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What is the usual method of dealing with many uncertain mean returns in portfolio optimization?
For example say you have a 3 asset portfolio with assets A, B and C. All the correlations and variances are the same.
You can't reject the null that the assets have the same mean return for (A and B) and (B and C), but you can reject the null for (A and C).
It seems there is no nonbiased way to choose the means?
Your question accurately addresses of the practical problems of applying Modern Portfolio Theory (i.e. mean-variance optimization) in practice.
Generally the correlations are considered much more difficult to accurately estimate in this context.
I am not sure I understand the question. Isn't $E(r)$ is an unbiased estimator of $r$?
You may want to look at the Black-Litterman model. It is an optimization model developed to address the practical problems in applying portfolio optimization.
A simple way that Ray Dalio suggested is using Risk-Parity portfolio. Even though Modern Portfolio Theory with mean-variance optimization gives great framework for mathematically designing a tradeoff between alpha, risk and costs, some of the drawback of this framework is that one needs to come up with 1) correct alpha and 2) correct estimation of covariance matrix.
Assuming our estimate of covariance matrix is relatively stable, then the only problem that we'd need to deal with is Alpha.
You can have a look at risk-parity optimization where you can set 1) risk contribution of each asset is equal (equal risk contribution optimization) 2) marginal risk contribution of each asset is equal.
In these framework, it removes alpha from your utility function and let the optimizer to choose the optimal portfolio only using historical available data (historical realized volatility). This may not be optimal in realizing the best sharpe, however, it could be a good alternative to the approach that you are taking if you are unsure of your alpha estimation.
