Parametric VaR for a portfolio · RiskAtoms Academy

By the end you will be able to

1Compute the delta-normal VaR of a multi-factor book from its sensitivity vector and the covariance matrix.
2Split the VaR into marginal and component VaR by factor, by book and by position, and tell component from incremental VaR.
3Use marginal VaR to find the trades that cut risk most, and the best hedge in one factor.
4Say where delta-normal VaR breaks: options, fat tails and changing correlations.

Why it matters

Ask each of RiskAtoms Bank's four books for its own 99% one-day VaR and add them up: £3.46m. Ask for the desk's VaR and the answer is £1.94m. The gap is not an error. The books partly hedge each other, and the only way to see by how much, and who should get credit for it, is to put the whole desk into one covariance matrix.

Core explanation

One covariance matrix for the whole desk

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

Eight positions mapped onto eight factors in L3.2.02: Risk factors and position mapping. One day, 99%, mean zero, in pounds.

L2.5.06: Value at risk: the idea computed VaR for one fund as z × σ × value, and L3.1.14: Risk contributions and risk budgeting split a portfolio's VaR among assets held as weights. A trading desk is different in one way: its exposures are not weights but sensitivities. The book makes £40.73m per 1.00 log return of the UK index (so £407k for a 1% rise), −£92,489 per basis point of the 10-year rate, and so on. Multiply each by the factor's daily volatility and you have the size of a typical day's P&L from that factor alone:

Factor	δ (sensitivity)	Daily σ	\|δ\| × σ	Stand-alone 99% VaR
UK index	£40.73m per 1.00 log return	1.0%	£407k	£948k
US index	£30.00m per 1.00 log return	1.1%	£330k	£768k
2y rate	£35,143 per bp	6.0 bp	£211k	£491k
10y rate	−£92,489 per bp	7.0 bp	£647k	£1,506k
GBP/USD	−£10.00m per 1.00 log return	0.5%	£50k	£116k
Brent	£12.31m per 1.00 log return	2.0%	£246k	£573k
UK vol	£124,091 per vol pt	0.8 pts	£99k	£231k
IG spread	−£34,303 per bp	2.5 bp	£86k	£200k
Sum				£4,831k

The 10-year rate is the biggest single risk, then the UK index. But the factors do not move independently, and several positions are hedges, so the stand-alone figures add up to far more than the desk can lose. The delta-normal method combines them with the factor correlations (L3.1.06: Many assets and the covariance matrix), turned into a covariance matrix $\Sigma$ :

Calm daily correlations of the eight factor moves
	UK index	US index	2y rate	10y rate	GBP/USD	Brent	UK vol	IG spread
UK index	1.00	0.60	0.10	0.15	−0.20	0.30	−0.75	−0.50
US index	0.60	1.00	0.10	0.15	0.10	0.30	−0.55	−0.50
2y rate	0.10	0.10	1.00	0.75	0.20	0.10	−0.10	−0.10
10y rate	0.15	0.15	0.75	1.00	0.10	0.15	−0.15	−0.15
GBP/USD	−0.20	0.10	0.20	0.10	1.00	0.10	−0.10	−0.15
Brent	0.30	0.30	0.10	0.15	0.10	1.00	−0.25	−0.30
UK vol	−0.75	−0.55	−0.10	−0.15	−0.10	−0.25	1.00	0.55
IG spread	−0.50	−0.50	−0.10	−0.15	−0.15	−0.30	0.55	1.00

The quadratic form (M-E02: Matrix multiplication and quadratic forms) does the rest: $\sigma^2 = \delta^\top\Sigma\,\delta$ gives a daily P&L standard deviation of £835,845, and with the normal quantile (M-C06: The normal distribution) the 99% VaR is 2.326 × £835,845 = £1,944,466. At 95% it would be £1,374,843.

Who owns the VaR?

