Bootstrapping economic scenarios : applying a multivariate tail correction factor while preserving tail dependence

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Master’s Thesis

Bootstrapping economic scenarios

Applying a multivariate tail correction factor while preserving

tail dependence

Kevin Weltevreden

Student number: 5935423

Date of final version: January 18, 2015 Master’s programme: Econometrics

Specialisation: Financial Econometrics Supervisor: Dr. N. P. A. Van Giersbergen Second reader: Prof. dr. C. G. H. Diks

Abstract

In this thesis, an economic scenario generator is developed by a bootstrap method with adjustments. These adjustments consist of an AR-GJR-GARCH and a multivariate tail correction factor while preserving (tail) dependence with a Student-t copula. The tail cor-rection factor enables the model to generate a reasonable number of extreme values if the dataset does not contain these extreme values, i.e. due to small samples. It adds some statistical variation in the tails of a distribution while maintaining the overall variance. As a re-sult, a distribution backtest could not reject the null hypothesis of the same underlying distribution for the historical returns and generated returns at a 1% significance level.

Keywords: backtest, bootstrap, copula, economic scenario genera-tor, tail correction facgenera-tor, tail index

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1 Introduction

Since the Great Financial Crisis (GFC) in 2008 there is a lot more attention to risk management and their models for risk estimation. These models must sufficiently capture the potential losses and profits to give risk managers the correct information they need to make important decisions. A wrong decision could lead to extremely high losses or even bankruptcy as in the GFC of 2008 and earlier crises.

Most of these models simulate future economic scenarios, the so-called economic scenario generators (ESG), to give insights in the risks that risk managers are facing. This could be done parametrically, by fitting a proper model to the data, or non-parametrically, i.e. by using the bootstrap method. The latter is done by M¨uller, B¨urgi and Dacorogna (2004), where they use the bootstrap method in combination with a few adjustments to generate future economic scenarios. These adjustments are made to properly handle the possible trends, contemporaneous dependence and serial dependence.

As (financial) risk is often seen as the probability of suffering from the highest potential losses, a closer look at these probabilities is necessary. These probabilities can be found in the left tail of the distribution. It could be the case that the used data sample does not exhibit enough extreme values and the probability of extreme values may be underestimated. In Müller, Bürgi and Dacorogna (2004) a tail correction factor is introduced, which gives a bit more variation in the tails for realistic simulations, while maintaining the overall variance. It is of great importance that this tail correction factor does not change the dependence or other characteristics in the tails. Müller, Bürgi and Dacorogna (2004) used a single tail index parameter, a parame-ter that measures tail behaviour, for all economic variables to compute the tail correction factor for these variables. In fact, every economic variable has a (slightly) different tail-index and hence a (slightly) different tail cor-rection factor. These tail-indices could be estimated parametrically or non-parametrically, for example by calculating the Hill estimator. This thesis will focus on extending the bootstrap method of Müller, Bürgi and Dacorogna (2004) with a multivariate tail correction factor instead of a univariate tail correction factor.

It is important to preserve the dependence between the different variables used in the ESG. From Hartmann, Straetmans and de Vries (2004) it follows that there is higher dependence in the tails during periods of crisis than outside these periods. It is likely to believe that if one market crashes other markets could also crash. One way to preserve tail dependence is by using copulas. This copula could be fitted to the data parametrically and could then be used to simulate.

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There are many different copulas to model dependencies. Probably the best known copula is the Gaussian copula. A major downside of the Gaus-sian copula is the absence of tail dependence and fat tails in the univariate distribution functions, which are two of the main characteristics of financial data. A copula much used in financial literature because of its fat tails and ability to capture tail dependence is the Student-t copula. A comparison be-tween the Gaussian copula and the Student-t copula is made in Jondeau and Rockinger (2006). They find that the Student-t copula is capable of fitting financial data very well, because of its ability to capture tail dependency. The Gaussian and Student-t copula have the key advantage of specifying different levels of correlation between its marginal distribution functions.

Another copula that is capable of capturing tail dependency very well is the Gumbel copula. This copula is an extreme value copula and captures tail dependence even better than the Student-t copula. However, following from Aas (2004), the Gumbel copula exhibits greater dependence in the positive tail than in the negative. Risk managers are often more interested in the potential losses than in potential profits when making decisions. Therefore, a greater dependence in the negative tail may be more desired. This is one of the properties of the Clayton copula. Both the Gumbel and Clayton copula are asymmetric copulas, which is also a stylized fact of financial data. There-fore, these copulas may be able to model financial returns better than the Gaussian or Student-t copula. These Archimedean copulas became widely used, because of their easiness to implement and their broad allowance of dependence structures.

This thesis will explain how the tail correction factor mentioned earlier could be extended to multiple tail correction factors while preserving tail de-pendence. This multivariate tail correction factor will then be incorporated in the ESG. To test if these adjustments improve the ESG, the generated returns of this new ESG will be compared to the generated returns of its pre-vious version with a single tail correction factor. Furthermore, an application in the area of pensions of the developed ESG will be given. The first section will provide the theoretical background. Results of the explained techniques will be given in the second section. It shows a comparison between the dif-ferent copulas and a comparison of the enhanced ESG with its predecessor. Last, the application of the ESG will be demonstrated.

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2 Theory

In this section the theoretical considerations will be discussed. First, some well-known characteristics of financial data will be stated. Second, the boot-strap method of M¨uller, B¨urgi and Dacorogna (2004) will be described. Third, an estimator for the tail-index will be outlined. Fourth, a brief overview of the used copulas and their properties will be given. Last, the backtesting procedures are explained.

2.1 Stylized facts financial time series

There is a lot of research done in the stylized facts of financial time series, see i.e. Tsay (2010). Throughout this thesis there will be referred to some of these well-known properties of financial data. For example,

• Prices display a (time-varying) trend

• Returns have a constant mean close to zero

• Returns have very little autocorrelation or serial correlation, which de-scribes the correlation of a variable with itself over time

• Returns have volatility clustering, meaning that periods with high and low variance alternate

• Returns do not have a Gaussian distribution and exhibit fat tails, latter is also known as excess kurtosis

2.2 Bootstrap method

Here, a brief overview of the bootstrap method and adjustments developed by M¨uller, B¨urgi and Dacorogna (2004) will be provided.

2.2.1 Concept of bootstrap

The concept of bootstrap is to generate a random sample from the original sample. If there is a vector X with elements Xi, i = 1, ..., n where n is the

number of observations, then a bootstrap sample Y of n elements could be generated by random selection of element Xl, l = 1, ..., n, from X and assign

Yj = Xl for j = 1, ..., n. Advantages of this non-parametric method are that

no debatable assumptions of a model need to be made. For example, one does not need to estimate the underlying distribution. Also, dependencies between the variables are maintained as a draw consists of element l from

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each variable. Furthermore, it is easy to add extra variables as it is not necessary to estimate a new model.

However, there are a few drawbacks of the conventional bootstrap method. First, as time series are always subject to serial correlation, these dependen-cies are interrupted by the bootstrap method as it generates new time series randomly. A way to solve this problem partly is by using a block boot-strap method. Instead of randomly select single elements one could also randomly select a block of single elements. In this way the serial correlation is maintained within blocks, but not between blocks. Second, the statistical variation within the data sample could be too small to generate (enough) extreme values. This could lead to underestimation of risks. In the following sections, these disadvantages are (partly) solved.

2.2.2 General overview

To use the bootstrap method explained in the previous section in a proper way for handling time series, a few adjustments need to be made. This leads to a semi-parametric economic scenario generator as estimation of parame-ters is necessary. Rather than using the bootstrap method directly on the variables, it makes more sense to define innovations of the variables. These innovations are less subject to dependence over time than their original prices and could be used to create new samples of the variables in a cumulative way. The bootstrap method of M¨uller, B¨urgi and Dacorogna (2004) will partly be altered, because this will lead to enhancements or could not be done by the original method. The method consists of the following steps, which will be explained in the subsequent sections:

• Transform the four assets by taking the logarithm.

