CFA Level I Exam · Yield-Based Bond Convexity and Portfolio Properties

Bond Convexity and Convexity Adjustment Explained

Updated 7 October 2026 · Fact-checked

Convexity measures the curvature of the price-yield curve, which is how much a bond's duration changes as yield changes. To estimate a price change, use %ΔPrice ≈ −ModDur × Δy + ½ × Convexity × (Δy)². The first term is the duration effect. The second is the convexity adjustment, positive for option-free bonds.

Understand Bond Convexity and Convexity Adjustment

Bond prices and yields do not move in a straight line. The price-yield curve bends. It is convex, shaped like a bowl, for an option-free bond.

Duration is the slope of that curve at the current yield. It gives a straight-line estimate of the price change. A straight line is only exact for tiny yield moves. For larger moves the line misses the curve.

Convexity measures the curvature. For an option-free bond, the actual price lies above the duration line on both sides. So duration alone underestimates the price gain when yields fall. It overestimates the price loss when yields rise. The convexity adjustment corrects this. It is always added, and for an option-free bond it is positive.

All else equal, higher convexity is good for the holder. The price rises more when yields fall and drops less when yields rise. Convexity is generally higher for longer maturity and lower coupon bonds.

Bonds with embedded options behave differently. A callable bond can show negative convexity at low yields, because the call caps its price gain. For these bonds you use effective duration and effective convexity. They are based on a shift in the benchmark yield curve, not on a change in the bond's own yield to maturity.

Key formulas to remember

Approximate modified duration: ApproxModDur = (PV− − PV+) ÷ (2 × PV0 × Δyield); PV− is the price when yield falls by Δyield. PV+ is the price when yield rises by Δyield. Use Δyield as a decimal.
Approximate convexity: ApproxCon = (PV− + PV+ − 2 × PV0) ÷ (Δyield² × PV0); Uses the bond's own yield-to-maturity change. Fits option-free bonds.
Effective convexity: EffCon = (PV− + PV+ − 2 × PV0) ÷ (ΔCurve² × PV0); ΔCurve is a parallel shift in the benchmark yield curve. Use it for bonds with embedded options.
Price change from duration and convexity: %ΔPV ≈ −ModDur × Δy + ½ × Convexity × (Δy)²; The first term is the duration effect. The second term is the convexity adjustment. Δy is a decimal.
Money convexity version: ΔPV ≈ −MoneyDur × Δy + ½ × MoneyConvexity × (Δy)²; Money convexity = Convexity × full price. The result is a currency amount, not a percentage.

How to solve Bond Convexity and Convexity Adjustment questions

Use this method for any question that asks you to estimate a price change or find convexity.

1Read what is given: modified duration or effective duration, convexity, yield change, or a set of prices at different yields.
2If prices at yield up and yield down are given, compute duration and convexity from the formulas first. Use the same Δ in both.
3Convert the yield change to a decimal. A 50 bp rise is 0.005.
4Compute the duration effect: −Duration × Δy. A rise in yield gives a negative number.
5Compute the convexity adjustment: ½ × Convexity × (Δy)². Square Δy before multiplying.
6Add the two effects to get the estimated percentage price change.
7If a currency amount is asked, multiply the percentage by the starting price.
8Check the sign and size. The adjustment should be positive for an option-free bond, and the total should be close to the duration effect alone.

Quickest way: Duration effect plus a small positive adjustment

When to use it: Use it when the question gives duration, convexity and a yield change, and you have about 90 seconds.

Write Δy as a decimal.
Compute the duration effect first and note its sign.
Compute ½ × Convexity × Δy² next. It is small and positive.
Add it to the duration effect. The answer moves slightly toward zero if yields rose, and further up if yields fell.
Use this to eliminate options. If yields rise, any option more negative than the duration effect alone is wrong, so you can often discard it without computing the rest.

