Pension risk modelling under the requirements of Basel II.

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Pension risk modelling under the

requirements of Basel II.

Marta Kwapien

s1655043

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Master Thesis

Econometrics, Operations Research and Actuarial Studies Faculty of Economics and Business

Rijksuniversiteit Groningen Supervision:

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Chapter 1 Introduction

In this chapter we give an introduction to the background of the problem treated in this work. First we provide a short description of the relation between a sponsoring company and the pension fund, followed by the requirements coming from this relation. Next we discuss the main research question together with a number of the subquestions which help to answer the main question.

1.1 Background of the problem

The role of the pension fund is to provide retirement benefits to the pensioners. The bene-fits represent an amount of money prespecified in the contract. They have to be projected in the long time horizon due to the system of payments. To be solvent, the investment of the fund should produce high enough returns to cover benefits payment. During 1980s and 1990s the equities that fund invested in were generating stable and high surplus, sufficient to pay the liabilities. However conditions began to change. The multiple crises on the market around the world created major problems in the financial stability of the pension funds assets. At the same time the liabilities started to rise due to aging populations and early retirements. The pension fund faced the problem of insolvency.

Nowadays, if the pension fund is not able to pay out its obligations toward the pensioners, it will look for additional funding. The most common procedure is to return to the spon-soring companies and ask for financial help. If the fund is in deficit, the sponsors have a responsibility to compensate the shortfalls from corporate cash flow. In this situation, a sponsoring company, which we assume in our project is a bank, has to be prepared for it. At every moment in time it needs to be able to pay a certain amount of capital on the account of the pension fund.

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uncertainty in the financial markets. Basel II adopted by the majority of financial institu-tions, provides a framework to estimate the sufficient capital. This accord helps to built an accurate risk management model. Following the Basel requirements, a bank has to clearly identify all risks that are expected to influence the future demand of capital.

Banks began to pay attention to Value at Risk as a method to quantify the risk. VaR has been recognized as a valuable risk management tool to estimate the future required capital for the funding purpose. Additionally, its ability to combine different types of risks fulfills the requirements of Basel II for the best applied model. With VaR bank can estimate the amount of money it will be asked to pay in the future.

1.2 Research questions

The aim of the research is to built and describe a statistical model which will estimate the capital required from a bank functioning as a sponsoring company. With it we will be able to answer the main question: ”‘what is the risk for this institution that it has to supplement its pension obligations?”’. The model will recognize all the risk factors which contribute to the shortfall of the pension fund. Additionally, it will allow for possible dependence between them. As a result, it will determine the additional funding needed to come back to the safe financial position for a pension fund.

In order to answer our main research question and proceed with modelling, we will have to answer the subquestions below.

Pension funds need an injection of capital in the situation when they do not have enough money to pay to the pensioners. What it follows, the risk for a bank to supplement the pension fund is equal to the risk of insolvency of the pension fund. Therefore to estimate the amount of capital needed we should first analyze separately the risks incorporated in assets and liabilities of the pension fund. The changes in both positions define the financial status of the fund.

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on the modelling of the yield curve. In the mentioned model we include the correlation between the yields themselves and inflation.

We use Value at Risk as a tool to measure the risk. It is described by a smallest number l such that the probability that the loss L exceeds l is no larger than (1-α), for α ∈ (0, 1).[1] We can rewrite it as follows:

VaRα = inf {l ∈ R : Pr(L > l) ≤ 1 − α} = inf {l ∈: FL(l) ≥ α}

Using VaR, we estimate which loss in portfolio value can occur with probability α. We will be able also to estimate the increase in value of the liabilities with probability α. First we compute VaR on assets and liabilities at the time horizon equal to one day, meaning for day 01/01/2008. Next, we try to estimate the risk projected one year ahead, on 31/12/2008. Together with VaR estimation, we analyse how sensitive our calculations are to modelling assumptions for assets and liabilities. We use three different modelling approaches. In the first one, we will follow the old fashion method, still used in certain companies, which assumes that the risk factors are normally distributed. Next we will fit the best marginal distribution to the returns and then, we find the best copula describing the dependence between the factors. The copulas under consideration will be Gaussian copula and t copula. We will find out that the last choice is the most appropriate.

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Chapter 2 Pension funds: regulations and

modelling of assets and liabilities

In this chapter we provide information concerning Dutch pension system and main require-ments for a pension fund and for a sponsoring company. Both institutions are regulated by Dutch Central Bank which adopted risk-based supervision in order to prevent under-funding of defined-benefit plans in the Netherlands. Next we shortly describe the approach we will use to model the assets and liabilities of the pension fund. Brief description of the required capital modelling is also given.

2.1 Pension system

Dutch pension funds are in the second pillar of the Dutch pension system. This pillar is formed by the occupational pensions. The general aim of the occupational pension funds in the Netherlands is to top up the premium received in the first pillar to around 70% of the income from work. Contributions to those pensions are normally made by employer and employee. They are managed by pension fund, where they are invested in stock and bonds. Those collected amounts of money and the future returns on investment compose the assets of the pension funds.

Dutch pension funds represent mainly Defined-Benefits plans. It is a plan in which partic-ipants make contribution to the pension fund which might change over the time. However the pension benefit repaid in the future to the pensioner is guaranteed and prespecified in the contract. Those projected benefits represent the liabilities of the pension fund.

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2000s the crash of the equity market combined with the drop in interest rates led several funds to become under-funded or only marginally funded.[2] The financial stability of the pension fund will depend on the investment performance of the portfolio and the changes in the future payment obligations.

2.2 Capital requirements

2.2.1 Requirements for a pension fund

Pension funds collect, pool and invest funds contributed by beneficiaries and sponsors to provide for the future pensions of beneficiaries. They are a means for individuals to accu-mulate savings over their working lives to finance their needs in retirement. Consequently, the ultimate risk for pension plans is that asset returns will not be sufficient to meet promised benefits and required household needs.[3]

The establishment of a pension fund requires the permission of Dutch National Bank. Such permission is granted only if the pension fund financial assets are managed by trustwor-thy and knowledgeable directors. Dutch National Bank monitors pension funds under the Pensions Act which introduced a system originated with a set of solvency standards. It includes a minimum solvency margin and solvency buffers designed to minimize the risk of under-funding of the pension plans due to longevity factor and fluctuations in interest rates, inflation and asset prices.(see [4]) All pension funds must comply with a minimum solvency requirement equivalent to:

Assets

Liabilities > 105%

Whenever this funding ratio is lower then 105%, the pension fund will look to raise the assets. It has 8 years to return back to the required level. We assume that in this situation a sponsoring company will provide the pension fund with capital. For the sake of our further computations, we assume that the injection of capital will be immediate, once the funding ratio will drop below the required level. We assume also that it will be a one-time payment with amount sufficient exactly to come back to the safe level above 105%.

2.2.2 Requirements for a bank

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Banking Supervision. It defines the requirements to maintain a minimum amount of capi-tal.

