CFA Level I Exam · Interest Rate Risk and Return

Bond Convexity and Price Change Estimation for CFA Level I

Updated 7 October 2026 · Fact-checked

Convexity measures how a bond's duration changes as yield changes, which is the curvature of the price-yield curve. Duration alone gives a straight-line estimate. Add the convexity adjustment: %ΔPrice ≈ −ModDur × Δy + ½ × Convexity × (Δy)². Use decimals for Δy and always add the convexity term for a bond with positive convexity.

Understand Bond Convexity and Price Change Estimation

The price-yield relationship of an option-free bond is a curve, not a straight line. Duration is the slope of a line tangent to that curve at the current yield. It gives a good estimate for small yield changes, but the curve bends away from the line as the change gets larger.

Convexity measures that bend. For an option-free bond the curve is bowed toward the origin, so the bond has positive convexity. Compared with the duration-only estimate, the actual price rises more when yields fall and falls less when yields rise. So duration alone understates a price gain and overstates a price loss. This is why positive convexity is good for bondholders. All else equal, investors would pay for it.

The convexity adjustment fixes the estimate. You add ½ × Convexity × (Δy)² to the duration estimate. The term has a squared yield change, so it is always positive when convexity is positive, whether yields rise or fall. It is small for small yield moves and grows quickly for large ones.

Callable bonds behave differently. When yields fall far enough, the issuer is likely to call the bond, so the price rises less and gets close to the call price. The curve flattens and bends the other way. That is negative convexity. Putable bonds have higher convexity than comparable option-free bonds. For bonds with embedded options, you use effective duration and effective convexity, which come from re-pricing the bond with a shift in the benchmark curve. All else equal, convexity is higher for longer maturity, lower coupon and greater dispersion of cash flows.

Key formulas to remember

Duration plus convexity price change: %ΔPV ≈ −(ModDur × Δy) + ½ × Convexity × (Δy)²; Δy is in decimals (50 bp = 0.005). Use effective duration and effective convexity for bonds with embedded options.
Approximate (effective) duration: (PV− − PV+) ÷ (2 × PV0 × Δcurve); PV− is the price when the yield decreases by Δcurve. PV+ is the price when the yield increases by Δcurve.
Approximate (effective) convexity: (PV− + PV+ − 2 × PV0) ÷ (Δcurve² × PV0); Use the same Δcurve for both prices. PV− is the price when the yield decreases and PV+ is the price when it increases. A negative result signals negative convexity, typical of a callable bond near its call price.

How to solve Bond Convexity and Price Change Estimation questions

Use this routine for any question that asks you to estimate a price change, find convexity or interpret it.

1Identify what is given: modified or effective duration, convexity, or the three prices PV0, PV− and PV+.
2If you only have prices, compute duration and convexity with the approximate formulas. Use the same yield change in both.
3Convert the yield change to a decimal. A 75 bp rise is +0.0075. Keep the sign, because it matters for the duration term.
4Compute the duration term: −ModDur × Δy. A rise in yield gives a negative number.
5Compute the convexity term: ½ × Convexity × (Δy)². Square the decimal before multiplying. This term is positive for positive convexity.
6Add the two terms to get the estimated percentage price change. Multiply by the starting price if the question wants a currency change.
7Sense-check. For a rise in yield, the convexity-adjusted loss should be smaller than the duration-only loss. For a fall in yield, the gain should be larger.
8If the bond is callable and yields are low, expect lower or negative convexity and a smaller price gain than for a similar option-free bond.

Quickest way: Duration estimate first, then add the small correction

When to use it: Use when the exam gives duration and convexity and three close numerical options.

Do the duration term first: −ModDur × Δy. Write it as a percentage.
Do the convexity term: 0.5 × Convexity × Δy², with Δy as a decimal. It is usually a small number such as 0.1% to 2%.
Add the convexity term to the duration term. Never subtract it for positive convexity.
Eliminate options: the duration-only value is a trap, and so is the value with the convexity term subtracted. For positive convexity the correct answer is the duration estimate plus the positive convexity term, so it is a better outcome for the holder than the duration-only figure.

