The maximum diversification portfolio: an alternative to mean-variance optimisation

Mean-variance optimisation requires expected return estimates that are notoriously hard to forecast. The maximum diversification portfolio sidesteps that problem by maximising a different objective entirely—the ratio of weighted average asset volatility to portfolio volatility, a single number that summarises how much of the available diversification benefit a portfolio is capturing.

What the maximum diversification portfolio is

Choueifaty and Coignard (2008) introduced the diversification ratio as DR = (Σwᵢσᵢ) / σₚ, where wᵢ and σᵢ are the weight and volatility of each asset and σₚ is the portfolio volatility. The numerator is the weighted average of individual asset volatilities; the denominator is the realised portfolio volatility, which is lower than the weighted average to the extent that the assets are imperfectly correlated.

A portfolio with DR = 1 is a single-asset portfolio (or a portfolio of perfectly correlated assets); there is no diversification benefit to capture. A portfolio with DR = √N, where N is the number of independent return drivers in the universe, is the theoretical maximum—every available diversification benefit has been captured. Real-world DR values for diversified multi-asset portfolios typically lie in the 1.3–2.0 range; values above 2.0 indicate either a large universe of low-correlation assets or, more commonly, sample-period correlation estimates that overstate the true diversification benefit.

The maximum diversification portfolio is the long-only portfolio with the highest possible diversification ratio given the asset universe and the covariance estimate. Choueifaty and Coignard show that this portfolio has tangency-portfolio-like properties under specific assumptions: if expected returns are proportional to volatility (a Sharpe-ratio-equivalent condition), the maximum diversification portfolio is the maximum-Sharpe portfolio.

How it works

Maximising the diversification ratio is a quadratic optimisation problem with linear constraints—structurally similar to mean-variance optimisation, but with the objective replaced. The inputs are the covariance matrix and the individual asset volatilities; expected returns are not required.

The intuition behind the construction is straightforward. The methodology favours assets with high individual volatility (which contribute more to the numerator) and low correlation with the rest of the portfolio (which keeps the denominator small). For a multi-asset universe, this typically produces a portfolio that allocates meaningfully to less-correlated asset classes—commodities, alternatives, and currencies—relative to a cap-weighted or even an equal-weight portfolio, where the equity allocation tends to dominate.

Like minimum variance and risk parity, maximum diversification is a heuristic that produces a specific portfolio without requiring expected return estimates. The three approaches differ in their objective: minimum variance minimises portfolio volatility; risk parity equalises risk contributions; maximum diversification maximises the diversification ratio. Each has periods of relative outperformance.

What the evidence shows

Choueifaty and Coignard's original work showed that maximum diversification portfolios applied to broad equity universes outperformed cap-weighted equivalents on a risk-adjusted basis over the 1990–2008 backtest. Subsequent work has documented similar patterns in multi-asset settings: the methodology tends to produce portfolios with lower drawdowns and higher Sharpe ratios than cap-weighted alternatives over long horizons, with the cost being underperformance during regimes of strong, narrow market leadership.

Empirically, maximum diversification portfolios sit close to risk-parity and minimum variance portfolios in their characteristics: low drawdown, moderate volatility, exposure to the small-size and low-volatility factors as a structural property of the construction. The three approaches' returns are highly correlated in practice, even though they optimise different objectives.

The methodology has been packaged into ETFs by several providers, particularly TOBAM, the firm Choueifaty and Coignard founded to commercialise the approach. Adoption has been gradual; the methodology is more common in institutional contexts than in retail portfolios.

Limitations and trade-offs

Maximum diversification depends on the covariance matrix, which is subject to estimation error. As with all covariance-based methods, the result is sensitive to the look-back window and the estimation technique. Shrinkage estimators and other robust techniques can mitigate this, but the underlying noise problem remains.

The methodology is silent on expected returns, which is its strength but also its weakness. An investor who has a genuine view on which asset class is likely to outperform cannot express that view directly through the maximum diversification framework. The methodology assumes the investor is choosing among assets without strong directional views, and it produces a portfolio consistent with that assumption.

Like risk parity and minimum variance, maximum diversification can produce concentrated portfolios in some asset universes—particularly universes where one or two assets have unusually low correlation with the rest. The optimiser will load up on those assets to push the diversification ratio higher, which can produce a portfolio that is, paradoxically, less diversified in the colloquial sense than a more equal-weighted alternative.

Maximum diversification in pfolio

Maximum diversification optimisation is not currently a built-in option in pfolio. The platform offers three optimisation methods: mean-variance optimisation (default), Hierarchical Risk Parity, and equal weight. HRP, in particular, is an alternative covariance-aware approach that is robust to estimation error, and is the closest practical analogue available in the platform.

Related articles

• Mean-variance optimisation: the algorithm behind optimal portfolio construction
• Hierarchical Risk Parity (HRP): a robust alternative to mean-variance optimisation
• The minimum variance portfolio: construction, trade-offs, and when it outperforms
• Portfolio diversification: why spreading risk across asset classes beats spreading across stocks