Quantitative Analysis
Probability, statistics, and models
Candidates must handle probability rules, Bayes' theorem, discrete and continuous distributions, moments, covariance and correlation, regression, hypothesis testing, and simulation. For a two-asset portfolio, variance is w1^2sigma1^2 + w2^2sigma2^2 + 2w1w2rhosigma1*sigma2; the covariance term is the most common omission. Correlation ranges from -1 to +1 and does not capture nonlinear dependence; copulas do.
Regression traps: R-squared measures explained variation, not causality; a high t-statistic on one coefficient does not validate the model; heteroskedasticity and autocorrelation distort standard errors. In hypothesis testing, the p-value is the probability of data at least as extreme given the null is true, not the probability that the null is true. A 5% significance level means a 5% Type I error rate, not a 5% chance of being wrong.
Worked example: Assets have weights 60%/40%, sigma1=10%, sigma2=20%, rho=0.3. Variance = 0.36(0.01) + 0.16(0.04) + 2(0.6)(0.4)(0.3)(0.10)(0.20) = 0.0036 + 0.0064 + 0.00288 = 0.01288; sigma = 11.35%. Ignoring covariance would give 10.9%, understating risk when rho is positive.
For simulation, Monte Carlo estimates distributions from assumed processes; historical simulation uses actual past data and is sensitive to the sample window. The central limit theorem says sample means approach normality as n grows, not that raw returns are normal.