Price-Change Approximation
Price-Change Approximation
Second-order Taylor approximation of bond price change: −ModDur × Δy captures the linear (duration) effect, and 0.5 × Conv × Δy² adds the curvature (convexity) refinement. Gives accurate price-change estimates for moderate yield shifts without recomputing the full bond price.
When to use: Use to estimate ΔP/P for any yield scenario without re-running the full bond pricer — fast and accurate enough for risk reports, scenario analysis, and back-of-the-envelope hedging.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| DeltaPpct | Price Change % | Approximate percentage price change | % |
| ModDur | Modified Duration | Price sensitivity coefficient: −(1/P)(dP/dy), in years | years |
| Conv | Convexity | Second-order price sensitivity to yield, in years² | integer |
| DeltaY | Yield Change | Change in yield as a decimal (e.g. 0.0050 for 50 bps) | % |
Real-Life Examples
Example 1: +100bp Yield Shock
Par bond: ModDur = 4.376, Convexity = 22.61, yield rises by 100bp (Δy = 0.01).
Given
Step-by-Step
Approximate price drop of 4.26% — close to the −4.38% pure-duration estimate, with 11 bps of convexity offsetting the linear approximation. The actual full-pricer answer is approximately −4.27%, so the second-order approximation is within a basis point.
Example 2: −100bp Yield Shock
Same bond, but yield falls 100bp (Δy = −0.01).
Given
Step-by-Step
Asymmetry: the bond gains 4.49% on a 100bp drop but only loses 4.26% on a 100bp rise. That asymmetry is convexity in action — and it always favors the bondholder (for option-free bonds).
Frequently Asked Questions
It is a second-order local approximation. Error depends on maturity, yield, cash-flow structure, embedded options and the size of the yield change. There is no universal five-basis-point error bound for a two-percentage-point yield move.
Reprice its actual cash flows for a precise scenario or a large yield change. Embedded options may also change the cash flows themselves.