Module 6
Portfolio risk: Value-at-Risk
Once you hold more than one bond, the question becomes: how much could this whole book lose on a bad day? Value-at-Risk (VaR)answers it with a single number — e.g. “we're 99% confident we won't lose more than $X tomorrow.” Its companion, Expected Shortfall, answers the follow-up: “and on the days we doblow through VaR, how bad is it on average?”
There are three ways to compute it, and they disagree in instructive ways:
- Parametric — assume a bell curve and use the portfolio's volatility (from each position's risk, the factor vols, and their correlations). Fast, but thin-tailed.
- Monte-Carlo — simulate thousands of random days and read the loss off the distribution. Flexible.
- Fat-tailed — same, but with a Student-t that has bigger tails, because real interest-rate moves crash harder than a bell curve predicts. It reports a larger, more honest loss.
And the most important free lunch in finance shows up here too: diversification. Because your positions aren't perfectly correlated, the portfolio's VaR is lessthan the sum of each position's standalone VaR. Lower the correlation and watch the benefit grow.
🎛 VaR lab
Your book — DV01 per factor ($/bp; negative = long, loses when rates rise)
Parametric VaR
$93,225
Expected Shortfall
$106,907
Monte-Carlo VaR
$92,308
Fat-tailed VaR
$102,235
Simulated daily P&L distribution (15,000 paths)
VaR is the loss you shouldn't exceed on a normal bad day (here 99% of days). Parametric assumes a bell curve; Monte-Carlo simulates it; the fat-tailed version uses a Student-t to reflect that real rate moves have bigger tails than a bell curve — so it reports a larger loss. Educational tool — not investment advice.
Which risk factors matter for fixed income
A bond book doesn't have “a price that moves” — it has exposure to a handful of underlying risk factors, and the P&L is whatever those factors do. Get the factor list right and VaR falls out of it. The ones that matter:
- 1. Interest-rate level (duration / DV01). The big one. A parallel shift in the whole curve. Measured by DV01 — dollars per basis point. For most bond books this is 60–80% of the risk.
- 2. Curve shape (slope & curvature). The 2s10s steepening/flattening and the belly's bow. Two books with the same total DV01 can behave very differently if one is a steepener and the other a flattener. Captured by key-rate DV01s.
- 3. Credit spread. For corporates, EM and structured credit — the extra yield over the risk-free curve, and how far it can widen. Measured by CS01. Largely independent of rates, which is exactly why credit diversifies a rates book.
- 4. Inflation / breakevens. For TIPS and linkers — the real-vs-nominal wedge. Measured by IE01 (inflation DV01).
- 5. Volatility (vega). For anything with optionality — swaptions, callables, MBS. You're short or long vol, and implied vol is its own risk factor.
- 6. FX. Any non-domestic bond carries currency risk on top of its rate risk; often the currency move dwarfs the coupon.
- 7. Basis & financing. Repo/SOFR-OIS spreads, swap spreads, CDS-bond basis, cross-currency basis — the “plumbing” spreads that blow out in a crisis.
- 8. Prepayment (for MBS). Homeowners refinancing changes your cashflows and caps your upside — negative convexity as its own factor.
- 9. Liquidity. Not a price factor but a P&L reality: the bid-offer you actually cross to exit, which widens precisely when you need to sell.
The lab below is built exactly this way: each position is expressed as a sensitivity to one of these factors, and VaR aggregates them. That's how a real risk system works — not bond-by-bond, but factor-by-factor.
The whole engine: from sensitivities to the covariance matrix to one number
Parametric VaR is four steps. Everything a risk desk does sits on top of them, so here is the entire chain with real numbers — a three-factor book (rates, credit, FX).
