Lesson 3 of 4 11–14% of the exam

By Daniil Tarakanov Updated Oct 21, 20262027 curriculum

How price, coupon, maturity and yield are related

Prices move opposite to yields. For the same yield change, lower-coupon and (usually) longer-maturity bonds move more in percentage terms, and a fall in yield lifts the price more than an equal rise cuts it. If the yield never changes, every bond's price drifts to par by maturity.

8 min cram, 36 min full read. Based on the 2027 CFA® Program Level I curriculum, Fixed Income, Learning Module 6. 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: Pricing a bond from its yield to maturity

On the exam

Identify the relationships among a bond’s price, coupon rate, maturity, and yield-to-maturity

Identify the relationships among a bond's price, coupon rate, maturity and yield to maturity. Expect a question on which bond is most sensitive, one on pull to par, and one on the asymmetry of price changes (convexity).

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 is a bond’s price linked to its yield?

A bond’s price is the present value of fixed cash flows, so a higher discount rate means a lower price. This is the inverse relationship between price and yield. How far the price moves depends on the bond’s coupon, its maturity and the direction of the move.

What moves a bond’s price

Bond pricethe present value of the cash flows at the yield

Yield to maturity Memorize price moves the opposite way
- Convexity effect a fall in yield lifts the price more than an equal rise cuts it
Coupon rate Memorize coupon effect: a lower coupon means a bigger percentage move
Time to maturity Memorize maturity effect: longer usually means a bigger move
The passage of time Memorize pull to par at a constant yield

InteractiveWhich bonds move most when yields change?

Try this 0 of 3

In the coupon effect, raise the yield by exactly 1 point
In the convexity view, move the yield 1 point up and 1 point down
On Pull to par, find the premium bond's price with 1 year left

New yield (start 6%)6.5%

Add a third bond of your own

10-year 6%
10-year zero

10-year 6%, +50 bps−3.59%

10-year zero, +50 bps−4.60%

10-year zero moves most: −4.60% for +50 bps.

Show working

Bond	Price at 6%	At 6.5%	Change
10-year 6%	100.000	96.406	−3.59%
10-year zero	55.839	53.273	−4.60%

10-year 6%: a 1-point fall in yield +7.72%, a 1-point rise −7.02%.

10-year zero: a 1-point fall in yield +9.94%, a 1-point rise −8.96%.

Percentage change = new price ÷ starting price − 1. For example 10-year 6%: 96.406 ÷ 100.000 − 1 = −3.59%.

10-year 6% at 6.5% on your calculator

2NDCLR TVMwith P/Y set to 1
10N6.5I/Y6PMT100FV
CPTPV−96.406

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

fCLEAR FIN
10n6.5i6PMT100FV
PV−96.406

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Low coupons and long maturities magnify yield moves; time alone pulls every price to par.

Why do low-coupon bonds move more?

The coupon effect : for bonds with the same maturity, a lower coupon means a larger percentage price change for the same change in yield. A low coupon leaves more of the bond’s value in the final payment, the cash flow furthest away and most affected by a change in the discount rate.

Worked example Two 10-year bonds at a 6% yield

Both bonds are priced at a 6% yield and then yields move 1 percentage point.

Bond	Price at 6%	Yield +1 point	Yield −1 point
6% coupon	100.000	92.976 (−7.02%)	107.722 (+7.72%)
Zero-coupon	55.839	50.835 (−8.96%)	61.391 (+9.94%)

The zero-coupon bond has no coupons to cushion the move, so it falls and rises more.

The zero-coupon bond at a 7% yield

2NDCLR TVM with P/Y set to 1
10N7I/Y100FV
CPTPV −50.835
+|-÷55.839−1= −0.0896, the percentage change from 55.839

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · FV future value · CPT compute · PV present value · +|- change sign

fCLEAR FIN
10n7i100FV
PV −50.835
CHS55.839÷1− −0.0896, the percentage change from 55.839

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · FV future value · PV present value · CHS change sign

Why do long-maturity bonds move more?

The maturity effect : for bonds with the same coupon, a longer maturity usually means a larger percentage price change. More of the cash flows sit far away, where a change in the discount rate compounds for longer.

Worked example Two 4% bonds, priced at par

Both bonds pay a 4% annual coupon and yield 4%, so each is priced at par. Yields then move 1 percentage point.

Bond	Yield +1 point	Yield −1 point
5-year	95.671 (−4.33%)	104.580 (+4.58%)
20-year	87.538 (−12.46%)	114.877 (+14.88%)

The 20-year bond at a 5% yield

2NDCLR TVM with P/Y set to 1
20N5I/Y4PMT100FV
CPTPV −87.538
+|-÷100−1= −0.1246

fCLEAR FIN
20n5i4PMT100FV
PV −87.538
CHS100÷1− −0.1246

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value · CHS change sign

Exam tip Combine the two effects. Lowest coupon and longest maturity make the most sensitive bond. A zero-coupon bond is the most sensitive for its maturity.

