Time to move on from mean-variance diversification

Diversification has been called the oldest trick in the investment book. So old that to construct a diversified portfolio, it is still common to apply mean-variance optimisation, a 70-year-old technique that outputs the portfolio with the least volatility for a given expected return.

Mean-variance optimisation is widely considered an outdated approach to diversification. This is because it aims to minimise volatility – which is not the same as diversifying.

“It’s been noticed that using mean-variance optimisation results in portfolios that invest in only a few securities,” says Gianluca Fusai, professor of finance at Cass Business School in London and Università del Piemonte Orientale in Italy.

But a portfolio containing only a few securities with low volatility is unlikely to be well-diversified.   

Fusai, together with Domenico Mignacca at the Qatar Investment Authority and Andrea Nardon and Ben Human of Sarasin & Partners, propose a different solution. “We developed a decomposition of portfolio risk into volatility risk plus a diversification component. Essentially, we isolate diversified and undiversified risk,” he explains.  

Based on this decomposition, they created the quantitative diversification index (QDX), which measures the contribution of each asset to portfolio diversification.

“Our simple idea is to measure the contribution to diversification of each individual security using the variance of the idiosyncratic risk,” Fusai says. “A well-diversified portfolio should have the idiosyncratic risk equally distributed among its components. To do that, we minimise the dispersion of the contribution to diversification of each security.”

The resulting equally diversified portfolio has a higher Sharpe ratio than the S&P 500 and equally weighted indexes over the 25-year period tested by the authors. Crucially, it outperformed not only US but also European and Asian indexes in drawdown periods – the ultimate measure of diversification. To test this, the authors compared the performance of the equally diversified portfolio to the indexes when market returns were in the fifth and first percentiles – that is, the tail of the distribution. They found the equally diversified portfolio had higher average returns and lower average volatility in these periods.

The authors also tested whether their approach produces a similar concentration of exposures as mean-variance strategies. To do this, they used Meucci’s entropy measure, which calculates the effective number of bets in a portfolio. This measure takes into account the correlation between assets, counting, for example, two perfectly correlated assets as just one. They found that the equally diversified portfolio had a much larger number of bets than the other portfolios.

Our simple idea is to measure the contribution to diversification of each individual security using the variance of the idiosyncratic risk

Gianluca Fusai, Cass Business School

The decomposition of portfolio risk – on which the QDX is based – is obtained by running a regression of weighted returns of portfolio constituents against the portfolio returns. The explained part of the regression represents the undiversified volatility, while the residuals are used to construct the index.

The variances of each asset’s residuals, or partial variances, over the portfolio variance indicate the contribution to diversification of that asset. These individual contributions are summed together to calculate the portfolio’s QDX.

The covariances of each asset over the portfolio variance, or partial covariances, are also summed together to quantify the diversification factor, or DIV. They found that in the case of homogeneous risk measures, such as volatility, the DIV cancels out the QDX. This demonstrates that the diversifiable component of portfolio volatility can indeed to be diversified away.

The research provides further insights into the concept of diversification. First, it highlights the usefulness of focusing on the residuals of the regressions rather than the returns themselves, so as to isolate the diversifiable and undiversifiable parts of the portfolio. Second, it confirms that a security with high correlation to the portfolio does not contribute much to diversification. Conversely, a security with low correlation to the portfolio contributes more to diversification. Third, the mathematical framework presented by the authors makes it possible to construct portfolios with different levels of diversification for a given level of volatility.

Some quants argue that a simple and interpretable measure of diversification should be used when communicating with investors. But for money managers themselves, a rigorous mathematical definition with clear practical implications is most welcome.