The marginal VaR of a factor is how much the VaR rises per unit of extra exposure to it; the component VaR is exposure × marginal VaR. As in L3.1.14: Risk contributions and risk budgeting, the components add up exactly to the total (the Euler allocation). By factor:

UK index£659k

US index£510k

2y rate−£110k

10y rate£628k

GBP/USD£13k

Brent£265k

UK vol−£115k

IG spread£93k

Desk VaR£1,944k

Three factors carry most of the risk: the UK index (34%), the 10-year rate (32%) and the US index (26%). The 2-year rate and UK vol have negative components: the pay-fixed swap and the puts' vega offset the rest. By book, with each book's stand-alone VaR for comparison:

Book	Stand-alone VaR	Component VaR	Share of desk VaR
Equities	£1,451k	£1,094k	56.2%
Rates	£877k	£364k	18.7%
FX and commodities	£639k	£239k	12.3%
Credit	£493k	£248k	12.7%
Desk	£3,460k	£1,944k	100.0%

The Rates book has a stand-alone VaR of £877k but owns only £364k of the desk's: rates tend to fall when equities fall, so the long gilt partly hedges the equity book. The sum of stand-alone book VaRs, £3.46m, minus the desk's £1.94m is the diversification benefit, which L3.2.13: Scaling and aggregating risk takes further.

Component or incremental?

Component VaR says whose VaR it is today. Incremental VaR says what happens if a position goes. They differ, because marginal rates change as a position shrinks:

Position	Stand-alone	Component	Incremental (VaR falls by)
UK index futures (long £60m)	£1,396k	£971k	£547k
US equity basket ($39m)	£811k	£549k	£427k
UK index puts (1,000 contracts)	£640k	−£427k	−£474k
10-year gilt (£100m face)	£1,297k	£541k	£112k
2-year swap, pay fixed (£300m)	£790k	−£177k	−£313k
Sell $26m forward	£233k	−£26k	−£40k
Brent futures (200,000 barrels)	£573k	£265k	£190k
Corporate bonds (£80m)	£493k	£248k	£195k

The puts and the swap have negative incremental VaR: taking them away would raise the desk's VaR.

Using marginal VaR

The UK index has a marginal VaR of 0.0162: each £1 of extra UK index delta adds about 1.6p of VaR. Selling £10m of UK index futures should cut the VaR by about £162k; the exact recomputed change is £154k, a little less because the marginal rate falls as the position shrinks. Taken to its end, the VaR-minimising UK hedge sells £58.2m of UK index delta, where its marginal VaR reaches zero, and leaves a VaR of £1.40m.

Where delta-normal goes wrong

The method is fast and it splits cleanly, which is why desks still use it for limits and attribution. It rests on three shortcuts. Positions are linear in their factors, so the puts' gamma is ignored. Moves are normal, so fat tails are missed. And one correlation matrix holds for every day. On this book, full-revaluation historical simulation gives £2.27m (L3.2.10: Historical simulation in practice) and full-revaluation Monte Carlo £1.85m (L3.2.11: Monte Carlo VaR). The correlations matter in both directions here: with all correlations set to zero the VaR would be £2.10m, and with the crisis correlations of the stressed spell (same volatilities) £1.89m, slightly lower, because in a crisis rates fall harder with equities and the book's hedges work better.

Sort itWhich measure answers the question?

Pick stand-alone, marginal, component or incremental VaR.

How much VaR would the Rates book have if it were a separate desk?
Which factor is the cheapest place to add £1m of exposure?
How should the desk's VaR limit be split among the four books so the parts add up?
What happens to the desk's VaR if the FX forward is closed out entirely?

The formal version

Let $x = (x_1, \dots, x_n)^\top$ be the day's factor moves and $\delta$ the book's sensitivities. Assume $x \sim N(0, \Sigma)$ and P&L $= \delta^\top x$ . Then

\begin{aligned} \mathbb{E}[\text{P\&L}] &= \delta^\top\,\mathbb{E}[x] = 0 \\ \operatorname{var}(\text{P\&L}) &= \mathbb{E}\big[(\delta^\top x)^2\big] = \mathbb{E}\big[\delta^\top x\,x^\top\delta\big] = \delta^\top\,\mathbb{E}[x\,x^\top]\,\delta = \delta^\top\Sigma\,\delta \\ &= \sum_i \sum_j \delta_i\,\delta_j\,\sigma_{ij} = \sum_i \delta_i^2\sigma_i^2 + 2\sum_{i<j} \delta_i\,\delta_j\,\rho_{ij}\,\sigma_i\sigma_j \\ \text{VaR}_\alpha &= z_\alpha\,\sigma, \qquad \sigma = \sqrt{\delta^\top\Sigma\,\delta} \end{aligned}

