The effects of macro-economic indicators on the funding rate of Dutch pension funds

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The effects of macro-economic indicators on the funding

rate of Dutch pension funds.

Laurens Voogd

5964148

BACHELOR OF SCIENCE

supervisor: D.H.J. Chen MSc

May 19, 2014

Abstract

Prior to the financial crises of 2008, Dutch consumers confidently placed their retirement funds in the hands of the 314 Dutch pension funds. Since the beginning of the crisis, this confidence and trust has steadily been decreasing, mainly due to the steep drop in the funding ratio of the pension funds. This research investigates the main drivers behind this steep decline in funding ratio’s over the past few years. It concludes that interestrate -and longevity -risk are key determinants in the state of the funding ratio’s of Dutch pension funds.

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1 Introduction

The Dutch pension sector contains 314 registered pension funds, which collectively own 1007 Billion euro, which equals 168 percent of the Dutch Gross Domestic Product (GDP). From a social welfare point of view, these enormous amounts of funds need to be allocated and invested in such a way that they will provide stable returns over time. This is however by nature conflicting with the financial goal to maximize short-term returns, or, in other words, to increase the funding ratio. One of many goals of a pension fund is indeed to maximize the participants wealth. This goal in the long run may correspond with stable returns, but in the short run is often associated with more riskier investment allocation, which can result in a volatile return pattern over the years.

When a pension fund falls below a funding rate of 105 percent, the rights of fund par-ticipants are promptly impaired, e.g. by withholding indexation. This may result in (in-tergenerational) asymmetric risk sharing, since higher premiums negatively affect active participants, while withholding indexation negatively affects retired participants. It has been shown by Bikker and Knaap 2011, that a nonlinear relationship exists between the pension funds policy response to changes in their funding rate. This can be either explained as unwillingness to react to changes in the funding rate, or as a simple lack of understanding of the behavior of the funding rate.

The Dutch pension system has been build on three pillars, the first being a public pen-sion scheme, which pays an individual arriving at retirement age a flat rate. The second is a scheme that sets apart a percentage of the salary earned by an individual during his working life, which will form an extra pension during his retirement years. Finally any individual is free to save funds for future spending.

Pension funds and insurers are operating in a climate of multiple financial requirements and frameworks, in which they seek an optimum balance between regulatory requirements and participants wealth. Reporting and valuation standards and requirements such as Interna-tional Financial Reporting Standards (IFRS) and Insurance Group Directive (IGD) heavily influence pension funds and insurers on their valuation methods, risk appetite and liquid-ity requirements. Due to large differences within the pension and insurance industry, with respect to risk appetite and financial goals, authorities have taken increasingly more rigid measures by creating liquidity requirements and maintaining close contact with firms, in order to monitor their risk mitigating strategies. Pension funds and insurers are required to report to the Dutch Central Bank (DNB) on a quarterly basis, and in case of financial dis-tress, they are obligated to share short- to medium term plans with the DNB. The purpose of these plans is to give insight in the pension fund or insurance’s strategy for maintaining and increasing the financial health of the company.

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1.1 the funding rate

The DNB, defines the funding rate -at time t- as {F Rt}_t>0= T otalAssets T otalLiabilities = At Lt t>0

The DNB differentiates between three classes of funding rates. • Insufficient funding ratio: {F Rt} < 105% (at time t)

• Tight funding ratio: 105% < {F Rt} < 130% (at time t)

• Sufficient funding ratio: 130% < {F Rt} (at time t)

The way in which pension funds value their assets and liabilities differs within the dif-ferent regulatory frameworks. It can occur that a pension fund has a funding rate of 110% in the IFRS framework, where it has a 115% funding rate in the IGD framework.

The differences between the various financial frameworks and requirements lead to an intrinsic hedge problem. Normally a pension fund seeks a prof it&loss (P&L)-neutral hedge. Due to these different frameworks, pension funds are required to valuate their assets and lia-bilities by different discount factors. A ”perfect” interest -duration- hedge in one framework, is likely to result in an inadequate hedge in the other. This creates a conflict of interest since investor?s normally higher value the IFRS framework, where in contrast, authorities prefer the IGD framework. It is often the case that pension funds choose to hedge an asset class with a liability class in one framework, and another asset- and liability class in another framework.

clearpage

1.2 the driving factors of the decline in the funding rate of

Dutch pension funds

Prior to the financial crisis, 94 percent of the Dutch pension fund participants, participated in a fund that had or more funding rate of at least 105 percent. In contrast, at the end of 2009 this percentage was reduced to only 3 percent of the Dutch fund participants. The figure below, shows the time series (1988-2012)of the weighted average funding rate of the Dutch Pension funds.

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The reasons for this decline in the funding rate can mainly be found in changes in the term structure, longevity and the downfall of the financial markets during the crisis. Naturally there exists a duration overhang in the portfolio of pension funds, since the maturities of their liabilities exceed those of their assets. As pension funds continuously try to duration hedge their assets and liabilities, the hedge will be insufficient when big changes in the term structure occur. ln(F Rt) = ln(At) − ln(Lt) ⇔ dln(F Rt) drt = dAt drt At − dLt drt Lt = −DA+ DL⇔ dF Rt drt = F Rt(−DA+ DL)

Where DA, DLand rt denote the duration of the assets, liabilities and the interest rate

respectively. As the interest rate with which the portfolio is discounted decreases, the values of the assets and liabilities increase. However, due to the maturity mismatch, liabilities will increase at a faster pass than assets will.

