Estimation of Value-at-Risk for bank portfolio by using Markov Chain Monte Carlo method

Abstract

Value-at-Risk models (VaR) are widely used in bank risk management practice. VaR models come from the field of “worst statistics” and help to understand the worst loss with a certain probability. After the last major financial crisis of 2007-2008, the regulators tried to modify existing models for risk assessment. VaR models were criticized by many researchers and new modified models, such as conditional VaR, Expected Shortfalls (ES), were proposed. One of the major drawbacks of VaR models is low sensitivity to the tails of the returns’ distribution. In this work, we propose Gibbs sampling for efficient sampling from the joint distribution of returns. Gibbs sampling belongs to the family of Markov Chain Monte Carlo (MCMC) methods which use Markov chains to create random walks while sampling from the target distribution. We considered multivariate distribution of returns for a portfolio consisting of several instruments. For this purpose, we simulated a situation of 3 instruments-portfolio with known Covariance matrix under the normality assumption of the marginal distributions. After calculating the conditional probabilities, we ran Gibbs sampler and succeeded to achieve marginal distributions allowing further VaR calculation.