• The logarithm transformation of the previous step is used to define innovations I as log returns, which is the first differences of the trans-formed variables. Log returns are chosen as it could be used in a cumulative way and it leads to positive economic variables. The latter is necessary, because the economic variables used in this thesis can not be negative.

• Fit a AR-GARCH process to treat serial correlation and volatility clus-tering and correct the innovations with this AR-GARCH process. • Bootstrap the modified innovations.

• Multiply the bootstrapped innovations by a tail correction factor to obtain realistic extreme values.

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After applying the tail correction factor, the AR-GARCH process will be reapplied, and the innovations will cumulatively lead to the bootstrapped samples of the economic variables.

2.2.3 AR-GJR-GARCH model

As described earlier, returns could exhibit serial correlation, however, this ef-fect is mostly very small. To perform a bootstrap correctly, independent and identical distributed (iid) variables are necessary. In case of serial correla-tion, the independence assumptions is violated. Therefore, an autoregressive (AR) model is fitted to each time series. An autoregressive process of order p, also AR(p), for time series Xt, for t = 1, ..., T can be described as follows

Xt= φ0 + p

X

l=1

φlXt−l+ t (1)

where φ1, ..., φp are the parameters of the model, φ0 is a constant and t

follows a white noise process with zero mean E(t) = 0, finite variance

Var(t) = σ2 and statistically uncorrelated Cov(t, t−l) = 0, l 6= 0. Due

to simplicity, no moving average (MA) process is estimated. Pure AR mod-els could be estimated by linear least squares, while MA often requires non-linear least squares based on a AR(∞) representation. However, there is a lot of literature regarding to other estimation techniques, but these techniques require also more (difficult) computations.

Also one of the stylized facts of financial data is that it exhibits volatil-ity clustering. To properly perform the bootstrap method, this volatilvolatil-ity clustering has to be treated first. This could be done by fitting a Gener-alized Autoregressive Conditional Heteroskedasticity (GARCH) process. In Müller, Bürgi and Dacorogna (2004) a GARCH(1,1) process is fitted at ev-ery iteration. A GARCH(1,1) model is vev-ery common in financial literature, because it could describe volatility clustering very well, see i.e. Ashley and Patterson (2010). Also, in Müller, Bürgi and Dacorogna (2004), innovations are computed by market expectations, which are updated at every iteration. Therefore, some simplicity of the GARCH model is needed and they use an automated GARCH estimation procedure to compute its coefficients every iteration. However, in this thesis, innovations are computed by log returns and a single estimation of the GARCH model is sufficient. This creates the possibility to select the optimal GARCH model that fits the data very well. A GARCH(m,s) process is defined as follows. Let t denote the return

residuals, then this could be split in a stochastic part ztand time-dependent

standard deviation σt.

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where σt is modelled by σ_t2 = α0+ m X i=1 αi2t−i+ s X j=1 βjσt−j2 (3)

and ztfollows a white noise process with zero mean E(zt) = 0, finite variance

Var(zt) = 1 and statistically uncorrelated Cov(zt, zt−l) = 0, l 6= 0. To

correct the time series for serial correlation and volatility clustering, t is

divided by σt, which results in zt. It is required that all GARCH parameters

are non-negative, otherwise it is possible that the bootstrap procedure does not provide real numbers, which does not make sense economically.

Furthermore, recent literature proposes asymmetric models, due to the beliefs that large negative shocks are expected to increase the volatility more than large positive shocks. This is also known as the leverage effect, see Tsay (2010). One of the many asymmetric GARCH models is the GJR-GARCH, proposed by Glosten, Jagannathan and Runkle (1993), which is essentially the same as a TGARCH model. An extra parameter is added to the original GARCH model to account for the leverage effect, resulting in a GJR-GARCH(m,s) model

σ_t2 = α0+ m

X

i=1

(αi+ γiI(t−i < 0)) 2t−i+ s

X

j=1

βjσt−j2 (4)

where I is an indicator function which equals 1 if the statement in parentheses is true and zero otherwise.

A more general model was proposed by Ding, Granger and Engle (1993). They proposed the Asymmetric Power ARCH (APARCH(m,s)) model, which is defined as σ_tδ = α0 + m X i=1 κi(|t−i| − λit−i)δ+ s X j=1 βjσt−jδ (5)

If δ = 2, this model leads to the GJR-GARCH model described earlier, see Ding, Granger and Engle (1993) for a derivation. It turns out that αi = κi(1 − λi)2 and γi = 4κiλi.

The models will be estimated by maximum likelihood (ML). In case of independent and identical distributed variables, the probability density func-tion is

f (x1, ..., xn|θ) = f (x1|θ) × · · · × f (xn|θ) (6)

then the likelihood function is defined as

L(θ; x1, ..., xn) = f (x1, ..., xn|θ) = n

Y

i=1

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Maximum likelihood could be done by optimizing the log-likelihood function ln L(θ; x1, ..., xn) = n X i=1 ln f (xi|θ) (8)

for all parameters θ.

Furthermore, the starting values are chosen to be equal to the uncondi-tional mean and uncondiuncondi-tional variance, respectively

φ0 1 −Pp l=1φl (9) and α0 1 −Pm i=1αi− Ps j=1βj (10) As multiple models could be fitted to the data, the best model needs to be selected. Selection of the models is based on four criteria Akaike Informa-tion Criterion (AIC), Schwarz or Bayesian InformaInforma-tion Criterion (BIC), Shi-bata Information Criterion (SIC) and Hannan-Quinn Information Criterion (HQIC). The model with the lowest values for all these information criteria, is regarded as the model that best fits the data. The four information criteria are calculated by AIC = 2k n − 2 ln L n ; (11) BIC = k ln(n) n − 2 ln L n ; (12) SIC = ln n + 2k n − 2 ln L n ; (13) HQIC = 2k ln(ln(n)) n − 2 ln L n ; (14)

where ln L equals the log-likelihood, k the number of parameters that need to be estimated and n the number of observations. The division by n is added to adjust the information criteria to yield a contribution per observation. It acts like a penalty term for the number of included observations.

2.2.4 Tail correction factor

The described tail correction factor in M¨uller, B¨urgi and Dacorogna (2004) is based on the fact that any fat-tailed distribution is (approximately) Pareto distributed far in its tails. It does not change the tail behaviour of the

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economic variables. It adds some small stochastic variation in both tails of a distribution by increasing or decreasing the bootstrapped innovations while maintaining the overall variance. They use a Pareto distributed random variable η to multiply the bootstrapped innovations. These innovations Ii

are equal to Zt = (z1t, ..., znt)0 from the GARCH processes and multiplying

will result in corrected innovations

I_i0 = ηIi (15)

where η follows a Pareto distribution

η = A + B(1 − u)−α1 (16)

where u ∼ U (0, 1) and A and B are defined by

A = ηmin− B (17)

B = 1 2

q η2

min(α − 2)2+ 2(α − 1)(α − 2)(1 − η2min) − ηmin(α − 2)

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1

α and α equals the tail-index, see Section 2.3. The tail

correction factor is based on the zero mean assumption of the returns, which is satisfied in the modified returns by fitting an AR model.

They proposed a simpler approach by the use of α = 4 instead of using different values for α for different variables. This implies using the same η for all economic variables. They find realistic tail simulations. However, in practice, different economic variables have different tail behaviour leading to different tail-indices and therefore different values of α. As different values of α implies different values of η, this could influence the dependencies between economic variables, where they need to be maintained. The most convenient way to maintain dependencies is by using copulas. A proper copula is needed and will be fitted to the data in order to maintain dependence within u and thereby within the tail correction factor η and subsequently the innovations.

