How Tower Research Capital's process runs, as candidates and the firm describe it, and the questions candidates report, each with a worked solution. Free, no sign-up.
Trader and analyst candidates report Fermi estimates, probability (making a fair coin from a biased one, correlation bounds), and for quant roles regression and machine-learning theory (lasso v ridge).
Try each one before opening the solution. Interviewers change the numbers, so learn the method.
Analyst (prop trading) · Interview · Statistics · reported by a candidate
corr(X, Y) = 0.2 and corr(Y, Z) = 0.5. Nothing else is known.
(a) What is the largest possible value of corr(X, Z)?
Answer and worked solution
Answer: 0.94853
Key idea. The correlation matrix must be positive semi-definite, which gives c ∈ ab ± √((1 − a²)(1 − b²)).
a = 0.2, b = 0.5: ab = 0.1, √(0.96 × 0.75) = √0.72 ≈ 0.849.
Max c = 0.1 + 0.849 ≈ 0.949.
Check. Geometric view: correlations are cosines of angles between vectors. Those angles are about 78° and 60°, so the X–Z angle ranges from 18° to 138°.
(b) The smallest?
Answer and worked solution
Answer: -0.74853
Key idea. Same interval, lower end.
Min c = 0.1 − 0.849 ≈ −0.749.
Check. Correlation isn't transitive: two positive correlations allow a strongly negative third.