Basel II Accord allows banks to use risk measures derived from their pension risk manage-ment models. However it imposes certain rules for the use of those models. First of all, the risk factors contained in a risk measurement system should be sufficient to capture the risk inherent in both: assets and liabilities of the pension fund. Additionally, the model should recognize empirical correlations within risk factors. Next, the dataset used in the model should be updated frequently. Finally, Basel II recognizes Value at Risk as a good risk measure. It should be computed daily using a 99th percentile with one-tailed confidence interval.[5]

As in was mentioned, a bank has to consider the same risk that a pension fund faces. It will be a risk of loss in assets and growth in the liabilities, until the moment that the funding ratio will be below 105%. We will build a statistic model which incorporates all those risks. It will estimate, using VaR, the minimum capital that a bank should hold. The daily capital required is represented by a number:

Ct= 1.05Lt− At

2.3 Risk factors and modelling

We estimate the future assets At+1 and liabilities Lt+1 in order to evaluate the required

future capital Ct+1. Following the Basel II regulations, it is important to recognize all the

dependences included in both positions.

We can look closer at the decomposition of the risk factors. On the side of assets the only relevant components for our modelling are investments in a chosen categories - domestic and foreign stocks, bonds and Real Estate. The only relevant liabilities are discounted future benefit payments that depend on interest rate, inflation and longevity factor. We use a highly stylized example of the pension fund. As we already mentioned, assets are additionally composed by the contributions paid by the active professionally individuals. However, we assume that they are a know cash flow, specified by the contract. With no risk included, we decided to omit them in our risk modelling. On the liabilities side, we can find service cost component. It is, by definition, the expense caused by the increase in pension benefits payable to employees during the current year. We will not include this payment into our computations since it is also evaluated once per year and the risk is insignificant.

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in the modelling of the future level of assets At+1. Next we will be able to compute the

loss function together with Value at Risk on assets.

As far as the position of liabilities is concerned, the daily value of liabilities depends on factors mentioned: interest rate, inflation and longevity. However, the longevity will not be included in calculations at the horizon of 1-day. We incorporate it only in the computations of liabilities projected one year ahead. It will be given by the probability of surviving obtained from the life tables. In estimation of Lt+1, we include the correlation

between the yields required to project the benefit payments and the inflation.

Following the described procedures we obtained daily independent observations for At+1

and Lt+1. The independence arise from the fact that we did not include the correlation

between the value of bonds and interest rate in the previous computations. To estimate the level of Ct+1 we should include this relation. It will be incorporated into our model

through copulas. We fit the bivariate copula to the historic series of At and Lt in order

to estimate its parameters. With the best fit we project the correlated values of At+1 and

Lt+1. Then we will be able to calculate the series of capital levels Ct+1.

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Chapter 3 Statistical theory

In this section we present the statistical theory used for the returns modelling. We in-troduce the possible marginal distributions and describe their characteristics. Next we talk about copulas and Monte Carlo simulation as methods of modelling financial series allowing for dependence. Finally we mention Value at Risk.

3.1 Returns and their marginal distributions

As it was mentioned in the introduction, the best practice in risk measurement is to work with returns. In our model we work with the arithmetic rate of return rt. It is constructed

using the following formula:

rt =

Pt− Pt−1

Pt−1

where Pt is the price of the position at the day t.

Once generated the series of returns, we fit the best distribution to them. We consider the normal distribution, Student-t distribution, NIG distribution and case of hyperbolic distribution. Here, we shortly present their characteristics.

-Normal distribution

Normal distribution is the most common distribution used for the returns. The general formula for the probability density function of the multivariate case is:

f (x) = 1 (2π)d/2_|Σ|1/2 exp −1 2(x − µ) 0 Σ−1(x − µ)

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of VaR using Variance-Covariance method. -Student–t distribution

Another choice for the return distribution can be Student–t distribution. The probability density function the multivariate t distribution is given by the formula:

f (x) = Γ( v+d 2 ) (vπ)d/2_Γ(v 2)|Σ|1/2 1 + (x − µ) 0_Σ−1_{(x − µ)} v −(v+d)/2

where v is the number of degrees of freedom , µ - mean, Σ - covariance matrix and Γ is Gamma function. The t-distribution becomes closer to the normal distribution as v → ∞. Also the overall shape of the probability density function of the t-distribution resembles the bell shape of a normal distribution, except that it is wider. We can suspect that it will explain in a better way the observations situated in the tails.

-Generalized hyperbolic distribution

More advanced distribution fitted to the series of returns can be the generalized hyperbolic distribution. It is defined as the normal variance mixture. We say that Y has a normal variance mixture distribution when we can represent it as follows:

Y = µ +√W AZ

where A is a constant, Z has a standard normal distribution and W is a random variable independent of Z. If we condition Y on W we obtain the normal distribution with mean µ and variance wAA0. The distribution of W we call the mixing distribution. The random variable Y is generalized hyperbolic distributed if the mixing distribution is the generalized inverse Gaussian distribution.

The generalized hyperbolic density function is given by: f (x) = cKλ−(d/2)(p(χ + (x − µ)

0_Σ−1_{(x − µ))(ψ + y}0_Σ−1_y)e(x−µ)0Σ−1y

(p(χ + (x − µ)0_Σ−1_{(x − µ))(ψ + y}0_Σ−1_y))(d/2)−λ

where the normalizing constant c is: c = ( √ χψ)−λψλ_{(ψ + y}0_Σ−1_y)(d/2)−λ (2π)d/2_|Σ|1/2_K λ( √ χψ) and Kλ denotes a modified Bessel function of the third kind.

This distribution decreases exponentially, more slowly than the normal distribution. It is therefore suitable to model the series where numerically large values are more probable. The generalized hyperbolic distribution is extremely flexible. It contains many special cases, depending on the choice for the values of the parameters. We will distinguish 2 cases: λ = −1/2 which will give us the distribution known as an NIG distribution and λ = 1 which is known simply as a hyperbolic distribution.

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3.2 Interpolation

To estimate the future liabilities, we will be asked to project the returns on yields with maturities from 1 until 80 years. Those yields form a big dataset that might be difficult to analyze. The method that could simplify it is interpolation.

We use spline interpolation. It consists on estimation of the unknown numerical series in the interval (a, b) using the polynomials. We pick up numbers xi belonging to this interval

that have a following relation: a = x0 < x1 < x3 < .... < xn = b. With it, we create a

series of small intervals (x0, x1), (x1, x2), ...(xn−1, xn) called knots.

For numbers belonging to each interval, we are trying to find a specific function S(x) that interpolates them the best. In this way we obtain a series of functions for each interval. Hence, the spline function will have a following form:

S(x) =                S0(x) for x ∈ [x0, x1] S1(x) for x ∈ [x0, x1] . . Sn−1(x) for x ∈ [xn−1, xn]

We interpolate the data using function spline in R: A Language and Environment for Statistical Computing software. This function performs cubic spline interpolation of given data points. The given functions Si(x) are polynomials of degree three which means that

they join continuously each pair of knots with continuous first and second derivatives. [14]

3.3 Modelling dependence for future returns

We presented the possible choices of the return distributions. Next we need to generate the future observations for the risk factors returns to project the future assets and liabilities. In this projection we need to include the factors dependence. The best way is to simulate the vector of financial correlated observations using multivariate distribution and copulas. Here, we provide description of both methods.

