Analysing Indexation Uncertainty and Future Inflation and
Interest Rates
J.T.H. L¨
ossbroek
Analysing Indexation Uncertainty and Future Inflation and
Interest Rates
Jeroen L¨
ossbroek
1st May 2008
Abstract
Preface
In August 2007 the final project of my study Econometrics at the Groningen University com-menced: my Master Thesis. For six months, I investigated for Aon (among other things an Employee Benefits Adviser) an interesting and a present-day issue. I had six pleasant months and I learned a lot. I want to thank all my colleagues for the good time. Especially I want to thank my supervisors, Frank Bosman and Rien Buikema, for their help. Moreover, I want to thank Gert Maarsen for his assistance in several technical issues I dealt with.
For the assistance at the university I want to express my gratitude to Professor Ruud Koning; his comments and assistance were very valuable.
Special thanks to my girlfriend Gianna for, among other things, motivating me at the right times. Finally, I want to thank my parents for giving me the opportunity and for giving me all the help to complete my study.
Jeroen L¨ossbroek
Contents
1 Introduction and problem formulation 1
2 Theoretical background 2
2.1 Pension . . . 2
2.2 Inflation and Indexation . . . 3
2.3 Legislation and supervision . . . 5
3 Asset Liability Management (ALM) 7 3.1 ALM general . . . 7
3.2 ALM Light . . . 8
3.3 Mathematical description of a pension fund . . . 9
4 Time series model 11 4.1 Model selection . . . 11
4.2 Choosing the historical dataset . . . 12
4.3 Description Yule-Walker method . . . 14
4.4 Why Yule-Walker is used . . . 15
4.5 Results . . . 17
4.6 Some remarks . . . 19
4.7 Subconclusion . . . 20
5 Modelling indexation 21 5.1 Indexation based on the funding ratio . . . 21
5.2 Indexation based on excess returns on assets . . . 23
6 Uncertainty of future indexation 25 6.1 Measures of uncertainty . . . 26
6.2 Subconclusion uncertainty . . . 34
7 Sensitivity analysis 36 7.1 Sensitivity based on the case study fund . . . 36
7.2 Testing the results on other funds . . . 48
8 Conclusion and recommendations for further research 51
A Nelson-Siegel 52
1
Introduction and problem formulation
In the Netherlands, every employee over 21 years is legally bound to participate in some pension scheme to ensure benefits after retirement. However, many participants in pension funds have absolutely no idea about the details of their scheme. According to a study of DNB (De Ned-erlandsche Bank) (Van Els, Van den End, and Van Rooij (2003)) 44% does not know whether their pension is based on a final pay or an average pay scheme. In addition, interest in details is often quite low as proved in another DNB study (Kakes and Broeders (2006)). When asked for their attitude on their pension, 44% gave the answer ‘I don’t care about my pension, we will see later’ and another 42% answered ‘my pension should be well organized, but I am not interested in details’. An explanation for this uninterested attitude and poor knowledge might be the difficulty of all regulations and details, which can be improved by clear and easily understandable communi-cation to participants. New regulations prescribe pension funds to inform participants about their pension. For instance, information about future indexation, compensation for inflation, must be provided. Indexation usually depends on the financial position of a pension fund. As we will see later, the financial position depends on future inflation rates and interest rates. The first part of the thesis consists of the derivation of a suitable econometric time series model to predict these inflation rates and interest rates. The subject of the second part of the thesis is future indexation. First, the modelling procedure of calculating future indexation of several different types of indexa-tion policy is explained. Second, several statistics to measure uncertainty of future indexaindexa-tion are compared. Finally, the influence on future indexation of a different financial position or different policies of a pension fund is investigated: a sensitivity analysis is carried out.
Summarizing, the questions in this thesis are as follows:
• Which econometric time series model is most suitable to estimate future inflation and interest rates in the analysis of pension funds?
• How is future indexation modelled?
• Which statistical measures can describe the uncertainty of this future indexation and which of them are suitable for communication to participants?
• How sensitive is the future indexation for changes in the input parameters?
2
Theoretical background
2.1
Pension
A pension is a periodical benefit paid from a certain age, usually 65. A pension comprises three parts. The first part is the ’old-age pension act’ (AOW): the government pays a fixed benefit to everyone of 65 and over. The second part is relevant in this thesis. It consists of benefits remitted by pension funds and are paid from premiums (and asset returns) during the working career. This part also includes benefits paid to widows and widowers and to orphans. Finally, in the third part, people can buy a voluntary individual life assurance. This is especially for people without a steady job (entrepreneurs for instance). From now on, we mean the second part when we talk about ’pension’.
Three parties are involved in a pension plan: an employer, the employees, and a pension fund. The employer makes an arrangement with the employees about the details of a pension plan. The employer and the pension fund make an arrangement for the execution of the pension plan.
In general, two different pension plans are distinguished: Defined Benefits (DB) and Defined Contribution (DC).
Defined Benefits
In the Defined Benefits plan the accrual rate of the pension is fixed, whereas the premium the sponsor pays can fluctuate over the years. The employer of the participants usually pays this premium or sometimes the employee also pays part of the premium. Unless an exceptionally bad financial position of the pension fund, there is, except for uncertainty about indexation, no risk for the employee in the level of the accrued benefits.
Every year, a fixed percentage of the pension base (total salary minus a part that is related to the AOW, called offset), is accrued as pension. For instance, if for 40 years every year 1.75% of the pension base is accrued, the total pension accrual is 70% of the pension base. Unfortunately, in these 40 years the pension base is not equal every year. Wages usually increase over the years. Wages increase as a result of compensation for inflation and wages increase because of promotion and other career developments. The pension base increases therefore over the years. To deal with this problem pension funds can use two different systems, an average pay plan and a final pay plan. In a final wage plan, the total accrued pension is a fixed percentage of the pension base in the last worked year, multiplied by the number of service years. For the example above (1.75% of the pension base every year, for 40 years), this implies a total pension of 70% of the pension base in the last worker year. In an average pay plan the accrual is not based on the past. Every year a percentage of the pension base of that year is accrued. To overcome the prob-lem that pension accrued in earlier years becomes less valuable because of inflation the accrued pension can be adjusted for inflation, in other words indexation (see section 2.2 for an explanation).
Defined Contribution
The employer pays a fixed premium while the pension is unknown. The risk is completely for the employee. In years with low return on shares, the value of the pension can decrease considerably. When the employee retires the pension amount is calculated from the total amount of money available (all premium payments and returns on shares and bonds in the past). In the pension amount, a yearly increase to compensate for inflation may be considered.
2.2
Inflation and Indexation
For simplicity, we assume the euro existed ten years ago. After all, the exchange rate with the guilder will never change.
One Euro was ten years ago worth more than it is nowadays, meaning that prices were lower ten years ago than they are today. The overall increase in prices is a consequence of inflation, although inflation is frequently defined as this overall increase in prices (for instance in Mankiw (2003)). Rothbard (1990) stated that inflation may be defined as any increase in the economy’s supply of money not consisting of an increase in the stock of the money metal. The supply of money is the total amount available in the economy. In the past, the total supply of money was equal to the amount of gold held by the central bank. Money of this sort is called commodity money. Coins and notes were widely accepted as instrument of payment, because everyone knew the coins and notes could always be exchanged for gold.
