A Study on Effect of Single Additive Outlier on Estimation of Value at Risk and Expected Shortfall under Peaks over Threshold Framework

Value at Risk (VaR) is one of the most popular measures of risk associated with financial instruments. The generalized Pareto distribution (GPD) is useful in modeling values exceeding a high threshold in estimation of VaR and Expected Shortfall (ES). It has been observed that none of the existing methods for estimating parameters of GPD performs uniformly better than others (P. Z. Bermudez and S. Kotz, 2010), and even if one develops a method, it would suffer from heavy computational requirements. Recently, P.Chen et.al (2017) proposed a method of estimating parameters of GPD based on minimum distance approach and M-estimation. In literature survey it has been observed that a study on effect of outliers on estimation of parameters of GPD has not been carried out, though presence of outliers is common in financial data. In this regard this paper focuses on the effect of outliers on estimators of parameters of GPD, on estimators of VaR and estimators of ES under Peaks over Threshold (PoT) framework obtained from different methods. A simulation study is carried out to compare performances of five robust and seven non-robust methods for estimating parameters of GPD, VaR and ES in the presence of a single additive outlier under PoT set up and it is found that robust methods suggested by P.Chen et.al (2017) for estimating GPD parameters perform on par with other estimators when shape parameter k > 0 (for thin tailed distributions), in estimating VaR in presence of outlier, especially when sample size is moderately large. © 2020 Elixir All rights reserved. Elixir Statistics 146 (2020) 54836-54843 Statistics Available online at www.elixirpublishers. com (Elixir International Journal) Ananda KS and J.V. Janhavi / Elixir Statistics 146 (2020) 54836-54843 54837 In literature several threshold selection methods have been suggested; (Embrechts et al. 1999b). It is often seen that the number of exceedances is small in peaks-over-threshold approach and if there is an outlier, abnormally large value present in excedances, it may distort estimates. Hence, our objective of this study is to see the effect of outlier on various estimators and also to compare their performance in estimation of parameters of GPD, VaR and ES, through a simulation study using bias and root mean square error criterion in presence of outlier. So, for this purpose, in the next the section, we discuss various estimation procedures for parameters of GPD, estimation of VaR and ES under PoT framework. In Section 3, using a simulation study, we evaluate performance of estimation methods for parameters of GPD, VaR and ES using bias and root mean square error criteria. Finally in Section 4, the summary of results of the simulation study is presented. 2. Estimation Of Parameters Of Gpd, Value At Risk And Expected Shortfall 2.1: Literature review: Estimation of parameters of GPD Several parameter estimation methods have been studied for generalized Pareto distribution in literature and various methods have been compared under different conditions for estimating the GPD parameters. However, there are no universally accepted methods for estimating GPD parameters. Even if few methods are better than others over certain range of shape parameter k, they suffer from various constraints and convergence problems (P.Z Bermudeza & S. Kotz 2010, Part I & II). Among these, maximum likelihood method (MLE) is preferred, due to its asymptotic optimality properties and has been studied by Davison (1984), Smith (1984, 1985), Grimshaw (1993). Hosking and Wallis (1987) compared maximum likelihood estimates with method of moments (MOM) and probability weighted moment (PWM) estimates over small ranges of k, |k|≤ 1⁄2 as it is common to observe k between -1 and 1⁄2 (Zhang and Stephen, 2009) and found that probability weighted method performs well for 0 ≤ k ≤ 1 and very good for k ≤ 1⁄2. Castillo and Hadi (1997) introduced elemental percentile method (EPM) and compared it with the MOM and the PWM methods, using root mean square error criterion when |k| ≤ 2, through simulation study and showed that the PWM estimator performs well in small samples for k ≤ 1⁄2. Zhang and Stephens (2009) and Zhang (2010) developed empirical Bayes method (EBM) based on the likelihood and which uses a data-driven prior to estimate parameters of GPD. They showed that EBM performs better than MLE, MOM, PWM and Likelihood Moment Estimator (LME) with respect to bias and mean square error when – 1⁄2 < k < 1⁄2. Luceño (2006) proposed estimators based on minimum distance approach by minimising the squared differences between empirical and model distribution functions, given in terms of various goodnessof-fit statistics, including the Cramer–von Mises statistic (CM), the Anderson– Darling statistic (AD) and the right-tail weighted Anderson– Darling statistic (ADR) and compared few maximum goodness-of-fit (MGF) estimators with Quasi Maximum Likelihood (QML), MLE, MOM, PWM and EPM over k = -2, -1, 0, 1, 2. Recently P.Chen et. al (2017) proposed two new robust estimators for the GPD parameters using the minimum distance approach and M-estimation, where the Tuckey biweight function is used as the distance measure which minimizes the distance between the empirical distribution and the family of GPDs. They compared proposed methods with Maximum Likelihood (ML), the Elemental Percentile Method (EPM), and a method proposed by Zhang (2010) (EBM). They claim that as the distance measure is borrowed from robust estimation, the proposed methods are robust to outlier contamination and the breakdown point is as high as 50%. It is seen that many simulation studies for comparing different methods of estimation of GPD parameters, have been conducted by many authors but these simulation studies are somewhat difficult to compare, as they have been performed under different conditions. Moreover, some of the methods have never been compared via a simulation study. For more details refer Bermudeza & S. Kotz (2010) part I & II. As our objective is to study the effect of outliers on estimation of VaR, we consider comparison of some robust methods with some traditional methods available in literature through a simulation study. Hence, we considered five robust methods and seven non robust methods which are available in R-environment. Among these three robust method are based on minimum distance approach (PZ, WPZ, MGF) Table 1: List of estimation methods considered in the study Robust Methods Non Robust Methods 1. Robust methods proposed P.Chen et.al (2017), (PZ and WPZ) 2. Median Estimator, (MED), Peng and Welsh, (2001) 3. Minimum Density Power Divergence, (MDPD), Juárez and Schucany (2004) 4. Maximum Goodness-of-fit Estimators, (MGF), (Based on Anderson Darling statistic) Alberto Luceño (2006) 5. PICKANDS (Pickands) Pickands, J. (1975) 6. Maximum Likelihood Estimator, (MLE), Smith (1984). 7. Maximum Penalized Likelihood, (MPLE), Coles and Dixon (1999) 8. Probability Weighted Moments,(PWMU, PWMB), Hosking and Wallis (1987) 9. Likelihood Moment Estimator, (LME), Zhang (2007) 10. Empirical Bayes Method, (EBM or ZJ), Zhang (2010), 2.2. Value at Risk and Expected Shortfall Generally returns on investments on a portfolio follow normal distribution and as a result, under typical conditions, VaR is thought to be almost as effective as expected shortfall (ES) in capturing risk. However, the financial crisis in 2007 highlights the importance of measuring the risk associated with non-normal returns. In this connection we are using generalized Pareto distribution to model returns exceeding a sufficiently high threshold value, in case of violation of normality assumption and for heavy tailed distribution. The procedure is to estimate parameters of GPD and then use them for estimation of VaR and ES. Value at Risk (VaR) If X denotes the return random variable with distribution function ( ), then for each ( ) the VaR with 100 (1– ) % confidence coefficient is defined as: ( )( ) ( ( ) ) = ( ( ) ) ( ) then ( )( ) under PoT framework is given by (Ref: Mc. Neil. A., 1998)