Identifying Broad and Narrow Financial Risk Factors with Convex Optimization

Factor analysis of security returns aims to decompose a return covariance matrix into systematic and specific risk components. To date, most commercially successful factor analysis has been based on fundamental models, although there is a large academic literature on statistical models. While successful in many respects, traditional statistical approaches like principal component analysis and maximum likelihood suffer from several drawbacks. These include a lack of robustness, strict assumptions on the underlying model of returns, and insensitivity to narrow factors such as industries and currencies, which affect only a small number of securities, but in an important way.We apply convex optimization methods to decompose a security return covariance matrix into low rank and sparse parts. The low rank component includes the market and other broad factors that affect most securities. The sparse component includes narrow factors and security specific effects.We measure the variance forecasting accuracy of a low rank plus sparse covariance matrix estimator on an equally weighted portfolio of 100 securities simulated from a model with two broad factors and 21 narrow factors. We find that the low rank plus sparse estimators are more accurate than estimates made with classical principal component analysis, in particular, at forecasting risk due to narrow factors. Finally, we illustrate a low rank plus sparse decomposition of an empirical covariance matrix of 100 equities drawn from 21 countries.