11.3 Convexity and the second-order approximation

Motivation

For large moves, the duration estimate is biased. In the 2022 sell-off and the 2026 sell-off, 10-year yields moved more than 150 bp. At that size the second-order term matters.

11.3.1The second-order expansion#

Definition

C = \frac{1}{P}\frac{d^2P}{dy^2}. \tag{11.7}

Proposition 11.4

(a) $C = \frac{1}{P}\sum_{k} t_k\left(t_k+\tfrac1m\right) CF_k\,(1+y/m)^{-mt_k-2}. \tag{11.8}$ (b) For a yield change $\Delta y$ ,

\frac{\Delta P}{P} = -D_{\text{mod}}\,\Delta y + \tfrac12 C\,(\Delta y)^2 + O\big((\Delta y)^3\big). \tag{11.9}

(c) The first-order approximation always understates the price after the move, whether yields rise or fall. Its error is $\tfrac12 C(\Delta y)^2 + O(\Delta y^3)$ .

Proof

(a) Differentiate $dP/dy = -\sum_k t_k CF_k(1+y/m)^{-mt_k-1}$ once more:

\frac{d^2P}{dy^2} = \sum_k t_k\,CF_k\,(mt_k+1)\frac1m(1+y/m)^{-mt_k-2} = \sum_k t_k(t_k+\tfrac1m)\,CF_k\,(1+y/m)^{-mt_k-2}.

(b) $P$ is analytic in $y$ on $(-m,\infty)$ . Taylor's theorem with Lagrange remainder gives $P(y+\Delta) = P + P'\Delta + \tfrac12 P''\Delta^2 + \tfrac16 P'''(\xi)\Delta^3$ . Divide by $P$ and use (11.5) and (11.7). (c) The first-order estimate is $P + P'\Delta$ , the tangent line. A strictly convex function lies above its tangent, so $P(y+\Delta) > P + P'\Delta$ for every $\Delta\neq0$ .∎

For a zero-coupon bond, (11.8) reduces to $C = T(T+1/m)/(1+y/m)^2$ . Convexity grows roughly with the square of maturity, while duration grows linearly.

11.3.2Accuracy of the approximations#

Worked example 11.3

The table compares repricing with the two approximations for $\pm100$ bp and $+50$ bp moves. The numbers come from m11_bonds.py, per ₹100 face.

Bond	$P$	$D_{\text{mod}}$	$C$	$\Delta y$	True $\Delta P$	1st order	2nd order
6.50% 10y at par	100.00	7.270	66.49	+100 bp	−6.948	−7.270	−6.937
				−100 bp	+7.614	+7.270	+7.602
				+50 bp	−3.553	−3.635	−3.552
7.00% 30y at par	100.00	12.472	251.23	+100 bp	−11.312	−12.472	−11.216
US-style 4.75% 30y at 5.37%	90.81	15.197	344.08	+100 bp	−12.365	−13.800	−12.238

Three observations, all tested in test_m11_bonds.py:

The second-order error is an order of magnitude smaller.
Halving $\Delta y$ cuts the first-order error by about 4 (it is $O(\Delta y^2)$ ), and the second-order error by about 8 (it is $O(\Delta y^3)$ ).
Convexity is worth more on long bonds. For the 30-year, the duration estimate overstates a 100 bp loss by ₹1.16 per ₹100.

Practitioner notes

Positive convexity has value: a long position gains more from a fall in yields than it loses from an equal rise. Markets price this. Holding volatility fixed, a more convex bond of the same duration trades at a slightly lower yield.
Callable bonds and mortgage-backed securities have negative convexity. Their price is capped because the issuer refinances when rates fall. US mortgage hedging flows amplified the 2022 and 2026 Treasury sell-offs for this reason. That effect is well documented in general but was not measured here.
Options are the cleanest source of convexity. M12 · 12.6 uses that idea for geopolitical hedges.