoven algorithm (see appendix A for more details). The mortality rates are only available for age 0 till 94. For ages above 94 the table will be closed using the logistic mortality law of Kannist¨o as stated in Kannist¨o (1992) and described in appendix B.

The next step is analysing the historic mortality rates. After analysing the mortality rates the AG recognised two trends. A long-term trend based on the data period 1988 till 2011. Since the first Prognosetafel model came in use the new realisations of mortality rates have improved more than the long-term trend predicted. Therefore the AG introduced the short-term trend based on the data period 2001 until 2011, which is steeper than the long-short-term trend. The short-term trend is given by:

fshort(x) = 9

s q_x,2010/2011

q_x,2001/2002 (3.18)

The long-term trend is given by:

flong(x) = 23

s qx,2010/2011

qx,1987/1988

(3.19)

For the higher ages there is a different approach than the short-term and long-term trend factors. The mortality rates for the ages from 100 year old and onwards are assumed to stay the same. Hence the trend factors are equal to one. Between the ages 94 and 100 the trend factors are interpolated linearly to one.

Using the two trend factors the start and goal table will be derived. The start table is

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3.4. AG PROGNOSETAFEL 2012-2062

derived by the following formulas:

q_x,2011start = qx,2010/2011

p

fshort(x) (3.20)

The goal table is derived from the start table and the long-term trend using the following formula:

qgoal_x,2062= qstart_x,2011(flong(x))51 (3.21)

The last step in setting the goal table is smoothing the goal table with the Van Broekhoven algoritm. There is a restriction stated that the mortality rate of the goal table for a certain age can’t be higher than the mortality rate of the start table for that same age.

The forecast model is based on the start and goal table. The model is given by the next two formulas (where t + T = 2062):

qx,t+n = qx,tfshort(x)nexp αxn(n + 1) 2 (3.22) αx= ln qx,t+T − ln qx,t− T ln(fshort(x)) 0.5T (T + 1) (3.23)

3.4.2 Transformation to a stochastic table

To transform the AG Prognosetafel 2012-2062 model into a stochastic model the mortality rates are first transformed into the force of mortality:

µx,t= − log(1 − qx,t) (3.24)

After the transformation an error term is added to the force of mortality to model the uncertainty. The first step is to determine the uncertainty for each age x. We choose an error term εx which is normally distributed with zero mean and variance σx2. This variance σx2 is

estimated with the following formula:

ˆ σx= v u u t 1 N − 1 N X t=1 (ln(µx,t) − ln(µx,t−1))2− ( 1 N N X t=1 (ln(µx,t) − ln(µx,t−1)))2 (3.25)

The distribution for the error is now defined. The period over which σx is determined

is 1950, ..., 2011, which is in line with the other mortality models since they have the same data period. The last step is to transform the force of mortality with a logarithm to keep the simulated mortality rates between zero and one. The new model for the force of mortality becomes:

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3.5. MORTALITY MODEL PLAT

The last step is to transform the force of mortality back into mortality rates.

qx,t= 1 − exp(−µx,t) (3.27)

The chosen method to implement the stochasticity in the model is similar to the way the stochasticity is handeled in the Lee-Carter model. This method is chosen to keep the models in line with each other.

3.4.3 Remarks

The AG Prognosetafel 2012-2062 is the most often used mortality table for pension funds and is specifically designed for the Dutch market. The fact that the model originally is a deterministic model must be kept in mind when studying the results.

3.5 Mortality model Plat

The third stochastic mortality model that will be used is the model as stated in Plat (2009). This model uses the positive features of the Lee Carter model and the CBD model eliminating most of the remarks for the Lee-Carter model. Furthermore Plat fitted this model for the Dutch population too.

3.5.1 Model

The mortality model of Plat models the central mortality rate. The central mortality rate combines Dx,t which is the number of death of age x during year t and the exposure to risk

Ex,t which is the average population of age x during year t:

ˆ

µx,t= mx,t=

Dx,t

Ex,t

(3.28)

The model is stated in the following formula:

log(µx,t) = αx+ κ1t + (¯x − x)κ2t + max(¯x − x, 0)κ3t + γt−x (3.29)

The αx are similar to the αx in the Lee Carter model, in which they represent the average

shape of the historical mortality rates. The κ1_t term changes the level of mortality for all ages. The κ2_t term allows changes in mortality to vary between ages. The κ3_t term models the dynamics of mortality rates at the lower ages, which can differt sometimes due to AIDS, drugs etc. The γt−xterm is there to model the cohort effect. The parameter ¯x is a constant

that is estimated from the data.

