Convexity: The Curve Duration Misses

Last lesson, duration handed you a clean one-liner: a bond’s price moves about minus-duration times the change in yield. Tidy. Linear. And — for anything bigger than a baby rate move — a little bit wrong. The true price–yield relationship isn’t a straight line at all; it’s a gently bowed curve, and duration is just the straight ruler we laid tangent to it at today’s yield. The further yields wander from today, the more the real curve peels away from that ruler. Here’s the lovely part: it always peels away in the holder’s favour. That bend has a name — convexity — and once you can see it, you’ll never trust a bare duration estimate on a big rate move again.

Duration is a tangent line, not the truth

Picture a smooth hill in profile. Stand at one spot and lay a straight plank flat against the slope so it just kisses the ground at your feet — that plank is a tangent line. Right where you’re standing, the plank and the hill agree almost perfectly. Take a few steps in either direction, though, and the hill curves away while the plank keeps going dead straight; the gap between them grows the further you walk.

That’s exactly the relationship between a bond’s price and its yield. The price–yield curve — actual price plotted against yield — is a downward-sloping curve that bends (it’s convex, meaning it bows so it always sits above its own tangent line). Duration is the tangent: the straight-line approximation that’s spot-on at today’s yield and drifts off as yields move. Formally, duration’s estimate of the fractional price change is

$\%\Delta P \approx -D \cdot \Delta y$

where $D$ is modified duration (roughly, the bond’s percentage price sensitivity per 1-point yield change) and $\Delta y$ is the change in yield as a decimal. It’s the equation of a straight line — so by construction it can’t capture a curve.

Before you read — take a guess

Guess before reading. Duration says a bond will lose exactly 14% if yields jump. Compared with what the bond's price ACTUALLY does, the straight-line duration estimate of the loss is most likely…

An underestimate of the loss — the real price falls more than 14%Exactly right — duration is the true price changeAn overestimate of the loss — the real price falls a bit less than 14%Impossible to tell without the coupon rate

Duration is only exact at one point

The single most common slip is treating the duration line as the bond’s true price curve everywhere. It isn’t — it’s only exact at today’s yield, where the line touches the curve. For tiny yield wiggles the line and the curve are practically the same, so duration alone is fine. For a big move the curve has visibly peeled away, and a duration-only number is measurably off.

When it matters

For pocket-change yield moves (a few basis points), the tangent hugs the curve and you can stop at duration. The mismatch only becomes worth fixing once yields move a lot — exactly the scenarios (rate shocks, long bonds) where you most need the number to be right.

The curve always bows the holder’s way

Here’s the asymmetry that makes convexity worth caring about. Because the price curve sits above its tangent on both sides, the actual price is always above the duration estimate — no matter which way yields move.

Read off what that means in each direction:

Yields rise → both line and curve predict a loss, but the curve is higher, so the real loss is smaller than duration claimed. Duration overestimates your loss.
Yields fall → both predict a gain, but the curve is again higher, so the real gain is bigger than duration claimed. Duration underestimates your gain.

Lose a little less when rates go against you; gain a little more when they go your way. That’s an unambiguously good deal for the bondholder, and it’s why positive convexity is a desirable property: the bend bails you out on the downside and pads you on the upside. The curve below makes it literal — drag the slider and watch the brand-coloured marker (the true price on the curve) stay above the accent-coloured marker (duration’s straight-line guess) on both sides of today’s yield. The shaded bar between them is the convexity effect.

Convexity corrects durationD 7 · C 70

True price (convex)Duration estimate (tangent)

Yield change (Δy)+2.0%

Actual Δprice (curve): -12.60%
Duration only (−D·Δy): -14.00%
Duration + convexity: -12.60%
Duration-only error: -1.40%

Duration is the straight tangent line. The real price–yield curve bows above it, so duration overstates losses when yields rise and understates gains when they fall. That gap is convexity — and it works in the holder’s favour.

Sort each statement by which direction of yield move it describes.

Place each item in the right group.

Duration underestimates the size of the gain
Convexity sweetens the upside
Duration overestimates the size of the loss
The real price is above the straight-line estimate, so you gain more than predicted
Convexity softens the blow
The real price is above the straight-line estimate, so you lose less than predicted

When it matters

The asymmetry is the whole reason convexity gets priced: investors will pay a hair more (accept a slightly lower yield) for a bond with more convexity, because that favourable bend is genuinely valuable. When two bonds look identical on duration, convexity is the tiebreaker — more on that below.