Nowadays, fiat money is the standard in most countries. Fiat money cannot be exchanged for gold, because it is money without intrinsic value. Fiat money is only valuable, because everyone knows that everyone else values it. The government and the central bank through its monetary policy control the total supply of money. The central bank has the monopoly in printing new money. The total amount of money and the price level are linked by the ’quantity equation’ of Friedman (1957):
Money × Velocity = Price × Transactions
M × V = P × T
M is the money supply, V the transaction velocity of money, which is the rate at which money circulates in the economy, P the price level and T the total number of transactions. T is usually replaced by the total output of the economy Y, because it is difficult to measure. If the central bank increases the total supply of money by printing money (M increases on the left side of the equation) the price level will also increase (P on the right side), because V is assumed to be fixed and Y is fixed in the short run. For instance, the FED, the central bank of the United States, recently decreased the rate at which banks can borrow money from the central bank. To lend this extra money to banks (the total amount of money the FED lends to banks will increase because the interest rate is lower), the central bank needs to print extra money: M increases. This results in inflation: P increases. The central bank for the Eurozone is the European Central Bank (ECB). The primary objective of the ECB is to maintain price stability, which in fact means an inflation below or nearby 2%1.
Besides the monetary policy, an overall increase in prices can result in demands for higher wages to compensate these increased prices. However, higher wages will imply even higher prices, because salary costs of firms are increasing, etc. The name for his phenomenon is wage-price spiral. In addition, prices can increase as a result of an increase in demand. A recent example of this is the continual increase in oil prices. Finally, firms can increase their prices due to an increase in production costs.
Strictly speaking, inflation has to do with increasing prices and increasing wages. The latter is called wage inflation. Wage inflation is the average increase in wages of all employees.
The price inflation in the Netherlands is determined by Statistics Netherlands (CBS). CBS uses some basket of goods which represents an average consumption pattern and collects data about the prices of that basket at several times. The total price inflation is calculated as the weighted average of the change in prices of the basket of goods. It can be argued that the weights CBS uses for calculating the overall price inflation is not equal for different groups of people, for ex-ample, and important in this thesis, pensioners. Pensioners have a different consumption pattern compared with students, for instance. It is reasonable to assume that pensioners spend a bigger percentage of their income to costs for medical treatment than students do while students spend more on textbooks. Unfortunately, CBS does not make a distinction between social groups for the price inflation.
The impact of inflation on price level can be seen in figure 1 where the cumulative price level Λ in the Netherlands of 60 years (1947-2006) is plotted. The cumulative price level in year t, Λt, is defined as Λt= t Y i=1 (1 + λi)
with λi the price inflation in year i. A 60-year period seems long, but a 40-year career and a 20-year retirement benefits inflation, for instance, affects pensions for a very long time. Further, in figure 1 the price inflation a year is plotted. Note the large differences in price inflation over the years.
Figure 1: left: cumulative price index 1947-2006 (1946 = 1), right: price inflation per year
We see that from 1946 to 2006 prices increased to about 1000% in relation to the 1946 prices. This means a 90% decrease in the value of the euro in this period.
Indexation is compensation for inflation. If salaries, social benefits etc. are not compensated for inflation the purchasing power of the salaries and benefits decreases in the course of time as we saw in figure 1. Salaries and social benefits are therefore usually yearly corrected to compensate for inflation. For accrued pension rights the same ’decreasing value’ problem holds. In the final pay plan this is not a big issue for employees (the active members of the pension fund) since their salaries as well as their pension rights are corrected for inflation anyway. However, to preserve purchasing power, indexation is important for the economically inactive people. This group com-prises the retired members and early leavers; the latter group includes former employees, such as dismissed workers and employees who changed employer. As pension is usually the only type of income for pensioners, increased prices that are partly or not all compensated for inflation have, especially for pensioners, the immediate consequence of a decreased purchasing power: pensioners can buy fewer goods than the year before.
In an average pay plan, indexation is an issue for every pension fund member. An another com-plication for them is that pension funds have their own, different policies for active and inactive members.
policy of the fund: sometimes indexation is unconditional and sometimes conditional. Secondly, indexation depends on the measure that indexation is based on. Adjusting with the wage infla-tion or adjusting with the price inflainfla-tion are most common. Financing indexainfla-tion is usually done through a surcharge on the premium, through excess return on assets or from own assets if the financial position is good. For example, ABP, the largest pension fund for the government and educational sector in the Netherlands, lately announced that its financial position at the end of 2007 was good enough for full indexation: it will increase all its pensions with the average wage inflation2.
2.3
Legislation and supervision
Employers, employees and pension funds, the three parties involved in pension plan, (see section 2.1) are bound to extensive regulation by law. Since January 2007 the new Pension Act has been effective. Many regulations were changed compared with the old pension legislation, which was from the early fifties.
An important aspect in the Pension Act for pension funds is the obligation of market valuation of the liabilities of the fund. Before the new Pension Act was introduced, pension funds valuated the liabilities at a fixed 4% discount rate, regardless of the level of market interest rates.
Another point is various regulations for solvency to guarantee a solid financial position ((Dietvorst, Dilling, and Stevens, 2007)). A sufficiency test measures this financial position of a pension fund. The sufficiency test contains three parts: the minimum test, the solvency test and a continuity analysis. The minimum test and the solvency test are used for the financial position at balance date: the financial position at a point in time is investigated. More specific, the tests both focus on the financial buffer of the pension fund compared to the buffer required by law. The minimum test and solvency test must be carried out at least every year. The continuity analysis is an analysis of the long-term financial position of the pension fund. In the long-term much more is uncertain and a more extensive analysis is therefore necessary:
• How will the stock market perform the next years? • What is the impact of changes in interest rates? • In what way will benefits paid to pensioners develop? • etc.
The continuity analysis must be completed at least every three years. The analysis in this thesis is related to the continuity analysis.
Finally, an interesting factor of the new rules is communication to participants according to index-ation of pension rights. The Pension Act states that in case of conditional indexindex-ation expectindex-ations, financing and realisation of conditional indexation should be consistent with each other 3. This means that a clear and consistent policy is necessary; otherwise the indexation is not allowed to be conditional. The ’indexation matrix’ 4 is a guideline for this information consistency. In the indexation matrix is for each type of indexation policy a guideline for indexation ambition, financing method and communication to participants.
To make the above information about the indexation matrix clearer, we will give an example of the information in the indexation matrix. The indexation matrix is divided into six categories, ordered from the first category indicating no indexation policy at all to the last category for un-conditional indexation. For this category, the ambition or aim is equal to the realisation: full compensation (100%) depending on some measure, usually the price or wage inflation. Here, the expectation of participants is equal to the ambition. For other categories, the expectation may be lower than the ambition, for instance 80%. For unconditional indexation, the indexation must be paid from a surcharge on the premium for indexation; for conditional indexation this is not
2http://www.abp.nl/abp/abp/u_ontvangt_pensioen/indexatie/. 3Pension Act, article 95.
always the case. Moreover, some specific sentences about the unconditional indexation must be entered in the official pension regulations. Finally, some specification in the way of communication to participants is given.
Predictions of indexation related to inflation percentages in future years must also be given; the guideline in the indexation matrix describes the expectation and a - one-sided - 95% confidence interval. The specifications for communication to participants in the indexation matrix are de-signed only for 2007 and possibly 2008; in 2008 or 2009 an ’indexation label’ will replace the communication specifications in indexation matrix. The indexation label is more extensive and more explicit in communication requirements and is in design phase at the moment.
