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Solve Market-Risk and Probability Questions

Morgan StanleyData Scientist · HR Screen
MediumData ScientistHR ScreenStatistics & Math

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Morgan Stanley data-scientist HR-screen quant set covering four sub-questions: design a WTI/Brent crude stress-testing framework using heavy-tailed marginals and a t-copula for joint tail dependence; the variance of a sum of correlated normal variables and when the sum is normal; the Monty Hall switching probability; and the eight-balls one-heavier puzzle in two weighings. It probes both conceptual understanding and practical application of dependence modeling, scenario generation, and probabilistic reasoning.

Question

Answer the following quantitative interview questions for a market-risk / data-scientist screen at Morgan Stanley.

  1. Stress testing WTI and Brent crude. You manage risk for positions exposed to WTI and Brent crude oil. Design a stress-testing framework for the joint behavior of the two series over a chosen horizon (e.g. 1 day for the trading book, 10 days for portfolio/regulatory stress). Assume you have historical daily returns for both series and want to generate joint downside scenarios. Describe:
  • what data you would use;
  • how you would model each series marginally;
  • how you would capture the dependence structure and tail risk between them;
  • whether and why a t-copula is appropriate (and why it may be preferable to a Gaussian copula);
  • the full t-copula calibration and simulation workflow;
  • how you would aggregate to a multi-day (e.g. 10-day) horizon;
  • how you would generate joint stress scenarios (historical, hypothetical, model-based, conditional, reverse);
  • how you would validate the framework and communicate its limitations.
  1. Variance of a sum of normal variables. Let X and Y be normally distributed random variables with variances σ_X^2 and σ_Y^2 and correlation ρ.
  • If X and Y are independent, what is Var(X + Y)?
  • If they are dependent with correlation ρ, what is Var(X + Y)?
  • Under what condition is X + Y itself normally distributed?

  1. Monty Hall problem. There are 3 doors. One door hides a car and the other 2 hide goats. You choose one door. The host, who knows where the car is, opens a different door that always reveals a goat and offers you the chance to switch to the remaining unopened door. Should you switch, and what is the probability of winning if you stay versus switch?

  2. Eight balls, one heavier. You have 8 visually identical balls, but exactly one is heavier than the other seven. Using a balance scale at most 2 times, give a strategy that always identifies the heavier ball.