Convexity: Definition & Examples

Convexity measures how much a bond's price–yield curve bends. It is the second-order term beyond duration: how fast the bond's price sensitivity to yield changes as yield changes. With

y

the yield as a decimal, the convexity is

C=\frac{1}{P}\frac{d^2P}{dy^2}

, and the price change for a yield move

\Delta y

is approximated by

\frac{\Delta P}{P}\approx -D_{\text{mod}}\,\Delta y+\tfrac12\,C\,(\Delta y)^2 ,

the first two terms of a Taylor series in $\Delta y$ , with $D_{\text{mod}}$ the modified duration. For a bond with fixed cash flows $C$ is positive, so for equal-sized moves the bond gains more than it loses.

The duration page computes this for one bond: a four-year 4% semiannual bond with

D_{\text{mod}}=3.639102

and

C=15.600570

, where adding the convexity term moves the +100 bp estimate from 929.06 to 929.82 against an exact 929.80. This page asks what convexity is worth, who pays for it, and when it turns negative. Conventions: yields are annual decimals, so 100 bp is

\Delta y=0.01

; a price change is an instantaneous repricing with cash flows unchanged unless stated.

What the number is made of

For fixed cash flows $CF_t$ at times $t$ , with annual compounding, $P=\sum_t CF_t(1+y)^{-t}$ . Differentiating twice gives $\sum_t t(t+1)\,CF_t\,(1+y)^{-t-2}$ , and every term is positive. A sum of convex curves is convex, which is why a fixed-cash-flow bond cannot have negative convexity. For a zero-coupon bond maturing in $T$ years,

D_{\text{mod}}=\frac{T}{1+y},\qquad C=\frac{T(T+1)}{(1+y)^2}.

Duration grows roughly in proportion to maturity, convexity roughly with its square. Equivalently, convexity is how fast dollar duration changes: as yield falls, the tangent steepens and duration rises, which accelerates gains, and as yield rises, duration shrinks and slows losses (Fabozzi, Chapter 4). Spreading cash flows over a wider range of dates, at unchanged duration, raises convexity.

Check the convention before using any reported number:

Yield basis. With $m$ payments a year and period index $k$ , the factor $t(t+1)$ becomes $k(k+1)/m^2$ and discounting uses $(1+y/m)^{-k-2}$ . The duration page's bond uses $m=2$ .
Units of $\Delta y$ . The formula needs a decimal, as in the first display.
Scaling. Fabozzi notes that some writers define the measure as half the second derivative divided by price, so the price term has no ½. Others scale it by 100 so that $\Delta y$ can be entered in percentage points. The 10-year zero below, $C=99.77$ , could appear as 49.89 or 0.9977. A bare figure means little until you know which formula it feeds.

Same duration, more convexity

Two portfolios start at 1,000,000 USD on a fictional flat 5% annual curve. The holdings are zero-coupon bonds, shifts are instantaneous and parallel, and no time passes.

	Holdings (market value → face)	$D_{\text{mod}}$	$C$
Bullet	10-year zero: 1,000,000 → 1,628,894.63	9.5238	99.77
Barbell	2-year zero: 714,285.71 → 787,500.00; 30-year zero: 285,714.29 → 1,234,840.68	9.5238	244.90

The barbell weights, 20/28 and 8/28, are chosen so $\tfrac{20}{28}\cdot\tfrac{2}{1.05}+\tfrac{8}{28}\cdot\tfrac{30}{1.05}=9.5238$ matches the bullet. Portfolio convexity is the value-weighted average, $\tfrac{20}{28}\cdot5.44+\tfrac{8}{28}\cdot843.54=244.90$ , more than double the bullet's. Exact repricing after each shift:

Shift	Bullet	Barbell	Barbell − bullet
−200 bp	1,212,050.58	1,251,032.29	+38,981.71
−100 bp	1,100,422.84	1,108,812.45	+8,389.61
+100 bp	909,566.25	915,870.47	+6,304.22
+200 bp	828,047.43	850,050.46	+22,003.03

Duration alone gives both portfolios the same answer: 1,095,238.10 at −100 bp and 904,761.90 at +100 bp, and 1,190,476.19 and 809,523.81 at ±200 bp. Adding the convexity term separates them. At −100 bp the estimates are 1,100,226.76 (bullet) and 1,107,482.99 (barbell); at +100 bp, 909,750.57 and 917,006.80. The estimated barbell edge, $\tfrac12(244.90-99.77)(0.01)^2\times1{,}000{,}000\approx7{,}256$ USD, sits between the exact edges of 8,389.61 and 6,304.22. The quadratic still misses at ±200 bp, mostly through the 30-year leg: the barbell estimate is 11,576.51 below the exact value at −200 bp, the bullet's 1,619.74.

