Maximum-Diversification Portfolio

When you combine assets that don't all move together, the volatility of the whole is less than the average volatility of the parts. That shrinkage is the entire benefit of Diversification, and the maximum-diversification portfolio (MDP) is the one that squeezes the most of it out. It does so by maximising a single number, the diversification ratio:

$$\text{DR}(w) = \frac{\sum_i w_i\,\sigma_i}{\sigma_p} = \frac{\text{weighted-average volatility of the holdings}}{\text{volatility of the portfolio}} .$$

The top is what your risk would be if the assets were perfectly correlated and offered no diversification at all — just the weighted average of each asset's own volatility $\sigma_i$. The bottom, $\sigma_p$, is the portfolio's actual volatility, which is lower whenever the assets are less than perfectly correlated. Their ratio measures how much diversification you captured. If everything moved in lockstep, DR would equal 1 (no benefit); the more the assets offset each other, the higher DR climbs. The MDP is the set of weights that pushes DR to its maximum.

Worked example

Three assets, each with volatility 20%, every pair correlated at $\rho = 0.3$. With equal vols and equal correlations, the most-diversified mix is equal weight, $1/3$ each. The weighted-average volatility is just 20%. The portfolio variance for $n$ equally weighted, equally correlated assets is

$$\sigma_p^2 = \sigma^2\!\left[\frac{1}{n} + \left(1 - \frac{1}{n}\right)\rho\right] = 0.04\!\left[\tfrac{1}{3} + \tfrac{2}{3}(0.3)\right] = 0.04(0.533) = 0.0213,$$

so $\sigma_p = 14.6\%$. The diversification ratio is

$$\text{DR} = \frac{20\%}{14.6\%} = 1.37 .$$

Diversification turned three 20%-vol assets into a 14.6%-vol portfolio — a 27% cut in risk for free, with no drop in the assets' own return potential. Notice the ceiling: with a shared correlation of 0.3, no reshuffling of weights beats 1.37 here. Push the correlations toward zero and the MDP would capture far more; push them toward 1 and DR collapses back to 1, with nothing left to diversify. When the assets have different volatilities, equal weight is no longer optimal — the MDP tilts toward the low-volatility, low-correlation names, the same instinct behind the Minimum-Variance Portfolio and Risk Parity.

The core property

The MDP has an elegant signature: at the optimum, every asset has the same correlation with the final portfolio. If one holding were more correlated with the book than another, you could shift weight away from it and improve diversification — so at the best mix, all holdings are equally "useful" as diversifiers. It's the correlation cousin of Risk Parity's equal-risk-contribution idea.

Where it misleads

• Correlation estimates are fragile. DR is built entirely from the covariance matrix, and correlations are notoriously unstable. A regime shift that lifts correlations toward 1 collapses your realised diversification ratio at the worst possible moment — the "diversification" evaporates in the crash you bought it for.
• Concentration in quiet corners. Maximising DR can pile weight into a cluster of low-vol, low-correlation assets that merely looked independent in-sample, much like a raw Minimum-Variance Portfolio. Constraints and shrinkage help.
• It ignores return. Like min-variance and risk parity, the MDP optimises a pure risk quantity. It implicitly assumes the assets are worth holding on a risk-adjusted basis; feed it a low-return, low-correlation asset and it will happily overweight junk.