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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?

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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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