Encyclopedia of Financial Models, 3 Volume Set

Moving Average Models for Volatility and Correlation, and Covariance Matrices

CAROL ALEXANDER, PhD

Professor of Finance, University of Sussex

Abstract: The volatilities and correlations of the returns on a set of assets, risk factors, or interest rates are summarized in a covariance matrix. This matrix lies at the heart of risk and return analysis. It contains all the information necessary to estimate the volatility of a portfolio, to simulate correlated values for its risk factors, to diversify investments, and to obtain efficient portfolios that have the optimal trade-off between risk and return. Both risk managers and asset managers require covariance matrices that may include very many assets or risk factors. For instance, in a global risk management system of a large international bank all the major yield curves, equity indexes, foreign exchange rates, and commodity prices will be encompassed in one very large dimensional covariance matrix.

Variances and covariances are parameters of the joint distribution of asset (or risk factor) returns. It is important to understand that they are unobservable. They can only be estimated or forecast within the context of a model. Continuous-time models, used for option pricing, are often based on stochastic processes for the variance and covariance. Discrete-time models, used for measuring portfolio risk, are based on time series models for variance and covariance. In each case, we can only ever estimate or forecast variance and covariance ...