Duration & Convexity

When a central bank raises interest rates, bond prices fall. When rates fall, prices rise. That much is common knowledge. What is less obvious is how much they move — and why the relationship isn't a straight line.

Two numbers capture that sensitivity. Duration is the first-order answer: for every 1% rise in yield, a bond with duration $D$ years loses approximately $D$% of its value. A 10-year Treasury with a modified duration of 8 will drop roughly 8% if yields jump by a full percentage point.

But "approximately" hides something important. The true price-yield relationship is a curve, not a line. Convexity measures that curvature — the second-order term. Because the curve bows toward the investor, convexity is almost always a gift: it means the bond falls less when rates rise than duration alone predicts, and gains more when rates fall.

Together, duration and convexity are the two derivatives of the price-yield function — the same Taylor-series logic that appears in dynamic programming and numerical optimization, applied to the oldest financial instrument in the world.

Adjust the bond parameters and drag the yield slider. The chart shows three lines: the exact price (blue curve), the duration-only linear approximation (dashed), and the duration + convexity quadratic correction (dotted).

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Notice that the duration line is tangent to the price curve at the current yield. As you move the yield away, the gap grows — that gap is exactly what convexity corrects. The correction is always positive: convexity bends the approximation back toward the true price, in both directions.

The price of a plain coupon bond paying coupon $C$ each period, with face value $F$, $n$ periods to maturity, and yield $y$ per period is:

$$P(y) = \sum_{t=1}^{n} \frac{C}{(1+y)^{t}} + \frac{F}{(1+y)^{n}}$$

Macaulay duration is the weighted average time to receive each cash flow, where the weight is the present-value fraction of that flow:

$$D_{Mac} = \frac{1}{P} \sum_{t=1}^{n} t \cdot \frac{CF_t}{(1+y)^{t}}$$

Modified duration converts this into a price-sensitivity measure — it is literally $-\frac{dP}{dy} / P$:

$$D_{mod} = \frac{D_{Mac}}{1+y}$$

So the first-order approximation is $\Delta P \approx -D_{mod} \cdot P \cdot \Delta y$.

Convexity is the normalized second derivative $\frac{1}{P} \frac{d^2P}{dy^2}$:

$$C_{vx} = \frac{1}{P(1+y)^{2}} \sum_{t=1}^{n} t(t+1) \cdot \frac{CF_t}{(1+y)^{t}}$$

The full second-order approximation (the Taylor expansion to two terms) is:

$$\Delta P \approx -D_{mod} \cdot P \cdot \Delta y + \tfrac{1}{2} \cdot C_{vx} \cdot P \cdot (\Delta y)^{2}$$

Three facts that fall out of this algebra:

• Zero-coupon bonds have Macaulay duration exactly equal to their maturity — every cent arrives at the end, so the weighted average time is the time.
• Convexity is always positive for plain bonds (every cash-flow weight is positive and $t(t+1) > 0$), which is why bondholders love it: the second-order term always helps.
• Longer maturity and lower coupon both raise duration and convexity, making the bond more sensitive to rates in both directions.

Duration and convexity are not theoretical curiosities — they drive decisions worth trillions of dollars every year:

• Immunization: a pension fund matches the duration of its bond portfolio to the duration of its liabilities. If rates move, assets and liabilities shift by the same amount, leaving the funding ratio intact.
• Hedging: a trader who owns a bond with duration 7 can sell Treasury futures with an equivalent dollar duration (DV01) to neutralize rate risk while keeping other exposures.
• Convexity trading: two portfolios can have the same duration but different convexity. The higher-convexity portfolio outperforms in either direction of rate moves — investors pay a premium for it, compressing yields.
• Mortgage-backed securities: homeowners refinance when rates fall, shortening the MBS duration just when investors want it to be long. This negative convexity is the defining risk of the asset class and the reason MBS spreads are wider than comparable Treasuries.
• Central banks: the Fed's quantitative easing removed enormous duration from the market, compressing the term premium — the extra yield investors demand for holding long bonds. Measuring that effect requires exactly the tools above.

Understanding duration and convexity also connects to optimization: immunization is a linear-programming problem over duration constraints, and convexity turns it into a quadratic one.

Bond pricing looks like arithmetic — discount each cash flow, add them up. But sensitivity to interest rates is calculus. Duration is the slope of the price-yield curve at a given point; convexity is the curvature that keeps the slope from being the whole story.

The practical punchline is the same as the mathematical one: the price-yield relationship always bows toward the bondholder. Rates rise — you lose less than duration says. Rates fall — you gain more. That asymmetry, measured by convexity, is one of the few genuine free lunches in fixed income, and it is why long-dated, low-coupon bonds command a convexity premium in every deep bond market in the world.