Volatility Models with Asymmetric Dynamics and their Application to Risk Estimation

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Volatility Models with Asymmetric

Dynamics and their Application to Risk

Estimation

Stefanos Charalampopoulos

(S2832305)

A thesis submitted in partial fulfillment for the

degree of Master of Science Econometrics, Operations

Research and Actuarial Studies

in the

Faculty of Economics and Business

University of Groningen

Supervisor: prof. dr. T.K. (Theo) Dijkstra

Co-assessor: prof. dr. C. (Kees) Praagman

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Acknowledgements

First and foremost, I would first like to thank my supervisor prof. dr. T.K. (Theo) Dijkstra of the Faculty of Economics and Business at University of Groningen. His patient guidance, encouragement, and advice made me a more complete student and individual. I would also like to thank prof. Kevin Sheppard from the University of Oxford for his valuable comments on the collection of the dataset.

I would like to express my gratitude to my great family for the support, and constant encour-agement I have received all these years. In particular, I would like to thank my parents, Theo and Monique, my brother George and my uncle and aunt, Rob and Simone. You are the salt of the earth, and I undoubtedly could not have done this without you.

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Abstract

We compare multivariate GARCH models under different distributional assumptions with respect to fit and usefulness in risk management. Our analysis extends the work by Cappiello, Engle and Sheppard (2006) in three ways: our sample adds the period 2002-2016, we allow for non-normal distributions for the innovations, and we calculate and backtest Value at Risk and Expected Shortfall. We find clear evidence of contagion in 2008, the onset of the financial crisis. Allowing for asymmetry in conditional moments enhances the model’s statistical power. Alternatives to the Gaussian distribution clearly improve risk measurement.

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1 Introduction

Over the past 20 years, financial markets were often in turmoil, contributing to or even onsetting economic crisis, resulting in undesirable outcomes for private investors and financial institutions. Economists showed strong linkages between markets under stress which is highly relevant for asset and liability management and risk management. As a result, multivariate modeling of financial assets is significantly improved in financial econometrics over the past decade. Understanding the dynamic interdependence of components of portfolios may lead to better risk assessment, asset allocation and hedging.

Unfavorable market movements can be due to several risk factors. That is, the level of interest rates changes, exchange rates, equity prices and volatility of these rates and prices. Such movements are seen as being random. While the randomness concept is not generally questioned, the absence of distorting effects and the normality assumption, while offering flexibility and stylishness to the mathematical modeling setup, do not hold in reality. The stylized facts of asset returns are widely argued to depart from those of the Gaussian distribution, as such quantile-based methods that are used to control and manage financial risk should be able to account for fat tails and skewness in volatility clustering. This gave rise to multivariate models and distributions able to account for non-normal shapes and asymmetries in conditional moments. However, most of these models can not be used in practice (curse of dimensionality) and many distributions are complex with undesirable attributes (infinite variance) for financial work.

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a stylized fact is of major importance and has to be incorporated in financial and econometric analysis.

This thesis’s main target is the investigation of the dynamic evolution of conditional correla-tions across regions and their potential use in risk management. The hypotheses that we examine are: Does allowing for asymmetric responses to news increase the quality of the dynamic model? What is the effect of the use of non-normal distributions on conditional correlations? For the empirical analysis, a sample of national share and bond indices is employed which includes most events that had a significant impact on financial markets during 1987–2016.

This thesis adds to the literature in several ways. First, we update previous research of Cappiello et al. (2006)by extending their sample up to 2016 to examine whether their conclusions still hold. Second, we check the robustness of the Asymmetric Dynamic Conditional Correlation model by considering different distributions for the innovations. Third, we evaluate the out of sample performance of the dynamic models by backtesting risk measures.

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Chapter

2 Literature Review

Anticipating uncertainty by the forecasting volatility of assets is crucial for financial decisions. The novel class of conditional volatility models introduced by Engle (1982) and generalized by Bollerslev (1986), namely generalized autoregressive conditional heteroskedastic (G)ARCH models are able to capture the volatility clustering phenomenon. The main idea behind these models is that they allow volatility to operate as a dynamic function of previous squared returns and variances. They are flexible, easy to fit and are widely applicable in financial modeling. The GARCH models are extended to many dimensions of which some of the most relevant for our purpose are the Exponential GARCH (EGARCH) Nelson (1991) and GJR-GARCH by Glosten et al. (1993). Both models, EGARCH, and GJR-GARCH named after Lawrence.R. Glosten, Ravi Jagannathan and David E. Runkle, can capture the asymmetric effect of unpredictable events. Other extensions such as the Asymmetric Power ARCH (APARCH) by Ding et al. (1993) and Component GARCH (CGARCH) by Lee and Engle (1999) are also able to mimic different stylized facts of financial returns such as the long-memory property and the long-run dependence in volatility.

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Bollerslev (1990) introduced a model able to deal with the ‘curse of dimensionality’ by allowing for a ‘breakdown’ in the likelihood function. The author considered a sample of Foreign Exchange rates and found high comovements between currencies. The Constant Conditional Correlation (CCC) model was argued to be unrealistic because the correlation matrix is held constant over time. A far superior extension is the Dynamic Conditional Correlation (DCC) model developed by Engle (2002) and Engle and Sheppard (2001) able to account for time-varying correlations by maintaining the estimation decomposition. The DCC model, whose main target is the estimation of large correlation and covariance matrices, uses the standardize residuals to update the correlation at each point in time. They prove parameter consistency of the model and employ equities and equity indices to show the powerful performance of this estimator. However, this model is symmetric, in a sense that both negative and positive news have a similar effect on conditional correlations and covariances. Cappielo et al. (2006) generalizes the model to the Generalized Asymmetric DCC (AGDCC) able capture the leverage effect. They consider a sample of international share and bond indices, and provide substantial evidence of asymmetric effects on the conditional volatility and correlation of these assets. Furthermore, they indicate that markets show high linkages across regions especially during financial distress and that the introduction of the fixed currency regime in 1999 leads to bond market conversion. Based on their results and considering the worst financial crisis of 2008 since the Great Depression, we extend their dataset to examine the effects of this great turmoil.

Some of the most recent examinations on financial contagion using these models are Syri-opoulos and Roumpis (2009), Kenourgios et al. (2011) and Lahrech and Sylwester (2011). Syriopoulos and Roumpis (2009)use the DCC and ADCC models to investigate for correlation and volatility dependence across The Balkans and developing markets. They find correlation linkages between The Balkan stock markets. Kenourgios et al. (2011) used the ADCC model and a Gaussian Copula to investigate for financial contagion between BRIC1_{countries and the}

U.S and U.K markets during periods of industrial stress. They reveal significant evidence of correlation spill-overs. Lastly, Lahrech and Sylwester (2011) use the DCC framework to examine the level of integration of Latin American markets to the U.S market during 1988-2004. They also report dependencies between markets.

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Chapter

3 Methodology

1 _{The purpose of this chapter is to briefly introduce conditional correlation measurement in}

financial time series and then describe the DCC and ADCC models and their benefits.

3.1 Multivariate GARCH and Dependence

Definition 1 :If Xi and Xj are two random variables, then the correlation coefficient between

Xiand Xj is

ρ(Xi, Xj) =

cov(Xi, Xj)

pvar(X_i)var(Xj)

given that the second moment exist and is finite. The numerator denotes the covariance between two random variables and is written as cov(Xi, Xj) = E(Xi, Xj) − E(Xi)E(Xj).

In time series, conditional correlation refers to correlation between Xitand Xjt given their

history up to t − 1 and is defined as ρt(Xi,t, Xj,t) =

covt−1(Xi,t, Xj,t)

pvart−1(Xi,t)vart−1(Xj,t)

.

