Robust Monte Carlo Simulation for Approximate Covariance Matrices and VaR Analyses

Abstract

Value at Risk (VaR) analysis plays very important role in modern Financial Risk Management. The are two very popular approaches to portfolio VaR estimation: approximate analytical approach and Monte Carlo simulation. Both of them face some technical difficulties steaming from statistical estimation of covariance matrix decribing the distribution of the risk factors. In this paper we develop a new robust method of generating scenarios in a space of risk factors consistent with a given matrix of correlations containing possible small negative eigenvalues, and find an estimate for a change in VaR. Namely, we prove that the modified VaR of a portfolio VaR’ satisfies the inequality |VaR ² — VaR ²| ≤ K · μ, where μ is the maximum of the absolute values of negative eigenvalues of the approximate covariance matrix and K is an explicitly expressed constant, closely related to the market value of the portfolio.

This paper was prepared when Alexander Levin was with Risk Lab, University of Toronto.