2.3 Tail-index

The tail correction factor mentioned in the previous section is mainly based on the tail-index α, which is a measure for tail behaviour. It is a well-known fact that the empirical distribution of financial returns have fatter tails than a normal distribution. This means that financial returns have more extremely high or extremely low values than a normal distribution. Tail fatness is one of the subclasses of heavy tails and is defined by

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where X is a random variable exhibiting fat tails. When modelling financial returns it is important to know at what level tail fatness occurs amongst the different assets. This tail behaviour is measured by the tail-index α. It turns out that the tail-index is closely related to the shape of a distribution. 2.3.1 Relation of tail-index and fat tailed distributions

Another notation of the tail-index is called γ, where the relation γ = _α1 holds. A normal distribution has tail-index α = ∞. In fat tailed distributions, such as the Student-t distribution, α is often similar to (one of) the parameter(s) of these distributions. For example, in a Student-t distribution α equals the number of degrees of freedom. In general, a smaller value of α indicates fatter tails than higher values of α.

2.3.2 Original Hill Estimator

There are several estimators to estimate this tail-index. One of the most used estimators of the tail-index is the Hill estimator proposed by Hill (1975). This well-known estimator is easy to implement and asymptotically unbiased. The Hill estimator is defined as

ˆ γ(k) = 1 k k X i=1 ln(x(i)) − ln(x(k + 1)) (20)

where a sample of n positive independent observations is drawn from the financial returns, x(j) is the jth-order statistic in reverse order (to simplify formulas), such that x(j) ≤ x(j − 1) with j = 2, ..., n and suppose that k observations are included from the right tail.

2.3.3 Modified Hill Estimator

However, Huisman, Koedijk, Kool and Palm (2001) sketched two problems with the original Hill estimator. First, the difficulty of choosing k. They showed that a small k leads to a less biased estimator (they noted that there will always be a biased estimator for k > 0), while a large k leads to a more efficient estimator. This gives us a trade-off between bias and efficiency. Second, they stated that the original Hill estimator suffers from small-sample bias. In practice, it is not always possible to gather large samples or the sample may be split due to testing objectives. Huisman, Koedijk, Kool and Palm (2001) also find that γ estimates increase almost linearly for values of k up to a certain threshold κ. To improve the original Hill estimator they suggest a linear regression equation to estimate the tail-index. They propose

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to estimate γ(k) for k = 1, ..., κ. These estimates could be used to compute the linear regression equation

γ(k) = β0 + β1k + (k) (21)

They prefer a weighted least squares (WLS) approach to correct the er-rors (k) for heteroskedasticity. The proposed weighting matrix W has √

1,√2, ...,√κ on the diagonal and zeros elsewhere. The first element of the resulting estimator for β contains the estimated tail-index.

2.4 Copulas

Next to the tail-index estimates, also a copula has to be estimated to gain a multivariate tail correction factor. In this section, copulas will be explained and several widely used copulas will be given. A copula is a multivariate cumulative distribution function (CDF) and was introduced by Sklar (1959). For a random vector X of size (or dimension) d and distribution function F a distribution function C is called a copula if

F = C(F1, ..., Fd) (22)

where Fi = Fi(Xi), i = 1, ..., d are uniform marginals on [0,1] and Xi has

marginal distribution function Fi, thus Xi ∼ Fi. In case of continuous

marginal distribution functions the representation of a copula can be written as C(u1, ..., ud) = P (F1(X1) ≤ u1, ..., Fd(Xd) ≤ ud) = P X1 ≤ F1−1(u1), ..., Xd≤ F_d−1(ud) = F F₁−1(u1), ..., F_d−1(ud) (23)

Copulas are very popular due to their capability to model the marginal distribution function separately from their dependence. In finance, copulas are often used to preserve the dependence within financial data. Especially when proper modelling of tail dependence is necessary the so-called extreme value copulas could be helpful. They describe the behaviour of both vari-ables when exhibiting extremely low or high values. In this thesis, four differ-ent copulas will be compared, namely the Gaussian, Studdiffer-ent-t, Gumbel and Clayton copula. Following the studies of Kole, Koedijk and Verbeek (2005) and Aas (2004), both their mathematical properties as their advantages and disadvantages will be discussed.

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2.4.1 Gaussian copula

Probably the most used copula is the Gaussian copula and this copula is given by

C_dΦ(u; Ω) = Φd(Φ−1(u1), ..., Φ−1(ud); Ω) (24)

where u is a vector of marginal probabilities, Φd is the multivariate CDF of

the normal distribution with correlation matrix Ω and Φ−1 is the inverse of the univariate CDF of the standard normal distribution.

The Gaussian copula belongs to the class of elliptic copulas, which have the key advantage of specifying different levels of correlation between the marginal distribution functions. Unfortunately, it does not perform very well in describing tail dependence and does not produce extreme values. This is also visible from its tail-index, which is α = ∞.

2.4.2 Student-t copula

Also part of the elliptic copulas is the Student-t copula and this copula is defined as

C_dΨ(u; Ω, ν) = Ψd(Ψ−1(u1; ν), ..., Ψ−1(ud; ν); Ω, ν) (25)

where Ψd is the multivariate CDF of the Student-t distribution with

cor-relation matrix Ω and ν degrees of freedom and Ψ−1 is the inverse of the univariate CDF of the Student-t distribution.

However both the Gaussian copula and Student-t copula are belonging to the same class, the Student-t copula is better capable to capture tail dependence, while preserving the dependence in the center of the distribution. As was mentioned earlier a higher value for ν (equals the tail-index in case of the Student-t copula) decreases the probability of extreme values. When ν → ∞, the Student-t copula converges to the Gaussian copula. Moreover, for ν > 30, the differences between the Student-t and Gaussian copula are negligible.

2.4.3 Gumbel copula

In case of tail dependence, the Gumbel copula is a copula that is even better in capturing this tail dependence and is described by

C_dG(u; α) = exp  − d X i=1 (− log ui)α !α1  (26)

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where α is the dependence parameter with α ≥ 1 and α = 1 implies inde-pendence.

The logistic copula belongs to the class of Archimedean copulas and cop-ulas from this class have a simple closed form formula. It is also an extreme value copula, which means that it performs very well with tail dependencies. Although the logistic copula only exhibits upper tail dependence, it could also exhibit lower tail dependence by using the survival copula.

2.4.4 Clayton Copula

Instead of using the survival copula of the Gumbel copula, one may model lower tail dependence by the Clayton copula, which is expressed as

C_dC(u; δ) = d X i=1 u−δ_i − d + 1 !−1δ (27)

where δ is the dependence parameter with δ ≥ 0 and δ = 0 implies indepen-dence.

Lower tail dependence may be more interesting due to risk of losses, therefore one could choose to use a Clayton copula if it wants to be sure that this loss risk is modelled properly. Also, the Clayton copula is part of the class of Archimedean copulas.

2.4.5 Test the fit of copulas

One of the four mentioned copulas will be chosen to model the dependence. Which copula will be used, is determined by several tests. The first test is a goodness-of-fit test. Kole, Koedijk and Verbeek (2005) used modifications of the Kolmogorov-Smirnov test and the Anderson-Darling test to compare the four copulas to the observed data. Under the null hypothesis the sample of observations is drawn from the hypothesized copula. A Kolmogorov-Smirnov test has a better focus on the dependence in the center of the distribution, while the Anderson-Darling test pays more attention to tail dependence.

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where FE is the empirical CDF and FH the hypothesized CDF. Under the

null hypothesis that the observed returns are from the hypothesized CDF, FE

will converge to FH almost surely. The first and the third statistic measure

the greatest distance between the empirical and hypothesized CDFs, while the second and the fourth measure the average distance between the CDFs. The empirical CDF based on sample X is given by

CE(u; X) = 1 T T X t=1 I(x1,t ≤ xu1 ·T 1 ) · ... · I(xn,t≤ xunn·T) (32)

where I is an indicator function which equals 1 if the statement in parentheses is true and zero otherwise, xui·T

i is the kth order statistic where k is the

largest integer not exceeding ui · T . Furthermore, the statistics are not the

original statistics and are modified by Kole, Koedijk and Verbeek (2005). As a result, the distributions of the statistics under the null hypothesis are not the same as the original statistics. Therefore, these distributions are simulated to gain the critical values at certain significance levels. This is done by simulating 10,000 samples of the hypothesized copula and calculate the four test statistics based on these samples.

