# Portfolio optimization with monte carlo sampling from predictive distribution

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.

• Even though the predictive distribution is not normal, can it be modeled analytically or is it purely empirical? Aug 4 '11 at 14:45
• It is purely empirical and cannot be modeled analytically. Aug 4 '11 at 15:03
• I am not sure how exactly to solve your problem, but have you read Axioma's research paper on Real World Case Studies in Portfolio Construction Using Robust Optimization? Aug 9 '11 at 15:25
• This is an excellent paper, thank you. Another paper that solves for weights given some expected return vector and uncertainty dispersion is Robust Bayesian Allocation by Meucci: papers.ssrn.com/sol3/papers.cfm?abstract_id=681553 Aug 9 '11 at 20:07
• You should post whatever approach you ultimately go with as your answer. At least 7 people besides me also thought your question is very interesting. Aug 9 '11 at 20:25