CFA Level I Exam · Yield-Based Bond Duration Measures and Properties

Convexity and Price Change Estimation for Bonds

Updated 7 October 2026 · Fact-checked

Convexity measures how a bond's duration changes as yield changes. It corrects the straight-line error in duration. To estimate the percentage price change, use −ModDur × Δy + ½ × Convexity × (Δy)². Duration gives the first-order estimate. The convexity term is always positive for an option-free bond, so it adds to the estimate.

Understand Convexity and Price Change Estimation

Duration is a straight-line estimate of a bond's price change. The real price-yield curve is curved. For large yield moves, the straight line is wrong. It understates the price rise when yields fall and overstates the price fall when yields rise.

Convexity measures that curvature. It is the second-order effect: how fast the bond's duration itself changes as yield changes. An option-free bond has positive convexity. Its price-yield curve bends upward, away from the duration line.

That is why convexity benefits bondholders. For the same size yield move, a more convex bond gains more when yields fall and loses less when yields rise. All else equal, investors will pay for it. Convexity is higher for longer maturity, lower coupon and lower starting yield. Bonds with the same duration can have different convexity.

You combine the two in one estimate: the duration term plus a convexity adjustment of ½ × convexity × (Δy)². Because Δy is squared, the adjustment is positive whether yields rise or fall. For small moves it is tiny. For a 100 bp move it matters.

Money convexity is the same idea in currency terms: convexity × full bond price. It pairs with money duration, and lets you estimate the price change in currency units instead of percent. Callable bonds can show negative convexity when yields are low. For those you use effective measures, based on shifting the benchmark curve.

Key formulas to remember

Percentage price change (duration + convexity): %ΔPV ≈ −ModDur × Δy + ½ × Convexity × (Δy)²; Write Δy as a decimal, so 100 bps = 0.01. Squaring a negative Δy still gives a positive number.
Convexity adjustment: ½ × Convexity × (Δy)²; Always positive for positive convexity. It is the second term above.
Approximate convexity: (PV₋ + PV₊ − 2 × PV₀) ÷ [(Δy)² × PV₀]; PV₋ is the price after yield falls by Δy. PV₊ is the price after yield rises by Δy. Use the same Δy both ways.
Approximate modified duration: (PV₋ − PV₊) ÷ (2 × Δy × PV₀); Needed alongside approximate convexity. For bonds with embedded options, this gives effective duration.
Money convexity: Money convexity = Convexity × PV (full price); Money duration = ModDur × PV. Use the full price, including accrued interest.
Money price change estimate: ΔPV ≈ −MoneyDur × Δy + ½ × MoneyConvexity × (Δy)²; Gives the change in currency units. Divide by PV to get the percentage change.
Price value of a basis point: PVBP = MoneyDur × 0.0001; Duration-only estimate of the price change for a 1 bp yield move.

How to solve Convexity and Price Change Estimation questions

Use this order for any question that asks for a price change, a convexity figure or a comparison of bonds.

1Identify what is given: modified duration (or effective duration), convexity, the yield change and whether the bond price is per 100 par or a total position.
2Convert the yield change to a decimal. A move of 75 bps is 0.0075. Keep the sign: negative if yields fall, positive if they rise.
3Compute the duration term: −ModDur × Δy. Its sign is opposite to the yield move.
4Compute the convexity term: ½ × Convexity × (Δy)². Square Δy first. The result is positive for positive convexity.
5Add the two terms to get the percentage price change. Convert to a currency amount by multiplying by the starting price if needed.
6If convexity must be found from prices, use the approximate convexity formula with PV₋, PV₊ and PV₀. Check that the numerator is small and positive.
7Sanity-check: with positive convexity, the true price change should be better than the duration-only estimate. A fall in yield should give a gain larger than duration alone implies.

Quickest way: Duration first, then add the small positive term

When to use it: Use it for any MCQ with given duration, convexity and a yield change. With only three options, a quick check removes two of them.

Compute −ModDur × Δy in your head or on the calculator. This is the duration-only answer.
Estimate the convexity term: ½ × Convexity × Δy². For a 100 bp move it equals Convexity × 0.00005.
Add the term to the duration answer. It always moves the result in the favourable direction: up for a gain, toward zero for a loss.
Scan the options. Reject the one equal to the duration-only answer. Reject the one that moves the wrong way. Confirm the last option by checking that it matches your computed total.
On the BA II Plus, enter Δy as a decimal and use the x² key before multiplying.