As a descriptive measure, the graph below depicts the 3-month’s, yearly average interest rate, included in this research.

It clearly depicts a, though volatile, steady decrease in the interest rates. In context of this research, does it already give an explanation to why the funding ratios of the Dutch pension funds declined. The duration mismatch in combination with the downfall in the interest rates, resulted in a higher book value of the liabilities of the pension funds, which hence resulted in lower funding ratios.

Figure 2: The development of the yield on investment over the sample period of 1988-2012 This research looks at the longevity development over the years as a second explana-tion for funding rate development. There are two phenomena that heavily influenced the development of the funding rate over the last few years. First, the birth rate burst that occurred short after the WWII resulted in a large flux of active workers becoming retired. More mature pension funds result in an imbalance between premiums received and contribu-tions paid by the funds. As a second reason, the life expectancy, is continuously increasing. Shown in the figure below, this development is occurring at a fairly constant rate. Since the discount function is a combination of both the term structure and mortality rates, does this development negatively influence the funding rate of pension funds.

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Figure 3: The development of average male+f emale₂ life expectancy in years over the sample period of 1988-2012

The influence of longevity developments is being enforced by the fact that pension funds only recently started using dynamic mortality tables. These tables dynamically estimate life expectancy instead of doing this statically and refreshing the table every few years. By relying on static mortality tables, pension funds mispriced the costs of the defined benefit (DB) contracts they offered their customers. Due to a large peak in post WWII birth the decline in the funding rates decreased even more.

Finally, and perhaps the most obvious reason, was the beginning of the financial crisis. Large financials came under financial distress, stock markets took a steep dive downwards and money markets froze. The sub-prime mortgage crisis in the US was one of the primary reasons causing this crisis. Securitizations of US mortgages heavily toxicities the derivatives market. Unknowing the portfolio structure of counter parties, financials became unwilling to lend money, except to economic stable countries.

Each of the above factors listed above could individually cause a downward spiral of the funding rates. As a group, the three factors had a dramatic impact on the funding rates of the Dutch pension funds.

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1.3 Counter measures

The dramatic drop in funding rates over the years required authorities and pension funds to take action. In principle, pension funds can take three actions to remedy this.

• Increase the premiums paid by their active participants.

• Make workers unions deposit large amounts, to improve the funding rate • Withhold indexation and reduce the rights of retired participants

An increase in the premium of active participants seems a logic measure to resolve funding rate shortages. However, this measure only makes the active participants worse off, while all other parties of interest such as sleepers -which are active workers who switched to a other pension fund, but remain having rights in the pension fund- and retired participants remain unaffected by the increase in the premium. And although this counter measure helps recovering the funding rate position of the pension funds, it takes a lot of time and is therefore not a short term solution for the funding rate problem. Alternatively, pension funds can request workers unions to make large deposits into the financially troubled pension fund.

Pension funds can also withhold indexation. This measure, however, has a direct negative impact on the wealth fair of the retired participants of the pension fund. They immediately find themselves being less able to maintain their consumption habits.

The chapters above aim to explain what a funding rate is, what might cause it to change and what can be done to change it. The goal of this paper is to determine whether there exists a relationship between macro economic indicators and the funding rate. If this is the case, it will hopefully result in a better understanding of the behavior of the funding rate. Currently, pension funds remain confronted with a non-linear long response time compared to changes in their funding ratio. If the underlying behavior of the funding rate can be explained by macro economic indicators, it might help to stabilize the funding rate in the sense that pension funds can adjust their policy much faster (e.g. at a more suitable point in time).

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2 Literature review

In this paper, the macro economic indicators are defined as being consumer price index (Chit), economic growth (Get), average life expectancy (Let) and (one-year) interest rates (rt).Based on option theory, it can be shown that the interest rate and longevity, play a

vital role in the level of the funding rate. Since they both often constitute the underlying of derivatives, owned by pension funds to hedge their risks.

2.1 Interest Rate Theory

As mentioned in the introduction of this article, the steady decline of the term structure as a whole not only resulted in an increase of the value of the short to medium term assets of pension funds but also, and even more, impacted the long run liabilities (when valued market consistently).1 _{Prior to 2004, Dutch pension funds were relatively free in choosing}

their preferred discount rate for both assets and liabilities. Inconsistencies between the rates utilized to value the assets and liabilities result in a distorted insight into the future financial situation.

