Portfolio Diversification Measures from Return Covariance and Weights

Summary

The document addresses how to quantify diversification for a portfolio with more than two securities. It presents portfolio return variance as a function of asset weights, individual return variances, and pairwise correlations. This captures how co-movement among holdings affects total portfolio risk and provides a covariance-based measure relevant to mean-variance analysis.

It also names two alternative measures: normalized variance, which scales portfolio variance by the average variance of the holdings, and a sum of squared deviations between portfolio weights and market-index weights. These measures reflect different notions of diversification, including risk relative to constituent volatility and deviation from a market allocation. The response does not compare their strengths, specify implementation details such as weight constraints, or recommend one universal score; the appropriate measure depends on what aspect of diversification is being assessed.

Key ideas

• Portfolio variance combines weighted asset variances with pairwise covariance contributions.
• Correlations among holdings help determine whether their risks offset or reinforce one another.
• Normalized portfolio variance scales total variance by the average variance of portfolio constituents.
• A squared weight-deviation measure compares portfolio allocations with market-index weights.
• Different measures capture different definitions of diversification and need not rank portfolios identically.

Tags

# How to score a portfolio's diversity based on security returns?


# How to score a portfolio's diversity based on security returns?












What is the best way to score a portfolio's diversity based on it's returns covariance matrix?

I know that if my portfolio has two securities and their returns' correlation coefficient is -1 that is a good diversified portfolio. Now I would like to know how do I score a portfolio with more than 2 securities.

If the theory is valid, should I calculate the correlation coefficient of N variables ? What's the formula for that?

## Answer by user1157 (score 4, accepted)

https://quant.stackexchange.com/a/10130

There are several measures discussed in the literature, the classical approach is Markowitz mean-variance portfolio optimization.

The formula for portfolio return variance is $$\sigma_p^2 = \sum_i w_i^2 \sigma_{i}^2 + \sum_i \sum_{j \neq i} w_i w_j \sigma_i \sigma_j \rho_{ij}$$ where $\rho_{ij}$ are the correlations betweent the assets.

Others suggeste measures are:

- Normalized portfolio variance (NV), which is obtained by dividing the portfolio variance by the average variance of stock returns in the portfolio: $$ NV = \frac{\sigma^2_p}{\bar{\sigma}^2}$$

- Sum of squared portfolio weights (SSPW), where $w_i$ is the portfolio weight assigned to stock $i$ in the portfolio and w_m is the portfolio weight assigned by the market (i.e. an index): $$\sum_N (w_i-w_m)^2 $$

For more references, see for example: Goetzmann and Kumar, Equity portfolio diversification, Review of Finance, 2008 Google will give you a lot of results, I found this Minimum Correlation Algorithm interesting.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.