What is component VaR and how does it decompose portfolio risk?
FRM Part II covers component VaR as a way to understand how each position contributes to total portfolio VaR. How is it calculated and what makes it different from individual VaR?
Component VaR (CVaR) tells you how much each position contributes to the total portfolio VaR. It's one of the most powerful risk decomposition tools because it answers the question: if I removed this position, how much would my VaR decrease?
Key property: Component VaRs sum to total portfolio VaR
Σ CVaRᵢ = Portfolio VaR
This makes it an additive risk decomposition — you can see exactly where your risk comes from.
Calculation:
CVaRᵢ = βᵢ × wᵢ × Portfolio VaR
Or equivalently:
CVaRᵢ = wᵢ × (Cov(Rᵢ, Rp) / σp²) × Portfolio VaR
Where:
- wᵢ = weight of position i
- βᵢ = beta of position i with respect to the portfolio
- Rᵢ = return of position i
- Rp = portfolio return
- σp = portfolio standard deviation
Example — Three-Asset Portfolio:
Silverstone Capital manages a $100M portfolio:
| Asset | Weight | Individual VaR | Component VaR | % of Portfolio VaR |
|---|---|---|---|---|
| US Equities | 50% | $6.2M | $4.8M | 60% |
| US Bonds | 30% | $1.5M | $0.8M | 10% |
| EM Equities | 20% | $5.0M | $2.4M | 30% |
| Portfolio | 100% | — | $8.0M | 100% |
Key observations:
- Individual VaR doesn't add up: $6.2 + $1.5 + $5.0 = $12.7M > $8.0M (because of diversification)
- Component VaR adds up perfectly: $4.8 + $0.8 + $2.4 = $8.0M
- US Bonds have low component VaR ($0.8M vs. $1.5M individual) because they are negatively correlated with equities — they actually REDUCE overall risk
- EM Equities contribute disproportionately (20% weight but 30% of risk) because they are volatile and positively correlated with US equities
Negative component VaR:
A position can have a negative component VaR — meaning it acts as a hedge and reduces portfolio risk. This happens when the position's correlation with the rest of the portfolio is sufficiently negative.
Individual VaR vs. Component VaR vs. Marginal VaR vs. Incremental VaR:
| Metric | Question It Answers |
|---|---|
| Individual VaR | What's the standalone risk of this position? |
| Component VaR | How much does this position contribute to portfolio VaR? |
| Marginal VaR | What's the rate of change in VaR per unit increase in position? |
| Incremental VaR | What happens to portfolio VaR if I add/remove this entire position? |
Exam tip: FRM Part II tests the calculation, the additive property, and the interpretation — especially why individual VaRs don't sum to portfolio VaR but component VaRs do. Know how to identify diversifying vs. concentrating positions.
Practice risk decomposition on AcadiFi's FRM materials.
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