To prevent this from happening and creating a more stable insight into the actual funding rate of a pension fund, Dutch pension funds have, since 2006, been allowed (or required) to discount their liabilities with a market consistent swap curve, instead of a constant 3- or 4 % rate. Assets, who were already valued market consistently, did not experience many differences after the more scrutinized regulatory measures. This fair value approach, gives a more transparent insight into the financial position of the pension fund, in the sense that a market consistent valuation should contain all available information in the market. While the fair value approach was more consistent, it became heavily related to the volatility of the financial markets.2 This volatility and the fact that the long-term market is only modestly liquid, caused the Dutch authorities to propose an alternative in the name of the Ultimate Forward Rate (UFR). The UFR proposes to discount long run liabilities with a discount factor based on expert opinion instead of a market consistent rate. This rate was set at approximately 4.2 %. This proposal has been heavily debated and resulted in a lot of alternative methodologies to overcome this problem. The main topic of argument was the convergence rate at which the UFR and swap curve, converge. Several suggestions have been made, ranging from 10 years3, to 40 years4. Furthermore the introduction of the UFR will most likely result in a bimodal swap curve, since the UFR will dramatically increase the demand for the last liquid point (LLP) on the swap curve.

2.2 Longevity

This effect is not only statistically highly significant but also economically: each year of additional life expectancy would increase private U.S. DB pension plan liabilities by as much as 84-USD billion.5 Longevity risk is the risk that life expectancy will be greater then expected. Traditionally, pension funds have been using static mortality tables. However, it

1_{Sundaresan and Zapatero, 1997; Lucas and Zeldes, 2006, 2009; Benzoni, Collin-Dufresne, and Goldstein,}

2007)

2_{L. Rebel, The Ultimate Forward Rate Methodology to Valuate Pensions Liabilities:}

A Review and an Alternative Methodology, 2012

3_{European Parliament, 2012} 4_{EIOPA, 2012}

5_{Kisser. M, J. Kiff, The Impact of Longevity Improvements on U.S. Corporate Defined Benefit Pension Plans,}

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has been shown, that over a 20-year forecast, these mortality tables on average underestimate the life expectancy by 3-years.

While there are multiple alternatives to hedge longevity risk, these types of hedge strate-gies are complex and pension funds may easily over- or under hedge these risks. DB pension plan providers can hedge longevity risk using market-based solutions such as pension buy-ins, buy-outs, securitization, longevity swaps or longevity bonds. Bikker and Blake (2009) provide a detailed comparison of the various tradeoffs involved across the different method-ologies and products.6

The figure below, shows an 24-year forecast based on the sample used in this paper. The figure clearly indicates that there is a large variability in the realizable, actual life expectancy.

Figure 4: A dynamic forecast of life expectancy

Despite the hedge opportunities, pension funds are currently switching to dynamic mor-tality tables, which account for the stochastic dynamics of life expectancy.

2.3 Macro Economic Indicators

The pension sector as a whole, largely influences macro economic conditions. Political measures taken by the Dutch government can have a huge impact on the pension sector. For example changes in the retirement age. Changes in these types of standards cause major changes in consumption, tax receivables and the economic life cycle as a whole.

The Dutch central government plays an important role and has a large interest since its decisions greatly affect the tax amount receivable by the government. Up to now, little research has been done on the macro economic indicators and their relationship with the funding rates of pension funds. While a lot of research is done within the macro economic field, and the relationship between indicators, they are not considered in a pension funding rate framework.

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3 Methodology

The methodology used to analyze the relationship between macro economic indicators and the funding rates of Dutch pension funds, will be based on econometric methods. This anal-ysis will concern several types of methods. The analanal-ysis will consider vector auto regression model (of order n) (V AR(n)), auto regression models (of order n) (AR(n)), autoregressive moving average models (of order nap) (ARM A(n, p)) and finally Feasible Generalized least squares (F GLS). The AR(n) and ARM A(n, p) are good starting points, due to the nature of the data used in this research. We will take a closer look into the stationareity of each time series data on its own and also briefly consider the statistical properties of the pooled time series data. A follow up to this analysis is the V AR(n) analysis. In the paragraph V ariables, this research will focus on whether or not there exists a large codependent struc-ture within the sample. Finally in the F GLS part of the analysis we will take a more static approach to the model estimation and furthermore will test for heteroskedacity, endogeneity and serial correlation.

3.1 Data

The data used in this paper, consist of the following variables. • {F Rt}t>0 The funding rate at time t

• {EGt}t>0 The economic growth at time t

• {LEt}t>0The life expectancy at time t

• {CP It}t>0The consumer price index at time t

• {Tt}t>0 The time at time t

The sample size runs from 1988 to 2012. In the analysis, we consider F Rt to be the

dependent variable, where all other variables are considered as independent variables. Also will a pooled time series analysis be discussed, in which a simple AR(n) model will be estimated of the form Y et = α +Pn

i=1φiYt−i+ t, where Yt= (LEt+ rt+ CP It+ EGt).

Data Collection The {F R}t data set is collected from the Dutch Central Bank. This

data is not publically available and was provided by J. Bikker.7

The one-year interest rates {r}t, which are publically available, also were collected from

the DNB.8. They are represented as a yearly-observed time-series. The single observation was obtained by averaging all the one-year interest rates, revealing in that particular year. All other data was collected from The Central Bureau for statistical data (CBS).9There have been performed any alterations to the data.10

Data Summary To create a better understanding of the data this research is concerned with, the table below obtains some key statistics of all relevant variables.

