CFA Level I Exam · Curve-Based and Empirical Fixed-Income Risk Measures

Effective Convexity and Embedded Option Bonds Explained

Updated 7 October 2026 · Fact-checked

Effective convexity measures how a bond's effective duration changes when the benchmark yield curve shifts, using prices recomputed with an option model. Callable bonds show negative convexity at low yields because the call caps price gains. To estimate price change, use −effective duration × Δcurve + ½ × convexity × (Δcurve)².

Understand Effective Convexity and Embedded Option Bonds

Duration is a straight-line estimate of how a bond's price reacts to a yield change. Real price-yield curves are curved, not straight. Convexity measures that curvature. For an option-free bond, the curve bends upward, so the price rises more when yields fall than it drops when yields rise by the same amount. That is positive convexity, and it helps the bondholder.

Embedded options change the cash flows, so you cannot hold them fixed. Approximate convexity uses a change in the bond's own yield to maturity and assumes the cash flows do not change. Effective convexity shifts the benchmark yield curve up and down, then reprices the bond with a valuation model that lets the cash flows change when the option would be exercised. For bonds with embedded options, effective convexity is the right measure.

A callable bond gives the issuer the right to buy the bond back. When rates fall far enough, the issuer is likely to call, so the price stops rising much above the call price. The curve flattens or bends downward. This is negative convexity. When rates are high, the call is unlikely to matter and the bond behaves like an option-free bond with positive convexity.

A putable bond gives the investor the right to sell it back at a set price. When rates rise, the put sets a floor under the price. The put bond therefore has positive convexity, and it is at least as convex as a comparable option-free bond. Compared with an option-free bond, a callable bond has lower price gains when rates fall, and a putable bond has smaller price losses when rates rise.

To estimate a price change, combine both measures. Duration gives the first-order effect. Convexity adds a correction for the curve. The correction is positive when convexity is positive and negative when convexity is negative. It matters most for large yield moves.

Key formulas to remember

Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve); PV₋ is the price when the benchmark curve falls by ΔCurve, and PV₊ is the price when it rises. ΔCurve is a decimal, so 25 bps = 0.0025.
Effective convexity: EffCon = (PV₋ + PV₊ − 2 × PV₀) ÷ [(ΔCurve)² × PV₀]; Prices come from a model that allows cash flows to change with the option. The result is negative when PV₋ + PV₊ is less than 2 × PV₀.
Approximate convexity: ApproxCon = (PV₋ + PV₊ − 2 × PV₀) ÷ [(ΔYTM)² × PV₀]; Uses the bond's own yield to maturity and fixed cash flows. Suitable for option-free bonds.
Percentage price change estimate: %ΔPV ≈ (−EffDur × ΔCurve) + ½ × EffCon × (ΔCurve)²; Use ΔCurve as a decimal. For option-free bonds, use modified duration and ΔYTM with approximate convexity.
Convexity of embedded-option bonds: Callable: positive at high yields, negative at low yields | Putable: positive; Putable convexity is at least that of a comparable option-free bond. Callable convexity is lower than that of a comparable option-free bond.

How to solve Effective Convexity and Embedded Option Bonds questions

Use this method for any question on effective convexity, option bonds or price change estimates.

1Identify the bond type: option-free, callable or putable. Note whether yields are high or low relative to the call or put price.
2Check what is given: prices at curve shifts (PV₋, PV₊, PV₀) or duration and convexity already calculated.
3Convert the curve shift to a decimal: 50 bps = 0.005, 1% = 0.01.
4If computing, find effective duration first with (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve). Then find convexity with (PV₋ + PV₊ − 2 × PV₀) ÷ [(ΔCurve)² × PV₀].
5If estimating a price change, compute the duration term −EffDur × ΔCurve, then add ½ × EffCon × (ΔCurve)². Keep the sign of convexity.
6Convert the percentage change to a price change by multiplying by the starting price, if asked.
7Sanity check: with positive convexity, the estimate should be better than duration alone for both rises and falls. With negative convexity, the correction reduces the gain from falling yields.
8Eliminate options that have the wrong sign, ignore convexity, or use a decimal that is off by a factor of 100.

Quickest way: Sign check, then plug in

When to use it: Use this under time pressure when options differ in sign or when one option equals the duration-only answer.

Compute the duration-only effect first: −EffDur × ΔCurve. This is one of your distractors.
Decide the sign of the convexity term from the bond type and yield level. Positive convexity adds to the price change. Negative convexity subtracts from it.
Compute ½ × EffCon × (ΔCurve)². Use the decimal shift squared. For a 1% shift, (0.01)² = 0.0001.
Add the two terms and pick the matching option. If the convexity term is small, the answer is close to the duration-only figure, so check the options carefully.
For the convexity formula, check the numerator first: if PV₋ + PV₊ is less than 2 × PV₀, the answer is negative, and you can remove positive options.

