CFA Level I Exam · Curve-Based and Empirical Fixed-Income Risk Measures

Interest Rate Risk and the Bond Price-Yield Relationship

Q: Why do bond prices fall when yields rise?

A bond's price is the present value of its fixed cash flows. A higher yield means a higher discount rate, so the present value is lower.

Q: What does convexity mean for the price-yield curve?

The curve bends rather than being a straight line. For an option-free bond, this means a yield fall raises the price by more than an equal yield rise lowers it.

Q: How do coupon and maturity affect interest rate risk?

Other things equal, a longer maturity and a lower coupon both raise sensitivity to yield changes. A zero-coupon bond is the most sensitive for a given maturity.

Q: Does a lower yield level change sensitivity?

Yes, for an option-free bond. At a lower starting yield, the same yield change gives a larger percentage price change, because duration and convexity rise as yield falls and the curve is steeper at low yields.

Updated 7 October 2026 · Fact-checked

Interest rate risk is the risk that bond prices change when market yields change. Price and yield move in opposite directions along a convex curve. Longer maturity and lower coupon raise sensitivity; a lower starting yield does too. For an option-free bond, price rises from a yield fall exceed price falls from an equal yield rise.

Understand Interest Rate Risk and Bond Price-Yield Relationship

A bond is a set of fixed cash flows. Its price is the present value of those flows at the market yield. When the yield goes up, you discount the same cash flows more heavily, so the price falls. When the yield goes down, the price rises. This inverse relationship is the core of interest rate risk.

If you plot price against yield, you do not get a straight line. You get a curve that bends towards the origin. This shape is called convex. Because of the bend, a yield fall of 1 percentage point raises the price by more than a yield rise of 1 percentage point lowers it. Convexity is good for the bondholder: gains are larger than losses for equal yield moves. This applies to an option-free bond.

Three features drive how sensitive a bond's price is to yield changes, holding the others constant:

Maturity: a longer maturity means more distant cash flows, which are hit harder by discounting. Longer bonds are more sensitive.
Coupon: a lower coupon means more of the value sits in the final principal payment, which is far away. Lower-coupon bonds are more sensitive. A zero-coupon bond is the most sensitive for its maturity.
Yield level: for an option-free bond, at a lower starting yield the same yield change causes a larger percentage price change. Duration and convexity both rise as yield falls, so the curve is steeper at low yields.

The bond's price also relates to its coupon rate. If the coupon rate equals the yield, the bond trades at par. If the coupon is above the yield, it trades at a premium. If below, at a discount.

For exams, you must state the direction, compare two bonds, and recognise that the price-yield curve is convex, so a linear estimate from duration alone is imperfect. Duration gives the slope of the curve; convexity captures the curvature.

Key formulas to remember

Bond price: PV = Σ [PMT ÷ (1 + r)^t] + FV ÷ (1 + r)^N; r is the yield per period, N the number of periods. Higher r gives lower PV.
Price and yield relationship: Yield ↑ ⇒ Price ↓; Yield ↓ ⇒ Price ↑; The relationship is inverse and convex for an option-free bond.
Coupon versus yield: Coupon = YTM: par; Coupon > YTM: premium; Coupon < YTM: discount; Uses the yield per period on the same compounding basis.
Sensitivity rules (other things equal): Longer maturity ⇒ more sensitive; Lower coupon ⇒ more sensitive; Lower yield ⇒ more sensitive; Compare one feature at a time. Do not state these as guaranteed if two features change together.
Percentage price change: % change = (New price − Old price) ÷ Old price; Use this to compare the size of gains and losses.

How to solve Interest Rate Risk and Bond Price-Yield Relationship questions

Use this routine for any question on interest rate risk or the price-yield relationship.

1Identify what changes: yield up or down, and by how much.
2State the direction: yield up means price down, and the reverse.
3If the question compares two bonds, list the differences in maturity, coupon and starting yield.
4Apply the sensitivity rules one feature at a time: longer maturity, lower coupon and lower yield mean more sensitivity.
5If the features point in opposite directions (e.g., longer maturity but higher coupon), you cannot rank the bonds without a calculation or a stated constant.
6For a size comparison of up and down moves, use convexity: the price gain from a yield fall exceeds the loss from an equal yield rise.
7If a calculation is needed, discount cash flows at the new yield with the calculator and compute the percentage change.
8Check that the answer has the right direction before choosing an option.

