Lesson 2 of 3 11–14% of the exam

By Daniil Tarakanov Updated Oct 20, 20262027 curriculum

Estimating price change with duration and convexity

Add the two effects: %ΔP ≈ −ModDur × Δy + ½ × Convexity × (Δy)². The first term is the straight-line duration effect. The second, always positive for an option-free bond, corrects for curvature. In currency, use money duration and money convexity in the same formula.

8 min cram, 34 min full read. Based on the 2027 CFA® Program Level I curriculum, Fixed Income, Learning Module 12. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Modified Duration, Money Duration and PVBP, Convexity and the convexity adjustment

On the exam

Calculate and interpret convexity and describe the convexity adjustment
Calculate the percentage price change of a bond for a specified change in yield, given the bond’s duration and convexity

Calculate the percentage price change for a given yield change using duration and convexity, and calculate money convexity and the currency change. Expect a large move (100 to 200 bp). Wrong options forget the ½, subtract the convexity term or stop at duration.

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 combine duration and convexity?

Duration gives the straight-line estimate of the percentage price change for a yield change. Convexity adds the curvature that the line misses, and that second term is the convexity adjustment . Put together, they give the standard estimate, and the exam asks for it constantly.

Price change with duration and convexity Memorize

\%\Delta PV^{Full} \approx (-\text{AnnModDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta y)^2\right]

Work it as a three-step routine, and write Δy as a decimal (150 bp = 0.0150):

Duration effect = −ModDur × Δy.
Convexity effect = ½ × Convexity × Δy².
Add them.

Two effects, two units

Estimated price changeduration effect plus convexity effect

Percentage form Memorize per 100 of par, uses annualized figures
- −AnnModDur × Δy Memorize sign opposite to the yield move
- +½ × AnnConvexity × (Δy)² Memorize always positive for option-free bonds
Currency form Memorize for a position, uses money measures
- −MoneyDur × Δy Memorize MoneyDur = AnnModDur × full price or position value
- +½ × MoneyCon × (Δy)² Memorize MoneyCon = AnnConvexity × full price or position value

InteractiveDuration plus convexity: the three-step estimate

Try this 0 of 3

With Bond K, double the move to +200 bp
At +100 bp, open the common mistakes
Show the EUR 20,000,000 position at +100 bp

Modified duration (annual)Convexity (annual)

Yield change+100 bp

Show it for a position (EUR) Common mistakes

1. Duration effect−5.009%

2. Convexity effect+0.1585%

3. Estimated %ΔP, +100 bp−4.851%

Bond K actual−4.854%

Duration effect (blue when yields fall)
Convexity effect
Total
Bond K actual (dot)

Duration only
Duration + convexity
Bond K actual

Duration gives −5.009%; convexity adds +0.1585%, 3.2% of the duration effect. Estimate −4.851%, Bond K actually moves −4.854%.

Show working

Step 1, duration effect = −ModDur × Δy = −5.0091 × 0.01 = −5.009%.

Step 2, convexity effect = ½ × Convexity × (Δy)² = ½ × 31.7 × 0.01² = +0.1585%.

Step 3, add them: −5.009% + 0.1585% = −4.851%. Repricing Bond K at 7% gives −4.854%.

The three steps

0.5×31.7×0.01x²=0.001585, step 2: convexity effect
STO1
5.0091×0.01=0.050091, duration effect (a fall)
+|-+RCL1=-0.048506, step 3: −4.851%

x² square · STO store · +|- change sign · RCL recall a stored number

31.7ENTER0.01ENTER××2÷0.001585, step 2: convexity effect
5.0091ENTER0.01×0.050091, step 1: duration effect
−-0.048506, step 3: −4.851%

ENTER enter (on the HP, puts the number on the stack)

Duration does the heavy lifting; convexity is the always-positive correction that matters more as the move gets bigger.

Worked example Bond K with a 100 bp and a 200 bp move

Bond K (6 years, 5% annual coupon, yield 6%) has a modified duration of 5.0091 and a convexity of 31.70. For a +100 bp rise:

Duration effect: −5.0091 × 0.01 = −0.050091, or −5.009%.
Convexity effect: ½ × 31.70 × 0.01² = 0.001585, or +0.1585%.
Total: −0.048506, or −4.85%.

