Importance of Correlation in a Portfolio of Many Assets
You've seen the calculation for the variance of a portfolio of risky assets both with explicit covariance terms:
$$\displaystyle \sigma^2_P = \displaystyle \sum_{i=1}^N w^2_i \sigma^2_i + \sum_{i,j=1, i \ne j}^N w_i w_j Cov_{ij}$$
...and without:
$$\displaystyle \sigma^2_P = \displaystyle \sum_{i,j=1}^N w_i w_j Cov_{ij}$$
Based on this general formulation for the variance of a portfolio of N securities, consider what happens if Corey decided to add more and more securities, taking N to a very large number. What does the fraction containing the individual asset variances approach?
Exactly! This is the key thing to remember: the coefficient on the individual asset variances becomes close to zero, so that Corey doesn't have to worry about the effect of these individual asset variances on the portfolio variance.
No, consider the fraction 1/N with N getting very large.
Now consider the second part of the equation: the covariances (or equivalently, the correlation multiplied by the individual standard deviations, which are assumed right now to be identical). As Corey chooses a higher N, this fractional coefficient approaches what value?
No, try plugging in larger and larger numbers in the fraction (N-1)/N.
To summarize this discussion:
[[summary]]
Now suppose Corey's 100 stocks each had the same weighting in the portfolio, the same variance, and each pair of stocks at the same correlation. In this case, the portfolio variance wouldn't need to be indexed, and could be simplified to:
$$\displaystyle \sigma^2_P = \frac{\sigma^2}{100} + \frac{99}{100} \rho \sigma^2$$
Or more generally:
$$\displaystyle \sigma^2_P = \frac{\sigma^2}{N} + \frac{N-1}{N} \rho \sigma^2$$
__Note:__ Recall that covariance can be expressed as correlation multiplied by the standard deviation of each asset. That substitution is used here.
Corey inherited two stocks from his grandmother, then he added a third. He has been spending more time reading and learning about individual investments, and now is up to 100 stocks in his portfolio. In watching his portfolio's performance, Corey has started to wonder if individual asset variances are even something to be concerned with anymore.
His intuition is sound. As more and more stocks are added to a portfolio, individual asset variances become less and less important. The correlations are key.
Precisely! Now not every correlation is the same in a portfolio, but as the portfolio gets sufficiently large, the individual variances move toward having zero impact, while the correlations' impact remains.
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