7_{http://www.dnb.nl/onderzoek-2/onderzoekers/overzicht-persoonlijke-paginas/dnb150116.jsp} 8_{http://www.statistics.dnb.nl/cgi-bin/grafieken/grafiek.cgi}

9_{http://www.cbs.nl/nl-NL/menu/cijfers/statline/informatie/default.htm}

10_{The data that was provide by J. Bikker from the DNB, has been altered before receiving the data. The F R} t

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Statistics FR LE EG CPI R Mean 159.59 78.54 2.13 2112. 3.89 Median 154.00 78.06 2.20 2082.90 3.29 Maximum 265.42 81.01 4.70 2678.80 9.27 Minimum 95.46 76.80 -3.70 1570.80 0.23 Std. Dev. 44.93 1.42 1.99 345. 2.54 Skewness 0.43 0.51 -1.04 -0.01 0.68 Kurtosis 2.52 1.82 4.08 1.71 2.73 Jarque-Bera 1.02 2.53 5.75 1.71 2.03 Probability 0.59 0.28 0.05 0.42 0.36 Observations 25 25 25 25 25

Table 1:Descriptive statistics

Data Type The data collected is time series data. It runs over 24-periods. This is a relatively small sample, and hence deserves some extra attention when performing for example tests that are only asymptotically distributed chi-squared. Each variable in the data set will be briefly discussed on its nature below.

• {F R}t : Over a period of 25-years, a weighted average of pension fund’s their funding

ratios.

• {r}t: Over a period of 25-years, the 4-moonths average interest rate is observed once

every year.

• {CP I}t: The base year of 1980, started at 1000. Since 1988,

the CPI is observed once every year.

• {EG}t : Economic growth is observed yearly for 25-years. It is obtained by _GDPGDPt

t−1,

where GDP stand for Gross Domestic Product.

• {LE}t : A average of male and female life expectancy was observed over 25-years

Sample Here we consider possible biases that may exist due to the selection of our sample. The funding rate data was obtained from a weighted average of public pension funds.

3.2 Variables

As already slightly touched upon, the explanatory time series data are related to the ex-plained funding rates, but likely also related to each other. This might cause a multicolin-earity problem, which will be tested for further on in this Research. Constructing a simple correlation matrix, our attention immediately goes to the large negative correlation between LEt and F Rt of −0.87, the large positive correlation between rt and F Rt of 0.80 and

be-tween CP Itand F Rt of −0.87. These are some hopeful preliminary results for our analysis

later. j = f r le eg cpi r ρf r,j 1 -0.87 0.47 -0.87 0.80 ρle,j -0.87 1 -0.45 0.96 -0.76 ρeg,j 0.47 -0.45 1 -0.44 0.37 ρcpi,j -0.87 0.96 -0.44 1 -0.80 ρr,j 0.80 -0.76 0.37 -0.80 1

Table 2: Correlation matrix

In contrast however, is there also a large correlation within the explanatory variables. For example between CP Itand LEtthe correlation is 0.95. Especially when we are applying

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ΩF R = ΩXβ + Ω, stability of the matrix (ΩX0ΩX)−1requires additional attention.11

Next to the endogeneity within the F GLS setting, stationary within the dynamic esti-mation is also a concern. Especially for the LEt data is the series believed to diverging.

This is perhaps contra intuitive, since no one can live forever, but from the sample used in this paper, it is not possible to determine a mean reversion point.

11_{When the nature of the variances of the error terms are unknown, but assumed to be heteroskedastic. The}

Ordinary Least Squares (OLS) method is ”extended”, by an additional least squares estimation, namely the one of

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4 Analysis

In this analysis, several models will be specified and tested. Each specification will, when appropriate, be tested on hetereskedacity, endogeneity, serial correlation, stability, omitted variables and the distribution of the error terms will be reviewed.

4.1 The Model

AR(1) Model for F Rt The argument that the current funding rate F Rt is a best

estimate of the future funding rate F Rt+1, seems reasonable. It seems more likely that if

the average funding rate is currently 100 %, the funding rate in the next period will be close to 100 % instead of for instance 200%. To check whether the current F Rt is a sufficient

estimator for the future, we are interest in the behavior of the error terms, t= F Rt−F Rr−1.

Testing whether t∼ N (0, σ2), the Jarque Bera12

is performed. First robustly estimating the model, yields dF Rt= 0.97F Rt−1+ t. Based

on the sample has a J B − statistic equal to 2.12 with corresponding p − value of 0.34. Clearly there is not sufficient evidence to reject the normality assumption of the error terms. Next a Ljung Box Test13 is performed. This test determines whether there exists serial correlation within the timeseries, which will determine whether t can be considered to

be white noise. The LB − statistic is 16.95. There is not enough evidence to reject the H0: ρ0= ρ1· · · = ρn= 0. Finally we are interested if there is a deterministic or stochastic

trend in {F Rt}t>0. Using the Unit root test (Dikey Fuller test)14, a deterministic trend

is found. However, due to the size of the sample, these results may not be accurate. In summary, we can, based on our tests, assume that the innovation process, consist of white noise 15, and furthermore do we assume that {F Rt}_tt>0 is a stationary timeseries. The

nature of our trend is (weakly) proven to be deterministic. This result should not be built upon, since the unit root test is fairly inaccurate when performed on small samples.

