# Methods for superior estimates of returns in m.v. portfolio optimization

Leaving aside the aspects related to the estimation of the variance component (all the latest techniques to compute a stable covariance matrix of a given set of assets such as simple shrinkage, Ledoit-Wolf shrinkage, Oracle shrinkage, MCD (minimum covariance determinant), EWMA covariance etc.) how is it possible to improve the optimization results considering the estimation of the mean return component or rather what methods happened to be more effective and efficient in estimating the returns (both from the empirical and practical point of view)?

Of course the simple linear model (expectation = mean return over the last n years) is quite simplistic.

## 1 Answer

Expected returns are very difficult to estimate reliably without incurring estimation error as found out by Merton (1980) "On estimating the expected return on the market". This is why estimating volatility/the covariance matrix has become the default approach in the mean-variance model because volatility is easier to predict than returns. Even the global minimum variance portfolio, often considered to outperform other frontier portfolios, can be solved without the asset means:

$$\omega = \frac{\Sigma^{-1}\iota}{\iota^{\top}\Sigma^{-1}\iota}$$

The classical CAPM and Fama-French 3, 4, or 5 factor models, when re-arranged, are available though for those who insist that superior estimates of expected returns can be consistently measured well from empirical (heteroskedastic) data, but even these CAPMs rely on the assumption that the market portfolio is observable (find Roll's critique and the article Beta is dead). Then again, beta itself is computed from the covariance matrix.

• Thank you for the reply. Would you mind tell me if you support or not the possibility to consistently compute superior estimates of expected returns? I read a lot about CAPM, Fama-French 3,4,5 and APT but very little is written about practical application of these models (while applying each one I consistently get only awful results). Moreover, what you mean with "beta is computed from the covariance matrix"? Shouldn't be calculated applying linear regression between the historical returns of an asset and historical returns/changes of a factor (market portfolio, macroeconomic factors etc.)? Jun 16 '19 at 10:43
• supporting the possibility to compute superior estimates of expected returns is not as important as being able to compute superior estimates of expected returns. Expected returns will likely remain difficult to estimate no matter how many financial models claim to be better.. nevermind about the beta from the covariance versus beta from linear regression. the first approach is just the analytical solution to the second since linear regression implicitly optimizes for the covariance making both approaches equivalent. Aug 4 '20 at 10:52