Bond
What is bond convexity?
Convexity measures curvature in the bond price–yield relation, or second-order sensitivity to yield.
Definition
Convexity measures curvature in the bond price–yield relation, or second-order sensitivity to yield.
Intuition
Bond price is not linear in yield; convexity improves a duration-only estimate for modest rate moves.
Formula
Convexity sums the discounted second-order yield sensitivity of each cash flow, divided by bond price.
Model assumptions
The model assumes fixed coupons, regular periods, valuation on a coupon date, and a parallel YTM shift. CFₜ is period-t cash flow, m payments per year, and P the full price. Accrued interest, credit risk, and embedded options are excluded.
Example
- Enter face 1,000, coupon 5%, YTM 6%, five years, and semiannual coupons.
- Each coupon is 25 and periodic yield 3%; discount ten coupons plus principal to find price.
- Weight discounted cash flows for modified duration and positive convexity; approximate a small price change with −Dmod×Δy + ½×C×(Δy)².
Common mistakes
Do not treat the second-order approximation as an exact quote; large moves, credit changes, and call features can invalidate it.
Frequently asked questions
Why is duration alone insufficient?
Duration is a linear approximation; curvature matters more for larger yield changes.
What does positive convexity mean?
For a plain fixed-cash-flow bond, price is convex in yield; an equal yield decline usually raises price more than a rise lowers it.
Does this apply to callable bonds?
Not directly. Call options change the cash-flow schedule and require an option-adjusted model.