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

Key Rate Duration and Price-Yield Behavior of Bonds with Options

Updated 7 October 2026 · Fact-checked

Key rate duration measures a bond's price sensitivity to a change in one point on the par yield curve while all other points stay fixed. Summing key rate durations gives effective duration. Callable bonds show negative convexity at low yields; putable bonds keep positive convexity and gain value as yields rise.

Understand Key Rate Duration and Price-Yield Behavior

Effective duration assumes the whole yield curve moves in parallel. Real curves also steepen, flatten and twist. Key rate duration (also called partial duration) handles this. You shift one key maturity on the par curve, for example the 2-year, 5-year or 10-year point, and hold the others fixed. Then you measure the bond's percentage price change for that single shift.

A bond has one key rate duration per chosen maturity. A bullet bond pays most of its cash flow at maturity, so its key rate duration is concentrated at the point nearest its maturity. A bond with coupons spreads sensitivity across earlier points too. The sum of the key rate durations equals the bond's effective duration. Use the profile to see where on the curve the exposure sits, and to estimate the price effect of a non-parallel shift.

Now the price-yield curve. An option-free bond has a convex curve: positive convexity. A callable bond equals an option-free bond minus a call option, because the issuer owns the call. When yields fall, the issuer is more likely to call, so price gains are capped near the call price. The curve flattens and bends the wrong way. This is negative convexity. Effective duration falls as yields fall, and is lower than the option-free bond's duration when the call is in or near the money.

A putable bond equals an option-free bond plus a put option, because the investor owns the put. When yields rise, the put gives a price floor near the put price. Price losses are limited. The bond keeps positive convexity and its duration shortens as yields rise. Its value is at least that of an otherwise identical option-free bond.

Interest rate volatility matters because option value rises with volatility. Higher volatility raises the value of both the call and the put. So higher volatility lowers a callable bond's value (value of option-free bond minus a bigger call) and raises a putable bond's value (option-free value plus a bigger put).

Key formulas to remember

Key rate duration: KRD_k = (PV₋ − PV₊) ÷ (2 × PV₀ × Δy_k); PV₋ and PV₊ are prices after the key rate falls or rises by Δy_k, other key rates unchanged. Express Δy_k as a decimal, e.g. 0.0001 for one basis point.
Sum of key rate durations: Effective duration ≈ Σ KRD_k; The sum approximates effective duration, which is defined for a parallel shift; use individual KRDs for non-parallel shifts.
Price change from a shift in one key rate: %ΔPrice ≈ −KRD_k × Δy_k; For several shifts, add the effects: %ΔPrice ≈ −Σ (KRD_k × Δy_k).
Callable bond value: V_callable = V_option-free − V_call; The issuer holds the call, so the investor's bond is worth less.
Putable bond value: V_putable = V_option-free + V_put; The investor holds the put, so the bond is worth more.
Effective duration: (PV₋ − PV₊) ÷ (2 × PV₀ × Δcurve); Uses a parallel shift of the benchmark curve with the OAS held constant.

How to solve Key Rate Duration and Price-Yield Behavior questions

Read the vignette for the bond type, the key rates given and the direction of each rate change. Then match the question to one of three tasks: key rate math, price-yield shape, or volatility effect.

1Identify the bond: option-free, callable or putable. Note who owns the option.
2Find the data in the exhibit: key rate durations by maturity, or prices at shifted yields.
3For key rate questions, multiply each KRD by its own rate change (in decimals), change the sign, and add the products.
4Check the sign: a rate fall gives a price rise for positive durations. A steepening or twist can give gains at one point and losses at another.
5For shape questions, ask where the option is in the money. Callable: low yields cap price and give negative convexity. Putable: high yields floor price.
6For volatility questions, decide the option's value direction first (higher volatility means higher option value), then apply minus for a call and plus for a put.
7Sanity-check: a callable bond's price should not rise far above its call price; a putable bond's price should not fall far below its put price.

Quickest way: Option-owner shortcut

When to use it: Use for any qualitative question on convexity, duration, or volatility for bonds with embedded options.

Ask who owns the option. Issuer owns a call; investor owns a put.
Callable = option-free minus call. Putable = option-free plus put.
Callable: low yields bring negative convexity and shorter duration. Putable: high yields shorten duration and convexity stays positive.
Higher volatility raises option value: callable value falls, putable value rises.
For key rate math, compute −Σ(KRD × Δy) in one line and check the sign.