The institution that acts as supervisor of pension funds, and verifies therefore whether pension funds have a healthy financial position and do not break the law, is De Nederlandsche Bank (DNB). DNB inspects the sufficiency tests of the pension funds and has the right to oblige a fund to hand in a plan of measures that are taken in case of solvency problems. Also, DNB’s task is to supervise the financial stability of the sector as a whole.
3
Asset Liability Management (ALM)
3.1
ALM general
The Netherlands is the country with the largest amount of pension assets per capita in the world, with more than $400 billion pension assets (Ziemba and Mulvey (1998); chapter 22 is an article written by G.C.E. Boender, P.C. van Aalst and F. Heemskerk with the title ‘Modelling & Man-agement of assets & liabilities of pension plans in the Netherlands’). This is a result of an early founding of the Dutch pension funds, the generous pension schemes, the relatively small public pensions (the AOW) and a full capitalisation of the liabilities. Using these enormous amounts of money in the right way is a very important question for pension funds.
The board of a pension fund faces the question to determine a long-term strategy. Firstly, the board satisfies the interest of the stakeholders of the pension fund (participants, sponsor) and, secondly, it complies with the law regulations to ensure a good financial position in the future with a small risk of financial problems. All available policy instruments have to be determined in an integrated fashion. Carrying out this objective is called Asset Liability Management, abbrevi-ated ALM (Ziemba and Mulvey (1998)). The name is chosen because it deals with managing the assets of a pension fund - all available, invested, money - and the liabilities - the present value of all future payments to participants. For a review of the working of Asset Liability Management is referred to figure 2.
Figure 2: Review of the working of Asset Liability Management (Ziemba and Mulvey (1998))
The strategy of a pension fund ALM investigates is a long-term strategy with a long-term pre-diction of the future financial position of the pension fund. The time horizon is usually 15 years, sometimes 20.
position caused pension funds to do better predictions about the future. Finally, together with the second argument, the declined market interest rates in the last few years caused lower funding ratios.
All received premiums will be invested in different financial categories with the aim of an ac-ceptable rate of return without experiencing too much risk. The most important measure of the financial position of a pension fund is the ratio of the assets and the liabilities, called the funding ratio. A funding ratio smaller than 1 means financial problems, as this means that the present value of all benefits that should be paid is higher than the available capital to pay these benefits. In this situation the pension rights cannot therefore be guaranteed anymore. A funding ratio more than 1 does not automatically imply a good financial position. If the funding ratio is close to 1, the probability that the funding ratio drops below 1 in the future will probably be unacceptably high. A pension fund must take measures to return to a healthy position if a low funding ratio occurs.
The funding ratio is a point of time measure, while ALM investigates the possible fluctuations in the financial position of the pension fund in the future. The value of the assets changes in the course of time and is not known for the future: future returns on assets are uncertain; the assets are increased by premium payments etcetera. Consequently, the development of the liabilities in the future is uncertain as well: how many retired participants will die, how many employees will quit their job etc. Furthermore, the above introduced obligation of market valuation of the liabilities (see section 2.3) causes a changing value of the liabilities: when the market interest rates increase or decrease, the liabilities may decrease or increase substantially. ALM tries to predict all these unsure parameters in terms of probabilities and expectations. In section 3.3 a mathematical description of a pension fund investigated with ALM is shown.
3.2
ALM Light
A good ALM study takes into account all financial risks a pension fund bears. All characteristics of the (participants of the) pension fund, the pension plan the fund executes and the current policy of the fund with respect to for example premium payments or investment policy are used as input in an ALM model. With this situation as a starting point, the development of the assets and the liabilities is simulated for every analysed future year. Besides, different policies are calculated to compare their consequences for all stakeholders. The development of a pension fund is a complicated issue, many things must be taken into account: not only the many characteristics of the pension fund, but also the development of the future world economy is an important factor in the future situation of the fund. Because is it complicated, it is very time-consuming and thus costly to carry out a full ALM study. The ALM model that is used in this thesis is ALM Light, developed by Aon. ALM Light is a model that is a simplified version of a full ALM study and is therefore much cheaper than a full ALM study. The goal of ALM Light is to get more insight in the influence of investment policy, indexation policy and contribution policy into the development of the financial position of the fund. The time horizon in ALM Light is 15 years, which is usual for ALM studies.
As just mentioned, ALM Light is simplified compared to a full ALM study. The main differences of ALM Light compared with a full ALM study are as follows:
• In ALM Light, only two asset categories can be chosen: shares and bonds. Some percentage of the assets is invested in shares and the rest in bonds. In a full ALM study many more asset categories can be chosen, such as mortgages, hedge funds, real estate and so on. Possible investments of a pension fund in these asset categories are classified as bonds or shares.
• A full ALM study can deal with swaps to reduce the risk according to interest change and reducing currency risk. ALM Light cannot.
• In a full ALM study, different time series to predict inflation, market interest rates and so on can be chosen while in ALM Light only the implementation of a predetermined economic model (see section 4) can be used.
Ren´e (2007) investigated the reliability of ALM Light compared to a full ALM study and concluded that the concept of ALM Light has potential, but need some adjustments to become more reliable. The recommended adjustments were mainly related to the calculation of future liabilities. In the current version of ALM Light these recommendations are implemented.
3.3
Mathematical description of a pension fund
The future financial position of a pension fund heavily depends on the development of the world economy. Future asset returns, interest rates and price and wage inflation are economic variables that must be estimated accurately to predict about the financial position of the fund. It is difficult to predict in what way the economy develops. To deal with this problem, ALM studies usually use a scenario based analysis. Based on some model, for every future year (usually 15 years, sometimes 20) a prediction of all economic variables is made. This is repeated many times, say 500 or 1000 times. This set of predictions of the economic variables is used to analyse the risks of the pension funds according to the economic variables. Every scenario can be seen as a possible development of the financial position of the pension fund.
For every scenario s (s = 1, . . . , S) the complete financial position in every analysed year t (t = 1, . . . , T ) of the pension fund is calculated. The funding ratio in scenario s on time t, F Rs,t, is determined as the quotient of the assets and the liabilities in the same year and scenario: F Rs,t= ALs,t
s,t. The analysis starts with the initial values of the assets and the liabilities (A0 and
L0), both input parameters of the model. The liabilities on time 0 must be calculated using market interest rates.
We start with the assets. Every year, the assets A are increased with premium income P r, this is multiplied by the total return on assets TR and then pension benefits PB are paid. Finally a percentage κ of the premium is subtracted from the assets because of execution costs.
As,t= (As,t−1+ P rs,t) × (1 + TRs,t) − PBs,t− κ × P rs,t
In ALM Light, bonds and shares are the only two assets categories, if we define φ as the fraction of the assets invested in shares (consequently, 1 − φ is the fraction invested in bonds), the total return on assets TR is defined as TRs,t= φ × Ss,t+ (1 − φ) × Bs,t, with S is the return on shares and B the return on bonds.
Another remark is about the benefits PBs,t: the benefits yearly increase with the indexation given the year before. High inflation rates have therefore a negative effect on the value of the assets.