However, these four statistics may not always be very useful in selecting the best copula. As it determines the fit of the four copulas, it may not always show significant differences of one copula with regard to the other copula, i.e. if the values of the mentioned statistics differ only slightly. Therefore, also the Rivers-Vuong test is calculated. This test by Rivers and Vuong (2002) uses the log-likelihood function of each of the copulas and compares these outcomes. Its hypotheses is defined as

H0 = E[ln Lit(θ∗i) − ln Ljt(θj∗)] = 0;

H1 = E[ln Lit(θ∗i) − ln Ljt(θj∗)] > 0;

H2 = E[ln Lit(θ∗i) − ln Ljt(θj∗)] < 0;

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where H0 is tested versus H1 and H2, ln Lit(θ∗i) equals the log-likelihood of

copula i, i 6= j at time t and θ_i∗ are the pseudo-true values of the parameters of copula i. Under the null hypothesis, the following result holds

√ Tln LiT( ˆθi) − ln LjT( ˆθj) ˆ σijT d − → N (0, 1) (34) where ln LiT( ˆθi) = _T1 PT

t=1ln Lit( ˆθi), ˆθi contains the estimated parameters of

copula i and ˆσ2_ijT is a Newey and West (1987) Heteroskedasticity and Auto-correlation Consistent (HAC) estimator for Vh√T ln LiT( ˆθi) − ln LjT( ˆθj)

i . If the null hypothesis is rejected with respect to a right-sided alternative, then the first copula performs significant better than the second copula. In case the null hypothesis is rejected with respect to a left-sided alternative, then the second copula would outperform the first copula.

Whereas the Rivers-Vuong test requires non-nested models, the likelihood-ratio (LR) test requires nested models. Therefore, the Gaussian copula is compared to the Student-t copula by a LR test. In nested models, one model is a special case of the other model. In the case of the Gaussian copula and the Student-t copula, the Gaussian copula is a special case of the Student-t copula, because if the degrees of freedom parameter of the Student-t copula equals infinity, the Student-t copula equals the Gaussian copula. The test statistic is given by

D = −2 ln(Lmodel underH0) + 2 ln(Lmodel underHa) (35)

where L is the likelihood of the copula, the model under H0 is the

Gaus-sian copula and the model under Ha equals the Student-t copula. The test

statistic follows a chi-square distribution with the degrees of freedom equal to the number of parameters of the alternative model minus the number of parameters of the null model. In this case, this equals 7 − 6 = 1 degree of freedom.

2.4.6 Conclusion

Kole, Koedijk and Verbeek (2005) find that the Gaussian copula underesti-mates the risks and the logistic copula overestiunderesti-mates the risks. This may be due to the fact that the Gaussian copula does not capture tail dependence and the Gumbel copula does not capture dependence in the center very well. They could not find evidence to reject the Student-t copula and show that the Student-t copula is capable to capture the risks accurately. However, it is always necessary to test which copula provides the best fit to the data.

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2.5 Backtesting

At last, the gathered generated returns should be tested with the histori-cal observed returns. One of the methods that is capable of such a test is backtesting. Backtesting could be done in several ways. Here, is chosen for out-of-sample backtesting and to measure by using a Value at Risk (VaR) and a distribution backtest. This means that the first part of a time series is used to estimate and the second part is used to test the validity of these estimations. Generated returns will be compared to the observed historic returns. This is when the VaR and the distribution of a generated returns are corresponding with the VaR and distribution of the observed returns. 2.5.1 Distribution backtesting

The distribution backtest is based on Christoffersen (2012) and M¨uller, B¨urgi and Dacorogna (2004). This entails the following. The marginal empirical cumulative distribution function (ECDF) of the generated return of the same day over all the simulations is determined. Thus, if there are 1,000 simula-tions, the ECDF is calculated over 1,000 generated returns of variable X on day i. Now, the calculated ECDF is evaluated at the observed historic return of variable X on day i. If this is done for all days of the sample that is used for backtesting, one would expect these outcomes to follow a U (0, 1) distribution by the Inverse CDF Transformation. Unfortunately, it is hard to test the hypothesis of a U (0, 1) distribution and therefore the outcomes are transformed to a N (0, 1) distribution by using the inverse CDF of the standard normal distribution. This result can be tested by a Jarque-Bera test, for example.

2.5.2 Value at Risk backtesting

In Boucher, Dan´ıelsson, Kouontchou and Maillet (2014) a VaR backtesting procedure is explained. This backtest method contains three elements of which two are used in this thesis. These two elements are testing the fre-quency of exceeding the VaR and the independence of these exceedances. The first test is based on the unconditional coverage test by Kupiec (1995) and the second is proposed by Christoffersen (1998). Both tests use a LR statistic to test the null hypothesis.

In case of the unconditional coverage test, the null hypothesis states that the percentage of exceedances equals the significance level α at which the

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VaR is calculated. An exceedance is defined as I_tVaR(α) =

(

1 if rt< VaR(α)t−1

0 otherwise (36)

where t = 1, ..., T . The number of exceedances equals TI =

PT t=1I

VaR t (α)

and ˆα = TI

T is the unconditional coverage. Then, under the null hypothesis

LRU C = 2 ln ˆαTI 1 − ˆαT −TI − ln αTI 1 − αT −TI

d −

→ χ2(1) (37) The second test, tests if clustering of exceedances is present, where there is no clustering under the null hypothesis. The LR statistic for this test is given by

LRIN D = 2 (ln L(ˆπ01, ˆπ11) − ln L(ˆπ, ˆπ)) d

−

→ χ2₍₁₎ ₍₃₈₎

where L(πa, πb) = (1 − πa)T00πaT01(1 − πb)T10πbT11, πij is the probability of

state j in the current period while in state i for the previous period, πij =

PI_tVaR(α) = j|I_t−1VaR(α) = i. Tij is the number of observations in state j in

the current period while in state i for the previous period. This results in ˆ

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3 Results

This section will show us some of the results of the techniques from the previous section applied to data. First, the data and its characteristics that are used throughout this thesis are discussed. Second, the results of the tail-index estimates will be given. Third, the AR-GJR-GARCH models with the best fit are shown and the four copulas are compared. Next, the output of the backtesting methods are stated and last, a comparison between the various ESGs will be made.

3.1 Data

For convenience there is chosen for four different assets in which one could invest, namely equity, real estate, corporate bonds and government bonds. These assets are represented respectively by the four indices MSCI World Index, MSCI World Real Estate Index, Dow Jones Corporate Total Index, FTSE Global Government All Maturities Index. Each of these indices contain a wide (global) coverage of its asset. The data is gathered from DataStream over the period of 3 January 2000 to 29 August 2014 and consist of 3,825 returns per index. The first 1,911 returns are used for estimation purposes and the remaining 1,914 returns are used for out-of-sample backtesting. 3.1.1 Characteristics

In Table 1 below, a few characteristics of the log returns of the four indices can be found.

