Michaud resampled optimization: more stable portfolio weights through simulation

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Resampled optimization following Richard Michaud takes the weakness of classic mean-variance optimization seriously: its inputs are only estimates. Instead of optimizing once, it simulates many plausible datasets, calculates an efficient frontier for each and averages the weights. The result is more diversified and more stable portfolios. Here you will learn the procedure step by step, recalculate a small example and find out what research has found for and against the method.

The problem: optimizers love estimation errors

The Markowitz efficient frontier assumes that expected returns, volatilities and correlations are known. In practice, they are estimated from the past, and the optimizer treats these estimates as certainties. Michaud (1989) therefore called mean-variance optimizers error maximizers: they overweight precisely the assets whose return happened to be overestimated and whose risk was underestimated (error maximization).

The size of the estimation error is often underestimated. The standard deviation of an estimated mean falls only with the square root of the number of years. For two equity markets whose return difference fluctuates by just over 10% a year, the standard error of the mean difference is still more than 3 percentage points after ten years. A measured lead of one percentage point is then statistically barely distinguishable from zero. Yet the optimizer still bets everything on the winner. According to Chopra and Ziemba (1993), errors in returns weigh many times more heavily than errors in variances and covariances, and Best and Grauer (1991) show how small changes in a single expected return can knock half of the assets out of the optimum.

The idea: many plausible worlds instead of one

Resampling literally means drawing again. The estimate from the real data is treated as one of many histories that could just as well have happened. Richard and Robert Michaud describe the procedure in detail in their book (Michaud and Michaud 2008) and patented it in the US in 1999 (Michaud and Michaud 1999). The procedure:

1. Estimate the expected returns μ̂ and the covariance matrix Σ̂ from T monthly returns, as in classic optimization.
2. Simulate S new datasets of T months each, usually from a normal distribution with exactly these μ̂ and Σ̂ (Monte Carlo simulation). Alternatively, draw whole months of the real data with replacement (bootstrap, following Efron (1979)).
3. Re-estimate μ and Σ from each simulated dataset and calculate the efficient frontier under the same weight limits, at K points from the minimum variance portfolio (rank 1) to the portfolio with the highest return (rank K).
4. Average the weights for each rank across all S simulations. The averaged portfolios form the resampled efficient frontier.
5. Evaluate the return and risk of the averaged portfolios with the original μ̂ and Σ̂ and compare them with the classic frontier.

The rank links portfolios with a similar position on different frontiers: rank 1 is always the lowest-risk efficient portfolio of the respective simulation, rank K the one with the highest return. Michaud and Michaud market the procedure under their own name; we refer to it neutrally as Michaud resampling.

Worked example with two equity markets

Two equity markets, estimated from ten years of monthly returns: Europe with an expected return of 7% and 16% volatility, the US with 8% and 17%, correlation 0.8. Weights between 0 and 100%, no short sales. With two assets, the classic frontier can be calculated by hand: it starts at the minimum variance portfolio and ends at 100% in the asset with the higher estimated return; the ranks lie in between in equal weight steps.

For the resampling calculation, we draw 1,000 datasets of 120 months each from a normal distribution with these values (monthly mean and covariance = annual values / 12, fixed seed 2026), re-estimate each time and average the weights by rank:

In 64.0% of the simulated datasets, the US was ahead, close to the analytical 61.9%; the rest is Monte Carlo error. Exactly this share ends up in the highest-return rank: the resampled portfolio with the highest return holds 64.0% US and 36.0% Europe instead of 100% US. At the low-risk end, almost nothing changes, because the minimum variance portfolio needs no return estimate.

Properties of the resampled frontier

• More diversified: assets that win in only some of the simulations receive a medium weight instead of zero or everything. Corner solutions disappear.
• More stable: small changes in the inputs shift the weights only slightly, which reduces turnover when re-optimizing.
• Shorter and somewhat lower: the resampled frontier does not reach the single asset with the highest return. Evaluated with the original estimates, it lies on or below the classic frontier, because the classic one is by definition optimal for this estimate. With two assets, every mix between the minimum variance portfolio and a single asset lies on the classic frontier; with more assets, the resampled frontier usually lies slightly below it.
• Not an optimum in the strict sense: the averaged portfolio does not maximize any utility function. It is an average of solutions, not the solution to a decision problem under uncertainty.
• Dependent on the limits: a large part of the effect comes from the ban on short sales: in the example, every simulation jumps to 100% in one market at its end, and only the average mixes the two.

