Statistics interview questions for trading and quant roles

Trading interviews lean on probability. Quant research and quant trading interviews add statistics: can you tell a real signal from noise, and do you know how your tools fail?

Variance and correlation

• Var(aX + b) = a² × Var(X). Constants add nothing.
• Var(X + Y) = Var(X) + Var(Y) + 2 × Cov(X, Y). For a difference, the covariance term is subtracted.
• Cov(X, Y) = ρ × σx × σy.
• Uncorrelated does not mean independent. If X is standard normal and Y = X², their correlation is zero but Y is determined by X.

Question. Two assets each have 10% volatility and zero correlation. What is the volatility of a 50/50 portfolio? √(0.25 × 100 + 0.25 × 100) = 7.07%. Diversification reduces risk by a factor of √2.

Estimation

• The standard error of a mean is σ / √n. Four times the data halves the error.
• Sample variance divides by n − 1 because one degree of freedom is spent estimating the mean.
• A 95% confidence interval is roughly the estimate plus or minus two standard errors.

Hypothesis tests

• A p-value is the probability of seeing data at least this extreme if the null hypothesis is true. It is not the probability that the null is true.
• A Type I error is a false positive, a Type II error a false negative.
• Multiple testing. Test 100 random signals at 5% and about 5 will look significant. Ask how many ideas were tried before the one being presented.

Regression

• Slope = Cov(X, Y) / Var(X) = ρ × σy / σx. A stock with 30% volatility, 0.6 correlation to a market with 15% volatility has a beta of 1.2.
• R² never falls when you add a variable, so it cannot be used alone to choose a model.
• Multicollinearity inflates standard errors. Coefficients become unstable although the fit is good.
• Heteroskedasticity leaves coefficients unbiased but makes the usual standard errors wrong.
• Overfitting shows as low error in sample and high error out of sample.

Sharpe ratio

Excess return divided by volatility. To annualise a daily Sharpe ratio multiply by √252, about 15.9. A daily Sharpe of 0.1 is an annual Sharpe of about 1.6.

Question. How many years of data do you need to be confident a Sharpe of 1 is real? The t-statistic is roughly Sharpe × √years, so about four years for a t-statistic of 2.

Backtest pitfalls

• Look-ahead bias: using information that was not available at the time of the trade.
• Survivorship bias: testing only on companies that still exist.
• Overfitting: tuning parameters until the history looks good.
• Ignoring costs: spread, fees and market impact.

Distributions to know

Normal (68 / 95 / 99.7 rule), binomial (mean np, variance np(1 − p)), Poisson (mean and variance λ), exponential (memoryless), uniform on [0, 1] (mean 1/2, variance 1/12).

Related: probability interview questions and expected value questions.