Common mistakes in Bond Convexity and Convexity Adjustment

Leaving out the ½ in the convexity adjustment.
The convexity formula itself has no ½, so students forget it appears in the price-change formula.
Fix: Memorise the price-change formula as one line: −ModDur × Δy + ½ × Convexity × (Δy)².
Using Δy in percent instead of decimal.
Yield changes are quoted as 1% or 50 bp, so students plug in 1 or 50.
Fix: Convert first: 1% = 0.01, 50 bp = 0.005. Then square the decimal.
Subtracting the convexity adjustment when yields rise.
Students think a yield rise means every term is negative.
Fix: Only the duration term changes sign with the yield move. Since (Δy)² is always positive, the adjustment is positive for an option-free bond in both directions.
Squaring incorrectly, for example using (Δy × 2) or forgetting to square.
Time pressure and calculator slips.
Fix: Compute (Δy)² on its own line, then multiply by convexity and by 0.5.
Treating approximate convexity and effective convexity as the same.
The formulas look identical.
Fix: The formula is the same, but approximate convexity uses a change in the bond's own yield, while effective convexity uses a shift in the benchmark curve. Use the effective version for callable or putable bonds.
Assuming convexity is always positive.
Option-free bonds always show positive convexity, so students generalise.
Fix: A callable bond can have negative convexity when yields are low enough that the call is likely to be exercised.

Worked examples

Example 1

An option-free bond has a modified duration of 7.2 and convexity of 60. Yield rises by 100 bps. The estimated percentage price change is closest to: A. −7.50%, B. −7.20%, C. −6.90%.

Show the solution

Δy = 0.01.
Duration effect = −7.2 × 0.01 = −0.072, or −7.20%.
Convexity adjustment = ½ × 60 × (0.01)² = 0.5 × 60 × 0.0001 = 0.003, or +0.30%.
Total = −7.20% + 0.30% = −6.90%.
Check: A and B are at or below the duration effect alone. The adjustment must be positive, so only C fits.

Answer: C. −6.90%

Example 2

A bond is priced at 100.00. If its yield falls by 0.50%, its price is 103.5625. If its yield rises by 0.50%, its price is 96.5625. Using these prices, estimate the percentage price change for a 200 bp rise in yield. Options: A. −14.00%, B. −13.00%, C. −12.00%.

Show the solution

Δ = 0.005 for the approximation prices.
Modified duration = (103.5625 − 96.5625) ÷ (2 × 100 × 0.005) = 7.0 ÷ 1.0 = 7.0.
Convexity = (103.5625 + 96.5625 − 200) ÷ (0.005² × 100) = 0.125 ÷ 0.0025 = 50.
Now the 200 bp rise: Δy = 0.02.
Duration effect = −7.0 × 0.02 = −0.14, or −14.00%.
Convexity adjustment = ½ × 50 × (0.02)² = 0.5 × 50 × 0.0004 = 0.01, or +1.00%.
Total = −14.00% + 1.00% = −13.00%.
Calculator: key 0.02, press x², multiply by 50, multiply by 0.5 to get 0.01.

Answer: B. −13.00%

Exam tips

Questions often give duration and convexity directly. Do the two-term formula and check the sign.
Wrong options are usually the duration effect alone, the adjustment with the wrong sign, or a missing ½. Compute your own answer and match it.
Expect conceptual items: which bond has higher convexity, why callable bonds can show negative convexity, and why convexity helps investors when yields move a lot.
Watch for effective versus approximate wording. If the bond has an embedded option, the effective measures apply.
Keep Δy in decimals throughout and do not round until the final step.

Practice questions from Yield-Based Bond Convexity and Portfolio Properties

Bond Convexity and Convexity Adjustment 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 Convexity Adjustment: frequently asked questions

What is the convexity adjustment in bond pricing?

It is the term ½ × Convexity × (Δy)² added to the duration effect. It corrects the straight-line duration estimate for the curve in the price-yield relationship. For an option-free bond it is positive whether yields rise or fall.

Why does duration underestimate the price when yields fall?

Duration is a straight line tangent to a curved price-yield relationship. For an option-free bond the curve lies above the tangent on both sides. So the actual price is higher than the line predicts.

What is the difference between effective convexity and approximate convexity?

The formulas have the same structure. Approximate convexity uses a change in the bond's own yield to maturity. Effective convexity uses a parallel shift in the benchmark yield curve, which suits bonds whose cash flows change with rates, such as callable bonds.

Can convexity be negative?

Yes. An option-free bond has positive convexity. A callable bond can show negative convexity when yields are low and a call becomes likely, because its price rise is capped.