Common mistakes in Key Rate Duration and Price-Yield Behavior

Adding key rate durations when only some rates shift and calling the sum effective duration.
The sum equals effective duration only for a parallel shift of all key rates.
Fix: Multiply each KRD by its own shift. Use the plain sum only for a parallel move.
Entering basis points as whole numbers, such as 50 instead of 0.0050.
Speed under time pressure.
Fix: Convert to decimals first. A 50 bp change is 0.005, so a 50 bp rise with KRD of 4 gives about −2.0%.
Saying higher volatility raises the value of every bond with an option.
Students remember that option value rises with volatility and stop there.
Fix: Add the sign. The call is a liability to the bondholder, so callable value falls; putable value rises.
Saying a putable bond has negative convexity at low yields.
Confusing the two options.
Fix: The put is out of the money at low yields, so the putable bond behaves like an option-free bond there, with positive convexity.
Forgetting that a rate fall can hurt if the KRD is negative or if the option is exercised.
Applying a fixed positive duration everywhere.
Fix: Use each KRD with its sign, and remember a callable bond's duration shrinks as yields fall.

Worked examples

Example 1

A bond has a price of 100. Its key rate durations are 0.5 at 2 years, 1.5 at 5 years, 3.0 at 10 years and 2.0 at 20 years. (1) What is its effective duration? (2) By how much does the price change approximately if the 5-year rate rises 20 bps and the 10-year rate falls 10 bps, with other rates unchanged?

Show the solution

Effective duration = sum of KRDs = 0.5 + 1.5 + 3.0 + 2.0 = 7.0.
5-year effect: −1.5 × 0.0020 = −0.0030, which is −0.30%.
10-year effect: −3.0 × (−0.0010) = +0.0030, which is +0.30%.
Total = −0.30% + 0.30% = 0.00%.

Answer: (1) Effective duration is 7.0. (2) The approximate price change is 0.00%, so the price stays about 100.

Example 2

A callable bond has a call price of 102 and trades at 101.5 when yields are low. An analyst says: 'If expected interest rate volatility rises, this bond's value will rise because option values rise.' (1) Is the analyst correct? (2) What happens to the bond's effective duration if yields fall further? (3) Which bond would gain value in the volatility rise: this bond or an otherwise identical putable bond?

Show the solution

(1) Callable value = option-free value − call value. Higher volatility raises call value, so the callable bond's value falls. The analyst is wrong.
(2) As yields fall further, the bond's price approaches the call price. The call becomes more likely to be exercised, so price gains are capped. Effective duration falls and convexity is negative.
(3) Putable value = option-free value + put value. A higher put value raises the bond's price. The putable bond gains.

Answer: (1) No: the value falls. (2) Effective duration decreases, and convexity is negative. (3) The putable bond gains value.

Exam tips

Look for the option owner in the vignette first. Most qualitative answers follow from callable = minus call and putable = plus put.
For key rate calculations, convert basis points to decimals and keep the sign of each rate change.
Expect a rate exhibit with several shifts (steepening, flattening, twist). Compute each product separately, then add.
Test items often ask about convexity at high versus low yields. Callable: negative at low yields. Putable: protection at high yields.
There is no penalty for wrong answers, so answer every question even if you must eliminate options and guess.

Key Rate Duration and Price-Yield Behavior in other exams

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

Key Rate Duration and Price-Yield Behavior: frequently asked questions

What is key rate duration in CFA Level II?

It is the percentage price sensitivity of a bond to a change in one maturity point on the par yield curve, with all other points held constant. It lets you measure the effect of non-parallel shifts such as steepening, flattening and twists.

How do you compute key rate duration?

Shift one key rate down and up by the same small amount, find the prices PV₋ and PV₊, then apply (PV₋ − PV₊) ÷ (2 × PV₀ × Δy). Repeat for each key maturity. Their sum approximates effective duration.

Why does a callable bond show negative convexity?

When yields fall, the issuer is likely to call the bond, so the price cannot rise much above the call price. The price-yield curve flattens and bends downward at low yields, which is negative convexity.

How does interest rate volatility affect callable bond value?

Higher volatility raises the value of the call option the issuer owns. Since callable value equals option-free value minus call value, the bond's value falls. A putable bond gains value instead.