Measures of Portfolio Diversification and Concentration
Summary
The document compares several ways to describe how diversified a portfolio is. The Diversification Ratio relates the weighted average volatility of holdings to total portfolio volatility; for a long-only portfolio it is at least one, reaching one for a single holding. Implied correlation expresses the portfolio’s aggregate risk as the common correlation that would reconcile component risks with portfolio risk. The Herfindahl-Hirschman Index instead uses asset weights directly, with equal weights indicating less concentration than a single-asset allocation.
The discussion notes that correlation or volatility-based measures capture how holdings interact, while weight-based concentration measures capture allocation spread. It also raises downside co-risk measures as alternatives when the goal is protection from tail events. These measures answer different questions, so the choice depends on whether diversification means lower combined volatility, less concentrated weights, or reduced exposure to joint losses. The document offers definitions and conceptual explanations, but no empirical comparison or guidance on which measure performs best in a particular strategy.
Key ideas
- The Diversification Ratio compares weighted component volatility with portfolio volatility.
- Implied correlation summarizes the common correlation consistent with component and portfolio risk levels.
- The Herfindahl-Hirschman Index measures concentration from portfolio weights alone.
- Equal weighting minimizes the stated weight concentration index for a fixed number of assets.
- Downside co-risk measures may be more relevant when diversification is intended to address tail losses.
Tags
Full text
# What can I use to measure of diversification?
# 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?
## Answer by Alex C (score 4, accepted)
https://quant.stackexchange.com/a/21700
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
## Answer by Kiwiakos (score 5)
https://quant.stackexchange.com/a/21702
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.
## Answer by dnl (score 4)
https://quant.stackexchange.com/a/21762
You can also use the Herfindahl-Hirschman-Index (HHI) as a measure for concentration.
In portfolio analysis, you can calculate it as $$\frac{1}{N} \leq HHI(x) = \sum_{i=1}^N x_i^2 \leq 1$$ where $x$ is a vector of $N$ portfolio asset weights.
One can easily see that $HHI(x) = 1$ if 100% is invested in a single asset, and $HHI(x) = 1/N$ if the portfolio is perfectly diversified (equally-weighted portfolio).
In contrast to Diversification Ratio or Diversification Index, the HHI works directly on portfolio weights.
The Herfindahl-Index can be normalized between 0 and 1 by $$NHHI(x) = \frac{N \times HHI(x) - 1}{N-1}$$
## Answer by Dr. Christian Zimmer (score 1)
https://quant.stackexchange.com/a/21715
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.
## Answer by Kritz (score 0)
https://quant.stackexchange.com/a/21698
The correlation between the assets in the portfolio will give you a measure of the diversification in the portfolio.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.