For t = 1, . . . , T , La
s,tis determined as follows:
Las,t= ((Las,t−1+ CSs,t+ CIas,t) × (1 + rs,t(1)) − zs,t) × δs,t
Here,
• CS are the costs of the coming service, the costs of the pension accrual in the specific year • CIa are the costs of indexation for the actives. CIa (and CIia for the inactives) depends
on inflation rates. If the inflation in some scenario s and year t was very high, the costs to increase all pensions with this percentage are high also. The costs of indexation are a percentage of the liabilities of the year before. Section 5 gives detailed information about modelling indexation.
• rs,t(1) is the one-year (market) interest rate. This factor is implemented because the lia-bilities this year are one year less discounted than the lialia-bilities last year. See for more information about the calculation of rs,t(1) appendix A.
• zs,tis a conversion factor which determines the amount of money that flows from the active liabilities to the inactive liabilities
• δs,tis a correction factor according to changes in interest rates. This is because the liabilities must be valuated according to market interest rates and therefore the changes in interest rates result in a different value of the liabilities.
Lia
s,t is determined quite similar to Las,t:
Lias,t= ((Lias,t−1+ CIs,tia) × (1 + rs,t(1)) − PBs,t+ zs,t) × δs,t
Compared to the La
s,t, CSs,t is removed (inactives have no coming service), the conversion factor is of course of opposite sign and the benefits are subtracted from the liabilities (actives have no benefits).
4
Time series model
In section 3.3 was explained why ALM studies usually use a scenario based analysis: mainly be-cause of the uncertainty of the future economic situation. Future return on shares, interest rates and price and wage inflation are variables that must be estimated accurately. In this section an econometric model to predict the economic variables in an ALM study will be derived.
A special case in the economic variables is the interest rate. It is too time consuming to predict for every future year the interest rate for every time horizon. Only the short-term and long-term interest rate is therefore simulated and a model is used to estimate every possible time horizon in between. This, widely used, model is known as the Nelson-Siegel Model (Nelson and Siegel (1987)). The definition of this model and how it works is explained in appendix A.
The set of scenarios of the economic variables is fixed in ALM Light. However, the underlying econometric model of this dataset is unknown in the current version. A new dataset with scenarios is created by means of time series analysis. The goal is to find a clear and understandable model which makes a good prediction of the economic variables and is easy to use for users of ALM Light. The model should not be restricted to ALM Light: the derived model should also be useful for other ALM studies.
In the model we restrict ourselves to the price and wage inflation and the interest rates. The return on shares is omitted in the model. In section 4.6 is explained why we made this choice. Concluding, the economic variables used in the model are
• price inflation (pi) • wage inflation (wi)
• short-term interest rate (si) • long-term interest rate (li)
4.1
Model selection
The model we choose is a Vector Autoregressive (VAR) model, mainly because the advantage of a VAR model is that (auto)covariances and (auto)correlations of different time series are included. Another option would be an Autoregressive-(Integrated)-Moving Average (AR(I)MA) model to estimate all equations separately, but in an AR(I)MA model correlations between the different variables are ignored. Moreover, Lurvink (2007) concluded that the VAR model gives better pre-dictions than a AR(I)MA model.
The order of the VAR model will be equal to 1. First, this is because this is most common among comparable (ALM) studies. Second, the number of estimated parameters increase substantially with a higher order model. This could have a negative effect on the predictions that are made with the model estimates.
The VAR(1) model is defined as:
Yt= c + AYt−1+ t (1) with Yt = Y1,t Y2,t Y3,t Y4,t = pi(t) wi(t) si(t) li(t)
Furthermore, c ∈ <4 and A ∈ <4×4. t∈ <4is a white noise process (t∼ N (0, Σ)). This means thas a multivariate normal distribution with mean 0 and variance-covariance matrix Σ.
method is derived and the working is explained. In section 4.4 the question why the Yule-Walker method is preferred to standard OLS will be answered.
4.2
Choosing the historical dataset
The choice of an appropriate dataset is not as trivial as may be thought. Several problems arise when it is decided which historical data should be used for creating the model. First is described which data source is used and then it is explained why this data is chosen to estimate the model. The data are captured from Bloomberg. The following data are available:
• Yearly price inflation in the Netherlands • Yearly wage inflation in the Netherlands • Yearly short-term interest rate (three months)
• Yearly long-term interest rate (10 years’ government bonds)
The interest rate on 10 years government bonds is used as measure of the long-term interest rate, because historically this appears to be a good indicator. The data is available from 1970 until 2006. In this way, the number of available data points are not very large. Nevertheless, yearly data is chosen and preferred to data with a shorter time interval, such as monthly data. The reason behind this choice is that the forecasted period in an ALM study is also in years. If we create scenarios with the economic variables for the analysis of the future financial position of pension funds the scenarios will be created in years because an ALM study only deals with yearly changes. Monthly changes are therefore completely irrelevant and are omitted in the model.
In figure 3 all historical data from 1970 are plotted.
We now have the difficult decision to decide which observations we prefer to use for the time series analysis. First, we notice the extreme high inflation rates in the early seventies. The main reason for this very high inflation rates was the oil crisis. If we include these data points, from 1970 until about 1976, the model gives too high estimates of expected inflation rates. To deal with this situation, we have two options: introduce a dummy variable or exclude the observations. We decided to exclude this set of observations.
Figure 3: upper left: price inflation, upper right: wage inflation, lower left: short-term interest rate, lower right: long-term interest rate
category Average standard deviation correlations
pi 2.21% 0.77% 1
wi 2.59% 1.06% 0.62 1
si 3.84% 1.62% 0.27 0.43 1
li 5.27% 1.28% 0.31 0.20 0.74 1
Table 1: averages, standard deviations and correlations of the historical dataset 1992-2006
4.3
Description Yule-Walker method
The Yule-Walker method is based on (co)variances and (auto)covariances. From equation (1) we obtain the expectation vector µ, the variance-covariance matrix Γ(0) and the matrix of auto(co)variances with lag k, Γ(k). µ = E(Yt) = E(c + AYt−1+ t) = c + AE(Yt−1) = (I − A)−1c (2) Γ(0) = V ar(Yt) = V ar(c + AYt−1+ t) = V ar(AYt−1) + V ar(t) = AV ar(Yt−1)A0+ Σ = AΓ(0)A0+ Σ (3) Γ(k) = Cov(Yt, Yt−k) = Cov(c + AYt−1+ t, Yt−k)
= Cov(c, Yt−1) + Cov(AYt−1, Yt−k) + Cov(t, Yt−k) = 0 + ACov(Yt−1, Yt−1) + 0
= ACov(Yt−1, Yt−k)
= AΓ(k − 1) (4)
To estimate c, A and Σ from the sample data the moments of the equations (2), (3) and (4) are calculated for the sample data in the following way:
The historical average of the categories i = 1, . . . , 4 in the course of the years t = 1, . . . , T (T = 15 in our sample): b µi = Yi= 1 T T X t=1 Yi,t (5)
The elements of the matrix of auto(co)variances with lag k, bΓ(k) are defined quite similar to bΓ(0): b Γ(k)i,j= 1 T T X t=k+1 (Yi,t− ˆµi)(Yj,t−k− ˆµj) (7)
Now we substitute these estimates in the moments of the equations (2), (3) and (4) calculated above (k = 1 is used since our model is a VAR(1) model) to obtain the estimates ˆc, ˆA and bΣ.
