Credit spread risk · RiskAtoms Academy

By the end you will be able to

1Compute a position's CS01 and spread duration, and tell spread risk from interest rate risk.
2Use bucketed CS01 to see where on the spread curve the risk sits, and estimate a spread VaR.
3Measure jump-to-default risk and explain why spread VaR misses it.
4Hedge spread and default risk with a CDS, and name what is left over.

Why it matters

On day 486 of the desk's history, investment-grade spreads widened 25bp in one day. The corporate bonds lost £852k from the wider spread, while the rally in swap rates gave back £673k. A risk report that showed only interest rate risk would have called this book well hedged. And if one issuer had defaulted that night, the loss would have been £6.0m, about 3 times the whole desk's 99% VaR.

Core explanation

Two ways to lose money on a corporate bond

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.

The credit book is £80m of corporate bonds, modelled as one 5-year bond bought at par with a 5.35% annual coupon. Its yield is the 5-year swap rate, 4.15%, plus the investment-grade spread of 120bp.

L1.4.07: Credit ratings and credit spreads defined a credit spread as the extra yield a bond pays over a safe benchmark. A trading desk splits a corporate bond's yield into those two parts and manages them separately, often on different desks: the benchmark rate belongs to the rates traders (L3.2.04: Interest rate risk in the trading book), the spread to the credit traders. The desk's bonds yield 5.35% = 4.15% + 1.20%. A corporate bond can lose money in two quite different ways: its spread can widen as the market charges more for the issuer's credit, or the issuer can default.

CS01: the spread's DV01

CS01 is the change in value for a one-basis-point widening of the spread, with the benchmark rate unchanged. For the desk's bonds it is −£34,303 per basis point. Dividing by the value gives the spread duration: £34,303 ÷ (£80m × 0.0001) = 4.29 years.

The same bond's DV01 to swap rates, split over the 2-year and 10-year points the book maps it to, is −£21,440 + −£12,864 = −£34,303 per basis point: the same number. That is no accident. Rate and spread enter the yield in the same way, so a basis point of either moves the price by the same amount. The two risks are still different, because rates and spreads do not move together: in the book's calm correlations, the 10-year rate and the spread have a correlation of just −0.15.

Where the two come apart

A floating-rate note has almost no interest rate risk, because its coupon resets with SONIA. Its margin over SONIA is fixed, though, so its spread duration is about as long as a fixed-rate bond's. Hedge a fixed-rate bond with a pay-fixed swap and you have built much the same thing: the rate risk is gone and the spread risk is all that is left.

Which spread?

“The spread” can be measured against several benchmarks. The book uses the simplest that matches its rate curve. Open each card for what it is and when desks use it.

Spread curves and bucketed CS01

An issuer's spreads differ by maturity: the spread curve. For investment-grade issuers it usually slopes upwards, because there is more time for the credit to weaken. An illustrative curve, with the book's 120bp at five years:

Maturity	1y	3y	5y	7y	10y
Spread over swaps	70bp	95bp	120bp	135bp	150bp

Bump one pillar of the curve at a time and you get bucketed (key-rate) CS01, just as L3.2.04: Interest rate risk in the trading book did for swap rates. The bonds pay a coupon every year, so a little of their CS01 sits at the 1-year and 3-year pillars, but most is at five years with the principal:

1-year pillar−£752

3-year pillar−£2,068

5-year pillar−£31,484

7-year pillar£0

10-year pillar£0

Total (parallel)−£34,303

The buckets let a desk see curve trades. A desk long five-year bonds and short ten-year CDS protection could have a parallel CS01 near zero and still lose if the curve flattens. For the desk's bonds, the three reshapes below each widen spreads by up to 40bp; the bucket figures predict the full revaluation closely:

Reshape (bp at 1/3/5/7/10y)	Σ bucket CS01 × move	Full revaluation
Parallel widening (20/20/20/20/20)	−£686k	−£682k
Front end widens (flattener) (40/25/10/5/0)	−£397k	−£395k
Long end widens (steepener) (0/5/10/25/40)	−£325k	−£324k

