Convexity · RiskAtoms Academy

By the end you will be able to

1Improve a duration estimate of a price change with a convexity adjustment.
2Explain why duration alone misjudges price changes for large moves in yield.
3Calculate convexity with the table method.
4Tell positive convexity from negative convexity, and name bonds that have each.

Why it matters

In L1.4.08, duration predicted 97.28 for our 3-year bond after a 1% rise in yields. The real price was 97.33. A gap of 0.05 looks harmless. On a 20-year bond after a 3% move, the same gap is worth about 9 points. That gap has a name.

Core explanation

Bending the line back towards the curve

Duration measures the slope of the price-yield curve at today's yield, and uses it to draw a straight line. But the real curve bends (L1.4.06). For small moves the line and the curve stay close. For large moves they drift apart, and the line always ends up below the curve.

That means duration alone always gets it slightly wrong in the same direction: it overstates the loss when yields rise, and understates the gain when they fall. The fix is a second term that measures the bend. That measure is convexity.

A driving analogy

Duration is your speed: it tells you how far you will go in the next second. Convexity is your acceleration. Over a second, speed alone is fine. Over a minute, you need acceleration too.

Why investors like it. A bond with more convexity gains more when yields fall and loses less when they rise than a bond with the same duration but less bend. That is valuable, so, all else equal, investors accept a slightly lower yield for it.

Positive and negative convexity. Ordinary bonds have positive convexity. Some bonds carry an option that flips this. A callable bond (L1.4.02) can be repaid early by the issuer, which it will do when rates fall, so its price struggles to rise much above the call price. A mortgage-backed security behaves the same way, because homeowners refinance when rates fall and repay their loans early. Both can show negative convexity: they gain little when yields fall but still lose when they rise.

For such bonds, duration and convexity are measured by repricing the bond with the yield bumped down and up, allowing for the option: effective duration and effective convexity.

Sort itPositive or negative convexity?

5 items

A 10-year government bond with a fixed coupon and no options.
A 20-year zero-coupon bond.
A corporate bond the issuer can call at 100 from next year, when market yields are well below its coupon.
A pool of fixed-rate home loans, where borrowers refinance when rates drop.
A bond the investor can sell back to the issuer at 100.

The formal version

Convexity adds a squared term and a new symbol, so let's build it slowly. We write $\text{Conv}$ for convexity, because $C$ is already the coupon.

The formulas, for reference

\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2}\,\text{Conv}\,(\Delta y)^2 \qquad \text{Conv} = \frac{1}{P\,(1+y)^2}\sum_{t=1}^{T} t(t+1)\,PV_t

The table form is for annual coupons. With options, use effective convexity from bumped prices:

\text{Conv}_{\text{eff}} = \frac{P_{-} + P_{+} - 2P_0}{P_0\,(\Delta y)^2}

Optional: where the ½ comes from (Math Atom: Taylor expansion)

A Taylor expansion approximates a curve near a point with its slope and its curvature:

P(y + \Delta y) \approx P(y) + P'(y)\,\Delta y + \tfrac{1}{2}\,P''(y)\,(\Delta y)^2

Divide by $P$ . The slope term gives $-D_{\text{mod}}$ (L1.4.08), and $P''/P$ is the convexity. Differentiating the price twice brings down $t$ and then $t+1$ , which is where $t(t+1)$ and $(1+y)^2$ come from.

Try it

It opens on the 3-year 5% bond with a +100 bp move. Push the yield change to ±300 bp, then stretch the maturity to 30 years. Watch how far the duration line falls below the real curve, and how much of the gap the convexity term closes.

Try itLine, curve and correction

Open the full bond pricer →

Coupon rate5.00%

Years to maturity3

Yield today5.0%

Yield change+100 bp

Actual new price

97.33

Modified duration2.723

Convexity10.21

Duration only97.28

error0.05

+ convexity97.33

error0.00

actual price duration line duration + convexity

The straight line always sits below the real curve, so duration alone always underestimates the price. The convexity term closes most of the gap.

