Macaulay duration is the weighted average time to receive a bond's promised cash flows, where each cash flow is weighted by its present value as a share of the bond's price. For CFA Level I, you need both sides of it: how to calculate the number and what it means once you have it. The result is measured in years, and it underpins modified duration, which estimates how a bond's price reacts to a change in yield.
Quick Answer
Macaulay duration is the weighted average time to receive a bond's cash flows, with each cash flow weighted by its present value as a share of the full price. For CFA Level I, read it as a timing measure in years, and remember that modified duration is built from it to estimate price sensitivity to yield changes.
Key Takeaways: Macaulay Duration
Macaulay duration is measured in years.
Each cash flow is weighted by its present value, not by its face amount.
A higher coupon generally lowers Macaulay duration, all else equal.
A longer maturity generally raises Macaulay duration.
A higher yield generally lowers Macaulay duration, all else equal.
Modified duration uses Macaulay duration to estimate price sensitivity to yield changes.
What You Need to Know for CFA Level I
You should be able to define Macaulay duration, write and explain the formula, identify each input, calculate duration from a small set of present-valued cash flows, interpret the result as a weighted average timing measure, and separate it from modified and effective duration.
What Is Macaulay Duration?
Macaulay duration tells you, on a present-value basis, how long you wait on average to receive a bond's cash flows. A plain average of the cash-flow dates would ignore that early money is worth more than late money. Macaulay duration fixes that by weighting each date by the present value of the cash flow received then.
Because most of a coupon bond's value sits in the final principal payment, that last cash flow carries the heaviest weight, which is why Macaulay duration usually lands close to, but below, the bond's maturity.
Macaulay Duration Formula
The formula sums each cash flow's timing, weighted by its present value, then divides by the bond's full price.
Formula Breakdown
Symbol | Meaning |
|---|---|
Macaulay Duration | |
Time period when the cash flow is received | |
Cash flow received at time | |
Present value of the cash flow at time | |
Sum of the present values of all the bond's cash flows, including accrued interest where it applies |
When the cash-flow periods are annual, the output is in years. The weights, which are each always add up to one.
How to Calculate Macaulay Duration
To calculate Macaulay duration, present-value each cash flow, turn each present value into a weight by dividing it by the full price, multiply each weight by its time period, and add the results.
Consider a four-year bond with a 5% annual coupon, a $1,000 face value, and a yield of 4%. The annual cash flows are $50 in years one through three and $1,050 in year four.
Year (t) | Cash flow | PV at 4% | Weight | Weight × t |
|---|---|---|---|---|
1 | $50.00 | $48.08 | 0.0464 | 0.0464 |
2 | $50.00 | $46.23 | 0.0446 | 0.0892 |
3 | $50.00 | $44.45 | 0.0429 | 0.1287 |
4 | $1,050.00 | $897.54 | 0.8661 | 3.4644 |
Total | $1,036.30 | 1.0000 | 3.73 |
Adding the final column gives a Macaulay duration of about 3.73 years. The four-year maturity sits above it, because the early coupons pull the weighted average timing in.
How to Interpret Macaulay Duration
Read Macaulay duration as the bond's average wait, in years, for its present-valued cash flows. A duration of 3.73 years on a four-year bond says that, weighted by value, you receive your money a little before the final maturity date. It is a timing measure, not a direct price-change figure.
That timing reading also explains the standard relationships. Larger coupons return more value earlier, which shortens duration. Longer maturities push value further out, which lengthens it. A higher yield discounts distant cash flows more heavily, which shifts weight toward earlier dates and shortens duration.
Macaulay Duration vs Modified Duration
Macaulay duration is a timing measure, while modified duration is a price-sensitivity measure built from it. You convert one to the other by dividing Macaulay duration by one plus the yield per period. In the example above, modified duration is 3.73 divided by 1.04, or about 3.59. That figure estimates the percentage price change for a 1% change in yield, which is a different question from how long you wait for cash flows.
Common Exam Traps
Forgetting to weight cash flows by present value and averaging the dates instead.
Treating Macaulay duration as the same as maturity.
Confusing Macaulay duration with modified duration.
Leaving the final principal repayment out of the calculation.
Reading the result as a price-sensitivity figure when it is a timing measure.
Practice Question
A three-year bond pays an annual coupon and has a Macaulay duration of 2.8 years. Which statement is most accurate?
The bond's price will fall 2.8% if its yield rises by 1%.
On a present-value-weighted basis, the average time to receive the bond's cash flows is 2.8 years.
The bond matures in 2.8 years.
Macaulay duration equals the bond's modified duration.
Answer: B
Macaulay duration is the present-value-weighted average time to receive the bond's cash flows, here 2.8 years.
Option A describes modified duration.
Option C confuses duration with maturity.
Option D ignores that modified duration is Macaulay duration divided by one plus the yield per period.
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FAQs About Macaulay Duration
What is Macaulay duration?
It is the present-value-weighted average time, in years, to receive a bond's cash flows. Each cash flow is weighted by its present value as a share of the bond's price.
What is the Macaulay duration formula?
Macaulay duration equals the sum of each period's time multiplied by the present value of that period's cash flow, divided by the bond's full price.
Is Macaulay duration the same as modified duration?
No. Macaulay duration is a timing measure, while modified duration estimates price sensitivity to yield. You get modified duration by dividing Macaulay duration by one plus the yield per period.
What affects Macaulay duration?
Coupon, maturity, and yield. A higher coupon and a higher yield generally shorten it, while a longer maturity generally lengthens it.