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CFA Level I Exam · Interest Rate Risk and Return

Bond Price-Yield Relationship for CFA Level I

Updated 7 October 2026 · Fact-checked

A bond's price is the present value of its coupons and principal, discounted at the market yield. When yields rise, the discount rate rises and price falls, and the reverse. The curve is convex, so price rises more for a yield fall than it falls for an equal yield rise. Solve by discounting cash flows or using the calculator.

Understand Bond Price-Yield Relationship

A bond promises fixed cash flows: coupons and a final principal. Its price today is what those cash flows are worth when discounted at the yield investors require. The coupons do not change. The required yield does. So when the market yield rises, each cash flow is discounted more heavily and the price falls. When the yield falls, the price rises. This is the inverse relationship between price and yield.

The link to par is simple. If the coupon rate equals the yield, the bond prices at par. If the coupon rate is below the yield, the bond trades at a discount. If the coupon rate is above the yield, it trades at a premium.

The price-yield curve is not a straight line. It is convex, meaning it bows toward the origin. For the same size of yield change, the price gain when yields fall is larger than the price loss when yields rise. This is why duration alone, which is a straight-line estimate, is not enough for large yield moves. Convexity corrects it.

Price sensitivity to yield changes is interest rate risk. Holding other factors constant, sensitivity is greater when the coupon rate is lower, when maturity is longer, and when the yield level is lower. A low-coupon bond pays more of its value late, so more of its value is hit by discounting. A zero-coupon bond has the most sensitivity for its maturity. The maturity effect is a general rule and you should treat it as 'all else equal'.

These are the same ideas that sit behind duration and convexity. If you know why the curve slopes down and bows, those later topics become much easier.

Key formulas to remember

Bond price
PV = PMT ÷ (1+r) + PMT ÷ (1+r)² + … + (PMT + FV) ÷ (1+r)^N
r is the yield per period, N the number of periods, PMT the coupon per period. For semiannual bonds, halve the annual yield and coupon and double the years.
Coupon-yield rule
Coupon rate = yield → price = par; coupon rate < yield → discount; coupon rate > yield → premium
Holds when the yield is measured on the same periodic basis as the coupon.
Inverse relationship
Yield ↑ → price ↓; yield ↓ → price ↑
For option-free bonds the curve is convex. Callable bonds can show negative convexity at low yields.
Convexity effect
Price gain for a yield fall of Δy > price loss for a yield rise of Δy
Holds for an option-free bond with positive convexity.
Price change estimate
%ΔPrice ≈ −(Modified duration × ΔYield) + ½ × Convexity × (ΔYield)²
The first term is the straight-line estimate. The second is the convexity adjustment, which is positive for option-free bonds.
Sensitivity rules (all else equal)
Lower coupon → more sensitive; longer maturity → more sensitive; lower yield level → more sensitive
Treat maturity as a general rule, not a law for every bond.

How to solve Bond Price-Yield Relationship questions

Use this method for any question on how a bond price responds to yield, or how to compare two bonds.

  1. 1Identify what is asked: a price level, a direction of change, a size of change, or which bond is more sensitive.
  2. 2Check the coupon rate against the yield to decide whether the bond is at a premium, discount or par.
  3. 3For a price, match the coupon frequency and the yield frequency. Set N, I/Y, PMT and FV for that period length.
  4. 4Compute the price with the calculator (CPT PV) or by discounting each cash flow.
  5. 5For a direction question, apply the inverse rule: yield up means price down, yield down means price up.
  6. 6For a comparison, rank the bonds by coupon (lower is more sensitive), maturity (longer is more sensitive) and yield level (lower is more sensitive).
  7. 7For a size question with a large yield move, remember convexity: the gain from a fall is bigger than the loss from a rise.
  8. 8Check your answer with a sanity test: a discount bond must price below par, and a premium bond above par.

Quickest way: Direction and ranking shortcut

When to use it: Use this when the question asks which bond reacts most, or whether the price goes up or down, without needing an exact price.

  1. Decide the direction first from the yield move. Cross out any option with the wrong direction.
  2. Pick the bond with the lowest coupon and longest maturity as the most sensitive, if both features point the same way.
  3. If a yield fall and a yield rise of the same size are compared, choose the bigger price move for the fall.
  4. Compare the coupon to the yield to eliminate options that call a premium bond a discount bond.
  5. For a numeric price, enter N, I/Y, PMT and FV, then CPT PV. Ignore the negative sign.

Common mistakes in Bond Price-Yield Relationship

  • Treating the price-yield link as a straight line

    Duration is taught first and gives a linear estimate, so students forget the curve bows.

