Use modified duration to estimate how a bond's price will change
The idea
Modified duration measures how much a bond's price moves for a small change in yield: the approximate percentage price change is minus modified duration times the change in yield. It rises with maturity and falls with coupon, because a bond that returns more of its value sooner has less exposure to a change in the rate those distant cash flows are discounted at. The estimate is a straight line drawn against a curved relationship, so it is accurate only for small moves; convexity is the correction for the curvature, and for an option-free bond it works in the holder's favour — prices rise more than duration predicts and fall less.
Formulas
Price change estimated from modified duration
%Δprice ≈ −ModDur × Δyield
- %Δprice
- the percentage change in the bond's full price
- ModDur
- modified duration
- Δyield
- the change in yield, as a decimal — 25 bps is 0.0025
A straight-line estimate of a curved relationship, so it is close for small moves and increasingly wrong for large ones. It understates the price both ways: the true curve lies above the line, so prices rise more than this predicts and fall less.
A practice question
The approximate percentage price change of a bond is minus modified duration multiplied by the change in:
- yield — correct
- couponWhy not: The coupon is fixed in the bond's terms and does not change. It is an input to duration, not the variable duration multiplies.
- maturityWhy not: Maturity shortens by itself with the passage of time, which is a separate effect from a yield move.
The minus sign carries the inverse relationship: yields up, prices down.
A harder one
A bond has modified duration of 6. A 50 basis point rise in yield changes its price by approximately:
- +3.0%Why not: The right magnitude with the wrong sign. Yields rising means prices falling.
- -0.5%Why not: The yield change itself, unmultiplied by duration. Duration of 6 magnifies a 0.5% move into roughly 3%.
- -3.0% — correct
Six multiplied by 0.005 is 3%, and the sign is negative because yields and prices move in opposite directions.
The hardest tier
A duration-based estimate is least accurate for:
- a small yield changeWhy not: The most accurate case. Over a small move the tangent barely departs from the curve.
- a zero-coupon bondWhy not: Its convexity is high, so the error is larger for a given move — but the question asks what makes the estimate least accurate, and that is the size of the move.
- a large yield change — correct
The straight-line approximation drifts further from the curve the further the yield moves.
In the app: 21 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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