What are the three main VaR calculation methods and when should each be used?

Question

AcadiFi · Accepted Answer

Value at Risk (VaR) is the cornerstone risk metric in FRM Part I, and understanding the three calculation methods — their mechanics, strengths, and weaknesses — is essential for the exam.

**The Three VaR Methods:**

```mermaid
graph TD
  A[VaR Calculation Methods] --> B[Parametric / Variance-Covariance]
  A --> C[Historical Simulation]
  A --> D[Monte Carlo Simulation]
  B --> B1[Assumes normal distribution]
  B --> B2[Fast - analytical formula]
  B --> B3[Poor for non-linear instruments]
  C --> C1[Uses actual historical returns]
  C --> C2[No distribution assumption]
  C --> C3[Limited by historical window]
  D --> D1[Simulates thousands of scenarios]
  D --> D2[Handles any distribution or payoff]
  D --> D3[Computationally expensive]
  B1 --> E{Portfolio contains options or non-linear payoffs?}
  E -->|No - bonds, equities, FX| B
  E -->|Yes - options, structured products| F{Need speed or accuracy?}
  F -->|Speed| C
  F -->|Accuracy| D
  style A fill:#c9a84c,color:#0a0a0f
  style B fill:#1e3a5f,color:#fff
  style C fill:#1a3c2a,color:#fff
  style D fill:#2d1b4e,color:#fff
```

**1. Parametric (Variance-Covariance) VaR**
- Formula: VaR = Portfolio Value x z-alpha x sigma x sqrt(t)
- Assumes returns are normally distributed
- Best for: linear portfolios (equities, bonds, FX forwards)

**Example:** Thornton Capital holds a $50 million equity portfolio with daily volatility of 1.4%. The 99% 1-day parametric VaR is:
VaR = $50M x 2.326 x 0.014 = **$1,628,200**

This means there is a 1% probability of losing more than $1.63 million in a single day.

**2. Historical Simulation VaR**
- Collects actual historical returns (e.g., 500 trading days)
- Applies each historical return to the current portfolio
- Sorts the resulting P&L and picks the percentile corresponding to the confidence level
- Best for: capturing fat tails, regime shifts, non-normal behavior

**Example:** Using Thornton's last 500 daily returns, sorted from worst to best. For 99% VaR, the 5th worst return (500 x 1% = 5th observation) was -3.8%. Historical VaR = $50M x 3.8% = **$1,900,000** — higher than parametric because it captures actual tail events.

**3. Monte Carlo Simulation VaR**
- Specifies a stochastic process for each risk factor
- Generates 10,000+ random scenarios
- Reprices the full portfolio under each scenario
- Best for: options, structured products, path-dependent instruments

**Example:** Thornton also holds $15 million in barrier options. Monte Carlo generates 50,000 paths for the underlying, reprices each option, and calculates portfolio P&L. The 99th percentile loss is **$2,340,000** — highest because the option's non-linear payoff amplifies tail losses.

**Comparison Table:**

| Feature | Parametric | Historical | Monte Carlo |
|---|---|---|---|
| Distribution assumption | Normal | None (actual data) | User-specified |
| Non-linear instruments | Poor | Moderate | Excellent |
| Computation speed | Very fast | Fast | Slow |
| Data requirement | Covariance matrix | 1-3 years daily returns | Calibrated model |
| Fat tails captured | No | Yes (if in history) | Yes (if modeled) |
| Forward-looking | No | No | Yes |

**Exam Tip:** If a question describes a portfolio of plain vanilla bonds and equities, parametric VaR is appropriate. If the portfolio contains options or the question emphasizes tail risk, lean toward historical simulation or Monte Carlo. If the question mentions path-dependent or exotic instruments, Monte Carlo is the correct choice.

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What are the three main VaR calculation methods and when should each be used?

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