3.3.1 Multivariate distributions

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compa-The formula defining the multivariate normal distribution was given in the previous sec-tion. It is clear that univariate margins are normal. Hence to model future assets and liabilities, we assume that returns on risk factors are normally distributed. The important component of multivariate modelling is the covariance matrix which shows the dependence between the variables.

3.3.2 Copulas

Copula is also a concept of multivariate modelling that allows for dependence between the random variables. It is useful for our project since we can investigate the correlation together with modelling of various marginal distributions for the returns. The distributions of portfolio returns will probably differ between themselves so we can combine them using copulas in order to simulate the future assets observations.

A d-dimensional copula is a distribution function on [0, 1]d_{with standard uniform marginal}

distributions. It is a mapping denoted by C : [0, 1]d→ [0, 1].[1]

By definition C(u1, ..., ud) is an n-dimensional copula if the following three proprieties are

fulfilled.

• C(u1, ..., ud) is increasing in each component of u ∈ [0, 1]n

• C(1, .., 1, ui, 1.., 1) = ui if u ∈ [0, 1]d has all the components equal to 1 except the ui

• For all (a1, ..., ad), (b1, ..., bd) ∈ [0, 1]d with ai ≤ bi we have 2 X i1=1 ... 2 X id=1 (−1)i1+...+id_{C(u) ≥ 0}

where uj1 = aj and uj2 = bj for all j ∈ 1, ...d

With copula we can generate at the same time series of random returns which follow different distributions and are correlated. It is a time saving method which allows for simulation including dependence between the observations.

We use different copulas in the modelling of future returns. Our first choice - the Gaussian copula will be given by:

C_PGa(u) = P r(Φ(X1) ≤ u1, ..., Φ(Xd) ≤ ud)

= ΦP(Φ−1(u1), ..., Φ−1(ud))

where Φ denotes the standard univariate normal density function and ΦP denotes the joint

density function of vector consisting of Xi. The measure P is the correlation matrix of Xi.

The second choice - t-copula will be given by

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where tv is the density function of standard univariate t distribution and tv,P is the joint

density function of the vector consisting of Xi. Here the parameter P is also the correlation

matrix of Xi.

Both copulas are perfect to model high dimensional series. The difference lies in the coef-ficient of tail dependence. It is a measure of the dependence in the tails. For the Gaussian copula the value is zero which means that it is asymptotically independent in both tails. No matter how high is the correlation between the series of random variables, the extreme events will always occur independently. For the t copula the coefficient is non zero. It signifies that this copula allows for the phenomenon of dependent extreme values which is often observed in financial data.

Next we present the possible copulas that might be used to model the dependence be-tween assets and liabilities series. We are interested in the family of Archimedean copulas: Gumbel copula and Clayton copula. We chose them since they are mainly modelling two dimensional dataset. They are described by the following formulas:

C_θGu(u1, u2) = exp −((− ln u1)θ+ (− ln u2)θ)1/θ 1 ≤ θ < ∞

C_θCl(u1, u2) = (u1−θ+ u−θ2 − 1) −1/θ

0 < θ < ∞ where θ is a measure of dependence. If θ=0 we have independence copula.

The Clayton copula represents the lower tail dependence and the Gumbel copula - up-per tail dependence. It means that both copulas generates joint extreme values in the respectively lower and upper corners.

3.4 Value at Risk

The loss on the portfolio or rise in the liabilities of the pension fund are unpredictables. Value at Risk is an useful statistical tool to construct the model that assess the risk of the extreme random events. It measures the worst financial loss under normal market conditions over a specific time interval at a given confidence level 1 − α.

By definition, Value at Risk is described by a smallest number l such that the probability that the loss L exceeds l is no larger than (1-α), for α ∈ (0, 1). We can rewrite it as follows:

V aRα = inf {l ∈ R : P (L > l) ≤ 1 − α} = inf {l ∈: FL(l) ≥ α}

In probabilistic term, VaR is simply a quantile of the loss distribution.[1]

In order to estimate VaR we need to define the loss distribution. For the sake of simplicity we will do it on the example of the value of portfolio: V . Hence the value of portfolio at time t, which can be any time interval chosen, is given by Vt and we assume that random

variable Vtis observable at time t. The loss of portfolio over the period [t, t+1] is described

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Following standard risk-management practice, the value of portfolio Vt can be represented

also as a function of a d-dimensional vector of risk factors: Zt=(Zt,1, , Zt,d) which is given

by:

Vt= f (t, Zt)

In our project the risk factors are the market prices of the components of the portfolio. What follows, the loss distribution on the portfolio is determined by the distribution of the risk factor changes defined as returns rt.[1]

First we compute Value at Risk at the horizon of one day. Since we dispose the dataset of risk factors for time period from 01/01/2004 until 31/12/2007, we can find VaR for day 01/01/2008. We just simply fit the best model to describe the behavior of the returns rt and next we project them, using Monte Carlo simulation, for the next day rt+1. We

generate 1000 observations representing future returns rt+1 followed by 1000 observations

representing losses over one day. With this, we straight forward compute VaR. We can order the series of losses from the higher to the lowest and pick out the result corresponding to the desired confidence level.

Value at Risk at 99% confidence interval: V aR99 gives an estimation of the loss, while

the rare events causing it happen (the events with a 1% of probability). Constructing the V aR99 for a portfolio will give us an estimation of the amount of money we can lose,

investing in a group of assets, with a probability 1%. On the other hand, constructing V aR99on the liabilities will tell us how much the liabilities can grow with this probability.

3.5 Monte Carlo simulation and returns aggregation

The Monte Carlo method describes the simulation of a parametric model for risk-factor changes. It consists on random sampling from the defined model in order to receive ad-ditional series of observations. We use this method to simulate the future assets and liabilities.

The first step of this method is the choice of the model describing the historical observa-tions for the returns rt−n+1, ..., rt. It should be a model from which we can readily simulate,

since in the second stage we generate an important number m of correlated realizations of the future returns which will be denoted as _er_t+1(1), ...,_er(m)_t+1.

With Monte Carlo simulation we generate returns projected 1 day ahead. Next we are able to estimate the future value of assets and liabilities.

Using Monte Carlo simulation we can also easily estimate VaR at the horizon of one year. To compute it, we use the returns projected one-day ahead. Assuming that a year has 249 trading days, we can estimate one year return _ert+1:

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Hence first, we simply simulate 249 daily future returns rt+1from the describing model and

compute the one year return using the formula above. Next we repeat the procedure 1000 times, in order to receive the series of 1000 future returns _erT +1. With estimated yearly

returns, we compute the series of yearly losses followed by VaR.

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Chapter 4 Data description

In this chapter the dataset used in the document is presented. We describe the data rep-resenting the components of assets and liabilities. We use the statistical software R: A Language and Environment for Statistical Computing.

