Bond Duration and Convexity

A bond is a promise of fixed future cash flows, and its price is just the present value of those payments. When market interest rates rise, those fixed payments are discounted harder, so the bond is worth less. Rates fall, the bond is worth more. Everyone knows the direction. Duration and convexity answer the quantitative question: given a move in yields, exactly how much does the price change?

Think of it as a two-term Taylor approximation. Duration is the slope of the price-yield relationship — a first-order, straight-line estimate of the price move. But the relationship isn't a straight line; it's a curve that bows upward. Convexity measures that curvature and corrects duration's estimate, especially for larger yield moves. If you've met the option Greeks, duration is the bond's delta and convexity is its gamma.

The two-term formula

The fractional change in a bond's price for a change in yield $\Delta y$ is:

$$\frac{\Delta P}{P} \approx -D\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2 .$$

The first term uses modified duration $D$: price moves opposite to yield (hence the minus sign), and $D$ is roughly how many years' worth of sensitivity the bond has. The second term uses convexity $C$: it's always positive because the price-yield curve bends upward, so it adds to your return whichever way yields move — softening losses when rates rise and boosting gains when rates fall. For small moves the duration term dominates; for big moves the convexity correction earns its keep.

Worked example

Take a bond with modified duration $D = 7$ and convexity $C = 60$. Suppose yields jump by a full percentage point, $\Delta y = +0.01$ (100 basis points).

• Duration term: $-D\,\Delta y = -7 \times 0.01 = -0.07$, i.e. duration alone predicts a $7\%$ price drop.
• Convexity correction: $+\tfrac12 C (\Delta y)^2 = \tfrac12 \times 60 \times (0.01)^2 = \tfrac12 \times 60 \times 0.0001 = +0.003$, i.e. $+0.3\%$.
• Total: $-0.07 + 0.003 = -0.067$, about a $6.7\%$ loss — a bit less than the straight-line estimate.

Now run it the other way: yields fall by a point, $\Delta y = -0.01$. The duration term is $+7\%$, and the convexity term is still positive — $\tfrac12 \times 60 \times (0.01)^2 = +0.3\%$ — because it depends on $(\Delta y)^2$. So the gain is $+7.3\%$. That asymmetry is the beauty of convexity: your upside ($+7.3\%$) beats your downside ($-6.7\%$) for the same size move. High convexity is a feature investors will pay for, and it's why long-dated and zero-coupon bonds, which are very convex, behave so differently from short ones.

Where it misleads

• Duration is only a tangent line. For small yield moves the straight-line estimate is fine, but for large moves it drifts off the true curve. That's the entire reason convexity exists — always add the second term when yields move a lot.
• It assumes a parallel shift. Both numbers assume every yield across the curve moves by the same $\Delta y$. In reality the yield curve twists and steepens; the short end and long end rarely move together, so a single duration can mislead about a curve reshaping.
• Embedded options break it. Callable bonds and mortgages can have negative convexity: when rates fall, the borrower refinances and your bond gets called away, so the price stops rising. There the convexity term flips sign and works against you.
• Duration is not a maturity. A common beginner error is to read "duration 7" as "matures in 7 years." It's a sensitivity, measured in years only by analogy; a high-coupon 10-year bond has a shorter duration than a zero-coupon 10-year bond.

Duration and convexity are the fixed-income analogues of the The Option Greeks, and they're the raw material of interest-rate hedging: match the duration of your assets and liabilities to immunize against rate moves, then watch convexity for the second-order effects. How rates differ across maturities — the input that determines these sensitivities in the first place — is the subject of the yield curve, and swapping fixed for floating exposure is the job of Interest Rate Swaps.