Go deeper When the maturity effect can fail

For a long, low-coupon bond trading at a deep discount, a longer maturity can occasionally mean a smaller percentage price change: past a certain point, adding years no longer pushes the weight of the cash flows further out. The case is rare. It never applies to zero-coupon bonds, par bonds or premium bonds, and the exam treats longer maturity as more sensitive unless it says otherwise.

Why do prices rise more than they fall?

The convexity effect : for the same size of yield change, the percentage price gain when yields fall is larger than the percentage loss when yields rise. The price–yield curve bends toward the origin instead of following a straight line. Every row in the tables above shows it (for the 20-year bond, +14.88% against −12.46%).

Worked example A 9-year 5% bond at par, yield moves 0.6 points

Yield falls to 4.4%, with par as the FV (future value): N = 9, I/Y = 4.4, PMT = 5, FV = 100 gives 104.381, a gain of +4.38%.
Yield rises to 5.6%: I/Y = 5.6 gives 95.847, a loss of −4.15%.

Equal yield moves, unequal price moves.

The 4.4% case

2NDCLR TVM with P/Y set to 1
9N4.4I/Y5PMT100FV
CPTPV −104.381
+|-÷100−1= 0.0438

fCLEAR FIN
9n4.4i5PMT100FV
PV −104.381
CHS100÷1− 0.0438

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value · CHS change sign

Percentage price change Use it

\Delta P\% = \frac{P_1}{P_0} - 1

Rearrange itPercentage price change: solve for the change, the new price or the old price

The question asks for

A bond's price moves from 100.00 to 104.38 when its yield changes. The percentage price change is closest to:

Starting priceNew price

Δ P % = \frac{P _{1}}{P _{0}} - 1

104.38 ÷ 100.00 − 1 = 4.381%

Percentage price change4.38%

On your calculator

104.381÷100−1=0.04381

104.381ENTER100÷1−0.04381

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

Go deeper Where this goes next

Duration is one number that captures the coupon and maturity effects: it estimates the percentage price change for a small yield change. Convexity measures the curvature that causes the asymmetry. Both come later in Fixed Income, and neither is needed here: you only need to recognize the direction and rank the bonds.

Where does the price go if the yield never changes?

Time alone moves a bond’s price. As maturity approaches, the remaining cash flows shrink in number and the price is pulled to par, provided the issuer does not default. This is the constant yield-price trajectory , and the pull itself is called pull to par .

Where the price goes at a constant yield

Price at a constant yield, no default

Discount bond Memorize price rises toward par
Par bond Memorize price stays at par
Premium bond Memorize price falls toward par

Price at a constant yield, with time Use it

PV_{N-k} = \sum_{j=1}^{N-k} \frac{PMT}{(1 + r)^{j}} + \frac{FV}{(1 + r)^{N-k}}

Worked example Two 8-year bonds at a constant 5% yield

A 3% coupon bond starts at a discount and a 7% coupon bond at a premium. Recompute each price as the remaining years shrink, holding the yield at 5%.

Years left	8	6	4	2	1	0
3% coupon	87.07	89.85	92.91	96.28	98.10	100
7% coupon	112.93	110.15	107.09	103.72	101.90	100

The discount bond climbs and the premium bond slides, and both reach 100. The two prices converge.

The 3% bond with 4 years left

2NDCLR TVM with P/Y set to 1
4N5I/Y3PMT100FV
CPTPV −92.908

fCLEAR FIN
4n5i3PMT100FV
PV −92.908

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Go deeper Pull to par is not a gain you can bank for free

The price drift is already built into the yield: a discount bond’s YTM includes the gain as the price rises to par, and a premium bond’s YTM includes the loss as it falls. If the yield stays unchanged, the gain or loss arrives as scheduled. If the yield moves, the price moves with it.

Memorize this

MemorizeWhich bond is more sensitive

Compare	More sensitive to a yield change	Why
Same maturity, different coupon	Lower coupon	More of the value sits in the last payment
Same coupon, different maturity	Longer maturity (almost always)	More distant cash flows
Rise vs equal fall in yield	The fall (price gain is bigger)	Convexity

Memory hook Value far in the future is shaky. Zero coupon and long maturity push the value furthest away.

Know it cold A zero-coupon bond is the most sensitive bond of its maturity. Exception: some long, low-coupon, deep-discount bonds; never zeros, par or premium bonds.

MemorizePull to par

Memory hook Everything ends at 100: discount bonds climb, premium bonds slide.

Know it cold At an unchanged yield the price moves to par as maturity approaches, if the issuer does not default.

