My question: How can I compare the Resampled Frontier (REF) to the standard MVO frontier when I have been provided with $\mu$, $\Omega$, and don't have access to true future data to test real out of sample performance (i.e. these parameters are "current").

I'm looking for some way to determine which one is superior (based on whatever measure(s) of "superiority" is(are) used in the literature).

  • $\begingroup$ If you want to just compare the frontiers, why not just plot both of them? $\endgroup$
    – John
    Sep 28, 2012 at 14:47
  • $\begingroup$ @John Good point. See my edit. I'm looking for determining the superiority of one over the other. Open to a very broad range of possible definitions of "superiority". I have already shown how the weights are much more stable to estimation error under resampling and that the REF is much more diversified at every point, so I'm looking for something other than this. $\endgroup$
    – user2921
    Sep 28, 2012 at 15:46

2 Answers 2


Bernd Scherer has done exactly this test in his text "Portfolio Construction and Risk Budgeting 4th Edition". There is an SSRN paper by Scherer called "Resampled Efficiency and Portfolio Choice (2004)" you can take a look at as well.

I would suggest you skip re-sampling (especially if you have a long-only portfolio) and take a look at Meucci's Robot Bayesian Allocation.

In summary there are a couple pitfalls:

  • there is no theoretical basis for re-sampling. It looks a Bayesian bootstrap procedure but upon closer inspection it is not. For example, uncertainty about the the mean shows up a shortening of the maximum expected return (and hence the efficient frontier). This is not theoretically consistent with a Bayesian approach

  • The interaction of the averaging rule and the long-only constraint results in higher weights to more volatile assets. So an asset with an inferior Sharpe ratio can have a higher weight. The reason why is that in a long-only portfolio assets are either in or out but never have a negative weight. The long-only rule creates an "optionality" for the more volatile security when averaging over long-only portfolios

  • The re-sampled efficient frontier can have upward bending components. This is also a serious theoretical problem

  • The re-sampling changes the structure of the maximum Sharpe ratio portfolio because of re-sampling tendency to weight more volatile assets. One way to see this is that in modern portfolio theory, the tangency portfolio will never contain cash. However, the re-sampled portfolio will always contain cash (as it sampled in some trials)

  • Biggest critique is that the there is no statistical foundation -- all re-sampling are derived from the same vector and covariance matrix. And since the true distribution is unknown all re-sampled protfolios suffer from the deviation in estimated return vector and covariance matrix in the same way. Averaging will not remove this bias and so all portfolio will inherit this estimation noise.

  • $\begingroup$ I really like that last point. $\endgroup$
    – John
    Sep 28, 2012 at 16:34
  • $\begingroup$ This is a great answer. Do you know of any references that deal test REF's advantages over MVO? This is purely focused on its weaknesses in general. $\endgroup$
    – user2921
    Sep 28, 2012 at 17:08
  • $\begingroup$ The best is probably Michaud's original text. Markowitz himself has some positive remarks: pionline.com/article/20031222/PRINTSUB/312220715 Of course, Markowitz's comments were before Scherer's systematic take-down so take it with a grain of salt. My view - there are better alternatives out there. $\endgroup$ Sep 28, 2012 at 18:01
  • 2
    $\begingroup$ not true quant guy. markowitz never paid any attention to scherer and the markowitz study was done after scherer. please see my comment below. $\endgroup$ Sep 28, 2012 at 19:07
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    $\begingroup$ @richardmichaud I'm not sure that Meucci being a fan (no matter how much I like his work) or there being a patent are good reasons to want to use it. The best evidence you have going for resampling is papers showing it is comparable to Bayesian portfolio construction. $\endgroup$
    – John
    Sep 28, 2012 at 20:25

There is a great deal of misinformation and out-of-date information on this site. Many of the references in this discussion and elsewhere have serious research flaws.

The Michaud efficient frontier was invented and patented by Robert Michaud and Richard Michaud, U.S. patent # 6,003,018. The alternatives discussed here are not patented nor in many cases refereed.

To answer the original question, Monte Carlo simulation tests are the standard method in modern statistics used to determine the superiority of one statistical procedure over another. Such a study was used to prove that Michaud optimization is superior to MVO. These are mathematical proofs. There is no doubt of the result. But if you need reassurance you can listen to the world’s greatest authority on portfolio optimization, Harry Markowitz, who has said in print: Michaud optimization beats Markowitz optimization.

The simulation tests for Michaud vs. MVO are published in Michaud (1998, Harvard, Chs. 6 and 9) and Michaud and Michaud (2008, Oxford Chs. 6 and 9). The tests were reproduced in Markowitz and Usmen (2003, Journal Of Investment Management). It is important to note that simulation tests are far superior to back tests. A back test proves absolutely nothing. It only tells you what happened during some time period. A different time period may have very different results.

The Markowitz-Usmen tests took for granted that the resampling process we introduced (note: this is not the Morningstar Encorr procedure), is superior to MVO. What Markowitz and Usmen wanted to prove is whether better information beats a better optimizer? They tested MVO and better information against Michaud with much less information. They performed 30 tests and found that their superior information never beat the superior optimizer.

People who have played with Michaud optimization have noted features about the resampled optimizer that are different from MVO. In almost all cases these anomalies are actually very consistent with real investor behavior once properly understood.

I recommend visiting www.newfrontieradvisors.com for much more information on the procedure, various tests and many extensions of the resampling procedure, rejoinders to flawed studies and ideas, and how extensively it is being used around the world in practice today. Many billions of dollars are being managed with Michaud optimization globally. Michaud optimization remains the only portfolio optimizer with a rigorous mathematical proof of investment effectiveness in the world today.


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