Fixed Income · Reading 62

Bond Convexity

CFA Level I · Fixed Income · Reading 62: Yield-Based Bond Convexity and Portfolio Properties · about 28 min

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What you'll learn

LOS 62.a Calculate and interpret convexity (cash-flow and approximate methods) and describe the convexity adjustment.
LOS 62.b Estimate a bond's percentage (or money) price change for a given yield change using duration and convexity.
LOS 62.c Calculate portfolio duration and convexity as weighted averages and explain the limitations of these measures.

Module 62.1

Yield-Based Bond Convexity and Portfolio Properties

This reading adds convexity to duration. It shows how convexity is measured, how the convexity adjustment improves a duration-based estimate of a bond's price change, and how duration and convexity are combined for a portfolio.

LOS 62.a — Convexity and the convexity adjustment

Modified duration provides a straight-line (first-order) estimate of how a bond's full price responds to a change in its yield to maturity. The true price-yield relationship of an option-free bond is curved. It is convex toward the origin, so the straight line lies below the actual curve. For small yield changes the gap is negligible; for large changes a duration-only estimate becomes noticeably wrong. Convexity measures the curvature of the price-yield relationship, and the convexity adjustment (a second-order term) corrects the duration estimate.

Convexity from the cash flows

Convexity can be built cash flow by cash flow, exactly like Macaulay duration. For the cash flow paid at period $t$ , with periodic yield $r$ (the annual YTM divided by the periods per year):

convexity_{t} = \frac{t ( t + 1 )}{( 1 + r ) ^{2}}

The bond's convexity is the weighted average of these figures, where each weight is the cash flow's present value divided by the bond's price (the same weights used for Macaulay duration). If coupons are paid more than once a year, the result is in periods squared, so annualize by dividing by the periodicity squared (by $2^{2} = 4$ for a semiannual bond, by $4^{2} = 16$ for a quarterly bond).

Worked example. A 3-year, 5% annual-pay bond yields 7%, so its price is 94.751.

t	Cash flow	PV at 7%	Weight	$t (t + 1) /1.0 7^{2}$
1	5	4.673	0.0493	1.747
2	5	4.367	0.0461	5.241
3	105	85.711	0.9046	10.481

Convexity $= 0.0493 \times 1.747 + 0.0461 \times 5.241 + 0.9046 \times 10.481 \approx 9.81$ . No annualizing is needed because the bond pays annually.

Approximate convexity

A pricing model can reprice the bond after a yield decrease ( $V_{-}$ ) and an equal yield increase ( $V_{+}$ ):

Key concept

approximate convexity = \frac{V _{-} + V _{+} - 2 V _{0}}{( Δ Y T M ) ^{2} V _{0}}

The numerator, $V_{-} + V_{+} - 2 V_{0} = (V_{-} - V_{0}) - (V_{0} - V_{+})$ , is how much larger the price rise is than the price fall. That gap exists only because the slope of the curve changes, which is what curvature means. The companion measure uses the same three prices: approximate modified duration $= (V_{-} - V_{+}) / (2 V_{0} Δ Y T M)$ . When the shifts are applied to a benchmark yield curve rather than to the bond's own YTM, the same formula gives effective convexity (Reading 63).

Worked example. $V_{0} = 104.20$ ; after a 30 bp fall in yield the price is 105.48, after a 30 bp rise it is 102.95.
Approximate convexity $= (105.48 + 102.95 - 208.40) / (0.00 3^{2} \times 104.20) = 0.03/0.00093780 \approx 31.99$ ; approximate modified duration $= 2.53/ (2 \times 104.20 \times 0.003) \approx 4.05$ .

What drives convexity

The same features that raise duration raise convexity: a longer maturity, a lower coupon rate and a lower YTM all increase convexity. Between two bonds with the same duration, the one whose cash flows are more dispersed over time has more convexity (a barbell of cash flows is more convex than a bullet).

Positive and negative convexity

	Option-free bond (positive convexity)	Callable bond at low yields (negative convexity)
Shape	Curve bends away from the tangent line	Price is capped near the call price
Equal yield fall vs. rise	Price gain exceeds price loss	Price gain is smaller than price loss
As yields fall	Price rises at an increasing rate	Price rises at a decreasing rate

Callable bonds (and mortgage-backed securities, because of prepayments) exhibit negative convexity at low yields: the issuer is likely to call, so the price cannot rise much above the call price. At higher yields the call is unlikely and the callable bond behaves like an option-free bond with positive convexity. Option-free bonds, including government and zero-coupon bonds, always have positive convexity.

