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Resampling is a popular method for portfolio optimization. We repeatedly draw samples from a distribution, compute the optimal mean-variance portfolio and finally average over all allocations.

However, from a mathematical point of view, I do not understand why we would gain anything from this procedure. Have we not put all of the uncertainty into our distribution already? Why would we distinguish between the risk of bad allocations and the risk of bad estimation?

Perhaps more rigorously: If the mean-variance portfolio maximizes the utility of an investor with a risk aversion of $\lambda$, then what does a resampled portfolio optimize?

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  • $\begingroup$ joim.com/wp-content/uploads/emember/downloads/p0048.pdf $\endgroup$
    – fes
    Commented Apr 24, 2022 at 15:10
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    $\begingroup$ It seems like a heuristic way to deal with parameter uncertainty that (apparently) works well in practice. Formally incorporating parameter uncertainty through a Bayesian treatment is tricky especially if you have many assets. See section 5 in the link above. $\endgroup$
    – fes
    Commented Apr 25, 2022 at 8:54
  • $\begingroup$ Thanks. That was an interesting read, but in terms of a theoretical justification I am still not convinced $\endgroup$
    – Claudio Moneo
    Commented Apr 26, 2022 at 14:24
  • $\begingroup$ I agree that there is not much of a theoretical justification. $\endgroup$
    – fes
    Commented Apr 26, 2022 at 14:34

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The "estimation problem" in Portfolio Optimization is a serious one. The parameters (returns and covariances) are known very imprecisely. For example the covariance between stocks and bonds for the next 10 years is going to be different from the one that we measure today using data from the past 10 years. And this is true even if the structure of the economy does not change, which is unlikely (think of the current inflation scare).

The uncertainty in the parameters is substantial and calls into question the whole procedure. According to Richard Michaud (personal communication) the resampling procedure initially was just an attempt to illustrate the issue: by solving the problem several time with randomly varied inputs and comparing the solutions one can get a sense of how far from optimal the allocations will be ex-post. The client can be shown a few alternatives rather than a single one, avoiding overconfidence in a single result.

In a second step Michaud realized that the proper approach to a problem where we do not know the parameters would be a Bayesian one, and he proposed modeling the uncertainty explicitly and using a Monte Carlo approach to find a compromisec solution. This is how resampling is understood today.

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    $\begingroup$ But wouldn't a Bayesian approach compute a posterior and then do mean-variance optimization w.r.t this posterior? $\endgroup$
    – Claudio Moneo
    Commented Apr 24, 2022 at 8:02
  • $\begingroup$ That is more or less what resampling is doing, but with discrete samples rather than with a continuous probability distribution. AFAIK we can only solve the Markowitz problem for specific input values, not with prob distr as inputs. Need to ask a Bayesian expert, which I am not. $\endgroup$
    – nbbo2
    Commented Apr 24, 2022 at 12:14
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    $\begingroup$ What I would expect from a Bayesian framework is optimizing w.r.t the posterior directly. Not sampling and combining the solutions. But nevertheless thanks for your answer $\endgroup$
    – Claudio Moneo
    Commented Apr 24, 2022 at 12:30
  • $\begingroup$ @ClaudioMoneo the CAPM, Black-Scholes, etc do not survive in a Bayesian framework for many reasons. The two biggest ones is that most Bayesian axiomatic systems do not allow countable additivity, and the integrals diverge. An assumption of the underlying calculus is that the parameters are known. Parameters are random variables in a Bayesian framework. Parameters are fixed points in the Frequentist one and models like the CAPM inherit the axioms that undergird Frequentist methods. Without them, you will not get the same answer. $\endgroup$
    – Dave Harris
    Commented Apr 27, 2022 at 4:02
  • $\begingroup$ @ClaudioMoneo also, models like the CAPM, etc violate the converse of the Dutch Book Theorem unless it is a true statement that the parameters are known. Because of that, the absence of arbitrage opportunities is excluded by theorem. $\endgroup$
    – Dave Harris
    Commented Apr 27, 2022 at 4:05
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True, Mean Variance gives you a mathematical solutions. But resampling, especially for Monte Carlo simulations allows you to specify any kind of distribution you want, and repeat for a large number of trials to see what an expected outcome would be. An important part of Monte Carlo simulation is computing a probability of success or achieving a specific outcome over a longer time frame. Both the mathematical and simulation methods are both useful. It's not that one is better than the other.

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    $\begingroup$ I don't really understand. Perhaps more rigorously: Classic mean variance maximizes some utility function. What do we maximize by averaging allocations each corresponding to one sample? $\endgroup$
    – Claudio Moneo
    Commented Apr 24, 2022 at 20:43
  • $\begingroup$ You can realize expected returns and variance in various ways. Sequence of returns are very important. Hypothetical return sequence (in percent) of -10,-10,-10,-10,-10,10 has the same mean and variance as 10,-10,-10,-10,-10. However they will have different outcomes. Monte Carlo simulation will account for that in terms of averaging. $\endgroup$
    – Ralph Winters
    Commented Apr 24, 2022 at 22:19

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