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

Callable, Putable and Option-Free Bond Price Behavior

Q: What is negative convexity in a callable bond?

It is the region where the price rises by less than duration predicts when yields fall. The issuer is likely to call the bond, so the price is capped near the call price. It happens at lower yields, where the call option is near or in the money.

Q: How does a callable bond price-yield curve differ from an option-free curve?

At high yields the two curves are nearly the same. At low yields the callable curve flattens and sits below the option-free curve, because the call option value is subtracted from the price. The callable curve shows negative convexity in that region.

Q: Does a putable bond have positive or negative convexity?

It has positive convexity. The put gives the investor a floor when yields rise, so the price falls less than for an option-free bond. At low yields the put is unlikely to be used and the bond acts like an option-free bond.

Q: Why use effective duration instead of modified duration for callable bonds?

Modified duration assumes the cash flows do not change when yields change. An embedded option can change the cash flows. Effective duration reprices the bond with the option under shifted curves, so it captures that effect.

Updated 7 October 2026 · Fact-checked

An option-free bond has a convex price-yield curve (positive convexity). A callable bond matches it at high yields but flattens and bends down as yields fall, giving negative convexity. A putable bond has a price floor at high yields, so it keeps positive convexity and loses less than an option-free bond when yields rise.

Understand Bond Price Behavior: Callable, Putable and Option-Free

Start with the option-free bond. Its price-yield curve slopes down and is bowed toward the origin. This shape is positive convexity. When yields fall, the price rises by more than duration alone predicts. When yields rise, the price falls by less. Both effects help the holder.

A callable bond is an option-free bond minus a call option held by the issuer. The issuer can redeem the bond early at the call price. When yields are high, the call is unlikely to be used, so the callable bond behaves like an option-free bond. When yields fall, the issuer is likely to call, so the price is capped near the call price. The curve flattens and bends down. This region has negative convexity: the price rises by less than duration alone predicts when yields fall.

A putable bond is an option-free bond plus a put option held by the investor. The investor can sell the bond back at the put price. When yields rise, the put becomes valuable and sets a floor under the price. When yields fall, the put is unlikely to be used, so the bond behaves like an option-free bond. The putable bond keeps positive convexity. At high yields its price is higher than the option-free bond's, so it loses less than the option-free bond when yields rise.

In price terms: callable price = option-free price − call option value. Putable price = option-free price + put option value. The option value is largest when the option is near or in the money. That is why the callable curve sits below the option-free curve at low yields, and the putable curve sits above it at high yields.

For embedded-option bonds, the standard measures are effective duration and effective convexity. They come from price changes when the benchmark yield curve shifts up and down. Yield-to-maturity based measures such as modified duration and convexity ignore the option's effect on cash flows, so they are not appropriate here.

Key formulas to remember

Callable bond value: Callable bond price = Option-free bond price − Call option value; The issuer owns the call, so the investor's bond is worth less. The gap is biggest when yields are low.
Putable bond value: Putable bond price = Option-free bond price + Put option value; The investor owns the put, so the bond is worth more. The gap is biggest when yields are high.
Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve); PV₋ and PV₊ are prices after the curve falls and rises by ΔCurve. Use for bonds with embedded options.
Effective convexity: EffCon = (PV₋ + PV₊ − 2 × PV₀) ÷ (PV₀ × ΔCurve²); A negative result means negative convexity. ΔCurve is a decimal, such as 0.0025.
Price change estimate: %ΔPV ≈ (−EffDur × ΔYield) + (½ × EffCon × ΔYield²); Use the same yield change in decimal form. Convexity adds a positive term if EffCon > 0 and a negative term if EffCon < 0.
Convexity by bond type: Option-free: positive. Putable: positive. Callable: positive at high yields, negative at low yields.; Callable bonds show negative convexity only when the call option is near or in the money.

How to solve Bond Price Behavior: Callable, Putable and Option-Free questions

Use this sequence for any question on price-yield behavior of option-free, callable and putable bonds.

1Identify who owns the option. Issuer owns a call. Investor owns a put.
2Write the price relationship: callable = option-free − call; putable = option-free + put.
3Decide the yield region. Low yields mean the call is near or in the money. High yields mean the put is near or in the money.
4Match the curve shape. Callable at low yields: flat or bending down (negative convexity). Putable at high yields: price floor. Option-free: always positive convexity.
5Pick the right measures. For embedded options use effective duration and convexity, not modified duration or YTM-based convexity.
6If numbers are given, plug prices into the effective formulas. Convert basis points to decimals first.
7Check the sign and direction. A callable bond's price gain from a yield fall should be smaller than the option-free bond's.
8Eliminate the two wrong options by testing them against the direction in step 7.

Quickest way: Who holds the option, and where are yields?

When to use it: Use for conceptual questions with no calculation, which are most questions on this topic.

