Modified duration from Macaulay duration

How hard a yield move hits the price

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.

When this calculation is used

• Estimating the percentage price impact of an expected or feared move in yields on a bond position.
• Comparing the interest-rate exposure of two bonds whose maturities or coupons make them otherwise hard to rank.
• Sizing a hedge — choosing how much offsetting exposure cancels the sensitivity of a holding.
• Turning a Macaulay duration, computed on its own page, into the working risk multiplier a decision needs.
• Judging when a move is large enough that the linear estimate needs its convexity correction.

Worked example

The pack’s declared reference chain: the Macaulay duration of the cluster’s annual-coupon reference bond, carried here with the same yield and schedule it was computed under — the hand-off the two pages are designed to make.

The output is the modified duration: a figure just below the Macaulay input, in years. Read it as the seesaw’s tilt — the approximate percentage the reference bond’s price gives up for a unit rise in its yield, and gains for a unit fall.

Raise the yield input and watch the gap between the two durations widen; compress it toward nothing and watch them nearly meet. Then recall the estimate is a tangent line: generous to losses and stingy to gains on large moves, which is convexity’s cue to enter. Every figure is computed by the certified engine after the page mounts — the page stores no answers of its own.

What each input represents

Assumptions and limits

• The Macaulay input and the yield-and-schedule pair describe the same bond under the same compounding convention.
• The sensitivity is first-order: accurate for small yield moves, progressively flattered or punished by large ones as the price curve bows away from its tangent.
• The yield curve moves in parallel — one yield shifts, and with it the whole discounting of the bond; twists and butterflies are outside the measure.
• The bond’s cash flows do not themselves change when yields move — callable and prepayable instruments break that premise and need their own machinery.

What the guards protect against

• A zero or negative Macaulay duration is refused — a weighted wait cannot be negative, so such an input describes no bond and the conversion declines it.
• A zero or negative yield is refused: the per-period growth factor the division depends on must be built from a positive rate, so the degenerate case is declined rather than answered wrongly.

Provenance

Modified duration relation

Macaulay duration divided by one plus the per-period yield — the annual yield split across the compounding schedule — giving the first-order percentage price sensitivity to a move in yield.

Educational reference, not investment advice. The signed pack carries its own citation; the page reports the verification state of the release it mounted rather than asserting one.