The set of observations is given from day 1/01/2004 until 31/12/2007. It gives 995 ob-servations for each trading day in a year. The series were mainly taken from the website: http://finance.yahoo.com/. They correspond to the prices of the components of the port-folio and the indexes of the financial factors like inflation and interest rate.

4.1 Assets

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First we plot the logarithmic prices of the portfolio components to have the first idea about their characteristics.

Figure 4.1: Plot of the logarithmic prices.

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In the table below we briefly describe some properties of the dataset: Statistics AEX SP500 Bond Real Estate

Minimum 5.739 6.726 1.115 7.064

Mean 6.041 6.902 1.351 7.454

Maximum 6.331 7.0426 1.552 7.864

Table 4.1: Summary statistics for market prices. Next, we generate the returns and present the plots in Figure 4.2:

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We observe the returns of AEX relatively volatile through the whole time period. Contrary, the SP500 and Real Estate returns show the big jumps only during the year 2007. Bonds seem to increase their volatilities in the middle of 2005.

Globally, the standard deviation of the returns have similar magnitude. The volatilities of AEX and SP500 returns are equal to around 0.008. For returns on Real Estate the standard deviation is 0.01 and for bonds it is 0.008.

Below we present the boxplots for the series in question:

Figure 4.3: Boxplots of the returns on assets components.

Bond returns are mainly concentrated around the mean. We see fewer observations defined as outliers comparing to other boxplots. We observe lots of outliers of AEX, SP500 and specially of Real Estate returns. Most of them are negative. This could significantly influence our future modelling. We need to find a model that describes in a good way the observations situated in the tails of the future distributions.

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distribution will be right-skewed. As far as kurtosis is concerned, we have the following results for the returns: AEX 1.35, SP500 1.34, Bond 1.05 and Real Estate 4.63. This measure gives and indication about the ”‘peakedness”’ of the distribution. We conclude that it will be relatively high for the Real Estate returns.

4.2 Liabilities

Liabilities of the pension fund are mainly the projected future benefits paid to the pen-sioners. The amount of liabilities is a series of payments that must be made to retirees far into the future. From the point of view of the pension fund, liabilities are subject to uncertainty in real interest rate, future inflation developments and longevity factor. -Interest rate

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Figure 4.4: Plot of the logarithm of the yields.

We observe that the yields of 5, 10 and 15 years represent close graphical characteristics. The jumps appear in exactly the same time periods. It gives us a confirmation that the remaining yields for years 16-80 could be also very similar. The same idea we can have observing the basic statistics for the yields. For each yield they are very similar:

Statistics 1-yr 5-yr 10-yr 15-yr Minimum 1.33 1.37 1.36 1.34

Mean 1.56 1.56 1.54 1.53

Maximum 1.77 1.78 1.76 1.74 Table 4.2: Summary statistics for yields.

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Figure 4.5: Plot of the growth factors.

As we can see, the variation of the growth factors for year 1 and 5 is high. We observe also that the factors 10-yr and 15-yr behave similarly through the whole time period. The higher jumps appear in the same moments. The standard deviation increases with the year of maturity. We have volatility equal to 0.0037 for 1-year yield, 0.0063 for 5-year, 0.0084 for 10-year and 0.0097 for 15-year.

To investigate how the returns are distributed around the mean we plot the boxplots in Figure 4.6. We see that all the returns have number of outliers. In the boxplot of 5-year yield we observe one extreme point corresponding to the huge loss in value in the middle of year 2005. We can distinguish that most of outliers are positive. As we mentioned already in the assets analyse, we need to consider this fact during the adjustment of the model. Hence, the best models for growth factors are the ones which allow for the fat tails of the distribution.

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it is equal to 13.89, 8.09 for 5-year, 1.47 for 10-year and 1.99 for 15-year. It gives an idea that the growth factors distributions will have high ”‘peaks”’.

Figure 4.6: Boxplots of the growth factors.

-Inflation rate

Due to inflation, the rights accumulated during working years will have a much lower purchasing power upon retirement. Pension funds try to correct the pension rights for inflation by offering indexation of the benefits.

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We present the plot of CPI In Figure 4.7:

Figure 4.7: Plot of daily inflation.

The daily inflation observations end up to be very small numbers with mean equal to 0.00015, minimum = 0.000033 and maximum = 0.00029. As we suspect, the returns on the inflation will be numbers very close to zero. Additionally, they are regrouped in the multiple classes over the period 2004-2007, which we can observe on the plot above. -Life expectancy

Increasing life expectancy among the pensioners can result in situation where payout of the benefits should be higher than what a company or fund originally expects. The present value of the annuity payments will go up with raise of the number of years a pensioner will survive. The estimation of evolution of life expectancy is important to asses rightly the future liabilities of the fund.

Because we are trying to project liabilities only one year in advance, we will not simulate the future observations describing the life expectancy. To make easier our calculations we introduce the assumption concerning the retired population. We assume that at day 31/12/2007 we have only 65-years old pensioners, both men and women. Then, for estima-tion, we simply use the probability of surviving, once reached 65 years. This probability can be computed from the life tables accessible for each country. In our project we use the Dutch life tables. The probability estimated is equal to p65 = 0.987. It describes the

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Chapter 5 Marginal models

In this chapter we try to find the best descriptive models for the returns on risk factors. We proceed with fitting the marginal distributions to the series of returns. We use the graphical representations to find the best models. Next we present the log-likelihood test results and QQ-plots.

5.1 Assets

We print the histograms together with the densities of various fitted distributions. The graph reveals the best fit for the returns.

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Figure 5.2: Histogram with plotted densities.

We observe the heavy left tails for most of the histograms which were revealed by the skewness in the previous section. Those observations correspond to the losses generated by the components of portfolio. It is very important to asses the best distribution describing the behavior of the returns in the left tail. The accuracy of our further computation of Value at Risk will depend, to some extant, on this choice.

The asymmetric hyperbolic distribution seems to be the best fit for the returns on the stocks and on the bond. However the returns on Real Estate could be described in the best way by the NIG distribution which is one of the cases of the hyperbolic distribution. Additionally we can see that for all 4 cases, the normal distribution gives the worst fit. The estimated models can be compared using the log-likelihood value. The largest value of log-likelihood, the best is the model. In the tables 5.1 and 5.2 we present the results for the returns together with parameters of the fitted distributions:

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AEX SP500

Parameters norm t NIG hyp norm t NIG hyp

µ 0.0004 0.0007 0.0005 0.00086 0.00017 0.0004 0.00018 0.00058 σ 0.00074 0.00072 - - 0.00069 0.0072 - -df - 6.13 - - - 8.43 - -ψ - - 29240 29240 - - 3170 3170 χ - - 0.0014 0.00019 - - 0.00013 0.00098 γ - - -0.006 -0.301 - - -0.009 -0.0328 loglik 3307.1 3330.2 3330.2 3331.1 3345.1 3365.9 3368.3 3369.6 AIC -6610.3 -6654.4 -6652.4 -6654.2 -6686.2 -6725.9 -6728.6 -6731.2 Table 5.1: Table with estimated parameters and loglikelihood values for returns on assets components.

