Folio Lab/Documentation

Maximum Diversification

Choueifaty & Coignard's "Most Diversified Portfolio" maximises the diversification ratio: the weighted average of individual volatilities divided by the portfolio volatility. The result is the long-only portfolio that extracts the most out of the correlation structure of the universe.

Overview

The diversification ratio (DR) is defined as: the numerator is the average asset volatility taken at the portfolio weights, the denominator is the portfolio volatility itself. For an asset that is perfectly correlated with the rest of the universe DR collapses to 1; for a portfolio of uncorrelated assets DR can grow as large as where is an effective number of bets.

Choueifaty and Coignard (2008) introduced the "Most Diversified Portfolio" (MDP) as the long-only weight vector that maximises DR. Choueifaty, Froidure and Reynier (2013) proved a number of attractive properties for the MDP: it is volatility-invariant (rescaling any asset's volatility leaves the optimum unchanged), it is the long-only portfolio whose constituents have the highest correlation to it, and on empirical grounds it tends to deliver high risk-adjusted returns relative to mean-variance and equal-weight baselines.

FolioLab implements MDP via skfolio's MaximumDiversification estimator. The optimisation is convex and solves quickly, and like all volatility-invariant methods it does not require expected-return inputs.

Mathematical Formulation

Notation

  • — vector of per-asset standard deviations
  • — covariance matrix
  • — long-only portfolio weights, summing to 1

Diversification ratio

The numerator is the weighted average of asset volatilities ignoring correlations; the denominator is the portfolio's realised volatility given correlations. Their ratio is exactly 1 when all assets are perfectly correlated and grows as correlations fall.

Optimisation problem

Although the raw objective is non-convex (a ratio of a linear and a square root quadratic), the homogeneity of the ratio in allows a standard reformulation. Define ; then maximising DR is equivalent to the convex quadratic programme subject to and ; recovered weights are . skfolio uses this convex reformulation under the hood.

Properties

Choueifaty, Froidure and Reynier (2013) prove the following: (a) the MDP is invariant under per-asset volatility rescaling, so DR depends only on the correlation matrix and the weight vector; (b) every constituent of the MDP has the same correlation to the MDP itself, equal to ; (c) the MDP is the long-only portfolio whose constituents are most equally spread along the principal-component directions of the correlation matrix.

For Indian equity universes with sectoral concentration (banking and IT dominate weight in cap-weighted indices), MDP tends to upweight cyclicals, consumer staples and small-cap names that have low correlation to the financial-services cluster. Practitioners often use it as a diversifier overlay on top of a benchmark-tracking allocation.

Advantages & Limitations

Advantages

  • Vol-invariant: Avoids the low-vol concentration of min-variance.
  • No return estimates: Robust to the hardest part of MVO.
  • Convex after reformulation: Solves quickly and deterministically.
  • Sound theoretical basis: All constituents have the same correlation to the portfolio.

Limitations

  • Correlation-noise sensitivity: Sample correlations are noisy in high-N regimes.
  • Ignores returns: Cannot tilt toward high-conviction names.
  • Concentration in vol-uncorrelated assets: A small subset may dominate.
  • Crisis fragility: Correlations spike toward 1 in stress, eroding the diversification premise.

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.
  • Maillard, S., Roncalli, T., & Teiletche, J. (2010). "The Properties of Equally Weighted Risk Contribution Portfolios." The Journal of Portfolio Management, 36(4), 60-70.
  • 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.MaximumDiversification.