Regression & correlation: interview questions and key results

OLS, beta, R². The results to know for quant trading interviews, then a worked example of each of the 5 question types. Free, no sign-up.

Key results

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

Beta from covariance

The market has annual volatility 25%. A stock's covariance with the market is 0.0938. What is the stock's beta?

Answer and worked solution

Answer: 1.5008

β = Cov(r_s, r_m)/Var(r_m) = 0.0938/0.0625 = 1.5008. This is exactly the OLS slope of stock returns on market returns.

OLS slope by hand

Fit OLS y = α + βx to the points (1, 12), (2, 10), (3, 13), (4, 18), (5, 20). What is β?

Answer and worked solution

Answer: 2.4

x̄ = 3, ȳ = 14.6. Σ(x−x̄)² = 10, Σ(x−x̄)(y−ȳ) = 24. β = 24/10 = 2.4.

Slope from correlation

ρ(X,Y) = 0.4, σ_X = 1, σ_Y = 6. What is the OLS slope when regressing Y on X?

Answer and worked solution

Answer: 2.4

β = ρ·σ_Y/σ_X = 0.4·6/1 = 2.4.

R² from two slopes

Regressing Y on X gives slope 0.9. Regressing X on Y gives slope 0.42. What is R² of either regression?

Answer and worked solution

Answer: 0.378

The slopes are ρσ_Y/σ_X and ρσ_X/σ_Y, so their product is ρ² = R² = 0.9·0.42 = 0.378. The slopes are not reciprocals unless |ρ| = 1.

Intercept through the means

In an OLS fit, x̄ = -0.3, ȳ = 0.6, and the slope is -0.5. What is the intercept?

Answer and worked solution

Answer: 0.45

The OLS line passes through (x̄, ȳ): α = ȳ − βx̄ = 0.6 − (-0.5)(-0.3) = 0.45.