Step 1 — turn each position into a daily $ volatility
Multiply each factor sensitivity by that factor's daily volatility. That gives sᵢ, the one-standard-deviation daily P&L from factor i on its own:
Rates: DV01 $75k/bp × 2.0 bp/day = s₁ = $150,000 Credit: CS01 $30k/bp × 2.0 bp/day = s₂ = $ 60,000 FX: $/% $67k/% × 0.6 %/day = s₃ = $ 40,000
Step 2 — the correlation matrix R
How the factors move together. Diagonal is always 1 (a factor is perfectly correlated with itself); off-diagonals are the pairwise correlations. Here rates and credit are slightly negatively correlated (risk-off rallies govvies while widening credit):
Rates Credit FX Rates [ 1.0 −0.2 0.1 ] Credit [−0.2 1.0 0.3 ] FX [ 0.1 0.3 1.0 ]
Step 3 — build the variance-covariance matrix Σ
This is the “meat.” Each cell is Σᵢⱼ = ρᵢⱼ · sᵢ · sⱼ — the correlation times the two $ vols. The diagonal is just each variance sᵢ²; the off-diagonals are the covariances that make diversification work. In matrix form Σ = D · R · D, where D is the diagonal matrix of the sᵢ:
Σ (in $²) Rates Credit FX Rates [ 22.50e9 −1.80e9 0.60e9 ] Credit [ −1.80e9 3.60e9 0.72e9 ] FX [ 0.60e9 0.72e9 1.60e9 ] e.g. Σ₁₂ = ρ₁₂·s₁·s₂ = −0.2 × 150,000 × 60,000 = −1.80e9 Σ₁₁ = s₁² = 150,000² = 22.50e9
Step 4 — collapse Σ to the portfolio volatility, then scale to VaR
Portfolio variance is the sum of every cell of Σ (equivalently the quadratic form sᵀ R s). Add the three variances on the diagonal and twice the three covariances above it:
σ²_P = Σᵢ Σⱼ Σᵢⱼ = (22.50 + 3.60 + 1.60)e9 + 2·(−1.80 + 0.60 + 0.72)e9 = 27.70e9 − 0.96e9 = 26.74e9 $² σ_P = √26.74e9 = $163,524 (portfolio daily vol) VaR₉₉ = 2.326 × σ_P = $380,356 (1-day, 99%)
The 2.326is the 99% z-score of the normal distribution; for 95% it's 1.645. For a 10-day horizon multiply by √10 (the square-root-of-time rule).
Why it's less than the sum of the parts
Each factor's standalone VaR is 2.326 · sᵢ: $348,900 + $139,560 + $93,040 = $581,500. But the portfolio VaR is only $380,356. The $201,144 difference is the diversification benefit— it comes entirely from the negative and small off-diagonal covariances. If every correlation were 1.0, Σ's cells would sum to (s₁+s₂+s₃)² and the benefit would vanish. That is the single most important idea in portfolio risk, and it lives entirely in the off-diagonal of Σ.
The three ways to actually compute it
Step 4 above is the parametric (variance-covariance) method — elegant, but it assumes a bell curve. The other two drop that assumption:
- Historical simulation. Don't assume any distribution — replay history. Take the last ~250–500 actual trading days of factor moves, apply each day's moves to today'spositions, and you get 250 hypothetical P&Ls. Sort them; the 99% VaR is the 3rd-worst of 250 (the 1st percentile). It captures fat tails and real correlations for freebecause it uses what actually happened — including the days a bell curve says can't occur. Its weakness: if the scary event isn't in your window, it isn't in your VaR.
- Monte-Carlo. Assume a distribution (normal, or fat-tailed Student-t), then draw thousands of correlated random days from it using the covariance matrix (via its Choleskyfactor). Most flexible — it can price non-linear positions like options that the parametric method can't — but it's only as good as the distribution you assume, and it's slow.
On the same book, historical and fat-tailed Monte-Carlo VaR usually come out larger than parametric — because real markets have fatter tails than the bell curve. That gap is the honest part of the number.
🎛 Multi-factor VaR — every input on the table
The same engine, across the fixed-income risk factors, with nothing hidden: edit each position's sensitivity and see its daily $ vol and standalone VaR; the factor vols and the full correlation matrix are shown; and VaR is computed all three ways — parametric, historical and Monte-Carlo — with the diversification benefit spelled out.