Quickest way: Direction first, then rank sensitivity

When to use it: Use for conceptual three-option questions with two bonds or a yield change.

Fix the direction from the yield move. This often removes one option.
Rank bonds by coupon and maturity: zero coupon and long maturity are most sensitive.
For equal yield moves of opposite sign, the price rise is larger than the price fall.
Pick the option that matches, and skip any calculation unless the options are numbers.

Common mistakes in Interest Rate Risk and Bond Price-Yield Relationship

Thinking higher-coupon bonds are more sensitive to yield changes.
Bigger coupons feel like bigger cash flows, so students assume bigger exposure.
Fix: Higher coupons return cash sooner, so the value is less tied to distant principal. Lower coupon means more sensitivity, other things equal.
Treating the price-yield curve as a straight line.
Duration-based estimates are linear and students carry that over.
Fix: The curve is convex for an option-free bond and lies above its tangent line. A duration-only estimate therefore understates the new price after a yield fall and after a yield rise. Add convexity to correct this.
Assuming a yield rise and an equal yield fall change the price by the same amount.
Students think symmetry applies because the yield moves are equal.
Fix: Convexity makes the price gain from a fall larger than the price loss from a rise.
Saying a longer maturity always means greater interest rate risk.
The rule is learned without the condition.
Fix: It holds with other features equal. A long, high-coupon bond can be less sensitive than a shorter, low-coupon one, so check what the question holds constant.
Mixing up premium and discount.
Students compare coupon with the wrong rate or forget the direction.
Fix: Coupon above the market yield gives a premium price. Coupon below the yield gives a discount.

Worked examples

Example 1

A 2-year annual-pay bond has a 5% coupon and a face value of $1,000. Its yield is 5%. The yield then rises to 6%. What is the new price? A) $981.67 B) $1,000.00 C) $1,018.52

Show the solution

At a 5% yield equal to the coupon, the price is par, $1,000.
Cash flows: $50 in year 1 and $1,050 in year 2.
At 6%: 50 ÷ 1.06 = 47.17.
1,050 ÷ 1.1236 = 934.50.
Add: 47.17 + 934.50 = 981.67.
A yield rise should lower the price below par, which matches option A.

Answer: A) $981.67

Example 2

Two option-free bonds have the same maturity and yield. Bond X has a 3% coupon and Bond Y has a 7% coupon. Yields fall by 50 basis points. Which statement is correct? A) Bond X rises by a larger percentage B) Both bonds rise by the same percentage C) Bond Y rises by a larger percentage

Show the solution

Yield falls, so both prices rise. This confirms direction.
Maturity and yield are equal, so only the coupon differs.
Lower coupon means more of the value is in the distant principal, so more sensitivity.
Bond X, the 3% coupon bond, therefore has a larger percentage price rise.

Answer: A) Bond X rises by a larger percentage

Exam tips

Settle the direction first. Many options fail on direction alone.
Learn the three sensitivity rules with their condition: other things equal.
For up versus down moves of equal size, choose the larger price gain from a yield fall.
In numerical questions, the answer must sit on the correct side of par: premium if the coupon exceeds the yield.
On the BA II Plus, use N, I/Y, PMT, FV, then CPT PV, and make sure PMT and FV carry opposite sign conventions to PV.

Practice questions from Curve-Based and Empirical Fixed-Income Risk Measures

Interest Rate Risk and Bond Price-Yield Relationship in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Interest Rate Risk and Bond Price-Yield Relationship: frequently asked questions

Why do bond prices fall when yields rise?

A bond's price is the present value of its fixed cash flows. A higher yield means a higher discount rate, so the present value is lower.

What does convexity mean for the price-yield curve?

The curve bends rather than being a straight line. For an option-free bond, this means a yield fall raises the price by more than an equal yield rise lowers it.

How do coupon and maturity affect interest rate risk?

Other things equal, a longer maturity and a lower coupon both raise sensitivity to yield changes. A zero-coupon bond is the most sensitive for a given maturity.

Does a lower yield level change sensitivity?

Yes, for an option-free bond. At a lower starting yield, the same yield change gives a larger percentage price change, because duration and convexity rise as yield falls and the curve is steeper at low yields.