Macaulay duration of a level-coupon bond

Picture the bond’s cash flows laid along a timeline, each one sized by its present value, and ask where the arrangement balances. That balance point is the Macaulay duration: the average waiting time for the bond’s money, with early coupons pulling it toward today and the redemption payment anchoring it near maturity. It is measured in years, and for any coupon-paying bond it lands strictly inside the time remaining — the coupons see to that.

This page evaluates the closed-form identity for a level-coupon bond rather than summing the stream term by term: the per-period coupon rate, the per-period yield and the count of periods determine the balance point exactly, and the pack’s own vectors re-verify the identity against the direct definition. The result in periods is then converted onto the yearly footing the rest of the cluster speaks.

What moves it is worth internalising. Richer coupons shift weight toward the near payments and shorten the wait; leaner coupons leave the redemption dominant and stretch it. A higher yield discounts the far payments harder, which also shortens the wait; more time to run lengthens it, though ever more slowly. A bond paying nothing along the way would have its wait equal its maturity — the limiting case this closed form deliberately excludes, since it divides weight among coupons that must exist.

The classic use is matching. An obligation due at a known horizon is best funded by a bond whose duration sits on that horizon, because at that alignment the two ways a rate move bites — reinvested coupons earning differently, and the bond being worth a different amount if sold — roughly offset. That discipline, immunisation, begins with the number computed here; what the number implies for price swings is the next page’s subject.

Start with the question in plain terms: on average, how long must this bond’s owner wait to be paid? Averaging the payment dates equally would be absurd — a small coupon and a large redemption would count the same. Averaging by payment size is better but still wrong, because a large amount arriving far away is not worth its face today. The only weighting that respects both size and distance is present value, and that is precisely the weighting Macaulay duration uses.

So each payment is discounted first, exactly as the pricing lesson discounted them, and the resulting present values are turned into shares of the price. Each payment’s timing is then counted in proportion to its share. Add the weighted timings and the result is one number in years: where the stream balances. The pricing page asked what the cash flows are worth; this page asks the same flows when — same discounting, different question.

For any bond that pays coupons, the answer lands strictly inside the remaining term. It has to: some of the money arrives before maturity, so the average of all arrivals cannot equal the last one. How far inside is the whole information content of the measure, and it is why two bonds maturing on the same day can carry very different durations.

Three inputs move the wait, and their directions are worth internalising because they explain most of what practitioners mean when they talk about duration. A richer coupon shifts weight toward the near payments and shortens the wait. A leaner coupon leaves the redemption dominant and stretches it toward maturity. Between two bonds maturing on the same day, the one paying more along the way is always the shorter instrument in the only sense that matters for rates.

A higher yield also shortens the wait, and the reason is quietly elegant: harsher discounting punishes the far payments more than the near ones, so weight migrates forward without a single cash flow changing. The same bond therefore carries a shorter duration in a high-yield world than in a low-yield one, which is one reason durations are recomputed rather than remembered.

More time to run lengthens the wait, but with diminishing returns — each additional year of maturity contributes less than the one before, because the payments it adds are the most heavily discounted ones in the stream. On a long coupon bond the duration flattens out well below the maturity, which is why very long bonds are less differentiated by maturity than their dates suggest.

The classic application is matching. An obligation falling due at a known horizon is conventionally funded with assets whose duration sits on that horizon, because at that alignment the two ways a rate move bites tend to offset: coupons get reinvested at different rates than expected, and the holding is worth a different amount if it has to be sold. Aligning the balance point with the obligation is the first step of immunisation, and it is why pension and insurance work is duration work.

The comparative use is simpler and more common. Given two bonds that maturity alone cannot separate — same date, different coupons — duration ranks them by the thing that actually determines rate exposure. It is the single most useful timestamp a bond has, and it travels: fund fact sheets quote an average duration for exactly this reason.

The number is also raw material. Divided once by the per-period growth factor it becomes modified duration, the working measure of price sensitivity — the third lesson in this arc. That conversion is why the timing question is worth answering carefully: an error here propagates straight into every exposure estimate downstream.

This page evaluates a closed-form identity for a level-coupon bond rather than summing the stream payment by payment, and the identity is algebraically equal to the direct weighted definition — the pack’s own vectors re-verify one against the other. What the identity requires is a genuine coupon stream to distribute weight across, which is why a coupon at or below nought is refused rather than answered: with no stream there is nothing to weight, and the zero-coupon case, whose wait simply equals its maturity, sits outside the form by construction. A yield at or below nought is refused on the same principle, since the discounting the weights depend on has to be built from a positive rate.

Two further limits belong on the page rather than in the footnotes. The measure assumes one flat yield discounts every payment; a sloped curve would weight the timeline differently at each point, and a duration computed against a flat yield is an approximation of the exposure it summarises. And it assumes the cash flows are what the contract says — callable, prepayable and defaultable instruments have flows that change when rates do, which breaks the premise that the timeline is fixed while the discounting varies.

Finally, duration is a snapshot. It drifts as time passes and as yields move, so it is a quantity to recompute, not a property to record. A duration quoted without the yield and schedule it was computed under is barely a number at all.

The engine below arrives pre-filled with the arc’s anchor: the same annual-coupon reference bond the pricing lesson carried at a discount to par, now asked when rather than how much. Read the result against the years remaining first — it should land close to the term but visibly inside it. Then run the two experiments that teach the measure: fatten the coupon and watch the wait shorten as weight migrates to the near payments, and stretch the maturity and watch the wait lengthen by less with each added year.

Read the output as a timestamp in years — the average moment this bond’s money arrives, weighted by present value. It is not a date, not a recommendation, and not valid apart from the yield and schedule you computed it under. Every figure is computed live by the verified engine; this page stores none.