Modified duration from Macaulay duration

Modified duration answers the trader’s question rather than the actuary’s: not when the money arrives, but what happens to the position if yields shift right now. Read it as a multiplier — the approximate percentage decline in price per unit rise in yield, and symmetrically the approximate gain when yields fall. A position with twice the modified duration of another carries roughly twice the exposure to the same rate move.

The conversion from Macaulay duration is a single division by the per-period growth factor, built from the yield and the compounding schedule. The result always sits slightly below the Macaulay figure it came from, and the two drift apart as yields rise — at negligible yields they nearly coincide, which is why the distinction is easy to forget and occasionally expensive to have forgotten.

It is a slope, and slopes are local. Modified duration is the first-order sensitivity, exact only for an infinitesimal move; a real price curve bows away from its tangent line, so the estimate overstates losses and understates gains as moves grow. That bow is convexity, and the pack ships a companion calculator that layers the correction on when the linear estimate stops being enough.

The number scales portfolio thinking. A holding’s sensitivity is its modified duration weighted by its value; a portfolio’s is the value-weighted blend of its holdings; a hedge is a position engineered to cancel that blend. All of it — risk budgeting, hedge sizing, the currency loss per basis point a desk quotes — starts from the multiplier this page computes.

Macaulay duration answers the actuary’s question: when, on average, does the money arrive. Modified duration answers the trader’s: what happens to this position if yields shift right now. Remarkably, the second is the first divided once by the per-period growth factor built from the yield and the compounding schedule. The arithmetic is trivial; the reinterpretation is not.

The reason the same number does both jobs is that discounting is exponential in time. How sensitive a present value is to the discount rate depends on how long it is being discounted for — so a stream whose weight sits far out is inherently more rate-sensitive than one whose weight sits near. The weighted-average wait already encodes exactly that, and the division rescales it from a time into a percentage per unit of yield.

The result always sits slightly below the Macaulay figure it came from, and the two drift further apart as yields rise, because the growth factor doing the dividing grows with the yield. At very low yields the two nearly coincide — which is precisely why the distinction is easy to forget and occasionally expensive to have forgotten.

The pricing lesson had you sweep the yield and watch the price trace a bowed curve. Modified duration is the slope of that curve at the yield you are standing on — the tangent line, drawn at one point. For a small move the tangent and the curve are indistinguishable and the estimate is excellent. For a large move they part company, and they part company in a consistent direction.

Because the curve bows away from its tangent, the linear estimate errs in the holder’s favour on both sides: when yields rise a lot the price falls by less than duration predicted, and when they fall a lot it rises by more. The estimate overstates the loss and understates the gain. That asymmetry is convexity, and it is the reason a duration-only view of a large rate move is systematically wrong rather than randomly wrong.

The practical rule follows from the geometry. Use modified duration for the moves that actually happen day to day, treat it as an approximation for the moves that make headlines, and reach for a convexity correction when the size of the move starts to matter more than the direction. The pack ships a companion convexity calculator for exactly that layer.

Modified duration is quoted as a percentage per unit of yield, which makes it a property of the instrument rather than of the holding. Multiply it by the value actually held and it becomes an amount of money at stake per unit of rate move — the form a risk report uses. Scale the unit down to a basis point and you have the desk’s working quantity: currency lost per basis point of yield rise.

The measure blends. A portfolio’s modified duration is the value-weighted average of its holdings’ durations, which is what lets a fund quote one figure for a hundred bonds and what makes hedging arithmetic tractable: a hedge is a position engineered so that the blend of the whole book lands where it is meant to. All of that scaling starts from the multiplier this page computes for a single instrument.

The blend inherits every assumption underneath it. It presumes each component was computed on the same compounding convention, and that the yields of every holding move together — the parallel-shift premise. Real curves twist, and a book that is duration-neutral against a parallel move can still be exposed to a change in the curve’s shape. Recognising that limit is part of using the number honestly.

This calculation takes a Macaulay duration as its main input, and the commonest error in using it has nothing to do with the arithmetic: it is feeding in a figure computed under one compounding convention and converting it under another. A duration derived from a semiannual schedule and divided by a growth factor built on an annual one yields a number that is wrong by a factor nobody notices, because it is wrong by a plausible amount. The yield and frequency entered here must be the same pair the Macaulay figure was computed under — the workspace assumes an annual schedule when unstated, declared as an illustration rather than a market fact.

The guards enforce the rest. A Macaulay duration at or below nought is refused, because a weighted wait cannot be negative and such an input describes no bond; a yield at or below nought is refused, because the growth factor the division depends on has to be built from a positive rate. In both cases the engine declines rather than returning a figure the method cannot stand behind.

Two structural premises complete the picture. The bond’s cash flows are assumed not to change when yields move — callable, prepayable and defaultable instruments break that assumption exactly when it matters, and need effective-duration machinery this page does not carry. And the measure describes price sensitivity only; it says nothing about whether a rate move is likely, whether a position should be held, or what any of it means for a particular investor. This is an educational reference, not investment advice.

The engine below arrives pre-filled with the arc’s hand-off: the Macaulay duration of the same annual-coupon reference bond the previous lesson measured, carried here with the yield and schedule it was computed under. Read the output against that input first — it sits just below it. Then raise the yield and watch the gap between the two durations widen; compress the yield toward nothing and watch them nearly meet.

Read the output as the seesaw’s tilt: the approximate percentage the price gives up for a unit rise in yield, and gains for a unit fall. It is a tangent-line estimate, reliable for small moves and generous to losses on large ones. Every figure is computed live by the verified engine; this page stores none.