The model still has an identifiability problem, hence there will be some restrictions on the

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parameters. The following equations will lead to the same solutions for the model:

˜ γt−x= γt−x+ ψ1+ ψ2(t − x) (3.30) ˜ κ1_t = κ1_t − ψ1− d¯xψ2 (3.31) ˜ κ2_t = κ2_t + dψ2 (3.32) ˜ αx = αx+ (1 − d)ψ2x (3.33)

To solve this problem the constraints will be set following Cairns et al. (2009):

X c γc= 0 (3.34) X c cγc= 0 (3.35) X t κ3_t = 0 (3.36)

The first two constraints are chosen this way to ensure that ˆψ1 = ˆψ2 = 0 if γt−x is fitted

as the function ψ1+ ψ2(t − x). By using these constraints the γt−x will have no increasing or

decreasing trend and will fluctuate around 0. The third constraint normalizes the estimates for κ3_t.

3.5.2 Estimation

The Plat model is assumed to be Poisson log-bilinear model too, following the same reasoning as with the Lee-Carter model. The number of deaths at age x in calendar year t are therefore assumed to follow a Poisson distribution:

Dx,t ∼ P oisson(Ex,tµx,t) (3.37)

where µx,t is given by model (3.29). The αx, κ1t, κ2t, κ3t and γt−x will be estimated

with maximum likelihood methods. The log-likelihood function L(θ; D, E), where θ is the parameter set (αx, κ1t, κ2t, κ3t, γt−x), is given by:

L(θ; D, E) =X

x,t

(Dx,tln[Ex,t µx,t(θ)] − Ex,t µx,t(θ) − ln(Dx,t!)) (3.38)

Solving the equation above with maximum likelihood gives the estimates for ˆαx, ˆκ1t, ˆκ2t,

ˆ

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3.5.3 Forecasting

The next step is to determine the ARIMA(p,d,q)-models for ˆκ1_t, ˆκ2_t, ˆκ3_t and ˆγt−x using the

methodology Box-Jenkins described in Box and Jenkins (1976) and make a projection to forecast future mortality rates. Plat defined random walk with drift as a model for ˆκ1_t. The random walk with drift is given by:

ˆ

κ1_t = ˆκ1_t−1+ θ + ε_t1 with ε1_t ∼ N (0, σ2)i.i.d. (3.39)

In which σ is the standard deviation and θ is a constant drift term. The estimators of ˆκ2_t, ˆ

κ3

t and ˆγt−x are specified as an AR(1)-process in Plat (2009):

ˆ κ2_t = φ1κˆ2t−1+ ε2t (3.40) ˆ κ3_t = φ2κˆ1t−1+ ε3t (3.41) ˆ γt−x= φ3γˆt−x−1+ εt4 with εit∼ N (0, σ2)i.i.d. (3.42)

These models fitted the time series best because all the time series of ˆκ1

t, ˆκ2t and ˆκ3t

show some autoregressive behavior. The models for ˆκ1_t, ˆκ2_t and ˆκ3_t are correlated, therefor these models are calibrated using Seemingly Unrelated Regression. The model for γt−x is

uncorrelated from the other three models.

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Chapter 4

Longevity

With the different mortality models defined, the next step is to discuss longevity. In the recent history many articles about longevity risk and hedging longevity risk have been written. In this chapter some of these methods will be discussed. The goal of this chapter is to find a way to assign a monetary value to longevity risk so that pension funds can value that risk. In the first paragraph the current market products will be looked at. In the second and third paragraph the q-forward and s-forward from Loeys et al. (2007) will be discussed. In the fourth paragraph a method to derive risk neutral prices based on q-forwards described as in Plat (2009) will be discussed.

