Lesson 5 of 8
Duration
Macaulay duration is the average year a bond's value arrives in. Modified duration turns that into the price move a rate change implies, computed by hand.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- Macaulay duration is the weighted average time a bond's value arrives, with each year weighted by its share of the price.
- Modified duration is that figure divided by one plus the yield, and it estimates the percentage price move per point of rate change.
- Duration is a slope at one point, so it is accurate for small moves and progressively wrong for large ones.
A five-year bond does not deliver its value in year five. It delivers a little each year and most of it at the end. Duration is the honest average of those arrival dates, and it is the single most useful number for anticipating what a rate move does to a price.
Where a five-year bond's value actually sits
- Year 1$47.72
- Year 2$45.54
- Year 3$43.46
- Year 4$41.48
- Year 5$831.38
Macaulay duration, computed
Take each year's present value, divide by the total price to get its weight, and multiply the weight by the year number. Add the five results.
The five-year 5 percent bond, priced at 1,009.58 to yield 4.78 percent
- Year 1: 47.72 of 1,009.58 is a weight of 0.0473. Multiplied by 1 gives 0.0473.
- Year 2: 45.54 is a weight of 0.0451. Multiplied by 2 gives 0.0902.
- Year 3: 43.46 is a weight of 0.0431. Multiplied by 3 gives 0.1292.
- Year 4: 41.48 is a weight of 0.0411. Multiplied by 4 gives 0.1644.
- Year 5: 831.38 is a weight of 0.8235. Multiplied by 5 gives 4.1174.
- Add the five: 4.5484 years.
A five-year bond has a Macaulay duration of 4.55 years. The gap between 5 and 4.55 is the four coupons pulling the average forward. A zero-coupon bond has a duration exactly equal to its maturity, because there is nothing to pull.
Modified duration, and what it predicts
Modified duration equals Macaulay duration divided by 1 + yield.
Example4.5484 / 1.0478 = 4.3409
Modified duration is the percentage the price moves for a one percentage point change in yield, with the sign reversed. This bond has a modified duration of 4.34, so a one point rise in rates implies a price fall of about 4.34 percent.
What duration is for
Three practical uses, none of them a forecast of rates. It ranks two bonds by how exposed each is to the same rate move. It converts a rate scenario you want to test into a price effect. And it says how long a portfolio's value takes to arrive, which is what makes it comparable to a liability with a date on it.
The property that makes it work is also its limit. Duration is a straight line drawn tangent to a curved relationship, at one particular yield. Move far enough along the curve and the line is somewhere else entirely.
Check yourself
Two bonds mature in ten years. One pays a 2 percent coupon and the other 8 percent. Which has the longer duration?
- The 2 percent bond
- The 8 percent bond
- They are equal, because the maturities match
Show the answer
Answer: A.
A larger coupon returns more value early, which pulls the weighted average arrival date forward. The smaller the coupon, the closer duration sits to maturity.A bond has a modified duration of 7. Rates rise by half a point. What price move does duration imply?
- A fall of about 7 percent
- A fall of about 3.5 percent
- A fall of about 14 percent
Show the answer
Answer: B.
Modified duration is the move per one percentage point, so half a point implies half the effect. The estimate is approximate and, for a rise, slightly too pessimistic.What is the duration of a zero-coupon bond maturing in eight years?
- Less than eight years
- Exactly eight years
- More than eight years
Show the answer
Answer: B.
All of the value arrives on one date, so the weighted average arrival date is that date. There are no intermediate payments to pull the average forward.Sources
- Damodaran, Investment Valuation: Macaulay and modified duration and their use as an interest rate sensitivity.
- US Department of the Treasury, Daily Treasury Par Yield Curve Rates, session of 11 September 2026: the 4.78 percent yield the worked bond is priced at.
What this leaves out
- The bond is invented so that every weight in the table can be recomputed. Only the yield is a real published figure.
- Duration assumes a parallel shift in the whole curve. Real curves change shape as well as level, which duration alone does not capture.
- Callable bonds, floating rate notes and mortgage securities have durations that change with the rate itself, and none of them is covered here.
- Nothing here is advice about any security.