ˆ A = Γ(1)bb Γ(0)−1 (8) ˆ c = (I − ˆA)ˆµ (9) b Σ = Γ(0) − ˆb AbΓ(0) ˆA0 = Γ(0) − bb Γ(1)bΓ(0)−1bΓ(0)bΓ(0)−1bΓ(1)0 = Γ(0) − bb Γ(1)bΓ(0)−1bΓ(1)0 (10)
4.4
Why Yule-Walker is used
Very important in the structure of the Yule-Walker model is the calculation of bΓ(k) in equation (7). Instead of just dividing by T − k, the summation is divided by T . This has an interesting consequence: the estimated model is always stationary.
The stationarity restriction can be seen by the fact that, for increasing k, bΓ(k) damps out, since it is divided by the same number T while the number of terms in the summation decreases. The autocovariances are therefore guaranteed to converge to 0, which corresponds to stationarity (Stee-houwer (2005)).
More formally, stationarity implies that the eigenvalues of ˆA are strictly smaller than 1 in absolute value (Johnston and DiNardo (1997)). For instance, this is a necessary condition for existence of the matrix (I − A)−1. For the Yule-Walker method, the eigenvalues of ˆA are always smaller than 1. A formal proof can be found in Boender and Romeijn (1990). Furthermore, Steehouwer (2005) proved that the matrix bΓ(0) is positive definite and is therefore invertible.
necessarily stationary because the estimated model can be non-stationary if the sample data is also non-stationary. This can result in an invalid model.
In our model, T = 15 which is even smaller than the by Steehouwer investigated sample sizes (25, 50 and 100). This does not mean that the smaller the sample size the better the estimates, but it does mean the smaller the sample the greater the difference between the two models. We therefore conclude that the estimates of the Yule-Walker method will be considerably better than the estimates of OLS.
An interesting and useful (actually the characterising) property of the Yule-Walker method lies in the covariances and the autocovariances of the model. If the estimated model is used to generate time series of future values of in this case inflation rates and interest rates, by definition of the model the vector of mean values, the matrix of covariances and the matrix of autocovariances of the generated time series converge to the historical values of these properties. For example, in the generated time series the autocovariance of the price inflation in year t and the wage inflation in year t − 1 converge to bΓ(1)1,2, the autocovariance of the price inflation in year t and the wage inflation in year t − 1 of the historical dataset, calculated with equation (7). A proof of this property can be found in Boender and Romeijn (1990).
In OLS the long-term expectation ˆµ of the model is fixed, since ˆc is estimated from the data and then ˆµ is calculated similar to equation (9). In the Yule-Walker model ˆµ can be chosen manu-ally. The model is based on covariances and autocovariances, the estimates of ˆA and bΣ in equation (8) and (10) do not depend on ˆµ, only the constant ˆc of equation (9) is changed. Of course, the historical average is the most logical choice for ˆµ. However, according to predictions of economists of future developments of the economy another long-term expectation than the historical aver-age may be better. Furthermore, DNB has regulations according to the values of the long-term average of the inflation and the interest rates. In the Yule-Walker model this can be adjusted if the historical average conflicts with these regulations. See section 7 for more information of the situation with other long-term inflation rates and long-term interest rates.
Concluding, the Yule-Walker model is more flexible than OLS en is therefore more useful for ALM studies.
To summarise, all arguments discussed above to use Yule-Walker are listed below.
• Stationarity restriction
• Better estimates in small samples
• The model has the same characteristics as the historical dataset • Flexible according to long-term expectations
4.5
Results
For the sample data (summarised in table 1) the estimates ˆc, ˆA and bΣ are calculated according to equations (8), (9) and (10) and rounded to 8 decimals:
ˆ A = −0.06213469 0.42946088 −0.11049911 0.09201971 0.06442126 0.55901768 −0.04467849 −0.00105601 −0.48758184 0.25409999 0.39863453 0.07455789 −0.63978270 0.24000570 0.24418886 0.41726138 (11) ˆ c = 0.00841695 0.01368635 0.02162542 0.02253954 (12) b Σ = 0.00004374 0.00002766 0.00002472 0.00002477 0.00002766 0.00007695 0.00005984 0.00001977 0.00002472 0.00005984 0.00020085 0.00008873 0.00002477 0.00001977 0.00008873 0.00008619 (13)
The structure of the model can be shown by generating many scenarios of model (1) with the estimates calculated above. In figure 4 the average and the 5%- and the 95% percentiles of the four categories are shown, based on generating 1001 scenarios and using the last year of historical data, 2006, as a starting point: Y0:= Y2006= (0.0110, 0.01760, 0.0367, 0.0399)T.
Figure 4: upper left: price inflation, upper right: wage inflation, lower left: short-term interest rate, lower right: long-term interest rate
here equal to the historical averages (see table 1 for the values).
Another property of the Yule-Walker method, the sample correlations converge to the historical correlations, is indeed visible. For example, for the two largest correlations of table 1, the correla-tion between the price inflacorrela-tion and the wage inflacorrela-tion and the correlacorrela-tion between the short-term interest rate and the long-term interest rate, the convergence is clearly present, see figure 5
4.6
Some remarks
In section 4.4 was argued why the Yule-Walker method was chosen as the method to estimate our VAR model and in section 4.2 the reasoning behind the choice of the historical dataset was discussed. In this section some remarks of our estimated model are given.
First, the number of observations is very small. We used only 60 observations (4 variables, each 15 years). This can have negative effects on the estimated parameters. It can be argued that we should use monthly data or a longer historical time period to increase the number of observations. However, as mentioned in 4.2, this has some serious drawbacks too.
The model is estimated with the purpose to make yearly simulations. An ALM study is after all done with years as unit of time. For a time horizon of 15 years the model has 15 iterations for yearly data. For monthly data, there are 180 iterations of which only 15 are used. This is not useful for ALM studies.
A longer time period will increase the number of observations, which is positive for the quality of the estimates. However, the argument that the observations of a long time ago are not consistent with the economic situation of the last one or two decades faces the question whether more years are may even be worse. We decided to remove observations that are not relevant.
Another remark is about the stationarity restriction, is it a valid restriction or not and why do we impose the stationarity restriction here?
Steehouwer (2005) argued: “Stationarity is often needed to estimate or even construct a sensible model. From this perspective the stationarity restriction in the (biased) Yule-Walker estimation technique will rarely be considered an invalid one or be a limiting factor in some sense”. So, stationarity is required to estimate a sensible model.
The use of the stationarity restriction faces the question why the restriction is imposed here. Is the underlying process stationary? First, we notice that the sample data do not reject a unit root in the Augmented Dickey Fuller (ADF) test. Fortunately, this does not imply a nonstationary underlying process. First, the ADF test statistic becomes less reliable when the test is based on less observations. Second, the policy of the ECB stimulates a long term average: the ECB actually tries to create a stationary process! Although some fluctuations in the long term average is inevitable, the ECB tries to maintain the long term average: exactly the characteristics of a stationary process. The aim of a stationary process is not bounded to the price inflation. Some other studies (such as Yakoubov, Teeger, and Duval (1999)) uses the price inflation as the main driver of the economic variables. On their part, the other variables will converge to a stationary process either, but probably with some time lag. Support for this theory can be found in the matrix ˆA of equation (11) where the coefficients ˆA(3,1) and ˆA(4,1) are relatively big (in absolute value).
