Dependence Modelling of Risks using Copulas in Financial Risk Management

Abstract

Financial Risk Management is a study to calculate and manage different risks across diversified portfolios of financial sectors. It is a crucial part of the financial world with all organisations wanting to lower their exposure to various risks involved with their investments and assets. Risk measurement is integral part of Financial Risk Management and with the different type of risks involved in a vast portfolio of investors, they all are to be modelled jointly to measure the aggregate risk. The effect of dependencies between risks across different sectors, departments and activities plays a crucial role and to study those dependencies of risks in a portfolio, where varieties of stock are included, is of utmost importance. Classical approaches to portfolio theory rely on concepts of correlation and the assumption of the underlying distribution being an elliptical one. Linear correlation is a good and easy implementable dependence measure, when random variables are multivariate normally distributed but that's usually not the case in the risk management world. Few type of risks like credit and operational risks, have more skewed distributions and are heavy-tailed risks, due to extreme and occasional losses, so correlation underestimates and doesn't give precise information about dependence, specially about the tails. So the underestimation of effects on aggregated risks, providing inexact risk measure, can lead to millions of loss for the organisation. Copulas are considered for such heavy-tailed risks, to develop the dependence structure of multivariate random variables and analyse, as an alternative approach, to be able to compute risks measures under extremal dependence structures. They are better suited to determine the dependence among such variables, as they join the individual marginal distributions to form joint distribution, giving all information about scaling and shape from the marginals. This thesis examines the situations where linear correlation goes beyond its limits and could result in underestimation of or have counterintuitive effects on aggregate risk for each type. Different Copulas and tail dependence coefficients are considered as alternatives to correlation and are estimated and analysed using VaR and ES risk measures. For Market risk, multivariate data (stocks) to create a portfolio is taken, linear correlation coefficients are calculated and dependence structure is modelled by fitting data for different copulas and using Monte Carlo simulations empirical distribution is calculated by simulating and calibrating and finding out the parameters for each copula model. All the distribution plots are checked for best fit, to see the performance of each model by goodness of fit (log likelihood) and VaR is calculated for each model and is backtested using the historical occurrences from the data. Similarly for credit risk, Bernoulli mixture model is used for credit default loss function and is simulated conducting Bernoulli's trials to calculate empirical distribution and by estimating required parameters, it is simulated for different copula distributions and VaR is derived. For operational risk, loss function taking into aggregated losses caused by various event and business types are simulated. Correlation coefficient and copulas are fitted for each event types loss, and empirical and fitted copula distributions are plotted and are checked for performance using Max-likelihood estimation, taking into account correlation tail dependence and goodness of fit measures.