Optimal shrinkage of means in the Markowitz model

This paper shows that portfolio optimization operates with low volatility in the portfolio weights and out-the-sample Sharpe ratio higher than the 1 / n rule by shrinking the vector of averages to its grand mean and minimizing the mean square error of the portfolio weights. It is also shown that shrinking the vector of means to its grand mean and minimizing the mean square error of the vector of means can obtain out-of-sample Sharpe ratios that exceed the 1 / n rule; however, this leads to high values of the portfolio weights’ volatilities. That is because minimizing the mean square error of the portfolio weights also optimizes the estimation error of the model’s decision variables. Therefore, this method can be part of the predict-and-optimize approach. In contrast, minimizing the mean square error of the vector of means involves incorporating the estimated values of the means to calculate the values of the decision variables (portfolio weights). That is, it minimizes the estimation error of the vector of means, not the decision variables. This method can be considered within the estimate-then-optimize approach. • The shrinkage of the vector of means that minimizes the mean square error of the portfolio weights overcomes the out-of-sample performance of the 1/n rule. • The shrinkage of the vector of means that minimizes the mean square error of the vector of means also overcomes the out-of-sample performance of the 1/n rule. • The shrinkage of the vector of means that minimizes the mean square error of the portfolio weights generates portfolios that have fluctuations well below those obtained with traditional methods. • Mean vector shrinkage methods that minimize the mean square error of the mean vector can be considered as part of the estimating and then optimizing approach. • Mean vector shrinkage methods that minimize the portfolio weights’ mean square error can be considered part of a simultaneous estimation and optimization approach.