Inputs — your book's sensitivity to each risk factor
| Risk factor | Your sensitivity | Factor vol | Daily 1σ P&L | Standalone VaR |
|---|---|---|---|---|
| Rates (DV01) | 25,000 $/bp | 7 bp/d | $175,000 | $407,050 |
| Curve 2s10s | -8,000 $/bp | 3 bp/d | $24,000 | $55,824 |
| Credit (CS01) | 12,000 $/bp | 4 bp/d | $48,000 | $111,648 |
| Inflation (IE01) | 5,000 $/bp | 3 bp/d | $15,000 | $34,890 |
| Rate vol (vega) | 3,000 $/pt | 1.5 pt/d | $4,500 | $10,467 |
| FX | 6,000 $/% | 0.6 %/d | $3,600 | $8,374 |
Correlation matrix (the other input)
| Rates | 1.0 | -0.4 | -0.2 | 0.5 | -0.1 | 0.1 |
| Curve | -0.4 | 1.0 | 0.2 | -0.1 | 0.1 | 0.0 |
| Credit | -0.2 | 0.2 | 1.0 | -0.1 | 0.5 | 0.2 |
| Inflation | 0.5 | -0.1 | -0.1 | 1.0 | 0.0 | 0.1 |
| Rate | -0.1 | 0.1 | 0.5 | 0.0 | 1.0 | 0.2 |
| FX | 0.1 | 0.0 | 0.2 | 0.1 | 0.2 | 1.0 |
Parametric VaR
$441,969
Historical VaR
$715,201
Monte-Carlo VaR
$435,209
Historical P&L distribution (250 days)
A fixed-income book's risk lives in a handful of factors — rates, curve shape, credit spread, inflation breakevens, volatility and FX. Express each position as a sensitivity, multiply by each factor's daily volatility, then aggregate through the correlation matrix. Parametric assumes a bell curve; historical replays the last 250 actual days (fat tails and all); Monte-Carlo simulates thousands. Educational tool — not investment advice.
VaR is a one-day snapshot — but risk plays out over time
Everything so far gives a single number for tomorrow. In reality a book lives for weeks and years, and its P&L is a path — up some days, down others. The simulator below runs that path forward: each day it draws a correlated, fat-tailed shockto all six risk factors, revalues your positions, and books the day's P&L. Chain the days together and you can watch the account equity wander, see its drawdowns, and count how often it actually breaches the 99% VaR line — which, over 252 days, should happen ≈2–3 times. That count is a VaR backtest.
Then flip to the deterministic panel: instead of random days, youshock the factors — rates +100bp, credit +175bp, a risk-off scramble — and read the instant P&L, factor by factor, with the total dropped onto the path so you can see the exact P&L difference the shock makes.
🎛 P&L path simulator — 1 week to 10 years
Cumulative P&L — one random path of daily returns
Total P&L
$1,583,829
Best day
$568,745
Worst day
-$750,291
Annualised vol
$2,705,771
Max drawdown
-$1,790,899
VaR breaches
2 / 2.5 exp
Over 252 days you expect ≈2.5 days worse than the 1-day 99% VaR ($441,969). Re-roll the path a few times — sometimes you get more, sometimes fewer. That variability is exactly what backtesting a VaR model measures.
Shock the risk factors — see the instant P&L
| Factor | Position | Shock | P&L |
|---|---|---|---|
| Rates (DV01) | 25k $/bp | 0 bp | $0 |
| Curve 2s10s | -8k $/bp | 0 bp | $0 |
| Credit (CS01) | 12k $/bp | 0 bp | $0 |
| Inflation (IE01) | 5k $/bp | 0 bp | $0 |
| Rate vol (vega) | 3k $/pt | 0 pt | $0 |
| FX | 6k $/% | 0 % | $0 |
| Net scenario P&L | $0 | ||
That's a 0.0σ day for this book — inside the 99% VaR. The dashed purple line above shows the same path with this shock dropped in on day 101; the gap between the two lines isthe P&L difference.
Each day the six risk factors get a correlated random shock(fat-tailed, via the same covariance matrix behind the VaR number); your positions revalue and that's the day's P&L. Stringing the days together gives the P&L path, its drawdowns, and how often it breaches VaR. The purple scenario shows a deterministicshock instead — you choose the factor moves and read the P&L straight off. Educational tool — not investment advice.
How a hedge fund reads VaR — and what it really models
A bank computes VaR to satisfy a regulator: back-test it, colour it green, hold capital against it, and try to keep it low. A hedge fund flips that on its head. VaR isn't a constraint to minimise — it's the raw material of return. The job is to take risk efficiently, so a fund cares less about the single number and more about the questions around it:
- Where is the risk actually coming from? (component & marginal VaR) Standalone VaR double-counts and hides hedges. Funds decompose total VaR into each position's contribution — which sums to the total — and watch marginal VaR: how much risk the next trade adds. That's how they find the trade that adds return without adding risk, and spot hidden concentration (“we think we're diversified but 85% of our risk is one rates bet”).
