Convexity

Chapter 6 gave duration and named its limit: "the relationship is slightly curved, so for a big rate move duration overstates the fall and understates the rise. The curvature is called convexity and it works mildly in your favour." This chapter measures it.

What duration leaves out

Duration gives a straight-line estimate of the price change:

$\frac{\Delta P}{P} \approx -D_{\text{mod}} \times \Delta y$

A straight line through a curve is exact at one point and wrong everywhere else. Convexity is the second-order term that corrects it:

$\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2$

Three things to read off that formula before any numbers.

The convexity term is squared, so it is tiny for small moves and grows fast for large ones. That is chapter 6's "for moves of a percentage point or so, ignore it."

It is positive regardless of direction, because $(\Delta y)^2$ is positive whether yields rise or fall. So convexity adds to the price in both cases — it reduces the loss when rates rise and increases the gain when they fall.

For an ordinary bond, $C$ is positive. Not all instruments have positive convexity, and the ones that do not are the ones to be careful with.

How wrong duration gets

Take a real instrument: a ten-year government bond priced at par, at the 10-year par yield the RBI reports for the week ended 2 October 2026 — 7.23%. Its modified duration is 7.03 and its convexity 63.32.

Yield move	Exact price	Duration only	Duration + convexity	Duration error	Remaining error
−300 bp	124.2571	121.0985	123.9479	+3.1587	+0.3092
−200 bp	115.4211	114.0656	115.3321	+1.3555	+0.0891
−100 bp	107.3603	107.0328	107.3494	+0.3274	+0.0108
−25 bp	101.7782	101.7582	101.7780	+0.0200	+0.0002
+25 bp	98.2614	98.2418	98.2616	+0.0196	−0.0002
+100 bp	93.2735	92.9672	93.2838	+0.3063	−0.0103
+200 bp	87.1207	85.9344	87.2008	+1.1864	−0.0800
+300 bp	81.4876	78.9015	81.7510	+2.5861	−0.2633

At 25 basis points, duration is wrong by two paise on a hundred rupees. Chapter 6's advice to ignore it holds completely.

At 300 basis points, duration is wrong by ₹2.59 to ₹3.16 on a hundred — two to three percent of the bond. That is no longer ignorable, and adding one term reduces it to about a quarter of a rupee.

And notice the sign of the duration error column: it is positive in every row. Duration understates the price whichever way yields move. That is the single most useful fact about convexity, and the next section is why.

Why gains exceed losses

Take the same bond and move yields 100 basis points each way:

	Price change
Yields rise 100 bp	−6.7265
Yields fall 100 bp	+7.3603
Difference	+0.6338

Duration says these should be equal and opposite. They are not: the gain is 63 paise larger than the loss.

The mechanism is in the discounting. Price is a sum of cash flows discounted at $(1+y)^t$. As $y$ falls, each discount factor rises, and the long-dated ones rise proportionally more — so the price accelerates upward. As $y$ rises, the same factors shrink, and shrinking has a floor: a bond's price cannot go below zero however high yields rise, while it can rise without an obvious ceiling as yields approach zero.

So the price-yield relationship must bend, and it bends in the holder's favour. Convexity is not a quirk of the formula; it is a consequence of discounting being a reciprocal.

And that is why convexity is worth paying for. Two bonds with the same yield and the same duration are not equivalent: the one with more convexity does better whichever way rates move. In a competitive market that difference is priced, which means the more convex bond usually offers a slightly lower yield — you are buying optionality, and it is not free.

Which bonds have the most

Convexity grows with maturity, and faster than duration does:

Maturity	Duration	Convexity
2 years	1.898	4.341
5 years	4.284	20.779
10 years	7.287	63.321
20 years	10.869	161.169
30 years	12.629	242.061

(All at a 7.23% coupon priced at par.)