Common exam traps

Forgetting to divide by periodicity squared when annualizing a semiannual convexity.
Using $V_{-} - V_{+}$ (the duration numerator) in the convexity formula. With the prices in the example this gives about 2,698 instead of 31.99.
A negative approximate/effective convexity is a legitimate answer for a callable bond.
The larger the convexity, the larger the error of a duration-only estimate; short-term bonds have little convexity.

LOS 62.b — Price change using duration and convexity

Key concept

%Δ P V^{F u l l} \approx - AnnModDur \times Δ Y T M + \frac{1}{2} \times AnnConvexity \times (Δ Y T M)^{2}

The first term is the duration effect (sign opposite to the yield change). The second is the convexity adjustment; because $(Δ Y T M)^{2}$ is always positive, the adjustment is positive for both increases and decreases in yield when convexity is positive. Duration alone understates the price gain when yields fall and overstates the price loss when yields rise, and the positive convexity term improves both estimates. Enter $Δ Y T M$ as a decimal (40 bp = 0.0040).

Worked example. A bond has ModDur 5.8 and convexity 42, and its yield falls 80 bp: duration effect $= - 5.8 \times (- 0.008) = + 4.64%$ ; convexity effect $= 0.5 \times 42 \times 0.00 8^{2} = + 0.1344%$ ; estimate $= + 4.7744%$ .

How much the convexity term adds. A 10-year, 5% annual-pay bond priced at par has a modified duration of 7.72 and a convexity of 75.0. Its estimated and actual percentage price changes are:

Yield change	Actual	Duration only	Duration + convexity
+100 bp	−7.36%	−7.72%	−7.35%
−100 bp	+8.11%	+7.72%	+8.10%
+200 bp	−14.05%	−15.44%	−13.94%
−200 bp	+17.06%	+15.44%	+16.94%

A duration-only estimate is symmetric, but the actual price rises more than it falls. The convexity term is +0.375% for a 100 bp move and +1.50% for a 200 bp move in either direction. Because it grows with the square of the yield change, doubling the move roughly quadruples the error of a duration-only estimate.

Money convexity

Money convexity is the currency counterpart of money duration:

MoneyCon = AnnConvexity \times P V^{F u l l} of the position

Δ P V^{F u l l} \approx - (MoneyDur \times Δ Y T M) + \frac{1}{2} \times MoneyCon \times (Δ Y T M)^{2}

Maturity, par value and coupon do not enter the money convexity formula directly. Only the annual convexity and the full value of the position do. The money-based estimate and the percentage-based estimate are the same calculation expressed in different units, so they give equivalent estimated prices (apart from rounding).

Worked example. A position with full value $3,900,000, ModDur 5.8 and convexity 42: MoneyDur = $22,620,000; MoneyCon = $163,800,000. For an 80 bp fall in yield: $22, 620, 000 \times 0.008 = $180, 960$ plus $0.5 \times 163, 800, 000 \times 0.00 8^{2} = $5, 241.60$ , a gain of about $186,202 (4.7744% of $3,900,000).

Worked example: estimating a price change from three prices

An 8-year, 4% annual-pay bond yields 4.40% and is priced at 97.351 on a coupon date, so its full price equals its flat price. A pricing model gives 98.996 if the yield falls 25 bp and 95.739 if it rises 25 bp. Estimate the percentage price change for a 75 bp rise in yield and the dollar change for a position of $5,000,000 par.

Step 1. Approximate modified duration $= (98.996 - 95.739) / (2 \times 97.351 \times 0.0025) = 6.691$ .

Step 2. Approximate convexity $= (98.996 + 95.739 - 2 \times 97.351) / (0.002 5^{2} \times 97.351) = 54.24$ .

Step 3. For a 75 bp rise in yield: duration effect $= - 6.691 \times 0.0075 = - 5.0183%$ ; convexity adjustment $= 0.5 \times 54.24 \times 0.007 5^{2} = + 0.1526%$ ; estimate $= - 4.8657%$ .

