Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks

Markowitz (1952) portfolio selection requires estimates of (i) the vector of expected returns and (ii) the covariance matrix of returns. Many proposals to address the first question exist already. This paper addresses the second question. We promote a new nonlinear shrinkage estimator of the covariance matrix that is more flexible than previous linear shrinkage estimators and has ‘just the right number ’ of free parameters (that is, the Goldilocks principle). In a stylized setting, the nonlinear shrinkage estimator is asymp-totically optimal for portfolio selection. In addition to theoretical analysis, we establish superior real-life performance of our new estimator using backtest exercises. KEY WORDS: Large-dimensional asymptotics, Markowitz portfolio selection, nonlinear shrinkage