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.

  • $\begingroup$ Even though the predictive distribution is not normal, can it be modeled analytically or is it purely empirical? $\endgroup$ Aug 4, 2011 at 14:45
  • $\begingroup$ It is purely empirical and cannot be modeled analytically. $\endgroup$ Aug 4, 2011 at 15:03
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    $\begingroup$ 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? $\endgroup$ Aug 9, 2011 at 15:25
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    $\begingroup$ 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 $\endgroup$ Aug 9, 2011 at 20:07
  • $\begingroup$ 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. $\endgroup$ Aug 9, 2011 at 20:25

1 Answer 1


Tools from the field of stochastic optimization are best suited for these problems. In particular, attached is a paper on non-parametric density estimation for stochastic optimization that describes an algorithm if state variables can be associated with draws from the predictive distribution.

Here's another approach by Kuhn. These are all one-period solutions. Multi-period solutions would require dynamic programming and its solutions cannot be resolved within the expected lifespan of the universe. At least that bounds the problem space!

  • $\begingroup$ Did you mean stochastic programming? $\endgroup$
    – jub0bs
    Aug 9, 2013 at 9:45
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    $\begingroup$ could the broken links be updated, and does this answer still stand with advances made since? $\endgroup$
    – develarist
    Jul 17, 2020 at 11:11
  • $\begingroup$ @develarist Did you find the papers online? The link doesn't work $\endgroup$ Apr 8, 2022 at 13:59
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    $\begingroup$ I believe the Kuhn paper was Kuhn; Parpas; Rustem Bound-based decision rules in multistage stochastic programming, Kybernetica, Jan 2008 docplayer.net/… or dml.cz/bitstream/handle/10338.dmlcz/135840/… $\endgroup$
    – nbbo2
    Apr 10, 2022 at 20:05

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