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Figure 5.3: QQ-plots for returns on assets components.

5.2 Liabilities

-Interest rate

Next we try to fit distributions to the growth factors. Below we present their histograms plotted together with densities of chosen distributions. We observe that the evaluated high kurtosis in the previous chapter is confirmed by the high ”‘peaks”’ around return equal to 1.

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This hypothesis seems to be confirmed by the loglikelihood outcome. Both distributions: NIG distribution and t-distribution have very good results. However we choose the best model looking up at the AIC criterion. What it follows, we will model all the growth factors using Student t distribution.

1-year 5-year

Parameters norm t NIG hyp norm t NIG hyp

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Additionally, we include QQ-plots. We present the plots for 4 yields in Figure 5.5. They are relatively linear hence we can assume that distributions fitted are accurate for the series.

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-Inflation

Because of the nature of the data a problem will appear while we fit the distribution. As we mentioned before the returns are very small to represent them graphically hence we skip the plotting of the histogram. We use only the loglikelihood outcome to choose the distribution for inflation returns. The best fit is NIG distribution.

Inflation

Normal Student-t NIG Hyperbolic

Loglik 944.8 5494.9 6004.0 1797.8

AIC -1885.6 -10983.8 -12000.0 -3587.6

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Chapter 6 Copulas and multivariate

distributions

This section treats the methods of future assets and liabilities simulations. We describe the dependence relations between the components of both positions. We shortly describe the simulations of the future returns on assets and liabilities. Finally we discuss the methods of fitting the best copulas and multivariate normal distribution.

6.1 Future assets and liabilities

6.1.1 Assets

As we know, the level of assets is mostly defined by the portfolio investments. Hence the value of assets at day t is given by:

At = 31%(XAEX× AEXt) + 10%(XSP 500× SP 500t) + 45%(Xbond× Bondt)

+14%(Xre× REt)

where Xcis the amount of money invested in the portfolio component c. And the projected

future level of assets at day t + 1 will be denoted by: At+1 = At(1 + rt+1assets)

= 31%(XAEX × AEXt)(1 + rAEXt+1 )) + 10%(XSP 500× SP 500t)(1 + rSP 500t+1 )

+45%(XBond× Bondt)(1 + rBondt+1 ) + 14%(XRE × REt)(1 + rREt+1))

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computed based on the historical observations: Cor(Assets) =     1 0.47 0.23 0.52 0.47 1 0.12 0.32 0.23 0.12 1 0.11 0.52 0.32 0.11 1    

In this matrix, the first row elements give the correlation between AEX returns and re-maining returns, the second row correspond to correlation between SP500 returns and other returns, the third row to Bond returns and the last one to Real Estate returns. As we can see the AEX stock index is strongly correlated with all other components. Also Real Estate position shows strong dependence on the prices of stocks and bonds.

6.1.2 Liabilities

The present value of the liabilities at day t is given by the equation: Lt = Benefits Paymentst

where future benefits payment are subject to uncertainty in inflation rate and yields with maturities from 1 until 80.

Lt+1= Benefits Paymentst(1 + r yields

t+1 )(1 + r

inf lation t+1 )

Hence to estimate the future observations for liabilities, we need to model growth factors that will be useful to discount expected future benefit payments. We generate also the inflation returns since pension funds offer indexation of the benefits in order to correct the pension rights for inflation. However, this method is not compulsory for the fund. The pension funds aim to index the pension rights to prices, but this indexation is conditional on the financial performance of the fund. It depends on the position of the investments of the pension funds. Very often the indexation procedure depends on the funding ratio. Above 135% of the ratio, there is a full indexation. Between 135% - 105% we have partial indexation, estimated with the specific formula. Below 105% the indexation is 0%.[6] In our document, we simplify the calculations. We assume the full indexation in every case. This allows us to observe the full effect of inflation on future Value at Risk. We will compare it to the interest rate’s contribution to VaR.

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15-year yields and the inflation returns. The yields and inflation are positively correlated between themselves. Cor(Liabilities) =       1 0.31 0.19 0.14 0.05 0.31 1 0.78 0.63 0.03 0.19 0.78 1 0.95 0.01 0.14 0.63 0.95 1 0.01 0.05 0.03 0.01 0.01 1      

Next step is to find the best descriptive model for our dataset of risk factors. Assuming that growth factors have similar characteristics, we model only the ones for 1-year, 5-year, 10-year and 15-year. Next we use the spline interpolation to find the estimated growth factors for maturities between 1-15 year. For maturities above 15 year, we use the estimates of 15-year growth factor. We follow this procedure since in the previous chapter we showed that we can suspect the close similarities between those yields and their returns.

6.2 Multivariate distribution

We fit the multivariate normal distribution to the series of growth returns, inflation and portfolio components in order to proceed with simulation the future assets and liabilities.

6.2.1 Assets

The first method we use to simulate the future assets is the multivariate normal distribu-tion. What it follows, we assume that the returns on portfolio components are normally distributed. The loglikelihood test shows a relatively high result equal to 13289.51. With fitted multivariate normal distribution and estimated mean vector together with correlation matrix we can generate the future values of returns. They are projected one day ahead. Using Monte Carlo simulation, we generate 1000 observations for each of the portfolio components: AEX, SP500, Bond and Real Estate. Next, we simply compute the assets projected one day ahead.

6.2.2 Liabilities

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6.3 Gaussian copula with empirical margins

In this section we present the results of fitting Gaussian copulas to returns on risk factors influencing assets and liabilities.

6.3.1 Assets

Before fitting the copula, we transform the observations into the right format. For each data point, we subtract the mean and divide by the standard deviation : (X − µ)/σ and we apply the empirical distribution function. Once we achieve this, we can proceed with copula estimation.

The likelihood of our fit is equal to 262.11 which is not very high. We will see in the next section that t–copula is better choice for our dataset. We can also examine the produced Spearmans estimate of the correlation matrix. It gives the outcome close to the true one:

d Cor(Assets) =     1 0.41 0.22 0.46 0.41 1 0.11 0.23 0.22 0.11 1 0.13 0.46 0.23 0.13 1    

Once estimated the parameters of the model, we proceed with Monte Carlo simulation of future observations. We generate 1000 observations for each of the components of the portfolio using the estimated copula. Those numbers will be uniformly distributed hence we will transform them in order to obtain the true values for the returns. The quantile function of the best fits of distributions is used to achieve it. Next, using the mentioned equation we compute the future assets. We will obtain 1000 observations for one day ahead value of At+1.