4.1 Current market products and transactions

Despite the demand for hedges for longevity risk there is no real liquid longevity market. The lack of a longevity market is partly a consequence of the fact that there is no standard for measuring mortality rates. To create some kind of a standard J.P. Morgan launched the LifeMetrics Index. The LifeMetrics Index consists of estimated mortality rates and life expectancies for different countries which can be used as a basis for valuation of longevity-linked and mortality-longevity-linked exposures and to determine the pay-off of longevity derivatives and bonds. Some examples of current longevity products or transactions in the market:

• In the UK a number of transactions have taken place between insurers and pension funds. For example Berkshire county for 1.3 billion US dollar, BMW for 4.6 billion US dollar and Rolls Royce for 4.7 billion US dollar.

• There are some products like mortality catastrophe bonds and the life settlements mar-ket.

• In 2012 one of the biggest longevity swap contracts in recent history is agreed upon between AEGON Group and Deutsche Bank with a value of 12 billion euro. Deutsche

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4.2. Q-FORWARDS

Bank has cut the contract in smaller packages and sold those to the market again. The expectation of Deutsche Bank is that the market will grow in the future.

4.2 q-forwards

The q-forward was first mentioned in Coughlan et al. (2007) and later on in Loeys et al. (2007). A q-forward is an agreement between two parties to exchange at a future date (the maturity of the contract) an amount proportional to the realized mortality rate of a given population, in return for an amount proportional to a fixed mortality rate that has been mutually agreed at the beginning of the contract. Hence a q-forward is a zero coupon swap that exchanges fixed mortality for realized mortality at maturity.

A pension fund that hedges its longevity risk expects to be paid by the other party if mortality rates decrease more than expected and is willing to pay if mortality rates end up higher, because its own cash outflows will then be less. This is illustrated in the following formula:

Net settlement = Floating amount − Fixed amount (4.1)

Here the Fixed amount = Notional×fixed mortality rates and Floating amount = Notional × realized mortality rates. The pension fund will receive the floating amount and pays the fixed amount. For the fixed q-forward rate the following approximation is proposed by Loeys et al. (2007):

q_x,tf or = (1 − t × Sharpe ratio × q_xvol) × q_x,texp (4.2)

Here q_x,texpis the expected mortality rate based on the chosen model and qvol_x is the volatility of the historic mortality rates. q_xvol is estimated as follows:

q_xvol= s X t 1 N − 1(qx,t− ¯qx,t) 2 _(4.3)

Here ¯qx,t is the average rate over t of mortality. The hard part is to find a fair mortality

curve that the pension fund and the investor agree upon. In the longevity market there are a lot more participants that want to hedge longevity risk than participants that want to invest in longevity risk. Hence to let investors enter the market the expected mortality rate should be decreased to introduce a so-called risk premium when calculating the q-forward. In the next section a method is described to derive a risk neutral mortality table. This mortality table is calibrated using q-forwards.

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4.3. S-FORWARDS

4.3 s-forwards

The s-forward was first mentioned in Coughlan et al. (2007) and later on in LLMA (2010). A s-forward is an agreement between two parties to exchange an amount at the end of the contract proportional to the realized survival rate of a given population, in return for a fixed survival rate that has been mutually agreed at the beginning of the contract. The net settlement at the end of the contract is illustrated in the following formula:

Net settlement = Notional × (sprealizedx,t −spf ixedx,t ) (4.4)

In this formula the survival ratespix,t with i ∈ {realized, f ixed} is given by:

Tpix,t= T

Y

t=0

1 − qi_x+t,t (4.5)

Here is T the maturity of the contract. A pension fund that hedges its longevity risk expects to be paid by the counterparty if survival rates increase more than expected and is willing to pay if survival rates end up lower than expected, because his own cash outflows will than be less.

4.4 Why longevity risk measurement matters: Solvency II

The current European regulation is stated in Solvency I. The goal is to have the same rules in all countries of the European Union. In 2009 Solvency II was approved by the European Par-liament. However Solvency II still has to be implemented. Under the Solvency II legislation insurers should have enough assets so that the probability that the insurer’s minimal funding ratio is at least met over a year is 99.5%. At this moment it is still unknown if the Dutch pension funds will be held to Solvency II rules as well. At this moment the Dutch pension funds are using the FTK (as stated in chapter 2) which handles a percentage of 97.5 instead of 99.5%. The uncertainty measure implies that the funding ratio should be prepared for a shock in for example the mortality rates. The Solvency II shock for longevity is assumed to be a decrease of the mortality rates by 20% (qx,t→ 0.8qx,t).