- How big should the book be? (vol targeting & leverage) Most funds run to a target volatility — say 10% annualised — and lever up or down to hit it. A relative-value fixed-income fund might run 5–20× gross because each trade is low-vol; VaR tells them how much leverage that target implies.
- Who gets how much rope? (risk budgeting) Multi-manager “pod shops” hand each PM a VaR and drawdown budget, and auto-cut a PM who draws down ~5–7.5%. Total-fund VaR is built bottom-up from pods, netting their diversification.
- Is the return worth the risk? (Sharpe, return-on-VaR) The goal is return per unit of risk. Leverage can't manufacture that — it scales return and vol together, so the Sharpe is invariant. Skill is the only thing that moves it.
- What does VaR miss? (stress, liquidity, financing, crowding) Funds distrust VaR's thin tails, so they lean on scenario stress (2008, the 2013 taper tantrum, COVID March 2020, the 2022 rate shock), liquidity-adjusted VaR (days to actually exit — crowded trades gap), and financing risk (a repo-haircut hike forces deleveraging at the worst moment). The most famous relative-value blow-up — a giant, Nobel-laureate-run fund that failed in 1998 — is the cautionary tale: its VaR looked fine; leverage, illiquidity and crowding are what killed it — exactly the things a single VaR number doesn't show.
The lab below models the first four directly on our factor book: it splits VaR into component contributions, computes the leverage needed to hit a vol target on a given NAV, and shows the Sharpe that risk buys.
🎛 The hedge-fund lens — risk budgeting, leverage & return-on-risk
1 · Where is the risk — component VaR, not standalone
Standalone VaR double-counts, because it ignores hedges. Component VaR splits the actualportfolio VaR into each factor's true contribution — and it sums exactly to the total. A negatively-correlated position shows a negativecontribution: it's a hedge.
| Factor | Position | Standalone | Component | % of total risk |
|---|---|---|---|---|
| Rates | 25k | $407,050 | $390,764 | 88.4% |
| Curve | -8k | $55,824 | $25,104 | 5.7% |
| Credit | 12k | $111,648 | $5,682 | 1.3% |
| Infl | 5k | $34,890 | $18,446 | 4.2% |
| Vol | 3k | $10,467 | $513 | 0.1% |
| FX | 6k | $8,374 | $1,459 | 0.3% |
| Portfolio VaR (1-day 99%) | $441,969 | 100% | ||
2 · Size the book — vol targeting & leverage
Current vol
6.0%
of NAV, ann.
Leverage to target
1.66×
gear up
VaR at target
$732,621
1-day 99%
VaR / NAV
1.47%
daily loss cap
3 · Is it worth it — return per unit of risk
Sharpe ratio
1.50
Respectable — allocators will look. Note the Sharpe doesn't change with leverage — gearing up scales bothreturn and vol, so it can't manufacture skill.
A bank uses VaR to satisfy a regulator; a fund uses it to size and price risk. It hunts the marginaltrade that adds return without adding much risk, runs to a vol target with leverage, hands each PM a VaR/drawdown budget, and judges everything by return per unit of risk — never by minimising VaR. Educational tool — not investment advice.
Putting it all together — a real portfolio, end to end
Everything so far has used stylised factor positions. The lab below runs the whole workflow on an actual book of six bonds — two Treasuries, an investment-grade and a high-yield corporate, a TIPS and a EUR Bund. It shows what each was bought at versus today's market(and the unrealised P&L), then maps every bond to its risk factors from its own duration, spread duration and currency, and runs Monte Carlo VaR— the most flexible method, because a mixed book of rates, credit, inflation and FX doesn't fit one neat formula.
From there it computes the Sharpe ratio step by step, breaks the risk down to each position's carry per unit of VaR, and spells out how a PM would act on it — trim the line that's eating risk without paying for it, add the one that earns cheap risk, and size the whole book to a vol target. Resize any position and watch it all recompute.
