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CFA Level I Exam · Fixed-Income Bond Valuation: Prices and Yields

Bond Price-Yield Relationship and Convexity Explained

Updated 7 October 2026 · Fact-checked

Bond price and yield move in opposite directions because a bond's price is the present value of fixed cash flows discounted at the yield. The curve is convex, so a yield fall raises price more than an equal yield rise cuts it. Solve questions by finding direction first, then size using duration and convexity.

Understand Price-Yield Relationship and Convexity

A bond promises fixed coupons and a principal repayment. Its price is the present value of those cash flows, discounted at the market yield. When the yield rises, you divide each cash flow by a bigger number, so the present value falls. When the yield falls, the present value rises. That is why price and yield are inversely related.

If you plot price on the vertical axis and yield on the horizontal axis, you do not get a straight line. You get a curve that bends toward the origin. This shape is convex. The result: for the same size of yield change, the price gain when yield falls is larger than the price loss when yield rises. Convexity works in your favour on an option-free bond.

The relationship also tells you how a bond is priced against its coupon. If the coupon rate is above the yield, the bond trades at a premium (price above par). If the coupon rate equals the yield, it trades at par. If the coupon rate is below the yield, it trades at a discount. As time passes with a constant yield, the price moves toward par at maturity.

Price sensitivity depends on bond features, with other things equal. A longer maturity means more sensitivity. A lower coupon means more sensitivity, because more of the value comes late. A lower yield level also means more sensitivity, so the same 100 bp move changes price more when yields are low than when they are high.

A straight-line estimate using modified duration works for small yield changes. For larger changes the line misses the curve. Convexity is the correction. For option-free bonds it is positive. Callable bonds can show negative convexity at low yields, because the price is capped as the call becomes likely.

Key formulas to remember

Bond price
PV = Σ [PMT ÷ (1 + r)^t] + FV ÷ (1 + r)^N
r is the yield per period. Higher r means lower PV, which is the inverse relationship.
Coupon vs yield rule
Coupon rate > YTM → premium; coupon rate = YTM → par; coupon rate < YTM → discount
Compare the coupon rate with the yield per period on the same basis.
Duration-only price change
%ΔPV ≈ −ModDur × ΔYield
Linear estimate. It understates the price rise for a yield fall and overstates the price fall for a yield rise on an option-free bond.
Duration plus convexity
%ΔPV ≈ −ModDur × ΔYield + ½ × Convexity × (ΔYield)²
Enter ΔYield as a decimal, e.g. 0.01 for 100 bp. The convexity term is always positive when convexity is positive.
Sensitivity rules (other things equal)
Longer maturity ↑ sensitivity; lower coupon ↑ sensitivity; lower yield level ↑ sensitivity
Use these for ranking questions. They are comparisons, so keep the 'other things equal' condition.

How to solve Price-Yield Relationship and Convexity questions

Use the same sequence for any question on price-yield behaviour. Decide direction first, then size, then check the traps.

  1. 1Read what changes: yield up or down, and by how much. Note whether the question asks for direction, relative size or a number.
  2. 2Set the direction. Yield up means price down. Yield down means price up. This alone can remove one or two options.
  3. 3Classify the bond by comparing coupon rate with yield: premium, par or discount.
  4. 4For comparison questions, apply the sensitivity rules: longer maturity, lower coupon and lower yield mean a bigger price change. Check that only one feature differs.
  5. 5For asymmetry questions, remember that on an option-free bond the price rise from a yield fall exceeds the price fall from an equal yield rise.
  6. 6For numbers, compute with the formula. Use the calculator for exact prices, or duration and convexity for estimates. Convert bp to decimals (50 bp = 0.005).
  7. 7Check that the sign and size make sense, then pick the option. The convexity term should be added, and it should be small.

Quickest way: Direction, then ranking, then convexity add-on

When to use it: Use this when you have about 90 seconds and the question is conceptual or asks for an estimate from duration and convexity.

  1. Mark direction: yield up means price down, yield down means price up.
  2. If the options are descriptions, eliminate any that say convexity makes gains smaller than losses for an option-free bond.
  3. For rankings, find the bond with the longest maturity and lowest coupon. It moves most.
  4. For estimates, compute −ModDur × ΔYield first. Then add ½ × Convexity × ΔYield², which is a small positive number.
  5. Check that the answer is slightly better than the duration-only estimate. If it is not, you made a sign error.

Common mistakes in Price-Yield Relationship and Convexity

  • Treating the price-yield relationship as a straight line

    Duration gives a linear estimate, so students forget the actual curve bends.

    Fix: Remember that duration is the tangent and the curve lies above it for option-free bonds. Convexity adds back the gap.

