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CFA® Level I

Bond Duration: Macaulay, Modified Duration, and a Worked Example

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Worked exampleInteractive check5 min read

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What duration measures

Bond duration turns a fixed-income question into a price-sensitivity question: if yields move, how much might the bond price move? For a small yield change, modified duration gives a first-order estimate.

Approximate % price change ≈ −Modified duration × change in yield

The minus sign matters: yields and prices generally move in opposite directions.

One bond, three related measures

Scroll horizontally to see every column.

MeasureWhat it describesExam cue
Macaulay durationPresent-value weighted average time to receive the bond’s cash flows.Time dimension
Modified durationApproximate percentage price sensitivity to a change in the bond’s own yield.Price change estimate
PVBP / DV01Approximate dollar price change for a 1 basis-point yield move.Dollar sensitivity

Worked example: a 3-year coupon bond

Consider a $100 par bond with a 6% annual coupon and a 6% yield to maturity. Its price is $100. The present-value weighted cash flows give a Macaulay duration of approximately 2.83 years. With an annual yield of 6%:

Modified duration = Macaulay duration / (1 + y) = 2.83 / 1.06 ≈ 2.67

If the yield rises by 0.50 percentage points, the first-order estimate is:

%ΔPrice ≈ −2.67 × 0.005 = −1.34%
Estimated price ≈ $100 × (1 − 0.0134) = $98.67

The estimate is linear. The actual bond price change will differ because the price-yield curve has convexity.

Move the yield

$100 starting price · modified duration 2.67

Yield move+0.50 percentage points
Approximate price change-1.34%
Estimated price$98.67

First-order duration estimate; convexity is not included.

What changes duration?

  • Higher coupon bonds usually have shorter duration than otherwise similar lower coupon bonds.
  • Longer maturity usually increases duration.
  • At the same maturity and coupon, a lower yield generally increases modified duration.
  • For bonds with embedded options, effective duration is more appropriate because cash flows can change when rates move.

Check your understanding

A bond has modified duration of 4. Yield rises by 25 basis points. What is the approximate price change?

Common mistakes

  1. Using Macaulay duration directly as a percentage price sensitivity.
  2. Forgetting that 25 basis points is 0.0025, not 0.25.
  3. Applying the linear estimate to a very large yield move without discussing convexity.
  4. Assuming effective duration and modified duration are interchangeable when cash flows can change.

FAQ

Does duration tell me the exact new bond price?

No. It is a first-order approximation. Convexity improves the estimate, especially for larger yield changes.

Why is there a minus sign?

For a plain option-free bond, a higher discount rate lowers the present value of its cash flows.

What is PVBP?

PVBP expresses the approximate dollar price change for a one-basis-point yield move. It is useful when the question asks for money rather than a percentage.

Curriculum context and further reading

Quizara's CFA Level I 2027 curriculum places this concept in V6 Fixed Income → Module 11 → Modified Duration, alongside separate topics for Money Duration and Price Value of a Basis Point and Properties of Duration. The CFA Institute refresher reading distinguishes Macaulay, modified, money duration and PVBP. Use the official Yield-Based Bond Duration Measures and Properties reading to check definitions and assumptions.