Common mistakes in Bond Convexity and Price Change Estimation

Leaving out the ½ in the convexity adjustment
Students remember the term as 'convexity times Δy squared'.
Fix: Write the full formula first: −ModDur × Δy + ½ × Convexity × (Δy)². The ½ comes from the second-order term of the price expansion.
Using Δy in percent instead of decimals
A move of 1% feels natural as '1', so the squared term becomes 1 instead of 0.0001.
Fix: Convert to decimals before squaring. 100 bp = 0.01, so (Δy)² = 0.0001.
Subtracting the convexity term when yields rise
Students think the sign follows the yield change, as it does for duration.
Fix: The convexity term uses (Δy)², which is always positive. For positive convexity it always adds to the estimate.
Saying callable bonds always have negative convexity
Students over-generalise from the negative convexity idea.
Fix: A callable bond has positive convexity when yields are high and the call is unlikely. It shows negative convexity when yields fall near or below the level where the call becomes attractive.
Using different yield shifts for PV− and PV+, or the wrong order in the duration formula
The two formulas look alike, and the order of PV− and PV+ is easy to swap.
Fix: PV− is the price when the yield decreases and PV+ is the price when the yield increases. Duration uses PV− minus PV+. Convexity uses PV− plus PV+ minus 2 × PV0. Use the same Δcurve in both, and check duration comes out positive for an option-free bond.
Applying modified duration and convexity to a bond with embedded options
Modified duration assumes cash flows do not change when yields change.
Fix: For callable, putable or other option bonds, use effective duration and effective convexity, which re-price the bond when the benchmark curve shifts.

Worked examples

Example 1

A bond has a price of 100.00. If its yield falls by 1.00%, its price is 108.00. If its yield rises by 1.00%, its price is 92.80. Using approximate duration and convexity, the estimated percentage price change if the yield rises by 50 bps is closest to: A) −3.90%, B) −3.80%, C) −3.70%.

Show the solution

Approximate duration = (108.00 − 92.80) ÷ (2 × 100 × 0.01) = 15.20 ÷ 2 = 7.6.
Approximate convexity = (108.00 + 92.80 − 2 × 100) ÷ (0.01² × 100) = 0.80 ÷ 0.01 = 80.
Δy = +0.005. Duration term = −7.6 × 0.005 = −0.038 = −3.80%.
Convexity term = ½ × 80 × 0.005² = 40 × 0.000025 = 0.001 = +0.10%.
Total = −3.80% + 0.10% = −3.70%.
Option B is the duration-only trap. Option A subtracts the convexity term: −3.80% − 0.10% = −3.90%.

Answer: C) −3.70%

Example 2

A bond has modified duration 7.6 and convexity 80. If the yield falls by 200 bps, the estimated percentage price change is closest to: A) 13.6%, B) 15.2%, C) 16.8%.

Show the solution

Δy = −0.02.
Duration term = −7.6 × (−0.02) = +0.152 = +15.2%.
Convexity term = ½ × 80 × (0.02)² = 40 × 0.0004 = 0.016 = +1.6%.
Total = 15.2% + 1.6% = 16.8%.
Option B ignores convexity. Option A subtracts it, which is wrong for positive convexity.

Answer: C) 16.8%

Exam tips

Expect three-option items that give duration, convexity and a yield change. Two options are usually the duration-only value and the value with the wrong sign on convexity.
Know the conceptual points cold: positive convexity helps holders in both directions, callable bonds show negative convexity at low yields, and convexity rises with maturity and with cash-flow dispersion.
When a question gives PV−, PV0 and PV+, check that both formulas use the same yield shift. Then do the arithmetic with decimals.
If you need a bond price, the TI BA II Plus works with N, I/Y, PMT, FV, then CPT PV. Enter the yield per period and remember the sign of the PV result.
Ask whether the bond has embedded options. If it does, the answer is about effective duration and effective convexity, not modified duration.

Practice questions from Interest Rate Risk and Return

Bond 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.

Bond Convexity and Price Change Estimation: frequently asked questions

Why is positive convexity good for bondholders?

With positive convexity the price rises by more than duration predicts when yields fall, and falls by less when yields rise. Two bonds with equal duration and yield, but different convexity, will differ in performance, and the higher-convexity bond does better when yields move a lot. Investors usually accept a lower yield for that benefit.

How do I calculate approximate convexity?

Price the bond at the current yield (PV0), at a lower yield (PV−) and at a higher yield (PV+), using the same shift each time. Then compute (PV− + PV+ − 2 × PV0) ÷ (Δy² × PV0). Keep Δy as a decimal.

What is negative convexity in a callable bond?

A callable bond can show negative convexity when yields fall to a level where the issuer is likely to call. The price then rises by less than it would for a similar option-free bond, so price gains are capped. Effective convexity captures this because it re-prices the bond with the option's effect included.

Do I use modified or effective duration in the duration plus convexity formula?

For option-free bonds, modified duration and convexity are standard. For bonds with embedded options, use effective duration and effective convexity, because their cash flows change with yields.