4.4.1 Remarks

This remark is written with upcoming changes in the Dutch pension system and the research question kept in mind. The adjustment mechanism in which the pension entitlements are lowered when the life expectancy increases is invented to protect the assets of a pension fund. The idea of Solvency II is holding an extra buffer to cover for a shock in for example mortality with a specified uncertainty. This will still decrease the assets if the life expectancy increases. It is not necessary to hold an extra buffer for life expectancy when lowering pension

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4.5. MARKET PRICES OF RISK

entitlements is an option. In case a buffer is held for longevity risk the uncertainty still remains and it stays unclear which part of the buffer is saved for the life expectancy increases. The pension fund still bears the longevity risk. Therefore this method is not implemented in this thesis.

4.5 Market prices of risk

Along with his model Plat (2009) a method to price longevity through risk neutral prices. The method is based on theory of Loeys et al. (2007) and Cairns et al. (2006a). The models described in chapter 3 are defined using the real world measures P. Plat defines a risk neutral measure Q to make it is possible to price mortality. In this section the same theory is applied on the Lee-Carter model and the AG Prognosetafel 2012-2062.

4.5.1 Plat model

Model

The ARIMA equations of the variables κ1_t, κ2_t, κ3_t and γt−x under the real world measure P

can be written in matrix notation as:

Kt= Θ + ΦKt−1+ ΣZtP (4.6)

In this formula Kt is the vector containing the following variables: κ1t, κ2t, κ3t and γt−x,

Φ contains the parameters of the AR(1)-processes, Θ is the vector with constants for the equations if applicable, Σ0Σ is a covariance matrix and Z_tP is a vector with standard normal random variables which are independent over time. Formula 4.6 can therefor also be written as:       κ1_t κ2_t κ3_t γt−x       =       θ 0 0 0       +       1 0 0 0 0 φ1 0 0 0 0 φ2 0 0 0 0 φ3             κ1_t−1 κ2_t−1 κ3_t−1 γt−1−x       +       a b c 0 d e f 0 g h i 0 0 0 0 1             ε1_t ε2_t ε3_t ε4_t       (4.7)

Under the risk neutral measure Q the dynamics can be written as:

Kt= Θ + ΦKt−1+ Σ(ZtQ− λ) (4.8)

where λ is a vector with the market price of risks which is assumed to be constant and Σ is the upper triangular matrix of the covariance matrix of κ1_t, κ2_t, κ3_t and γt−x, and ZtQ again

standard normal variables, independent over time.

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Calibration

Since there is no real longevity market Plat chooses the earlier introduced q-forwards of Loeys et al. (2007) as a basis for the longevity market. In line with Plat (2009) and Cairns et al. (2006a) the market price of risk λ is assumed to be contant over time. The risk neutral dynamics will be matched with the market using q-forwards.

The q-forward is defined the same as the formula in section (4.2).

qf or = (1 − horizon × Sharperatio × qvol) × qexp (4.9)

In line with Plat the Sharpe ratio is assumed to be equal to 25%, the horizon is chosen to be 10 years, the qvol is based on historical data and the qexp is based on the Plat model

(3.29). The mortality model of Plat is defined with forces of mortality. Hence qexp and qvol

are rewritten to 1-year mortality rates using formula (3.5).

E(µf orward) = gE(µf orward) (4.10)

Following the reasoning of Plat (2009) the E(µf orward) is assumed to be the expectation

under the risk neutral measure Q whereas the E(µexpected) is assumed to be the expectation

under the real world measure P.