All correlations, conditional and unconditional, lie between [−1, 1]. Important to note is that this class of correlations is invariant under linear transformations. If Xi and Xj are assumed

or transformed to have zero mean and unit variance, then conditional correlation equals the conditional covariance of the standardized values.

The Multivariate GARCH (MGARCH) models are a simple extension of the univariate GARCH to the multivariate domain.

Definition 2 : Let Xt{t = 1, 2, ...T } be a random process of financial returns with dimension

k × 1 and mean vector µt, given the history Ft−1:

Xt|Ft−1 = µt+ Σ 1/2

t Zt

1_{For most of the notation and definition of the models used in this research, we follow the book Quantitative}

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where Σ1/2_t is an k × k positive definite matrix with Σtbeing the conditional covariance matrix

of Xt, and Ztan k × 1 i.i.d random vector with moments:

E(Zt) = 0,

V ar(Zt) = Ik,

with Ikbeing the identity matrix. The conditional covariance matrix Σtof Xtis defined as:

V ar(Xt|Ft−1) = V art−1(Xt) = V art−1(Σ 1/2 t Zt) = Σ1/2_t V art−1(Zt)(Σ1/2t ) > = Σt

3.2 The Dynamic Conditional Correlation model

Among all MGARCH models described in the literature (Bauwens et all (2006)), those that model conditional correlation are considered to be most useful for financial modeling. The basic idea of these models is that they allow for a log-likelihood decomposition making practical work more feasible by easing the estimation process. The covariance matrix is separated in conditional volatilities and conditional correlations, allowing for a 2-stage estimation procedure where the conditional standard deviations are estimated in the first stage and then the conditional correlation matrix, based on the first stage volatilities, is estimated in the second stage. One of the first well know conditional models, the Constant Conditional Correlation (CCC) is introduced by Bollerslev (1990) for which the covariance matrix is given by

Σt= ∆tPc∆t

where ∆t= ∆t(Σ) = diag(σt,1, ..., σt,k) and Pcis the positive constant conditional correlation

matrix. The main critique of this model is that constant correlations are not realistic when dealing with financial markets. Below we discuss models with time-varying correlations.

These models build upon the time-invariant CCC model by making the correlation specifi-cation dynamic through time. The below model is introduced by Engle (2002) and Engle and Sheppard (2001).

Suppose Xt is an MGARCH k-dimensional process of financial returns. The Dynamic

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where σ2_t,k= αk0+ pk X i=1 αkiXt−i,k2 + qk X j=1 βkjσt−j,k2

and αk0 > 0, αki ≥ 0 for i = 1, ..., pk and βkj ≥ 0 for j = 1, ..., qk.

3. Ptis the conditional correlation matrix of the form

Pt= P((1 − p X i=1 αi− q X j=1 βj)Pc+ p X i=1 αiYt−1Yt−1> + q X j=1 βjPt−j) where Pc= PT t=1YtYt> T ,

2_{P is the operator that extracts a correlation matrix, Y}

t= ∆−1t Xt

is the devolatized process and αi ≥ 0 βi ≥ 0 withPp_i=1αi+Pq_j=1βj < 1.

The diagonal elements of ∆tcan follow different GARCH processes. The univariate models

tested in this project are the following:

1. GARCH (Bollerslev (1986)): The generalized autoregressive conditional heteroskedastic (GARCH) model allows the variance of a process to be linearly dependent on previous squared returns and previous squared volatilities.

σ_t2 = α0+ p X i=1 αiXt−i2 + q X j=1 βjσ2t−j with α0 > 0, αi ≥ 0 and βj ≥ 0.

2. EGARCH (Nelson (1991)): The Exponential GARCH model has the attractive possibility to capture asymmetric effects of shocks to volatility. Moreover, it does not require any restrictions on the parameters since the log variance is considered.

log σ_t2 = α0+ p

X

i=1

αi(|zt−i| − E(|zt−i|) + γizt−i+ q

X

j=1

βjlog σt−j2

3. GJR-GARCH (Glosten et al (1993)): This model, similar to EGARCH, has the ability to capture the leverage effect in volatility. The GARCH model is a restricted class of the GJR-GARCH model, for γ = 0.

σ_t2 = α0+ p X i=1 (αi+ γiIt−1)Xt−i2 + q X j=1 βjσ2t−j

where It−i= 1 if Xt−i < 0 and It−i= 0 if Xt−i > 0.

4. APARCH (Ding et al (1993)):The Asymmetric Power ARCH model, besides the fact that captures the asymmetric effect in volatility clustering, it also delivers the long memory property in a sense that the autocorrelation function has a hyperbolic decay. Considering the parameter δ, it can be seen as a parent model of most of the models described previously.

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5. CGARCH (Lee and Engle (1999)) The Component GARCH model separates the con-ditonal variance into two components to investigate short and long run dependence in volatility. The intercept now is time varying following autoregressive dynamics.

σ_t2 = qt+ p X i=1 αi(Xt−i2 − qt−i) + q X j=1 βj(σt−j2 − qt−j) where qt= α0+ ρqt−1+ φ(Xt−12 − σ2t−1)

Moreover, for both univariate and multivariate GARCH processes, we restrict to unit or-ders. Unit orders will favor computational effort and perceive high levels of volatility more parsimoniously.

To check that Σt= ∆tPt∆tis positive definite we establish that Qtthe argument of P(·):

Qt= (1 − p X i=1 αi − q X j=1 βj)Pc+ p X i=1 αiYt−1Yt−1> + q X j=1 βjPt−j

is positive definite. This is sufficient since all elements in the diagonal matrix ∆tare positive.

If we assume that Pt−q > 0, ..., Pt−j > 0 and take u 6= 0 with u ∈ Rkthen:

u>Qtu = (1 − p X i=1 αi− q X j=1 βj)u>Pcu + p X i=1 αiu>Yt−1Yt−1> u + q X j=1 βju>Pt−ju > 0

This holds sincePp

i=1αi− Pq j=1βj < 1 implying that (1 − Pp i=1αi− Pq j=1βj)u>Pcu > 0.

Moreover note that αi ≥ 0 βi ≥ 0 are sufficient to ensure thatPp_i=1αiu>Yt−1Yt−1> u ≥ 0 and

Pq

j=1βju

>_P

t−ju ≥ 0.

The matrix Qtcan be also written as a covariance stationary GARCH:

Qt= A + p X i=1 αiYt−1Yt−1> + q X j=1 βjPt−j

This would definitely increase the number of parameters to be estimated from 2 to3_{2 + k}k+1

2 .

This is called variance targeting (Engle and Mezrich (1995)). Correlation targeting on the other hand, which is considered in our model specification, is achieved by replacing A with A = Pt(1 − Pp i=1αi− Pq j=1βj). The conditionPp i=1αi+ Pq

j=1βj < 1, not only ensures positivity of the covariance matrix,

it also guarantees that Σtwill be mean-reverting. The structure of the DCC approach allows the

direct estimation of conditional correlations through the first step devolatized residuals. Thus, Yt

is used to update the conditional correlation each point in time.

3.3 Extension of the DCC model: Conditional

Asymmetry

Although DCC models have been a significant pillar for research and development within financial econometrics and other related sectors, they are symmetric in a sense that both negative

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and positive news have same effects on correlation and volatility. Below we describe an extension of the DCC model which can capture the leverage effect.