  • Subtracting the convexity term instead of adding it

    The duration term has a minus sign in front, and the sign gets carried over to the convexity term.

    Fix: Write −ModDur × ΔY + ½ × Conv × (ΔY)². The squared yield change makes the second term positive for either direction of change.

  • Using 100 bp as 100 or 1 in the formula

    Mixing basis points, percentages and decimals.

    Fix: Convert first: 100 bp = 1% = 0.01. Square the decimal, so 0.01² = 0.0001.

  • Saying a higher-coupon bond is more rate-sensitive

    Students think larger cash flows mean larger price moves.

    Fix: Higher coupons return cash sooner, which lowers sensitivity. Lower coupon bonds, with more value in the final payment, are more sensitive, other things equal.

  • Assuming every bond has positive convexity

    The convex shape is taught first using option-free bonds.

    Fix: State the condition. Option-free bonds have positive convexity. A callable bond can have negative convexity when yields fall low enough that a call becomes likely.

  • Confusing premium and discount using the price alone

    Students forget that the label depends on coupon rate versus yield.

    Fix: Compare coupon rate with YTM. Coupon above yield is a premium bond, and its price is above par.

Worked examples

Example 1

A 5-year annual-pay bond has a 4% coupon and is priced at par with a YTM of 4%. Compare the price change when the yield rises 100 bp with the price change when it falls 100 bp. How much larger, in price per 100 of par, is the gain from the fall than the loss from the rise? A. 0.00 B. 0.25 C. 4.33

Show the solution
  1. Price at 5%: N = 5, I/Y = 5, PMT = 4, FV = 100. On the BA II Plus: 5 N 5 I/Y 4 PMT 100 FV CPT PV, giving about −95.67 (the sign is a cash-flow sign). Price ≈ 95.67.
  2. Price at 3%: change I/Y to 3, CPT PV. Price ≈ 104.58.
  3. Loss from a yield rise: 100.00 − 95.67 = 4.33.
  4. Gain from a yield fall: 104.58 − 100.00 = 4.58.
  5. Difference: 4.58 − 4.33 ≈ 0.25. The price gain is larger because the price-yield curve is convex.

Answer: B. About 0.25. The gain from a 100 bp fall is larger than the loss from a 100 bp rise.

Example 2

An option-free bond has modified duration of 6.5 and convexity of 55. The yield falls by 100 bp. Estimate the percentage price change using duration and convexity. A. +6.50% B. +6.78% C. +7.05%

Show the solution
  1. ΔYield = −0.01.
  2. Duration effect: −6.5 × (−0.01) = +0.065, or +6.50%.
  3. Convexity effect: ½ × 55 × (0.01)² = 0.5 × 55 × 0.0001 = 0.00275, or +0.275%.
  4. Total: 6.50% + 0.275% = 6.775%, which rounds to 6.78%.
  5. Check: the answer is above the duration-only estimate, as expected for a yield fall on an option-free bond.

Answer: B. About +6.78%.

Exam tips

  • Expect three-option conceptual items asking which bond is most or least sensitive. Check that only one feature (coupon, maturity or yield) differs before you apply a rule.
  • If an option says price falls by the same amount for equal rises and falls in yield, treat it as wrong for an option-free bond.
  • In estimate questions, the options are listed from smallest to largest. Compute the duration-only figure first. The correct option is usually just beyond it.
  • Always convert basis points to decimals before squaring. Many wrong options come from using 1 instead of 0.01.
  • Watch for the words callable or putable. If the question mentions a call option, think about negative convexity, and do not apply option-free rules automatically.

Practice questions from Fixed-Income Bond Valuation: Prices and Yields

Price-Yield Relationship and Convexity in other exams

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

Price-Yield Relationship and Convexity: frequently asked questions

Why is bond price inversely related to yield?

The price is the present value of fixed coupons and principal. A higher yield is a higher discount rate, so each cash flow is worth less today. A lower yield does the opposite.

Why is the price-yield curve convex?

Each cash flow is discounted by (1 + r) raised to a power, which makes price a curved, decreasing function of yield. The slope is steeper at low yields and flatter at high yields. So a yield fall adds more price than an equal yield rise removes.

What is the difference between premium, par and discount bonds?

A premium bond has a coupon rate above its yield, so its price is above par. A par bond has a coupon rate equal to its yield. A discount bond has a coupon rate below its yield, so its price is below par.

Do all bonds have positive convexity?

No. Option-free bonds have positive convexity. Callable bonds can show negative convexity at low yields, since the call limits how far the price can rise.

Which bond is most sensitive to a change in yield?

With other things equal, the bond with the longest maturity and the lowest coupon. Sensitivity is also larger when the yield level is lower. Compare only when a single feature differs, or compare using duration.