Worked example

Closing the gap on the 3-year bond

The bond from L1.4.08: face value 100, annual coupon 5, three years, yield 5%, price 100. Modified duration 2.723. Try each step before revealing it.

Convexity and the improved estimate

Step 1 of 6

1
Add a t(t + 1) × PV column
Year t PV at 5% t(t + 1) t(t + 1) × PV
1 4.7619 2 9.5238
2 4.5351 6 27.2109
3 90.7029 12 1,088.4354
Total 100.0000 1,125.1701

Year t	PV at 5%	t(t + 1)	t(t + 1) × PV
1	4.7619	2	9.5238
2	4.5351	6	27.2109
3	90.7029	12	1,088.4354
Total	100.0000		1,125.1701

On a short bond, convexity is a small correction. On a long bond it matters a lot. Here is a 20-year 4% bond at 4% (modified duration 13.59, convexity 240.7):

Yield change	Actual	Duration only	+ Convexity
−3%	154.14	140.77	151.60
−2%	132.70	127.18	131.99
−1%	114.88	113.59	114.79
+1%	87.54	86.41	87.61
+2%	77.06	72.82	77.63
+3%	68.22	59.23	70.06

Duration alone is off by 13.37 points for a 3% fall in yields; adding convexity cuts the error to 2.53. Even then it is not exact: for very large moves, higher-order terms start to matter.

Regulatory lens

Regulators know that straight-line sensitivities miss the bend. Under the FRTB market risk standard, banks add a separate curvature charge for positions with options, on top of the charge based on first-order sensitivities (L4.3). For banking-book interest rate risk, the Basel IRRBB standard revalues positions under full rate shocks rather than relying on duration alone (L3.5).

Sparks

Spark1994: when convexity bit back

In February 1994 the Federal Reserve began raising rates, and bond yields jumped through the year. As yields rose, fewer homeowners refinanced, so mortgage-backed securities lengthened: their duration grew just as prices fell. Holders who hedged by selling Treasuries added to the sell-off. The episode is a textbook case of negative convexity.

SparkConvexity is not free

Between two bonds with the same duration, the one with more convexity tends to yield a little less, because the market charges for its better behaviour in big moves.

Key takeaways

Duration is a straight line; the price-yield curve bends. For ordinary bonds the line always sits below the curve.
Convexity measures the bend. ΔP/P ≈ −D_mod Δy + ½ Conv (Δy)² corrects the duration estimate.
The convexity term grows with the square of the yield change, so it matters most for big moves and long bonds.
Table method: Conv = Σ t(t + 1) PV ÷ [P(1 + y)²]. With options, use effective convexity from bumped prices.
Ordinary bonds have positive convexity. Callable bonds and MBS can have negative convexity.

Reaction

Check what stuck

Glossary

Convexity: A measure of how much a bond's price-yield curve bends. It corrects the straight-line duration estimate.
Positive convexity: The price rises more when yields fall than it drops when yields rise by the same amount. True of ordinary bonds.
Negative convexity: The price gains less when yields fall than it loses when they rise, because an option caps the upside. Callable bonds and mortgage-backed securities can show it.
Effective duration: Duration measured by repricing the bond with the yield bumped down and up, so it allows for options.
Effective convexity: Convexity measured by repricing with the yield bumped down and up. Used for bonds with options.
MBS: A mortgage-backed security: a bond paid from a pool of home loans (L1.4.11).

Deep Dives & Origins

Deep Dives

Bruce Tuckman and Angel Serrat, Fixed Income Securities: Tools for Today's Markets, 3rd edition, Wiley, 2011, chapter on convexity. Intermediate.

Origins

Frank J. Fabozzi and Francesco A. Fabozzi, Bond Markets, Analysis, and Strategies, 10th edition, Pearson, 2021.
Tuckman and Serrat (2011), as above.
Basel Committee on Banking Supervision, Minimum capital requirements for market risk, January 2019 (FRTB curvature risk).

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