Sample Questions
Question 1
Easy
Which short-rate model can produce negative interest rates with positive probability?
Solution
C is correct. The Vasicek model assumes the short rate follows a normal Ornstein-Uhlenbeck process with support on the entire real line, so the rate has positive probability of going below zero. CIR uses a square-root diffusion whose variance vanishes as the rate approaches zero, keeping rates non-negative. Black-Karasinski and Black-Derman-Toy use lognormal dynamics, which keep rates strictly positive.
Question 2
Medium
Which statement most accurately describes empirically observed properties of equity-return correlations documented in long-horizon studies of pairwise correlations?
Solution
D is correct. Empirical work on equity-pair correlations documents two robust regularities. First, realized correlations revert toward a long-run average over multi-year horizons, motivating mean-reverting specifications. Second, the realized correlation series is itself highly volatile, with month-to-month swings comparable in magnitude to return volatility, which is why stochastic-correlation extensions of static models are needed.
Question 3
Hard
A portfolio's daily P&L is normally distributed with zero mean. The one-day 95% parametric VaR is $1 million. Using , , and assuming square-root-of-time scaling, the 10-day 99% expected shortfall is closest to:
Solution
D is correct. First back out the dollar standard deviation from the given 95% VaR: . Next, the daily 99% expected shortfall multiplier under normality is , where is the standard normal density. Then daily 99% ES . Scaling to a 10-day horizon by gives million.
