Price impact of spread changes
A change in spread moves a bond's price just like a change in yield: %ΔP ≈ −ModDur × ΔSpread + ½ × Convexity × (ΔSpread)². Longer-duration bonds are more sensitive to spread changes, which is why a downgrade hurts long bonds most.
Before this: Estimating price change with duration and convexity, What drives credit spreads
On the exam
- Describe macroeconomic, market, and issuer-specific factors that influence the level and volatility of yield spreads
Calculate the price impact of a spread change given duration and convexity. Expect a convexity figure quoted as a small decimal (0.48) that must be rescaled (48), a large spread move, and a comparison of a short and a long bond after a downgrade.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
Why does a spread change move the price?
A corporate bond’s yield is a benchmark yield plus a spread. Price depends on the total, so a change in the yield can come from either part. For an option-free fixed-rate bond, the same modified duration and convexity apply whichever part moves. Investment-grade investors, who rarely face default, worry about this spread risk : the effect on price and return when spreads change.
The size of the effect depends on two things: the bond’s modified duration and how large the spread change is. Because the formula is the same as for a yield change, some people call the duration used here spread duration .
For a small, instant change, duration alone does well. A wider spread lowers the price and a narrower spread raises it. For larger moves, add the convexity term:
- Duration effect Memorize −ModDur × ΔSpread; widening lowers price
- Convexity effect Memorize +½ × Convexity × (ΔSpread)²; always added
- Rescale quoted convexity Memorize 0.48 becomes 48; ΔSpread as a decimal
-
- What makes it bigger Use it size of the move and the bond's duration
- Longer duration Use it more price change for the same spread move
- Bigger spread move Use it convexity matters more as the move grows
-
Watch the scale of convexity. Data services often quote convexity divided by 100, for example 0.48. The formula needs convexity scaling that matches duration squared, so rescale first: multiply by 100 (0.48 becomes 48). A quick test: convexity should be about the same order of magnitude as duration squared. With a duration of 6, a convexity of 0.48 is far too small and 48 looks right. And enter ΔSpread as a decimal (75 bp = 0.0075).
Try this 0 of 3
- At +75 bp, switch convexity to 'as quoted'
- Add the 15-year bond at +75 bp
- Narrow the spread by 100 bp
- Benchmark
- Credit spread
- Liquidity spread
- 6.2-duration bond
The yield goes from 6% to 6.75% because the spread widens 75 bp. Price falls about 4.52%.
Illustrative yields: benchmark 3.5%, credit spread 2.2%, liquidity spread 0.3%. Prices are duration-plus-convexity estimates from a start of 100.
Show working
Formula: %ΔP ≈ −ModDur × ΔSpread + ½ × Convexity × (ΔSpread)², with ΔSpread as a decimal (+75 bp = 0.0075).
Rescale convexity: reported 0.48 × 100 = 48.
Duration effect = −6.2 × 0.0075 = −4.650%.
Convexity effect = ½ × 48 × 0.0075² = +0.1350%.
Total = −4.650% + 0.1350% = −4.515%.
- 0.5×48×0.0075x²=0.00135, convexity effect
- STO1
- 6.2×0.0075=0.0465, duration effect (a fall)
- +|-+RCL1=-0.04515, −4.515%
x² square · STO store · +|- change sign · RCL recall a stored number
- 48ENTER0.0075ENTER××2÷0.00135, convexity effect
- 6.2ENTER0.0075×0.0465, duration effect
- −-0.04515, −4.515%
ENTER enter (on the HP, puts the number on the stack)
A bond has a modified duration of 6.2 and an annualized convexity of 48. If its credit spread changes by +75 bp, its estimated price change is closest to:
−(6.2 × 0.0075) + ½ × 48 × 0.0075² = -0.04515 = −4.52%
Estimated change in full price-4.52%
- 6.2×0.0075=+|-STO1
- 0.5×48×0.0075×0.0075+RCL1=as a decimal
+|- change sign · STO store · RCL recall a stored number
- 6.2ENTER0.0075×CHS
- 0.5ENTER48×0.0075×0.0075×+as a decimal
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Exam tip Spreads and signs: wider spread, price down. Narrower spread, price up. The convexity term is positive either way.
Go deeper Why this is an instant effect
The formula estimates the price change right after the spread moves. Over a longer holding period, a bond’s coupons and its shortening duration also matter, so the realized return can differ from this estimate. Investment-grade investors focus on spread risk like this. High-yield investors focus more on default and recovery.
Why do downgrades hurt long bonds most?
When an issuer suffers a downgrade , its spreads widen. Duration multiplies the spread change, so a bond with twice the duration loses about twice as much for the same move. Longer bonds usually also see a larger spread widening, because investors want more compensation for uncertainty about an issuer that grows with time.
Go deeper The same formula in reverse
Given a price change, you can solve for the spread move. The formula is a quadratic in ΔSpread, with two roots. Take the root close to zero (the other is a huge move the other way). If the price rose, the spread narrowed. The solver above does the algebra for you.
Memorize this
MemorizeSpread move, price move
| Step | Rule |
|---|---|
| 1. Rescale convexity | Quoted 0.48 → 48 (match duration squared) |
| 2. Duration effect | −ModDur × ΔSpread (decimal) |
| 3. Convexity effect | +½ × Convexity × (ΔSpread)² |
| 4. Add | Widening: price falls. Narrowing: price rises |
Memory hook Convexity should look like duration squared. Duration 6 means convexity in the tens, so 0.48 must really be 48.
Know it cold ΔSpread as a decimal (75 bp = 0.0075). The convexity term is always added.
MemorizeWho is hurt most by a downgrade
Memory hook The same spread shock multiplies by duration, so the longest bond loses most.
