Price change with effective duration and convexity
Use the same two-term formula as for yield duration, with the benchmark curve shift in place of the yield change: %ΔP ≈ −EffDur × ΔCurve + ½ × EffCon × (ΔCurve)². When effective convexity is negative, the second term subtracts, deepening losses and trimming gains.
Before this: Effective duration and effective convexity, Estimating price change with duration and convexity
On the exam
- Calculate the percentage price change of a bond for a specified change in benchmark yield, given the bond’s effective duration and convexity
Calculate the percentage price change for a benchmark curve shift, given effective duration and effective convexity. Expect a callable bond with negative convexity and a 100 to 200 bp move. Wrong options treat convexity as positive, ignore it, or use basis points instead of decimals.
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.
How do you estimate a price change from a curve shift?
The estimate is the formula from the convexity module with three substitutions: effective duration for modified duration, effective convexity for convexity, and the benchmark curve shift ΔCurve for the change in yield. The shift is a parallel shift : every maturity on the benchmark curve moves by the same amount.
Work it in three steps, with ΔCurve as a decimal (100 bp = 0.0100):
- Duration term = −EffDur × ΔCurve. Its sign is opposite to the shift.
- Convexity term = ½ × EffCon × (ΔCurve)². Its sign is the sign of EffCon, because the shift is squared.
- Add them.
- Duration term Memorize −EffDur × ΔCurve
- Sign opposite to the shift curve up, price down
-
- Convexity term Memorize +½ × EffCon × (ΔCurve)²
- Same sign as EffCon, up or down ΔCurve is squared
- Callable at low rates Memorize EffCon negative: hurts both ways
- Putable, option-free Memorize EffCon positive: helps both ways
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A bond has an effective duration of 4.74 and an effective convexity of −158.1. If the benchmark curve shifts by +100 bp, the estimated percentage change in its full price is closest to:
(−4.74 × 0.01) + (½ × (−158.1) × 0.01²) = −0.055305 = −5.53%
Estimated change in full price-5.53%
- 4.74×0.01=+|-STO1
- 0.5×158.1+|-×0.01×0.01+RCL1=as a decimal
+|- change sign · STO store · RCL recall a stored number
- 4.74ENTER0.01×CHS
- 0.5ENTER158.1CHS×0.01×0.01×+as a decimal
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Try this 0 of 3
- Press "Use the worked example prices"
- Set effective convexity to +30 at ±100 bp (an option-free bond)
- With negative convexity, push the shift to 200 bp
Model prices (per 100) with the curve shifted ± 25 bp:
Negative convexity: you lose 5.53% if the curve rises but gain only 3.95% if it falls. The convexity term is −0.79% in both directions.
Show working
Effective duration = (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀) = (102.35 − 99.95) ÷ (2 × 0.0025 × 101.2) = 2.4 ÷ 0.506 = 4.7431.
Effective convexity = (PV₋ + PV₊ − 2 × PV₀) ÷ ((ΔCurve)² × PV₀) = (102.35 + 99.95 − 202.4) ÷ (0.0025² × 101.2) = −0.1 ÷ 0.0006325 = −158.10.
%ΔP ≈ −EffDur × ΔCurve + ½ × EffCon × (ΔCurve)², with ΔCurve as a decimal (100 bp = 0.01).
Curve up: −4.74 × 0.01 + ½ × (−158.1) × 0.01² = −4.740% − 0.791% = −5.53%.
Curve down: −4.74 × (−0.01) + ½ × (−158.1) × (−0.01)² = +4.740% − 0.791% = +3.95%.
- 0.5×158.1+|-×0.01x²=STO1−0.007905, convexity term
- 4.74+|-×0.01=+RCL1=×100=−5.530, so −5.53% for +100 bp
+|- change sign · x² square · STO store · RCL recall a stored number
- 0.5ENTER158.1CHS×0.01ENTER××−0.007905, convexity term
- 4.74CHSENTER0.01×+100×−5.530, so −5.53% for +100 bp
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
- 0.5×158.1+|-×0.01+|-x²=STO1−0.007905, convexity term
- 4.74+|-×0.01+|-=+RCL1=×100=3.950, so +3.95% for −100 bp
+|- change sign · x² square · STO store · RCL recall a stored number
- 0.5ENTER158.1CHS×0.01CHSENTER××−0.007905, convexity term
- 4.74CHSENTER0.01CHS×+100×3.950, so +3.95% for −100 bp
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
- 102.35−99.95=÷2÷0.0025÷101.2=4.7431, effective duration
- 102.35ENTER99.95−2÷0.0025÷101.2÷4.7431, effective duration
ENTER enter (on the HP, puts the number on the stack)
- 2×101.2=STO1
- 102.35+99.95−RCL1=÷0.0025x²÷101.2=−158.10, effective convexity
STO store · RCL recall a stored number · x² square
- 102.35ENTER99.95+2ENTER101.2×−0.0025ENTER×÷101.2÷−158.10, effective convexity
ENTER enter (on the HP, puts the number on the stack)
Exam tip Check the sign last. The duration term takes the opposite sign to the shift. The convexity term takes the sign of EffCon and never flips.
How do you read the result?
The two signs tell you who wins and who loses.
- Callable bond. The upside is limited when rates fall. The call is the issuer’s refinancing option: the issuer buys the bond back at the call price and borrows again more cheaply. The investor is left with a smaller gain and has to reinvest the proceeds at lower rates.
- Putable bond. The downside is limited when rates rise. The investor can sell the bond back at the put price, usually par, and reinvest at the higher rate. Effective duration and convexity both fall as rates rise, but convexity stays positive.