The q-forward contract has a maturity of 10 years, hence the defined µx+10,t+10 is the

force of mortality at the end of the contract. When logarithms are applied to the equation 4.10 and if the assumption is made that the only difference between the underlying models for the risk neutral and real world measure is the drift term Plat argues that the equation can be rewritten as:

log[EQ(µx+10,t+10)]−log[EP(µx+10,t+10)] = EQ(log[µx+10,t+10])−EP(log[µx+10,t+10]) = log[g]

(4.11) Because the difference in the drift term is given by the matrix −Σλ the following will hold for a horizon k:

log[g] = EQ(log[µx+10,t+10]) − EP(log[µx+10,t+10]) = − k

X

t=1

WtΣλ (4.12)

Rewriting and solving this equation results in the following equation that leads to the market prices of risk:

ˆ λ = ( k X t=1 WtΣ)−1h (4.13)

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for κ1_t, κ2_t, κ3_t and γt−x to determine log(µx,t) and are given by:

Wt=       1 (¯x − 45) (¯x − 45)+ 1 1 (¯x − 55) (¯x − 55)+ ₁ 1 (¯x − 65) (¯x − 65)+ ct 1 (¯x − 75) (¯x − 75)+ 0       (4.14)

Following the reasoning of Plat (2009) the γt−x is only fitted for specific years of birth

the ct is 0 when t is less than 5 and equal to 1 otherwise. The model and calibration above

is developed for the Plat model in the following subsections the modelling and calibration is done for the Lee-Carter model and the AG Prognosetafel 2012-2062 too. The theory remains the same, but weights and covariation matrices will differ.

4.5.2 Lee-Carter model

Model

The ARIMA equation for Lee-Carter of the variable κt of the real world measure P can be

written as:

κt= κt−1+ θ + σεt (4.15)

in which σ is the standard deviation of model 3.13 and εtis error term which is distributed

standard normal, independently over t.

κt= κt+ θ + σ(εt− λ) (4.16)

where λ is the constant market price of risk.

Calibration

If the same methodology is followed and the same assumptions are made as stated in 4.5.1, then the equation for the market prices of risk in case of the Lee-Carter model remains the same in general. Hence the market price of risk is given by:

ˆ λ = ( k X t=1 WtΣ)−1h (4.17)

Since there is only one time series which is used to forecast the mortality rates the Σ and Wt both consist of only one element. The weight Wt is given by βx for a certain age x and

Σ is given by the σ of κt. Since the market price of risk is assumed to be constant over time

the equation should hold for each x. This assumption will be checked by estimating the λ for

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age 35, 55 and 75.

4.5.3 AG Prognosetafel 2012-2062

Model

The stochastic equation of the AG Prognosemodel 2012-2062 under the real world measure P is written as:

ln(µs_x,t) = ln(µs_x,t−1) + ln(µ_x,tAG) − ln(µAG_x,t−1) + σxεt (4.18)

in which σ is the standard deviation as given in equation (3.25) and εtis error term which

is distributed standard normal, independently over t.

ln(µs_x,t) = ln(µs_x,t−1) + ln(µAG_x,t) − ln(µAG_x,t−1) + σx(εt− λ) (4.19)

where λ is the constant market price of risk.

Calibration

If the same methodology is followed and the same assumptions are made as stated in paragraph 4.5.1, then the equation for the market prices of risk in case of the AG Prognosetafel 2012-2062 remain the same in general. Hence the market price of risk is given by:

ˆ λ = ( k X t=1 WtΣ)−1h (4.20)

In case of the AG Prognosetafel 2012-2062 there is only one stochastic trend serie defined as well which is used to forecast the mortality rates hence the Σ and Wtboth consist of only

one element. The weight Wt is given by 1 and Σ is given by the σx as defined in equation

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Chapter 5

Results

In this chapter the research and the results will be presented. The research is based on the theory described in the previous chapters. The calculations are done using the program R from the R Development Core Team (2011), the Lifemetrics 1 function fit701 for R for the Lee-Carter model and the R code function fit709 for the Plat model. The first section of this chapter states the assumptions made before and during the calculations. The second section discusses the results in a similar order as the previous chapters.

5.1 Assumptions

A couple of assumptions and simplifications have been made for the calculations which have been done. Simplifications are only made in case the simplifications are not expected to influence the outcome of the results. In this section the participant dataset, the demographic assumptions and the economic assumptions are discussed.

