Macaulay Duration

Bonds & Fixed Income
Updated Apr 2026 • Has calculator

The weighted-average time (in years) to receive all of a bond's cash flows, used as a measure of interest rate sensitivity.

What is Macaulay Duration?

Macaulay duration is the present-value-weighted average of the times at which a bond's cash flows are received. Each cash flow is weighted by its present value as a share of the bond's total price. A higher duration means more of the bond's value is concentrated in later cash flows, making it more sensitive to interest rate changes. A zero-coupon bond's Macaulay duration always equals its maturity, because there is only one cash flow at maturity. Coupon bonds have a shorter duration than their maturity. Macaulay duration is the theoretical foundation for modified duration and convexity.

Formula

D_mac = Σ [t × PV(CF_t)] / Price

Worked Example

Worked example — Hypothetical 8% Annual Coupon Bond

5-year maturity, at-par (YTM = 8%)

Step 1  Face: $1,000 | Annual coupon: $80 | YTM: 8% | Maturity: 5 yrs
Step 2  PV of coupons: $74.07 + $68.59 + $63.51 + $58.80 + $54.45 = $319.42
Step 3  PV of face: $680.58 | Total price: $1,000 (at par)
Step 4  D_mac = (1×74.07 + 2×68.59 + 3×63.51 + 4×58.80 + 5×734.03) / 1000
Step 5  D_mac ≈ 4.31 years — average cash flow arrives in year 4.31

Source: CFA Institute — Fixed Income Analysis, 3rd ed., Ch. 5 (2023-01-01)

Calculate Macaulay Duration

Par value repaid at maturity

Total annual coupon in dollars

Annual YTM used as the discount rate

Remaining years until maturity

1 = annual, 2 = semi-annual

Macaulay Duration

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Not investment advice.

How to Interpret Macaulay Duration

< 2
< 2 yrs: Very short — money-market or short-term note; low rate risk
2 – 5
2–5 yrs: Short-to-medium — moderate sensitivity to rate changes
5 – 10
5–10 yrs: Intermediate — meaningful rate risk; typical investment-grade bond
> 10
> 10 yrs: Long — high interest rate sensitivity; long-dated bonds

📚 Bond Risk — Complete the path

  1. Macaulay Duration
  2. Modified Duration
  3. Effective Duration
  4. Convexity
  5. Duration Price Approximation
Sources
  1. CFA Institute — Fixed Income Analysis, 3rd ed., Ch. 5
  2. Investopedia — Macaulay Duration