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

CFA® Level I · Portfolio Management · 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 a portfolio of n assets, how many covariance terms appear in the variance calculation?

  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.

Each ordered pair of distinct assets contributes a term. The count grows roughly with the square of n while the variance terms grow only linearly.

A harder one

Why does adding more securities eventually stop reducing portfolio risk?

  1. Because the average covariance remains — correct
  2. Because transaction costs offset the benefitWhy not: Costs make marginal diversification uneconomic but do not stop the risk falling. The floor is there even with free trading.
  3. Because correlations rise as portfolios growWhy not: Adding names does not change how the existing ones move together. What changes is the weight of variance against covariance.

Each asset's own variance contribution falls toward zero, but what the securities have in common does not. The irreducible floor is set by that shared exposure, since only the idiosyncratic part is diversifiable.

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

The variance terms are scaled by 1/n and vanish; the covariance terms are scaled by (n-1)/n and converge to the average covariance. That limit is systematic risk.

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