Folio Lab/Documentation

Maximum Decorrelation

Construct the long-only portfolio that minimises the quadratic form of the correlation matrix. By replacing the covariance matrix with the correlation matrix, the method isolates pure diversification effects from individual volatility levels.

Overview

The classical minimum-variance portfolio minimises . When the asset universe contains both very low-volatility and very high-volatility names, the minimum-variance solution loads almost entirely on the low-vol assets, regardless of how correlated they are with one another. Maximum Decorrelation strips out the individual volatility effect and asks a different question: which long-only portfolio is, in correlation space, the most spread out?

The optimisation is identical in form to minimum-variance but uses the correlation matrix in place of the covariance matrix . The result is invariant under per-asset volatility rescaling: doubling one asset's volatility leaves the optimal weights unchanged. This makes the method a natural sibling of Maximum Diversification (Choueifaty and Coignard, 2008), which maximises the diversification ratio.

Maximum Decorrelation is most useful when the analyst genuinely cares about decorrelation and is happy to tolerate higher absolute volatility, or when the universe has been pre-filtered for vol (for example, the portfolio is constructed on standardised returns).

Mathematical Formulation

Notation

  • — sample covariance matrix
  • — diagonal matrix of asset standard deviations
  • — correlation matrix (unit diagonal)
  • — long-only portfolio weights

Optimisation problem

This is a convex quadratic programme. is positive semi-definite by construction, so the problem is well-posed. The solution is the long-only portfolio that minimises the average pairwise correlation under the simplex constraint.

Connection to Min-Variance and Max-Diversification

Writing in vol-scaled coordinates, the minimum-variance problem becomes on the rescaled simplex, while Maximum Decorrelation works directly in the original simplex with as the kernel. So Max Decorrelation is the "volatility-stripped" version of minimum-variance. Maximum Diversification (Choueifaty and Coignard, 2008) is algebraically related: it maximises , which is also invariant to rescaling of the volatility vector and which has a closed form in terms of .

Advantages & Limitations

Advantages

  • Vol-invariant: Rescaling one asset's volatility leaves the optimum unchanged.
  • Pure diversification: Captures correlation information without being dominated by low-vol assets.
  • Convex QP: Solves quickly and deterministically.
  • No return estimates: Sidesteps the hardest part of MVO.

Limitations

  • Ignores volatility: A high-vol but low-correlation asset can receive a large weight.
  • Correlation noise: Sample correlation matrices are noisy in high-N, low-T settings.
  • No expected-return view: Cannot tilt toward high-conviction names.
  • Sensitive to regime changes: A breakdown in correlations during crises can leave the portfolio with unexpected joint exposure.

References

  • Choueifaty, Y., & Coignard, Y. (2008). "Toward Maximum Diversification." The Journal of Portfolio Management, 35(1), 40-51.
  • Choueifaty, Y., Froidure, T., & Reynier, J. (2013). "Properties of the Most Diversified Portfolio." Journal of Investment Strategies, 2(2), 49-70.
  • Christoffersen, P., Errunza, V., Jacobs, K., & Langlois, H. (2012). "Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach." Review of Financial Studies, 25(12), 3711-3751.
  • Goetzmann, W. N., & Kumar, A. (2008). "Equity Portfolio Diversification." Review of Finance, 12(3), 433-463.
  • Palomar, D. P. (2025). Portfolio Optimization: Theory and Application. Cambridge University Press, Chapter 6 (Portfolio Basics), Section 6.5 (Risk-Based Portfolios).
  • skfolio documentation — skfolio.optimization.MaximumDecorrelation.