The mean, standard error and significance follow from the AR-GJR-GARCH models, see Section 3.1.3. Also, all the indices are negatively skewed and do not follow a normal distribution as all the P-values from the Jarque-Bera test are below 0.05, which reject the normality assumption. To test whether the mean of the indices differ significantly from zero, a t-test could be used. Only for Real Estate, there is evidence for a non-zero mean, whereas for the other three assets no evidence against a zero-mean can be found. 3.1.2 Tail-index

Also, all the values of the kurtosis are greater than 3, which indicates fat-ter tails than the standard normal distribution. The same is visible when estimating the tail-index α. In Table 2 is shown that the four assets exhibit fatter tails than the standard normal distribution as was expected, because the estimated tail-index is much lower than infinity. Also, the Equity and

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Table 1: Characteristics log returns Equity Real Estate Corporate Bonds Government Bonds Mean 1.4 × 10−4 4.6 × 10−4 4.5 × 10−5 5.8 × 10−5 Std. Err. 1.5 × 10−4 1.6 × 10−4 6.9 × 10−5 6.5 × 10−5 P-value1 3.7 × 10−1 5.1 × 10−3 5.2 × 10−1 3.7 × 10−1 Skewness -0.028 -0.41 -0.14 -0.28 Kurtosis 5.5 5.3 4.2 4.7 P-value J-B2 _{<1.0 × 10}−16 _{<1.0 × 10}−16 _{<1.0 × 10}−16 _{<1.0 × 10}−16 1 _{Test H} 0: φ0= 0

2 _{Jarque-Bera test for normality under the null hypothesis}

Real Estate index have fatter tails than Corporate and Government Bonds. As the latter are less risky assets they have less fatter tails and larger values of α. Furthermore, in M¨uller, B¨urgi and Dacorogna (2004) a tail correction factor is used, which is based on α = 4. To justify the need for different tail correction factors and therefore different values of α, it is necessary to test if the estimated tail-index differs significantly from α = 4. Not all the reported P-values are lower than the 5% significance level at which the t-test rejects the null hypothesis of α being equal to 4 for all assets. If both tails of a distribution are considered, then the null hypothesis for Equity and Real Estate could not be rejected. However, for Corporate Bonds and Govern-ment Bonds the null hypothesis is rejected and the use of a multivariate tail correction factor is justified.

3.1.3 AR-GJR-GARCH model

Furthermore, AR-GJR-GARCH models are fitted to each of the four assets to account for possible serial correlation and volatility clustering. They were estimated by using the R1 package fGarch. Different models up to order 2 were fitted, because higher orders did not pass the statistical significance tests. From the models that did pass the significance tests, the AIC, BIC, SIC and HQIC were calculated. The models with the lowest values for all

1_{R is a free software environment for statistical computing and graphics.} _See http://www.r-project.org/ for more information.

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Table 2: Tail-index α of log returns Equity Real Estate Corporate Bonds Government Bonds Left-side Tail-index 5.0 3.5 7.2 4.9 Std. Err. 7.9 × 10−1 4.2 × 10−1 1.7 7.8 × 10−1 P-value1 1.1 × 10−1 1.2 × 10−1 3.1 × 10−2 1.2 × 10−1 Right-side Tail-index 4.5 6.1 7.5 7.2 Std. Err. 6.7 × 10−1 1.0 1.7 1.5 P-value1 _{2.2 × 10}−1 _{2.0 × 10}−2 _{1.7 × 10}−2 _{1.6 × 10}−2 Both sides Tail-index 4.6 4.5 7.1 5.8 Std. Err. 4.8 × 10−1 4.3 × 10−1 1.0 6.8 × 10−1 P-value1 _{1.1 × 10}−1 _{1.3 × 10}−1 _{1.2 × 10}−3 _{4.1 × 10}−3 1 _{Test H} 0: α = 4

(or most) information criteria were selected. Their coefficients can be found in Appendix A. The models with the best fit are given in Table 3.

Table 3: AR-GJR-GARCH models

Asset Model with best fit Equity AR(1)-GJR-GARCH(1,1) Real Estate AR(1)-GJR-GARCH(1,2) Corporate Bonds AR(0)-GARCH(1,1) Government Bonds AR(0)-GARCH(1,0)

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3.2 Comparison copulas

In order to preserve (tail) dependence in the multivariate tail correction fac-tor, the best copula to generate tail correction factors needs to be selected. All the copulas are fitted by ML and tested on significance of the estimated parameters. The parameters and their t-test values can be found in Ap-pendix B. Next, the fit of the copula is tested by the Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) statistics. These results can be found in Table 4.

Table 4: K-S and A-D statistics of the four estimated copulas

Gaussian Student-t Gumbel Clayton Max K-S 0.0525 0.0549 0.0392 0.0529 P-value 0.974 0.917 1.00 0.946 Mean K-S 0.00678 0.00659 0.00943 0.00841 P-value 0.214 0.105 0.185 0.0074 Max A-D 0.115 0.120 0.248 0.107 P-value 1.00 1.00 1.00 1.00 Mean A-D 0.0265 0.0245 0.0476 0.0286 P-value 0.997 0.902 0.998 0.671

At a 5% significance level the four copulas could not be rejected by the four tests, except for the Clayton copula in case of the mean of the K-S statistic. Also, at a 10% significance level the result remains the same. This means that the observed data could be drawn from all of these four copulas. All four statistics lead to similar distances for the Gaussian and Student-t copula. The ClayStudent-ton copula gives similar disStudent-tances as Student-the Gaussian and Student-t copula, except for the mean of the K-S statistic. The Gumbel copula only performs well in the maximum of the K-S statistic and is much worse in the other three statistics. Based on the maximum values of the K-S and A-D statistics, the Gumbel copula and Clayton copula have the best fit respectively, because it has the lowest maximum value. However, based on the means of both statistics, the Student-t copula fits very well. Unfortunately, this does not lead to a clear choice of a copula in generating

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the tail correction factors, because none of the copulas is rejected by all four tests.

Therefore, a Rivers-Vuong test is performed. Three scenarios are calcu-lated:

1. Student-t copula vs Gumbel copula 2. Student-t copula vs Clayton copula 3. Gumbel copula vs Clayton copula

Table 5: Rivers Vuong test

Scenario 1 2 3

Statistic 16.1 16.3 -0.53 P-value <1.0 × 10−16 <1.0 × 10−16 0.30

From Table 5 it is visible that the Student-t copula performs significantly better than the other two copulas. The statistic is clearly rejected at the right side of the normal distribution. Due to the fact that the Rivers-Vuong does not allow for nested models, also a LR test is calculated. The calcu-lated statistic equals D = 173.96 with corresponding P-value smaller than 1.0 × 10−16. This rejects the null hypothesis of the Gaussian copula and selects the Student-t copula as significantly better.

All three tests are in (slight) favour of the Student-t copula. This outcome does not come as a surprise as Kole, Koedijk and Verbeek (2005) had the same result. They concluded that the Gaussian copula was underestimating the tail dependence, whereas the Gumbel copula was overestimating the tail dependence. The Clayton copula was not part of their research.

3.3 Backtesting

Now, the model is completed and consists of a multivariate tail correction factor with a Student-t copula. Next, the validity of the generated returns needs to be tested. Therefore, two backtests are performed. The first is a distribution backtest and the second is a VaR backtest. The latter contains two tests, a frequency test and an independence test.

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3.3.1 Distribution backtesting

Distribution backtesting entails using the whole distribution of the returns. The result of this test should follow a N (0, 1) distribution. This is tested by Jarque-Bera test. The results are visible in Table 6. For the assets Real Estate, Corporate Bonds and Government Bonds, the null hypothesis of normality could not be rejected. Unfortunately, the null hypothesis is rejected for Equity at a 5% significance level. This means that the generated returns do not correspond with the observed returns. An explanation could be that the sample for backtesting contains the crisis of 2008, which is very unlikely to correspond with the period before the crisis. As a result, this does not justify the use of the tail correction factor, which is designed to generate more extreme values. However, at a 1% significance level, the null hypothesis could not be rejected and the tail correction factor could be justified.

Table 6: Distribution backtest

Equity Real Estate Corporate Bonds Government Bonds Statistic 8.04 0.329 0.508 1.90 P-value 0.0179 0.848 0.776 0.387

Also, the P-values of the distribution backtest per year are shown in Ta-ble 7. It is easy to see that none of the assets could be rejected at a 5% significance level for the years 2008 to 2013. This is not in line with the out-come of the overall backtest, because Equity is rejected at a 5% significance level. A reason could be that the distribution changes over time and the overall distribution is not sufficient to describe the historical returns.