Step 4. Repricing the bond at 5.15% gives 92.612, a change of −4.8680%. The duration-only estimate misses by about 0.15 percentage points; with the convexity adjustment the miss is about 0.002 percentage points.

Step 5. A position of $5,000,000 par is worth $5, 000, 000 \times 0.97351 = $4, 867, 550$ , so its estimated loss is $0.048657 \times 4, 867, 550 \approx $236, 840$ .

Common exam traps

Leaving out the $\frac{1}{2}$ , or subtracting the convexity term. For the 80 bp fall in the example, leaving it out gives +4.9088% and subtracting the term gives +4.5056%, against +4.7744%.
Getting the sign of the duration effect wrong: yields down means price up.

LOS 62.c — Portfolio duration and convexity

Two approaches exist:

Approach	How	Comment
Aggregate cash flows	One duration/convexity from all portfolio cash flows, discounted at the portfolio's cash flow yield	Theoretically more correct, harder to use; cannot handle bonds whose cash flows are uncertain
Weighted average	$portfolio duration = \sum w_{i} D_{i}$ , $w_{i}$ = full market value of bond $i$ / portfolio value	Used in practice: easier, and can use effective durations of bonds with embedded options

Portfolio convexity is weighted the same way, with weights based on full market values.

Worked example. Consider a portfolio holding $3.5 million par of Bond P (full price 100.00, duration 3.2, convexity 12) and $7.0 million par of Bond Q (full price 92.86, duration 8.6, convexity 88).

Market values: $3.5 \times 1.0000 = $3.50$ million and $7.0 \times 0.9286 = $6.50$ million, a total of $10.00 million.
Weights: 35% in P and 65% in Q (par weights would give 33.3% and 66.7%).
Portfolio duration $= 0.35 (3.2) + 0.65 (8.6) = 6.71$ ; portfolio convexity $= 0.35 (12) + 0.65 (88) = 61.4$ .

Limitations. The weighted-average measure is a valid estimate of the percentage value change per 1% change in yield only if every bond's yield changes by the same amount, which is a parallel shift of the yield curve. Real curves steepen, flatten and twist. Portfolio duration also inherits the straight-line limitation from LOS 62.a: without the convexity term it is accurate only for small yield changes.

To estimate a dollar change in portfolio value: $Δ V \approx - D_{p or t} \times Δ y \times V$ .

Price value of a basis point. The price value of a basis point (PVBP) is the change in a bond's price when its yield changes by one basis point (0.01%): $PVBP \approx modified duration \times V \times 0.0001$ . Convexity is ignored because a 1 bp change is too small for curvature to matter. The product of value and modified duration (without the 0.0001) is money duration.

Common exam traps

Averaging durations equally, or with par-value weights, instead of with market-value weights. For bonds P and Q, equal weights give 5.90 and par weights give 6.80, against 6.71.
Calling the weighted-average approach the theoretically correct one. It is the practical choice; the aggregate cash-flow approach is the theoretically correct one.

Exam shortcuts

For an option-free bond the convexity adjustment is positive whether yields rise or fall.
The convexity term grows with the square of the yield change, so doubling the change roughly quadruples it.

Bottom line

Convexity measures the curvature of the price-yield relationship; for an option-free bond, duration alone understates the price gain when yields fall and overstates the price loss when yields rise.
$%Δ P V^{F u l l} \approx - AnnModDur \times Δ Y T M + \frac{1}{2} \times AnnConvexity \times (Δ Y T M)^{2}$ .
Approximate convexity is $(V_{-} + V_{+} - 2 V_{0}) / [(Δ Y T M)^{2} V_{0}]$ .
A longer maturity, a lower coupon and a lower yield raise convexity; with the same duration, more dispersed cash flows give more convexity.
Callable bonds and mortgage-backed securities can have negative convexity at low yields; option-free bonds always have positive convexity.
Money convexity is annual convexity × the full value of the position.
Portfolio duration and convexity come either from the aggregate portfolio cash flows, the theoretically correct approach, or in practice as averages weighted by full market value, which give the portfolio's percentage change only for a parallel shift of the yield curve.

Quick check

Question 1Core

A 1.5-year option-free bond pays an 8% coupon semiannually and is priced to yield 6.00%. Using the present values of its cash flows as weights, its annualized convexity is closest to:

Show answer and explanation

Correct answer: A

Convexity is the present-value-weighted average of each cash flow's convexity, $t (t + 1) / (1 + r)^{2}$ , using the periodic yield. For a semiannual bond the result is in half-years squared and must be divided by $2^{2} = 4$ to annualize.

