FINC6023 · Financial Risk Management
Portfolio VaR: Variance, Diversification & Component VaR
Portfolio VaR: Variance, Diversification & Component VaR moves from a single position to a book. Portfolio variance is w′Σw — for two assets, w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ — and the portfolio VaR is W·z·σ_p·√T. The gap between the sum of standalone VaRs (undiversified) and the portfolio VaR (diversified) is the diversification benefit. You then decompose: marginal VaR is the sensitivity to a $1 change, component VaR splits the total across positions and sums back to it, and a negative component VaR marks a position that hedges the book.
What this chapter covers
- 01Two-asset variance σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂
- 02Matrix form σ_p² = w′Σw
- 03Portfolio VaR = W · z · σ_p · √T
- 04Diversified vs undiversified VaR and the diversification benefit
- 05Cross-check VaR_p = √(VaR_A² + VaR_B² + 2ρ·VaR_A·VaR_B)
- 06Marginal VaR (sensitivity to a $1 change)
- 07Component VaR (sums to total; negative component = a hedge)
- 08Incremental VaR (effect of adding a whole position) and beta to the portfolio
Two-asset portfolio VaR and the diversification benefit
- 1 markFind the weights: total W = 1,000,000, so w_A = 0.4, w_B = 0.6.
- 2 marksPortfolio variance σ_p² = 0.4²(0.018)² + 0.6²(0.010)² + 2(0.4)(0.6)(0.25)(0.018)(0.010) = 0.00005184 + 0.000036 + 0.0000216 = 0.00010944.
- 1 markPortfolio volatility σ_p = √0.00010944 = 0.0104614 (about 1.046%/day).
- 2 marksPortfolio VaR = W · z · σ_p = 1,000,000 × 2.326 × 0.0104614 = $24,333.
- 2 marksStandalone VaRs: VaR_A = 400,000 × 2.326 × 0.018 = $16,747; VaR_B = 600,000 × 2.326 × 0.010 = $13,956; undiversified sum = $30,703.
- 1 markDiversification benefit = 30,703 − 24,333 = $6,370.
Key terms
- Portfolio variance (w′Σw)
- The matrix form of portfolio variance, where w is the weight vector and Σ the covariance matrix (Σ_ij = ρ_ij σ_i σ_j). For two assets it expands to w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.
- Diversification benefit
- The undiversified VaR (sum of standalone VaRs) minus the diversified portfolio VaR. It reflects imperfect correlation and is larger when assets are less correlated.
- Marginal VaR
- The change in portfolio VaR from a $1 increase in a position, ∂VaR/∂xᵢ. It measures how much risk the next dollar in that position adds.
- Component VaR
- The part of total portfolio VaR attributable to a position; component VaRs sum exactly to the portfolio VaR. A negative component VaR identifies a position that hedges (reduces) overall risk.
- Incremental VaR
- The change in portfolio VaR from adding or removing a whole position (not just $1). It captures the discrete impact of a trade, unlike the marginal (per-dollar) measure.
Portfolio VaR: Variance, Diversification & Component VaR FAQ
What does a negative component VaR mean?
It means that position is reducing the book's overall risk — it acts as a hedge. Because component VaRs must sum to the total portfolio VaR, a hedging position contributes a negative slice, so adding more of it lowers total VaR.
How can I check my portfolio VaR without redoing the matrix?
Use the component-VaR identity VaR_p = √(VaR_A² + VaR_B² + 2ρ·VaR_A·VaR_B). It reproduces the matrix answer and isolates the cross term, which is where most arithmetic slips happen.
What is the difference between marginal and incremental VaR?
Marginal VaR is the sensitivity to a $1 change in a position (a derivative); incremental VaR is the change from adding or dropping an entire position. Marginal is for fine-tuning, incremental is for go/no-go on a whole trade.
Exam move
Drill the two-asset variance and the standalone-vs-portfolio VaR until you can do it under a minute, and always finish with the √(VaR_A² + VaR_B² + 2ρ·...) cross-check. Be able to explain component VaR summing to the total and why a hedge shows as a negative component.
