Two assets: covariance, the minimum and the spread
The variance of a sum is the sum of the variances plus twice the covariance. Attach weights and every portfolio calculation is that identity, including the one where a weight is negative.
- Turn a correlation into a covariance, in %²
- Compute a two-asset portfolio's variance and volatility
- Find the long-only minimum-variance weight, and say when it is a corner
- Compute the volatility of a long-short spread and the correlation a portfolio's volatility implies
The intuition
Asset A has 20% volatility, B has 30%, and their correlation is 0.5. Covariance is correlation with units: 0.5 × 20 × 30 = 300 %². A 60/40 portfolio has variance 0.6² × 400 + 0.4² × 900 + 2 × 0.6 × 0.4 × 300 = 432 %², so 20.8% volatility: less than the 24% a weighted average of the two vols would give, because the cross term is smaller than it would be at a correlation of one.
Variance is a quadratic in the weight, so it has a minimum, and the weight that reaches it depends only on the two variances and the covariance, not on expected returns: here 86% in the calmer asset, for 19.6% volatility. Flip B's sign and the cross term flips with it. A $1 long A, $1 short B spread has variance 400 + 900 − 600 = 700 %², 26.5% volatility. Between two highly correlated assets a spread can be calmer than either leg; between uncorrelated ones it is noisier than both.
cov(A, B) = ρ σ_A σ_B. Var(w_A A + w_B B) = w_A² σ_A² + w_B² σ_B² + 2 w_A w_B cov. Minimum-variance weight in A: w* = (σ_B² − cov) ÷ (σ_A² + σ_B² − 2 cov). Var(A − B) = σ_A² + σ_B² − 2 cov. Implied ρ = (σ_p² − w_A² σ_A² − w_B² σ_B²) ÷ (2 w_A w_B σ_A σ_B).
Why it works
- The conventions here: two assets with annual volatilities and one correlation. Variances and covariances are in %², so a 20% volatility enters as 20 and 20 × 30 × 0.5 is 300. Long-only weights sum to one. The spread is $1 long A and $1 short B per $1 of capital.
- Covariance is correlation with units. ρ is unit-free, between −1 and 1. Multiply by both volatilities and you have the cross term the variance formula needs.
- Why the cross term matters. At ρ = 1 the portfolio's volatility is the weighted average of the two vols; nothing is diversified. Every point below one shrinks the cross term and the portfolio's variance with it.
- The minimum is a property of risk alone. Differentiate the variance in w_A and set it to zero: w* = (σ_B² − cov) ÷ (σ_A² + σ_B² − 2 cov). Expected returns never entered. For a long-only pair the optimum leans toward the calmer asset; the correlation decides by how much.
- When the optimum is a corner. If cov exceeds σ_B², that is ρ σ_A above σ_B, the unconstrained w* is negative: the formula wants to short A. The long-only answer is then all of the calmer asset. Above one, the same in reverse.
- Shorting flips the sign. Var(A − B) replaces +2cov with −2cov. A pair trade between two 0.9-correlated stocks can run at a fraction of either leg's volatility; with zero or negative correlation, shorting one leg adds risk instead of removing it.
| Covariance: 0.5 × 20 × 30 | 300 %² |
| Variance: 0.36 × 400 + 0.16 × 900 + 2 × 0.6 × 0.4 × 300 | 432 %² |
| Volatility: √432 | 20.78% |
| Minimum-variance weight in A: (900 − 300) ÷ (400 + 900 − 600) | 85.7% |
| Minimum volatility | 19.64% |
| Spread, $1 long A and $1 short B: √(400 + 900 − 600) | 26.46% |
At a correlation of 0.9 the spread's variance is 400 + 900 − 1,080 = 220 %², 14.8% volatility: below both legs. At −1 a mix of 60% A and 40% B has zero volatility.
The formulas
Correlation scaled up by both volatilities; in %² here.
Two own terms and a cross term.
The minimum-variance weight in A; no expected returns needed.
Shorting B flips the sign of the cross term.
The variance formula solved for the correlation.
Worked example
Two own terms, one cross term, add, square-root. The covariance is given in %², so keep everything in percent.
See it move
Same two assets. Change the weight in A, the correlation, and each asset's volatility.
- Raise the correlation. The covariance and the portfolio's volatility rise; the spread's volatility falls.
- Move the weight in A. Watch the portfolio's volatility trace the bowl: it falls toward the minimum-variance weight and rises past it.
- Raise one asset's volatility. Watch: the portfolio usually gets riskier, but with a negative correlation a little more of a riskier asset can lower the total, and the spread can get calmer when the riskier leg is the one it is long.
Run it backwards
A mix of the two assets ran at a known volatility. What correlation does that imply?
Square the portfolio volatility, subtract the two own-variance terms, and divide what is left by 2 w_A w_B σ_A σ_B.
A result above one or below minus one means the inputs are inconsistent, usually volatilities and a portfolio figure measured over different windows.
Traps
Say it in the interview
“Two stocks at 20% and 30% vol, correlation 0.5. How risky is a 60/40 mix, and a pair trade?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- cov = ρ σ_A σ_B, in %². Variance = own terms + 2 w_A w_B cov.
- The minimum-variance weight needs only variances and covariance; outside 0–100% it is a corner.
- Shorting flips the cross term: spreads are calm between close substitutes, noisy otherwise.
- Run the formula backwards for the correlation a portfolio implies.