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

Effective Duration Formula and Curve Duration Measures

Updated 7 October 2026 · Fact-checked

Effective duration measures a bond's price sensitivity to a parallel shift in the benchmark yield curve. You reprice the bond after the curve moves down and up, then compute (PV₋ − PV₊) ÷ (2 × PV₀ × Δcurve). It works for bonds with embedded options, where cash flows change with rates.

Understand Effective Duration and Curve Duration Measures

Duration estimates how much a bond's price changes when interest rates change. Macaulay and modified duration assume the bond's cash flows are fixed. That works for option-free bonds. It fails when the cash flows depend on the interest rate level.

A callable bond is the standard example. When rates fall, the issuer may call the bond, so the expected cash flows shorten. A putable bond works in the opposite direction. You cannot compute modified duration from a single yield to maturity here, because there is no single fixed cash flow stream.

Effective duration fixes this. You shift the benchmark yield curve down and up by the same small amount. You then reprice the bond with a valuation model that lets cash flows change, for example when the call becomes likely. The price changes in the two directions give the duration.

The key difference: modified duration comes from a bond's own yield to maturity and fixed cash flows. Effective duration comes from curve shifts and a pricing model. The curve shift is not the bond's yield change. A callable bond's yield does not move one-for-one with the benchmark curve.

For a callable bond, effective duration is less than or equal to that of an otherwise identical option-free bond, and it falls as rates drop because the call becomes more likely.

For a putable bond, effective duration is also at most that of an otherwise identical option-free bond. It decreases as rates rise, because the put becomes more likely to be exercised.

Key formulas to remember

Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve); PV₋ is the price when the benchmark curve falls by ΔCurve. PV₊ is the price when it rises. ΔCurve is a decimal, so 25 bps = 0.0025.
Approximate price change: %ΔPV ≈ −EffDur × ΔCurve; Use for small parallel benchmark shifts. Add a convexity term for larger moves.
Modified duration from Macaulay: ModDur = MacDur ÷ (1 + YTM per period); Applies only to option-free bonds with fixed cash flows. Use periodic yield with the periodic compounding frequency.
Callable bond duration ranking: EffDur(callable) ≤ EffDur(option-free); The call option shortens expected cash flows when rates fall, so the callable has the lower effective duration.

How to solve Effective Duration and Curve Duration Measures questions

Follow these steps for any effective duration question, numerical or conceptual.

1Check whether the bond has an embedded option. If yes, use effective duration, not modified duration.
2Identify the three prices: PV₀ (the original price), PV₋ (curve down) and PV₊ (curve up). Read the labels carefully.
3Convert the curve shift to a decimal. 50 bps = 0.0050. Remember the formula divides by 2 × the shift.
4Compute the numerator PV₋ − PV₊. It should be positive for a normal bond, since the price is higher when the curve falls.
5Divide by 2 × PV₀ × ΔCurve.
6Sanity-check. A callable bond's duration should be below the option-free equivalent.
7If asked for a price change, multiply −EffDur by the change in the benchmark curve and apply the result to PV₀.

Quickest way: Shortcut for effective duration

When to use it: Use when the question gives three prices and a curve shift and you need to move fast.

Subtract the up-shift price from the down-shift price in your head or on the calculator.
Divide by PV₀ first. This gives the percentage price swing as a decimal.
Divide by 2 times the shift in decimal form. 2 × 0.0025 = 0.005.
Use size checks only to catch gross decimal errors. A duration far above the bond's maturity usually signals one, and a callable bond's duration above an otherwise identical option-free bond is wrong. Options can be close or far apart, so always compute the answer rather than pick by size.
On the TI BA II Plus: enter (PV₋ − PV₊) ÷ PV₀ ÷ (2 × shift) with brackets, then press =.