Cappiello et al (2006)developed an extension of the DCC model able to capture the phe-nomenon of asymmetry in volatility clustering. The Asymmetric Dynamic Conditional Correla-tion (ADCC) model, enriches the DCC specificaCorrela-tion and is able to capture persistent high levels of volatility during riotous periods. Making a comparison with the specification of the DCC model described above the correlation matrix is given by :

Pt= P((Pc−Pc p X i=1 αi−Pc q X j=1 βj−Nc s X k=1 γk)+ p X i=1 αiYt−1Yt−1> + q X j=1 βjPt−j+ s X k=1 γknt−1n>t−1)

where γk ≥ 0 and nt = I[Yt < 0] ◦ Yt with "◦" denoting the Hadamard product4. To ensure

positive definitness of the correlation matrix it is required thatPp

i=1αi+

Pq

j=1βj+λ

Ps

k=1γk <

1 with λ being the maximum eigenvalue of Pc−1/2NcP −1/2

c where Nc = E[ntn>t]. Again we

restrict attention to unit orders. This model, given correlation targeting as described previously, has only one additional parameter γ to be estimated. As such, the ADCC model is able to allow for specific information effects on correlations and covariances while maintaining all the attractive properties of the dynamic model.

3.4 Estimating and Forecasting the dynamic models

3.4.1 Estimation

Continuously compounded returns at weekly frequency are calculated as rjt = log ( Pjt Pjt−1 ) − 1 T T X s=1 log ( Pjs Pjs−1 ), j = 1, ..., n

where Pjt stands for the price of the index j at time t. For the analysis and estimation of

the models we use the rmgarch package of A.Ghalanos (2014) implemented in the statistical software R. As mentioned earlier the main benefit of the covariance decomposition Σt= ∆tPt∆t

is that allows of a two-stage model estimation, thereby significantly reducing computational costs. In the first step different univariate GARCH processes are fitted and the diagonal matrix ∆tis estimated. After that, the devolatized process Ytis used for the estimation of conditional

correlation and covariances. Engle (2002) and Engle and Sheppard (2001) use conditional normality as a working hypothesis.

The Quasi log-likelihood (QML) function is derived as: l = 1 2 T X t=1 (n log(2π) + log |Σt| + Xt>Σ −1 t Xt) = 1 2 T X t=1 (n log (2π) + 2 log |∆t| + Xt>∆ −1 t ∆ −1 t Xt) − 1 2 T X t=1 (YtYt>+ log |Pt| + Yt>P −1 t Yt) = lV(θ1) + lP(θ1, θ2)

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where we used Yt= ∆−1t Xteffectively in a function of θ1 and a function of θ1and θ2, with θ1

containing the volatility parameters and θ2 containing the correlation parameters. Moreover the

first log-likelihood can be interpreted as the sum of the univariate GARCH log-likelihoods or lV(θ1) = 1 2 T X t=1 n X i=1

(log 2π + log(σi,t) +

X_i,t2 σi,t

)

White (1996) proves asymptotic normality and consistency of the estimators. Although the log-likelihood flexibility makes the estimation of multidimensional portfolios feasible, it only holds for distributions without shape parameters. In the case of a Student-t distribution where the degrees of freedom enter the density, that is

l = T X t=1 (log [Γ(ν + n 2 )]−log Γ[ ν 2]− n 2 log [π(ν − 2)]− 1 2log |∆tPt∆t|− ν + n 2 log [1 + Y_t>R−1_t Yt ν − 2 ]) where ν are the degrees of freedom and Γ is the Gamma function, the estimation has to be implemented in a single step by jointly maximizing all the parameters. This feature gives rise to the ‘curse of dimensionality’ but for our sample size, it is still feasible. The degrees of freedom of a Student t distribution are linked to the shape of its tails. Depending on the ultimate goal of the research, we can either fix the degrees of freedom to ensure the presence of fat tails or treat them as an additional unknown parameter and estimate it. As the goal of this thesis contains the comparison of alternatives against the normal distribution the shape parameter will be fixed at a value of five to allow for excess kurtosis5_{. We will consider three different multivariate density}

functions of the following distributions: 1. Gaussian

2. Student-t

3. Laplace (double exponential)

Student-t and Laplace densities are widely used in financial modeling as their shape can better capture the stylized facts of asset returns distribution. For these distributions, density specifica-tions are provided in Appendix A. Note that the Laplace distribution does not contain any shape parameters and can be maximized in a similar way as the Gaussian. Thus, the first step will assume a normal distribution and for the second step the multivariate Laplace will be used to derive conditional correlations and covariances.

3.4.2 Forecasting

In order to forecast portfolio losses it is necessary to predict the covariance matrix Σt. We will

focus on forecasting ∆tand Ptone period ahead. For ∆t= diag(σt,1, ..., σt,k) the prediction of

univariate zero-mean GARCH models is as follows:

Et(Xt+12 ) = Et(σt+12 ) = α0+ α1Xt2+ β1σt2

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The prediction of the random process X_t+12 is based on information available at time t and thus the value of Xtis known.

The conditional correlation ρtbetween assets i and j at t + 1 is given as follows:

ρt+1,t= Et(ρt+1 q σ2 i,t+1σ2j,t+1) q Et(σi,t+12 )Et(σj,t+12 ) where ρt+1= Qi,j,t+1 pQi,i,t+1Qj,j,t+1 with Qt+1 = (1 − p X i=1 αi− q X j=1 βj)Pc+ p X i=1 αiYtYt>+ q X j=1 βjPt

Thus by forecasting ∆t and Pt one period ahead we get a one period ahead prediction for

Σt = ∆tPt∆t. We use a rolling window of 800 observations. Since our total number of

observations is 1520, we thus have N = 1520 − 800 + 1 = 721 rolling window subsamples. Now, for each subsample in N we calculate the one period ahead variable of interest resulting in 799 rolling forecasts. The general approach of the rolling forecast includes periodically model re-estimation. Given the fact that the dynamic models are computationally intensive, we re-estimate our model once a year.

3.5 Value at Risk and Expected Shortfall

Value at Risk(VaR) pioneered by JP Morgan, provides risk managers an aggregated level of uncertainty with a single number. This quantile-based method gives the size of a potential loss under a prespecified time horizon and a fixed confidence level. Regulators, especially after the 2008 liquidity crisis, in order to increase the certainty across markets and avoid any more bail-outs, imposed restrictions on the internal models used by banks. One of them, expected to be implemented by 2018, is the use of Expected Shortfall (ES) which answers the question: ‘What is the expected loss if VaR is violated?’.

Consider a definite period of time t where the portfolio’s value at the end of this period is Vt+1. The portfolio’s (linearized) loss is defined at the end of period t as Lt+1= −(Vt+1− Vt).

The loss distribution is defined by FL(l) = P (L ≤ l).

Given a confidence level α ∈ (0, 1), the VaR of a portfolio is defined as the probability that the loss L will not exceed a threshold l with confidence 1 − α.

V aRα(L) = infl ∈ R : P ((L > l) ≤ 1 − α)

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Parametric approach, Monte Carlo Simulation, etc. In our project and given the conditional dynamic models we make use of the parametric approach. Consider the return of our portfolio as: rp = k X j=1 wjrj = w>r where w1 + .. + wk = 1.

Under a parametric approach,that is an distributional assumption with fitted parameters, VaR at a confidence level α is the upper case limit of

α = Z V aR

−∞

frp(x)dx

where frp(x) is the probability density function of rpand thus Value at Risk can be defined as

6_{V aR}α

t = (w

>

t Σtwt)qa

where qais the quantile of the assumed standard distribution. Finally, based on the parametric

V aR specification Expected Shortfall can be defined as ES_tα = E(L|L > V aRα_t)

It is the expected loss given that the loss exceeds the VaR threshold. It looks deeper in the tails of the distribution by averaging all VaR violations.