🎛 Fixed-income portfolio — Monte Carlo VaR, Sharpe & the PM's view
1 · The book — cost vs. market
| Position | Sector | Face $M | Bought | Market | Mkt value | Unreal. P&L | Wt |
|---|---|---|---|---|---|---|---|
| UST 4.25% 2034 | Govt | 25 | 99.50 | 101.20 | $25.3M | $425,000 | 25% |
| UST 4.50% 2027 | Govt | 30 | 100.10 | 99.80 | $29.9M | -$90,000 | 29% |
| Apple 4.5% 2033 | IG Corp | 15 | 97.00 | 98.40 | $14.8M | $210,000 | 14% |
| Ford 6.75% 2030 | HY Corp | 10 | 101.00 | 99.20 | $9.9M | -$180,000 | 10% |
| TIPS 1.75% 2032 | TIPS | 12 | 96.00 | 97.80 | $11.7M | $216,000 | 11% |
| Bund 2.3% 2034 | EUR Govt | 10 | 98.50 | 100.10 | $10.8M | $172,800 | 11% |
| Portfolio | $102.5M | $753,800 | 0.7% | ||||
Drag a position's slider to resize it (0–2× face) and watch every number below move — that's the PM rebalancing.
2 · How much could it lose? — Monte Carlo VaR
The book is mapped to four risk factors Rates, Credit, Inflation, FX(from each bond's duration, spread duration and currency). We draw 20,000 correlated, fat-tailed days, revalue the book on each, and read the loss off the distribution — the most flexible and realistic method for a mixed portfolio.
MC 99% VaR
$1,027,982
1-day
Expected shortfall
$1,378,903
avg. of worst 1%
Parametric VaR
$940,489
normal, for compare
VaR / NAV
1.00%
of market value
3 · Risk vs. return, position by position
Component VaRis each position's true share of portfolio risk (it sums to the total). Divide the position's annual carry by it and you get carry-per-VaR — how much return each unit of risk actually buys. That, not the headline yield, is what a PM optimises.
| Position | Yield | Ann. carry | Component VaR | % of risk | Carry / VaR |
|---|---|---|---|---|---|
| UST 4.25% 2034 | 4.10% | $1,037,300 | $323,532 | 34.4% | 3.21× |
| UST 4.50% 2027 | 4.55% | $1,362,270 | $129,218 | 13.7% | 10.54× |
| Apple 4.5% 2033 | 4.70% | $693,720 | $156,119 | 16.6% | 4.44× |
| Ford 6.75% 2030 | 7.05% | $699,360 | $64,434 | 6.9% | 10.85× |
| TIPS 1.75% 2032 | 4.15% | $487,044 | $118,986 | 12.7% | 4.09× |
| Bund 2.3% 2034 | 2.25% | $243,243 | $148,200 | 15.8% | 1.64× |
4 · Return per unit of risk — the Sharpe ratio
Sharpe = (expected return − cash) ÷ annualised volatility = (7.00% − 4.3%) ÷ 6.26% = 2.70% ÷ 6.26% = 0.43Running yield
4.41%
Annualised vol
6.26%
Sharpe
0.43
Return on VaR
7.0×
Why each piece is there:
• Expected return − cash. You only get credit for return above what cash (T-bills) pays for free. That gap is the excess return, the reward for taking risk. Here 7.00% − 4.3% = 2.70%.
• ÷ annualised volatility. We scale that reward by how much the book bounces around. Volatility starts as a daily dollar number from the VaR engine (σ_day = √(sᵀ·R·s) = $404,337), gets annualised by ×√252 (252 trading days, and variance grows with time so vol grows with its square root) = $6,418,657, then divided by the $102M book to become 6.26%.
• The ratio is return per unit of risk. Above ~1 is good, above ~2 is elite. It's unit-free and leverage-invariant— doubling the book doubles both excess return and vol, so Sharpe can't be faked with leverage.
Notice this unlevered book's carry barely clears cash — which is exactly why a fund levers a high-Sharpe book to a vol target rather than holding it raw.
▸ Show the math — every metric, with today's numbers
Worked below on UST 4.25% 2034 (the current top-risk line). Every number updates as you move the sliders.