Common mistakes in Convexity and Price Change Estimation

Subtracting the convexity term when yields rise
Students think a yield rise means every term is negative.
Fix: Δy is squared, so the convexity adjustment is always positive for a bond with positive convexity. Only the duration term changes sign.
Forgetting the ½ in the convexity adjustment
The approximate convexity formula has no ½, so students carry it over.
Fix: The price change formula uses ½ × Convexity × (Δy)². Write the ½ before you substitute numbers.
Using basis points or percent instead of decimals
Δy is given as 100 bps or 1%, and it is entered as 100 or 1.
Fix: Convert to a decimal first: 0.01. A wrong Δy gets squared, so the error is large.
Using different yield shifts for PV₋ and PV₊ in approximate convexity
Students rush when reading price tables.
Fix: The formula needs equal and opposite shifts of Δy. Check both prices come from the same size of shift.
Using clean price for money convexity
Quoted prices are clean, so they are used by habit.
Fix: Money duration and money convexity use the full price, which includes accrued interest.
Saying convexity is always positive
Option-free bonds are the usual example.
Fix: Option-free bonds have positive convexity. A callable bond can have negative convexity when yields are low, near or below the call price. Use effective convexity for bonds with embedded options.

Worked examples

Example 1

A bond has a modified duration of 7.20 and a convexity of 68.0. Yield rises by 100 bps. Which estimate of the percentage price change is closest? A: −7.54% B: −7.20% C: −6.86%

Show the solution

Δy = +100 bps = +0.01.
Duration term = −7.20 × 0.01 = −0.0720, or −7.20%.
Convexity term = ½ × 68.0 × (0.01)² = 34 × 0.0001 = 0.0034, or +0.34%.
Total = −7.20% + 0.34% = −6.86%.
Option A applies the convexity term in the wrong direction (−7.20% − 0.34%). Option B ignores convexity.

Answer: C: −6.86%.

Example 2

A 5-year, 4% annual-pay bond (par 100) is priced at 95.6705 at a 5.00% yield. If yield falls by 25 bps the price is 96.7303. If yield rises by 25 bps the price is 94.6254. Find approximate modified duration, approximate convexity, and the estimated percentage price change for a 100 bp yield increase.

Show the solution

Δy = 0.0025, PV₀ = 95.6705, PV₋ = 96.7303, PV₊ = 94.6254.
Approximate modified duration = (96.7303 − 94.6254) ÷ (2 × 0.0025 × 95.6705) = 2.1049 ÷ 0.4783525 = 4.400 (about 4.40).
Approximate convexity numerator = 96.7303 + 94.6254 − 2 × 95.6705 = 191.3557 − 191.3410 = 0.0147.
Denominator = (0.0025)² × 95.6705 = 0.00000625 × 95.6705 = 0.00059794.
Approximate convexity = 0.0147 ÷ 0.00059794 = 24.58 (about 24.6).
Rounding check: the numerator is a tiny difference of large numbers. If the prices were rounded to two decimals (96.73, 94.63, 95.67), the numerator would be 0.02 and convexity would come out near 33. That is why you keep four decimals in prices here.
Now use Δy = +0.01. Duration term = −4.400 × 0.01 = −0.04400.
Convexity term = ½ × 24.58 × (0.01)² = ½ × 24.58 × 0.0001 = 0.001229.
Total = −0.04400 + 0.00123 = −0.04277, or about −4.28%.

Answer: Approximate modified duration ≈ 4.40, approximate convexity ≈ 24.6, and estimated price change ≈ −4.28%.

Exam tips

Look at the sign of the answer. With positive convexity, adding the adjustment always moves the estimate in the bondholder's favour. This often eliminates one option immediately.
Check the unit of Δy. Questions often give basis points, then ask for a percentage. Convert to decimals before squaring.
When two bonds have the same duration, the one with higher convexity is better for a given yield change, all else equal. Expect this as a conceptual question.
For approximate convexity, the numerator is a small difference of large numbers. Keep four decimals in the prices, or use unrounded calculator values, or the answer will be wrong.
Know that convexity rises with maturity and falls with coupon rate. Questions test the direction, not the number.

Practice questions from Yield-Based Bond Duration Measures and Properties

Convexity and Price Change Estimation in other exams

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

Convexity and Price Change Estimation: frequently asked questions

What is the convexity formula in CFA Level I?

The price estimate formula is %ΔPV ≈ −ModDur × Δy + ½ × Convexity × (Δy)². If you must compute convexity from prices, use (PV₋ + PV₊ − 2 × PV₀) ÷ [(Δy)² × PV₀]. Both appear in the curriculum.

Why is convexity beneficial to bondholders?

For the same yield move, a bond with higher convexity gains more when yields fall and loses less when yields rise. The convexity term adds to the estimate in both directions. Investors usually pay a higher price for it.

What is money convexity?

Money convexity is convexity multiplied by the full bond price. It lets you estimate the price change in currency units: ΔPV ≈ −MoneyDur × Δy + ½ × MoneyConvexity × (Δy)². Use the full price, including accrued interest.

Can convexity be negative?

Yes, for bonds with embedded options. A callable bond can show negative convexity when yields are low, because its price is capped near the call price. Option-free bonds have positive convexity.