Market Risk Analysis Volume II: Practical Financial Econometrics

II.6

Introduction to Copulas

II.6.1 INTRODUCTION

Portfolio risk is a measure of the uncertainty in the portfolio returns distribution. We use some measure of the dispersion about the mean of this distribution as the risk metric. But if we depart from the classical assumption that mean deviations are independent and symmetric with identical elliptical distributions, it is not possible to summarize uncertainty by a simple figure such as portfolio volatility.¹ Similarly, correlation is a measure of dependence that is very commonly applied in financial risk management, but it can only represent a certain type of risk. Each asset return must follow an i.i.d. process and the joint distribution of the variables must be elliptical. In practice very few assets or portfolios satisfy these assumptions, so we can use neither portfolio volatility as a measure of risk, nor the correlation of returns as a measure of association.² Instead we must work with the entire joint distribution of returns.

Classical theories of portfolio management and risk management have been built on the assumption of multivariate normal i.i.d. returns distributions. It is important to include classical theories in a text of this type, but financial markets do not behave according to these idealized assumptions. Assuming multivariate normal i.i.d. returns distributions is convenient not only because it allows one to use correlation as a measure of dependence, but also because linear value at risk (VaR) is a coherent ...