The same three steps for all four moves, against the price from repricing the bond exactly:

Yield change	Duration only	Convexity term	Duration + convexity	Actual (repriced)
+100 bp	−5.009%	+0.1585%	−4.851%	−4.854%
−100 bp	+5.009%	+0.1585%	+5.168%	+5.172%
+200 bp	−10.018%	+0.6340%	−9.384%	−9.414%
−200 bp	+10.018%	+0.6340%	+10.652%	+10.685%

Duration alone misses by about 15 bp at ±100 bp. With convexity the miss is under 1 bp. At ±200 bp the miss is still about 3 bp. Notice the convexity term is the same positive number whether yields rise or fall.

+100 bp on your calculator

5.0091×0.01=+|-STO1 −0.050091, the duration effect
0.5×31.7×0.01×0.01+RCL1= −0.048506, so −4.85%

+|- change sign · STO store · RCL recall a stored number

5.0091ENTER0.01×CHS −0.050091, the duration effect
0.5ENTER31.7×0.01×0.01×+ −0.048506, so −4.85%

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Rearrange itPrice change with duration and convexity: solve for any input

The question asks for

A bond has an annualized modified duration of 5.0091 and an annualized convexity of 31.7. If its yield changes by +100 bp, the estimated percentage change in its full price is closest to:

Annualized modified durationAnnualized convexityYield change (basis points)

%Δ P V^{F u l l} \approx (- AnnModDur \times Δ y) + [\frac{1}{2} \times AnnConvexity \times (Δ y)^{2}]

(−5.0091 × 0.01) + (½ × 31.7 × 0.01²) = -0.048506 = −4.85%

Estimated change in full price-4.85%

On your calculator

5.0091×0.01=+|-STO1
0.5×31.7×0.01×0.01+RCL1=as a decimal

+|- change sign · STO store · RCL recall a stored number

5.0091ENTER0.01×CHS
0.5ENTER31.7×0.01×0.01×+as a decimal

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Exam tip Check the sign last. The duration effect takes the opposite sign to the yield move. The convexity effect is always plus.

Go deeper Why the error of duration alone grows with the square of the move

The convexity term is proportional to $(\Delta y)^2$ . Double the move and the miss of a duration-only estimate roughly quadruples. Bond K misses by 0.16 percentage points at ±100 bp and by 0.63 at ±200 bp (0.1585 × 4 = 0.634). That is why convexity is a footnote for a 10 bp move and essential for 200 bp.

Go deeper Convexity scaling conventions

Some sources quote convexity divided by 100, or build the ½ into the figure. The result is the same only if the formula’s terms are consistent with the number you are given. Exam questions state the convexity statistic to use with the ½ form, so take the figure as given and apply the formula on this page.

Go deeper Spread duration and spread convexity

The same two-term formula prices a change in a credit spread, with spread duration and spread convexity in place of the yield measures. You meet it in the credit risk module.

How do you estimate a change in currency?

Percentages need a position size before they mean money. The currency version swaps the percentage measures for money measures. You know money duration from the duration module: annualized modified duration times the full price. Money convexity is built the same way.

Money convexity Memorize

\text{MoneyCon} = \text{AnnConvexity} \times PV^{Full}

For a position, $PV^{Full}$ is its market value (full price × par), never the par amount. Then the estimate has the same structure as before:

Change in value with money duration and money convexity Memorize

\Delta PV^{Full} \approx (-\text{MoneyDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{MoneyCon} \times (\Delta y)^2\right]

Worked example A EUR 20 million position in Bond K

An investor holds EUR 20,000,000 of par of Bond K. At a full price of 95.0827 the market value is 20,000,000 × 0.950827 = EUR 19,016,540.

Money duration = 5.0091 × 19,016,540 = 95,255,751.
Money convexity = 31.70 × 19,016,540 = 602,824,318.
For +100 bp: duration effect −95,255,751 × 0.01 = −952,557.51; convexity effect ½ × 602,824,318 × 0.0001 = +30,141.22.
Estimated change: −952,557.51 + 30,141.22 = −EUR 922,416.

Repricing the bond at 7% gives an actual change of −EUR 923,151, so the estimate is off by about EUR 700. Duration alone would have been off by about EUR 29,400.