Callable means issuer's option. Yields fall, price is capped, negative convexity.
Putable means investor's option. Yields rise, price is supported, still positive convexity.
Option-free is the benchmark. Always positive convexity.
If an option says a callable bond has the greatest price rise when yields fall, reject it.
If an option says a putable bond has negative convexity, reject it.

Common mistakes in Bond Price Behavior: Callable, Putable and Option-Free

Saying a callable bond always has negative convexity.
Students remember the label but not the condition.
Fix: Negative convexity appears only when yields are low enough for the call to matter. At high yields the callable bond looks like an option-free bond.
Thinking a putable bond has negative convexity because it has an embedded option.
Students link any option to the callable case.
Fix: The put gives the investor a price floor, so the putable bond keeps positive convexity. Ask who owns the option.
Adding the call option value to the option-free price.
Students forget the investor is short the call.
Fix: Callable = option-free − call value. Putable = option-free + put value.
Using modified duration for a callable or putable bond.
Modified duration is the familiar measure.
Fix: Modified duration assumes fixed cash flows. Embedded options change cash flows, so use effective duration.
Entering basis points straight into the effective convexity formula.
Students forget to convert to decimals.
Fix: Convert 25 bps to 0.0025 before squaring. Check that the answer has a sensible size.

Worked examples

Example 1

A callable bond with a 3% coupon and an otherwise identical option-free bond with a 3% coupon both yield 3%, so both trade near par. The call price is par. Yields on both fall to 1%. Which statement about the price change is most accurate? A) The callable bond rises by more than the option-free bond. B) The callable bond rises by less than the option-free bond. C) Both rise by the same amount.

Show the solution

The issuer owns the call option.
At a 3% yield the bonds trade near par, so the call is near the money at the par call price.
When yields fall to 1%, the 3% coupon is well above market rates. The option-free bond trades well above par, so the call at par is well in the money. The issuer is likely to exercise it and the call option has high value.
Callable price = option-free price − call value. As yields fall, call value rises sharply.
So the callable bond's price rises by less than the option-free bond's.
This is negative convexity at low yields.

Answer: B

Example 2

A putable bond has price 100.00 at the current curve. If the benchmark curve falls 25 bps the price is 100.65. If it rises 25 bps the price is 99.55. Calculate effective duration. Options: A) 1.10 B) 2.20 C) 4.40

Show the solution

PV₋ = 100.65, PV₊ = 99.55, PV₀ = 100.00, ΔCurve = 0.0025.
Numerator: PV₋ − PV₊ = 100.65 − 99.55 = 1.10.
Denominator: 2 × 100.00 × 0.0025 = 0.50.
EffDur = 1.10 ÷ 0.50 = 2.20.
Check the sign: the price rises when yields fall, so duration is positive.
Check the shape: the gain from a fall (0.65) is larger than the loss from a rise (0.45). PV₋ + PV₊ = 200.20, which is above 2 × PV₀ = 200.00, so effective convexity is positive: 0.20 ÷ (100 × 0.0025²) = 0.20 ÷ 0.000625 = 320. This fits a putable bond.

Answer: B

Exam tips

Most questions are conceptual. Decide who owns the option first, then decide the yield region.
Expect statements about which bond has the greatest price rise when yields fall, or the smallest fall when yields rise. Option-free and putable gain more than callable on yield declines.
If effective convexity is asked, convert bps to decimals and watch the sign.
Do not use YTM-based modified duration for embedded-option bonds. Effective duration is the answer in these cases.
With three options and no penalty, eliminate any option that gives the putable bond negative convexity or the callable bond a larger gain than option-free when yields fall.

Practice questions from Yield-Based Bond Convexity and Portfolio Properties

Bond Price Behavior: Callable, Putable and Option-Free in other exams

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

CFA Level IIValuing Option-Free, Callable and Putable Bonds

Bond Price Behavior: Callable, Putable and Option-Free: frequently asked questions

What is negative convexity in a callable bond?

It is the region where the price rises by less than duration predicts when yields fall. The issuer is likely to call the bond, so the price is capped near the call price. It happens at lower yields, where the call option is near or in the money.

How does a callable bond price-yield curve differ from an option-free curve?

At high yields the two curves are nearly the same. At low yields the callable curve flattens and sits below the option-free curve, because the call option value is subtracted from the price. The callable curve shows negative convexity in that region.

Does a putable bond have positive or negative convexity?

It has positive convexity. The put gives the investor a floor when yields rise, so the price falls less than for an option-free bond. At low yields the put is unlikely to be used and the bond acts like an option-free bond.

Why use effective duration instead of modified duration for callable bonds?

Modified duration assumes the cash flows do not change when yields change. An embedded option can change the cash flows. Effective duration reprices the bond with the option under shifted curves, so it captures that effect.