How do historical, variance-covariance, and Monte Carlo VaR differ?

Question

AcadiFi · Accepted Answer

The three methods differ mainly in how they create the loss distribution. Historical VaR reorders actual past returns and reads a tail percentile from that empirical history. Variance-covariance VaR assumes a distribution, often normal, and uses volatility and correlation estimates to produce a parametric tail number. Monte Carlo VaR simulates many future scenarios from a chosen model and then reads the loss percentile from the simulated outcomes.

Suppose Northgate Market Risk holds a $20 million multi-asset book. Historical VaR might use 500 past daily return vectors and find the 5th worst loss at the 99 percent level. Parametric VaR might use the covariance matrix and assume returns are approximately normal. Monte Carlo VaR might simulate 50,000 joint return paths, revalue the book, and rank the losses.

```mermaid
flowchart LR
  A[Historical returns] --> D[Empirical VaR]
  B[Volatility and covariance] --> E[Parametric VaR]
  C[Simulated scenarios] --> F[Monte Carlo VaR]
```

The FRM exam point is that no method is universally best. Historical VaR is simple but backward-looking. Parametric VaR is fast but assumption-heavy. Monte Carlo is flexible but depends on model quality and computation. 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.

How do historical, variance-covariance, and Monte Carlo VaR differ?

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