Line chart of the value of two portfolios, each worth 1,000,000 USD today, against an instant parallel yield shift from minus 300 to plus 300 basis points on a flat 5 percent annual curve, with a straight duration-only tangent through the common starting point. Both portfolios have modified duration 9.5238. The bullet (one 10-year zero, convexity 99.77) is worth 1,212,050.58 at minus 200 basis points and 828,047.43 at plus 200. The barbell (2-year and 30-year zeros, convexity 244.90) is worth 1,251,032.29 at minus 200 and 850,050.46 at plus 200. The barbell curve lies above the bullet curve away from zero shift, and both lie above the tangent.

Original fictional illustration. Flat 5% annual-compounding curve, zero-coupon bonds, instantaneous parallel shifts, no time passing, no transaction costs. The barbell advantage shown is before any yield give-up for convexity.

Convexity is valuable, so it is priced

The table looks like a free lunch. The barbell costs the same as the bullet today and is never worth less than the bullet after any parallel shift. If a flat curve could only move in parallel, buying the barbell and selling the bullet would be riskless profit, so real curves cannot behave that way. This is Tuckman's point in the barbell-versus-bullet discussion: greater convexity does better whether rates rise or fall, but bond prices reflect that advantage.

It is paid for in yield. In the models of Tuckman's Chapter 10, convexity lowers a bond's yield by about $\tfrac12 C\sigma^2$ a year, where $\sigma$ is the annual volatility of rates. With the gap above and an assumed $\sigma=1\%$ , that is $\tfrac12(145.13)(0.01)^2\approx0.73\%$ of value, the same 7,256 USD per million as the quadratic edge. An investor who holds the barbell therefore earns less carry while rates stay put. The barbell wins when rates move a lot, the bullet when they barely move; Fabozzi makes the same point, that investors who expect low rate volatility should be willing to sell convexity. A positive-convexity position is long volatility, which connects to gamma: both are second derivatives, both pay off on large moves in either direction, and both are financed by a lower running return.

Convexity also does not fix the other assumption in the example. The parallel-shift model sees only the level of the curve. Of the barbell's 9.5238 years of modified duration, the 30-year leg carries $\tfrac{8}{28}\times\tfrac{30}{1.05}=8.16$ . If the 30-year yield rises 50 bp while the 2-year and 10-year yields stay put, a first-order estimate has the barbell losing about 4.08% (40,816 USD), the bullet nothing, far more than the convexity edge. A steepening hurts the barbell and a flattening helps it; key-rate duration measures that exposure.

When convexity turns negative

A callable bond lets the issuer redeem early. Economically, the investor owns the straight bond and is short a call option, so the price is the straight bond's value minus the option's value (Tuckman, Chapter 5). When yields are well above the coupon the call is unlikely and the price follows the usual convex curve. As yields fall toward the level where calling becomes attractive, the price flattens toward the call price: Fabozzi calls this price compression. Price gains from a fall in yield are then smaller than losses from an equal rise, which is negative convexity. The CFA Institute summary of embedded-option analysis says the convexity of a callable bond is negative when the option is near the money. Agency mortgage-backed securities behave alike because borrowers refinance when rates fall, so prepayment is the embedded option.

Schematic price-yield curves with price on the vertical axis and yield on the horizontal axis. A fixed-cash-flow bond has a convex curve that rises ever more steeply as yield falls. A stylised callable bond follows the same curve at high yields, then flattens and bends the other way as yield falls toward the level where the price approaches the 102 call price, so price gains are capped near 102. Marked as an illustrative stylised curve, not a model output.

Illustrative, stylised curves, not a model output and not calibrated to any bond. The callable curve is drawn to show price compression near a 102 call price.