To conclude the discussion about the Yule-Walker method and stationarity, we cite Steehouwer (2005) again: “Besides the sample data, there is nothing from preventing an AR model estimated by OLS from being close or even over the non-stationary boundary. Note that his observation can become especially important when estimating large unrestricted VAR models on small samples of macroeconomic data.”
The last remark is about the return on shares. The model is restricted to the price and wage inflation and the interest rates. This choice is made for the following two major reasons: the qual-ity the estimates (i.e. the prediction qualqual-ity: do the predictions made with the model correspond to the reality?) of the other variables and the small relation between the return on the shares and the other variables.
Quality of the estimates:
vari-ables. Moreover, the volatility of the return on shares (historical data of the Dutch stock market gives a standard deviation of about 20%) may create relations that in reality do not exist. Finally, the number of parameters in the model will increase with 50% (15 parameters) when the return on shares in added! This has negative consequences for the quality of the estimates of the other variables.
Relations with the other variables:
First, the correlations between historical data of the return on shares and the other variables are very low (lower all other crosscorrelations). Only the price inflation shows some, negative, corre-lation. Second, the autocorrelation (which is the principle of the VAR model) of the return on shares is negligible (about 0.07 for a long historical time period). The negligible autocorrelation of the historical dataset fits perfect in the ‘efficient market hypothesis’, which (among other things) states that all available information is already implemented in the market prices. Modelling the return on shares with a model that predicts the return on shares next year from the value of the return on shares (and the value of the inflation rates etc.) this year is in contradiction with the efficient market hypothesis. The negligible autocorrelations is also an argument that adding the return on shares may create relations that in reality do not exist. Third, even though we add the return on shares, results of the new model show some arguments in favour of not including the return on shares. The coefficients of the other variables barely change. For instance, the first row of the matrix ˆA changes to (−0.09561708; 0.43931501; −0.11405505; 0.10238981). Compared to the old values in equation (11) the change is negligible. Moreover, for every row of the matrix
ˆ
A the coeffient in the last column is in absolute terms the smallest which confirms the conclusion of a small difference in the estimates of the two models. Finally, the extremely high coefficients in the last row of ˆA (-9.16699116; -6.53510574; 3.19112694; 3.07308628) supports the presumption of creating not existing relations.
Concluding, for the return on shares the existing dataset is used for the ALM study. A recom-mendation for further investigation is to do research about a (separate) new model for the return on shares, because the model behind the dataset in ALM Light is still unknown for the return on shares.
4.7
Subconclusion
5
Modelling indexation
Predictions about indexation are made with the use of the scenario based analysis created with the economic model derived in section 4. For every scenario and every analysed future year the indexation percentage is determined, which is partly based on the observation of the quantities in the economic model in that scenario. The dataset this analysis creates is used to investigate the uncertainty of this indexation in section 6 and the sensitivity analysis of section 7. This section explains the model behind the creation of the indexation dataset in ALM Light.
The indexation in a named scenario in a specified year is a percentage of the realisation of the price or the wage inflation (dependent on the type of indexation), created with the economic model. If no indexation is given, the percentage is 0% and if full indexation is given it is 100%. Some pension funds have the catch-up indexation option, which means that in years of an excellent financial position not given indexation in the years before can be catched up. In this case, the indexation percentage is over 100%.
For a final pay plan, explained in section 2.1, indexation is always unconditional and related to the wage inflation. This is true because the total pension accrued is based on the last wage, and the wage increases every year with on average the wage inflation. This means in every scenario the indexation is 100% of the wage inflation for every year.
Final pay plans have become more and more uncommon in the last few years. An average pay plan does not automatically compensate for inflation, although unconditional indexation is still possible. Of course, only conditional indexation is interesting to investigate, in contrast to uncon-ditional indexation it can vary over the years and is unknown in advance. The two most common types of conditional indexation policy are indexation dependent on the funding ratio and indexa-tion by means of excess return on assets.
5.1
Indexation based on the funding ratio
For every scenario s and for every year t the assets As,tand the liabilities Ls,tare calculated, and then the funding ratio F Rs,tis determined: F Rs,t=
As,t
Ls,t. Dependent on the policy of the pension
fund, indexation Is,t+1is given dependent on the value of F Rs,t, since the indexation percentage given at the end of the year is determined at the beginning of the specific year. If F Rs,t < 1 the fund has not enough assets to meet the liabilities and stopping indexation one year later is one of the first measures to be taken. Furthermore, a buffer of at least 5% of the liabilities is minimal required by law. Of course, not meeting this minimum buffer requirement could happen since the future financial position is unknown. When this happens, conditional indexation will also be stopped to increase the probability that one year later the funding ratio exceeds 105% as required. For conditional indexation based on the funding ratio we can conclude the following: F Rs,t< 105% ⇒ Is,t+1= 0%.
The two most common used policy for indexation at higher funding ratios are listed below and shown graphically in figure 6:
1. For a fixed funding ratio x%(≥ 105%) (for instance 120%) the following indexation policy is used:
F Rs,t< x% ⇒ Is,t+1= 0% F Rs,t≥ x% ⇒ Is,t+1= 100%
x% and y%: F Rs,t≤ x% ⇒ Is,t+1= 0% x% < F Rs,t< y% ⇒ Is,t+1= F Rs,t− x% y% − x% × 100% F Rs,t≥ y% ⇒ Is,t+1= 100%
Figure 6: left policy 1, right policy 2
Besides these two common types other policy is of course possible. An example of another possibil-ity is a step by step improvement of the indexation percentage with steps at several funding ratios.
catch-up indexation
If the indexation policy includes catch-up indexation, the calculation of the indexation percentage is extended a bit. If catch-up indexation is included in the policy, it usually exists only with the funding ratio based indexation policy. If the funding ratio is higher than some value F Rcu all lost indexation in the past few years will be catched up.
The indexation deficit ds,tis defined as the cumulative wage inflation or price inflation divided by the cumulative indexation minus one:
ds,t= Qt
i=1(1 + pis,i) Qt
i=1(1 + inds,i) − 1
Here, inds,i is the indexation factor in year i, meaning the factor where all pension rights are increased with. The relation between the indexation percentage Is,t and the indexation factor inds,tis
Is,t= inds,t
pis,t
Again, wi is used instead of pi for indexation based on the wage inflation.
If F Rs,t > F Rcu, the deficit in year t − 1 is added to the indexation factor in year t, multiplied by the catch-up indexation of year t itself, it results in a deficit of 0 after year t:
5.2
Indexation based on excess returns on assets
The return on all invested assets is an important factor in the financial stability of a pension fund. Pension funds can use high return on assets for indexation. This can be done in several different ways.
First we define the total return on assets in scenario s in year t, T Rs,t, as follows:
T Rs,t= φ × Ss,t+ (1 − φ) × Bs,t
The fraction φ of the assets invested in shares with return Ss,tadded to the fraction of the assets in bonds 1 − φ with return Bs,t. Remember from section 3.2 that only this two assets categories are used.
Besides reasons as new entrants or the coming service of the accrued pension rights, as the future benefits are one year less discounted one year later, the liabilities increase every year. This is called the liability return. The return on assets must be at least as large as the liability return to compensate this higher liabilities. The asset return above this needed return to meet the increas-ing liabilities is called the excess return. This excess return is used for indexation.