We will consider three different vectors of weights for our portfolio:

w1 =        w1 = 0.02941 .. . w34 = 0.02941        w2 =                        w1 = 0.03571 .. . w21 = 0.03571 w22 = 0.01923 .. . w34 = 0.01923                        w3 =                        w1 = 0.01190 .. . w21 = 0.01190 w22 = 0.05769 .. . w34 = 0.05769                       

6_{The mean value is not considered in the VaR specification as we assume that it will be much less significant}

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The first vector, w1, refers to an equally weighted portfolio. w2 weights equity indices by 75% and bond indices by 25% whereas w3 weights equity indices by 25% and bonds indices by 75%. Considering the two dynamic models (DCC&ADCC) and the three distributional assumptions we will analyze eighteen portfolios in total.

3.5.1 Back-Testing

“Disclosure of quantitative measures of market risk, such as Value-at-Risk, is enlightening only when accompanied by a thorough discussion of how the risk measures were calculated and how they related to actual performance"

—Alan Greenspan (1996) Back-Testing is used to evaluate the performance of a model or procedure such as Value at Risk or Expected Shortfall against the realized observations. We consider the unconditional coverage test of Kupiec (1995), the conditional coverage test of Christoffersen (1998), the duration-based approach of Christoffersen and Pelletier (2004) and the Expected Shortfall test of McNeil and Frey (2000).

Let us define a dummy variable which counts the number of VaR violations as

It+1=    1, if Lt+1 > V aRαt 0, otherwise

The unconditional coverage test (uc) examines whether the true violations are in accordance with the expected breaches, defined as N × (1 − α) where N is the total number of observations. Under the null of a correctly specified model we should obtain m =P It+1 ∼ B(n, 1 − α). A

likelihood ration test is used to check whether the model is adequate and is given by : LRuc = − log (α)m(1 − α)N −m (m_N)m_{(1 −} m N) N −m _H0 ∼ χ2 1.

The unconditional coverage test (uc), although it is able to check for correct exceedances it doesn’t take into account the independence between violations. That is, past violations do not hold information about future violations. The conditional coverage test (cc) of Christoffersen (1998) is able to test for both, correct exceedances and independent. Christoffersen (1998) considers a binary first-order Markov chain, It+1, with

Π = "

π00 π01

π10 π11

#

being the transition probability matrix, with πij = P r(It = j|It−1 = i) where π00 = 1 −

π01, π10= 1 − π01. π01denotes a violation after a non-violation whereas π11denotes a violation

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As a result given a sample of length N the likelihood statistic will be L(Π) = π₀₀NπN₀₁π₁₀NπN₁₁.

Now by maximizing the likelihood function with respect to π01and π10we can come up with an

estimated matrix ˆΠ.

Combining the tests we obtain a conditional coverage likelihood test by LRcc = LRuc+ LRind or LRcc = − log (α)m(1 − α)N −m (m_N)m_{(1 −}m N)N −m + (1 − π) π00+π10_{+ π}π01+π11 (1 − π0)π00π0π01(1 − π1)π10ππ111 _H0 ∼ χ2₂ where π0 = _π00+π01π01 , π1 = _π10+π11π11 and π = _{π00+π01+π10+π11}π01+π11 . This test can jointly account for

correct exceedances and independent.

Christoffersen and Pelletier (2004) proposed a modification of the independence test, the duration-based test. In this test under the null, the time duration between violations should have the no memory property. If we define Di = ti− ti−1 the time (in weeks) between two

exceedances over the VaR threshold and take into account that the only distribution with the memory free property is the exponential then under the null we should obtain

f (D|α) = α exp (−alphaD)

Although a test statistic can not be conducted since the exponential distribution does not allow for dependence as an alternative. Thus, the Weibull distribution is used for which the exponential distribution is derived for b = 1, the scale parameter. The likelihood test statistic is distributed as χ2₁.

A test for Expected Shortfall is given by McNeil and Frey (2000). This test is based on VaR violations and under the null, violations should have a mean of zero. If we consider that ES_tα = µt+1+ σt+1EStα(Z), with EStα(Z) being the expected shortfall of the innovation

distribution, and define:

Kt+1 =    Lt+1−ESα t ESα t−µt+1, if Lt+1> V aR α t 0, otherwise

where µt+1= E(Lt+1|Ft). We can calculate that

Kt+1 =

Zt+1− ESα(Z)

ESα_(Z) I{Lt+1>V aRαt}

so that Ktforms a process of i.i.d variables with mean zero. Then in the event that there is a

VaR violation {Lt+1 > V aRαt}, Kt+1compares the actual size of the violations Lt+1with its

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Chapter

4 Data Analysis

For the applicability of all theory and model specification discussed above a proper portfolio needs to be employed. We use an updated version of the data considered in Capiello et al. (2006), namely FTSE All-World Indices for 21 countries and DataStream-constructed five-year average maturity government bond indices for 13. The states for share indices are clustered as Europe, Australasia, and the Americas. European countries considered are Austria, Belgium, Denmark, France, Germany, Ireland, Italy, the Netherlands, Norway, Spain, Sweden, Switzerland, and the United Kingdom. Australasia consists of Australia, Hong Kong, Japan, New Zealand, and Singapore whereas the Americas include Canada, Mexico, and the United States. Data on government bonds are also identically clustered including the countries mentioned above except Australia, Italy, Norway, Spain, Hong Kong, New Zealand, Singapore, and Mexico.

The FTSE All-World Indices are well valued, and bond indices consist of the most liquid government bonds. As mentioned in Capiello et al. (2006) there is never a time when all 21 markets are open. For that reason, they calculate continuously compounded weekly returns using Thursday-to-Thursday closing prices. Market trading time intervals are not homogeneous and nonsynchronicity in returns may cause misleading conclusions. Furthermore, the sample covers the period from 08/01/1987 to 11/02/2016 a total of 1520 observations including multiple periods of financial distress. The most important are:

• Black Monday (1987)

Started in Hong Kong on October 19, 1987, producing tremendous declines in financial stock markets. At that time, Dow Jones Industrial Average (DJIA) faced a decay of 22.61%.

• Gulf War (1990-1991)

In August 1990 Iraq invaded and conquered Kuwait, which marked the start of the Gulf War. Western stock markets fell by approximately 25% in a short period.

• Russian Financial Crisis (1998)

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80 110 140 -18.00% -13.00% -8.00% -3.00% 2.00% 7.00% 12.00% 17.00%

Weekly Stock Returns Weekly Bond Returns 5Y Gov.Bond Prices

Credit Crisis Black Monday Gulf War Recession Russian Financial Crisis

Figure 4.1: Graph showing the weekly cumulative returns for stock and bond indices, and the average 5Y datastream-constructed government bond prices.

As commodity prices had severely declined (partly due to less demand from Asia), Russia, being a significant exporter of commodities such as oil, metals and natural gas, ran into huge problems.

• Recession (2000-2003)

The three years of recession that followed (2000-2003) were marked by declining stock prices worldwide. This feature was partly driven by the subsequent war on terrorism that was initiated after September 11.

• Liquidity Crisis (2007-2008)

The problems started when the Fed decided to raise interest rates significantly again to slow down the economy and mitigate inflation. A series of collapses and nationalizations followed in 2008.

We herewith conclude our overview of main historical events that have significantly impacted on global financial markets during 1987-2016. This historical understanding is mentioned to explain the multiple periods of volatility clustering that are taking place in our sample.

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close to zero. However, univariate characteristics can not adequately describe the density of a multivariate distribution which is of paramount importance when dealing with large portfolios.

