A GARCH-VaR Investigation on the Brazilian Sectoral Stock Indices/Uma Investigacao GARCH-VaR sobre os Indices de Acoes Setoriais Brasileiros.
| Autor | Bernardino, Wilton |
| Cargo | Texto en ingles - Ensayo |
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Introduction
The advances in theory and application involving the risk management in stock markets have received significant attention from academics, practitioners (e.g, Webby et al. (2007), Bi and Giles (2009), Candelon et al. (2010), Gaglianone et al. (2011), Asfaha et al. (2014)) and regulators agencies (on Banking Supervision, 2009, 2013). In Brazil, for example, there is a set of central roles in several financial applications, including pricing and asset allocation, among others (see Banco Central do Brasil (2009)). One widely used measure to quantify and regulate risk is the Value-at-risk (VaR) which is an indicator of the extremes within which the stock trading returns of an investment in a specific asset or portfolio fluctuates (Giot and Laurent, 2004). In other words, the VaR represents a potential extreme capital loss incurred under a small probability of occurrence. The investment loss is expected to exceed the VaR with a small probability. VaR can be seen as a natural approach to understand volatility given that it directly depends on the forecast of the conditional standard deviation.
The literature has been shown that the family of GARCH models (see e.g., Francq and Zakoian (2011), Bollerslev et al. (1994), Ghysels et al. (1996), Engle and Patton (2001), Hansen and Lunde (2005)) provides much empirical regularity associated with the volatility of returns on financial assets such as leptokurtic distributions, volatility clustering, leverage effects, persistence and asymmetric volatility, among others. Using the conditional mean and the conditional variance obtained from the estimated GARCH family models, the quantiles of the conditional distribution can be easily obtained for the calculation of VaR (Tsay, 2005).
This paper proposes to use GARCH approach to evaluate the most important Brazilian sectoral stock indices in a Value at Risk point of view. On this way we estimated the [VaR.sub.99%] (1% VaR) and [VaR.sub.1%] (99% VaR) by using the most common GARCH specifications, namely, the Normal-GARCH (NGARCH), the tGARCH, and the eGARCH (Nelson, 1991a) approaches. In this sense, although new strategies to estimate VaR have been being proposed (e.g., Chernozhukov (2005), Paolella and Polak (2015)), we understand that the usual GARCH specifications are enough to generate confident estimates focusing on an easy implementation, which does not enforce an exhaustive computational effort. In our study, we highlighted the VaR measure as a risk indicator able to help investors on making a portfolio selection on the Brazilian stock market.
Beyond this introduction, the paper is organized as follows. Section 2 provides a theoretical background of the used methods. Section 3 presents a brief motivation of choosing our modeling strategy and the results of our VaR analysis on the Brazilian sectoral stock indices. Finally, in Section 4 we present the conclusions of the paper. 2 2
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Theoretical Background
* Volatility measure: Volatility can be informally defined as a measure of fluctuation in a given stochastic processes, as for example, the time series of an asset prices (Brockwell and Davis, 2002). The most simple method to estimate volatility is by evaluating the standard deviation of a sequence of stock returns (called as Historical Volatility). Historical Volatility is efficient only when the returns are normally distributed. However, many empirical data indicate that the returns are generally non-normal distributed with presence of clusters, fat tails and nonlinear patterns (Cont, 2001, Engle and Patton, 2001, Danielsson, 2011). Volatility models can provide both conditional and unconditional volatility estimates. Unconditional volatility is simply the sample standard deviation ([??]) over the entire time range. Conditional volatility is the standard deviation estimate for a specific point in time given previous values. The number of previous values used in the model may vary. In this case, it is common to use a parametric statistical modeling to estimate the standard deviation at t-period ([[??].sub.t]). Morgan (1996) introduced the Exponentially Weighted Moving Average models (EWMA) and RiskMetrics[TM] software to evaluate asset returns. EWMA are non-parametric: no input besides the observation window is needed. This makes implementation easy and facilitates comprehension, but limits their power to deal with the challenges imposed by real volatility patterns such as non-normality and clustering (Danielsson, 2011).
