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Expected Shortfall Estimation Using Extreme Theory

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Expected Shortfall Estimation Using Extreme Theory Global Journal of Finance and Management. ISSN 0975-6477 Volume 8, Number 1 (2016), pp. 75-87 © Research India Publications http://www.ripublication.com Expected Shortfall Estimation Using Extreme Theory Kebba Bah1, Dr. Joseph Munga'tu2 and Dr. Antony Waititu2 1Pan African University, Institute of Basic Sciences, Technology and Innovation (PAUISTI 2Jomo Kenyatta University of Agriculture and Technology Abstract In this paper the Extreme Value Theory and GARCH model are combined to estimate conditional quantile and conditional expected shortfall so as to estimate risk of assets more accurately. This hybrid model provides a robust risk measure for the Nairobi 20 Share index by combining two well known facts about return time series: which are volatility clustering, and non-normality leads to fat tailedness of the return distribution. We first fit GARCH models to our return data using pseudo maximum likelihood to estimate the current volatility and use a GPD-approximation proposed by EVT to model the tail of the innovation distribution of the GARCH model. Keywords: Risk, Value at Risk, Expected Shortfall Extreme Value Theory, GARCH estimation, Peak Over Threshold. 1. INTRODUCTION In recent years, both practitioners and academics from the financial community have become interested in extreme events analysis particularly concerning financial risk management. The quantification of market risk for derivative pricing, portfolio choice and for market risk management has been of crucial interest to financial institutions and researchers especially during the last two decades. Since the early 1990s VAR has been the leading tool for measuring risk. Indeed, the ability to estimate extreme market movements can be particularly useful for detecting risky portfolios. Supervisors increase the control on banks to make sure they have enough capital to survive in bad 76 Kebba Bah, Dr. Joseph Munga'tu and Dr. Antony Waititu markets. While risk is associated with probabilities about the future, one usually uses risk measures to estimate the total risk exposure. A risk measure summarizes the total risk of an entity into one single number. While this is beneficial in many respects, it opens up a debate regarding what risk measures that are appropriate to use and how one can test their performance. VAR quantifies the maximum loss for a portfolio under normal market condition over a given holding period with a certain confidence level. Despite been universal, conceptual simple and been easy to evaluate VAR has been criticised for not been able to account for tail risk. It only tells us the maximum we can lose if a tail event does not occur, but if tail event occurs, we can expect to lose more. It is also criticised for its lack of subadditivity (Artzner et al 1997). Because of the above limitations Artzner et. al (1997) propose a more coherent risk measure, expected shortfall to overcome the shortcoming of VaR. Expected Shortfall quantifies the expected value of the loss if a VaR violation has occurs. The Basel committee on bank supervision in 2012 raised the prospect of replacing VAR with expected shortfall as a risk measure. The greatest challenge confronting the implementation of ES as the leading measure of market risk is the unavailability of simple tools for backtesting it. In fact, Gneiting (2011) prove that ES is not elicitable, unlike VAR. This result spark a lot of debate, some scholars believe that since ES lack such an important mathematics property it is not backtestable. However, 2014, Szekely et al. propose three methods for backtesting ES without exploiting it backtestability. The approaches use to estimate VaR and Expected shortfall can be classified in three. First, we have the non-parametric historical simulation, which estimate the quantiles base on available past data. The second is the is the fully parametric models base on econometric model for volatility and the conditional normality (most models from the GARCH family and J.P Morgan Riskmetrics) describe the entire distribution of returns including possible volatility dynamic. Finally, we have extreme value theory approach, which model only the tails of the return distribution. This approach is more effective since VaR and ES estimate are only related to the tail of the probability distribution. Due to the appealing aspects of EVT, it have been widely use in literature for the calculation of VaR and ES over the past decade. It has been applied many fields such as finance (Gencay and Selcuk (2004); Embrenchts (1999); Daneilsson and deVries (1997); and McNeil and Frey (2000)), insurance (Tajvidi (1997); McNeil (1997)) and from hydrology (Davison and Smith (1990); Katz et al. (2002)). The EVT provide the fundamental framework for model rare events.It model the tail of the distribution and, hence have to potential to perform better than other approaches in predicting unexpected extreme changes (Dacorogna et al.