FINC6023 Chap.5 Portfolio VaR: Variance, Diversification & Component VaR
Portfolio VaR: Variance, Diversification & Component VaR
Portfolio VaR: Variance, Diversification and Component VaR moves from one position to a portfolio whose risks interact. Portfolio volatility depends on each position's weight and volatility plus every covariance. For two assets, the cross term contains correlation and determines how much the individual risks reinforce or offset one another.
In matrix form the same calculation becomes the weight vector multiplied by the covariance matrix and the weight vector again.
Once portfolio volatility is known, the usual normal quantile and horizon conversion produce portfolio VaR. The sum of standalone VaRs is an undiversified benchmark. Portfolio VaR is generally smaller when correlation is below one, and the gap is the diversification benefit.
Correlation must remain within its valid range, and the answer should move toward the standalone sum as correlation rises.
Risk allocation asks a different question from risk measurement. Marginal VaR is the change in total VaR caused by a small increase in one position. Component VaR multiplies that marginal effect by the position size and allocates total VaR across holdings. Component amounts sum to portfolio VaR.
A negative component identifies a hedge because increasing that position reduces total risk. Incremental VaR instead measures the effect of adding or removing an entire trade, so it need not equal a local derivative.
The available final paper includes a portfolio VaR multiple-choice calculation. The quickest independent check uses standalone VaRs and their correlation rather than rebuilding the covariance matrix.
In written work, show the cross term, take the square root only after summing variance terms, and state the diversification benefit in dollars. A plausible number without those steps does not demonstrate that the correlation channel was handled correctly.
What this chapter covers
- 01
Two-asset variance σ_p² = 2ρ(w₁σ₁)(w₂σ₂) + (w₁σ₁)² + (w₂σ₂)²
- 02
Matrix form σ_p² = w′Σw
- 03
Portfolio VaR = W · z · σ_p · √T
- 04
Diversified vs undiversified VaR and the diversification benefit
- 05
Cross-check VaR_p = √(VaR_A² + VaR_B² + 2ρ·VaR_A·VaR_B)
- 06
Marginal and component VaR; components sum to total and a negative component is a hedge
- 07
Incremental VaR and beta to the portfolio
Worked example · free
Two-asset portfolio VaR and the diversification benefit
- 1Find the weights: total W = 1,000,000, so w_A = 0.4, w_B = 0.6.
- 2Portfolio 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.
- 1Portfolio volatility σ_p = √0.00010944 = 0.0104614 (about 1.046%/day).
- 2Portfolio VaR = W · z · σ_p = 1,000,000 × 2.326 × 0.0104614 = $24,333.
- 2Standalone 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.
- 1Diversification 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 can be written as 2ρ(w₁σ₁)(w₂σ₂) + (w₁σ₁)² + (w₂σ₂)².
- 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.
- Undiversified VaR
- Undiversified VaR is the sum of standalone position VaRs, a benchmark that ignores covariance benefits and equals portfolio VaR only under particular dependence conditions.
- Covariance matrix
- A covariance matrix places asset variances on the diagonal and pairwise covariances off the diagonal, allowing portfolio variance to be written compactly as w prime sigma w.
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 standalone-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. Also compare the result with the larger standalone VaR and their sum to confirm that the diversification direction is plausible.
What is the difference between marginal and incremental VaR?
Marginal VaR is the sensitivity to a $1 change in a position, while incremental VaR is the change from adding or dropping an entire position. Marginal analysis supports fine adjustments; incremental analysis supports a decision about a whole trade and can differ when the change is large or the risk measure is nonlinear.
Why must component VaRs sum to total VaR?
For a positively homogeneous VaR measure, Euler allocation decomposes the portfolio measure into each position size multiplied by its marginal contribution. Those products are component VaRs, and their sum recovers total portfolio VaR. This property makes component VaR useful for limits, performance evaluation, and identifying positions that hedge the rest of the book.
How does the available final test portfolio VaR?
The paper supplies two standalone VaRs and a correlation, so the efficient route is the standalone-VaR aggregation identity. Square each VaR, add twice their product times correlation, and take the square root. The result should be below the standalone sum when correlation is below one and should remain economically plausible.
Exam move
Practise each portfolio problem twice. First use weights and the covariance formula to obtain portfolio variance, volatility, and VaR. Then recompute with the standalone-VaR identity. If the answers disagree, inspect the factor of two in the cross term, the conversion from percentages to decimals, and whether you took a square root too early.
Before calculating, predict the result: with correlation below one, portfolio VaR should be below the sum of standalone VaRs but usually above the larger standalone VaR. For marginal, component, and incremental VaR, write a one-line definition beside a concrete management question. Marginal asks about the next small dollar, component allocates the current total, and incremental evaluates a whole trade.
Rehearse the identity that component amounts sum to total portfolio VaR and explain why a negative component represents hedging. Finish timed practice by calculating and interpreting the diversification benefit rather than stopping at the portfolio number or formula.
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