Macaulay duration
The average time, in years, until a bond pays its cash, with each payment weighted by its share of the bond's present value.
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How it is computed here
Each remaining payment is discounted at the bond's estimated yield; its time in years, counted from the settlement date, is weighted by the share of the full price that payment represents, and the weights add up to one. A zero-coupon bond's duration is its time to maturity; a coupon bond's is shorter, because coupons bring money back earlier.
Worked example
- Each of the 20 payments of UST 4.625% 2036-08-15 is discounted at its estimated yield of 5.28% and weighted by its share of the full price of 95.611.
- Weighting the time of every payment, in years, by that share gives 7.9284 years.
- The last payment arrives in 9.87 years: the coupons before it bring part of the money back earlier, which is why the average is shorter.
Real Treasuries valued today on the Treasury's par curve: our estimates, not dealer quotes, and no yield here is a return anyone is owed. The demo holds none; signed in with a Treasury of your own, this is worked on yours.
Where it misleads
It is a measure of time, not of risk by itself: the price sensitivity is the modified duration, which divides it by (1 + y/2). It also uses one yield for every payment and describes small, parallel moves of that yield. The figures are estimates from the Treasury curve, not dealer quotes.
The formula
- DMac — the Macaulay duration, in years;
- tj — the time in years from settlement to the j-th payment;
- PVj — the present value of the j-th payment, discounted at the yield y;
- P — the full price: the sum of all the present values;
- CFj, y, w — the j-th payment per 100, the yield (compounded twice a year) and the part of the current coupon period still to run, as in the yield lesson.
A worked example on a real Treasury
The Treasury note UST 4.625% 2036-08-15 at its estimated yield of 5.2394 %: 9.87 years remain until maturity, yet its Macaulay duration is 7.9348 years. Its last payment, 102.3125, is worth 61.402 today (64.0 % of the full price of 95.896) and arrives in 9.87 years; the nineteen coupons before it, 36.0 % of the price, arrive earlier. Weighting the time of every payment by its share of the price gives 7.93 years, 1.94 years less than the time to maturity, because the coupons bring part of the money back sooner. Every figure is our estimate from the Treasury curve of 1 October 2026, not a dealer quote.
What this page does not do
It does not say how much the price moves, and it does not say when you will get your money back: the principal of this bond returns in 9.87 years, whatever the average. It only summarizes the timing of all payments in one number.
Compared with
- Modified duration — the same figure turned into price sensitivity
- Convexity — how the sensitivity itself changes as the yield moves
- Yield to maturity — the rate whose discounting it uses
Questions people ask
Is a duration of 7.9 years the time until I get my money back?
No. It is the average time of all the payments, principal and coupons, weighted by their present value. The principal of the example bond comes back in 9.87 years; the coupons arrive earlier and pull the average down.
Does a Macaulay duration of 7.93 mean the price moves 7.93 % for a point?
About, but not exactly: the price sensitivity is the modified duration, the Macaulay duration divided by (1 + yield ⁄ 2). For this bond it is 7.73.
Sources
Last reviewed 2026-10-02 by Sphinx Risk.