In the new pension act, in which liabilities must be valuated with market interest rates, the liabil-ity return is defined as rs,t(1), the one year interest rate calculated in year t with the Nelson-Siegel model (see appendix A).
Finally, a correction factor is applied to the difference between asset return and liability return. Valuation according to market interest rates can cause big differences in the value of the liabilities. To avoid a situation in which the pension fund has a poor funding ratio due to a large drop in the market interest rate but still must give full indexation because the asset return was acceptable, excess return used for indexation is corrected for this change in market interest rate. The correc-tion factor δs,t is defined as:
δs,t= 1 + r s,t(durL) 1 + rs,t−1(durL) durL − 1
durL is the duration of the liabilities. The duration of the liabilities is not only the weighted average years from now of all future benefits paid to participants, but it also measures the interest rate sensitivity of the liabilities. A duration of x means, besides the liabilities are equivalent to a one-time payment of all benefits x years from now, that a 1% increase/decrease in interest rate causes a x% decrease/increase in the value of the liabilites. Furthermore, rs,t(durL) is the durL year market interest rate in scenario s in year t, again calculated with Nelson-Siegel.
Pension funds can use excess return for indexation in different ways. excess return per year
Every year, the indexation percentage (minimal equal to 0) in the specific year is determined such that just the excess return one year ago is given as indexation, usually maximized on the wage inflation (wi) or price inflation (pi):
Is,t=
M IN (pis,t; M AX(0; T Rs,t−1− rs,t−1(1) + δs,t−1)) pis,t
× 100%
If the aim is to give indexation based on the wage inflation, this happens only for active partici-pants, pis,t is replaced with wis,t
average excess return in five years
The indexation percentage in the specific year is the average of the excess return in the last five years:
Is,t=
M IN (pis,t; M AX(0;15P5i=1[T Rs,t−i− rs,t−i(1) + δs,t−i])) pis,t
× 100% excess return depot
from this depot. Unused excess return in years with high excess return is saved to pay indexation of it in a later year with low or even negative excess return.
The excess return ERs,t in scenario s in year t that is added to the depot Ds,t is defined as
ERs,t= (As,t−1+ Ps,t) × T Rs,t− rs,t(1) × Ls,t−1+ δs,t× Ls,t
The assets at the end of the year before are increased with the premium income Ps,t (assumed is that all premiums are paid at the beginning of the year). This is multiplied with the total return calculated before. From this, the liability return, multiplied with the liabilities one year earlier, is subtracted. Finally, the correction factor δs,tis applied to the liabilities of the specific year. Note that a negative excess return is possible.
Now the depot Ds,t, on which the indexation one year later, Is,t+1, is based, is determined:
Ds,t= Ds,t−1+ ERs,t− CIs,t
The excess return is added to the old depot while the costs of indexation CIs,tis subtracted from the depot.
Now we can define the indexation percentage Is,t as
Is,t= 1 pis,t × M IN (pis,t; Ds,t−1 Ls,t−1 ) × 100%
Again, pis,tis replaced with wis,tif the aim of the indexation policy is indexation with the wage inflation instead of the price inflation.
6
Uncertainty of future indexation
As every pension fund has his own characteristics and the financial position is dependent on many different factors, the set of parameters in an ALM model is extensive, so is in ALM Light. Conse-quently, showing results for every setup of the parameters is impossible. However, to show results for the statistical measures derived in this section, some setup of the model has to be inserted. Besides, to carry out the sensitivity analysis in section 7, a starting point from which the parame-ters will be changed to investigate the effect of different policies is needed. It is chosen to create a fictional pension fund which has ‘average’ parameters. This means that the parameters are chosen such that this case study fund is a realistic overview of existing pension funds. In this way, the effects of the sensitivity analysis are an average effect among all pension funds whereas it differs between different funds. For a list (and some explanation) of the most important variables, see appendix B.
After the parameters in appendix B have been implemented and for many scenarios s = 1, . . . , S the indexation percentage Is,t for time t = 1, . . . , T is calculated, we have a dataset which we use to analyse the uncertainty. Before our analysis started, the only statistics about indexation the model return are the average indexation percentage per year for all scenarios and the overall average: I.,t = 1 S × S X s=1 Is,t I.,. = 1 S × T × S X s=1 T X t=1 Is,t
For the case study fund the results can be seen in figure 7.
6.1
Measures of uncertainty
As the prediction of future indexation that must be communicated to participants is a long-term prediction, a more relevant statistic than the average per year is the average indexation percentage in T years (in ALM Light and most other ALM studies 15 years), for every scenario:
Is,. = 1 T × T X t=1 Is,t
An approximation of the distribution of this average indexation in 15 years over 1001 (see ap-pendix B) scenarios is given in figure 8. The probability is the number of scenarios with the specific average divided by the total number of scenarios.
Figure 8: Distribution of average indexation in 15 years
Figure 9: Some percentiles of the average indexation in 15 years
As the indexation percentage Is,t is just a percentage of the price or wage inflation, only the average indexation related to this price or wage inflation is not in itself a good measure to get in-sight into (uncertainty of) future indexation. It does not take into account differences in inflation. For instance, if 50% of the price or wage indexation is given as indexation, it makes a difference whether this is 50% of 1% inflation or 50% of 6% inflation. We need to find a measure that takes this into account.
The pension result measures the purchasing power of pensions and is therefore an excellent measure to solve the problem mentioned above. It is defined as the cumulative level of future pension payments related to the cumulative level of future pension payments if every year ex-actly the price inflation was given as indexation. The pension result prs,t for s = 1, . . . , 201 and t = 1, . . . , 15 is defined as follows:
prs,t = Qt
i=1(1 + inds,i) Qt
i=1(1 + pis,i)
Figure 10: Distribution of pension result after 15 years; above: inactive participants, below: active participants
Especially the distribution of the active participants shows an interesting pattern: on further inspection it looks like a normal distribution. We can apply a formal test to the hypothesis of normality for the case study fund. Among the several tests available to test for normality, we choose to apply the Anderson-Darling test (Anderson and Darling (1952)), because this test is, compared with other tests, very powerful (Stephens (1974)). The Anderon-Darling statistic has in our situation a null-hypothesis of normally distributed data against the alternative to not normally distributed. The test statistic A is defined as:
A = −n −1
n n X i=1
(2i − 1)(ln(Φ(Y(i))) + ln(1 − Φ(Y(n−i+1))))
does not really makes a difference with our sample of 201 observations:
A := A(1 + 0.75
n +
2.25 n2 )
For our dataset, the pension result of the active participants after 15 years, pr.,15, for the dataset with 201 scenarios, the test statistic A is equal to 0.327 and the p-value is 0.5173, which does not reject the null-hypothesis if tested on 95% confidence. The result of the test confirms the visual inspection that the pension result of active participants after 15 years is normally distributed with the current set of input-parameters. We can therefore now construct a confidence bound based on the normal distribution for pr.,15. A 95% confidence bound for pr.,15 is Φ−10.05(µpr, σpr) = 0.9237. In words, the formula gives the xvalue for which 5% of the mass of the normal distribution -with the mean and the standard deviation of the dataset - is left of this x-value and 95% is right. Concluding, the 5% worst scenarios are worse than this 0.9237 whereas with 95% centainty the pension result is at least 0.9237.
