Orthogonal Methods for Generating Large Positive Semi-Definite Covariance Matrices

Abstract: It is a common problem in risk management today that risk measures and pricing models are being applied to a very large set of scenarios based on movements in all possible risk factors. The dimensions are so large that the computations become extremely slow and cumbersome, so it is quite common that over-simplistic assumptions will be made. In particular, in order to generate the large covariance matrices that are used in Value-at-Risk models, some very strong constraints are imposed on the movements in volatility and correlations in all the standard models. The constant volatility assumption is also imposed, because it has not been possible to generate large GARCH covariance matrices with mean-reverting term structures. This paper introduces a new method for generating large positive semi-definite covariance matrices. It is based on univariate GARCH volatilities of a few, uncorrelated key risk factors to provide more realistic term structure forecasts in covariance matrices. Alternatively the method can be used with exponentially weighted moving average key risk factor volatilities, where the smoothing constant is automatically determined by the correlation in the system. In addition to implementing multivariate GARCH of arbitrarily large dimension without the need for constrained parameterizations, advantages of this method include: the ability to tailor the amount of noise in the system so that correlation estimates are more stable; and the volatility and correlation forecasting of new issues or illiquid markets in the system.