Portfolio value-at-risk by Bayesian conditional EVT-copula models: taking an Asian index portfolio for example

Abstract

Portfolio VaR (Value-at-Risk) is an important but difficult issue in risk management. Traditional approaches deal with single asset and usually fail to accurately estimate the threshold value under given probability level. This study proposed a new model which estimates portfolio VaR (Value-at-Risk) using a Bayesian conditional EVT (extreme value theory)-Copula based approach. First, GJR-GARCH model is used to model the time structure of each asset. Second, EVT is employed for modeling the residuals after GJR-GARCH. This study constructs the semi-parametric empirical marginal CDF (cumulative distribution function) for each residual using a Gaussian kernel estimate for the interior and a generalized Pareto distribution (GPD) estimate for the upper and lower tails. Our approach focuses on the entire distribution rather than the tail distribution only. Finally, a Student’s copula is fitted to the data and used to induce correlation between the simulated residuals of each asset. By using Bayesian MCMC (Markov chain Monte Carlo) to fit the GPD, our estimations do not rely on asymptotics; namely, our estimation gets rid of the influence of limited sample size of extreme events. In order to test the effectiveness of this model we backtest the estimated VaRs over a time period. Empirical results demonstrate that the Bayesian CEVT- Copula based approach outperforms traditional methods such as historical simulation (overestimate) or conditional normal (underestimate) model.