The position, step by step

20000000×0.950827=STO1 19,016,540, the market value
×5.0091×0.01=+|-STO2 −952,557.51, the duration effect
RCL1×31.7= 602,824,318, the money convexity
×0.5×0.0001+RCL2= −922,416.29

STO store · +|- change sign · RCL recall a stored number

20000000ENTER0.950827×STO1 19,016,540, the market value
5.0091×0.01×CHSSTO2 −952,557.51, the duration effect
RCL131.7× 602,824,318, the money convexity
0.5×0.0001×RCL2+ −922,416.29

ENTER enter (on the HP, puts the number on the stack) · STO store · CHS change sign · RCL recall a stored number

Rearrange itMoney convexity: solve for any input

The question asks for

A position has a market value of 19,016,540.00 and an annualized convexity of 31.7. Its money convexity is closest to:

Annualized convexityFull price or position value

MoneyCon = AnnConvexity \times P V^{F u l l}

31.7 × 19,016,540.00 = 602,824,318.00

Money convexity602,824,318.00

On your calculator

31.7×19016540=

31.7ENTER19016540×

ENTER enter (on the HP, puts the number on the stack)

Rearrange itChange in position value with money duration and money convexity

The question asks for

A position has a money duration of 95,255,750.51 and a money convexity of 602,824,318.00. If yields change by +100 bp, the estimated change in its value is closest to:

Money durationMoney convexityYield change (basis points)

Δ P V^{F u l l} \approx (- MoneyDur \times Δ y) + [\frac{1}{2} \times MoneyCon \times (Δ y)^{2}]

(−95,255,750.51 × 0.01) + (½ × 602,824,318.00 × 0.01²) = −922,416

Estimated change in value-922,416.29

On your calculator

95255750.51×0.01=+|-STO1
0.5×602824318×0.01×0.01+RCL1=

+|- change sign · STO store · RCL recall a stored number

95255750.51ENTER0.01×CHS
0.5ENTER602824318×0.01×0.01×+

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Mixing the two forms is a classic error. Percentage duration pairs with percentage convexity, and money duration pairs with money convexity. Never combine one of each.

Memorize this

MemorizeDuration plus convexity

Step	Percentage form	Currency form
1. Duration effect	−AnnModDur × Δy	−MoneyDur × Δy
2. Convexity effect	+½ × AnnConvexity × (Δy)²	+½ × MoneyCon × (Δy)²
3. Add them	%ΔPV	ΔPV in currency
Money measure	n/a	MoneyCon = AnnConvexity × full price

Memory hook Duration is the slope, convexity is the bend. Slope flips with the yield move; the bend never flips because Δy is squared.

Know it cold Δy as a decimal (150 bp = 0.0150). Keep the ½. Convexity effect is ADDED for option-free bonds, up or down.

Two effects, two units

Estimated price changeduration effect plus convexity effect

Percentage form Memorize per 100 of par, uses annualized figures
- −AnnModDur × Δy Memorize sign opposite to the yield move
- +½ × AnnConvexity × (Δy)² Memorize always positive for option-free bonds
Currency form Memorize for a position, uses money measures
- −MoneyDur × Δy Memorize MoneyDur = AnnModDur × full price or position value
- +½ × MoneyCon × (Δy)² Memorize MoneyCon = AnnConvexity × full price or position value

Price change with duration and convexity

\%\Delta PV^{Full} \approx (-\text{AnnModDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta y)^2\right]

Quick example

Modified duration 3.80, convexity 18.5, yield falls 75 bp. Duration: +3.80 × 0.0075 = 2.850%. Convexity: ½ × 18.5 × 0.0075² = 0.052%. Total +2.902%.

On your calculator

3.8×0.0075=STO1
0.5×18.5×0.0075×0.0075+RCL1= 0.02902

STO store · RCL recall a stored number

3.8ENTER0.0075×
0.5ENTER18.5×0.0075×0.0075×+ 0.02902

ENTER enter (on the HP, puts the number on the stack)

Money convexity

\text{MoneyCon} = \text{AnnConvexity} \times PV^{Full}

Quick example

AUD 10,000,000 par at a full price of 98.50 (market value 9,850,000) with convexity 40.0. MoneyCon = 40.0 × 9,850,000 = AUD 394,000,000.

On your calculator

10000000×0.985×40= 394,000,000

10000000ENTER0.985×40× 394,000,000

ENTER enter (on the HP, puts the number on the stack)

Change in value with money duration and money convexity

\Delta PV^{Full} \approx (-\text{MoneyDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{MoneyCon} \times (\Delta y)^2\right]

Quick example

MoneyDur 41,000,000, MoneyCon 394,000,000, yield +50 bp. −41,000,000 × 0.005 = −205,000; ½ × 394,000,000 × 0.005² = +4,925. Total −200,075.

On your calculator

41000000×0.005=+|-STO1
0.5×394000000×0.005×0.005+RCL1= −200,075

+|- change sign · STO store · RCL recall a stored number

41000000ENTER0.005×CHS
0.5ENTER394000000×0.005×0.005×+ −200,075

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

How it is asked

Percentage change, large rise

ModDur 8.20, convexity 85.0, yield +125 bp. Price change?