The second line uses $(\delta^\top x)^2 = \delta^\top x\,x^\top\delta$ (a number equals its own transpose) and $\mathbb{E}[xx^\top] = \Sigma$ because the mean is zero. A linear combination of jointly normal variables is normal, so the P&L is $N(0, \sigma^2)$ and its $(1-\alpha)$ quantile is $-z_\alpha\sigma$ .

Marginal and component VaR. Differentiate with respect to one sensitivity, as in L3.1.14: Risk contributions and risk budgeting:

\begin{aligned} \frac{\partial\,\text{VaR}}{\partial \delta_i} &= z_\alpha\,\frac{\partial \sigma}{\partial \delta_i} = z_\alpha\,\frac{1}{2\sigma}\,\frac{\partial\,(\delta^\top\Sigma\delta)}{\partial \delta_i} = z_\alpha\,\frac{(\Sigma\delta)_i}{\sigma} \\ \sum_i \delta_i\,\frac{\partial\,\text{VaR}}{\partial \delta_i} &= \frac{z_\alpha}{\sigma}\,\delta^\top\Sigma\,\delta = z_\alpha\,\sigma = \text{VaR} \end{aligned}

For a book $B$ with its own sensitivities $\delta^{B}$ (so $\delta = \sum_B \delta^B$ ), the component is $\sum_i \delta^B_i\, z_\alpha(\Sigma\delta)_i/\sigma$ , and the books' components add up to the VaR too. Setting a marginal VaR to zero gives the best hedge in factor $k$ : add $-(\Sigma\delta)_k/\sigma_{kk}$ of exposure.

The two equity indices in full (a 2 × 2 example)

Step 1 of 4

1
Sensitivities and covariance
$\delta = \begin{pmatrix} 40.732 \\ 30.000 \end{pmatrix}\ \text{£m}, \qquad \Sigma = \begin{pmatrix} 1.000 & 0.660 \\ 0.660 & 1.210 \end{pmatrix} \times 10^{-4}$

Try it

Resize the positions and watch the components move. Remove the hedges and see the VaR jump; then try the crisis correlations, and zero correlations, on today's book.

Try itResize the desk and watch its VaR

UK index futures (long £60m)100%

US equity basket ($39m)100%

UK index puts (1,000 contracts)100%

10-year gilt (£100m face)100%

2-year swap, pay fixed (£300m)100%

Sell $26m forward100%

Brent futures (200,000 barrels)100%

Corporate bonds (£80m)100%

Correlations

Split the VaR by

Equities£1,094k

Rates£364k

FX and commodities£239k

Credit£248k

Total VaR£1,944k

99% one-day VaR£1.944m

σ of daily P&L£835.8k

Sum of stand-alone VaRs£6.231m

Diversification£4.287m

Change from today£0.0k

Crisis correlations keep today's volatilities: only the co-movements change.

Worked example

The desk's 99% VaR, step by step

Eight factors, one number

Step 1 of 5

1
The sensitivity vector
From the book, in £ per unit move (UK index, US index, GBP/USD and Brent per 1.00 log return in £m; the rates and the spread per bp; UK vol per vol point):
$\delta = (40.73\text{m},\ 30.00\text{m},\ 35{,}143,\ -92{,}489,\ -10.00\text{m},\ 12.31\text{m},\ 124{,}091,\ -34{,}303)^\top$

Regulatory lens · looking ahead

The Basel Committee's 1996 Market Risk Amendment let banks use their own VaR models, at 99% over ten days, for market risk capital. FRTB has since replaced VaR with expected shortfall for capital and added its own sensitivities-based standardised approach (M4.3: Market risk capital (FRTB) and CVA). Inside banks, delta-normal VaR and its Euler split remain useful tools for limits (L3.2.16: Market risk limits) and for allocating capital (M7.1: Enterprise risk and economic capital). The covariance matrix tool runs the same algebra on your own inputs.