AR(1) Model for LEt The {LEt}t>0depicted in the introduction (p.8), shows a steady

increase in the life expectancy over 1988 − 2012.A (robust) estimation of the AR(1)-model, yields the following model LEt= 1.00LEt−1+ t.

The Coefficient is highly significant and has a p − value close to 0. Since pension funds are especially interested whether or not there is a point of convergence in the development of life expectancy, stationarity is tested. First considering the distribution of the innovation process, the J B − test is performed and results in the conclusion that the error terms are normal distributed, and based on the LB − static with no serial correlation. Performing a U R − test, there is significant evidence, that there exists an stochastic trend in {LEt}t>0.

The above results are surprising in the sense that the innovation process was expected to have large scenes rightward. The stochastic trend, as already discussed in section 2, has great complications for pension funds. The static mortality table approach, is clearly a insufficient one, especially due to the very significant evidence of a stochastic trend.

12_{Kazuyuki Koizumi,On Jarque Bera test on assesing multivariate normality. Under the H}

0: ∼ N (0, σ2),is

d

S() = _{(T −1)σ}1 3

PT

i=1(i− ¯)3 distributed as N (0.6/T , and dK() = _{(T −1)σ}1 4 PT i=1(i− ¯)4− 3 is distributed N (0.24/T ). Hence J B =S()d 2 6/t + d K()2 24/T ∼ χ 2

2. For large values of J B, H0: ∼ N (0, σ2) is rejected. 13_{http://www.burns-stat.com/pages/Working/ljungbox.pdf: The LB test tests H}

0 : ρ0 = ρ1· · · = ρn = 0,

against at least one ρi6= 0, i = 1.2, · · · , n

14_{http://faculty.washington.edu/ezivot/econ584/notes/unitroot.pdf:} _{The Unit root test, test H}

0 : ρ =

0(stochastictrend), againstH1: ρ < 0(deterministictrend) 15_{White noise are error terms,}

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AR(1) Model for rt The behavior of the term structure will hopefully confirm

economic-and mathematical finance theory.

In mathematical finance, the interest rate developments are normally modeled by rt =

k(µr− rt−1) + σrWt, where Wt∼ N (0.1)∀t > 0 .a16

The model is estimated to be rt= 0.14 + 0.98rt−1 (note rtis represented in percentages.

While the constant has a bigh P − value, is the estimated coefficient of rt−1 very significant

with a P − value near zero.

This estimate is significant and has a p−value approximately zero. Based on the sample, there is sufficient evidence to reject the stationarity assumption. The LB − statistic is equal to 36.55, which rejects the H0: ρ0= ρ1· · · = ρn= 0.This agrees with mathematical finance

theory, since in this theory it is assumed for interest rates to behave as Wiener processes17, whose volatility is time dependent.

AR(1) Model for CP It CP It, rtandEGtare closely related and likely to create a

lin-earity problem in the F GLS analysis. In this paragraph only the same relations are tested as in the paragraph of rt. We specify a AR(1) model, CP It = φCP It−1+ t. The model

is estimated by CP It = 1.02CP It−1+bt. Testing whether or not the innovation process is white noise, we again performed a J B − test , which resulted in a J B − statistic equal to 2.63. There is not sufficient evidence to reject the H0 of normality. If we further more

perform a LB − test, we find the LB = 78.44, which is enough evidence to reject the sta-tionarity assumption. If we finally determine the nature of the trend within the series the outcome equals U R = 0.38 , which is insufficient evidence to reject a stochastic trend.

AR(1) Model for EGt We end this section with the analysis of the economic growth

{EGt}t>0series. Estimating a AR(1) model, yields EGt= 0.74EGt−1+bt. The innovation process can be assumed to be white noise, since the J B−statistic = 3.36 andLB−statistic = 12.91, which is not sufficient evidence to reject the uncorrelated normality assumption. The series have a deterministic trend since the U R−statistic = −2.95, which is sufficient evidence to reject the H0 : EGt= stochastictrend.

AR(!) Model for Yt= (EGt+ CP It+ rt+ LEt)0 Pooling the explanatory timeseries

might become useful in later analysis. In this paragraph we obtain an idea of the behavior of this series. Estimating a AR(1) model yields Yt= 1.02yt−1+bt. The error terms bring forth a J B−statistic = 2.66 and a LB−statistic = 14.66, which confirms a white noise innovation process. Determining the nature of the trend of the series, we find U R − statistic = 0.25, which suggests a stochastic trend.

The purpose of the above paragraphs was to create a better understanding of the nature of each of the timeseries individually and combined as a group.

