Folio Lab/Documentation

Maximum Diversification

Choueifaty & Coignard's portfolio that maximises the diversification ratio — the ratio of the weighted average volatility of the constituents to the volatility of the portfolio.

Overview

Maximum Diversification (MaxDiv) seeks the long-only portfolio whose volatility is as small as possible relative to the weighted average volatility of its components. Intuitively, the method picks the allocation that benefits most from the diversification effect of imperfectly correlated assets.

When the universe consists of assets with identical Sharpe ratios, the MaxDiv portfolio coincides with the maximum-Sharpe portfolio. In practice, MaxDiv tends to allocate to assets that contribute uncorrelated risk and to under-weight redundant exposures.

Diversification Ratio

Let be the weight vector, the vector of asset volatilities, and the covariance matrix. The diversification ratio is:

The optimisation problem is then:

Folio Lab solves this through skfolio's MaximumDiversification estimator, which reformulates the ratio maximisation as a convex problem and accepts long-only constraints.

Advantages & Limitations

Advantages

  • Captures correlation: Unlike inverse volatility, it uses the full covariance matrix.
  • Concentration-aware: Penalises portfolios that load on redundant exposures.
  • No return estimates: Robust to forecast error, since DR depends only on second moments.
  • Convex: A unique optimum is guaranteed under standard conditions.

Limitations

  • Sensitive to covariance estimation: Noisy can produce extreme weights.
  • Ignores returns: The portfolio may avoid high-return but correlated exposures.
  • Cluster sensitivity: A new asset closely correlated to an existing position can dramatically reshuffle weights.

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.
  • skfolio documentation — skfolio.optimization.MaximumDiversification.