The parametric way is more simple to represent this type of series (in our case, the return series). It considers [R.sup.dm.sub.t] (demeaned-return) as the shock or innovation of the asset returns (return residuals, with respect to a mean process). The shocks are split into a stochastic piece [u.sub.t] and a time-dependent standard deviation ([[sigma].sub.t]) which is the process for the volatility such that volatility equation is given by [R.sup.dm.sub.t] = [u.sub.t] = [[sigma].sub.t][z.sub.t]. That is, conditional heteroskedasticity models are centered on the the dynamics of [[sigma].sub.t] with the random variable [z.sub.t] as a strong white noise process (Tsay, 2005). Using the Autoregressive Conditional Heteroskedasticity (ARCH) approach (Engle, 1982), the conditional variance ([[sigma].sup.2.sub.t]) can be modelled by
[[sigma].sup.2.sub.t] = [[gamma].sub.0] + [[gamma].sub.1][([R.sup.dm.sub.t-1]).sup.2] + ... + [[gamma].sub.q][([R.sup.dm.sub.t-q]).sup.2], (1)
where [[gamma].sub.i] > 0, i = 0, ..., q. The Generalized ARCH model (GARCH) was proposed by Bollerslev (1986) as alternative to the equation (1). The modeling strategy in a GARCH(p, q) is given by
[[sigma].sup.2.sub.t] = [[gamma].sub.0] + A([omega])[([R.sup.dm.sub.t]).sup.2] + B([omega])[[sigma].sup.2.sub.t], (2)
where A([omega]) = [[gamma].sub.1][[omega].sup.1] + ... + [[gamma].sub.q][[omega].sup.q] and B([omega]) = [[beta].sub.1][[omega].sup.1] + ... + [[beta].sub.p][[omega].sup.p] are polynomials of order p and q from the autoregressive and the move average components of GARCH(p, q), respectively, and [omega] denotes the lag order operator. In the eGARCH(p, q) (Nelson, 1991a) there is an asymmetric function for the conditional variance depending on the lagged disturbances ([[epsilon].sub.t-i]) according to the equation
[mathematical expression not reproducible], (3)
where E([[epsilon].sub.t]) denotes expected value of [[epsilon].sub.t]. The parameters [[gamma].sub.i] and [[alpha].sub.i] capture, respectively, the sign effect and the size effect of the asymmetric relation between stock returns and volatility. In financial assets (specially in high frequency data), the distribution of returns hardly ever is normal adjusted: extreme returns occur more frequently than expected under normality (fat tails) and extreme negative returns are frequent when comparing with the positive ones (negative asymmetry); there are also the volatility clusters and calendar effects (Schwert, 2003, Francq and Zakoian, 2011) to be considered.
Different variations on GARCH models can be found in literature (Angelidis et al., 2004). However, in many empirical applications the GARCH(1,1) is generally mentioned as that which outperforms others (Hansen and Lunde, 2005). Furthermore, it is computationally convenient and widely used by specialists to model volatility of daily returns. For these reason, in our work, the GARCH (1,1) specification was used to analyze VaR on the Brazilian sectoral stock indices. The fact that GARCH enables the amplitude of the demeaned-returns to depend upon previous values is of paramount importance, since it effectively models volatility clustering (Francq and Zakoian, 2011). Owning to its robust performance, superior estimation, overall simplicity and flexibility.
* Value-at-Risk (VaR) measure and its GARCH estimation: The use of VaR as a risk metric quickly spread during the 1990s. J.P. Morgan made available to the public its estimation software, RiskMetrics[TM], in 1994, which quickly became the industry benchmark. RiskMetrics was developed during the 1980s, starting as an aggregation of hundreds of risk factors and several VaR estimates calculated daily (Holton, 2002). RiskMetrics assigns an exponentially decaying weight ([lambda]) for each observation in the sample, so that, more recent returns have greater weight than old ones. In commercial applications, VaR has been a widely used risk metric. The 1988 Basel Committee signed by G-10 countries imposed risk management on banks, which have motivated the use of some VaR methodology to comply with regulations (Jorion, 2001). Several United States government agencies such as the Federal Reserve and the Security and Exchange Commision (SEC) use or advocate the use of VaR (Khindanova et al., 2000). In the Brazilian context, the Brazilian's Central Bank, requires financial institutions to manage risks based on VaR. Although not mentioned directly, requirements are based on the Basel Committees recommendations (Banco Central do Brasil, 2009). Other banks such as Credit Swiss, Chase Manhattan and Deutsche Bank followed J.P. Morgan. By 1999, over 80 different commercial vendors were offering VaR software (Christoffersen et al., 2001). Albeit used mostly by finance and actuarial-related clients, several studies were made using VaR in many different risk modeling, such as geology (Webby et al., 2007), the US movie industry (Bi and Giles, 2009), and agriculture (Asfaha et al., 2014). Basel regulations allow banks to develop their own methods to calculate VaR, stimulating research and development of new methodologies to understand and estimate this risk measure.
The concept of VaR is very simple: it is just a number representing how much loss may occur with a certain probability (1 - [alpha]) of occurrence of losses more severe than the VaR level (VaR[alpha]). In this sense, [alpha] and (1 - [alpha]) are, respectively, the confidence level and the coverage rate of VaR...
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