(1995); Longin (2000)). However, none of these studies has reflected the current volatility background. McNeil and Frey (2000) propose a combine approach which take into account the two stylized facts exhibited by most financial time series, namely fat- tailedness and stochastic. Within stock markets, implementing a risk measurement Expected Shortfall Estimation Using Extreme Theory 77 methodology based on the statistical theory of extremes is an important issue. The rest of this paper is organized as follows. In section 2 and 3 we present an overview of the theoretical frame work of EVT. While in section 4 we present the data analysis and finally in section 5 we present the conclusion. 2. GARCH-MODELS The ARCH models captures the stylized facts of real return data, but in order to have a good fit to real data we need a larger number of parameters, which reduce the data required for estimation. The Garch model introduce by Bollerslev (1986) added the concept for tomorrow volatility depends not only the past realizations but also depend on the errors of the volatility predicted. The Garch model has advantage over the ARCH model since it can capture the series correlation in squared residuals using a smaller number of parameters. The Garch have been extremely widely use since it integrate the two main characteristics of financial returns which are unconditional normalities and volatility clustering. In order to analyze data set of stock prices, we try to fit AR- GARCH to the log returns. Here we use the simplest possible ARGARCH model with conditional variance of the return as a GARCH(1,1) model and mean model as AR(1). Let Yt be the return at time t be defined by the following stochastic volatility model 푌푡 = 휇푡 + 휎푡푍푡 equ. 1 where 휇푡 is the expected return on day t and 휎푡 is the volatility and 푍푡 gives the noise variable with a distribution 퐹푍(푍) (commonly assumed to be standard normal). We assume that Yt is a stationary process. The most widely used suitable models are drawn from the ARCH/GARCH family. An autoregressive AR(1)-GARCH(1,1) process is given by 휇푡 = 훼0 + 훼1푦푡−1 and equ .2 2 2 2 휎푡 = 훿0 + 훿1휀 푡−1 + 훽휎 푡−1 Where 휀푡 = 푌푡−1 − 휇푡−1, 훿0, 훿1, 훽, 훼0, 훼1 and 훿1 + 훽 < 1 3. EXTREME VALUE THEORY: Theextremevaluetheoryplaysafundamentalroleinmodellingmaximaofarandomvariable just like the central limit does in modelling sums of random variables. In all the two cases, the theory tells us what the limiting distribution. There are two ways of identifying extreme in real data. Suppose we consider random variable representing daily losses or returns. The first method is done by dividing the data in to blocks and consider the maximum in each block as extreme event. The other approach is focusing 78 Kebba Bah, Dr. Joseph Munga'tu and Dr. Antony Waititu on the realizations exceeding a giving high threshold. Any observation above the selected threshold considered an extreme event. The block maxima method is the traditional method used to analyse data with seasonality as for instance meteorological data. But the threshold method uses data more efficiently for that reason, seems to become of choice in recent application (M.Gilli et. al 2006) The EVT relates to the asymptotic behaviour or the extreme observation of a random variable. It provides the fundamentals for the statistical modelling of rare events and is used to compute tail related risk measures. In this paper, we adopt the POT model to identify the extreme observations that exceed a high threshold u. 3.1 Generalized Pareto Distribution and The Peak over Threshold Model (POT) 3.1.1: Theory Theorem 1: (Pickands (1975), Balkema and de Haan (1974)). For a large class of underlying distributions F, the excess distribution function 퐹푢 can be approximated by GPD for an increasing threshold u 퐹푢(푦) ≈ 퐺휀,훽 , 푢 → ∞ equ. 3 퐺휀,훽 is the Generalized Pareto Distribution which is given by −1 휀푧 1 − (1 − ) 휀 푖푓 휀 ≠ 0 훽 퐺휀,훽 (푧) = { equ. 4 푧 1 − 푒푥푝 (− ) 푖푓 휀 = 0 훽 훽 For 푦휖[0, (푌 − 푢)] if 휀 > 0 and 푦휖 [0, − ] if 휀 < 0 . Here 휀 is the GPD shape 퐹 휀 parameter and 훽 is the GPD scale Definition (Distribution of exceedances): The distribution of excess over threshold τ for a random variable Y with df F is given by 퐹(푥+푢)−퐹(푢) 퐹(푦)−퐹(푢) 퐹 (푥) = 푃(푌 − 푢 ≤ 푦|푌 > 푢) = = equ. 5 푢 1−퐹(푢) 1−퐹(푢) For 0 < 푦 < 푦퐹 − 푢 where 푦퐹 < ∞is the right endpoint of F and 푥 = 푦 − 푢. 퐹푢 is the conditional excess. VaR and ES If there is an extreme distribution F with right endpoint 푌퐹 , we can assume that for some threshold u , 퐹푢(푦) = 퐺휀,훽 (푦) for 0 ≤ y < 푌퐹 − u and ε∈R and σ > 0. For y≥u, 퐹̂(푦) = 푃(푌 > 푢)푃(푌 > 푦|푌 > 푢) = 퐹̂(푢)푃(푌 − 푢 > 푦 − 푢|푌 > 푢) = 퐹̂(푢)퐹̂(푦 − 푢) −1 푦−푢 = 퐹̂(푢) (1 + 휀 ) 휀 equ.6 훽 Given F(u) the this gives formula for the tail probability.
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