Basic, lesson 1 of 3
Variance and covariance algebra
Every risk number in a research interview comes from one formula: the variance of a weighted sum. Learn it once and portfolio, hedge, and signal questions all follow.
10 min, 3 checks
1The variance of a sum
Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)
Cov(X, Y) = Corr(X, Y) x sd(X) x sd(Y). Constants added to X change nothing.
Standard deviations do not add; variances do, plus the covariance term. Two independent risks of 3 and 4 combine to 5, not 7.
Check 1
X has standard deviation 2 and Y has standard deviation 3, independent. What is the standard deviation of X + Y? Three decimals.
Check 2
Var(X) = 1, Var(Y) = 4, Cov(X, Y) = 1. What is Var(2X - Y)?
2Diversification and its floor
Average n assets that each have variance sigma^2 and pairwise correlation rho. The portfolio variance is sigma^2 x (rho + (1 - rho) / n). As n grows, the second term vanishes and you are left with rho x sigma^2. Correlation sets a floor that no amount of diversification removes.
Worked example
100 stocks, each 20% volatility, pairwise correlation 0.25, equal weights
- 1sigma^2 = 0.04.
- 2rho + (1 - rho) / n = 0.25 + 0.75 / 100 = 0.2575.
- 3Variance = 0.04 x 0.2575 = 0.0103.
Volatility = sqrt(0.0103) = 10.15%. The floor with infinitely many stocks is 20% x sqrt(0.25) = 10%, so 100 names already get almost all of the benefit.
Check 3
Two assets, each 20% volatility, uncorrelated, held 50/50. What is the portfolio volatility in percent? Two decimals.
Terms in this lesson: Standard deviation, Correlation.
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