Convexity

Duration is a fabulous liar. For tiny yield wiggles it’s spot-on, but the price–yield relationship is a curve, and duration is a straight line pretending to be that curve. The gap between the line and the curve has a name — convexity — and far from being an annoying error term, it’s a gift: convexity always works in the bondholder’s favour, making losses smaller and gains bigger than duration alone predicts. This lesson turns that gap into a number, folds it into a sharper price formula, and shows why two bonds with identical duration can be very different animals.

Convexity is the curve duration ignores

Analogy. Duration is like estimating a hill’s height by walking the tangent line at your feet — fine for the next step, but the hill curves away and your straight-line guess drifts off. Convexity measures how much the hill curves. For a bond, the price–yield curve bows upward (toward the origin), so the real price always sits above duration’s tangent line — and “above” means “better for you” on both sides.

Why it’s favourable. Look at the curve versus its tangent:

• Yields rise → the curve falls less steeply than the tangent → your actual loss is smaller than duration predicted.
• Yields fall → the curve rises more steeply than the tangent → your actual gain is bigger than duration predicted.

Either way the curve beats the line. That’s why traders say “convexity is good” — it’s a free improvement on duration’s estimate, and it’s the holder who pockets it.

Measuring convexity

The definition. Convexity is the second derivative of price with respect to yield, scaled by price — the rate at which duration itself changes as yields move:

Don’t be spooked by the formula — operationally, convexity weights each cash flow by roughly the square of its time, so far-out cash flows dominate even more than they do for duration. That’s why long-maturity, low-coupon bonds have the most convexity: their weight sits far out, where the $n(n+1)$ term explodes.

The drivers (same family as duration, amplified):

• Longer maturity → more convexity (far cash flows, squared time-weighting).
• Lower coupon → more convexity (less early weight pulling the curve straight).
• Lower yield → more convexity (far cash flows shrink less, so they dominate).

The full price approximation: duration + convexity

The idea. Take duration’s straight-line estimate and bolt on a curvature term. The second-order Taylor expansion of the price–yield relationship gives:

The convexity term has two beautiful properties: it’s multiplied by $(\Delta y)^2$, which is always positive regardless of the sign of $\Delta y$, so it always adds to the price change. That’s the mathematical statement of “convexity is good.” And because it’s squared, it’s negligible for small moves but kicks in hard for large ones — exactly where duration needs the help.

Worked example. A bond with $D_{\text{mod}} = 7$ and convexity $C = 100$. Yields rise 200 bp ($\Delta y = +0.02$):

• Duration term: $-7 \times 0.02 = -0.14 = -14.00\%$.
• Convexity term: $\tfrac{1}{2} \times 100 \times (0.02)^2 = \tfrac{1}{2}\times100\times0.0004 = +0.02 = +2.00\%$.
• Total estimate: $-14.00\% + 2.00\% = -12.00\%$.

Duration alone screamed a 14% loss; the convexity correction softens it to about 12%. On a yield drop of 200 bp, the duration term gives +14%, and the convexity term is still +2% (because $(\Delta y)^2$ is positive), for a total of +16%. Same magnitude move, but the gain (16%) beats the loss (12%) — the convexity asymmetry, quantified.

Two bonds, same duration, different convexity

The barbell vs. the bullet. Suppose you must build a portfolio with a target duration of, say, 7. You can do it two ways:

• A bullet — a single bond with duration 7.
• A barbell — a mix of a short bond (duration 2) and a long bond (duration 15), weighted to average duration 7.

Both have the same duration, so they react identically to small parallel yield moves. But the barbell has more convexity — its cash flows are spread to the extremes, and convexity rewards dispersion. For large or non-parallel yield moves, the barbell outperforms. The catch: the market knows this, so the barbell typically offers a lower yield — you pay for the convexity in give-up. There’s no free convexity lunch.

Negative convexity: when the gift is taken away

Analogy. Most bonds are like a trampoline that bounces in your favour. But a callable bond (or a mortgage-backed security) is a trampoline with a trapdoor: when yields fall enough, the issuer calls the bond (refinances) and snatches away your price upside. So instead of the price rising faster as yields fall, it flattens or bends down — the curve goes concave there. This is negative convexity, and it’s bad for the holder: you keep the downside (yields up → price down) but lose the upside (yields down → price capped).

Why it happens. The issuer’s call option is short to you. When rates drop and refinancing becomes attractive, that option moves in-the-money — for the issuer. The bond’s price can’t climb above the call price, so the price–yield curve caps out and bends the wrong way. Mortgage prepayments do the same: homeowners refinance when rates fall, handing you your principal back exactly when you’d least like it (to reinvest at the new, lower rates).

When to use it

Lead with duration for everyday rate-risk; it’s the headline number and it’s accurate for the small moves that dominate daily P&L. Add convexity when you face large moves (stress scenarios, rate shocks), when you hold long-dated bonds whose curvature is material, or when you’re choosing between two same-duration structures and convexity is the tiebreaker. And always flag negative convexity — callables, MBS — because there the usual “convexity is good” intuition flips, and a naïve duration hedge can blow up precisely when rates rally.

Putting it together

Convexity is the curvature duration ignores. The true price–yield curve bows above duration’s tangent, so duration overstates losses and understates gains — and convexity quantifies that always-favourable gap via the $\tfrac{1}{2}C(\Delta y)^2$ term, which is positive for any move and grows with its square. Long-dated, low-coupon bonds carry the most convexity; barbells out-convex same-duration bullets (at the cost of yield); and callables/MBS flip into negative convexity, where the issuer’s option caps your upside. For small moves, pocket convexity as a bonus; for shocks and long bonds, it’s essential.

Next — bootstrapping the yield curve — we stop treating “the yield” as one number and build the whole curve: stripping coupon bonds into pure spot (zero) rates, one maturity at a time, by no-arbitrage.