From CS01 to a spread VaR

The book measures credit spread risk with one factor: an investment-grade spread index with a calm daily volatility of 2.5bp. A first stand-alone spread VaR at 99% for one day is then z × |CS01| × σ = 2.326 × £34,303 × 2.5 = £199.5k. The whole credit book, with its rate risk added in, has a stand-alone VaR of £493k; L3.2.09: Parametric VaR for a portfolio builds that figure properly. Over the 750 days, the largest one-day moves in the spread were 25bp wider and 16.45bp tighter.

One index is a simplification. Real credit desks map each bond to spread curves by rating, sector and region, and keep an issuer-specific (idiosyncratic) part on top: an issuer's spread can widen 100bp on bad news while the index does not move.

Jump to default

Spread risk is about prices drifting. Default is a jump. Suppose the £80m is spread evenly over 8 issuers, £10m each (an illustrative split; the book's totals are unchanged). If one defaults and bondholders recover 40% of face, the jump-to-default loss is £10m − 40% × £10m = £6.0m. If all eight defaulted it would be £48m.

No 99% one-day VaR will show this. An investment-grade issuer defaulting on any given day is far rarer than 1 in 100, so the event never reaches the quantile, and the history behind the VaR contains no defaults. Desks therefore track JTD issuer by issuer, set limits on it, and look for concentrations: the largest JTD is often a bigger number than the whole desk's VaR of £1.94m.

Sort itSpread risk, rate risk or default risk?

Which risk does each event hit first?

A rating agency puts the issuer on negative watch and its bonds cheapen by 30bp.
The Bank of England cuts rates by 25bp unexpectedly; credit spreads do not move.
The issuer misses a coupon payment and enters administration.
A floating-rate note's price falls because the whole investment-grade index widens.
A recession scare widens every BBB spread by 40bp.

The formal version

CS01. With annual cash flows $CF_t$ , benchmark rate $r$ and spread $s$ , the price and its derivative with respect to the spread are

\begin{aligned} P(s) &= \sum_{t=1}^{T} CF_t\,(1 + r + s)^{-t} \\ \frac{\partial P}{\partial s} &= -\sum_{t=1}^{T} t\,CF_t\,(1 + r + s)^{-t-1} = -\frac{1}{1+y}\sum_{t} t\,CF_t\,(1+y)^{-t} = -P\,D_{\text{mod}} \\ \text{CS01} &= \frac{\partial P}{\partial s} \times 0.0001 = -P\,D_{\text{mod}} \times 0.0001 \end{aligned}

with $y = r + s$ and $D_{\text{mod}}$ the modified duration of L1.4.08: Duration. The derivative with respect to $r$ is the same expression, which is why CS01 equals the rate DV01 for a fixed-rate bond. For the desk's bonds: $D_{\text{mod}} = 4.2879$ , so $\text{CS01} = -\text{£}80\text{m} \times 4.2879 \times 0.0001 = -\text{£}34{,}303$ .

Big moves. A second-order Taylor expansion (L1.4.09: Convexity) adds convexity $C$ :

\Delta P \approx \text{CS01}\times \Delta s_{\text{bp}} + \tfrac12\,P\,C\,(\Delta s)^2, \qquad C = 23.57

Spread shock	CS01 × shock	+ convexity	Full revaluation	Error of CS01 alone
+10bp	−£343k	−£342k	−£342k	+£0.9k
+25bp	−£858k	−£852k	−£852k	+£5.9k
+50bp	−£1,715k	−£1,692k	−£1,692k	+£23.3k
+100bp	−£3,430k	−£3,336k	−£3,338k	+£92.3k
+200bp	−£6,861k	−£6,483k	−£6,499k	+£361.4k
+300bp	−£10,291k	−£9,442k	−£9,495k	+£796.2k

Bucketed CS01. Bump pillar $k$ by a triangle $w_k(t)$ that is 1 at the pillar and falls linearly to 0 at its neighbours. At every $t$ the triangles add up to one, $\sum_k w_k(t) = 1$ , so to first order $\sum_k \text{CS01}_k = \text{CS01}$ : here −£752 + −£2,068 + −£31,484 = −£34,303.

