Tipos de herramientas de comunicación en el área interna

3.7

Conclusion

In this chapter, we presented a theoretical review of M-GARCH(1,1) DCC model incor- porating copula functions and extreme value theory (EVT) analysis for forecasting VaR estimates. Financial asset return distributions have been proven to be leptokurtic, ex-

hibits volatility clustering, leverage effects, and auto-correlations of squared returns. To

capture these features, we have used M-GARCH(1,1) DCC volatility models to model the correlation structure amongst the asset returns which allows the correlation matrix to be time varying and thus reflects the current market conditions.

Because VaR models often focus on the behavior of asset returns in the left tail, we used copula functions to model the dependence structure between the asset returns and EVT

to model the left tail of the distribution of the noise variables to obtain theqthquantile for

VaR estimation. Results from back-testing suggest that the M-GARCH(1,1) DCC copula- EVT model captures VaR quite well at shorter and longer observation periods. However,

following the traditional method to obtain a cut offpoint known as the threshold on the

left tail of the distribution for GPD parameter estimation, the number of points beyond the threshold sometimes lie towards the center of the distribution. The threshold selection process is also very subjective. GPD is not a good approximation for the the center of a sample data and might result in poor approximation of parameter estimates and hence incorrect VaR estimates. Based on this fact, though back-testing results deemed the model to be reliable, the forecast VaR estimates might be inaccurate. In the next chapter, we introduce a more objective method for threshold selection that avoids the body of the distribution and restricts inferences only on the tails.

Chapter 4

Forecasting robust Value-at-Risk

estimates using Bayesian

GARCH(1,1) model, vine-copula

functions and Extreme Value Theory:

Evidence from UK banks

This chapter proposes an objective approach for threshold selection, which we term the

hybridmethod that will restrict inferences to the tails of asset return distributions when Extreme Value Theory (EVT) is employed in estimating VaR. Thus, our main goal is to improve the threshold selection method used in the POT method. As already seen in Chapter 3.7, ARCH LM test fail to reject the null hypothesis of no conditional het-

eroscedasticity in the standardised residuals;_η

i,t T_t=1, after the fitted DCC model. This is

a weakness of DCC models because it is not quite easy to ascertain that all correlations 68

4.1. Introduction 69

evolve in the same manner regardless of the assets involved, and diagnostic checks often rejects fitted DCC models (Tsay, 2013). Therefore, we employ a Bayesian GARCH(1,1)

model with student’s-tdistribution as the underlying volatility model. Many researchers,

for example Aas et al. (2009) and Ardia and Hoogerheide (2010) have shown that a simple GARCH(1,1) model is able to capture the dynamics of changes in asset returns. The mo- tivation of using Bayesian-GARCH(1,1) model is because Bayesian estimation methods provides reliable results even for finite samples, and are usually straightforward to obtain the posterior distributions of any non-linear function of the model parameters whereas for the classical maximum likelihood method, it is not easy to perform inferences on non-linear function of the model parameters, the convergence rate is slow, and presents limitations when the residuals are heavy tailed. The constraints on the GARCH parame- ters to guarantee a positive variance can be incorporated via priors whereas the classical maximum likelihood method may impede some optimization procedures (Virbickaite

et al., 2015; Hall and Yao, 2003). The motivation of Student’s-t distribution is because

it is able to account for the excess kurtosis in the conditional distribution common with financial time series processes (Ardia and Hoogerheide, 2010).

4.1

Introduction

Traditional VaR models such as the commonly used variance-covariance method and Monte Carlo simulation often assume asset returns in financial markets to be normally distributed. Numerous studies (see for example Berkowitz et al. (2011); Sheikh and Qiao (2010)) have shown that financial asset returns are in fact leptokurtic and heavy tailed with non-constant volatility. Normality assumptions in situations of non-normality will without doubt lead to inaccurate estimates of the probability of extreme events and hence wrong estimates of VaR. This is because a normal distribution has light tails, and VaR

4.1. Introduction 70

attempts to capture the behaviour of the portfolio return in the left tail. A model based on normal distribution where data is not will underestimates the frequency of the outliers and hence the true VaR (Jorion, 2007). Normality assumption also implies volatility is constant over time, and recent price changes which are based on current market information will be assigned weights in equal proportion to older ones. If the dependence characteristics

of the extreme realisations differ from all others in the sample, the consequences might

be dire (Poon et al., 2003). To avoid the normality assumption, most analysts now turn to use EVT to model the tail behaviour of asset returns. However, as stated earlier, EVT also assumes extreme events to be normally distributed which will probably not be the case in stressed periods (Wong, 2013).

Thus, this work is motivated by the work of McNeil and Frey (2000) who suggested applying EVT to the noise variable of the return series which are normally distributed to

obtain theqth quantile used to estimate conditional robust VaR estimates. By doing so,

the problem of volatility clustering and other related effects such as excess kurtosis are

accounted for. This approach was further investigated by Soltane et al. (2012) where they combined GARCH(1,1) model as the underlying volatility model with EVT to estimate

VaR and showed that the GARCH-EVT-based VaR approach appears to be effective and

realistic than the traditional VaR methods. Bob (2013); Hsu et al. (2012) also combined GARCH-EVT and copula functions (to model dependence) in estimating VaR. Their findings showed better performance compared to traditional VaR estimation methods, and also better estimates of VaR than copulas with conventional employed empirical distributions.

We construct and investigate the reliability of our VaR model, in line with Basel II and Basel III, and estimate VaR and minimum capital requirements (MCR) in some selected banks in the United Kingdom (UK) using actively traded stocks in the London Stock