Component VaR splits a diversified VaR into pieces that add back to the total. Standalone and incremental figures do not, and they point to the wrong sleeve.
Component VaR for a position is its weight times its covariance with the whole portfolio, divided by portfolio volatility, times the same multiplier that turns volatility into VaR. The components add up exactly to the total. On an 840M credit fund with a one-day 99 per cent VaR of 7,242,456, high yield is 35 per cent of assets but 3,346,615, or 46.2 per cent, of the VaR, while investment-grade credit is 30 per cent of assets and only 18.4 per cent of the risk.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The fund is Halverton, the open-ended credit fund of Financial Risk Management: 840,000,000 of net assets in four sleeves, annual volatilities by sleeve, the fund's reported correlation matrix, 252 trading days and a 99 per cent one-tailed multiplier of 2.3263. The allocation that follows is this article's; the inputs are the book's.
| Sleeve | Market value | Weight | Volatility |
|---|---|---|---|
| Investment-grade credit | 252,000,000 | 0.30 | 4.5% |
| High yield | 294,000,000 | 0.35 | 8.0% |
| Leveraged loans | 210,000,000 | 0.25 | 6.0% |
| Structured credit | 84,000,000 | 0.10 | 10.0% |
| Fund | 840,000,000 | 1.00 | 5.883% |
Reported correlations: investment grade with high yield 0.75, with loans 0.55, with structured 0.45; high yield with loans 0.80 and with structured 0.70; loans with structured 0.65. Portfolio variance is 0.0034615, volatility 5.883 per cent, and one-day VaR is 2.3263 × 5.883% / √252 × 840,000,000 = 7,242,456, or 0.862 per cent of net assets.
For each sleeve, multiply the covariance matrix by the weight vector. The result is the sleeve's covariance with the portfolio, which already contains every correlation that sleeve has with the other three. For high yield:
(Σw)HY = Σj wj × σj × σHY × ρj,HY
= 0.30 × 4.5% × 8.0% × 0.75 + 0.35 × 8.0%² + 0.25 × 6.0% × 8.0% × 0.80 + 0.10 × 10.0% × 8.0% × 0.70
= 0.000810 + 0.002240 + 0.000960 + 0.000560 = 0.004570
Excel, covariance matrix in a named range Cov and weights in a column W: =MMULT(Cov,W)
Divide each covariance by portfolio volatility to get the marginal volatility, the change in fund volatility per unit of weight. Multiply by the VaR multiplier, 2.3263 / √252 × 840,000,000 = 123,098,759, to get it in money. Component VaR is then weight times marginal VaR.
Component VaRi = wi × (Σw)i / σp × z / √252 × NAV
High yield: 0.35 × 0.004570 / 0.05883 × 123,098,759 = 0.35 × 7.768% × 123,098,759 = 3,346,615
Excel: =W*MMULT(Cov,W)/SQRT(MMULT(TRANSPOSE(W),MMULT(Cov,W)))*NORM.S.INV(0.99)/SQRT(252)*NAV
| Sleeve | Weight | Beta to fund | Marginal VaR per 1M | Component VaR | Share of VaR |
|---|---|---|---|---|---|
| Investment-grade credit | 30% | 0.614 | 5,296 | 1,334,618 | 18.4% |
| High yield | 35% | 1.320 | 11,383 | 3,346,615 | 46.2% |
| Leveraged loans | 25% | 0.890 | 7,670 | 1,610,800 | 22.2% |
| Structured credit | 10% | 1.312 | 11,315 | 950,422 | 13.1% |
| Fund | 100% | 1.000 | 7,242,456 | 100.0% |
The share of VaR divided by the weight is the sleeve's beta to the fund. High yield and structured credit both run at about 1.3 times their weight; together they are 45 per cent of assets and 59.3 per cent of the risk. Investment grade runs at 0.61 times its weight. The components add to 7,242,456 to the unit, because a risk measure that doubles when every position doubles can always be split this way (Euler's theorem).
The marginal column is the one a portfolio manager can use. Moving 21,000,000 from high yield to investment grade should change VaR by (5,296 − 11,383) × 21 = −127,826. Recomputing the full matrix gives 7,115,495, a fall of 126,961: the marginal estimate is within one per cent for a trade of 2.5 per cent of the fund.
The book re-estimates the matrix on stressed periods, with pairwise correlations from 0.82 to 0.95. Fund VaR rises to 7,925,849 with no trade. The allocation shifts too.
| Sleeve | Reported | Share | Crisis | Share | Standalone VaR |
|---|---|---|---|---|---|
| Investment-grade credit | 1,334,618 | 18.4% | 1,565,661 | 19.8% | 1,661,833 |
| High yield | 3,346,615 | 46.2% | 3,408,430 | 43.0% | 3,446,765 |
| Leveraged loans | 1,610,800 | 22.2% | 1,800,421 | 22.7% | 1,846,481 |
| Structured credit | 950,422 | 13.1% | 1,151,337 | 14.5% | 1,230,988 |
| Total | 7,242,456 | 100.0% | 7,925,849 | 100.0% | 8,186,067 |
In the crisis matrix every component moves towards its standalone figure, and structured credit, whose beta to the fund rises from 1.312 to 1.453, becomes the sleeve whose marginal VaR is highest: 13,706 per million. On the reported matrix it gained most from diversification, a component of 0.772 times its standalone VaR; in the regime that matters it adds most risk per unit. How much of the diversification benefit survives is covered in how much diversification survives a crisis.
Two shortcuts circulate, and both mislead. The first allocates the fund VaR pro rata to standalone VaRs. Those add to 8,186,067, the undiversified figure, and scaling them down gives high yield 42.1 per cent, or 3,049,455, understating the sleeve that drives the fund's risk by 297,160. The second uses incremental VaR, the fall in VaR if the sleeve were sold to cash: 3,260,254 for high yield, 1,252,211 for investment grade, 1,538,917 for loans and 901,973 for structured credit. Each is a correct answer to a different question, and together they sum to 6,953,355, leaving 289,101 of the risk owned by nobody. Only the Euler components both add up and tell you what a marginal trade does.
Component VaR inherits every weakness of the VaR it splits: the normal tail, the one-day horizon, and the assumption that the sleeve can be sold. A sleeve with a small component and thirty days to sell can still be the one that breaks a redemption.
The VaR construction this allocation starts from, with both correlation matrices side by side, is in the free workbook for this case; a blank version for your own book is the VaR and expected shortfall model.
Marginal VaR is the change in VaR per unit added to a position: 11,383 per million for high yield here. Component VaR is marginal VaR times the position, and the components add to the total, 7,242,456. Incremental VaR is the change from removing the position entirely: 3,260,254 for high yield. Incrementals do not add up; here they sum to 6,953,355.
Yes, for parametric VaR and any risk measure that scales linearly with position size, by Euler's theorem on homogeneous functions. On the fund here the four components, 1,334,618, 3,346,615, 1,610,800 and 950,422, sum exactly to 7,242,456. For historical simulation the same allocation is estimated from the scenarios around the VaR day and is noisier.
Yes, when a position's covariance with the rest of the portfolio is negative, typically a hedge. Its component is then a reduction of total VaR, and adding to it lowers risk. In a long-only credit fund with correlations between 0.45 and 0.95 every component is positive; in the crisis matrix here the smallest share is still 14.5 per cent for structured credit.
The parametric VaR decomposed here is built one visible step at a time in the workbook for chapters 4, 6 and 7 of Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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