Jump to default and a CDS hedge. For a position with market value $V$ , face $F$ and recovery $R$ , and for CDS protection with notional $N$ :

\begin{aligned} \text{JTD}_{\text{bond}} &= V - R\,F \\ \text{JTD}_{\text{CDS}} &= -N\,(1-R) \quad \text{(a gain for the protection buyer)} \\ \text{CS01}_{\text{CDS}} &\approx N \times \text{RPV01} \times 0.0001, \qquad \text{RPV01} = \sum_{q=1}^{4T} \tfrac14\, e^{-(r + \lambda)\,t_q}, \quad \lambda = \frac{s}{1 - R} \end{aligned}

RPV01 is the risky annuity: the value of 1 a year paid quarterly until maturity or default. The hazard rate $\lambda$ comes from the credit triangle of L1.7.10: Credit default swaps. A protection buyer gains about RPV01 × 1bp for each basis point the CDS spread widens, ignoring the small change in RPV01 itself.

Try it

Shock the spread of a par bond and compare the straight CS01 line with full revaluation. Try a ten-year bond at 400bp with a +300bp shock, then turn the recovery rate down and watch the default loss grow.

Try itShock a corporate bond's spread

A £10m bond bought at par today: its annual coupon equals the swap rate for its maturity plus its spread, so its price is 100. The swap rates come from the module's curve (4.00% at two years to 4.40% at ten). Illustrative.

Maturity5 years

Spread today120bp

Spread shock+100bp

Recovery rate if the issuer defaults40%

Full revaluationCS01 × shock (straight line)

Coupon (swap + spread)5.35%

Swap rate at 5y4.15%

Spread duration4.29 yrs

CS01 (per +1bp)−£4,288

P&L, full revaluation−£417.3k

P&L, CS01 × shock−£428.8k

P&L, CS01 + convexity−£417.0k

Error of CS01 alone+£11.5k

Jump to default−£6.00m

The bond gains when spreads tighten and loses when they widen. Because the price curve bends upwards (convexity), CS01 alone overstates losses from wide shocks and understates gains from tightening. Default is different again: the bond does not drift down by basis points but drops at once to the recovery value.

Worked example

Day 486 on the credit book, and a CDS hedge

Rates, spreads and default on one book

Step 1 of 6

1
The day's moves
Day 486 is the sharp risk-off day early in the stressed spell. The 2-year swap rate fell 18bp and the 10-year 22bp, so the book's 5-year rate (0.625 × 2y + 0.375 × 10y) fell 19.5bp. The investment-grade spread widened 25bp.

Regulatory lens · looking ahead

Basel's market risk framework (FRTB) treats the two risks separately. Credit spread risk enters through CS01-style sensitivities by bucket (credit quality and sector) and tenor in the standardised approach, and through spread factors in internal models; default risk gets its own default risk charge, built on jump-to-default amounts. M4.3: Market risk capital (FRTB) and CVA covers both. Credit spread risk in the banking book, outside the trading desk, returns in L3.6.12: Credit spread risk in the banking book and the SVB case, and market-implied default probabilities in L3.3.16: Market-implied default risk.

Spark Atoms

Spark AtomThe CDS Big Bang

In April 2009 the CDS market changed its plumbing in the “Big Bang”. ISDA's protocol wrote auction settlement after a credit event into the contracts, and at the same time North American contracts moved to standard fixed coupons of 100 or 500 basis points, with an upfront payment making up the difference. A trade's CS01 and its upfront value became easier to compare across dealers.

Spark AtomForty per cent is a convention, not a law

CDS prices are usually quoted with an assumed recovery of 40% for senior unsecured debt, and this lesson uses the same figure. Real recoveries vary widely. When Lehman Brothers failed, the CDS auction in October 2008 set a final price of 8.625 per 100 of face, so protection sellers paid out about 91% of notional.