3.3.2 Value at Risk backtesting

Furthermore, a VaR backtest is performed. For every day and every asset a 5% VaR is calculated from 1,000 simulations. The unconditional coverage, LRU C, and independence, LRIN D, are based on the exceedances of the

histor-ical returns in comparison to the VaRs of the generated returns. The result of each statistic is compared to the critical value χ2(1) at a 5% significance level.

In Table 8 the P-values for both tests and the four indices are shown. In both tests, only for Government Bonds the null hypothesis could not

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Table 7: P-values distribution backtest per year

Year Equity Real

Estate Corporate Bonds Government Bonds 2008 0.0633 0.185 0.722 0.195 2009 0.0869 0.194 0.408 0.166 2010 0.152 0.385 0.574 0.169 2011 0.0927 0.222 0.428 0.198 2012 0.105 0.298 0.541 0.236 2013 0.157 0.330 0.546 0.197

be rejected at a 5% significance level, or even at a 10% significance level. Therefore, the percentage of generated returns exceeding the VaR is not very different from the α that is used to determine the VaR of the historical returns. Also, the VaR exceedances are not clustered. The null hypotheses of the other three assets are all rejected at both a 5% and 1% significance level. This indicates significantly more or significantly less exceedances than expected at a 5% VaR and also clustering of these exceedances.

Table 8: P-values VaR backtest

Equity Real Estate Corporate Bonds Government Bonds Unconditional coverage <1.0 × 10 −16 _{<1.0 × 10}−16 _{5.6 × 10}−7 _0.776 Independence 1.4 × 10−5 <1.0 × 10−16 9.1 × 10−5 0.484

Furthermore, the P-values for both tests are shown per year in Table 9. In times of crisis, all the assets are rejected by the unconditional coverage test at a 1% significance level. The worst crisis years are 2008 and 2009. In the years 2010 to 2012 the effects of the crisis are still visible as some assets are rejected at a 1% significance level. In 2013 none of the assets could be rejected by the unconditional coverage test at a 5% significance level. Also, none of the four assets could be rejected by the independence test in case of

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Table 9: P-values VaR backtest per year Equity Real Estate Corporate Bonds Government Bonds 2008 Unconditional coverage 9.4 × 10 −15 _{<1.0 × 10}−16 _{1.9 × 10}−9 _{2.6 × 10}−3 Independence 5.0 × 10−3 <3.8 × 10−2 1.4 × 10−1 7.8 × 10−1 2009 Unconditional coverage 1.9 × 10 −7 _{<1.0 × 10}−16 _{1.3 × 10}−5 _{5.1 × 10}−3 Independence 9.9 × 10−2 2.2 × 10−2 2.0 × 10−1 3.3 × 10−1 2010 Unconditional coverage 0.0104 0.00111 0.113 0.550 Independence 0.0452 0.000217 0.592 0.325 2011 Unconditional coverage 4.3 × 10 −10 _{5.5 × 10}−8 _{1.9 × 10}−2 _{5.6 × 10}−1 Independence 5.7 × 10−1 4.5 × 10−3 1.3 × 10−1 7.2 × 10−2 2012 Unconditional coverage 0.790 0.989 0.0257 0.000631 Independence 0.208 0.145 0.595 0.792 2013 Unconditional coverage 0.0604 0.989 0.790 0.550 Independence 0.534 0.0193 0.196 0.325

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a 1% significance level for the years 2012 and 2013. One could argue that this ESG is not capable of generating a series of extreme values such as in the GFC of 2008.

3.4 Comparison of various ESG

To show a result of the economic scenario generator, Figure 1 shows what happens if e 100 is invested in Equity in 1,000 scenarios. It is easy to see,

Figure 1: Scenarios over time

0 10 20 30 40 50 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Years

that there are some very positive en very negative scenarios, but mainly a lot of scenarios in between. This result is very interesting for risk managers as they can calculate the risks they are facing when investing. In this section, a comparison is made between five variations of the ESG with a single tail correction factor, multivariate tail correction factor and no tail correction factor. In case of a tail correction factor, there is also a comparison between using a copula and a U (0, 1) distribution to generate these tail correction factors. The various ESG are

A. No tail correction factor

B. Single tail correction factor (based on α = 4 for all series) without Student-t copula

C. Multivariate tail correction factor without Student-t copula (assumes in-dependence between series by using same u in tail correction factor)

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D. Single tail correction factor (based on α = 4 for all series) with Student-t copula

E. Multivariate tail correction factor with Student-t copula

where the second ESG is the same as in M¨uller, B¨urgi and Dacorogna (2004) and the last ESG is the one used in this thesis. First, a discussion of the results of the distribution backtest will be given. Second, the results of the VaR backtest are reported. Last, an application of the various ESG can be found in section 4.

Table 10: P-values distribution backtest various ESG

Scenario Equity Real Estate Corporate Bonds Government Bonds A 0.0996 0.274 0.851 0.978 B 0.00881 0.706 0.0471 0.131 C 0.0170 0.810 0.579 0.473 D 0.00734 0.920 0.0652 0.0944 E 0.0179 0.848 0.776 0.387

In Table 10 the P-values of the distribution backtest of the variations of the ESG are shown. At a 5% significance only scenario A could not be rejected. For the other four scenarios Equity is rejected and in scenario B also Corporate Bonds is rejected. If there is chosen for a 1% significance level, also scenarios C and E could not be rejected. In scenarios B and D, Equity is still rejected. Thus, at 5% significance level only the ESG without a tail correction factor could not be rejected, at 1% significance level also the ESGs with multivariate tail correction factors could not be rejected and the ESGs with a single tail correction factor are rejected at both significance levels. Therefore, one may say that the tail correction factor does not perform very well in this dataset.

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Table 11: P-values VaR backtest various ESG in 2012 Equity Real Estate Corporate Bonds Government Bonds Scenario A Unconditional coverage 0.790 0.367 0.0257 0.00275 Independence 0.771 0.0465 0.595 0.724 Scenario B Unconditional coverage 0.763 0.550 0.00275 0.00275 Independence 0.282 0.0712 0.724 0.724 Scenario C Unconditional coverage 0.763 0.367 0.00930 0.000631 Independence 0.282 0.0465 0.659 0.792 Scenario D Unconditional coverage 0.418 0.367 0.0604 0.00930 Independence 0.984 0.0465 0.165 0.659 Scenario E Unconditional coverage 0.790 0.989 0.0257 0.000631 Independence 0.208 0.145 0.595 0.792

The P-values for the VaR backtest in 2012 are given in Table 11. The independence test could not be rejected for almost all scenarios and assets at a 5% significance level. It is rejected for Real Estate in scenarios A, C and D, but in case of a 1% significance level also in these scenarios the null hypothesis could not be rejected. Next, the unconditional coverage test is rejected at a 1% significance level in all scenarios for Government Bonds.

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Corporate Bonds could not be rejected in scenario D at a 5% significance level and also not in scenarios A, B and E at a 1% significance level. Equity and Real Estate perform well in the unconditional coverage test as both could not be rejected at 10% significance level.

Table 12: P-values VaR backtest various ESG in 2013

Equity Real Estate Corporate Bonds Government Bonds Scenario A Unconditional coverage 0.00275 0.989 0.989 0.224 Independence 0.724 0.0193 0.145 0.423 Scenario B Unconditional coverage 0.0257 0.790 0.588 0.989 Independence 0.595 0.0312 0.0481 0.243 Scenario C Unconditional coverage 0.00275 0.989 0.588 0.550 Independence 0.724 0.0193 0.0481 0.325 Scenario D Unconditional coverage 0.0604 0.989 0.588 0.550 Independence 0.534 0.0193 0.0481 0.325 Scenario E Unconditional coverage 0.0604 0.989 0.790 0.550 Independence 0.534 0.0193 0.196 0.325

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Also, a VaR backtest for 2013 is considered. These results are shown in Table 12. In this case, the independence test is rejected at a 5% significance level in all scenarios in case of Real Estate and in scenarios B, C and D in case of Corporate Bonds, but could not be rejected at a 1% significance level. Furthermore, almost all assets in all scenarios could not be rejected by the unconditional coverage test at a 5% significance level. Only Equity could be rejected in scenarios A, B and C. In case of a 1% significance level also scenarios B and C could not be rejected.