Go deeper Why the estimate is only local
A callable bond’s effective duration is not a constant. It shortens as rates fall and the call moves into the money, so an estimate built from today’s duration and convexity is accurate only for moves in the neighborhood of today’s rates. A very large fall can change the numbers the estimate rests on.
Go deeper Negative convexity is a cost the investor must be paid for
Suppose up and down moves are equally likely. For the callable bond above, the average of −5.53% and +3.95% is −0.79%, which is exactly the convexity term. A negatively convex bond has a negative expected price change from a symmetric shock, so it has to offer extra yield to compensate. A manager who expects rates to fall may still accept the trade, but only if the extra yield is worth the capped upside.
Memorize this
MemorizeEffective price change
| Bond | EffCon | Curve up | Curve down |
|---|---|---|---|
| Option-free | + | Loss trimmed | Gain boosted |
| Putable | + | Loss trimmed (floor) | Gain boosted |
| Callable at low rates | − | Loss deepened | Gain trimmed (cap) |
Memory hook Duration is the slope and flips with the shift. Convexity is the bend and does not flip, because ΔCurve is squared. Negative convexity: you lose more than you gain.
Know it cold ΔCurve as a decimal (150 bp = 0.0150). Keep the ½ and keep the sign of EffCon.
- Duration term Memorize −EffDur × ΔCurve
- Sign opposite to the shift curve up, price down
-
- Convexity term Memorize +½ × EffCon × (ΔCurve)²
- Same sign as EffCon, up or down ΔCurve is squared
- Callable at low rates Memorize EffCon negative: hurts both ways
- Putable, option-free Memorize EffCon positive: helps both ways
-
Quick example
Effective duration 7.20, effective convexity −210, benchmark curve up 75 bp. Duration: −7.20 × 0.0075 = −5.400%. Convexity: ½ × (−210) × 0.0075² = −0.591%. Total −5.99%.
- 7.2×0.0075=+|-STO1
- 0.5×210+|-×0.0075×0.0075+RCL1= −0.059906
+|- change sign · STO store · RCL recall a stored number
- 7.2ENTER0.0075×CHS
- 0.5ENTER210CHS×0.0075×0.0075×+ −0.059906
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
How it is asked
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Callable bond, curve up
EffDur 5.60, EffCon −90.0, curve +75 bp. Price change?
−4.20% − 0.25% = −4.45%. Duration alone gives −4.20%; treating the convexity term as positive gives −3.95%.
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Putable bond, curve up
EffDur 4.60, EffCon +45.0, curve +60 bp.
−2.76% + 0.08% = −2.68%. Convexity is positive, so it offsets part of the loss.
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Effect of negative convexity
Versus a duration-only estimate, a callable bond’s price rise when rates fall is?
Smaller: the negative convexity term is subtracted even when rates fall.
Exam traps
- Plugging in the size of EffCon and adding it as a positive term. Keep the negative sign.
- Entering ΔCurve in basis points (100) instead of as a decimal (0.01).
- Expecting the up and down moves to give mirror-image answers. They do not.
- Forgetting the ½ in the convexity term.
- Flipping the sign of the convexity term when the curve falls. Only the duration term flips with the shift.
Checkpoint
5 exam-style questions. About 90 seconds each.
1A callable bond has effective duration 6.20 and effective convexity −95.0. If the benchmark curve rises 150 bp, its price will most likely fall by an amount closest to:
2A putable bond has effective duration 3.80 and effective convexity 25.0. For a 200 bp upward shift in the benchmark curve, the estimated price change is closest to:
3Relative to a duration-only estimate, negative effective convexity makes a callable bond's estimated price rise when rates fall:
4A callable bond has effective duration 5.10 and effective convexity −120. If the benchmark curve falls 100 bp, the estimated price change is closest to:
5A putable bond has effective duration 3.10 and effective convexity 55.0. If the benchmark curve falls 150 bp, the estimated price change is closest to:
Key takeaways
- %ΔP ≈ −EffDur × ΔCurve + ½ × EffCon × (ΔCurve)².
- Keep the sign of EffCon: negative convexity hurts in both directions.
- Callables: capped upside. Putables: cushioned downside.
- The estimate is local: effective duration itself changes with the level of rates.
Quick example
Effective duration 7.20, effective convexity −210, benchmark curve up 75 bp. Duration: −7.20 × 0.0075 = −5.400%. Convexity: ½ × (−210) × 0.0075² = −0.591%. Total −5.99%.
- 7.2×0.0075=+|-STO1
- 0.5×210+|-×0.0075×0.0075+RCL1= −0.059906
+|- change sign · STO store · RCL recall a stored number
- 7.2ENTER0.0075×CHS
- 0.5ENTER210CHS×0.0075×0.0075×+ −0.059906
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
All Fixed Income formulas, every lesson
Practice exam-style questions on Effective Duration, Key Rate Duration and Empirical Duration
Questions about this lesson?
Where this comes back
- Mortgage-Backed Securities: RMBS, CMOs and CMBS Fixed Income
- Pricing and Valuing Options Derivatives
Questions candidates ask
How do you use effective duration to estimate price change?
Multiply effective duration by the curve shift and flip the sign, then add half the effective convexity times the shift squared. Enter the shift as a decimal.
What happens to a callable bond's price when rates fall?
It rises, but by less than an option-free bond's, because the call price caps it. With negative convexity, the convexity term reduces the gain further.
Can effective convexity be negative?
Yes, for callable bonds and mortgage-backed securities when rates are low. A putable bond's effective convexity is always positive.
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