6.3.2 Liabilities

For each series of returns on risk factors we perform the same data transformation as for the assets. Next we fit the Gaussian copula. The loglikelihood test gives a number 1766.345. For the sake of comparison we look at the estimates of the correlation matrix:

d Cor(Liabilities) =       1 0.26 0.19 0.15 0.05 0.26 1 0.80 0.68 0.03 0.19 0.80 1 0.95 0.01 0.15 0.68 0.95 1 0.01 0.05 0.03 0.01 0.01 1      

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using the Gaussian copula. With Monte Carlo simulation we generate 1000 observations for each growth factors and inflation. Next we use the quantile function to receive the true values of the projected returns and compute the series of Lt+1.

6.4 t copula with empirical margins

We present here the results from fitting t copulas to the returns with different marginal distribution. Those distributions present the best fit for the returns. They were specifically analysed in the previous chapter.

6.4.1 Assets

The last method to model future assets is to use different choice of copulas. We will simulate the matrix with columns corresponding to the future generated observations. Hence, the matrix will consists on 4 columns: AEX, SP500, Bond and Real Estate. Additionally, those observations will be correlated. The first step is to transform that dataset to the right format as we described in the previous section. Next we fit t copula. The loglikelihood value is equal to 277.01 which is higher than for Gaussian copula.

We can verify our fit with Spearmans estimate of the correlation matrix. We see below that the estimates of the correlation are relatively close to the true ones.

d Cor(Assets) =     1 0.39 0.22 0.44 0.39 1 0.11 0.25 0.22 0.11 1 0.11 0.44 0.25 0.11 1    

Next we proceed with Monte Carlo simulation of future observations. We will generate 1000 numbers for each risk factor from estimated copula. We transform them in order to obtain the true values for the returns using quantile function. The projected series of returns allow us to compute one day ahead value of At+1.

6.4.2 Liabilities

To find the future values of Lt+1we model t copula with dimension equal to 5. We transform

the growth factors and inflation into the right form. Next we fit t-copula and receive the results of loglikelihood equal to 1975.195.

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which allows us to make a judgment about accuracy of the fit. ˆ Cor(Liabilities) =       1 0.23 0.16 0.13 0.05 0.23 1 0.83 0.72 0.04 0.16 0.83 1 0.95 0.01 0.13 0.72 0.95 1 0.01 0.05 0.04 0.01 0.01 1      

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Chapter 7 Value at Risk on assets and liabilities

In this chapter we estimate Value at Risk. The series of projected assets and liabilities are evaluated using multivariate distribution and copulas. The results were presented in the previous chapter. Based on those computations, we estimate the possible loss in assets and gain in liabilities value. The VaR number will be given at the horizon of one day and one year.

We describe all the results together in order to have good comparison. First we present VaRs on separate risk factors. Next, we estimate the total VaR on assets and liabilities.

7.1 VaR at one day horizon

7.1.1 Assets

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Multivariate normal distribution V aR90% V aR95% V aR99% AEX -0.0104 -0.0135 -0.0197 SP500 -0.0097 -0.0129 -0.0205 Bond 0.0114 -0.0144 -0.0205 Real Estate -0.0126 -0.0170 -0.0244

Table 7.1: VaR on assets returns simulated with multivariate distribution Gaussian copula V aR90% V aR95% V aR99% AEX -0.0114 -0.0158 -0.0294 SP500 -0.0093 -0.0131 -0.0210 Bond -0.0109 -0.0142 -0.0219 Real Estate -0.0218 -0.0294 -0.0340

Table 7.2: VaR on assets returns simulated with Gaussian copula t-copula V aR90% V aR95% V aR99% AEX -0.0129 -0.0169 -0.0265 SP500 -0.0095 -0.0128 -0.0239 Bond -0.0124 -0.0163 -0.0256 Real Estate -0.0237 -0.0307 -0.0389

Table 7.3: VaR on assets returns simulated with t copula

We see that loss in value of returns is higher for t-copula case, specially for VaR99 %. The

marginal distributions like hyperbolic or Student t support better the heavy tails and generate more extreme observations.

We have information about the level of indexes for the day 31/12/2007. It is 515.77 for AEX, 1468.36 for SP500, 4.41 for bonds and 1916.34 for Real Estate. We assume that pension fund invests in portfolio the following amounts: 221,672,273 in AEX, 80,206,334 in SP500, 323,211,912 in bonds and 98,030,373 in Real Estate. Those numbers are given for the day 31/12/2007. The total value of portfolio at that day is equal to At= 723,120,891.[7]

With it we compute the loss function using the following formula: Losst+1 = −(Assetst+1− Assetst)

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VaR with multivariate distribution

V aR90% V aR95% V aR99%

5,660,874 7,635,400 10,331,174

(5,169,159 ; 6,151,435) (6,631,460 ; 8,420,568) (9,677,422 ; 11,235,839) Table 7.4: VaR on assets at the horizon of 1-day

VaR with Gauss copula

V aR90% V aR95% V aR99%

6,109,922 8,852,869 13,022,684

VaR with t-copula

V aR90% V aR95% V aR99%

7,167,606 8,948,490 13,137,807

Once again we obtain the highest results for numbers simulated with t-copula. We can conclude also from the high values in tables 7.5 and 7.6 that copula models allow for the extreme movements in the prices. This capacity helps to incorporate in the model the risk causing the huge losses in value of portfolio.

7.1.2 Liabilities

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Multivariate normal distribution V aR90% V aR95% V aR99% YR1 0.0055 0.0089 0.0115 YR5 0.0090 0.0121 0.0187 YR10 0.0098 0.0148 0.0216 YR15 0.0112 0.0172 0.0226 INF 0.0034 0.0071 0.0094

Table 7.7: VaR on liabilities returns simulated with multivariate distribution Gaussian copula V aR90% V aR95% V aR99% YR1 0.0067 0.0098 0.0149 YR5 0.0094 0.0126 0.0201 YR10 0.0124 0.0160 0.0212 YR15 0.01496 0.0191 0.0242 INF 0.0042 0.0082 0.0118

Table 7.8: VaR on liabilities returns simulated with Gaussian copula t copula V aR90% V aR95% V aR99% YR1 0.0032 0.0055 0.0187 YR5 0.0093 0.0134 0.0214 YR10 0.0115 0.0156 0.0227 YR15 0.0130 0.0176 0.0282 INF 0.0047 0.0107 0.0134

Table 7.9: VaR on liabilities returns simulated with t copula

Higher returns are generated with Gaussian and t copula. As it was explained, it is related to the characteristics of the marginal distributions that were used to generate the future returns. The tails are heavier comparing to the normal margins.

To estimate future gain on liabilities we use the formula for Lt+1. As we know, the future

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the funding ratio result.

We estimate the losses, using th starting value for discounted benefits payments for day 31/12/2007 equal to 572,444,320. [7] However, for liabilities we are interested how much this position can increase over 1 day, with a given probability.

