Convexity and the convexity adjustment

What does convexity measure?

Duration is a straight line laid against a curve. Convexity measures how far the curve bends away from that line. Duration is the first-order effect of a yield change on price, and convexity is the second-order effect. For an option-free bond the curve bends upward, which is called positive convexity : the curve always sits above its tangent line.

The practical result is an asymmetry. For the same size of yield move, the price gain when yields fall is bigger than the price loss when yields rise. Take Bond K (6 years, 5% annual coupon, yield 6%, full price 95.0827, modified duration 5.009). Repricing it at a 100 bp move gives:

Yield change	Duration says	Actual price change
−100 bp	+5.01%	+5.17%
+100 bp	−5.01%	−4.85%

Duration treats the two moves as mirror images. The real bond gains about 0.16 percentage points more than duration predicts when yields fall, and loses about 0.15 less when they rise. That gap is the convexity adjustment .

Price change for a yield changethe price–yield curve, approximated in two steps

• Duration Memorize first-order: the straight tangent line, the slope of the curve
• Convexity Memorize second-order: the curvature the line misses
  • Option-free bond Memorize always positive: the curve sits above its tangent
  • Bond with an embedded option Know the idea can turn negative (callable bonds, later module)

$$\text{Convexity adjustment} = \tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta y)^2$$

Because $(\Delta y)^2$ is positive whichever way yields move, and convexity is positive for an option-free bond, the adjustment is always added to the duration estimate. It is small for a 25 bp move and grows with the square of the move: double the yield change and the adjustment quadruples. Lesson 2 puts it to work.

Exam tip Convexity is extra price you collect for free whenever yields move a lot. It never works against an option-free bondholder.

How do you approximate convexity from prices?

Memorize

Approximate convexity uses the same three prices as approximate modified duration: today’s price $PV_0$, the price after a small yield fall $PV_-$ and the price after the same small rise $PV_+$. Duration looks at the difference between the outer prices. Convexity looks at how far their sum sits above twice the middle price, which is the bend.

$$\text{ApproxModDur} = \frac{PV_- - PV_+}{2 \times \Delta y \times PV_0}$$

$$\text{ApproxCon} = \frac{PV_- + PV_+ - 2 \times PV_0}{(\Delta y)^2 \times PV_0}$$

Compare the two denominators. Duration has a 2 and a single Δy. Convexity has no 2 and Δy squared. Always enter Δy as a decimal.

Go deeper The exact convexity calculation (not required on the exam)

Exact annualized convexity comes from a cash-flow table, like exact duration. For each cash flow at period $t$, take $t(t+1) \times PV$ of that cash flow, add them up, divide by the bond’s full price and by $(1 + r)^2$, then divide by the number of periods per year squared.

For Bond K (annual periods, $r = 6\%$) the cash flows at years 1 to 6 are 5, 5, 5, 5, 5 and 105. The sum of $t(t+1) \times PV$ is 3,386.7, and 3,386.7 ÷ 95.0827 ÷ 1.06² = 31.70. The exam hands you the figure or asks for the three-price approximation.

What makes a bond more convex?

Convexity responds to the same bond features as duration. A fixed-rate bond has more convexity the longer its maturity, the lower its coupon and the lower its yield. There is one extra lever: the dispersion of cash flows.

Higher convexitythe same levers as duration, plus one

• Longer maturity Memorize more years of cash flows to discount
• Lower coupon Memorize more of the value sits in the final payment
• Lower yield Memorize discounting is gentler, so distant cash flows count more
• More dispersed cash flows Memorize at equal duration: spread out beats bunched together

Two bonds can have the same duration yet different convexity. Dispersion of cash flows means how widely the payments are spread around their average timing. A bond that pays at many dates is more dispersed than one that pays everything at a single date, and at equal duration the more dispersed bond is more convex. A barbell portfolio (short and long bonds) is more convex than a bullet (all maturities in the middle) with the same duration.

Why investors pay for convexity

With equal duration, the more convex bond beats the less convex one when yields rise by a lot and when they fall by a lot. Nobody gives that away for nothing. If the market has priced it in, the convex bond trades at a higher price (a lower yield), and the buyer is paying for protection.

Go deeper Why a convexity pick-up is not a free lunch

Convexity has a price. Two bonds with equal duration and equal yield, one more convex than the other, cannot both be fairly priced: the convex one would dominate. In practice the more convex bond offers a lower yield, or costs more. The convexity pays off only if yields move enough to outweigh that lower yield. For a small move the less convex bond, with its higher yield, can come out ahead.

Go deeper Preview: negative convexity

A callable bond is the case where the “always positive” rule breaks. When yields fall far enough, the issuer can call the bond, so its price stops rising and the curve bends down. That is negative convexity, and it is measured with effective convexity in a later module. Everything in this lesson assumes an option-free bond.