Standard Excess

M ean Deviation Skewness Kurtosis M in M ax

Australia stocks 0.0009 0.0241 -2.1378 26.7641 -0.3385 0.1024 Austria stocks 0.0005 0.0326 -0.6940 6.0336 -0.2335 0.1971 Belgium stocks 0.0009 0.0292 -1.1245 9.3617 -0.2400 0.1655 Canada stocks 0.0013 0.0302 -0.9645 9.4503 -0.2702 0.2147 Denmark.stocks 0.0026 0.0265 -0.8411 6.2352 -0.2163 0.1576 France stocks 0.0008 0.0293 -0.4918 2.4871 -0.1463 0.1401 Germany stocks 0.0008 0.0314 -0.6316 3.2116 -0.1649 0.1465

Hong Kong stocks 0.0013 0.0410 -1.3914 13.6955 -0.3972 0.2373

Ireland stocks 0.0009 0.0351 -0.7017 5.1136 -0.2289 0.1981

Italy stocks 0.0000 0.0340 -0.3617 1.6090 -0.1471 0.1250

Japan stocks -0.0001 0.0316 -0.5048 3.6835 -0.2236 0.1501

Mexico stocks 0.0042 0.0379 -0.0121 4.8189 -0.2390 0.2567

Netherlands stocks 0.0008 0.0309 -0.7877 5.4692 -0.1666 0.1640 New Zealand stocks -0.0003 0.0259 -0.4055 4.0077 -0.1834 0.1172

Norway stocks 0.0017 0.0313 -0.9709 5.4449 -0.1987 0.1696 Singapore stocks 0.0006 0.0324 -1.3261 14.0724 -0.3213 0.1878 Spain stocks 0.0016 0.0386 -0.4878 4.5739 -0.2167 0.9495 Sweden stocks 0.0016 0.0304 -0.3786 2.6712 -0.1560 0.1556 Switzerland stocks 0.0004 0.0280 -1.0290 5.6896 -0.1774 0.1252 U.K. stocks 0.0009 0.0241 -0.9891 7.7314 -0.2184 0.1389 U.S. stocks 0.0013 0.0257 -1.5130 17.9170 -0.3152 0.1319 Austria bonds 0.0002 0.0045 -0.0883 4.7195 -0.0225 0.0351 Belgium bonds 0.0002 0.0051 0.4063 11.4767 -0.0318 0.0547 Canada bonds 0.0003 0.0066 -0.1660 4.5790 -0.0423 0.0511 Denmark bonds 0.0002 0.0054 0.0509 5.8723 -0.0358 0.0398 France bonds 0.0003 0.0054 0.3030 9.5429 -0.0427 0.0525 Germany bonds 0.0002 0.0046 -0.3015 2.089 -0.0266 0.0238 Ireland bonds 0.0006 0.0092 1.6604 7.4760 -0.1080 0.1263 Japan bonds 0.0000 0.0039 -0.0627 8.0463 -0.0239 0.0326 Netherlands bonds 0.0002 0.0047 0.1133 7.2050 -0.0311 0.0443 Sweden bond 0.0001 0.0061 -0.3052 8.0480 -0.0421 0.0532 Switzerland bonds 0.0000 0.0045 -0.3877 5.4772 -0.0304 0.0257 U.K.bonds 0.0002 0.0059 0.3543 4.7751 -0.0226 0.0447 U.S bonds 0.0002 0.0063 0.1761 3.5484 -0.0261 0.0514

Table 4.1: Descriptive statistics of equity and bond indices. There is evidence of asymmetry for both equity and bond share indices.

On the basis of asymptotic theory and by statistical inference we know that D2_i = (Xi− ¯X)>S−1(Xi− ¯X) f or i = 1, ...n

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K. V. (1970)) is implemented. This involves defining multivariate skewness and kurtosis by bd= 1 n2 n X i=1 n X j=1 D3_ij kd= 1 n n X i=1 D_i4,

respectively. Digiven above, is known as the Mahalanobis Distance between Xi and ¯X and Dij

as the Mahalanobis Angle between Xi− ¯X and Xj− ¯X both referring to the distance and angle

between a point and a distribution in the multivariate case. Under Mardias Test the following should hold asymptotically (n → ∞)

1 6nbd∼ x 2 d(d+1)(d+2)/6 kd− d(d + 2) p8d(d + 2)/n ∼ N (0, 1).

Table 4.2 clearly indicates that multivariate normality is rejected with a PV alueof 0 in all cases.

Figure 4.2 provides profound evidence of asymmetry where under a H0 these plots should be

roughly linear. 1 6nbd PV alue kd−d(d+2) √ 8d(d+2)/n PV alue

Stocks & Bonds 70880.18 0.00 474.70 0.00

Stocks 22860.08 0.00 334.12 0.00

Bonds 10663.78 0.00 403.18 0.00

Europe Assets 29809.08 0.00 419.22 0.00

Australasia Assets 2643.40 0.00 169.81 0.00

Americas Assets 603.47 0.00 87.39 0.00

Table 4.2: Mardias Test of joint normality estimates based on the formulae for multivariate skewness and kurtosis given above.

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20 30 40 50 60 70 0 200 400 600 Chi−sq(34) quantiles Ordered D data 10 20 30 40 50 0 100 200 300 400 Chi−sq(21) quantiles Ordered D data 5 10 20 30 0 100 200 300 Chi−sq(13) quantiles Ordered D data 10 20 30 40 50 0 100 300 500 Chi−sq(23) quantiles Ordered D data 0 5 10 15 20 25 0 50 100 200 Chi−sq(6) quantiles Ordered D data 0 5 10 15 20 0 20 40 60 80 Chi−sq(4) quantiles Ordered D data

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Average correlations Equity indices

M ean M inimum M aximum

0.5134 0.1977 0.8556

Australasia Europe America

Australasia 0.4745 0.4388 0.3925

Europe 0.6043 0.4803

America 0.4854

Bond indices

0.4282 0.1016 0.9004

Australasia Europe America

Australasia N/A 0.2246 0.2480

Europe 0.4923 0.3867

America 0.7190

Bond and equity indices

−0.0766 −0.2365 0.1043

Bonds

Equities Australasia Europe America

Australasia −0.0888 −0.0664 −0.1080

Europe −0.1069 −0.0681 −0.1496

America −0.0655 −0.0540 −0.0211

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Chapter

5 Results

In the previous sections, we pointed out the potential benefits of asymmetric models when analyzing asset returns in volatile periods. In this chapter, we present estimates using the dynamic models of Engle (2002) and Engle and Sheppard (2001) and Cappiello et al. (2006) illustrated in Chapter 3. Section 5.1 provides in sample estimates and compares models under different distributional assumptions. Conditional volatility and correlation across regions is plotted and analyzed. The robustness of the models is examined under different scenarios. Section 5.2 shows the forecasted covariance estimates of the dynamic models and their use in backtesting Value at Risk and Expected Shortfall.

5.1 In Sample Estimates

The in sample estimates are based on the full sample as described in Chapter 4. The dynamic models are fitted under three different distributional assumptions, namely the Gaussian distribu-tion, the Laplace distribution and the Student-t distribution resulting in a total of six models. That is, the Dynamic Conditional Correlation model with Gaussian innovations (DCC(MVN)),the Dynamic Conditional Correlation model with Laplace innovations (DCC(MVL)), the Dynamic Conditional Correlation model with Student-t innovations (DCC(MVT)), the Asymmetric Dy-namic Conditional Correlation model with Gaussian innovations (ADCC(MVN)), the Asymmet-ric Dynamic Conditional Correlation model with Laplace innovations (ADCC(MVL)) and the Asymmetric Dynamic Conditional Correlation model with Student-t innovations (ADCC(MVT)).

