Estimation of Value-at-Risk for bank portfolio by using Markov Chain Monte Carlo method
Igor Zavialov (1811335)
Value-at-Risk models (VaR) are powerful tools for financial risk management and are widely used by regulating Value-at-Risk models (VaR) are powerful tools for financial risk management and are widely used by regulating authorities and market players. VaR models come from the field of “worst statistics” and help to understand the worst loss with a certain probability. Because of its relative simplicity, VaR models have become popular among risk analysists working in a banking area. 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 were proposed such as conditional VaR, Expected Shortfalls (ES) and GARCH models. One of the major drawbacks of VaR models is low sensitivity to the tails of the returns’ distribution. Mathematically speaking, VaR is a quantile built on the Profit \& Loss distribution. Ability to accurately calculate worst outcome helps banks to keep the right amount of the allocated capital, i.e. to cover all the losses if needed and not to break the one’s obligations. Moving the focus from one particular bank to the whole financial system consisting of many interconnected participants, the risks’ calculation become more difficult. Regulators use the “systemic risk” term for distinguishing the risk responsible for overall system stability. Defined simply as the probability of the whole system’s default after breakdown of one of the participants, the systemic risk is difficult to calculate. The main reason is complexity of the system links and causal relationships between different nodes. Different mathematical theories, including chaos and complexity theories can be applied for the understanding of the concept of systemic risk. However, in simple approach regular risk management tools such as VaR can be used for systemic risk estimation which is implied by Basel capital requirements, main regulation guidelines in the banking field. The problem of less efficient work with tails of the distribution, is the significant drawback of VaR models. In this work, I propose Markov Chain Monte Carlo method (MCMC) for efficient sampling from the distribution. MCMC is a powerful tool using Markov chains to create random walks while sampling from the target distribution. To prove my approach I separate two different cases: 1) one-asset VaR estimation; 2) VaR estimation from multivariate distribution for portfolios consisting of several instruments. For the first case I do sampling using Metropolis-Hastings algorithm (MH), for the second – Gibbs sampling. Result is shown for the stock from S\&P 500 data set, MCMC (MH) worked almost the same with regular Monte Carlo method. For the multivariate case, I simulated a situation of 3 instruments-portfolio with known Covariance matrix under the normality assumption of the marginal distributions. After calculating the conditional probabilities, I ran Gibbs sampler and succeeded to achieve marginal distributions allowing further VaR calculation. MCMC methods proved to be applicable for VaR estimation and flexible for tackling the tail estimation problem.authorities. The performance of risk assessment is difficult to analyze due to complex market conditions; therefore, new models have been continuously developing. In this work we observe various directions towards VaR estimation. Finally, we present Markov Chain Monte Carlo (MCMC) method for VaR estimation.