Portfolio & Asset Pricingintermediate

Portfolio Variance

Portfolio risk is not the weighted average of the individual risks because of how assets co-move. For two assets the portfolio variance adds each weighted variance plus a cross term driven by their covariance. Extending to many assets gives the variance-covariance form, a double sum over every pair of holdings. As the number of assets grows, the covariance terms come to dominate, which is the algebraic reason individual variances matter less and co-movement matters more. This formula is the engine behind both diversification and mean-variance optimisation.

Try it yourself

The efficient frontier

Sweep the weight w on asset A to trace the frontier of risk and return. A lower correlation ρ bows it toward the return axis, the diversification benefit. The minimum-variance portfolio is its leftmost point.

Current portfolio (w_A = 0.50)11.0% · σ 13.5%

MVP risk σ 11.3%MVP weight w* 0.83

Weight on A, w0.50

Correlation ρ0.20

E(R) of A8.0%

σ of A12.0%

E(R) of B14.0%

σ of B22.0%

Risk is below the weighted-average volatility whenever ρ < 1. The frontier bends most as ρ falls toward −1, where the right mix drives risk close to zero. The MVP holds w* = 0.83 in A.

Try this. Set ρ = +1 — the frontier collapses to a straight line, so there is no diversification benefit. Now set ρ = −1 — it bends sharply and the right mix can push risk almost to zero.

Why it matters

If you naively averaged the volatilities you would always overstate portfolio risk, because some assets zig while others zag. The cross term is where that cancellation lives, and its sign is set by the covariance. In a large portfolio there are far more pairwise covariance terms than variance terms, so the average covariance, not the average variance, sets the risk floor. That is the precise sense in which a diversified portfolio inherits the market-wide co-movement of its members and sheds their individual quirks.

Formulas

Two-asset portfolio variance

\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2\,w_1 w_2\,\rho_{12}\,\sigma_1\sigma_2

Two weighted variances plus a covariance cross term, where

w_1+w_2=1

. The cross term is negative when

\rho_{12}<0

, lowering total risk.

N-asset variance-covariance form

\sigma_p^2 = \sum_{i=1}^{N}\sum_{j=1}^{N} w_i\,w_j\,\sigma_{ij} = \mathbf{w}^{\mathsf{T}}\,\boldsymbol{\Sigma}\,\mathbf{w}

A double sum over all pairs, compactly

\mathbf{w}^{\mathsf{T}}\boldsymbol{\Sigma}\mathbf{w}

with covariance matrix

\boldsymbol{\Sigma}

. Diagonal terms are variances, off-diagonal terms covariances.

Worked examples

Scenario

Asset 1 has $\sigma_1=20\%$ , asset 2 has $\sigma_2=30\%$ , correlation is 0.2, weights are 0.6 and 0.4. Find the portfolio standard deviation.

Solution

Compute $\sigma_p^2=0.6^2(0.2^2)+0.4^2(0.3^2)+2(0.6)(0.4)(0.2)(0.2)(0.3)$ . The three parts are 0.0144, 0.0144, and 0.00576, summing to $\sigma_p^2=0.03456$ , so $\sigma_p=\sqrt{0.03456}\approx 18.6\%$ . Notice the portfolio standard deviation sits below either asset on its own, under even the 20 percent of asset 1, because the low correlation cancels part of the risk.

Scenario

Why does the N-asset variance involve $N$ variance terms but $N(N-1)$ covariance terms?

Solution

The double sum has $N^2$ entries. The $N$ diagonal entries are variances and the remaining $N^2-N=N(N-1)$ off-diagonal entries are covariances. For large $N$ the covariance terms vastly outnumber the variance terms, so average covariance dominates portfolio risk, which is the formal basis for diversification.

Common mistakes

✗Portfolio variance is the weighted average of the individual variances. That ignores the covariance cross terms, which is exactly the part that makes diversification work, so the naive average overstates risk.
✗Portfolio standard deviation is the weighted average of the individual standard deviations. This equality holds only when every pair is perfectly positively correlated, otherwise the true figure is lower.
✗Covariance terms can be ignored when there are many assets. The opposite is true, since in a large portfolio the pairwise covariances outnumber the variances and come to dominate total risk.
✗Adding a volatile asset always raises portfolio variance. If that asset has low or negative covariance with the rest, it can lower portfolio variance despite its own high standard deviation.

Revision bullets

•Two-asset variance adds two weighted variances and a covariance cross term
•Risk is below the weighted-average volatility unless correlation is plus one
•N-asset risk is the double sum, compactly the quadratic form in the covariance matrix
•Diagonal entries are variances, off-diagonal entries are covariances
•For large N the covariance terms dominate, the algebra behind diversification

Quick check

In the two-asset variance formula, the cross term lowers portfolio risk when

In an N-asset portfolio, why do covariances matter more than individual variances as N grows?

Connected topics

Return & Risk Diversification Mean-Variance Efficient Frontier Asset Allocation

Sources

Brailsford, Heaney & Bilson (2015), Ch. 7
Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
Derives the two-asset and N-asset portfolio variance and the variance-covariance matrix.
Bodie, Kane & Marcus (2021), Ch. 7
Bodie, Z., Kane, A., & Marcus, A. J. Investments. 12th ed. McGraw-Hill Education, 2021.
Reference derivation of portfolio variance and the dominance of covariance terms.

How to cite this page

Dr. Phil's Quant Lab. (2026). Portfolio Variance. Derivatives Atlas. https://phucnguyenvan.com/concept/im-portfolio-variance

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