Covariance estimation and related problems in portfolio optimization

This overview paper reviews covariance estimation problems and related issues arising in the context of portfolio optimization. Given several assets, a portfolio optimizer seeks to allocate a fixed amount of capital among these assets so as to optimize some cost function. For example, the classical Markowitz portfolio optimization framework defines portfolio risk as the variance of the portfolio return, and seeks an allocation which minimizes the risk subject to a target expected return. If the mean return vector and the return covariance matrix for the underlying assets are known, the Markowitz problem has a closed-form solution. In practice, however, the expected returns and the covariance matrix of the returns are unknown and are therefore estimated from historical data. This introduces several problems which render the Markowitz theory impracticable in real portfolio management applications. This paper discusses these problems and reviews some of the existing literature on methods for addressing them.