Feasible Generalized Least Squares (FGLS) The {F Rt}r>0 series, show some

rather volatile behavior the sample period. The sample range

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is approximately 170%.This large variation occurred only over the total span of the period. The funding rate started at a 200 % plus level, and steadily declined to below 100 %. We specify our model by

F R = α + βLELE + βCP ICP I + βrr + βEGEG + , ∼ N (0, σ2Ω)

F GLS − estimation, yields a model containing not significant coefficients. Removing the constant from the model and F GLS − estimating slightly improves the model making

16_{Vasicek, http://www.emis.de/journals/HOA/JAMDS/8/11.pdf} 17

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βLE significant. The next step is to remove additional variables from the model by reducing

it to F R = βLELE + βrR + βEGEG + . The result is however still a model containing

insignificant coefficients except for the interest rate and life expectancy. Using CP IandEG as instruments for r we Two Stage Least Squares (TSLS) estimate the model and obtain a significant model with coefficients α = 1234.63, βr(CP I,EG)ˆ = 982.43andβLE= −14.17.

This research does not further focus on this model, since, as suspected, it does not meet the standards of a valid model. Once again, this conclusion was reached by improving our understanding of the timeseries and their relationship with each other. We feel that a vector autoregressive model would be the suitable model specification for creating a valid model.

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V AR(n) Model Discussing the Vector Autoregressive Modell (VAR), first, it is noted that it makes more economics sense to assume that the funding ratio in period t + 1,F Rt+1 is

likely to be influenced by both the level of the funding ratio at time t,F Rt, and changes in the

included macro-economic parameters, ∆rt, ∆LEt, ∆EGt and ∆CP It19. Later on into the

VAR-analysis, the research will therefore focus on the dataset D∗ = {F Rt, ∆rt, ∆EGt, ∆LEt, ∆CP It}.

While the above is noted, does the analysis start off with the VAR-model, based on the standard variables, D = {F Rt, rt, EGt, LEt, CP It}. The degree of the VAR-model is based

on both the Aikaike information Criterion (AIC) and Schwarz Information Criterion (SIC). We consider three alternatives, VAR(1),VAR(2) and VAR(3). These models are evaluated on their AIC and SIC, the one with the smallest AIC and SIC will be further analysed. The AIC is defined as AIC = N ln(

PN i=1e2i

N ) + 2k

20_{, where the SIC is defined as SIC =}

−2ln(L(F Rt| β) + kln(N ), where N = #observations = 25 and k = #parameters + 1

The AIC and SIC evaluation, resulted in the following outcomes. D D∗

AIC-prefered 1 2 SIC-preferd 2 2

The analysis of AIC and SIC had a scope of VAR(1), VAR(2) and VAR(3). From the table above, it can be seen tat the two-lagged model is preferred in most of the cases. Hence from now on will the research focus on the VAR(2)-model.

As discussed above, there is a great codependent structure within the timeseries pool. As can be seen from the correllogram there is a great correlation between the explanatory and explained variable, also within the group of explanatory variables. For this reason a vector autoregressive model seems a suitable way to specify a model. The model is defined by

Πt= Ψ + Λ1Φt−1+ Λ2Φt−2+ t

Where Φt = (F Rt, rt, LEt, CP It, EGt)0, Ψ a vector containing constants, Λi a 5x5 −

matrix, i = 1.2 and a vector consisting of error terms t ∼ N (0, σ2Ω). The output

resulting from the model estimation is significant and intuitively understandable. Based on the J B − statistic and LB − statistic the error terms shown are normally distributed, with no serial correlation.

The model is described in the table below.

F Rt rt LEt EGt CP It C 515.94 4.14 21.86 101.53 -473.46 F Rt−1 -0.47 0.02 0.00 0.01 0.70 F Rt−2 -0.17 -0.00 3.72E-05 -0.00 0.47 rt−1 3.06 1.18 0.04 -0.39 9.71 rt−2 5.57 -0.79 -0.01 -0.11 -19.93 LEt−1 -12.40 2.09 0.47 1.01 -15.58 LEt−2 12.87 -2.09 0.19 -2.24 18.86 EGt−1 2.71 -0.25 -0.00 0.02 -8.69 EGt−2 -0.62 -0.01 -0.01 0.05 1.30 CP It−1 -0.42 -0.00 -0.00 -0.05 1.29 CP It−2 0.27 0.00 0.00 0.05 -0.23 Log likelihood -95.66 -20.57 15.74 -28.68 -81.39 Akaike AIC 9.27 2.74 -0.41 3.45 8.03 Schwarz SIC 9.81 3.28 0.13 3.99 8.57 Estimated V AR(2)-model

19_{The ∆-term is considered as a forward difference in the sense that it represents ∆(X}

t) = Xt+1− Xt 20_{Since the error term}

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In line with theory, the interest rate does heavily influence the funding rate of a pension fund. Surprisingly however, the second lag has even a greater impact then the first lag. The first interest rate lag however is more significant than the second lag. Both interest rate lags have a positive effect on the funding rate. The life expectancy lags on the other hand both have a negative influence on the funding rate. The EGt series have a positive effect

on the funding rate for the first lag, but a small, negative effect for the second lag. From an economic perspective, this conclusion seems reasonable explainable due to the cyclical nature of economic growth. Finally, the CP It series in both lags have little effect on the

funding rate. Perhaps unsuspected, the first lag does have a small negative effect while the second lag has a small positive effect. From an economic perspective, there is no straight forward explanation for this result, since the CP It series is not captured by any cyclical

motion.

