A Cauchy-Gaussian mixture model for Basel-compliant Value-at-Risk estimation in financial risk management

The Basel II accords require banks to manage market risk by using Value-at-Risk (VaR) models. The assumption of the underlying return distribution plays an important role for the quality of VaR calculations. In practice, the most popular distribution used by banks is the Normal (or Gaussian) distribution, but real-life returns data exhibits fatter tails than what the Normal model predicts. Practitioners also consider the Cauchy distribution, which has very fat tails but leads to over-protection against downside risk. After the recent financial crisis, more and more risk managers realized that Normal and Cauchy distributions are not good choices for fitting stock returns because the Normal distribution tends to underestimate market risk while the Cauchy distribution often overestimates it.;In this thesis, we first investigate the goodness of fit for these two distributions using real-life stock returns and perform backtesting for the corresponding two VaR models under Basel II. Next, after we identify the weaknesses of the Normal and Cauchy distributions in quantifying market risk, we combine both models by fitting a new Cauchy-Normal mixture distribution to the historical data in a rolling time window. The method of Maximum Likelihood Estimation (MLE) is used to estimate the density function for this mixture distribution. Through a goodness of fit test and backtesting, we find that this mixture model exhibits a good fit to the data, improves the accuracy of VaR prediction, possesses more flexibility, and can avoid serious violations when a financial crisis occurs.