Spark Atoms

Spark AtomGiving the matrix away

In October 1994 J.P. Morgan published its RiskMetrics methodology, with daily volatilities and correlations for hundreds of market factors, free of charge. Delta-normal VaR on a published covariance matrix quickly became the industry's common language, and the RiskMetrics Technical Document went through four editions by 1996.

Spark AtomA crisis that lowers VaR

You might expect crisis correlations to raise every VaR. For this desk, with today's volatilities, they lower it from £1.94m to £1.89m: when rates rally harder as equities fall, the gilt hedges the equity book better. Crisis volatilities, which are 2.5 times higher here, are another matter, as L3.2.11: Monte Carlo VaR shows.

Key takeaways

A desk's exposures are sensitivities δ (£ per unit factor move), not weights.
Delta-normal VaR = z √(δ′Σδ). For RiskAtoms Bank's desk: σ = £835,845, 99% one-day VaR = £1,944,466.
Marginal VaR = z (Σδ)ᵢ / σ; component VaR = exposure × marginal VaR, and the components add up to the VaR (Euler).
Hedges have negative components; incremental VaR (remove a position) differs from component VaR.
Setting a marginal VaR to zero gives the best one-factor hedge.
Delta-normal ignores gamma and fat tails and fixes the correlations; full revaluation methods follow.

Reaction

Check what stuck

Question 1 · objective 1

A book has δ = (£10m, −£4m) per 1.00 log return on two indices with daily volatilities 1% and 2% and correlation 0.5. What is the standard deviation of its daily P&L, in pounds?

£

Question 2 · objective 1

Same book. What is its 99% one-day delta-normal VaR, in pounds (z = 2.326)?

£

Question 3 · objective 2

Same book. What is the first index's component VaR, in pounds?

£

Question 4 · objective 2

The puts have a component VaR of −£427k. A trader proposes selling them because "they cost theta and add nothing". What does the negative component say?

AThe puts hedge the rest of the book: removing them would raise the desk's VaRBThe puts are losing moneyCThe puts' VaR is mis-measured; components cannot be negativeDThe puts should be doubled, whatever it costs

Question 5 · objective 3

Which of these does delta-normal VaR get wrong for this book?

AThe puts' gamma, because it treats every position as linearBThe diversification between rates and equitiesCThe size of the gilt's DV01DThe confidence level

Glossary

Delta-normal VaR: Parametric VaR that treats the book's P&L as linear in its risk factors (the deltas) and the factor moves as jointly normal: VaR = z √(δ′Σδ).
Sensitivity vector: The book's sensitivities to each risk factor, stacked into one vector δ: £ per unit move of each factor.
Stand-alone VaR: The VaR of one book, factor or position on its own, ignoring everything else the desk holds.
Marginal VaR: How much the VaR rises per unit of extra exposure to one factor (or one more unit of a position): z (Σδ)ᵢ / σ.
Component VaR: One factor's, position's or book's share of the VaR: its exposure times its marginal VaR. The shares add up to the VaR exactly.
Incremental VaR: The change in VaR when a whole position is removed (or added). It does not add up across positions.
Diversification benefit: The gap between the sum of stand-alone VaRs and the VaR of the whole: what the desk saves by holding positions that do not all lose together.

Deep Dives & Origins

Deep Dives

P. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd ed., McGraw-Hill, 2007, the chapters on portfolio VaR and VaR tools. Intermediate.
C. Alexander, Market Risk Analysis, Volume IV: Value-at-Risk Models, Wiley, 2008. Advanced.

Origins

J.P. Morgan/Reuters, RiskMetrics: Technical Document, 4th ed., 1996.
Basel Committee on Banking Supervision, Amendment to the Capital Accord to incorporate market risks, BIS, January 1996.
M. Garman, “Taking VaR to Pieces”, Risk 10(10), 1997.
D. Tasche, “Risk Contributions and Performance Measurement”, working paper, Technische Universität München, 1999.

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