Advanced: what resampling corrects and what it does not

The center remains the estimate. All simulated datasets are scattered around μ̂ and Σ̂. If the estimate itself is off, the simulations are also off on average. Resampling captures the uncertainty around the estimate but does not correct any bias. Methods that shrink the returns themselves toward a common value start earlier: the Bayes-Stein estimator of Jorion (1986), equilibrium returns in the Black-Litterman model following Black and Litterman (1992) or shrinkage of the covariance matrix following Ledoit and Wolf (2004) (covariance estimation).

Bayes as the counter-model. A consistent Bayesian investor optimizes against the predictive distribution: the distribution of future returns, which already contains the uncertainty about the parameters. This is an optimization with a clear objective function. Resampling, by contrast, averages the solutions of many optimizations, each of which ignores the uncertainty. The difference is not merely philosophical: weights that are optimal under uncertainty need not equal the average of the weights that would be optimal if the parameters were known.

Adjustable settings. The length of the simulated datasets controls how strongly the results are averaged: shorter series mean more dispersion in the estimates and more even weights, while longer ones approach the classic solution. Matching by rank is a convention; you can also link portfolios by the same risk aversion or the same target volatility and get slightly different frontiers. The number of simulations determines only the Monte Carlo error, not the logic; it falls with the square root of the number of datasets and, for the share in the example with 1,000 datasets, is about 1.5 percentage points.

Weight limits have a similar effect. Jagannathan and Ma (2003) show that a short-sale ban works like shrinkage of the covariance matrix. Resampling and limits therefore reinforce each other: both take away the optimizer's ability to build extreme bets on estimation errors.

What the research shows

Scherer (2002) classifies the procedure as a well-thought-out heuristic whose peculiarities are easily misinterpreted by the inexperienced. Empirically, Fletcher and Hillier (2001) found slightly higher Sharpe ratios with resampling than with classic optimization for most return estimators in international asset allocation from 1983 to 2000, but hardly any evidence of significantly positive risk-adjusted excess returns for either variant.

A contest between resampling and Bayes is particularly well known. In an experiment by Markowitz and Usmen (2003), the resampling player did better than a Bayesian player with a diffuse prior. Harvey et al. (2008) replayed the game: in the original, the Bayesian player had calculated with a crude approximation of the distribution. With a standard procedure, the results were markedly different, and in a game with a one-period forecast, which is closer to the situation of a normal investor, the Bayesian player always won.

Becker et al. (2015) compare Markowitz and Michaud in a large simulation study with many estimation methods and an empirical test. On average, classic optimization actually performed better; more important than the choice between the two, however, were the estimator and the weight limits. With high estimation uncertainty, they recommend limits or Ledoit and Wolf's covariance estimation. And DeMiguel et al. (2009) generally urge modesty: among 14 optimization models, none reliably beat equal weighting out of sample.

Limits and common mistakes

• Resampling is not a forecast: it does not make poor return estimates good, only less dangerous.
• Smooth weights are not automatically right: an asset that loses in every simulation receives no weight even when resampled. If an important asset is missing from the universe, averaging cannot replace it either.
• In sample, it looks worse: evaluated with its own estimates, the resampled frontier never lies above the classic one. Its advantage only shows out of sample, if at all.
• Normal distribution: the usual simulation knows no fat tails and no crisis correlations. A bootstrap of whole months preserves them, but only repeats the observed history.
• Many settings to adjust: the length of the simulated series, the number of ranks and the type of matching influence the result. Anyone who keeps tweaking them until the backtest looks good ends up with backtest overfitting.
• Fair comparison: resampling should compete against simple alternatives such as equal weighting, minimum variance with limits and hierarchical risk parity, not just against unconstrained classic optimization.

How Portfolio Lab calculates it

Optimization gets resampling as a third method alongside classic optimization and Black-Litterman. As there, the estimate starts from the monthly returns of the grid, with sample or Ledoit-Wolf covariance. From this, the app simulates many datasets of the same length (from the normal distribution as in Michaud, or by bootstrapping whole months), re-estimates each one and calculates its efficient frontier under your minimum and maximum weights at fixed ranks, evenly spaced in expected return from the minimum variance portfolio to the portfolio with the highest return.

The weights for each rank are averaged and evaluated with the original estimates; the classic frontier at the same ranks is shown alongside for comparison, and the dispersion of the weights across the simulations shows how uncertain they are. The point selected is the one with the highest Sharpe ratio or the highest return up to a target volatility. A fixed seed makes the result reproducible, and as with the other methods, the procedure can be checked without hindsight in the walk-forward test. Details and defaults are in the optimization methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.