CFA Level II Exam · Valuation and Analysis of Bonds with Embedded Options

Effective Duration and Convexity for Bonds with Embedded Options

Updated 7 October 2026 · Fact-checked

Effective duration measures a bond's percentage price change for a parallel shift in the benchmark yield curve, allowing cash flows to change when an embedded option is exercised. Effective convexity measures the curvature of that price response. You find both from three values: the price at the base curve, the price with the curve shifted down, and the price with it shifted up.

Understand Effective Duration and Convexity

A plain bond has fixed cash flows. When yields move, only the discount rate changes. Modified duration works for these bonds because it assumes the cash flows stay the same.

A bond with an embedded option breaks that assumption. A callable bond may be redeemed early when rates fall. A putable bond may be sold back to the issuer when rates rise. The cash flows depend on the interest rate path. So you cannot use the yield to maturity to measure rate risk. You need a valuation model, usually a binomial interest rate tree, that reprices the bond after the benchmark curve shifts.

Effective duration comes from repricing. You shift the benchmark curve down and up by the same small amount, reprice the bond each time with the model, and compare. Effective convexity uses the same three prices to measure how much the duration itself changes as rates change.

Callable bonds show negative convexity when yields are low. As yields fall, the call becomes more likely, so the price rises more slowly and is capped near the call price. The price-yield curve bends the wrong way. Effective duration also falls as yields fall. A putable bond behaves the other way. As yields rise, the put becomes valuable and sets a floor, so the bond keeps positive convexity.

One-sided durations show this asymmetry. The down-curve duration covers only the price response to falling rates. The up-curve duration covers only the response to rising rates. For a callable bond near its call price, the down-curve duration is lower than the up-curve duration. For a putable bond, the up-curve duration is lower. When the two differ a lot, the single effective duration (their average) hides important information.

Key formulas to remember

Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀); PV₋ is the price when the benchmark curve falls by ΔCurve. PV₊ is the price when it rises by ΔCurve. ΔCurve is in decimals, so 25 bps = 0.0025.
Effective convexity: EffCon = (PV₋ + PV₊ − 2 × PV₀) ÷ (ΔCurve² × PV₀); A negative result means negative convexity. Use the same ΔCurve as in the duration formula.
Price change approximation: %ΔPV ≈ (−EffDur × ΔCurve) + (½ × EffCon × ΔCurve²); ΔCurve carries its sign, so it is negative for a fall. The convexity term does not change sign with the direction of the move.
Down-curve (one-sided) duration: (PV₋ − PV₀) ÷ (ΔCurve × PV₀); Measures sensitivity to falling rates only.
Up-curve (one-sided) duration: (PV₀ − PV₊) ÷ (ΔCurve × PV₀); Measures sensitivity to rising rates only. The average of the two one-sided durations equals the effective duration.
Option value relationships: Callable = Straight − Call value; Putable = Straight + Put value; Use these to explain why callable bonds have lower price gains when yields fall and putable bonds have a price floor.
Duration versus a straight bond: EffDur(callable) ≤ EffDur(straight); EffDur(putable) ≤ EffDur(straight); Holds for otherwise similar bonds. The option shortens the effective life in the scenario where it is valuable.

How to solve Effective Duration and Convexity questions

Use this order for any item-set question on effective duration or convexity. Most marks are lost by picking the wrong price from the exhibit.

1Read the exhibit and label the three prices: PV₀ at the base curve, PV₋ for the curve down, PV₊ for the curve up. Note the size of the shift and convert basis points to decimals.
2Check which bond it is: callable, putable, or straight. Check whether the exhibit gives the price of a straight bond or option value, since you may need them.
3Compute effective duration as (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀).
4Compute effective convexity as (PV₋ + PV₊ − 2PV₀) ÷ (ΔCurve² × PV₀). Keep the sign.
5If asked for an estimated price change, apply the duration term and the convexity term with the correct sign for the shift in the question. The shift may differ from the one used to compute the measures.
6If asked about one-sided durations, compute the down-curve and up-curve values separately. Compare them to the option type to confirm the answer makes sense.
7Sanity check: callable bonds should show lower or negative convexity and a lower down-curve duration. Putable bonds should show positive convexity and a lower up-curve duration.

Quickest way: Three prices and a sign check

When to use it: Use this when the vignette gives the three prices and you must pick the best of three options quickly.

Write ΔCurve as a decimal first. 25 bps = 0.0025 and 50 bps = 0.005.
For duration, find the price gap (PV₋ − PV₊), then divide by 2 × ΔCurve × PV₀.
For convexity, add PV₋ and PV₊ and subtract 2PV₀. If this is negative, convexity is negative, and you can eliminate any positive option at once.
For callable bonds, expect down-curve duration below up-curve duration. For putable bonds, expect the reverse. Use this to eliminate options without further arithmetic.
Only do the full estimated price change if the sign check leaves two or more options.

