A Simple Method for Predicting Covariance Matrices of Financial Returns makes three contributions. First, it proposes a new method for predicting the time-varying covariance matrix of a vector of financial returns, building on a specific covariance estimator suggested by Engle in 2002. The second contribution proposes a new method for evaluating a covariance predictor, by considering the regret of the log-likelihood over some time period such as a quarter. The third contribution is an extensive empirical study of covariance predictors. The authors compare their method to other popular predictors, including rolling window, exponentially weighted moving average (EWMA) and generalized autoregressive conditional heteroscedastic (GARCH) type methods. After an introduction, Section 2 describes some common predictors, including the one that this method builds on. Section 3 introduces the proposed covariance predictor. Section 4 discusses methods for validating covariance predictors that measure both overall performance and reactivity to market changes. Section 5 describes the data used in the authors’ first empirical studies and the results are provided in Section 6. The authors then discuss some extensions of and variations on the method, including realized covariance prediction (Section 7), handling large universes via factor models (Section 8), obtaining smooth covariance estimates (Section 9), and using the authors’ covariance model to generate simulated returns (Section 10).