5.1.1 Model Comparison

In Chapter 3, section 3.3.1 we described the two-stage log-likelihood decomposition of the dynamic models. As such, the first stage estimation requires the fit of different univariate models to estimate ∆t and Yt. These GARCH models are chosen for each asset class based

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BIC value. Table 5.2 consists of the best univariate GARCH fitted models and their estimated parameters. It shows that 27 out of 34 processes contain asymmetric effects. All of the equity Univariate GARCH models for each asset class

Asset M odelSelected α0 α1 β γ or η1 δ or η2

Australia stocks APARCH 0.0053 0.1548 0.8020 0.5801 0.7282

Austria stocks APARCH 0.0047 0.1227 0.8623 0.4057 0.6505

Belgium stocks GJR-GARCH 0.0000 0.0164 0.0787 0.2581

Canada stocks EGARCH -0.3236 -0.1028 0.9544 0.2118

Denmark.stocks GJR-GARCH 0.0000 0.0491 0.7693 0.1670

France stocks EGARCH -0.6190 -0.1794 0.9143 0.1663

Germany stocks EGARCH -0.5603 -0.1186 0.9206 0.2544

Hong Kong stocks EGARCH -0.5603 -0.1186 0.9206 0.2544

Ireland stocks EGARCH -0.5188 -0.1084 0.9204 0.3540

Italy stocks APARCH 0.0245 0.0854 0.8726 0.6920 0.2457

Japan stocks EGARCH -0.3405 -0.0788 0.9500 0.2012

Mexico stocks GJR-GARCH 0.0000 0.1274 0.8234 0.0962

Netherlands stocks EGARCH -0.2657 -0.0604 0.9574 0.3826 New Zealand stocks EGARCH -0.5566 -0.1507 0.9230 0.2501

Norway stocks EGARCH -0.1372 -0.0262 0.9811 0.2150

Singapore stocks EGARCH -0.6034 -0.1014 0.9141 0.2948

Spain stocks EGARCH -0.2563 -0.0729 0.9617 0.3034

Sweden stocks EGARCH -0.2734 -0.0992 0.9617 0.1723

Switzerland stocks EGARCH -0.5318 -0.1324 0.9254 0.2194

U.K. stocks EGARCH -0.9264 -0.2068 0.8741 0.2977

U.S. stocks EGARCH -0.5262 -0.1786 0.9308 0.2039

Austria bonds GJR-GARCH 0.0000 0.0849 0.9019 0.0026

Belgium bonds GJR-GARCH 0.0000 0.1839 0.7979 0.0088

Canada bonds GJR-GARCH 0.0000 0.0540 0.9314 0.0151

Denmark bonds CSGARCH 0.0000 0.1052 0.8710 0.9998 0.0298

France bonds EGARCH -0.2981 0.0042 0.9707 0.2460

Germany bonds EGARCH -0.3405 -0.0484 0.9673 0.1867

Ireland bonds GJR-GARCH 0.0000 0.1043 0.8650 0.0415

Japan bonds CSGARCH 0.0000 0.1448 0.7918 0.9996 0.0580

Netherlands bonds GARCH 0.0000 0.1567 0.79344

Sweden bond CSGARCH 0.0000 0.0881 0.9014 0.9997 0.0319

Switzerland bonds GARCH 0.0000 0.0564 0.9268

U.K.bonds CSGARCH 0.0000 0.0956 0.8458 0.9998 0.0189

U.S bonds GARCH 0.0000 0.0550 0.9243

Table 5.1: The table provides the univariate GARCH models used for each asset class. They are used to devolatize each return series for the first stage of the DCC estimation.

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accordance with the descriptives of the sample where skewness is documented (Chapter 4). The Akaike information criterion (Akaike (1974)) is an alternative to BIC. Similar to BIC, AIC penalizes for complexity via the number of parameters, but less severely. Table 5.2 reports values of the criteria for the most favorable model. Overall, the AIC and BIC values slightly favor conditional asymmetry for all distributions. Moreover, nonnormal distributions for the innovations are preferred given the information criteria. The fatter tailed Student-t distribution captures better the patterns of the sample, thus making the Asymmetric DCC model with multivariate Student-t innovations the most qualified.

Model criteria and log-Likelihood values

M odel Log − likelihood AIC BIC

DCC(MVN) 166688.55 −218.35 −215.77 ADCC(MVN) 166751.54 −218.44 −215.85 DCC(MVT) 168120.55 −220.24 −217.65 ADCC(MVT) 168143.09 −220.26 −217.68 DCC(MVL) 167695.25 −219.68 −217.09 ADCC(MVL) 167730.07 −219.72 −217.14

Table 5.2: The table provides Log-likelihood, AIC and BIC values for the DCC and ADCC models under different distributional assumptions.

Model estimated parameters

M odel α β γ DCC(MVN) 0.0072 0.9892 ADCC(MVN) 0.0059 0.9888 0.0034 DCC(MVT) 0.0066 0.9881 ADCC(MVT) 0.0059 0.9877 0.0036 DCC(MVL) 0.0066 0.9883 ADCC(MVL) 0.0057 0.9879 0.0023

Table 5.3: The table provides the estimated parameters for DCC and ADCC specifications. All parameters are significant at 1%.

Table 5.3 reports the estimated parameters for the dynamic models. All values are different from zero and very similar for all distributions. The coefficients for joint bad news are close to the symmetric values indicating high correlation when both share indices are dropping. The condition for covariance stationarity,Pp

i=1αi+

Pq

j=1β1+ λ

Ps

k=1γk < 1 holds for all cases.

5.1.2 Volatility and Correlation dynamics

Figure 5.1 plots equally weighted annualized conditional volatility for equity share indices across regions. Cross-market dependence is evidenced during crisis periods such as Black Monday 1987, Russian financial crisis in 1998 and the liquidity crisis in 2008.

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1990 1995 2000 2005 2010 2015 0.1 0.3 0.5 Europe Time V olatility 1990 1995 2000 2005 2010 2015 0.1 0.3 0.5 Australasia Time V olatility 1990 1995 2000 2005 2010 2015 0.2 0.6 Americas Time V olatility

Figure 5.1: Annualized Conditional volatility plots of stock indices across markets. Dependence is profound during periods of financial turmoil.

correlated. There are strong linkages under turbulent periods with Europe and Americas being the most volatile regions. Moreover, linkages between equity and bond volatilities are distinguished, explained by the fact that bonds are used as substitutes to stocks as uncertainty increases, the so called ‘flight to quality phenomenon’.

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1990 1995 2000 2005 2010 2015 0.1 0.3 0.5 Europe Time V olatility 1990 1995 2000 2005 2010 2015 0.2 0.8 1.4 Australasia Time V olatility 1990 1995 2000 2005 2010 2015 0.2 0.6 Americas Time V olatility

Figure 5.2: Annualized Conditional volatility plots of bond indices across markets. Similar to equities, bonds show linkages under stress periods.

2000-2003 recession and the global crisis in 2008. United States stock markets faced tremendous declines producing shocks transmissions to other countries and markets.

In Figure 5.5, correlations between European countries, Germany with the US and the US with Singapore for equity share indices are plotted. As anticipated, developed countries such as Germany, US, and the Netherlands are found to be highly correlated and thus deeply integrated. This is mainly due to similar business cycles. Economic growth gave rise to companies, expanding across the world making markets highly dependent. Changes up to 20% in correlation values are shown during the sample period. The average correlation between Germany and the US is 0.65, sharply increasing after the introduction of the fixed currency regime. This evidence verifies the importance of correlation and covariance understanding for more prudent investment decisions. Interestingly, spikes are also reported between US and Singapore, with an average of 0.51, indicating the magnitude of such recessions by affecting markets all around the globe.