Common mistakes in Effective Convexity and Embedded Option Bonds

Using approximate convexity or YTM changes for a bond with an embedded option.
The formulas look the same, so students plug in yield changes without noticing the cash flows change with the option.
Fix: If the bond has a call or put, use effective measures based on benchmark curve shifts and model-based prices.
Saying callable bonds always have negative convexity.
Students remember the headline and forget that the call only matters when yields are low enough to make exercise likely.
Fix: Say a callable bond has positive convexity at high yields and negative convexity at low yields.
Using the shift in percent instead of decimal in the formulas.
A shift of 1% is entered as 1 instead of 0.01, which makes the convexity term far too large.
Fix: Convert every shift to a decimal before squaring: 100 bps = 0.01, so (ΔCurve)² = 0.0001.
Dropping the minus sign on duration or on negative convexity.
Students focus on the arithmetic and forget that price and yield move in opposite directions.
Fix: Write −EffDur × ΔCurve first. A fall in yields (negative ΔCurve) gives a positive duration effect. Carry the sign of convexity into the second term.
Forgetting the ½ in the convexity adjustment.
Students remember the convexity formula but mix up the price change estimate, which has a half.
Fix: Remember the second-order Taylor term: ½ × convexity × (shift)². Check it against a duration-only answer in the options.
Believing a putable bond has negative convexity because it has an embedded option.
Students treat all embedded options alike.
Fix: The put is held by the investor and limits price falls when rates rise. A putable bond has positive convexity. Only the issuer's call option creates negative convexity.

Worked examples

Example 1

A callable bond has a current price of 100.00. When the benchmark yield curve is shifted down by 100 bps, the model price is 102.00. When it is shifted up by 100 bps, the model price is 97.00. What is the effective convexity? A. −100 B. −1 C. 100

Show the solution

Write the inputs: PV₀ = 100.00, PV₋ = 102.00, PV₊ = 97.00, ΔCurve = 0.01.
Compute the numerator: PV₋ + PV₊ − 2 × PV₀ = 102.00 + 97.00 − 200.00 = −1.00.
Compute the denominator: (0.01)² × 100 = 0.0001 × 100 = 0.01.
Divide: −1.00 ÷ 0.01 = −100.
Interpret: the downward price gain (+2.00) is smaller than the upward price loss (−3.00), which shows negative convexity, as expected for a callable bond at low yields.
For reference, effective duration = (102.00 − 97.00) ÷ (2 × 100 × 0.01) = 5 ÷ 2 = 2.5.

Answer: A. The effective convexity is −100. Option B has the right sign but forgets the squared shift in the denominator, and C has the wrong sign.

Example 2

A bond has an effective duration of 7.2 and an effective convexity of 70. The benchmark yield curve falls by 100 bps in a parallel shift. What is the estimated percentage price change? A. 6.85% B. 7.20% C. 7.55%

Show the solution

Write the inputs: EffDur = 7.2, EffCon = 70, ΔCurve = −0.01.
Duration term: −7.2 × (−0.01) = +0.0720, or +7.20%.
Convexity term: ½ × 70 × (−0.01)² = 35 × 0.0001 = 0.0035, or +0.35%.
Add the two terms: 7.20% + 0.35% = 7.55%.
Check the distractors: 7.20% ignores convexity, and 6.85% subtracts the convexity term instead of adding it.
The sign of the convexity term does not depend on the direction of the shift, because the shift is squared. Convexity is positive, so it adds in both directions.

Answer: C. The estimated price increase is about 7.55%.

Exam tips

If the numerator PV₋ + PV₊ − 2 × PV₀ is negative, the convexity is negative. You can remove positive options quickly.
Many options are built from common errors: duration only, the wrong sign on convexity, or a missing ½. Calculate the duration term first and see which option it matches.
For a callable bond, check whether the question gives low yields or a price near the call price. That tells you whether to expect negative convexity.
Know the comparison cold: a callable bond has lower convexity than an option-free bond and a putable bond has higher or equal convexity. Questions often ask which bond has the highest price rise or the smallest loss.
On the BA II Plus or HP 12C, do the numerator and denominator separately and store results in memory. Divide at the end to avoid rounding errors.

Practice questions from Curve-Based and Empirical Fixed-Income Risk Measures

Effective Convexity and Embedded Option Bonds in other exams

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

Effective Convexity and Embedded Option Bonds: frequently asked questions

What is the effective convexity formula?

Effective convexity = (PV₋ + PV₊ − 2 × PV₀) ÷ [(ΔCurve)² × PV₀]. PV₋ and PV₊ are the prices when the benchmark curve falls and rises by ΔCurve, and PV₀ is the starting price. Use the shift as a decimal.

Why do callable bonds have negative convexity?

When yields fall, the issuer becomes more likely to call the bond, so the price rises less than an option-free bond would. The price-yield curve flattens and then bends downward. This only happens when yields are low enough for a call to be likely.

What is the difference between approximate convexity and effective convexity?

Approximate convexity uses a change in the bond's yield to maturity and assumes the cash flows stay fixed. Effective convexity uses a shift in the benchmark yield curve and prices from a model that lets cash flows change with the embedded option. Use effective convexity for bonds with options.

How do I estimate a bond's price change using duration and convexity?

Compute −duration × Δyield and add ½ × convexity × (Δyield)². Use the shift as a decimal. The result is the percentage price change, which you multiply by the starting price if you need a currency amount.