Why does a zero-coupon bond's duration equal its maturity?

A zero-coupon bond pays nothing until the end, then pays face value once. Run the Macaulay recipe on it and the answer falls out in a single line — and that single line makes the zero the most useful reference instrument in the toolkit.

The one-line proof

Macaulay duration is the PV-weighted average time of the cash flows. A zero has one cash flow. Its weight is therefore 100%, and the weighted average of one number is that number:

Macaulay duration of an n-year zero = n. Exactly. At any yield.

A 10-year zero has a duration of 10.00 years whether it yields 1% or 9%. No coupon bond can say that — for coupon bonds, duration shifts as yields move.

What that means for price

A 10-year zero, €1,000 face, 5% yield:

• Price = 1,000 ÷ 1.05¹⁰ = €613.91
• Modified duration = 10 ÷ 1.05 = 9.52
• Rates rise 100bp: duration estimate −9.52% → €555.45
• True price at 6%: 1,000 ÷ 1.06¹⁰ = €558.39, a fall of 9.04%

Duration overstated the loss by about half a percent — a bigger gap than the coupon bond showed, because a longer duration comes with more curvature. Unit 3 measures exactly that.

Zeros are the atoms

Here is the idea that makes the whole framework click. Every coupon bond is a bundle of zeros. Each coupon is a small zero maturing on its payment date; the principal is a large one maturing at the end. Its duration is nothing more than the value-weighted average maturity of that bundle — which is precisely what the table in the Macaulay lesson computed, one row per implied zero.

Markets take this literally: STRIPS are created by separating a government bond's coupons and principal into individually traded zeros.

Three consequences worth memorising

• A zero has the longest duration available for its maturity. A 30-year zero has duration 30 and modified duration about 28.6 at a 5% yield — the most rate-sensitive plain-vanilla instrument that exists. A coupon only shortens it, and the size of the coupon is the size of the shortening: a 30-year bond paying 5% has a duration near 16, while one paying 0.01% comes in at 29.88. "Coupons shorten duration" is the rule; "nothing with a coupon comes close" is not.
• A zero has no reinvestment assumption. Unit 1's loose end, tied: held to maturity, a zero's YTM is its realised return, exactly, because there is nothing to reinvest.
• A zero's duration falls exactly one year per year. A coupon bond's duration falls more slowly than the calendar, which makes matching drift. The zero stays matched by construction — and that is why it is the perfect instrument for the liability problem in Unit 3.

In the data

The high-quality corporate curve is published twice: as par yields, for bonds paying coupons, and as spot yields, the zero-coupon curve. A spot yield is the discount rate for a single payment at that date, with nothing in between. The table has both at ten years for the newest month.

Same issuer quality, same maturity, different number: in June 2026 the par yield was 5.18% and the spot yield 5.27%. Mixing the two, or averaging them, gives a curve that is neither.

Try it now

1. Price a 20-year zero at 4% (1,000 ÷ 1.04²⁰ = €456.39) and again at 5% (€376.89). That is a −17.4% move.
2. Now compute what modified duration alone predicted: 20 ÷ 1.04 = 19.23, so −19.2%. Note that duration was wrong by nearly two percentage points — and wrong in your favour. Hold that thought; the next unit explains it.
3. The high-quality corporate spot curve below is the zero-coupon curve itself, for the newest month. Each maturity is a zero, so its Macaulay duration is its maturity in years. Take the 2-year and the 20-year rows, compute each one's modified duration (maturity ÷ (1 + yield)), and estimate what a 100bp rise does to each price. Then state the rule aloud: "duration equals maturity only for a zero; for everything else it is strictly less."