    Fix: Remember that equal yield changes do not give equal price changes. The price rise from a yield fall is larger than the price fall from a yield rise.

  • Using the annual yield with semiannual coupons

    The question gives annual figures and students skip the periodic conversion.

    Fix: Divide the annual yield and coupon by the number of periods per year, and multiply the years by the same number, before entering N, I/Y and PMT.

  • Choosing par as the price whenever coupon is near yield

    Students round mentally and ignore that any difference moves the price.

    Fix: Only a coupon exactly equal to the yield gives par. If the coupon is lower than the yield, the price is below par, even if only slightly.

  • Saying a higher coupon bond is more sensitive

    Students think bigger cash flows mean bigger moves.

    Fix: A higher coupon returns cash sooner, so less value sits in distant cash flows. Lower-coupon bonds have more interest rate risk.

  • Forgetting that a lower starting yield raises sensitivity

    Students focus on coupon and maturity and ignore the yield level.

    Fix: At a lower yield level, the curve is steeper. The same yield change causes a bigger percentage price change.

  • Treating the maturity rule as always true

    The rule is stated without its 'all else equal' condition.

    Fix: Use it when other features such as coupon and yield are the same. Do not apply it to compare bonds that differ in several features without checking.

Worked examples

Example 1

A 3-year bond pays a 5% annual coupon on a par value of USD 1,000. The market yield to maturity is 6%. Which is closest to the bond's price? A) USD 947.13 B) USD 973.27 C) USD 1,000.00

Show the solution
  1. The coupon of 5% is below the yield of 6%, so the bond trades at a discount. Price must be below USD 1,000.00, so C is out.
  2. Coupon = 5% × 1,000 = USD 50 per year. N = 3, r = 6%.
  3. PV of coupon 1 = 50 ÷ 1.06 = 47.17. PV of coupon 2 = 50 ÷ 1.1236 = 44.50.
  4. PV of final payment = 1,050 ÷ 1.191016 = 881.60.
  5. Sum = 47.17 + 44.50 + 881.60 = 973.27.
  6. Calculator: N = 3, I/Y = 6, PMT = 50, FV = 1000, CPT PV = −973.27.

Answer: B) USD 973.27

Example 2

A 10-year zero-coupon bond with par USD 1,000 has a yield of 5% (annual compounding). The yield falls to 4%. The bond's price change is closest to: A) +9.04% B) +10.04% C) +11.04%

Show the solution
  1. Initial price = 1,000 ÷ 1.05^10 = 1,000 ÷ 1.628895 = 613.91.
  2. New price = 1,000 ÷ 1.04^10 = 1,000 ÷ 1.480244 = 675.56.
  3. Percentage change = 675.56 ÷ 613.91 − 1 = +10.04%.
  4. Check with convexity: if the yield had risen to 6%, the price would be 1,000 ÷ 1.790848 = 558.39, a fall of 9.04%.
  5. The gain for a 1% fall (10.04%) is bigger than the loss for a 1% rise (9.04%). This confirms the convex curve and rules out A for a yield fall.

Answer: B) +10.04%

Exam tips

  • Read the direction of the yield change before anything else. Many questions are solved by the inverse rule alone.
  • Watch for the 'same size' wording. If it compares a yield rise with a fall, the answer usually uses convexity.
  • For price calculations, check whether coupons are annual or semiannual and adjust N, I/Y and PMT. Clear the calculator memory first.
  • Sort answers by premium, discount or par before computing. This often eliminates two options in seconds.
  • There is no penalty for wrong answers, so always pick the most reasonable option even if time is short.

Practice questions from Interest Rate Risk and Return

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.

Bond Price-Yield Relationship: frequently asked questions

Why do bond prices fall when yields rise?

A bond's coupons and principal are fixed. A higher market yield means those cash flows are discounted at a higher rate, so their present value, which is the price, is lower. New bonds also offer higher coupons, so older bonds must sell cheaper to match.

How do I calculate a bond price from yield to maturity?

Discount each coupon and the principal at the yield per period and add them up. On the BA II Plus, enter N, I/Y, PMT and FV, then press CPT PV. Match the period length to the coupon frequency.

Which bond has the most interest rate risk?

All else equal, the bond with the lowest coupon, the longest maturity and the lowest yield level. A zero-coupon bond with long maturity is a typical example. Compare the features one at a time.

What does convexity mean for the price-yield curve?

It means the curve bows toward the origin instead of being a straight line. An option-free bond gains more in price when yields fall than it loses when yields rise by the same amount. A duration-only estimate misses this.