Macaulay Duration
Macaulay Duration
Macaulay duration, named for economist Frederick Macaulay who formalized it in 1938, is the weighted average time to receive a bond's cash flows, where each cash flow is weighted by its present value as a fraction of the bond's total price.
Key takeaways
- Macaulay duration is a time-weighted average: each year is weighted by the proportion of present value received in that year
- Calculated as the sum of (present value of each cash flow × years until that cash flow) divided by bond price
- Zero-coupon bonds have Macaulay duration equal to their maturity
- Coupon bonds have Macaulay duration less than maturity
- The formula accounts for both the size and timing of every cash flow
Historical context: Why Macaulay invented duration
Frederick Macaulay, working at the National Bureau of Economic Research in the 1930s, faced a practical problem. Bond managers and actuaries needed to compare bonds that looked similar on the surface—same maturity, same coupon—but behaved differently when interest rates moved. Macaulay wanted a single metric that would unify these observations.
He proposed measuring the "average period to receive cash flows" from a bond. Not the simple arithmetic average of years, but a weighted average where larger cash flows (received later, closer to maturity) count more than smaller cash flows (coupons early on). His insight was elegant: you can distill the entire cash flow schedule into one number that represents the average time your capital is at risk.
By 1938, Macaulay's approach was published and gradually adopted. It became the standard way to measure bond "duration"—a term that stuck because it literally means length of time.
The formula
Macaulay duration is calculated as:
Duration = Σ [t × PV(CFt)] / Bond Price
Where:
- t is the time (in years) until each cash flow
- CFt is the cash flow at time t
- PV(CFt) is the present value of that cash flow (discounted at the bond's yield to maturity)
- The bond price is the sum of all present values
Let's walk through a concrete example: a 3-year bond paying 4% coupon annually, with a yield to maturity of 5%.
Year 1: Cash flow $40
- Present value = $40 / (1.05^1) = $38.10
- Weight = $38.10 / (total price)
Year 2: Cash flow $40
- Present value = $40 / (1.05^2) = $36.28
- Weight = $36.28 / (total price)
Year 3: Cash flow $1,040 (coupon + principal)
- Present value = $1,040 / (1.05^3) = $898.88
- Weight = $898.88 / (total price)
Bond price = $38.10 + $36.28 + $898.88 = $973.26
Macaulay duration = [(1 × $38.10) + (2 × $36.28) + (3 × $898.88)] / $973.26 = [$38.10 + $72.56 + $2,696.64] / $973.26 = $2,807.30 / $973.26 = 2.88 years
The bond's maturity is 3 years, but its Macaulay duration is only 2.88 years. The coupon payments pull the weighted average time earlier than the final maturity date.
Macaulay duration for zero-coupon bonds
A zero-coupon bond simplifies the calculation dramatically. Since there is only one cash flow—at maturity—the weighted average time equals maturity exactly.
Example: A 10-year zero-coupon bond has no interim coupons. Its only cash flow is $1,000 (or par equivalent) at year 10. The Macaulay duration is exactly 10 years.
This makes intuitive sense: you have no choice about when you get your money. You wait the full decade.
Compare this to a 10-year coupon bond paying 5% annually. You receive $50 in each of years 1–9, plus $1,050 in year 10. The weighted average time is less than 10 years because those $50 payments reduce your average wait. Depending on the yield, the duration might be 7.5 or 8.2 years—well short of the 10-year maturity.
Macaulay duration and coupon rate
Higher coupons reduce Macaulay duration. Consider two bonds, both maturing in 10 years and yielding 4%:
- Bond A pays 2% coupon: You receive $20 each year plus $1,000 at year 10. Early cash flows are small relative to the final payment.
- Bond B pays 6% coupon: You receive $60 each year plus $1,000 at year 10. Early cash flows are much larger, pulling the weighted average earlier.
Bond B's Macaulay duration is shorter than Bond A's, even though both mature in 10 years. The larger coupons in Bond B front-load your cash recovery.
This relationship—higher coupon, lower duration—holds across all coupon bonds. It reflects the simple truth that if you're getting paid more often and sooner, you recover your capital faster on average.
Macaulay duration and yield to maturity
Interestingly, Macaulay duration also declines with higher yields. This is subtler than the coupon effect.
When yields rise, present values of future cash flows decline more sharply. The immediate coupon (year 1) loses value less than the distant final payment (year 10). Relatively speaking, the near-term cash flows become more important—their proportion of total bond price rises. The weighted average time shifts earlier.
Conversely, when yields fall, the distant final payment becomes more valuable relative to the nearer coupons, pulling the duration forward.
This effect is usually smaller than the coupon effect, but it's real. A bond trading at higher yield will have shorter Macaulay duration than the same bond at lower yield.
Computing Macaulay duration in practice
Modern bond analytics tools (Bloomberg terminals, spreadsheets, online calculators) compute Macaulay duration automatically. You plug in the coupon rate, maturity, current yield, and the tool returns the duration.
For U.S. Treasury bonds, BND index constituents, LQD corporate bonds, or MUB municipal bonds, you can look up the duration directly from fund fact sheets. The stated "duration" usually refers to effective duration (which we'll cover later), but for plain-vanilla bonds without embedded options, effective duration and Macaulay duration are nearly identical.
Macaulay duration's limitation
While elegant, Macaulay duration has one significant limitation: it only tells you the time-weighted average when you're paid, not how much your bond price changes when yields move. A 5-year Macaulay duration doesn't directly tell you "if yields rise 1%, my price falls 5%."
That gap—between timing and price sensitivity—is where modified duration comes in. Modified duration takes Macaulay duration and adjusts it to directly estimate price changes from interest rate moves.
Process
Why Macaulay duration still matters
Even though modified and effective duration are more practical for trading and risk management, Macaulay duration remains important. It's a pure time measure—unambiguous, transparent, comparable across bonds globally. When you need to understand the actual cash flow timing of your portfolio, Macaulay duration is the right tool.
Central banks, pension funds, and insurers often report holdings in Macaulay duration terms because it's independent of the current yield environment. A "3-year Macaulay duration bond" is the same conceptually in 2020 (when yields were low) or 2022 (when yields spiked). Modified duration would shift with the yield changes.
Next
Macaulay duration measures the timing of your cash flows. But traders care most about how prices move when rates change. Modified duration takes Macaulay duration and converts it to a price sensitivity metric, telling you directly how much your bond's value fluctuates per 1% change in yield.