Let's say we have a predictive distribution of expected returns for N assets. The distribution is not normal. We can interpret the dispersion in the distribution as reflection of our uncertainty (or estimation risk) in the expected returns.
Question - What is the set of optimal portfolio weights that is mean-variance efficient with some constraint for the predictive distribution? Of course, we do not know what possible realization from the predictive distribution Nature will select.
I will suggest some angles of attack and show their various flaws. This is a very real and challenging problem -- I'm hoping the community can crack this!
Approach 1 (Monte Carlo) : Sample from the multivariate predictive distribution. Find the weights corresponding to the mean-variance efficient portfolio for each sample. Repeat the procedure for 1,000 draws from the predictive distribution. Average the weights via some procedure (for example, average of the rank-associated mean variance portfolios per Michaud 1998).
The re-sampling approach has several major flaws as discussed by Scherer: if short-sales are allowed this will add noise to estimation; procedure will include mean-variance dominated assets in the presence of a long-only constraint; the shape of the efficient frontier excludes the feasible maximum return option which is not theoretically plausible; re-sampled allocations can produce portfolios that violate investment constraints; and so on.
Approach 2 (Black-Litterman) : This knee-jerk answer doesn't work since the posterior is already provided here.
Approach 3 (Mean of Posterior): Sample from the predictive distribution N times. Collapse the distribution of expected returns into a single vector corresponding to each security's mean return.
The flaw here is that estimation risk is assumed away. This is a robust optimization problem where the goal is to identify weights that are approximately optimal under various realizations of market returns.
What might work?
Approach 4 (Complex objective function). Perhaps we could extend the objective function by summing over the mean-variance objective for each draw from the predictive distribution.
The mean-variance objective function for a single draw is: weights*expected return vector + wEw, where the expected return vector is a sampling from the predictive distribution, and E is the sample covariance matrix, and w is the vector we are solving for.
For multiple draws, the objective function = (w*expected return draw 1 + wEw) + (w*expected return draw 2 + wEw) + ... (w*expected return draw n + wEw)
This would be a lengthy objective function that is computationally intensive assuming 500 assets and 1,000 draws. Maybe some fast genetic algorithms could work in parallel. I also have not seen the literature discuss this approach although intuitively it seems sound.
Please poke holes and let me know if there is another approach.