① Market value & unrealised P&L
MV = face × price/100 × fx = 25 × 101.20/100 = $25.30M P&L = MV − cost = $25.30M − $24.88M = $425,000
② Map the bond to risk-factor sensitivities
A bond's risk is its duration in dollars. Rate DV01 = dollars lost per +1bp; multiply by the factor's daily vol to get its daily 1σ P&L.
rate DV01 = −MV × modDur × 0.0001 = −25,300,000 × 8 × 0.0001 = -$20,240/bp daily 1σ (rates) = DV01 × 7bp/day = -$141,680this position's factor vector a = [-141,680, 0, 0, 0] ($ per factor)
③ Portfolio volatility — combine factors through the correlation matrix
Add every position's factor vector to get the book vector s, then σ = √(sᵀ·R·s) — the quadratic form that nets out diversification.
s (book) = [-422,410, -56,813, 15,210, 64,865] factors: Rates, Credit, Inflation, FXR = [ 1.00 -0.20 0.50 0.10
-0.20 1.00 -0.10 0.20
0.50 -0.10 1.00 0.10
0.10 0.20 0.10 1.00 ] σ_day = √(sᵀ·R·s) = $404,337 (daily portfolio $ vol)④ VaR & Expected Shortfall
parametric 99% VaR = 2.326 × σ_day = 2.326 × $404,337 = $940,489Monte Carlo 99% VaR = 1st-percentile of 20,000 correlated fat-tailed days = $1,027,982Expected Shortfall = average of the worst 1% of those days = $1,378,903
⑤ Component VaR — each position's true share
First R·s (how each factor co-moves with the whole book), then each position's contribution. They sum exactly to total VaR.
R·s = [-396,956, 39,121, -183,827, 12,782] component VaRᵢ = 2.326 × (aᵢ · R·s) / σ_dayUST 4.25% 2034 : 2.326 × (56,240,728,914) / 404,337 = $323,532 (34.4% of total) Σ over positions = $940,489 ✓ (= parametric VaR)
⑥ Carry-per-VaR, Sharpe & return-on-VaR
carryᵢ = MV × yield = $25.3M × 4.10% = $1,037,300carry / VaRᵢ = carryᵢ / |component VaRᵢ| = 3.21× ann. vol % = σ_day × √252 / MV = $6,418,657 / $102M = 6.26% Sharpe = (exp.ret − cash) / ann.vol% = (7.00% − 4.3%) / 6.26% = 0.43return on VaR = (exp.ret × MV) / VaR = 7.0×
5 · How the PM reads this
- Concentration check. UST 4.25% 2034 is carrying the most risk (34% of portfolio VaR) — the PM asks whether that single factor bet is intended, and hedges or trims if it isn't.
- Best use of risk. Ford 6.75% 2030 earns the most carry per unit of VaR (10.85×) — often a credit name that looksrisky standalone but diversifies the rates book, so it's cheap risk. Add here.
- Worst use of risk. Bund 2.3% 2034 earns the least per unit of VaR (1.64×) — a candidate to cut or hedge first. Trim its slider and watch VaR fall faster than carry.
- Sizing. Sharpe 0.43 at 6.3% vol. If the mandate targets a higher vol, the PM levers the whole book up; if VaR/NAV (1.00%) breaches the desk limit, they cut the top-risk line first.
Illustrative holdings and analytics for education — not investment advice or a real portfolio.
Things to try
- • Drop the correlation toward 0 — the diversification benefit jumps, and portfolio VaR falls well below the sum of parts.
- • Compare parametric vs fat-tailed VaR — the Student-t reports a bigger loss. That gap is why 2008 “shouldn't have happened.”
- • Switch to 10-day and 99% — how regulators size capital.
- • In the multi-factor lab, push correlation stress to 100% — the diversification benefit collapses and portfolio VaR climbs toward the sum of parts. That's what a crisis does: everything correlates to 1.
- • Compare the lab's historical vs parametric tiles — historical (fat tails) reports the bigger, more honest loss.
- • In the path simulator, set 1 year and re-roll a dozen times — the VaR-breach count bounces around 2–3. Then switch to 10 years and watch a real drawdown build.
- • Hit the “Credit crisis” preset — see which factors make money (short credit) and which lose, and how the net compares to a normal day (σ multiple).
- • In the hedge-fund lens, watch Credit's component VaR: its standalone VaR is large, but because it partly hedges rates its true contribution is tiny — that's the gap between how risky a trade looks and how risky it is.
- • Raise target vol from 10% to 20% and watch the leverage and VaR/NAV double — then note the Sharpe doesn't budge. Leverage isn't alpha.
- • In the portfolio lab, drag the Ford HY slider up — carry rises but VaR barely moves (it diversifies the rates book), so its carry-per-VaR is high. Then drag the 10Y UST up and watch VaR jump: same risk, less carry.