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Explain why, as more assets are added, covariance becomes the main driver of portfolio risk

CFA® Level I · Portfolio Construction · Portfolio Risk and Return: Part I · PM.1.c

The idea

In a portfolio of n assets there are n variance terms and n(n-1) covariance terms, so as n grows the covariances overwhelm the variances in determining total risk. In an equally weighted portfolio, the contribution of each asset's own variance falls with 1/n toward zero, while the average covariance term persists. This is the formal reason diversification has a limit: idiosyncratic risk can be diversified away almost entirely, but the shared exposure captured by average covariance cannot be, however many securities are added.

Formulas

Variance of an equally weighted portfolio

portfolio variance = (1 ÷ n) × average variance + (1 − 1 ÷ n) × average covariance

n
the number of assets, equally weighted
average variance
the mean of the individual assets' variances
average covariance
the mean covariance across every pair

As n grows the first term vanishes and the second does not. That limit is the reason diversification stops helping: what is left is the covariance the assets share, which is systematic risk and cannot be diversified away.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the variance of a portfolio of n assets, how many covariance terms appear in the double summation, where each ordered pair of different assets is counted separately?

  1. nWhy not: That is the count of variance terms, one per asset along the diagonal of the matrix.
  2. n(n-1) - correct
  3. 2nWhy not: Grows linearly with the number of assets, whereas the pairings grow roughly with its square - which is why covariance comes to dominate.

Portfolio variance for n assets adds up every pair of assets. The variance terms are the asset paired with itself, so there are n of them. Covariance terms are pairs of different assets. With n assets, each can be paired with n - 1 others. Formula: number of covariance terms = n x (n - 1), when each ordered pair is counted separately. Example: with 4 assets there are 4 variance terms but 4 x 3 = 12 covariance terms. Covariances grow roughly with n squared, so they come to dominate.

A harder one

Daniel Okafor, a portfolio analyst, assumes every stock in an equally weighted portfolio has the same variance and every pair of stocks has the same positive covariance. He compares portfolios of 10, 50 and 500 stocks. The standard deviation falls sharply from 10 to 50 stocks but hardly changes from 50 to 500. Which explanation is correct?

  1. Later stocks are more highly correlated with those already heldWhy not: Nothing in the assumptions says correlations rise as stocks are added; every pair has the same covariance. The flattening comes from the structure of the variance formula, not from changing correlations.
  2. Covariance terms shrink toward zero as the weights get smallerWhy not: Each weight does shrink, but the number of covariance terms, n(n-1), grows faster. In total the covariance terms approach the average covariance rather than zero.
  3. Own-variance terms fade toward zero but average covariance remains - correct

In an equally weighted portfolio of n stocks, each weight is 1/n. Portfolio variance = (Average variance / n) + ((n - 1) / n) x Average covariance. The first part, from each stock's own risk, shrinks towards zero as n grows. The second part approaches the average covariance and stays. So the first stocks added remove most of the stock-specific risk, while later ones add almost nothing. What remains is the shared, market-wide risk. Correlations do not need to rise, and the covariance terms do not vanish.

The hardest tier

In an equally weighted portfolio of identical assets, what does portfolio variance approach as n becomes very large?

  1. ZeroWhy not: That would require zero covariance between every pair. Shared exposures keep the average covariance positive.
  2. The average variance of the individual assetsWhy not: The own-variance contribution falls away in proportion to one over n; it is the covariance that remains.
  3. The average covariance between the assets - correct

Formula: Variance = (1/n) x average variance + ((n-1)/n) x average covariance. As n becomes very large, 1/n goes to zero, so the first term disappears. The fraction (n-1)/n goes to 1, so the second term becomes the average covariance. So portfolio variance approaches the average covariance between the assets. This leftover is systematic risk, the shared risk that cannot be diversified away. It is not zero, because the assets share some common exposure that keeps the average covariance above zero.

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