Duration & Convexity

Last time we priced bonds and noticed that longer-maturity bonds are more sensitive to yield changes. We gave that sensitivity a name, duration, and offered a rule of thumb: a duration of 7 means roughly a 7% price swing per 1% yield move.

Now let’s do the math.

Macaulay Duration

The rule of thumb needs a definition underneath it. Frederick Macaulay supplied one in 1938. Macaulay duration is the weighted-average time until a bond’s cash flows arrive, with each cash flow weighted by its present value as a fraction of the bond’s price:

D_{Mac} = \frac{1}{P} t = 1 \sum T t \times \frac{C _{t}}{( 1 + y ) ^{t}}

where $C_{t}$ is the cash flow at time $t$ (coupons plus face value at maturity) and $y$ is the yield per period.

Take a 3-year bond with a 5% annual coupon, 100 face value, priced at par (yield = 5%). The cash flows are 5, 5, and 105. We discount each one and weight it by the time it arrives:

Year	Cash Flow	PV at 5%	Weight	Year x Weight
1	5	4.76	0.0476	0.0476
2	5	4.54	0.0454	0.0907
3	105	90.70	0.9070	2.7210
		100.00		2.86 years

Macaulay duration is 2.86 years. Even though the bond matures in 3 years, the coupons pull the weighted average forward. We convert that to a modified duration of $2.86/1.05 = 2.72$ , so a 1% yield rise implies roughly a 2.7% price drop.

The intuition rests on timing. A bond that returns cash sooner has lower duration, and a bond that defers it has higher duration. A zero-coupon bond has one cash flow, at maturity, so its Macaulay duration equals its maturity. A coupon bond front-loads part of its value, so its duration is always less than its maturity.

Three rules of thumb follow from the weighting. The first two should feel familiar from the bond pricing post, and the third is less obvious.

Higher coupon → lower duration. More cash arrives early, pulling the average forward.
Longer maturity → higher duration. More of the value arrives far in the future.
Higher yield → lower duration. Higher discount rates shrink the present value of distant cash flows more than near ones, pulling the average forward.

Modified Duration

Macaulay duration is measured in years. Modified duration converts those years into a price sensitivity:

D^{*} = \frac{D _{Mac}}{1 + y}

This ratio gives us the first-order approximation of how a bond’s price reacts to a small yield change:

\frac{Δ P}{P} \approx - D^{*} \times Δ y

The negative sign is the price-yield seesaw from the bond pricing post: when yields rise, prices fall. A bond with modified duration of 7 will lose approximately 7% of its value if yields rise by 1%.

This linear approximation works well for small yield moves. But the actual price-yield curve is not a straight line. It bends, and for larger yield changes the linear estimate starts to miss.

Convexity

The shape of that curve is what the linear estimate ignores. The price-yield relationship is convex. It curves upward. Duration captures the slope at a point, but misses the curvature, and convexity is the second-order correction:

\frac{Δ P}{P} \approx - D^{*} \times Δ y + \frac{1}{2} C \times (Δ y)^{2}

Readers who have seen Taylor series will recognize the first two terms of an expansion, though the derivation is not needed to use the result. Convexity itself is given by:

C = \frac{1}{P ( 1 + y ) ^{2}} t = 1 \sum T t (t + 1) \times \frac{C _{t}}{( 1 + y ) ^{t}}

We are weighting each cash flow by $t (t + 1)$ instead of $t$ and dividing by $P (1 + y)^{2}$ instead of $P$ , so the structure mirrors Macaulay duration. Distant cash flows contribute disproportionately more to convexity than to duration.

Because the yield change enters the correction squared, and because convexity is positive for vanilla bonds, the correction is always positive. Duration alone overestimates losses when yields rise (the actual curve falls less steeply than the tangent line). It underestimates gains when yields fall (the actual curve rises more steeply than the tangent line).

We return to the 3-year, 5% coupon bond we started with (modified duration 2.72, convexity 10.21). Suppose yields jump from 5% to 6%. How much does the price drop?

Using duration alone, we get:

\frac{Δ P}{P} \approx - 2.72 \times 0.01 = - 2.72%

Adding the convexity term, we get:

\frac{Δ P}{P} \approx - 2.72 \times 0.01 + \frac{1}{2} (10.21) (0.01)^{2} = - 2.72% + 0.05% = - 2.67%

When we reprice the bond at 6%, we get 97.33, a drop of 2.67%. Duration alone overshoots by a small amount. On a short-maturity bond the convexity correction is modest. On a 30-year bond it is not: at a 5% coupon and a 5% yield, modified duration runs above 15, and the gap the correction closes grows with it.

Convexity is worth having. All else equal, a bond with higher convexity benefits more from rate drops and suffers less from rate rises. When two bonds offer the same yield and duration, the one with higher convexity is the better deal.

The Approximation Gap

How large the correction gets depends on the bond. Set coupon, maturity, and yield with the sliders below, and watch duration and convexity move with them. The chart compares the actual price change (solid line) against the duration-only approximation (dashed) and the duration-plus-convexity approximation. The gap between the dashed and solid lines grows with larger yield moves. That gap is the error convexity corrects.

COUPON: 5.0% MATURITY: 10yr YTM: 5.0% PRICE $100.00 MACAULAY D 8.11 yr MODIFIED D 7.72 CONVEXITY 75.00 -16% -10% -4% 2% 8% 14% -2.0% -1.5% -1.0% -0.5% 0.0% +0.5% +1.0% +1.5% +2.0% Yield Change Price Change % Actual Duration only Duration + Convexity

What’s Next

We’ve now fully equipped the fixed-income toolkit: pricing, yield, duration, and convexity. Next we turn to the other side of the balance sheet. Stocks promise neither fixed payments nor a maturity date. The cash flows are uncertain, the time horizon is theoretically infinite, and the valuation gets a lot more interesting.