The convexity correction term

So how do we put a number on the bend? We add one more term to the straight-line estimate. The improved formula is a second-order approximation — duration captures the straight slope; convexity captures the curvature:

$\%\Delta P \approx -D \cdot \Delta y + \tfrac{1}{2} \cdot C \cdot (\Delta y)^2$

Here $C$ is the bond’s convexity, a measure of how much the price–yield line bends. Two things to notice. First, the correction uses $(\Delta y)^2$ — yield change squared — so it’s tiny for small moves and grows fast for big ones; that’s exactly why convexity only matters when yields move a lot. Second, because $(\Delta y)^2$ is always positive (a negative times a negative is positive) and convexity is positive, the correction term is always a plus. It nudges the estimate up whether yields rose or fell — pushing the straight-line guess toward the higher true curve in both directions. That single always-positive nudge is the math behind “lose less, gain more.”

Worked example — yields rise 2%

Take a bond with modified duration $D = 7$ and convexity $C = 70$ , and shock yields up by $\Delta y = +0.02$ (a 2-percentage-point jump). Step through both pieces:

Duration-only piece:

$-D \cdot \Delta y = -7 \cdot 0.02 = -0.14 = -14.0\%$

The straight line says the bond loses 14%. Convexity correction:

$\tfrac{1}{2} \cdot C \cdot (\Delta y)^2 = \tfrac{1}{2} \cdot 70 \cdot (0.02)^2 = \tfrac{1}{2} \cdot 70 \cdot 0.0004 = +0.014 = +1.4\%$

Add them:

$\%\Delta P \approx -14.0\% + 1.4\% = -12.6\%$

So the real loss is about 12.6%, not the scary 14% duration alone shouted. Convexity bought back 1.4 percentage points — duration overestimated the loss by exactly that much.

Worked example — yields fall 2% (the mirror)

Now the flip side: same bond, yields down by $\Delta y = -0.02$ .

Duration-only piece:

$-D \cdot \Delta y = -7 \cdot (-0.02) = +0.14 = +14.0\%$

Convexity correction — note $(\Delta y)^2 = (-0.02)^2 = 0.0004$ , still positive:

$\tfrac{1}{2} \cdot 70 \cdot (-0.02)^2 = \tfrac{1}{2} \cdot 70 \cdot 0.0004 = +0.014 = +1.4\%$

Add them:

$\%\Delta P \approx +14.0\% + 1.4\% = +15.4\%$

The real gain is about 15.4%, bigger than the 14% the straight line promised. Same 1.4-point convexity bonus — but this time it’s added on top of a gain. Lose 12.6% instead of 14% when rates rise; gain 15.4% instead of 14% when they fall. Asymmetry, in your favour, both ways.

Fill in the blanks for the convexity correction.

Pick the right option for each blank, then check.

The better price estimate is minus-duration-times-Δy one-half-times-convexity-times-Δy-squared. Because Δy is , the correction term is always , so it pushes the estimate toward the true curve whether yields rose or fell. For a bond with D = 7 and C = 70, a 2% yield rise gives a duration-only estimate of −14% and a convexity correction of , for a combined estimate near −12.6%.

Reading it off a table

Lining up duration-only, duration-plus-convexity, and the actual price across several yield moves shows the pattern at a glance. Same bond ( $D = 7$ , $C = 70$ ):

Yield move $\Delta y$	Duration only ( $-D\,\Delta y$ )	Convexity term ( $\tfrac12 C \,\Delta y^2$ )	Duration + convexity	Actual (stated)	Duration-only error
$-0.02$ (down 2%)	$+14.0\%$	$+1.4\%$	$+15.4\%$	$\approx +15.4\%$	understates gain by 1.4
$-0.01$ (down 1%)	$+7.0\%$	$+0.35\%$	$+7.35\%$	$\approx +7.35\%$	understates gain by 0.35
$0$	$0.0\%$	$0.0\%$	$0.0\%$	$0.0\%$	exact
$+0.01$ (up 1%)	$-7.0\%$	$+0.35\%$	$-6.65\%$	$\approx -6.65\%$	overstates loss by 0.35
$+0.02$ (up 2%)	$-14.0\%$	$+1.4\%$	$-12.6\%$	$\approx -12.6\%$	overstates loss by 1.4

Three things jump out. The duration-only column is symmetric — flip the sign of $\Delta y$ and the magnitude is identical ( $\pm 7\%$ , $\pm 14\%$ ). But the actual prices are asymmetric: the upside ( $+15.4\%$ ) is bigger than the downside ( $-12.6\%$ ) for the same-size move. And the duration-only error grows with the square of the move — quadruple at $\pm 0.02$ what it is at $\pm 0.01$ (1.4 vs 0.35), because the correction rides on $(\Delta y)^2$ . Tiny moves: barely any error. Big moves: the gap is real money.