Gaint+1 = Liabilitiest+1− Liabilitiest

For this reason we order the dataset from the highest to the lowest and pick up the observations corresponding to the chosen confidence intervals. We present the results below:

V aR90% V aR95% V aR99%

4,994,742 6,877,894 8,834,181

(4,611,388 ; 5,451,020) (6,194,600 ; 7,293,664) (8,328,694; 11,236,259) Table 7.10: VaR on liabilities at the horizon of 1-day

V aR90% V aR95% V aR99%

5,431,342 7,271,053 10,534,816

(5,063,215 ; 5,967,177) (6,614,498 ; 8,013,553) (9,265,411 ; 11,030,451) Table 7.11: VaR on liabilities at the horizon of 1-day

VaR with t-copula

V aR90% V aR95% V aR99%

5,594,639 7,279,208 11,599,422

(5,141,857 ; 5,948,296) (6,661,116 ; 7,829,947) ( 9,265,411 ; 12,030,451) Table 7.12: VaR on liabilities at the horizon of 1-day

The maximum gain generated with t-copula is higher than the remaining numbers.

7.2 VaR at one year horizon

7.2.1 Assets

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generate 249 observations from each of the model which correspond to returns for each trading day in a year. Next we add the obtained numbers, using the function colSums in order to find the returns on the horizon of one year. We repeat the simulation 1000 times. Once estimated the series of yearly projected returns, we order them and pick the lowest return corresponding to the chosen 1 − α value. Below we present the results:

Multivariate normal distribution V aR90% V aR95% V aR99%

AEX -0.071 -0.126 -0.237

SP500 -0.123 -0.174 -0.215

Bond -0.179 -0.176 -0.222

Real Estate -0.098 -0.158 -0.281

Table 7.13: VaR on assets returns simulated with multivariate distribution.

Gaussian copula V aR90% V aR95% V aR99% AEX -0.1234 -0.1658 -0.2677 SP500 -0.1215 -0.1604 -0.2317 Bond -0.2001 -0.2452 -0.3220 Real Estate -0.2152 -0.2742 -0.3960

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t-copula V aR90% V aR95% V aR99% AEX -0.1038 -0.1572 -0.2745 SP500 -0.1269 -0.1788 -0.2424 Bond -0.1952 -0.2422 -0.3322 Real Estate -0.1870 -0.2827 -0.4199

Table 7.15: VaR on assets returns simulated with t copula.

Next we compute the loss function on portfolio, using the given value of portfolio on 31/12/2007 equal to 723, 120, 891. With yearly returns, we calculate how much we can loose the next year, on day 31/12/2008. The results are given in tables 7.16, 7.17 and 7.18 together with confidence intervals:

V aR90% V aR95% V aR99%

42,575,242 70,509,657 103,716,032

(33,022,371 ; 53,582,198 ) (60,783,884 ; 75,079,371) (91,841,851 ; 110,717,098) Table 7.16: VaR on assets at the horizon of 1-year

V aR90% V aR95% V aR99%

48,398,454 77,610,454 128,017,233

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VaR with t-copula

V aR90% V aR95% V aR99%

48,600,558 78,808,285 129,915,104

(37,404,145 ; 51,255,721) (63,792,844 ; 81,986,025) (102,591,357 ; 155,554,148) Table 7.18: VaR on assets at the horizon of 1-year

We observe the highest VaR numbers for t-copula model. As discussed before, it is con-nected with a good support of heavy tails by marginal NIG and t distributions but also with tail dependence for this copula. The highest loss occurred with 1% for t model: 129,915,104 gives around 18% loss of the starting value of the portfolio.

7.2.2 Liabilities

To estimate the possible future gain in liabilities value after 1 year, first we need to generate an amount of 249 observations for growth factors and inflation returns. We obtain the value for each trading day of the year. Next we sum it up into the yearly return.

We arrange our dataset in the ascending order. Next we pick up the numbers corresponding to defined confidence interval. Below we present our findings:

Multivariate normal distribution V aR90% V aR95% V aR99% YR1 0.1592 0.1844 0.2213 YR5 0.1835 0.2221 0.2793 YR10 0.1757 0.2190 0.3102 YR15 0.1727 0.2358 0.3315 INF 0.0978 0.1567 0.1972

Table 7.19: VaR on liabilities returns simulated with multivariate distribution.

Gaussian copula V aR90% V aR95% V aR99% YR1 0.2081 0.2479 0.3146 YR5 0.2108 0.2596 0.3406 YR10 0.1794 0.2274 0.3332 YR15 0.1947 0.2558 0.3778 INF 0.1280 0.1849 0.2245

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t copula V aR90% V aR95% V aR99% YR1 0.2089 0.2491 0.3148 YR5 0.2191 0.2646 0.3544 YR10 0.1801 0.2464 0.3667 YR15 0.1897 0.2672 0.4010 INF 0.1380 0.1949 0.2341

Table 7.21: VaR on liabilities returns simulated with t copula

We observe again that the numbers generated from t copula are slightly bigger than those generated with Gaussian copula. Multivariate normal model gives smaller gain on returns. The liabilities projected one year ahead will be computed using the given formula for Lt+1.

The possible gains with given probabilities are presented in the Tables 7.22, 7.23 and 7.24:

V aR90% V aR95% V aR99%

97,261,030 129,665,709 194,763,976

(90,631,083 ; 105,492,914) (113,289,378 ; 140,274,594) (174,526,523 ; 223,344,477) Table 7.22: VaR on liabilities at the horizon of 1-year

V aR90% V aR95% V aR99%

110,474,183 157,527,201 238,583,819

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VaR with t-copula

V aR90% V aR95% V aR99%

119,386,319 157,093,315 258,630,785

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Chapter 8 Estimation of the minimum capital

required

In this section we present the estimation of the future capital required to transfer to the pension fund. Those calculations are made from the point of view of the sponsoring company which needs to pay off the shortfall of the fund. The projections are made for one year in advance, on day 31/12/2008.

We model the future assets and liabilities with copula. This method is used regarding the probable dependence between both positions.

8.1 Modelling with copula

We suspect that the assets and liabilities are correlated. This reasoning is based on the existence of strong dependence between the value of the bond and the interest rate used to discount the benefits payment. Hence the modelling of the future capital Ct= 1.05Lt− At

should incorporate this relation which is given by the following correlation matrix: Passets,liabilities =

1 0.27 0.27 1

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Log-likelihood

Gumbel copula 9.59

Clayton copula 11.61

Table 8.1: Table with loglikehood results

It seems that Clayton copula gives better fit than Gumbel. Hence we chose it to model the future level of assets and liabilities. We generate 1000 observations from Clayton copula, using the estimated parameters of θ. We obtain the observations representing the uniformly distributed random variables. Next we perform quantile transformation in order to obtain the real values of assets and liabilities. For positions generated with multivariate normal distribution we use the normal quantile function using the mean and standard deviation of the assets and liabilities. For Gaussian and t copulas cases, we repeat the same procedure, using this time the mean and standard deviation of positions generated with respective copulas. For both series, we use the quantile of Student t distribution since it gave the best fit.

We obtained the projected one year ahead future level of assets and liabilities, correlated between themselves. The estimated correlations matrices for all simulation methods are given below: ˆ Pmult = 1 0.279 0.279 1 ˆ PGauss = 1 0.289 0.289 1 ˆ Pt = 1 0.281 0.281 1

All of them are accurate. The new series of the future positions will be used to evaluate the required amount of capital needed to come back to the safe level of the funding ratio.

