# What can I use to measure of diversification?

I have to come up with a measure of diversification for trade (this can tie in closely to diversification as regards portfolios).

Are there any well known measures of portfolio diversification?

• It's note clear to me what you mean by "diversification for trade". What do you mean by "trade"? It's certainly just a question of terminology, but to enhance the quality of the question, could you please add an example? – SRKX Nov 12 '15 at 1:23

In 2006 Choueifaty proposed a measure of portfolio diversification, called the Diversification Ratio (DR), which he defined as the ratio of the weighted average of the volatilities of the assets in the portfolio, to the portfolios overall volatility. The DR of a long only portfolio is greater than or equal to one, and equals unity for a single asset portfolio. In essence, the DR of a portfolio measures the diversification gained from holding assets that are not perfectly correlated.

Source: Choueifaty et al. : Properties of the most diversified portfolio, 2011 link

More details in Choueifaty et al. Towards Maximum Diversification, JPM 2008 link

• Im not sure how this can be used in trade though. I'll have to think about it. I understand the numerator, but what is the role of the denominator here? The numerator by itself will make sense right? – ChinG Nov 11 '15 at 20:58
• Do you mean using this in international trade? The concentration of exports for a country is usually measured with the Herfindahl Index, wich is basically the sum of squares of weights. – Alex C Nov 11 '15 at 23:59
• Could you please be more specific to the article you're referring to? adding title and a link maybe? You can even show the formula here. – SRKX Nov 12 '15 at 1:28

I use the 'implied correlation' defined as $$\rho = \frac{V^2_P-\sum V^2_j}{(\sum V_j)^2-\sum V^2_j}$$ for $V_p$ the VaR (or volatility) of the portfolio, and $V_j$ the VaRs (or volatilities) of the individual components.

Essentially it shows what would be the common correlation that I would need to use in order to aggregate the stand-alone risks to the risk of the portfolio.

• VaR usually stands for Value-At-Risk, I believe the most appropriate and natural convention is $\sigma_j$ to stand for the standard deviation of asset $j$. – SRKX Nov 12 '15 at 1:04

You can also use the Herfindahl-Hirschman-Index (HCI) as a measure for concentration.

In portfolio analysis, you can calculate it as $\frac{1}{N} \leq HCI(x) = \sum_{i=1}^N x_i^2 \leq 1$ where $x$ is a vector of $N$ portfolio asset weights. One can easily see that $HCI(x) = 1$ if 100% is invested in a single asset, and $HCI(x) = 1/N$ if the portfolio is perfectly diversified (equally-weighted portfolio).

In contrast to Diversification Ratio or Diversification Index, the HCI works directly on portfolio weights.

The Herfindahl-Index can be normalized between 0 and 1 by $NHCI(x) = \frac{N \times HCI(x) - 1}{N-1}$.

Alex C's and Kiwiakos' answers are definitely the most realistic approaches. If you are open to consider also other kinds of risk measures, further alternatives might be thought of. Variance / correlation based approaches interprete "diversification" as how much your assets are heterogeneous from the point of view of deviations from the historical mean. In case that you want to protect your portfolio against events in the tail, you might erhaps be interested in approaches like "co-downside risk" etc. The literature on risk measures is vast, many alternatives to covariance exist. The oint would be to apply the implicity trick from the implied correlation.

The correlation between the assets in the portfolio will give you a measure of the diversification in the portfolio.

• The "correlation between the assets in the portfolio" is a matrix with n(n-1)/2 entries, not a single number. – Alex C Nov 11 '15 at 20:09
• Exactly..correlation is pairwise – ChinG Nov 11 '15 at 20:56
• Yeah although it's not specifically mentioned in the question but kind of obvious. – SRKX Nov 12 '15 at 1:22