Concluding, in the distribution backtest, scenario A could not be rejected at a 5% significance level and furthermore scenarios C and E could not be rejected at a 1% significance level. In a VaR backtest for 2012, scenarios B and E perform well in the independence test and scenario D in the uncon-ditional coverage test. These results are (partly) in contradiction with the outcomes of the distribution backtest, where other scenarios perform better. In case of a VaR backtest for 2013, scenario E performs well in both tests. There is no clear outcome, but overall scenario E is performing well at a 1% significance level and often also at a 5% significance level. An application of these ESGs will be given in the next section.

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4 Application

The following section will show an application of economic scenario gener-ators in the area of pensions. One of the most advanced pension systems in the world is the one in The Netherlands. Currently, the Netherlands is in third place in the Melbourne Mercer Global Pension Index, a study of features of retirement income systems around the world. First, a brief ex-planation of the pension system will be given. Second, an application of the ESG developed in this thesis will be demonstrated. Last, an overview of the application of the various ESG from the previous section will be shown.

4.1 Dutch pension system

In the Dutch pension system it is common that employers pay monthly or yearly premiums for the old age pensions of its employees. This premiums are paid to pension funds, insurers or premium pension institutions (PPI). They will invest the paid premiums to gain a higher capital, which is meant to lead to higher old age pensions for the employees.

In The Netherlands there are a few kinds of pension plans. One of them is the defined contribution pension plan (DC plan). It differs from other pension plans in the way that every employee has its own pension capital, which will be filled with premiums and returns from investments. The height of the premium to be paid is based on the age of the employee and their salary (there are a few more variables, but these are the main two variables). The premiums are higher for older employees in comparison to younger employees with equal salaries, because they have a shorter time period to invest in order to gain the same capital at their retirement age. Furthermore, the capital will be invested according to the age of the employee. It is a well-known fact that higher risks equal higher profits, but also higher losses. Younger employees will invest at a much higher risk, such as equities or real estate, while older employees will invest at a much lower risk, such as corporate or government bonds. One reason is that the capital of older employees is much higher than that of younger employees (because there is a longer period in which premiums are paid and capital is invested) and could not afford big losses.

Employers could often choose between insurers or PPIs which one will provide the pension plan of its employees. When choosing an insurer or PPI it is always important to compare the costs and the contractual conditions. Moreover, the quality of the investments could be interesting to compare too, because insurers and PPIs invest in different funds and in different ways. For example, one insurer could use funds with much lower returns than the others.

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Also, one insurer could keep investing in riskier funds at higher ages than the others. To be able to compare these investment strategies one could use an ESG.

4.2 Application of the ESG

The use of the developed ESG will not capture all the characteristics or investment strategies, such as the quality of the different funds within an asset, as it is not possible to generate scenarios for every fund that insurers or PPIs are using. Here, a comparison will be made between two fictitious insurers/PPIs based on their strategy at which age they will decrease the number of risky assets and the allocation of the four assets: equity, real estate, corporate bonds and government bonds. This strategy is also called a LifeCycle (LC). An example of a LifeCycle can be found in Table 13. A LifeCycle states the investment allocation of an individual at a certain age. There are different types of LifeCycles ranging from defensive to offensive, resulting in risk averse respectively risk loving investments. In practice, LCs

Table 13: Example LifeCycle

Age Equity Real Estate Corporate Bonds Government Bonds 25 40% 40% 20% 0% 45 20% 20% 30% 30% 55 0% 0% 40% 60%

are much more extensive and probably also well considered. This example shows us that when an employee gets older his or her investment allocation will change to less risky assets. The LCs used (also fictitious, but based on real LCs) can be found in Appendix C.

As a result of investing by the LifeCycle methodology, one could give more insight by using a ESG. In Figure 2, there are two LifeCycles. The first is risk averse and the second is risk loving, which means that at higher ages the second LifeCycle is still highly allocated in Equity and Real Estate while the first LifeCycle has shifted the allocation to more riskless investments in Corporate Bonds and Government Bonds. This is visible in Figure 2 by a broader range of outcomes in LifeCycle 2. Due to the more risky investments one could obtain a higher or lower return after several years than in LifeCycle

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Figure 2: Boxplot of pension outcomes LifeCycle 1 LifeCycle 2 4 5 6 7 8 9 Scenario E P ension capital (x 10^5)

1. However, the quantiles indicate that almost all the outcomes of the 1,000 scenarios of the second LifeCycle are higher than the outcomes of the first LifeCycle. This would leave the option to a individual to choose between a possible higher old age pension with the risk of a lower old age pension against a more certain, but probably lower old age pension.

Figure 3: Boxplot of pension outcomes of scenarios LifeCycle 1

Scenario A Scenario B Scenario C Scenario D Scenario E

4.0 4.5 5.0 5.5 6.0 6.5 7.0 LifeCycle 1 P ension capital (x 10^5)

In Figure 3, the pension outcomes of LifeCycle 1 over the various ESGs are shown. The differences between the ESGs are very small and the medians

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are almost equal in all cases. Furthermore, the tail correction factor will have greatest influence in the outliers and the lower outliers are of most interest, because of the risk of low pension outcomes. Therefore, various VaRs of the five scenarios and two LifeCycles are reported in Table 14 and Table 15.

Table 14: VaRs of LifeCycle 1

(× 1,000) Scenario VaR A B C D E 5% 459 471 466 461 470 3% 451 462 456 453 461 2% 445 457 444 449 447 1% 431 441 433 436 438 0.5% 416 432 427 423 426

Table 15: VaRs of LifeCycle 2 (× 1,000) Scenario VaR A B C D E 5% 532 540 543 532 539 3% 520 523 527 517 524 2% 508 511 516 508 517 1% 492 494 489 495 496 0.5% 473 480 473 467 473

Also the results of the reported VaRs are very close. This may indicate very little influence of the tail correction factor, which could be due to the possibility that there is already enough statistical variation in dataset. In LifeCycle 1, scenarios B and E produce similar results and scenarios A and D produce also similar results. The first has higher results and the latter has lower results. This indicates that scenario A and D probably contain lower

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returns and these ESGs are probably producing more extreme values. In case of the risk loving LifeCycle 2, scenarios A and D and scenarios B and E are producing similar results, where the results of scenarios A and D are lower and the results of scenario C are higher. Again, scenarios A and D probably contain lower returns, as the reported VaRs are lower. The ESG of scenario A does not have a tail correction factor and the ESG of scenario D uses a single tail correction factor (based on α = 4 for all series) with a Student-t copula.

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5 Conclusion

In this thesis an economic scenario generator (ESG) is developed. The model is based on the one used in M¨uller, B¨urgi and Dacorogna (2004). They use a bootstrap method with a few adjustments, namely possible volatility cluster-ing effects are taken into account by modellcluster-ing an GARCH(1,1) process and a tail correction factor is introduced to add some statistical variation in the tails of a distribution, because datasets may contain not enough statistical variation to model extreme value properly. The tail correction factor is based on a single tail-index, while in practice, different economic variables will have different tail-indices due to different tail behaviour. This thesis extends this research by modelling a multivariate tail correction factor with a copula to preserve (tail)dependence. Furthermore, an AR-GJR-GARCH model is esti-mated to account for the leverage effect, the belief that large negative shocks increase volatility more than large positive shocks.