As was already predicted, does it make more economics sense to predict future funding ratios, based on the current one, and changes in macro economic factors. For this reason the research focuses on the following data set, D∗ = {F Rt, ∆rt, ∆EGt, ∆LEt, ∆CP It}.

Considering the information criteria for this data set, the AIC and SIC indicate that the model is best specified when it is of order 2. Therefore, the research directly focuses on this model.

Φt= Ψ + Λ10Φt−1+ Λ20Φt−2+ t

Where Φ = (F Rt, ∆(LEt), ∆(rt), ∆(CP It), ∆(EGt))T

This model gives even more significant parameters estimates and can be easily explained via economic theory. It is noted that the second lag L2∆rt has almost five times as much

effect on the funding rate then the first lag L∆rt. This result may be explained by the

pass-through speed at which financial institutions offer revealing rates. Especially upward shifts in the term structure are known to slowly make its way to other parties active on the financial markets. This pass-through effect can result in in slowly changing the value of the assets and liabilities, owned by the pension funds. This observation would suggest, that pension funds are not completely aware of the duration of the assets and liabilities in their portfolios. In contrast, the first lag longevity change is more than thirty times greater than the second lagged longevity change. This result may be accounted for by the fact that pension funds often revalue their liabilities after changes in mortality tables. In other words, the funding ratio is already corrected for changes in life expectancy that occurred two periods ago. From the statements above, it can be concluded that the effects caused by changes in life-expectancy and interest rates, differ in their transparency. Changes in life-expectancy, relatively quickly find their workings in the portfolio held by pension funds, while changes in interest rates take at least one period to work their way into the portfolio owned by the pension fund.

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F Rt rt LEt EGt CP It C 63.10 1.50 0.47 -2.49 57.33 F Rt−1 -0.05 0.01 0.00 0.03 0.20 F Rt−2 0.81 -0.02 -0.00 -0.02 -0.22 ∆Rt−1 2.50 0.49 -0.02 -0.40 13.50 ∆rt−2 12.33 -0.32 0.01 -0.62 -7.79 ∆LEt−1 -48.55 1.27 -0.13 3.50 -25.52 ∆LEt−2 -8.83 -1.67 -0.04 0.78 -33.80 ∆EGt−1 3.33 -0.14 0.02 -0.51 -4.16 ∆EGt−2 0.16 -0.10 -0.00 -0.16 -0.99 ∆CP It−1 -1.01 -0.01 -0.00 -0.03 0.34 ∆CP It−2 0.60 -0.01 0.00 0.02 -0.17 Log likelihood -97.29 -23.77 9.69 -32.22 -82.86 Akaike AIC 9.84 3.16 0.11 3.92 8.53 Schwarz SIC 10.39 3.70 0.66 4.47 9.07 Estimated ∆ VAR(2)-model

The graph below depicts the prediction power of the estimated model and on the other hand, obtains the behavior of the residuals. The graph solely depicts the first element of the VAR(2)- model in the sense that it only considers the prediction power of the series {F R}t.

{small Figure 5: Prediction power of the ∆V AR(2)-model

Weak aspect of the analysis Several weaknesses were already stated as risks but not further explored in the analysis. The correlation matrix for example shows high correlation within the group of explanatory variables, however, we did not further test for multicolin-earity. Furthermore, the possibility of endogeniety has not been further investigated in this research. It was shown by the T SLS − estimation, that both CP Itand EGt could be used

as instruments for rt, relevance or validity was never confirmed or rejected. The research

does not contain any cointegration − tests21. This is for obvious reasons, since the data set used, contains too little data points, to adequately perform these types of tests. Finally, the question whether or not the VAR-models may overfit 22. the actual relationship between the explained and regressor variables, is left untouched.

21_{Cointegration tests of order p, tests whether there is a relationship between a timeseries yet and a p-lagged}

timeseries Lp_x

t= xt−p.

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5 Conclusion

Theoretically, a relationship between longevity, interest rates and the funding rates has long been established. The research described in this thesis (based on the sample obtained from the Dutch central bank) indeed confirms the existence of these relationships as prescribed in the theory. In addition, this research addresses to what degree these relationships exist. The final model that is specified in this research Φt= Ψ + Λ10Φt−1+ Λ20Φt−2+ t, was most

significant and from a economic perspective, makes perfect sense. The funding rate level, reacts -heavily- in a negative direction when a upward change in life expectancy occurs. A second order time lag of the life expectancy change also negatively influences the funding rate. For interest rates, this relationship is opposite. A change in the interest rate is magnified five times in a upward movement of the funding rate. When bringing the funding rate in relationship with ∆(CP It) and ∆(EGt), we found a sign alternating relationship

between ∆(CP It) and F Rt. The first lag has a negative effect on the funding rate, while

the second has a positive effect. Both factors however, are small, in light hereof we conclude that either the relationship is weak, or endogeneity covers the relationship. The relationship between ∆(EGt) and F Rton the other hand, is larger then expected, with a first lag factor of

3.3. The second lag however is close to zero, and insignificant. The outcome of this research justifies the conclusion that both longevity and interest rates have a major influence on the funding rate of Dutch pension funds. The macro economic indicators, used in this research, in comparison have little influence on the funding rates. These factors therefore do not predict future funding rates.