Key takeaways

A corporate bond's yield is a benchmark rate plus a credit spread; desks measure and hedge the two separately.
CS01 = −value × spread duration × 0.0001. For the desk's £80m of bonds it is −£34,303 per bp, equal to their rate DV01, but spreads and rates move differently.
CS01 is a first-order estimate: convexity matters for shocks of 100bp or more.
Bucketed CS01 shows where on the spread curve the risk sits, and catches curve trades that a parallel CS01 hides.
Jump-to-default is a separate risk: value − recovery × face, £6m for one £10m issuer here. Spread VaR does not see it.
A CDS can hedge both CS01 and JTD, leaving basis risk between the bond and the CDS.

Reaction

Check what stuck

Question 1 · objective 1

A desk holds £50m of corporate bonds with a spread duration of 6 years. What is its CS01, in pounds per basis point? (Enter a negative number for a loss.)

£

Question 2 · objective 1

A floating-rate note pays SONIA plus a fixed 150bp margin for five years. Compared with a five-year fixed-rate bond from the same issuer, it has…

Aabout the same CS01 but almost no interest rate DV01Babout the same DV01 but almost no CS01Cneither DV01 nor CS01Dtwice the CS01

Question 3 · objective 2

A credit book's CS01 is −£20,000 per bp and its spread moves have a daily standard deviation of 4bp. What is its 99% one-day spread VaR, in pounds (z = 2.326)?

£

Question 4 · objective 3

A trader holds £25m face of one issuer's bonds, priced at par. Recovery after default is expected to be 30%. What is the jump-to-default loss, in £m?

£m

Question 5 · objective 3

Why does a 99% one-day VaR built on spread moves say little about default?

ADefault of one issuer in a day is far rarer than 1 in 100, and its loss is a jump, not a spread moveBVaR ignores corporate bondsCSpreads never move on the day an issuer defaultsDDefault losses are always smaller than spread losses

Glossary

CS01: Credit spread 01: the change in a position's value, in money, when its credit spread widens by one basis point, everything else unchanged. Negative for a bond holder.
Spread duration: The percentage fall in a bond's price for a one-percentage-point rise in its spread. For a fixed-rate bond it is close to its modified duration.
I-spread: The bond's yield minus the swap rate of the same maturity.
Z-spread: The constant spread that, added to every point of the benchmark zero curve, discounts the bond's cash flows to its market price.
Spread curve: Credit spreads for one issuer, sector or rating at each maturity, from short to long.
Bucketed (key-rate) CS01: The CS01 from bumping one point (pillar) of the spread curve, with the bump fading to zero at the neighbouring pillars. The buckets add up to the parallel CS01.
Idiosyncratic spread risk: Spread moves specific to one issuer (news about that company), on top of the moves of its sector or rating index.
Jump to default (JTD): Jump to default: the loss if an issuer defaults now, with no warning. For a bond: its market value minus what is recovered.
CDS-bond basis: The gap between an issuer's CDS spread and the spread on its bonds. It moves, so a CDS is not a perfect hedge for the bond.

Deep Dives & Origins

Deep Dives

D. O'Kane, Modelling Single-name and Multi-name Credit Derivatives, Wiley, 2008, the chapters on the CDS and its risk measures. Advanced.
J. C. Hull, Options, Futures, and Other Derivatives, 11th ed., Pearson, 2021, the chapters on credit risk and credit derivatives. Intermediate.

Origins

Basel Committee on Banking Supervision, Minimum capital requirements for market risk, BIS, January 2019 (now chapters MAR21 and MAR22 of the Basel Framework).
ISDA, 2009 ISDA Credit Derivatives Determinations Committees and Auction Settlement CDS Protocol (the “Big Bang” protocol), March 2009.
B. Tuckman and A. Serrat, Fixed Income Securities, 3rd ed., Wiley, 2011, on spreads and key-rate sensitivities.

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