Common mistakes in Effective Duration and Convexity

Using the yield to maturity change instead of the benchmark curve shift in the effective duration formula.
Modified duration uses a change in the bond's own yield, and candidates carry that habit across.
Fix: Effective duration uses a parallel shift in the benchmark curve, with the bond repriced by the model. Cash flows can change.
Entering basis points directly, for example 25 instead of 0.0025.
The exhibit quotes shifts in bps and the formula looks the same.
Fix: Convert to decimals before you substitute. A duration of several hundred means you forgot to convert.
Swapping PV₋ and PV₊ in the numerator.
The labels are easy to confuse under time pressure, and the denominator gives no clue.
Fix: The numerator is the price after rates fall minus the price after rates rise. For normal bonds it is positive. A negative duration is almost always a swap.
Subtracting the convexity term when the curve rises, or dropping it when convexity is negative.
Candidates link the sign of the convexity adjustment to the direction of the rate move.
Fix: The convexity term is ½ × EffCon × ΔCurve², and ΔCurve² is always positive. Its sign comes from the convexity itself. Negative convexity reduces the estimated price in both directions.
Saying a callable bond always has negative convexity.
The textbook picture of the call shows only the low-yield region.
Fix: Negative convexity appears when yields are low enough that the call is likely to be exercised. At high yields the call is out of the money and the bond behaves like a straight bond.
Treating one-sided durations as equal to effective duration.
The effective duration is the average of the two, so candidates assume they match.
Fix: They match only if the price curve is symmetric. For callable and putable bonds near the strike, they differ, and the exam tests the direction of that difference.

Worked examples

Example 1

A callable bond has a base-curve price of 100.00. If the benchmark curve falls by 25 bps, the model price is 100.90. If it rises by 25 bps, the model price is 98.90. (1) Calculate the effective duration. (2) Calculate the effective convexity. (3) Estimate the percentage price change if the benchmark curve falls by 50 bps.

Show the solution

ΔCurve = 0.0025. PV₀ = 100.00, PV₋ = 100.90, PV₊ = 98.90.
Effective duration = (100.90 − 98.90) ÷ (2 × 0.0025 × 100.00) = 2.00 ÷ 0.50 = 4.00.
Effective convexity = (100.90 + 98.90 − 200.00) ÷ (0.0025² × 100.00) = −0.20 ÷ 0.000625 = −320.
For a 50 bp fall, ΔCurve = −0.005. Duration term = −4.00 × (−0.005) = +0.02, or +2.00%.
Convexity term = ½ × (−320) × 0.005² = −160 × 0.000025 = −0.004, or −0.40%.
Total estimated change = 2.00% − 0.40% = +1.60%.

Answer: (1) Effective duration is 4.00. (2) Effective convexity is −320, which is negative. (3) The estimated price change is about +1.60%, less than the 2.00% a duration-only estimate suggests.

Example 2

A putable bond has a base-curve price of 102.00. If the benchmark curve falls by 50 bps, its model price is 103.60. If the curve rises by 50 bps, its model price is 100.70. A comparable straight bond has a base-curve price of 99.50. (1) Calculate the effective duration. (2) Calculate the effective convexity and comment on its sign. (3) Calculate the two one-sided durations and the value of the put option.

Show the solution

ΔCurve = 0.005. PV₀ = 102.00, PV₋ = 103.60, PV₊ = 100.70.
Effective duration = (103.60 − 100.70) ÷ (2 × 0.005 × 102.00) = 2.90 ÷ 1.02 = 2.84.
Effective convexity = (103.60 + 100.70 − 204.00) ÷ (0.005² × 102.00) = 0.30 ÷ 0.00255 = 117.6. It is positive, as expected for a putable bond.
Down-curve duration = (103.60 − 102.00) ÷ (0.005 × 102.00) = 1.60 ÷ 0.51 = 3.14.
Up-curve duration = (102.00 − 100.70) ÷ 0.51 = 1.30 ÷ 0.51 = 2.55. The average of 3.14 and 2.55 is about 2.84, which matches the effective duration.
Put value = putable price − straight price = 102.00 − 99.50 = 2.50.

Answer: (1) Effective duration is about 2.84. (2) Effective convexity is about 117.6, positive. (3) The down-curve duration is about 3.14 and the up-curve duration about 2.55. The up-curve duration is lower because the put limits losses when rates rise. The put value is 2.50.

Exam tips

Look at the exhibit first and write ΔCurve as a decimal before touching the formula. Unit errors are the commonest slip in these questions.
Expect conceptual options about direction: which one-sided duration is lower, whether convexity is negative, whether effective duration is below the straight bond's. Learn the callable and putable pattern as a pair.
When an item gives three prices and asks for duration, check the denominator uses 2 × ΔCurve × PV₀. Options with half or double the right answer are planted traps.
If a question changes the shift size for the price estimate, use the new shift in the approximation, not the one used to compute the measures.
There is no penalty for wrong answers, so never leave an option blank. If time is short, use the sign checks to cut the options to two.

Effective Duration 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.

Effective Duration and Convexity: frequently asked questions

What is the difference between modified duration and effective duration?

Modified duration measures price change for a change in the bond's own yield to maturity and assumes cash flows stay fixed. Effective duration measures price change for a shift in the benchmark curve and lets cash flows change as an embedded option is exercised. You must use effective duration for callable, putable and other option-embedded bonds.

Why do callable bonds have negative convexity?

When yields fall, the issuer is more likely to call the bond, so the price rises more slowly and is held near the call price. The price-yield curve flattens and bends the opposite way to a straight bond. This is called negative convexity and appears when the call option is close to or in the money.

What are one-sided durations and why do they matter?

One-sided durations measure price sensitivity to rate falls only (down-curve) and rate rises only (up-curve). They matter because option-embedded bonds respond asymmetrically. A callable bond has a lower down-curve duration, and a putable bond has a lower up-curve duration.

Is the effective duration of a callable bond lower than that of a straight bond?

For otherwise similar bonds, yes, it is lower or equal. The call shortens the expected life when rates fall, so the bond is less sensitive. The same holds for a putable bond, because the put shortens the expected life when rates rise.