5.1.1 Participant data

The participant dataset is created in Excel making use of prespecified characteristics. The characterics were translated with a random generator into the different participant data files. The pension funds are assumed to be closed, so no new entries are assumed. The participant datasets of three pension funds were created in which the average age of the participants in the pension funds was the distinctive characteristic.

• Young pension fund: The average age of the participants lies approximately around 30 years.

• Average pension fund: The average age of the participants lies approximately around 45 years.

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5.1. ASSUMPTIONS

• Old pension fund: The average age of the participants lies approximately around 60 years.

The participants accrue old-age pension and spouse pension. The characteristics of the participants information are presented in table 5.1.

Pension Fund Young Average Old

Actives Male Female Male Female Male Female

Number 250 221 214 229 161 169

Average Age 34.1 33.8 41.6 42.1 54.6 53.9 Average OP 12,656 11,844 17,193 17,367 31,242 30,132 Average SP 8,859 8,291 12,035 12,157 21,870 21,092 Pensioners Male Female Male Female Male Female

Number 0 0 6 5 48 46

Average Age 0.0 0.0 65.2 65.1 73.8 74.1 Average OP 0 0 35,272 43,939 28,635 28,176 Average SP 0 0 24,690 30,757 20,045 19,723 Spouses Male Female Male Female Male Female

Number 15 14 20 26 36 40

Average Age 31.9 32.5 41.9 41.1 60.1 58.7

Average OP 0 0 0 0 0 0

Average SP 7,010 5,533 12,656 10,221 21,933 19,818

Table 5.1: Characteristics of the young, average and old pension fund.

5.1.2 Demographic and economic assumptions

In this section the demographic and economic assumptions are explained that were made for some of the calculations.

• Mortality rates: The mortality rates are based on the theory as described in chapter 3. The calculation date for all the calculations is 31 December 2012, hence the first year of the mortality tables which are used is 2013. Furthermore mortality experience is not taken into account, this will probably not influence the results in such a way that it will lead to a different conclusion.

• Age difference: An age difference of three years between males and females is assumed: males are three years older than females.

• Marriage: Each participant is assumed to be married.

• Interest Rate: The interest rate curve published by DNB as of 31 December 2012 is used. This a three month average swap curve including the ultimate forward rate of 4.2%.

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5.2. RESULTS

• Pensionable Age: The pensionable age is 65.

• Indexation: There is assumed to be no indexation in the calculations.

5.2 Results

In this section the results are presented which follow from the theory specified in the previous chapters. All the calculations stated below are based on the Dutch population2. The first step is modelling and forecasting the mortality rates for the Dutch population following the theory in chapter 3.

5.2.1 Mortality tables

The first mortality table that is presented is the stochastic version of the AG Prognosetafel 2012-2062. The second table is based on Lee-Carter method and the third table is based on the model described by Plat.

AG Prognosetafel 2012-2062

The AG Prognosetafel 2012-2062 as described in section 3.4 is reproduced in this thesis3. The next step was to transform the current model into a stochatistic model. For this transforma-tion the historic data from 1950 until 2011 is used to define standard deviatransforma-tions for the error terms for each age. The standard deviations σx are published in Appendix C.

Figure 5.1: The mortality rates for a 45 year old male based on the AG Prognosetafel 2012-2062 model.

The development of the mortality rate of 45 year old male is shown in figure 5.1. The graph shows that the trend is steeper at the beginning and that it becomes more flat in

2_{While programming some checks were done using other populations to compare the results with published}

papers.

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5.2. RESULTS

the later years. The steeper trend at the beginning of the curve is modelled because the realized mortality rates of the Dutch population show a steeper trend in the more recent years especially for males. However for now the expectation is that on the long run the mortality rates will return to the long-term trend which is more flat.

Lee-Carter model

The Lee-Carter model is estimated for the Dutch population using maximum likelihood meth-ods. This results in the estimates ˆαx, ˆβx and ˆκt. The plots of these estimates can be found

in figure 5.2.