The fixed-cash-flow formulas above fail here because the cash flows change with rates. Practitioners reprice with a model after shifting the benchmark curve down and up by $h$ (decimal), holding the option-adjusted spread fixed, and use the finite differences

D_{\text{eff}}=\frac{P_{-}-P_{+}}{2P_0h},\qquad C_{\text{eff}}=\frac{P_{+}+P_{-}-2P_0}{P_0\,h^2},

where $P_-$ and $P_+$ are the repriced values after the down and up shifts. These are effective duration and effective convexity. They depend on the model's volatility and exercise assumptions, so report those with the number. A short-convexity holder loses whether rates gap up or down, Tuckman's description of a short option hedged with bonds, which is why mortgage and callable-bond investors treat convexity as a risk to be compensated by spread.

Practice

1. Calculation. A bond has modified duration 7.2 and convexity 65, both defined against the annual yield in decimals. Estimate its percentage price change when the yield rises by 50 bp, and compare with the duration-only estimate.

Answer. With

\Delta y=0.005

-7.2(0.005)+\tfrac12(65)(0.005)^2=-0.036+0.0008125=-0.0351875

, or

-3.51875\%\approx-3.52\%

. Duration alone gives

-3.60\%

; convexity trims the loss by about 0.08 percentage points. If a system quoted this convexity as 0.65 (divided by 100), rescale it to 65 before using the decimal formula.

2. Multiple choice. Two portfolios have the same market value, modified duration and yield. Portfolio X has convexity 240 and Portfolio Y has 100. Yields then jump 150 bp in parallel, instantly. Which portfolio does better?

(a) X
(b) Y
(c) Neither: same duration means the same price change
(d) It depends only on the direction of the jump

Answer. (a). To second order, X's value change exceeds Y's by

\tfrac12(240-100)(0.015)^2=0.01575

, or 1.575% of portfolio value, whichever way yields move, because the term is the square of

\Delta y

. Option (c) is the main trap: equal duration matches only the first-order term, so it gives equal results only for infinitesimal moves. Option (d) confuses duration, which depends on direction, with convexity, which does not. In practice the market usually charges for convexity through a lower yield, so X's advantage is paid for in carry.

3. Interview-style. A model reprices a callable bond, holding the option-adjusted spread constant, at

P_0=101.20

. After a downward shift of 25 bp the price is 101.90, and after an upward shift of 25 bp it is 100.30, so

h=0.0025

. Compute the effective duration and effective convexity and interpret them.

Answer.

D_{\text{eff}}=\dfrac{101.90-100.30}{2(101.20)(0.0025)}=3.16

and

C_{\text{eff}}=\dfrac{101.90+100.30-2(101.20)}{101.20\,(0.0025)^2}=-316.2

(decimal-yield convention). The price rises 0.70 when rates fall and drops 0.90 when they rise, which is the asymmetry negative convexity describes. As rates fall the issuer is more likely to call, so price gains are capped, while losses on a rate rise are not. Quoting

C_{\text{eff}}

divided by 100 would show

-3.16

. A positive-convexity bond with duration 3.16 would show the opposite asymmetry.

Continue learning

Three lessons in SelfMadeQuant's Fixed Income and Interest Rate Models Programme extend this page; each has a public preview and the full lesson requires membership.

Key Rate Exposures, Bucket Shifts, and Immunization splits duration across maturity buckets.
Expectations, Convexity, and Risk Premium in the Term Structure shows convexity lowering long rates in a small rate tree.
Agency Mortgage Pass-Through Securities explains negative convexity and duration extension for pass-throughs; it is still in progress.

Sources

Frank J. Fabozzi, Bond Markets, Analysis, and Strategies, 9th ed.: Chapter 4 (convexity, the convexity measure and its vendor scaling, the value of convexity, properties of convexity); Chapter 18 (callable-bond price–yield relationship, negative convexity, effective duration and effective convexity); Chapter 19 (negative convexity of residential mortgage-backed securities).
Bruce Tuckman, Fixed Income Securities: Tools for Today’s Markets, 2nd ed.: Chapter 5 (convexity, short convexity, callable bonds), Chapter 6 (convexity of zeros, the barbell versus the bullet), Chapter 10 (the value of convexity and its effect on yield).
CFA Institute, Valuation and Analysis of Bonds with Embedded Options, refresher reading summary.