Variance & covariance: interview questions and key results

The algebra behind risk. The results to know for quant trading interviews, then a worked example of each of the 4 question types. Free, no sign-up.

Practise endless questions like these

Key results

Var(X) = E[X²] − E[X]².
Var(aX + bY) = a²Var X + b²Var Y + 2ab Cov(X,Y).
ρ = Cov/(σ_Xσ_Y).

Worked examples (4)

Try each one before opening the solution. In the app, every type generates fresh numbers each time.

Var(aX + bY)

Var(X) = 2, Var(Y) = 5, Cov(X,Y) = -0.6. Find Var(3X − 1Y).

Answer and worked solution

Answer: 26.6

Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov = 9·2 + 1·5 + 2·(3)(-1)·-0.6 = 26.6.

Correlation from covariance

Var(X) = 36, Var(Y) = 36, Cov(X,Y) = 32.4. What is the correlation ρ(X,Y)?

Answer and worked solution

Answer: 0.9

ρ = Cov/(σ_Xσ_Y) = 32.4/(6·6) = 0.9. Watch out: you're given variances, so take square roots first.

Discrete variance

A trade pays -5 with probability 0.2, 2 with probability 0.5, 10 with probability 0.3. What is the variance of the payoff?

Answer and worked solution

Answer: 28

E[X] = 3, E[X²] = 37. Var = E[X²] − (E[X])² = 37 − 9 = 28.

Var(X − Y) for independents

X and Y are independent with Var(X) = 7 and Var(Y) = 5. What is Var(X − Y)?

Answer and worked solution

Answer: 12

Var(X − Y) = Var(X) + Var(Y) − 2Cov = 7 + 5 − 0 = 12. Variances add even when you subtract, so a spread is riskier than either leg alone unless the legs are correlated.

Variance & covariance: interview questions and key results

Key results

Worked examples (4)

Other topics