Figure 5.2: The estimates of ˆαx, ˆβx and ˆκt are shown for males and females

The trend parameter κ shows the earlier mentioned steeper trend in the more recent years. To make a projection of the mortality rates the estimates of ˆκt are modelled using the

Box-Jenkins method to determine which ARIMA model fits best. The random walk with drift appears to be a good approximation of the mortality trend for females. The mortality trend for males is more difficult to model due to the increasing decline in mortality in the most recent years. Another model might give a better fit however the development of the trend in the future is uncertain so for simplicity reasons the random walk with drift is chosen for males too. The estimation of the parameters is done in R and the estimates are shown in table 5.2.

Random walk with drift for κt Male Female

θ -1.715 -1.515

Standard error 0.339 0.272

Table 5.2: Estimations of the parameter for the model ˆκt

The development of the mortality rate of 45 year old male with and without bootstrap

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5.2. RESULTS

are shown in figures 5.3.

Figure 5.3: The mortality rates for a 45 year old male based on the Lee-Carter model without bootstrap is shown left and with bootstrap is shown right.

The confidence intervals of the Lee-Carter model are smaller than the intervals of the AG Prognosetafel 2012-2062. Furthermore the projected mortality rates ends at a higher rate compared to the AG Prognosetafel 2012-2062. In the Lee-Carter model without the bootstrap the parameter uncertainty is not taken into account and this is visible in the narrower confidence intervals. The confidence intervals at the beginning of the projection are due to the parameter uncertainty.

Plat model

The Plat model is estimated using maximum likelihood methods as well. Resulting in esti-mates for ˆαx, ˆκ1t, ˆκ2t, ˆκ3t and ˆγt−x. The plots of these estimated parameters for males and

females are shown in figure 5.4.

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5.2. RESULTS

The ˆκ1_t, ˆκ2_t and ˆκ3_t are estimated using Seemingly Unrelated Reggression since these pa-rameters show some correlation. The ˆγt−x is independent of the other parameters therefore

ˆ

γt−xis modelled separately. The estimation of the parameters is done in R and the estimates

are shown in table 5.3: The found models for each of the estimates are described in paragraph

Estimates for the different models Male Female Model κ1_t - θ -0.0107 -0.0072 Standard error - θ 0.0027 0.0026 Model κ2_t - φ1 0.9035 0.9543 Standard error - φ1 _0.0417 _0.0105 Model κ3_t - φ2 0.8904 0.9643 Standard error - φ2 0.0443 0.0175 Model γt−x- φ3 0.9545 0.9184 Standard error - φ3 0.0265 0.0382

Table 5.3: Estimations of the parameters for the models of ˆκ1t, ˆκ2t, ˆκ3t and ˆγt−x

3.5.3. The corresponding covariance and correlation matrix of the four models for males can be found in table 5.4. As seen from tabel 5.4 especially the estimates of κ2_t which is the term

Males κ1_t κ2_t κ3_t γt−x κ1_t 0.00000718 0.00000098 0.00000161 0 κ2_t 0.00000098 0.00174217 0.00112307 0 κ3_t 0.00000161 0.00112307 0.00196527 0 γt−x 0 0 0 0.00320600 Correlation κ1_t κ2_t κ3_t γt−x κ1_t 1 0.0087 0.0135 0 κ2_t 0.0087 1 0.6069 0 κ3_t 0.0135 0.6069 1 0 γt−x 0 0 0 1

Table 5.4: Covariance and correlation matrix for males

that allow changes of the mortality vary over the ages and κ3_t which is the term to model the mortality rates for younger ages are correlated. The correlation between the estimates of κ1_t and the other variables is small and as stated before the estimates of γt−x are not correlated

with the other variables. The corresponding covariance and correlation matrix of the four models for females can be found in table 5.5. The table shows that the estimates of the parameters show different correlations compared to the estimates for males. For females the estimates of κ1_t, κ2_t and κ3_t are all more or less correlated. γt−x is again not correlated with

the other variables. To give a more practical view to the found estimates and for the first comparison between the models the development of the mortality rate of 45 year old male

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5.2. RESULTS Females κ1_t κ2_t κ3_t γt−x κ1_t 0.00000686 -0.00001070 -0.00000853 0 κ2_t -0.00001070 0.0001108 0.0000883 0 κ3_t -0.00000853 0.0000883 0.0003459 0 γt−x 0 0 0 0.00320640 Correlation κ1_t κ2_t κ3_t γt−x κ1_t 1 0.3883 -0.1866 0 κ2_t 0.3883 1 -0.4810 0 κ3_t -0.1866 -0.4810 1 0 γt−x 0 0 0 1

Table 5.5: Correlation matrix for females

with and without bootstrap are shown in figure 5.5.