8.2 VaR estimation of the funding ratio and capital

With estimated future correlated positions, we can easily compute the funding ratio: At+1/Lt+1 projected one year in advance. Using VaR we can furthermore approximate

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V aR90% V aR95% V aR99%

0.14 0.19 0.24

(0.12 ; 0.16) (0.18 ; 0.20) (0.22 ; 0.25)

Table 8.2: VaR on funding ratio at the horizon of 1-year VaR with Gauss copula

V aR90% V aR95% V aR99%

0.18 0.23 0.34

( 0.17 ; 0.20 ) (0.22 ; 0.27 ) ( 0.33 ; 0.41) Table 8.3: VaR on funding ratio at the horizon of 1-year

VaR with t copula

V aR90% V aR95% V aR99%

0.19 0.26 0.38

(0.17 ;0.21 ) ( 0.24 ; 0.29) (0.36 ; 0.42 )

Table 8.4: VaR on funding ratio at the horizon of 1-year Following the regulations of Pension Act, we constate that the

With estimated assets and liabilities we compute the future required capital payment: Ct+1 = 1.05Lt+1 − At+1. All the positive numbers in the series show how much money

should be transfer to the pension fund. As we assume in the introduction of the problem, funding occurs immediately after the shortfall is spotted.

On the day 31/12/2007 pension fund is in a good financial situation. We computed already the funding ratio which was equal to 1.27. The starting demand of capital is equal to −122, 153, 663. It is negative since the pension does not need any funding. We compute the series of 1000 observations representing the projected future capital Ct+1. Next, our aim

is to find the number that represents the demand of capital which occurs with probability: 10%, 5% and 1%. To find those numbers we use the series of losses on funding level. We create the matrix 1000×2 with losses in the first column and projected capital Ct+1 in the

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We pick up the numbers corresponding to the desired confidence interval. We see that bank, on 31/12/2008 will be asked to provide, with probability 1%, the highest amount equal to 110, 551, 992, assuming that we model with t copula.

10% 5% 1%

-44,498,884 -10,270,858 40,674,170

Table 8.5: Estimated demand of capital with multivariate distribution

10% 5% 1%

-18,848,794 9,535,662 96,241,910

Table 8.6: Estimated demand of capital with Gauss copula

VaR with t-copula

10% 5% 1%

-16,862,018 34,681,095 110,551,992

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Chapter 9 Conclusion

During the analysis that we conducted in our document we found the best marginal fits for returns distributions. We saw that normal distribution supports badly the heavy tails and tend to underestimate the risk of extreme events. It showed the worst fit with log-likelihood and AIC criterion. Contrary Student t distribution and family of generalized hyperbolic distributions gave accurate results for both: asset and liabilities risk returns. The simulation of the future observations allowed for projection of higher losses or gains. Next we used multivariate normal distribution and copulas : Gaussian and t to gener-ate the future series of assets and liabilities. The best fit gave t copula. Comparing to multivariate Gaussian distribution which is still used in the major number of companies, with copulas we could construct models which go beyond the standard notions of correla-tion. We measured the dependence structure with tail dependence. It incorporated into the model the possible extreme joint co-movements of risk returns. This phenomenon of dependent extreme values is often observed in financial return data hence it is important to recognize it in capital estimation. The next advantage of copulas is that it enabled us to construct a multivariate distribution function from different marginal distributions of returns. Contrary, using multivariate Gaussian distribution we assumed normal margins for the returns. All those aspects are important to consider in order to precisely estimate the risk incorporated in movements of prices and market indexes. Sponsoring companies need to have an accurate forecast of the financial stability of the pension fund. This fore-cast needs to include all the extreme movements in returns, even those that occur with low probability. That is why we suspect that copulas will be gradually becoming an element of risk management practice of financial institutions.

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is the most important risk factor in the liabilities calculation. It constitutes around 95% of the total VaR on liabilities. The accurate estimation of the yield changes is an important tool to control the level of liabilities and consequently the funding ratio.

Estimation of Value at Risk revealed the probable losses and gains in values over one day and one year. It showed that portfolio value can lose in average over 11 mln euros over one night with 1%. The prognostics for one year are more considerable, with loss equal to 150 mln euros. On the other hand we can look closer at the possible gain in value of liabilities. With 1% of probability, the projected benefits payment can raise for about 10 mln euros during one day and 230 mln euros during one year. We observed high disparity in values between multivariate model and copulas concluding that Value at Risk is very sensitive to modelling assumptions.

The procedure we used to simulate the correlated series of assets and liabilities is not the simplest way to do. We could omit the modelling of separate assets and liabilities series and construct a copula including all the risk factors influencing both positions. Neverthe-less it would be difficult to fit copula to very high dimensional dataset. The model would probably be inexact. Due to Basel regulations, we were required to include all the recog-nized dependence between the components. Hence we modelled the assets and liabilities separately and next we used copula to include the correlation between them. We constate that this approach is better than the assumption about independence.

We observe that starting from the high relation assets/liabilities we have 1% of probability to end up around the critical value of 105% for the funding ratio, if we consider models with normal margins. Concerning the models with Student t and hyperbolic distribution, it appears that with 5% of probability we have chance to see the pension fund insolvent. And, looking at the current situation of the funds, we assume that this outcome is more realistic. Nowadays, practically all pension funds have suffered huge investment losses as a result of the current economic crisis. As a result, the coverage ratios long-term pension capital against long-term pension commitments of most pension funds have fallen below the critical limit of 105% to levels of 90% or lower.[13]

Funding ratio can be raised back to the safe level with the funding provided by the sponsor-ing company. We estimated this amount ussponsor-ing the relation: fundsponsor-ing level-required capital. The results differ with respect to the models. Multivariate normal distribution generates more friendly scenarios for the bank, however less accurate. With, as we assumed ”‘more realistic model”’, banks need to prepare for significant capital depletion.

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[3] OECD Secretariat., (n.d.), Developments in Pension Fund risk management in selected OECD and Asian countries,http://www.oecd.org, 2008.

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Chapter 10 Appendix

10.1 R programming code to analyse Assets

########################## ASSETS #####################

library(QRMlib) library(fBasics) library(fGarch)

###Importing the dataset###

series <- read.csv(’Data.csv’,header=T,sep=";"); order <- dim(series)[1] components <- as.vector(series); summary(components); ###Creating a timeseries### components.series <- as.timeSeries(components) par(mfrow=c(2,2)) plot(components.series[,1],type="l",xlab="Time", ylab="AEX", main="AEX" ); plot(components.series[,2],type="l",xlab="Time", ylab="SP500", main="SP500" ); plot(components.series[,3],type="l",xlab="Time", ylab="Bond", main="Bond" ); plot(components.series[,4],type="l",xlab="Time", ylab="Real Estate", main="Real Estate" );

###Constructing the log-observations### logcomponents <- log(components.series); par(mfrow=c(2,2))

Pension risk modelling under the requirements of Basel II.