What moves a bond’s price

Bond pricethe present value of the cash flows at the yield

Yield to maturity Memorize price moves the opposite way
- Convexity effect a fall in yield lifts the price more than an equal rise cuts it
Coupon rate Memorize coupon effect: a lower coupon means a bigger percentage move
Time to maturity Memorize maturity effect: longer usually means a bigger move
The passage of time Memorize pull to par at a constant yield

Where the price goes at a constant yield

Price at a constant yield, no default

Discount bond Memorize price rises toward par
Par bond Memorize price stays at par
Premium bond Memorize price falls toward par

Percentage price change

\Delta P\% = \dfrac{P_1}{P_0} - 1

Quick example

A 9-year 5% bond at par falls to 95.85 when its yield rises to 5.6%. 95.85 ÷ 100 − 1 = −4.15%.

On your calculator

95.847÷100−1= −0.0415

95.847ENTER100÷1− −0.0415

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

Price at a constant yield, with time

PV_{N-k} = \sum_{j=1}^{N-k} \dfrac{PMT}{(1 + r)^{j}} + \dfrac{FV}{(1 + r)^{N-k}}

Quick example

A 10-year 6% bond priced at a constant 5% yield is 107.72 today and 106.46 two years later (8 years left). Same formula, fewer periods.

On your calculator

2NDCLR TVM with P/Y set to 1
8N5I/Y6PMT100FV
CPTPV −106.4632

fCLEAR FIN
8n5i6PMT100FV
PV −106.4632

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

How it is asked

Which bond is most sensitive?

8-year 5%, 8-year 1% or 4-year 1%. Which falls most when yields rise 1 point?

Lower coupon and longer maturity: the 8-year 1%. The higher coupon (A) and shorter maturity (C) are the traps.
Price after a yield fall

9-year 5% bond at par, yield falls to 4.4%. New price?

N = 9, I/Y = 4.4, PMT = 5 gives 104.38. Mirroring the fall for a rise (104.15) ignores convexity.
Pull to par

Premium bond, constant yield, no default. Price over time?

Falls toward par. Staying put or rising is the trap.
Gain or loss for a ±1 point move

Which is larger, the gain from a 1-point fall or the loss from a 1-point rise?

The gain (convexity).

Exam traps

Picking the higher-coupon bond as more sensitive. The lower coupon moves more.
Thinking a premium bond's price stays above par until maturity at a constant yield. It falls to par.
Treating price changes for rises and falls in yield as symmetric. The gain from a fall is larger.
Applying the maturity effect to a par or premium bond or a zero without checking. It fails only for some long, low-coupon, deep-discount bonds.

Checkpoint

3 exam-style questions. About 90 seconds each.

1Which bond's price is most sensitive to a 1-point change in yield?
2A 10-year, 4% annual-coupon bond trades at par. If its yield falls to 3.5%, its price per 100 of par is closest to:
3A premium bond's yield stays constant until it matures, and the issuer does not default. Over time its price will:

Key takeaways

Price and yield move in opposite directions.
Lower coupon or longer maturity: larger percentage price changes.
Gains from falling yields exceed losses from equal rises: convexity.
At a constant yield, prices are pulled to par.

Percentage price change

\Delta P\% = \dfrac{P_1}{P_0} - 1

Quick example

A 9-year 5% bond at par falls to 95.85 when its yield rises to 5.6%. 95.85 ÷ 100 − 1 = −4.15%.

On your calculator

95.847÷100−1= −0.0415

95.847ENTER100÷1− −0.0415

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

Price at a constant yield, with time

PV_{N-k} = \sum_{j=1}^{N-k} \dfrac{PMT}{(1 + r)^{j}} + \dfrac{FV}{(1 + r)^{N-k}}

Quick example

A 10-year 6% bond priced at a constant 5% yield is 107.72 today and 106.46 two years later (8 years left). Same formula, fewer periods.

On your calculator

2NDCLR TVM with P/Y set to 1
8N5I/Y6PMT100FV
CPTPV −106.4632

fCLEAR FIN
8n5i6PMT100FV
PV −106.4632

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

All Fixed Income formulas, every lesson

Practice exam-style questions on Bond Valuation: Prices and Yields

Next lesson Matrix pricing

Questions about this lesson?

Where this comes back

Interest Rate Risk and Return Fixed Income
Modified Duration, Money Duration and PVBP Fixed Income
Convexity and Portfolio Duration Fixed Income

Questions candidates ask

Why are zero-coupon bonds more sensitive to interest rates?

All their value comes from the single payment at maturity, which is discounted the most. Coupon bonds return part of the value earlier, so changes in the discount rate affect them less.

Do longer bonds always fall more when rates rise?

Almost always. The rare exception is some long, low-coupon bonds trading at a deep discount, where the effect can reverse. For par, premium and zero-coupon bonds the longer bond always moves more.

What does pull to par mean?

As maturity approaches, a bond's price moves toward its par value if the yield does not change and the issuer does not default. Discount bonds rise and premium bonds fall.

Spotted an error? Tell us and we will fix it.