Know it cold Investment-grade investors focus on spread risk. High-yield investors focus on default and recovery.
- Duration effect Memorize −ModDur × ΔSpread; widening lowers price
- Convexity effect Memorize +½ × Convexity × (ΔSpread)²; always added
- Rescale quoted convexity Memorize 0.48 becomes 48; ΔSpread as a decimal
-
- What makes it bigger Use it size of the move and the bond's duration
- Longer duration Use it more price change for the same spread move
- Bigger spread move Use it convexity matters more as the move grows
-
Quick example
Modified duration 9.1 and a spread that tightens by 35 bp. −9.1 × (−0.0035) = +3.185%.
- 9.1×0.0035= 0.03185, so +3.185%
- 9.1ENTER0.0035× 0.03185, so +3.185%
ENTER enter (on the HP, puts the number on the stack)
Quick example
Modified duration 4.8, convexity 30 (already rescaled), spread widens 60 bp. −4.8 × 0.006 = −2.880%; ½ × 30 × 0.006² = +0.054%. Total −2.826%.
- 4.8×0.006=+|-STO1
- 0.5×30×0.006×0.006+RCL1= −0.02826
+|- change sign · STO store · RCL recall a stored number
- 4.8ENTER0.006×CHS
- 0.5ENTER30×0.006×0.006×+ −0.02826
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Quick example
A data service shows convexity 0.62 for a bond with modified duration 7.4. Used in the formula: 0.62 × 100 = 62, the same order as 7.4² ≈ 55.
- 0.62×100= 62
- 0.62ENTER100× 62
ENTER enter (on the HP, puts the number on the stack)
How it is asked
-
Quoted convexity
ModDur 4.6, reported convexity 0.29, spread +110 bp. Price change?
Rescale to 29: −5.060% + 0.175% = −4.88%. Unscaled 0.29 gives −5.06%.
-
Spread narrows
ModDur 7.5, convexity 70, spread −130 bp. Price change?
+9.75% + 0.59% = +10.34%. The convexity term is still added.
-
Downgrade, short versus long
Same issuer, two bonds, same spread widening. Which falls more?
The one with the longer duration.
-
Work back to the spread move
ModDur 7, convexity 60, price up 5.792%. Spread change?
Solve the quadratic and take the root near zero: −80 bp (sign: price up means spread down).
Exam traps
- Using convexity as quoted (0.48) instead of rescaled (48).
- Subtracting the convexity term. It is always added.
- Forgetting that a narrower spread raises the price.
- Entering the spread change in basis points (75) or percent (0.75) instead of a decimal (0.0075).
- Assuming the shorter bond falls more because its coupon or maturity looks safer.
Checkpoint
6 exam-style questions. About 90 seconds each.
1A bond has a modified duration of 5.5 and a reported convexity of 0.36. If its credit spread widens by 120 bp, its price will most likely fall by an amount closest to:
2An issuer is downgraded and spreads widen by the same amount on all its option-free bonds. The bond whose price falls the most is most likely the one with the:
3A bond with a modified duration of 8.0 and a convexity of 80 sees its credit spread narrow by 150 bp. Its estimated price change is closest to:
4A bond with a modified duration of 6.0 and a convexity of 45 rises in price by an estimated 6.225% after its credit spread changes. The spread change was closest to:
5A bond has a modified duration of 3.0 and a reported convexity of 0.12. If its credit spread widens by 250 bp, its estimated price change is closest to:
6Compared with high-yield investors, investment-grade bond investors are most likely more focused on:
Key takeaways
- %ΔP ≈ −ModDur × ΔSpread + ½ × Convexity × (ΔSpread)², for option-free bonds.
- Rescale convexity quoted per 100 (0.48 becomes 48) so it matches duration squared.
- Longer duration means more spread sensitivity, so downgrades hurt long bonds most.
- Wider spreads lower prices and narrower spreads raise them.
Quick example
Modified duration 9.1 and a spread that tightens by 35 bp. −9.1 × (−0.0035) = +3.185%.
- 9.1×0.0035= 0.03185, so +3.185%
- 9.1ENTER0.0035× 0.03185, so +3.185%
ENTER enter (on the HP, puts the number on the stack)
Quick example
Modified duration 4.8, convexity 30 (already rescaled), spread widens 60 bp. −4.8 × 0.006 = −2.880%; ½ × 30 × 0.006² = +0.054%. Total −2.826%.
- 4.8×0.006=+|-STO1
- 0.5×30×0.006×0.006+RCL1= −0.02826
+|- change sign · STO store · RCL recall a stored number
- 4.8ENTER0.006×CHS
- 0.5ENTER30×0.006×0.006×+ −0.02826
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Quick example
A data service shows convexity 0.62 for a bond with modified duration 7.4. Used in the formula: 0.62 × 100 = 62, the same order as 7.4² ≈ 55.
- 0.62×100= 62
- 0.62ENTER100× 62
ENTER enter (on the HP, puts the number on the stack)
Questions about this lesson?
Where this comes back
- Credit Analysis for Government Issuers Fixed Income
- Credit Analysis for Corporate Issuers Fixed Income
Questions candidates ask
How do you calculate the price impact of a credit spread change?
Use modified duration and convexity exactly as for a yield change, with the spread change as a decimal: minus duration times the change, plus half the convexity times the change squared.
Why rescale convexity?
Some sources quote convexity divided by 100. It must be on the same scale as duration squared for the formula to work, so 0.48 becomes 48.
Why are long bonds more sensitive to spread changes?
Their higher duration means a larger price change for the same spread move. Investors also want more spread for the greater uncertainty about an issuer far into the future.
Does a narrowing spread always raise the price?
Yes, for an option-free bond. A narrower spread lowers the yield, which raises the price, and the convexity term adds to the gain.
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