Portfolio construction
Maximum diversification portfolio
The mix that gets the most out of correlation: the highest ratio of the tickers' average volatility to the portfolio's volatility.
The idea in plain words
The diversification ratio compares the risk you hold piece by piece with the risk of the whole. A ratio of 1 means no diversification at all; 2 means the portfolio is half as risky as its parts. The maximum diversification portfolio maximises that ratio, so it leans towards tickers that are uncorrelated with the rest.
How it is calculated
Maximise the weighted average volatility divided by the portfolio volatility, long only, weights adding up to 1. Mathematically it is the maximum Sharpe portfolio with each ticker's "return" replaced by its volatility.
DR(w) = (Σ wᵢ σᵢ) ÷ √(wᵀΣw)
maximise DR(w) subject to Σwᵢ = 1, wᵢ ≥ 0How to use it
- When you believe returns are roughly proportional to risk and want the most diversified way to hold it.
- Read the diversification ratio of any portfolio in the study to compare models on this score.
Drawing conclusions
- A higher ratio than ERC means the model found more correlation benefit.
- Tickers left at 0% are well represented by others.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). Weights 18% US500, 11% GER40, 28% gold, 44% EURUSD. Diversification ratio 1.51 (equal weights: 1.43). GER40 is cut most because US500 already carries its risk; gold gets the most of the volatile tickers because it is the least correlated.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- Sensitive to small changes in correlations when several tickers are alike.
- It does not look at returns; a diversified portfolio can still be a poor earner.
A little history
Yves Choueifaty and Yves Coignard, "Toward Maximum Diversification" (Journal of Portfolio Management, 2008); the basis of TOBAM's "anti-benchmark" funds.
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.