# Predicting Covariance Matrices of Returns

Covariance plays an important role in portfolio construction as it measures the relationship between the returns of different assets in a portfolio. Understanding covariance helps investors to diversify their investments effectively by selecting assets that are not highly correlated with each other. Covariance also allows investors to assess the impact of adding or removing assets from their portfolio on its overall risk and return profile.

Predicting covariance matrices is important in portfolio management due to the significance of covariance. Reference [1] provides a thorough review of covariance prediction methods and introduces a novel covariance predictor. The authors pointed out,

First, we propose 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. Our method is a relatively simple extension that requires very little tuning and is readily interpretable…

Our second contribution is to propose a new method for evaluating a covariance predictor, by considering the regret of the log-likelihood over some time period such as a quarter. This approach allows us to evaluate how quickly a covariance estimator reacts to changes in market conditions.

Our third contribution is an extensive empirical study of covariance predictors. We compare our new method to other popular predictors, including rolling window, exponentially weighted moving average (EWMA), and generalized autoregressive conditional heteroscedastic (GARCH) type methods. We find that our method performs slightly better than other predictors. However, even the simplest predictors perform well for practical problems like portfolio optimization.

Briefly, the suggested predictor,  a so-called combined multiple iterated EWMA or CM-IEWMA for short, performs effectively.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Kasper Johansson, Mehmet Giray Ogut, Markus Pelger, Thomas Schmelzer, Stephen Boyd, Simple Method for Predicting Covariance Matrices of Financial Returns, 2024, https://arxiv.org/abs/2305.19484

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