Estimation of VaR Using Copula and Extreme Value Theory

Abstract

This paper proposes a method for estimating the VaR of a portfolio based on copula and extreme value theory. Each return is modeled by ARMA-GARCH models with the joint distribution of innovations modeled by copula. The marginal distributions are modeled by the generalized Pareto distribution in the left tail (large loss) and empirical distribution otherwise. The copula is estimated by an estimator which gives more weight to observations with large loss. The method is applied to a two-asset portfolio and compared to other traditional methods (JEL: C15, D81,G10).

Estimating value at risk of portfolio by conditional copula-GARCH method

“…Value at Risk (VaR) has become the standard measure used by financial analysts to quantify the market risk of an asset or a portfolio (Hotta et al, 2008). VaR is defined as a measure of how the market risk of an asset or asset portfolio is likely to decrease over a certain time period under general conditions.…”

“…However, VaR estimation is not difficult to compute if only a single asset in a portfolio is owned, and becomes very difficult due to the complexity of the joint multivariate distribution. Besides, one of the main difficulties in estimating VaR is to model the dependence structure, especially because VaR is concerned with the tail of the distribution (Hotta et al, 2008).…”

Research on structural correlation of HS 300 stock index based on AR (n)-XARCH-Copula model

“…Jondeau and Rockinger (2006) studied the conditional dependence model and its application in the stock market [3] . Hotta and Palaro (2008) connected Copula function theory with extreme value theory. The research showed that Copula extreme value theory was significantly superior to the traditional risk calculation method when describing risk of assets [4] .…”

The role of model risk in extreme value theory for capital adequacy

“…This approach can be seen in Longin (2000); McNeil and Frey (2000); Nekhili et al (2004); Marinelli et al (2007); Hotta et al (2008); Ghorbel and Trabelsi (2009). Traditional models usually use time conditional methods for the mean and volatility of time series process (mostly ARMA-GARCH models or similar structures) in combination with the normal, Student t or empirical distribution (see, e.g., Longin, 2000;Hotta et al, 2008;Ghorbel and Trabelsi, 2009).…”

“…Traditional models usually use time conditional methods for the mean and volatility of time series process (mostly ARMA-GARCH models or similar structures) in combination with the normal, Student t or empirical distribution (see, e.g., Longin, 2000;Hotta et al, 2008;Ghorbel and Trabelsi, 2009). All these studies show that univariate EVT methods perform better with respect to a correct risk assessment of the Value-at-Risk in almost all instances.…”