Convexity
Convexity
Second-order rate sensitivity: the curvature in the price-yield relationship. Captures the fact that bond prices rise more on a yield drop than they fall on an equal yield rise. Always positive for option-free bonds.
When to use: Use alongside modified duration to refine the price-change approximation: ΔP/P ≈ −ModDur × Δy + 0.5 × Conv × (Δy)². The convexity term gets material once yield moves exceed ~50 bps.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Conv | Convexity | Second-order price sensitivity to yield, in years² | integer |
| F | Face Value | Par value paid at maturity | $ |
| CR | Coupon Rate | Annual coupon rate as a decimal (e.g. 0.05 for 5%) | % |
| y | Yield | Annual yield as a decimal; periodic yield is y/m | % |
| N | Years to Maturity | Years remaining until the bond matures | years |
| m | Coupons per Year | Number of coupons paid per year (e.g. 2 for semi-annual) | integer |
Real-Life Examples
Example 1: Par Bond, 5y, 5%, Semi-Annual
Same 5-year par bond as the duration examples.
Given
Step-by-Step
For a 100bp yield change, the convexity contribution is 0.5 × 22.61 × (0.01)² = 0.113% — a small but positive add-on that always works in the bondholder's favor for option-free bonds.
Frequently Asked Questions
Each positive discounted cash flow has a positive second derivative with respect to yield, so their sum does too. Embedded options can change cash flows and can produce negative effective convexity.
Settlement is on a coupon date with a positive whole number of remaining coupon periods. Between-coupon valuation requires fractional discount times and accrued interest.