−10.25% + 0.6641% = −9.59%. Duration alone gives −10.25%; forgetting the ½ gives −8.92%.
Currency change on a position

GBP 400m par, full price 102.50, ModDur 8.00, convexity 90.0, yield −75 bp.

Market value 410m: 24.6m + 1.038m = +GBP 25.638m. Subtracting convexity gives 23.562m.
Effect of adding convexity

Compared with duration alone, a 200 bp rise: is the estimated decline larger or smaller?

Smaller: the convexity term is positive, so it offsets part of the loss.

Exam traps

Forgetting the ½ in the convexity term.
Subtracting the convexity term when yields rise. It is always added.
Using Δy in basis points (100) or percent (1) instead of a decimal (0.01).
Mixing money duration with percentage convexity, or the reverse.
Using par instead of market value to build the money measures.

Checkpoint

6 exam-style questions. About 90 seconds each.

1A bond has an annualized modified duration of 7.50 and an annualized convexity of 70.0. If its yield rises by 150 bp, its full price will most likely fall by an amount closest to:
2An investor holds JPY 2,000,000,000 par of a bond with a full price of 99.00, an annualized modified duration of 6.00 and an annualized convexity of 45.0. If yields fall by 100 bp, the estimated change in the position's value is closest to:
3Compared with the estimate using modified duration alone, adding the convexity adjustment for a 200 bp rise in yield makes the estimated price decline:
4An investor holds INR 50,000,000 par of a bond with a full price of 102.00 and an annualized convexity of 60.0. The position's money convexity is closest to:
5The position in the previous question has a money duration of INR 357,000,000 (modified duration 7.0 on a market value of INR 51,000,000) and a money convexity of INR 3,060,000,000. If yields rise by 50 bp, the estimated change in value is closest to:
6A bond has an annualized modified duration of 12.0 and an annualized convexity of 180. If its yield rises by 200 bp, the estimated percentage change in its full price is closest to:

Key takeaways

%ΔP ≈ −ModDur × Δy + ½ × Convexity × (Δy)².
The convexity term is always positive for an option-free bond.
In currency, use money duration and money convexity with the same structure.
The bigger the yield move, the more the convexity term matters.

Price change with duration and convexity

\%\Delta PV^{Full} \approx (-\text{AnnModDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta y)^2\right]

Quick example

Modified duration 3.80, convexity 18.5, yield falls 75 bp. Duration: +3.80 × 0.0075 = 2.850%. Convexity: ½ × 18.5 × 0.0075² = 0.052%. Total +2.902%.

On your calculator

3.8×0.0075=STO1
0.5×18.5×0.0075×0.0075+RCL1= 0.02902

STO store · RCL recall a stored number

3.8ENTER0.0075×
0.5ENTER18.5×0.0075×0.0075×+ 0.02902

ENTER enter (on the HP, puts the number on the stack)

Money convexity

\text{MoneyCon} = \text{AnnConvexity} \times PV^{Full}

Quick example

AUD 10,000,000 par at a full price of 98.50 (market value 9,850,000) with convexity 40.0. MoneyCon = 40.0 × 9,850,000 = AUD 394,000,000.

On your calculator

10000000×0.985×40= 394,000,000

10000000ENTER0.985×40× 394,000,000

ENTER enter (on the HP, puts the number on the stack)

Change in value with money duration and money convexity

\Delta PV^{Full} \approx (-\text{MoneyDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{MoneyCon} \times (\Delta y)^2\right]

Quick example

MoneyDur 41,000,000, MoneyCon 394,000,000, yield +50 bp. −41,000,000 × 0.005 = −205,000; ½ × 394,000,000 × 0.005² = +4,925. Total −200,075.

On your calculator

41000000×0.005=+|-STO1
0.5×394000000×0.005×0.005+RCL1= −200,075

+|- change sign · STO store · RCL recall a stored number

41000000ENTER0.005×CHS
0.5ENTER394000000×0.005×0.005×+ −200,075

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 Convexity and Portfolio Duration

Next lesson Portfolio duration and convexity

Questions about this lesson?

Where this comes back

Credit Risk Fixed Income
Effective Duration, Key Rate Duration and Empirical Duration Fixed Income

Questions candidates ask

How do you estimate bond price change with duration and convexity?

Multiply modified duration by the yield change and flip the sign, then add half the convexity times the yield change squared. Enter the yield change as a decimal.

What is money convexity?

Convexity multiplied by the bond's full price or position value, used to estimate the currency change in value.

Why is the convexity adjustment always positive?

Because the yield change is squared and an option-free bond's convexity is positive.

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