First, the tail-indices are estimated by a modified Hill estimator to justify the multivariate tail correction factor. It turns out that in the used dataset, the assets Corporate Bonds and Government Bonds differ significantly from a tail-index equal to 4, the tail-index where the single tail correction factor is based on. Although the other two assets, Equity and Real Estate, are not significantly different, this result still justifies the use of a multivariate tail correction factor.

Next, a copula that best fits the data is determined. Four copulas were es-timated, namely the Gaussian, Student-t, Gumbel and Clayton copula. The Gaussian and Student-t copula allow for separate modelling the dependence between variables, while the Gumbel and Clayton copula have simple closed forms. The Gaussian copula does not take tail dependence into account. The Student-t copula is performing better in terms of tail dependence, but the Gumbel and Clayton copula are in theory the best of these four copulas in capturing tail dependence. Both the Gumbel and Clayton copula also allow for asymmetries. The goodness-of-fit of all copulas is determined by modified Kolmogorov-Smirnov and Anderson-Darling statistics. Except for the Clay-ton copula in one of the statistics, all the copulas could not be rejected in their goodness-of-fit statistics at a 10% significance level and all the copulas fit the data well. Furthermore, to determine the best copula a Rivers-Vuong test is applied to test how copulas perform in comparison with the other copulas. The Rivers-Vuong does not allow for nested models and therefore the performance of the Gaussian and Student-t copula is measured by a LR test. The outcomes of both tests are in favour of the Student-t copula.

Last, the multivariate tail correction factor is applied. These results are tested by using backtest procedures. These procedures test if the generated

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returns are sufficient similar to the historical returns. This is done by a distribution backtest and a value at risk (VaR) backtest. The distribution backtest tests if the historical returns could be from the same distribution as the generated returns. The VaR backtest consist of two test, the uncondi-tional coverage and the independence test. The first part tests if the number of exceedances of the VaR of the generated returns is similar to the number of exceedances of the historical returns. The second part tests if the ex-ceedances are clustered or not. In case of the distribution backtest, only at a 1% significance level the null hypothesis of returns from the same underlying distribution could not be rejected. However, when considering the distribu-tion backtest per year over 2008 to 2013, then the null hypothesis could not be rejected at a 5% significance level. In case of an overall VaR backtest, only for Government Bonds the null hypotheses of both tests could not be rejected at a 10% significance level. The remaining assets could be rejected at a 1% significance level. Furthermore, the VaR backtest is applied per year over 2008 to 2013. At a 1% significance level, almost all assets could not be rejected in both tests in 2012 and 2013. This indicates significantly more or less exceedances in 2008 to 2011 and that they are subject to clustering, which both may be a reasonable phenomenon in times of crisis.

Furthermore, the (ESG) with a multivariate tail correction factor is com-pared to four other ESGs. The other ESGs are one without tail correction factor (scenario A), one with single tail correction factor and tail-index equal to 4 (scenario B), one with multivariate tail correction factor and without Student-t copula (scenario C), one with single tail correction factor (also tail-index equal to 4) and with Student-t copula (scenario D). The ESG used in this thesis is the one with multivariate tail correction factor and with Student-t copula (scenario E). Scenario A assumes enough statistical varia-tion within the dataset and no tail correcvaria-tion factor is needed. The scenarios with a single tail correction factor, scenarios B and D, assume similar tail behaviour of the economic variables and a single tail correction factor is suffi-cient. In case of dissimilar tail behaviour, a multivariate tail correction factor is needed, which is done in scenarios C and E. Also, scenarios D and E differ from scenarios B and C by using a Student-t copula. This copula is used to preserve the (tail) dependence in the tail correction factor.

There is no clear outcome from the comparison in ESGs. In the distri-bution backtest scenario A could not be rejected at a 5% significance level. Also, scenarios C and E could not be rejected at a 1% significance level. In the VaR backtest only 2012 and 2013 are considered. In 2012, scenarios B and E perform well in the independence test and scenario D in the uncon-ditional coverage test. These results are (partly) in contradiction with the outcomes of the distribution backtest, where other scenarios perform better.

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In case of a VaR backtest for 2013, scenario E performs well in both tests. Despite of an unclear outcome, scenario E is overall performing well at a 1% significance level and often also at a 5% significance level. Also, the boxplots of all the scenarios do not lead to clear results. The results are very close. Due to the interest in generating extreme values, various VaRs of the five scenarios and investing by two LifeCycles are compared. These outcomes are also very close. This may indicate very little influence of the tail correction factor, which could be due to the possibility that there is already enough statistical variation in dataset.

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Appendix

A

AR-GJR-GARCH models

Here, the coefficients of the selected AR-GJR-GARCH models are presented. First, the model for the asset Equity is a AR(1)-GJR-GARCH(1,1) model, which is given by

Xt= φ0+ φ1Xt−1+ t

t= σtzt

σ_t2 = α0+ (α1+ γ1I(t−1< 0)) 2t−1+ β1σ2t−1

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where the estimated parameters are in Table 16 below.

Table 16: Coefficients AR(1)-GJR-GARCH(1,1) model for Equity

Parameter Coefficient Standard

Error T-value P-value φ0 1.4 × 10−4 1.5 × 10−4 9.0 × 10−1 3.7 × 10−1 φ1 1.5 × 10−1 2.3 × 10−2 6.4 1.3 × 10−10 α0 9.4 × 10−7 2.1 × 10−7 4.4 9.4 × 10−6 α1 3.1 × 10−2 1.2 × 10−2 2.7 7.7 × 10−3 γ1 9.9 × 10−1 3.6 × 10−1 2.8 5.8 × 10−3 β1 9.2 × 10−1 10.0 × 10−3 9.3 × 101 <1.0 × 10−16

Second, the model for the asset Real Estate is a AR(1)-GJR-GARCH(1,2) model, which is given by

Xt= φ0+ φ1Xt−1+ t

t= σtzt

σ_t2 = α0+ (α1+ γ1I(t−1< 0)) 2t−1+ β1σ2t−1+ β2σ2t−2

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Third, the model for the asset Corporate Bonds is a AR(0)-GARCH(1,1) model, which is given by

Xt= φ0+ t

t= σtzt

σ2_t = α0+ α12t−1+ β1σt−12

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Table 17: Coefficients AR(1)-GJR-GARCH(1,2) model for Real Estate

Error T-value P-value φ0 4.6 × 10−4 1.6 × 10−4 2.8 5.1 × 10−3 φ1 2.1 × 10−1 2.4 × 10−2 8.9 <1.0 × 10−16 α0 3.4 × 10−6 9.2 × 10−7 3.7 1.8 × 10−4 α1 8.7 × 10−2 2.4 × 10−2 3.6 3.1 × 10−4 γ1 3.0 × 10−1 1.1 × 10−1 2.7 7.4 × 10−3 β1 4.2 × 10−1 1.8 × 10−1 2.3 2.2 × 10−2 β2 4.3 × 10−1 1.7 × 10−1 2.6 1.1 × 10−2

Table 18: Coefficients AR(0)-GARCH(1,1) model for Corporate Bonds

Error T-value P-value φ0 4.5 × 10−5 6.9 × 10−5 6.5 × 10−1 5.2 × 10−1

α0 9.2 × 10−8 4.2 × 10−8 2.2 2.8 × 10−2

α1 3.5 × 10−2 6.6 × 10−3 5.3 9.8 × 10−8

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Last, the model for the asset Government Bonds is a AR(0)-GARCH(1,0) model, which is given by

Xt= φ0 + t

t= σtzt

σ2_t = α0+ α12t−1

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Table 19: Coefficients AR(0)-GARCH(1,0) model for Government Bonds

Error T-value P-value φ0 5.8 × 10−5 6.5 × 10−5 9.0 × 10−1 3.7 × 10−1

α0 7.3 × 10−6 3.0 × 10−7 2.4 × 101 <1.0 × 10−16

Bootstrapping economic scenarios : applying a multivariate tail correction factor while preserving tail dependence

Master’s Thesis