Why can parametric VaR and Monte Carlo VaR disagree under a lognormal assumption?

Question

AcadiFi · Accepted Answer

They can disagree because the implementations may not be measuring exactly the same object. A parametric VaR may use a closed-form quantile for return or price under a simplified assumption. A Monte Carlo VaR simulates paths, converts them into portfolio values, and estimates a percentile from finite samples. Small differences in drift, horizon, compounding, and whether losses are defined in price or return terms can create visible gaps.

Suppose Granite Equity Risk models a stock-only portfolio with geometric Brownian motion. The parametric version computes a one-day 99 percent loss from log-return assumptions. The Monte Carlo version simulates 25,000 prices, converts them to simple returns, and ranks losses. If the analyst mixes log and simple returns or uses too few simulations, the numbers will not align.

The FRM answer is to reconcile definitions before judging methods. Are both using the same horizon, confidence level, volatility, drift treatment, and loss definition? Is the simulation sample large enough? If yes, the two estimates should converge within sampling error for a simple portfolio. If not, the difference is diagnostic information, not just noise. On the exam, also state the horizon, confidence level, data source, modeling assumption, and validation step, because those choices usually explain why two reasonable VaR answers differ materially. Practice more in our FRM question bank.

Why can parametric VaR and Monte Carlo VaR disagree under a lognormal assumption?

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