Figure 11: Some percentiles of the pension result during the years; above: inactive participants, below: active participants
In the beginning of this section we defined the average indexation percentage per year. We now take a look at the deviation from this average. An interesting measure for this deviation is the downward deviation. The downward deviation is a measure for the negative deviation from a prespecified reference value, it takes both the frequency of observations lower than the reference value and the magnitude of the difference with the reference value into account. The downward deviation measures, in contrast to the standard deviation, only the deviation of observations that are lower than the reference value. The downward deviation with reference value c, δc, is defined as: δc = v u u t 1 S × T T X t=1 S X s=1 (M IN (0; Is,t− c))2
The two most interesting reference values c for this subject are c = 100% and c = I.,.. Except in a policy with catch-up indexation, for δ100%all observations are used in the calculation, as the indexation percentage has a maximum of 100%. For δI
.,. only the negative deviations from the
overall average indexation percentage are taken into account. The value of the downward devia-tion on its own, for instance δI
.,. = 0.3, is not very useful. However, if the downward deviation
down-ward deviation and can conclude that the negative deviation from the average has increased or decreased. It indicates more or less uncertainty in the indexation percentage, if only looked to in-dexation percentages smaller than the average. The most important reason to prefer the downward deviation to the standard deviation is that, for pension funds and their participants, downward risk is far more important than upward risk. The standard deviation takes both risks into account.
In the case study fund, indexation is based on the funding ratio; linear interpolation between two fixed funding ratios (see appendix B) is used. If F Rs,t< x% or F Rs,t> y% where x% and y% are the left and right bound of the interpolation interval (see section 5.1), then Is,t= 0% or Is,t= 100% respectively. The probabilities related to these interpolation intervals give a look into the risk of not getting full indexation, P (Is,t< 100%), or getting no indexation at all, P (Is,t= 0%). Figures 12 shows these measures, calculated per year.
Figure 12: above: Probability no full indexation, below: probability no indexation
average change in indexation percentage? |∆I.,.| = 1 S × (T − 1) × S X s=1 T X t=2 |Is,t− Is,t−1|
This means that a participant can expect that the indexation next year differs on average |∆I.,.|(×100%) from the indexation this year, irrespective of an increase or a decrease.
In the calculation of the average absolute difference the occurrence of a change at all is not visible: a series of 100% → 80% → 60% → 40% → 20% → 0% indexation returns the same value as a series of 100% → 100% → 100% → 100% → 100% → 0%. Combined with the probability of a change in indexation percentage, P (I.,t− I.,t−1 6= 0), the average absolute difference is a better measure of uncertainty between years.
Just as with the downward deviation, negative differences are more important than positive dif-ferences. The probability of a negative difference means the probability that the indexation per-centage next year is lower than it is this year and is defined using the indicator function IA(ω) which is equal to 1 if ω ∈ A and equal to 0 if ω /∈ A for some set A:
P (I.,t− I.,t−1< 0) = 1 S × (T − 1)× S X s=1 T X t=2 I(−∞,0)(Is,t− Is,t−1)
Not every negative difference is a great difference, a decrease from 100% to 95% is not as bad as a decrease from 100% to 0%. Given the difference is negative, the average negative difference takes this into account and gives therefore more relevance to the probability of a negative difference calculated above. This conditional average negative difference is calculated by dividing the total negative difference by the number of scenarios and years a negative difference occurs:
I.,t− I.,t−1|(I.,t− I.,t−1< 0) = S X s=1 T X t=2 [(Is,t− Is,t−1) × I(−∞,0)(Is,t− Is,t−1)] ÷ S X s=1 T X t=2 I(−∞,0)(Is,t− Is,t−1)
We end this list of interesting statistics to measure uncertainty of indexation with two probabilities that give insight into the influence of a year without indexation on next year’s indexation and in the influence of a year with full indexation on next year’s indexation. Both conditional probabilities are calculated using the definition of conditional probability (P (A|B) = P (A∩B)P (B) , with A and B events and P (B) 6= 0). P (I.t< 100%|I.,t−1≥ 100%) = S X s=1 T X t=2 [I[100%,∞)(Is,t−1) × I[0%,100%)(Is,t)] ÷ S X s=1 T X t=2 I[100%,∞)(Is,t−1)
Now we focus on the other way around, given that the financial position last year was so bad that no indexation was given. What is the probability that one year later the financial position has been improved so much that that some indexation can be given?
P (I.t> 0%|I.,t−1= 0%) = S X s=1 T X t=2 [I{0%}(Is,t−1) × I(0%,∞)(Is,t)] ÷ S X s=1 T X t=2 I{0%}(Is,t−1)
6.2
Subconclusion uncertainty
Section 6.1 derives many statistics to measure uncertainty of future indexation. In this section, we give a summary and an indication of the relevance for different stakeholders, like pension funds, DNB and participants.
As the prediction of the future inflation that must be provided to participants is a long-term prediction, the average indexation in 15 years per scenario is a better measure than the average indexation per year. The overall average is an easy statistic to understand for participants as it just gives the expected indexation percentage on the long term and is therefore a good measure for communication to participants. A disadvantage of this statistic is that it is useless if the policy includes catch-up indexation. In that case the indexation percentage can get extremely high. For instance, if the price inflation is in some year equal to 0.1% and in the same year 4% indexation is catched up as a result of incomplete indexation in the years before, the indexation percentage in the specific year becomes, for the inactives, 4.1%0.1% = 4100%.
A possible better measure for long term expectation of indexation rights is the pension result (after 15 years). In contrary to the average indexation percentage, the pension result takes dif-ferences in inflation rates into account. The pension result is easy to understand and therefore useful to use for communication purposes, because it measures the purchasing power of pensions. Furthermore, the pension result is not affected with the possible extremely high indexation per-centage in case of catch-up indexation.
Besides the average in indexation percentage and pension result, communication about the situa-tion in bad scenarios must be provided. The interpretasitua-tion of the 5% percentile for participants is that participants can expect an average indexation in 15 years or a pension result of at least the value of this percentile with a probability of 95%. The pension result and the 5% percentile of it will probably be the required information that must be communicated to participants.
The downward deviation is too difficult and not interesting for participants. However, it can be a useful instrument for the board of a pension fund to decide about the policy of the pension fund. As the downward deviation takes both the frequency and the magnitude of the differences into account it is probably a better measure of the uncertainty of the indexation percentage than just the average of the average in 15 years (that is the overall average) combined with the 5% percentile of this average in 15 years. Moreover, the downward deviation is not sensitive to the ‘catch-up indexation problem’ described above. However, some caution with the use of the down-ward deviation is needed, as the number in itself is not useful and the downdown-ward devation can be calculated with the overall average as reference value and 100% indexation as reference value.
The probabilities of incomplete indexation (< 100%) and no indexation are useful to get insight into the distribution of the dataset as a whole, the magnitude of the cluster of observations at 0% and 100% become visible and the pension fund and their participants can observe the probability of not getting full indexation or getting no indexation at all. These two measures are easy to understand (and therefore probably suitable for communication to participants) and they are not sensitive to the ‘catch-up indexation problem’. These two measures can for instance be used in the analysis of the quality of the indexation policy in general.