Common mistakes in Effective Duration and Curve Duration Measures

Using modified duration for a callable or putable bond.
Modified duration is the formula students know best, and it seems to apply to any bond.
Fix: If the bond has an embedded option, cash flows change with rates. Use the curve-shift formula with model prices.
Forgetting the 2 in the denominator.
Students remember the shift but miss that the price difference spans a down move and an up move.
Fix: Write the denominator as 2 × PV₀ × ΔCurve every time. The total move across the two prices is twice the shift.
Entering the shift in percent instead of decimal.
The question says 25 bps or 0.25%, and the number is typed as 25 or 0.25.
Fix: Convert to decimal: 25 bps = 0.0025. A duration of several hundred is a warning sign.
Swapping PV₋ and PV₊.
The labels are easy to misread under time pressure.
Fix: Prices rise when the curve falls. The larger price goes first in the numerator, so a normal bond gives a positive duration.
Assuming the bond's yield changes by the same amount as the benchmark curve.
Modified duration is linked to the bond's own yield, so students carry that idea over.
Fix: Effective duration is tied to the benchmark curve shift. For bonds with options, the bond's yield may move by a different amount.
Saying a callable bond always has lower duration than any option-free bond.
Students overgeneralize the rule.
Fix: The comparison is with an otherwise identical option-free bond. Maturity and coupon differences can change the result.

Worked examples

Example 1

A callable bond is priced at 102.00. If the benchmark curve falls by 25 bps, the model price is 103.10. If it rises by 25 bps, the model price is 100.80. What is the effective duration? A) 1.13 B) 4.51 C) 9.02

Show the solution

PV₀ = 102.00, PV₋ = 103.10, PV₊ = 100.80, ΔCurve = 0.0025.
Numerator: 103.10 − 100.80 = 2.30.
Denominator: 2 × 102.00 × 0.0025 = 0.51.
EffDur = 2.30 ÷ 0.51 = 4.51.
Check the other options: 1.13 = 2.30 ÷ (2 × 102.00 × 0.01) = 2.30 ÷ 2.04, which uses a 1% shift instead of 0.25%. 9.02 = 2.30 ÷ (102.00 × 0.0025) = 2.30 ÷ 0.255, which omits the 2 in the denominator and doubles the correct answer.

Answer: B) 4.51

Example 2

A putable bond has price 98.00. A 50 bps fall in the benchmark curve gives a model price of 98.60, and a 50 bps rise gives 97.20. Estimate the percentage price change if the benchmark curve rises by 1.00% using effective duration. A) −2.86% B) −1.43% C) −0.71%

Show the solution

ΔCurve = 0.0050.
Numerator: 98.60 − 97.20 = 1.40.
Denominator: 2 × 98.00 × 0.0050 = 0.98.
EffDur = 1.40 ÷ 0.98 = 1.4286.
Price change ≈ −1.4286 × 0.0100 = −0.014286, or −1.43%.

Answer: B) −1.43%

Exam tips

If a question mentions a call, put, or other embedded option, expect effective duration to be the right tool.
In conceptual items, rank callable, putable and option-free durations. Callable bonds lose duration as rates fall. Putable bonds lose duration as rates rise.
Watch units. Basis points must become decimals before you divide.
Do the sanity check on duration size. A figure far above the bond's maturity usually signals a decimal error.
Pick answers that tie effective duration to curve shifts, and modified duration to the bond's own yield.

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

Effective Duration and Curve Duration Measures 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 Curve Duration Measures: frequently asked questions

What is the effective duration formula in CFA Level I?

Effective duration = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve). PV₋ is the price after the benchmark curve falls and PV₊ is the price after it rises. The shift is in decimal form.

What is the difference between modified duration and effective duration?

Modified duration uses the bond's own yield to maturity and assumes fixed cash flows. Effective duration uses benchmark curve shifts and a pricing model that lets cash flows change. So effective duration suits bonds with embedded options.

How do you calculate the effective duration of a callable bond?

Reprice the bond with the benchmark curve shifted down and up by the same amount, using a model that reflects the call option. Put the three prices into the effective duration formula. The result is normally lower than that of an otherwise identical option-free bond.

How does effective duration compare with Macaulay duration?

Macaulay duration is the weighted average time to receive the cash flows. Effective duration measures price sensitivity to curve shifts. They answer different questions, and Macaulay duration does not handle embedded options.