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Variance, Covariance, and Correlation

Variance and covariance
Var(X)=E[(X−μ)2]Cov(X,Y)=E[(X−μX)(Y−μY)]\text{Var}(X) = E\left[(X - \mu)^2\right] \qquad \text{Cov}(X,Y) = E\left[(X-\mu_X)(Y-\mu_Y)\right]

Variance is the spread of one variable. Covariance is how two move together, and it is the term diversification acts on.

ρX,Y=Cov(X,Y)σXσY\rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_X\sigma_Y}

Variance measures a single asset's risk. Covariance measures how two move together, and correlation makes that comparable across assets by stripping out units.

Portfolio variance is where this pays off

σp2=∑iwi2σi2+∑i≠jwiwjρijσiσj\sigma_p^2 = \sum_i w_i^2\sigma_i^2 + \sum_{i \neq j} w_iw_j\rho_{ij}\sigma_i\sigma_j

Two terms: individual risks, and every pairwise interaction.

The second term dominates as the portfolio grows. With nn assets there are nn variance terms and n(n−1)n(n-1) covariance terms, so for a large portfolio the covariances are almost the whole answer.

This has a clean limiting consequence. For nn equally weighted assets with equal variance σ2\sigma^2 and equal pairwise correlation ρ\rho:

σp2=σ2n+n−1nρσ2  ⟶  ρσ2as n→∞\sigma_p^2 = \frac{\sigma^2}{n} + \frac{n-1}{n}\rho\sigma^2 \;\longrightarrow\; \rho\sigma^2 \quad \text{as } n \to \infty

Idiosyncratic risk diversifies away entirely. Correlated risk does not. No amount of diversification removes it, which is why systematic risk carries a premium and idiosyncratic risk does not.

The rate at which it arrives matters as much as the limit, and it is faster than most people expect.

Worked example: how quickly diversification pays

Take assets with σ=25%\sigma = 25\% each and pairwise correlation ρ=0.2\rho = 0.2, equally weighted. Feeding nn through the formula above:

Holdings Portfolio volatility
1 25.00%
5 15.00%
20 12.25%
100 11.40%
∞\infty 11.18%

Almost all of it is done early. Going from 1 name to 20 captures 92% of the total reduction available; the next eighty names buy less than a percentage point, and no number of names ever gets below 11.18%.

Two practical readings follow. A concentrated book of twenty uncorrelated-ish positions is already close to fully diversified, so the argument for holding hundreds is about capacity and turnover rather than about risk. And the residual floor is set by ρ\rho alone, so the only way through it is finding genuinely different exposures, not more of the same ones.

Key takeaway

Diversification eliminates the first term and leaves ρσ2\rho\sigma^2. That residual is systematic risk, and it is why holding a thousand stocks still leaves you exposed to the market.

The estimation problem

Correlation is estimated, and the estimate is both noisy and unstable.

Noisy: with nn assets there are n(n−1)2\frac{n(n-1)}{2} correlations to estimate, quickly exceeding the data available. See covariance matrices.

Unstable: correlations change with regime, and they rise toward 1 in a crisis. A portfolio constructed to be well diversified under normal correlations can be highly concentrated under crisis ones, and it discovers this at the worst moment.

The linearity limitation

Correlation measures linear dependence only. Two variables can be perfectly dependent with zero correlation, the standard example being Y=X2Y = X^2 for symmetric XX.

This matters directly for anything with option-like exposure, where the relationship to the market is quadratic by construction. A book that looks uncorrelated by this measure can still be one large move from a serious loss, and mutual information is the measure that would reveal it.

Tip

Correlation is a summary statistic for the ordinary case. Before relying on it, ask whether the relationship could be non-linear and whether the estimate would survive a stressed market.

Test your knowledge

A portfolio holds equally weighted assets, each with volatility \( \sigma = 20\% \) and pairwise correlation \( \rho = 0.3 \) with every other. As the number of assets grows without limit, what does the portfolio volatility approach, as a percentage? Round your answer to two decimal places.
A book of option positions shows near-zero correlation with the index. Why is that a weaker reassurance than it appears?
NextLaw of Large Numbers and Central Limit Theorem

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