CRA Monte Carlo Simulation & VaR 2 — Questions and Answers
Question 1: Which variance reduction technique in Monte Carlo simulation uses pairs of random numbers that are negatively correlated to reduce estimation error?
- Importance sampling
- Antithetic variates (Correct answer)
- Stratified sampling
- Control variates
Correct answer: Antithetic variates
Antithetic variates generates pairs of random numbers (u, 1-u) that are negatively correlated, averaging their outcomes to reduce variance in the estimate.
Question 2: A 10-day 99% VaR of $5 million means there is a 1% probability that losses will exceed $5 million over:
- Any single trading day
- A 10-trading-day period (Correct answer)
- A calendar month
- A full trading year
Correct answer: A 10-trading-day period
The time horizon is explicitly stated in the VaR definition — a 10-day 99% VaR refers to losses over a 10-trading-day holding period.
Question 3: In Monte Carlo VaR, which distribution assumption is most commonly used to model daily equity returns as a starting point?
- Uniform distribution
- Lognormal distribution (Correct answer)
- Poisson distribution
- Exponential distribution
Correct answer: Lognormal distribution
Equity prices are typically modeled as lognormal, meaning daily log-returns follow a normal distribution, which is the standard starting point in Monte Carlo equity VaR models.
Question 4: What is the primary purpose of using a Cholesky decomposition in a multi-asset Monte Carlo simulation?
- To speed up random number generation
- To incorporate correlation structure between assets (Correct answer)
- To eliminate fat tails from return distributions
- To convert nominal returns to real returns
Correct answer: To incorporate correlation structure between assets
Cholesky decomposition of the correlation matrix is used to generate correlated random variables, ensuring simulated asset returns reflect observed correlations.
Question 5: Expected Shortfall (ES) at the 99% confidence level is best described as:
- The maximum loss observed in 1% of scenarios
- The average loss in the worst 1% of scenarios (Correct answer)
- The median loss across all simulation paths
- The loss at exactly the 99th percentile
Correct answer: The average loss in the worst 1% of scenarios
ES (also called CVaR) is the conditional expectation of losses given that the loss exceeds the VaR threshold — i.e., the average of the worst 1% of outcomes.
Question 6: Which of the following is a key advantage of Monte Carlo VaR over Historical Simulation VaR?
- It requires no distributional assumptions
- It can model complex, path-dependent instruments (Correct answer)
- It is always faster to compute
- It never requires correlation estimates
Correct answer: It can model complex, path-dependent instruments
Monte Carlo simulation can price path-dependent derivatives (e.g., Asian options, barrier options) by generating full price paths, which Historical Simulation cannot easily handle.
Question 7: If a firm scales its 1-day VaR to a 10-day VaR using the square-root-of-time rule, which critical assumption is being made?
- Returns are autocorrelated
- Daily returns are i.i.d. (independent and identically distributed) (Correct answer)
- Volatility is mean-reverting
- The portfolio is delta-neutral
Correct answer: Daily returns are i.i.d. (independent and identically distributed)
The square-root-of-time scaling rule is valid only when daily returns are i.i.d., so that variance scales linearly with time.
CRA Monte Carlo Simulation & VaR 2
Which variance reduction technique in Monte Carlo simulation uses pairs of random numbers that are negatively correlated to reduce estimation error?