Folio Lab/Documentation

Resampled Mean-Variance

Michaud's resampled efficient frontier — averages max-Sharpe weights across many bootstrapped samples of the historical data. The resulting portfolio is more stable and less sensitive to the specific sample at hand.

Overview

Mean-variance optimisation is unstable: tiny changes in the input sample produce large changes in optimal weights. Michaud (1998) proposed averaging the optimal portfolio across a large set of bootstrap resamples. The resulting Resampled Efficient Frontier is smoother and more diversified than any single MVO solution.

Folio Lab's implementation uses a block bootstrap that preserves short-horizon serial dependence (volatility clustering, autocorrelation) — critical for financial returns, which are not iid.

Algorithm

Step 1 — Block bootstrap

For each of bootstrap iterations, draw contiguous blocks of length with replacement to construct a synthetic return sample of length . The default uses bootstraps with trading days.

Step 2 — Optimise per sample

For each bootstrap, estimate (annualised mean) and (Ledoit–Wolf shrunk covariance), then solve the long-only max-Sharpe portfolio:

Step 3 — Average

The final portfolio is the equal-weight average of all successful bootstrap solutions, then renormalised:

Folio Lab requires at least 8 successful bootstraps to produce a result; otherwise the method raises so the failure is visible.

Advantages & Limitations

Advantages

  • Stability: Averages out single-sample idiosyncrasies.
  • Diversified: Naturally less concentrated than single-sample MVO.
  • Block bootstrap: Preserves serial dependence in returns.
  • Compatible: Works with any base optimiser (Sharpe, min-vol, etc.).

Limitations

  • Compute: Each bootstrap is a full optimisation.
  • Bias: Bootstrap inherits any structural bias in the original sample.
  • Block-size choice: Affects the dependence structure preserved.
  • Not coherent: The averaged portfolio is not necessarily on any single frontier.

References

  • Michaud, R. O. (1998). Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation. Harvard Business School Press.
  • Michaud, R. O., & Michaud, R. O. (2008). "Estimation Error and Portfolio Optimization: A Resampling Solution." Journal of Investment Management, 6(1), 8-28.
  • Politis, D. N., & Romano, J. P. (1994). "The stationary bootstrap." Journal of the American Statistical Association, 89(428), 1303-1313.