Convexity

Duration gave the desk a fast, honest answer for a small rate move. But 2008, 2020, and plenty of ordinary Tuesdays have seen yields move by 200 or 300 basis points, not 25 — and a straight-line estimate, stretched that far, starts telling a visibly wrong story. The bond's true price-vs-yield relationship is a curve, not a line, and the gap between the line and the curve has a name, a formula, and — for anyone holding the bond — a happy sign.

The second-order correction

Duration is the first derivative of price with respect to yield, rescaled. Convexity is the second:

Convexity: \displaystyle C = \frac{1}{B}\frac{d^2B}{dy^2} = \frac{1}{B}\sum_i t_i^2\,C_i\,e^{-y t_i} (continuous compounding) — a present-value-weighted average of the square of each payment time.
The full second-order Taylor expansion of the price-yield relationship around today's yield y_0 is \displaystyle \frac{\Delta B}{B} \approx -D\,\Delta y + \tfrac12\,C\,(\Delta y)^2.
The duration term flips sign with \Delta y (rates up, price down, and vice versa); the convexity term does not — it is +\tfrac12 C(\Delta y)^2 \geq 0 regardless of which way rates move, since it is squared.

That last bullet is the whole point of this lesson: convexity is a correction that always adds to the price, whichever direction yields move. It makes the true price curve sit above the straight tangent line drawn by duration alone, on both sides of today's yield.

Worked example: a 300 bp move puts duration alone to the test

Same bond as before — the 2-year, 6%-semiannual-coupon bond priced at B_0 = \$98.881 with yield y_0 = 6.5\%, Macaulay/modified duration D = 1.914. Its convexity, built from the same cash-flow table with an extra t_i^2 column:

t_i	PV_i	t_i^2	t_i^2\,PV_i
0.5	2.904	0.25	0.726
1.0	2.811	1.00	2.811
1.5	2.721	2.25	6.122
2.0	90.444	4.00	361.776
Sum			371.435

C = \frac{371.435}{98.881} \approx 3.756.

Now push yields up by a full \Delta y = +3\% (300 basis points) — big enough that duration alone should start to strain — and compare three answers: duration only, duration plus convexity, and the honest reprice from first principles (summing every discounted cash flow at the new yield 9.5\%).

Method	\Delta B/B	New price
Duration only: -D\Delta y	−5.74%	$93.21
Duration + convexity	−5.57%	$93.37
Actual reprice	−5.58%	$93.37

Duration alone misses by about 16 cents on a $100 bond — small here, but this is a plain vanilla 2-year note; on a longer, more convex bond the same-size miss can be many times larger. Duration-plus-convexity, using only two numbers computed once at today's yield, lands within a penny of the full reprice. That is the entire economic value of convexity: a cheap correction that rescues a first-order estimate from a large move.

The chart makes the same point visually: the true price curve is convex (it curves upward), the duration-only line is straight and tangent to it only at today's yield, and the duration-plus-convexity parabola hugs the true curve far longer before it, too, eventually drifts away.

Compare two equal-and-opposite rate moves of size \Delta y: a fall and a rise. Duration alone predicts symmetric effects — the price gain from the fall exactly cancels the price loss from the rise. But the actual, convex price curve does not behave that way: because it bends upward, the gain from a fall in yields is genuinely larger in dollar terms than the loss from an equal-sized rise. This is a direct instance of Jensen's inequality — for a convex function, the average of the function at two points exceeds the function at the average point — the exact same geometric fact that makes an option's payoff valuable purely from being convex, with no view on direction required.

The practical upshot: all else equal (same price, same duration, same yield), a bond holder always prefers more convexity — it is pure upside with no matching downside. Bonds with embedded call options (the issuer can redeem early) have the opposite property, negative convexity, over some yield range — which is precisely why they trade cheap relative to an otherwise-identical option-free bond, and why option-embedded fixed income needs its own, more careful treatment later in this course.

Two traps:

Trusting a linear estimate too far. The worked example above is the whole lesson: for a 25 bp wiggle, duration alone is essentially exact; for a 300 bp shock, it is off by a small but very real amount that compounds badly across a large book. Any risk report that quotes duration-only price sensitivity for a stress scenario several hundred basis points wide is quietly understating gains and overstating losses (or vice versa) — always in the direction that makes the linear line look worse than the truth on both sides.
Subtracting the convexity term. A common algebra slip is to write \Delta B/B \approx -D\Delta y - \tfrac12 C(\Delta y)^2, treating the second term like it must flip sign along with the first. It must not: the correct formula adds the convexity term regardless of the direction of the yield move, because (\Delta y)^2 is never negative.