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6 References

[1] Schmitz. C, E. van de Pol, Dekkingsgraden pensioen fondsen nader bekeken

http://www.cbs.nl/NR/ rdonlyres/CE7B8559-7D71-4548-B94C-0634C1A29FDA/ 0/2011dekkings-gradenpensioenfondsenart.pdf

[2] OECD (2011), DB Funding Ratios, in Pensions at a Glance 2011 Retirement-income Systems in OECD and G20 Countries, OECD

[3] A. Siegman, Minimum Funding Ratios for Defined-Benefit Pension Funds*, forth-coming, Journal of Pension Economics and Finance 2011

[4] Inkman. J, D. Blake, Pension Liability Valuation and Asset Allocation in the Pres-ence of Funding Risk Discussion Paper, 2007

[5] Draper. N, E. Westerhout, Defined Benefit Pension Schemes: A Welfare Analysis of Risk Sharing and Labour Market Distortions 2001

[6] J. Bikker, T. Knaap & W. Romp, Real Pension Rights as a Control Mechanism for Pension Fund Solvency 2011

[7] P. Broer, Social Security and Macroeconomic Risk in General Equilibrium, 2012 [8] F. Peters, W.Nusselder & J. Mackenbach, The longevity risk of the Dutch Actuarial Association’s projection model, 2012

[9] O. Ozcicek, et al. Lag length selection in Vector Autoregressive models

[10] Kisser. M, J. Kiff, The Impact of Longevity Improvements on U.S. Corporate De-fined Benefit Pension Plans, 2012

[11] . Rebel, The Ultimate Forward Rate Methodology to Valuate Pensions Liabilities: A Review and an Alternative Methodology, 2012

[12] EIOPA 2012 https://eiopa.europa.eu/about-eiopa/work-programme/multi-annual-work-programme-2012-2014/index.html

[13] Vasicek, 1977 ,http://www.emis.de/journals/HOA/JAMDS/8/11.pdf [14] Bongaarts and Bulatao 2000

http://www.un.org/en/development/desa/population/events/pdf/expert/4/bongaarts.pdf [15] E.H.M. Ponds 2012,

http://webwijs.uvt.nl/publications/562785 ext.pdf [16] D. Broeders, P Hilbers and D Rijsbergen 2013,

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7 Appendix

7.1 The estimated coefficients, belonging to the model

F R

t

= α + β

le

LE

t

+ β

r

t

+

Variable Coefficient Std. Error t-Statistic Prob. α 1652.06 359.71 4.59 0.00 LEt -19.30 4.47 -4.31 0.00

rt 6.05 2.73 2.21 0.03

R2 _0.80 _{Mean dependent var} _159.59 _{Adj R}2 _0.78

S.D. dependent var 44.93 S.E. of regression 20.61 AIC 9.00 Sum squared resid 9348.50 SIC 9.14 likelihood -109.52

7.2 The estimated coefficients, belonging to the model

F R

t

= α + β

le

LE

t

+ β

r

t

+ β

cpi

CP I

t

+ β

eg

EG

t

+

rt 5.18 3.36 1.54 0.13

EGt 2.10 2.75 0.76 0.45

CP It -0.03 0.05 -0.57 0.56

R2 _0.81 _{Mean dependent var} _159.59 _{Adj R}2 _0.78

S.D. dependent var 44.93 S.E. of regression 21.03 AIC 9.10 Sum squared resid 8853.12 SIC 9.35 likelihood -108.84

7.3 The estimated coefficients, belonging to the model

F R

t

= α + β

le

LE

t

+ β

r

t

+ β

eg

EG

t

+

rt 5.95 2.63 2.26 0.03

EGt 2.07 2.67 0.77 0.44

R2 0.81 Mean dependent var 159.59 Adj R2 0.78 S.D. dependent var 44.93 S.E. of regression 20.73 AIC 9.04 Sum squared resid 9024.39 SIC 9.24 likelihood -109.08

7.4 The estimated coefficients, belonging to the model

F R

t

= α + β

le

LE

t

+ β

r

t

+ β

cpi

CP I

t

+

rt 5.29 3.48 1.52 0.14

CP It -0.03 0.06 -0.51 0.61

R2 0.81 Mean dependent var 159.59 Adj R2 0.78 S.D. dependent var 44.93 S.E. of regression 20.91 AIC 9.06 Sum squared resid 9184.80 Schwarz criterion 9.25 likelihood -109.30

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7.5 The estimated coefficients, belonging to the model

F R

t

= α + β

le

∆(LE

t

) + β

r

∆(r

t

) +

Variable Coefficient StDelta. Error t-Statistic Prob. α 176.17 13.10 13.44 0.00

∆(rt) 3.85 7.16 0.53 0.59

∆(LEt) -97.77 51.43 -1.90 0.07

R2 0.18 Mean Dependent var 159.13 Adj R2 0.10 S.D. Dependent var 45.84 S.E. of regression 43.33 AIC 10.49 Sum squared resid 39438.58 SIC 10.63 likelihood -122.90

The effects of macro-economic indicators on the funding rate of Dutch pension funds