Skipping convexity on a big move hides the asymmetry

For small yield wiggles, ignoring convexity costs you almost nothing — the correction is a rounding error. The danger is using duration alone for a large rate move: you’ll get a symmetric, too-pessimistic-on-the-downside, too-stingy-on-the-upside picture and completely miss the favourable asymmetry. The bigger the move, the more the missing $\tfrac12 C (\Delta y)^2$ term matters — and it’s exactly the big moves where being wrong is expensive.

When it matters

Convexity earns its keep in three places: big rate moves (the squared term finally bites), long-maturity bonds (which carry both high duration and high convexity, magnifying everything), and comparing two bonds with equal duration — where convexity is the only thing left to tell them apart.

Match each term to what it means.

Pick a term, then click its definition.

Two bonds, same duration

Suppose two bonds have identical modified duration of 7 — so duration alone calls them interchangeable. Bond A has convexity 40; Bond B has convexity 90. Shock yields $\pm 0.02$ and watch them diverge:

	Bond A ( $C = 40$ )	Bond B ( $C = 90$ )
Convexity term at $\Delta y = 0.02$	$\tfrac12 \cdot 40 \cdot 0.0004 = +0.8\%$	$\tfrac12 \cdot 90 \cdot 0.0004 = +1.8\%$
Yields up 2% (loss)	$-14\% + 0.8\% = -13.2\%$	$-14\% + 1.8\% = -12.2\%$
Yields down 2% (gain)	$+14\% + 0.8\% = +14.8\%$	$+14\% + 1.8\% = +15.8\%$

Bond B loses less when rates rise and gains more when they fall — it dominates Bond A in both directions, purely on its higher convexity. That’s why, all else equal, more convexity is more desirable, and why the market quietly prices it in: a more-convex bond typically trades at a slightly richer price (slightly lower yield). You don’t get that favourable bend for free — but when you’re choosing between two equal-duration bonds, it’s the deciding feature.

A quick reality check on the price of convexity

Because convexity is valuable, you rarely find a high-convexity bond going cheap by accident — its appeal is already baked into its price. Convexity isn’t a free lunch; it’s a fairly-priced feature. The lesson is to recognise it so you’re comparing bonds on equal footing, not to expect to pocket it for nothing.

Putting it together

Duration draws the straight line; convexity restores the curve. Chunk the whole idea into one picture:

Big picture

Convexity

Convexity
- The geometry
  - Price–yield is a convex CURVE
  - Duration is its straight TANGENT line
  - Curve always sits ABOVE the line
- The asymmetry
  - Yields rise → real loss is SMALLER (duration overstates)
  - Yields fall → real gain is BIGGER (duration understates)
  - Lose less, gain more — good for the holder
- The formula
  - %ΔP ≈ −D·Δy + ½·C·(Δy)²
  - (Δy)² makes the correction always positive
  - D=7, C=70, Δy=+2%: −14% + 1.4% = −12.6%
- When it matters
  - Big yield moves (squared term bites)
  - Long bonds (high D and high C)
  - Equal-duration bonds → convexity decides
  - More convexity is desirable — and gets priced in

Duration is the straight tangent line to the true convex price–yield curve. Because the curve bows above the line, duration overstates losses and understates gains — and the ½·C·(Δy)² correction fixes both, always in the holder's favour.

A mixed recap — it pulls from everything above:

Question 1 of 70 correct

What is duration, geometrically, relative to the true price–yield curve?

The curve itself — duration is the exact price at every yieldA line that always sits above the curveThe straight tangent line that touches the curve at today's yieldThe area under the price–yield curve

Check your answer to continue.

Key Takeaways

What to remember

The true price–yield relationship is a curve, not a line. Duration is only the straight tangent to that curve at today’s yield — exact right there, off everywhere else.
The curve bows above its tangent, so the actual price is always above the duration estimate. Duration therefore overestimates losses when yields rise and underestimates gains when yields fall.
That favourable bend is convexity, and positive convexity is good for the holder — you lose a little less and gain a little more than the straight line says.
Better estimate: $\%\Delta P \approx -D \cdot \Delta y + \tfrac{1}{2} \cdot C \cdot (\Delta y)^2$ . The correction rides on $(\Delta y)^2$ , so it’s always positive and grows with the square of the move.
Worked anchor: $D = 7$ , $C = 70$ . Yields up 2%: $-14\% + 1.4\% = -12.6\%$ . Yields down 2%: $+14\% + 1.4\% = +15.4\%$ . Lose less, gain more.
When it matters: big rate moves, long bonds, and choosing between two equal-duration bonds — where more convexity wins. It’s valuable, so the market prices it in; it’s a feature, not a free lunch.