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1990 1995 2000 2005 2010 2015 0.50 0.55 0.60 0.65 0.70 Europe Time Correlation ADCC(MVN) ADCC(MVL) ADDC(MVT) 1990 1995 2000 2005 2010 2015 0.50 0.55 0.60 0.65 Australasia Time Correlation ADCC(MVN) ADCC(MVL) ADDC(MVT) 1990 1995 2000 2005 2010 2015 0.60 0.65 0.70 0.75 0.80 0.85 Americas Time Correlation ADCC(MVN) ADCC(MVL) ADDC(MVT)

Figure 5.3: Equally weighted conditional correlation plot for equity share indices across regions.

1990 1995 2000 2005 2010 2015 0.40 0.45 0.50 0.55 0.60 0.65 Europe Time Correlation ADCC(MVN) ADCC(MVL) ADDC(MVT) 1990 1995 2000 2005 2010 2015 0.80 0.85 0.90 Americas Time Correlation ADCC(MVN) ADCC(MVL) ADDC(MVT)

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1990 1995 2000 2005 2010 2015 0.65 0.75 0.85 Time Correlation France−Germany ADCC(MVN) ADCC(MVL) ADCC(MVT) 1990 1995 2000 2005 2010 2015 0.55 0.65 0.75 0.85 Time Correlation UK−Netherlands ADCC(MVN) ADCC(MVL) ADCC(MVT) 1990 1995 2000 2005 2010 2015 0.5 0.6 0.7 0.8 Time Correlation Germany−U.S ADCC(MVN) ADCC(MVL) ADCC(MVT) 1990 1995 2000 2005 2010 2015 0.35 0.50 0.65 Time Correlation US−Singapore ADCC(MVN) ADCC(MVL) ADCC(MVT)

Figure 5.5: Conditional correlation plots for equity share indices between European countries, between Germany and the US and between US and Singapore.

regime. The European and US bond markets seem to remain highly dependent reaching their maximum values during the 2007-2008 liquidity crisis. Similar to equities, bond correlations as shown in figure 5.6, tend to increase during 2000-2010. Beside the Credit Crunch in 2008, Australasia bond markets tend to be less integrated with the rest of the world, showing an average of 0.3.

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1990 1995 2000 2005 2010 2015 0.55 0.70 0.85 Time Correlation France−Germany ADCC(MVN) ADCC(MVL) ADCC(MVT) 1990 1995 2000 2005 2010 2015 0.4 0.5 0.6 0.7 Time Correlation UK−Netherlands ADCC(MVN) ADCC(MVL) ADCC(MVT) 1990 1995 2000 2005 2010 2015 0.3 0.4 0.5 0.6 0.7 Time Correlation Germany−U.S ADCC(MVN) ADCC(MVL) ADCC(MVT) 1990 1995 2000 2005 2010 2015 0.20 0.30 0.40 0.50 Time Correlation US−Japan ADCC(MVN) ADCC(MVL) ADCC(MVT)

Figure 5.6: Conditional correlation plots for bond share indices between European countries, between the Germany and the US and between US and Japan.

in Basel 3 guidelines, banks are much less willing to make loans leading to a lack of liquidity. Given the different distributions for the innovations, the correlation evolutions over time are very similar. As reported in table 5.3, this is driven by the similar values of the estimated parameters. There is not a straightforward explanation for this result, but it may be further tested. All figures explained so far are based on a full sample of 1520 observation, and thus may be biased. In order to assess the model performance, managers run ad-hoc analysis under different assumptions to investigate whether the conclusions still apply. In the following subsection, we test the models under different sample sizes and fixed parameters.

5.1.3 Model Validation

In model building there is a danger or tendency to ‘overfit’, to accommodate the idiosyncrasies of the data in attempting to get the best model. This might lead to a model that represents the data on which it is fitted, far better than new data. Here we refit our model to smaller subsamples. Subsample1contains data for the period of 1987 − 1996 a total of 500 observations,

Subsample2 consists of 500 observations for the period of 1997 − 2006 and Subsample3 refers

to the last 520 data points. In table 5.4 the estimated parameters for the models are reported. All the parameters remain highly significant.

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Subsample Model estimated parameters Subsample1 α β γ DCC(MVN) 0.0055 0.8009 ADCC(MVN) 0.0026 0.7601 0.0116 DCC(MVT) 0.0067 0.7637 ADCC(MVT) 0.0014 0.7019 0.0224 DCC(MVL) 0.0070 0.7311 ADCC(MVL) 0.0016 0.6508 0.0192 Subsample2 α β γ DCC(MVN) 0.0093 0.8419 ADCC(MVN) 0.0036 0.8958 0.0119 DCC(MVT) 0.0078 0.8232 ADCC(MVT) 0.0035 0.8718 0.0094 DCC(MVL) 0.0084 0.7890 ADCC(MVL) 0.0037 0.8531 0.0105 Subsample3 α β γ DCC(MVN) 0.0102 0.8726 ADCC(MVN) 0.0079 0.8395 0.0143 DCC(MVT) 0.0093 0.9004 ADCC(MVT) 0.0083 0.8919 0.0049 DCC(MVL) 0.0093 0.8942 ADCC(MVL) 0.0078 0.8796 0.0082

Table 5.4: The table provides the estimated parameters for DCC and ADCC specifications under three subsamples. All parameters are significant at 1%.

estimated parameters given the distributional assumptions are very close to each other. The information criteria favor on a higher level the asymmetric version of the model. The dynamic correlation degrees are mainly driven by two factors. That is the standardized residuals and the autocorrelation factors. To investigate which factor weights more, we refit the models by keeping each time one factor fixed.

For the first two subsamples, keeping the news or the autocorrelation factor fixed, does not make a material difference. Models under different distributions provide similar conclusions as before. Clearly, subsample3 is the most volatile. The Gaussian and the Laplace perform in a

similar way when keeping α or β fixed. The Student t distribution gives much larger values for both factors compared to the alternatives. This result comes from the fact, that the Student t with 5 degrees of freedom, has thicker quantiles than the alternatives. Thus, during turbulent periods where volatility increases it is more sensitive compared to the others.

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5.2 Risk Management Implications

To assess the out of sample performance of the dynamic models discussed in section 3, we provide forecasted covariances for a rolling window of 800 observations. We re-estimate our models once a year to reduce computational effort. Prediction of dynamics is of paramount importance for financial institutions to manage their risks. Risk managers try to predict possible losses for a better understanding of uncertain future outcomes. Quantile based methods such as Value at Risk discussed in Chapter 3 are often used, and under a parametric approach the estimated one period ahead forecast of comovements is crucial. Critical values presented in this thesis are based on 95% confidence level. This threshold is used in judging the adequacy of a proposed risk measure. Figure 5.7 shows the 800 out of sample covariance estimates for both DCC and ADCC models. The covariance matrix together with the quantile of the assumed innovations are the main parameters under a parametric-based VaR approach. Given the conditional densities, the covariance evolution through time is similar to all models. Although, the ADCC models provide slightly higher estimates compared to the symmetric ones under periods of distress. Volatilities are identical for both specifications, hence the result is only driven by correlations. As argued before, the significant positive asymmetric term of the ADCC models stipulates that bad news have adverse effects on correlations, making them higher. The memory of the dynamic correlation decays with exponential ratePp

i=1αi+

Pq

j=1βj in the symmetric

case, else it decays withPp

i=1αi +

Pq

j=1βj +

Ps

k=1γk. Moreover, the estimated values of

Pp

i=1αi+

Pq

j=1βj under the DCC model are larger for the Gaussian and Laplace distributions,

thus the DCC(MVT) converges fastest to the unconditional correlation. Interestingly, the estimated values ofPp

i=1αi+

Pq

j=1β1+

Ps

k=1γkunder the ADCC specification are smaller

for the multivariate Laplace innovations.