Figure 5.5: The mortality rates for a 45 year old male based on the Plat model without bootstrap is shown left and with bootstrap is shown right.

The model of Plat shows a wider range of possible scenarios for the mortality rates for a 45 year old male compared to the Lee-Carter model for the model with as well as without the bootstrap. The wider range of possible scenarios can be explained in the fact that the Plat model is based on the Lee-Carter model with some extra parameters to cover for some disadvantages of the original Lee-Carter model, the extra parameters that needs to be esti-mated and modeled could cause the extra uncertainty. The AG Prognosetafel 2012-2062 has the largest confidence intervals compared to the other models. A fact that should be kept in mind is that the AG Prognosetafel 2012-2062 is originally a deterministic table which is transformed into a stochastic table.

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5.2. RESULTS

5.2.2 Market prices of risk

The three stochastic mortality models are set. The next step will be to determine the market prices of risk as stated in section 4.5. The market price of risk will be calibrated using a q-forward for each of the three models.

AG Prognosemodel 2012-2062

The market price of risk for the AG Prognosemodel 2012-2062 will be determined using only one age. Since there is only one term influencing the trend of the model in combination with the assumption that the market price of risk is constant over time suggests that for each age the same result should come up. In this case the starting age is set to 35, 55 and 75. This is done to check whether the assumption that the market price of risk is constant over time is valid. In table 5.6 the indication of the q-forward is given for a ten year horizon for males as well as females.

Sex Age start Age end qvol(%) qexp(%) qf or(%) µexp(%) µf or(%) h

Male 35 45 1.18 0.110 0.107 0.110 0.107 0.030 Female 35 45 1.45 0.101 0.098 0.101 0.098 0.037 Male 55 65 0.64 0.963 0.948 0.968 0.953 0.016 Female 55 65 0.87 0.697 0.682 0.700 0.685 0.022 Male 75 85 0.73 8.756 8.597 9.163 8.989 0.019 Female 75 85 0.58 6.363 6.270 6.574 6.475 0.015

Table 5.6: Indication q-forward rate for horizon 10 and translation to µ-forward based on the AG Prognosetafel 2012-2062

Applying formula (4.20) to the results of table 5.6 and C.1 will give us the market prices of risk for the AG Prognosetafel 2012-2062. The market prices of risk are shown in table 5.7 for males and females.

Market price of risk λ Male Female AG model (age start 35) 0.0099 0.0110 AG model (age start 55) 0.0072 0.0084 AG model (age start 75) 0.0077 0.0069

Table 5.7: Market prices of risk for the AG Prognosetafel 2012-2062

The resulting market prices of risk for the different starting ages show that the assumption that the market price of risk is constant over time seems to be invalid. There is a small difference between the market prices of risk for each different starting age. This will make the further conclusions in this thesis less valid, since the used model and assumption may not hold for the AG Prognosemodel 2012-2062.

Longevity risk in the dutch pension : transfer of longevity risk from the pension fund to the individual participant.

University of Amsterdam, Amsterdam

Longevity Risk in the Dutch pension

system

Transfer of Longevity Risk from the Pension Fund to the

Individual Participant

Contents

Chapter 1

Introduction

1.1

Motivation

1.2

Research objective

1.3

Paper structure

Chapter 2

Pension system in the Netherlands

2.1

Current pension system

2.2

Developments of the pension systems

2.3

Adjustment mechanism for life expectancy changes

Chapter 3

Mortality Modelling

3.1

Data and Assumptions

3.2

Lee-Carter model

3.3

Cairns, Blake, Dowd model

3.4

AG Prognosetafel 2012-2062

3.5

Mortality model Plat

Chapter 4

Longevity

4.1

Current market products and transactions

4.2

q-forwards

4.3

s-forwards

4.4

Why longevity risk measurement matters: Solvency II

4.5

Market prices of risk

Chapter 5

Results

5.1

Assumptions

5.2

Results