From 10 years to 30, duration rises by 73% and convexity by 282%. Long bonds are not simply more sensitive to rates; they are more curved, so their behaviour in a large move departs much further from the linear estimate.

And for the same maturity, lower coupons mean more convexity:

10-year instrument	Duration	Convexity
Zero-coupon	10.000	97.801
7.23% coupon	7.287	63.321

A zero-coupon bond has the most of both, because all of its value sits at the far end. Chapter 6's point that coupons pull duration below maturity applies to convexity with more force.

When convexity turns against you

Positive convexity is a property of a bond with fixed cash flows. Instruments whose cash flows change when rates change can have negative convexity, and then the asymmetry reverses — the loss on a rise exceeds the gain on a fall.

The classic case is anything callable or prepayable. If the issuer can repay early when rates fall, the price stops rising, because the holder knows the bond will be taken away at the call price. The upside is capped and the downside is not.

The Option pricing subject names the position precisely: the holder of a callable bond is short an option. Chapter 9 of that subject, on selling options and tail risk, describes the payoff shape, and chapter 9's gamma discussion describes the curvature. A callable bond is an ordinary bond with a sold call attached, and the negative convexity is that sold option's gamma.

For a retail holder in India this matters mainly for bonds with issuer call options, which appear in corporate and some perpetual structures. A yield that looks attractive relative to a plain bond of the same duration is often compensation for a sold option — and chapter 7's discussion of credit risk is not the only thing to check in that comparison.

Working the problem

Par bond, modified duration 7.03, convexity 63.32, yields rise 200 basis points ($\Delta y = +0.02$).

Step 1 — the duration term.

$-D_{\text{mod}} \times \Delta y = -7.0328 \times 0.02 = -0.140656$

So the estimate is $100 \times (1 - 0.140656) = ₹85.93$.

Step 2 — the convexity term.

$\tfrac{1}{2} \times C \times (\Delta y)^2 = 0.5 \times 63.3206 \times 0.0004 = +0.012664$

Step 3 — combine.

$100 \times \left(1 - 0.140656 + 0.012664\right) = ₹87.20$

Step 4 — compare with the exact price of ₹87.12.

Method	Price	Error
Duration only	₹85.93	−₹1.19
Duration + convexity	₹87.20	+₹0.08
Exact	₹87.12	—

Adding one term cut the error by about 93%, from ₹1.19 to ₹0.08 on a hundred rupees.

Three observations worth taking away.

Duration alone was pessimistic by 1.4%. On a ₹1 crore holding that is ₹1.4 lakh of error, in the direction of overstating the damage. An investor who panics at the duration estimate is panicking at a number that is too bad.

The convexity correction is small in absolute terms — 1.27 rupees on 100, against duration's 14.07. Duration is still doing 92% of the work, which is why chapter 6 can stand alone for most purposes and why convexity is an advanced chapter rather than a basic one.

And the remaining 8 paise is the third-order term, which nobody computes. The series continues; two terms is where the accuracy stops being the binding constraint on a decision.

When I would actually bother with convexity. When the move being considered is large — a stress test rather than a forecast; when the maturity is long, since the 30-year bond's convexity is nearly four times the 10-year's; and when comparing two bonds of similar duration, where convexity is the tie-breaker and the yield difference between them is the price being charged for it.

The point

Duration's straight line understates a bond's price whichever way yields move, and the correction is $+\tfrac{1}{2}C(\Delta y)^2$ — squared, so negligible at 25 basis points and worth ₹1.19 on a hundred at 200. For a 10-year par bond at the 7.23% G-Sec yield, a 100 basis point fall gains 63 paise more than a 100 basis point rise loses, because discounting is a reciprocal: prices accelerate upward and decelerate downward. Convexity rises sharply with maturity and as coupons fall, so a 30-year bond has nearly four times a 10-year's and a zero-coupon has the most. It is a desirable property, so it is priced — and it reverses to negative convexity in callable bonds, where the holder is short an option.