Backtesting VaR and ES results is important for risk managers to evaluate their internal model performance. To do this three different weighted portfolios are tested. Tables 5.4, 5.5 and 5.6 report violations and critical values for the unconditional, conditional and durations tests. For portfolio w1 the following conclusions are drawn. In all cases, the asymmetric DCC model is superior to the symmetric DCC. The models with multivariate normal disturbances are strongly rejected under both unconditional and conditional coverage tests. The DCC model with Student t errors is also rejected, but less clearly. Asymmetry improves the performance of the model and the ADCC with Student-t disturbances pass the test for correct exceedances and independent. Both models under the Laplace distribution pass the coverage tests. For the riskier portfolio, w2, only the ADCC(MVL) is able to pass the conditional tests. Normality is rejected in all cases. This is due to the fact that alternatives provide higher quantiles and hence larger loss thresholds. Overall all six models tend to underestimate the risk of the portfolios. Interestingly, non of the models fails the duration based test. This shows that violations are evenly spread over the threshold avoiding clustering.

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2005 2010 2015 0.0000 0.0005 0.0010 0.0015 EW Portfolio Time Co v ar iance DCC(MVN) DCC(MVL) DDC(MVT) 2005 2010 2015 0.0000 0.0005 0.0010 0.0015 EW Portfolio Time Co v ar iance ADCC(MVN) ADCC(MVL) ADDC(MVT)

Figure 5.7: Forecasted covariance plots for the DCC and ADCC models under three different distributions, multivariate normal,multivariate Laplace and multivariate Student-t.

M odel V iolations LRuc Puc LRcc Pcc b PW eibull

DCC(MVN) 20 12.834 0.000 13.246 0.001 1.1238 0.513 ADCC(MVN) 19 11.023 0.001 11.549 0.003 1.0784 0.697 DCC(MVT) 15 4.92 0.027 6.09 0.048 1.0429 0.851 ADCC(MVT) 14 3.72 0.054 5.101 0.078 1.0429 0.851 DCC(MVL) 14 3.715 0.054 5.101 0.078 1.5219 0.092 ADCC(MVL) 13 2.655 0.103 4.285 0.117 1.4407 0.170 LRcritical|b 3.841 5.991 1

Table 5.5: VaR unconditional, conditional and Duration test results under portfolio w1.

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that violations do not come in clusters.

As discussed in Chapter 3, backtesting ES is based on VaR numbers. Under the null, the mean of excess violations of VaR should have the zero mean property. For the normal distribution, all models given the three weighted portfolios, fail to pass the expected shortfall test. However, models with Student t and Laplace innovations reject the alternative hypothesis of the mean of excess violations of VaR being greater that zero.

The results of the evaluation procedure show the importance of statistical testing. By applying more weights to riskier assets, such as portfolio w1 and w2, most models fail to capture extreme events and thus fail the conditional tests. In contrast, portfolio w3 indicates the benefits of diversification by applying a more appropriate weight vector to the weekly returns.

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Chapter

6 Conclusion and Further Development

In this thesis, we examined the in and out-of-sample performance of the Asymmetric Dynamic Conditional Correlation (ADCC) model against the symmetric DCC based on different distri-bution functions. We considered an extended version of the dataset used in Cappiello et al. (2006)that consists of highly liquid bonds and well-valued stocks. We compare the models by means of AIC and BIC criteria, we examine their robustness on subsamples of the sample and we assess their prediction performance by backtesting risk measures. Overall the ADCC model outperforms the DCC. The use of different quasi log-likelihoods lead to similar results given our data sample. We find significant evidence of asymmetry for bond share indices as opposed to Cappiello et al. (2006). This finding shows that investors become more risk averse and seek safer investments in exchange for lower profits, moving wealth from stocks to bonds. Cross-market dependence is documented during stress periods and since the fixed exchange rate, markets show strong linkages. This homogeneity across markets emerges governments and regulators to impose tighter conditions to reduce uncertainty. However, such regulatory frameworks need to be well structured to avoid liquidity black holes. The tails of the Student t and Laplace distributions mimic on a higher level the quantiles of the realized data as such providing superior risk management insights. Although, all models pass the duration-based test, the violations under a 99% confidence level are high, indicating that models should be improved. In addition, less risk-weighted portfolios, such as portfolio w3, allow models to perform better and maintain possible losses below thresholds.

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Appendix

A

Multivariate density functions and

Information Criteria

A.1 Distributions

1. Gaussian fx(x1, .., xk) = 1 p(2π)k_|Σ|exp(− 1 2(X − µ) > Σ−1(X − µ)) 2. Student t fx(x1, .., xk) = Γ[(ν + k)/2] Γ[ν₂]νk/2_πk/2_Σ−1/2_{[1 +} 1 ν(X − µ) >_Σ−1_{(X − µ)]}(ν+k)/2

where Γ is the gamma function. 3. Laplace fx(x1, ..., xk) = 1 (2π)(k/2) 2 λ K(k/2)−1 q 2 λq(X) q λ 2q(X) (k/2)−1 where q(X) = (X − µ)>Γ−1(X − µ)

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−4 −2 0 2 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Comparison of density functions

x Density Distributions Gaussian Student−t(df=5) Laplace

Figure A.1: Density plots for the Gaussian, Student-t and Laplace distributions.

A.2 AIC & BIC

1. Akaike Information Criterion: If M1, ..., Mm are models and kj are the parameters of

model j captured in θj = (θj1, ..., θjkj)0, the AIC is given by

AIC(Mj) = −2 ln Lj( ˆθj; X) + 2kj

where Lj( ˆθj; X) is the likelihood function. The criterion favor the model with the more

negative AIC value. It penalizes the model fit for model complexity. It typically aims for the best approximation amongst models neither of which is necessary correct.

2. Bayesian Information Criterion:Similar to AIC, BIC imposes a higher penalty to the number of parameters (complexity). It is computed as:

BIC(Mj) = −2 ln Lj( ˆθj; X) + kjln n

The parameter n refers to the sample size. It aims for the simplest ‘true’ model among candidates of which at least one is true.

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Appendix

B

Value at Risk Plots

2005 2010 2015

−0.10

0.00

0.05

Weekly Returns and Quantile Based Measures (Series: DCC(MVN), alpha=0.01)

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.10 0.00 0.05

Weekly Returns and Quantile Based Measures (Series: DCC(MVL), alpha=0.01)

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.15 −0.05 0.05

Weekly Returns and Quantile Based Measures (Series: DCC(MVT), alpha=0.01)

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.10 0.00 0.05

Weekly Returns and Quantile Based Measures (Series: ADCC(MVN), alpha=0.01)

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.10 0.00

Weekly Returns and Quantile Based Measures (Series: ADCC(MVL), alpha=0.01)

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.15 −0.05 0.05

Weekly Returns and Quantile Based Measures (Series: ADCC(MVT), alpha=0.01)

Time

W

eekly Log Retur

n

return < ES VaR ES

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2005 2010 2015

−0.10

0.00

0.10

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.15 −0.05 0.05

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.15 −0.05 0.05

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.10 0.00 0.10

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.15 −0.05 0.05

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.15 −0.05 0.05

Time

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eekly Log Retur

n

return < ES VaR ES

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2005 2010 2015

−0.04

0.00

0.02

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.04 0.00 0.02

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.04 0.00

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.04 0.00 0.02

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.04 0.00

Time

W

eekly Log Retur

n return < ES VaR ES 2005 2010 2015 −0.06 −0.02 0.02

Time

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eekly Log Retur

n

return < ES VaR ES

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Volatility Models with Asymmetric Dynamics and their Application to Risk Estimation