Periodic yield $r = 3%$ ; cash flows 4, 4, 104.

PVs: $4/1.03 = 3.8835$ ; $4/1.0 3^{2} = 3.7704$ ; $104/1.0 3^{3} = 95.1747$ ; price $= 102.8286$ .

Weights: 0.0378, 0.0367, 0.9256. Cash-flow convexities: $1 \times 2/1.0 3^{2} = 1.8852$ ; $2 \times 3/1.0 3^{2} = 5.6556$ ; $3 \times 4/1.0 3^{2} = 11.3112$ .

Periodic convexity $= 0.0378 (1.8852) + 0.0367 (5.6556) + 0.9256 (11.3112) \approx 10.748$ ; annualized $= 10.748/4 \approx 2.69$ .

Why the other options are wrong

B. 5.37 divides by the periodicity (2) instead of the periodicity squared (4).
C. 10.75 is the convexity in semiannual periods, before annualizing.

Key takeaway Annualize convexity by dividing by periodicity squared (4 for semiannual).

Practice Questions

Question 2Core

Three option-free bonds are priced at the same yield to maturity. Which one is most likely to have the highest convexity?

Show answer and explanation

Correct answer: B

Convexity is increased by the same features that increase duration: a longer maturity and a lower coupon rate (and a lower yield, which is equal here). The 25-year, 2% coupon bond combines the longest maturity with the lowest coupon, so it has the most convexity.

Why the other options are wrong

A. This bond has both the shorter maturity and the higher coupon, so it has the least convexity of the three.
C. The low coupon raises convexity relative to the 7% bond, but with a 10-year maturity it is less convex than the 25-year bond with the same coupon.

Key takeaway A longer maturity, a lower coupon and a lower yield all raise convexity. For equal duration, more dispersed cash flows give more convexity.

Question 3Core

Which statement about an option-free bond with positive convexity is most accurate?

Show answer and explanation

Correct answer: A

With positive convexity the price-yield curve is steeper at lower yields, so an equal-sized decline in yield produces a larger price gain than the price loss produced by an equal-sized rise in yield.

Why the other options are wrong

B. Bond prices and yields move in opposite directions; convexity does not change that.
C. Convexity describes the asymmetry between price gains and losses. It does not compare the rate of price change with the rate of yield change.

Key takeaway Positive convexity means bigger gains than losses for equal yield moves.

Question 4Core

A portfolio holds the three option-free bonds shown below (prices per 100 of par).

Portfolio holdings

Bond	Par value ($)	Price (per 100 of par)	Duration
Bond 1	1,000,000	112	3.1
Bond 2	1,200,000	86	10.8
Bond 3	800,000	101	6.0

Using the weighted-average method, the portfolio's duration is closest to:

Show answer and explanation

Correct answer: A

Portfolio duration is the average of the bonds' durations weighted by each bond's market value (price per 100 of par times par, divided by 100) as a proportion of the portfolio's total market value.

Market values: $1, 000, 000 \times 1.12 = 1, 120, 000$ ; $1, 200, 000 \times 0.86 = 1, 032, 000$ ; $800, 000 \times 1.01 = 808, 000$ . Total $= 2, 960, 000$ .

D_{p} = \frac{1 , 120 , 000 ( 3.1 ) + 1 , 032 , 000 ( 10.8 ) + 808 , 000 ( 6.0 )}{2 , 960 , 000} = \frac{19 , 465 , 600}{2 , 960 , 000} \approx 6.58

Why the other options are wrong

B. 6.63 is the simple (equal-weighted) average $(3.1 + 10.8 + 6.0) /3$ , which ignores the bonds' different market values.
C. 6.95 weights the durations by par value ( $1.0, 1.2, 0.8$ out of 3.0 million) instead of market value; the discount bond is overweighted.

Key takeaway Weight each bond by its market value. Par-value and equal weights give the wrong answer.

This reading has 26 questions in the full bank. Practice all of them.

Key Takeaways

Annualize convexity by dividing by periodicity squared (4 for semiannual).
A longer maturity, a lower coupon and a lower yield all raise convexity. For equal duration, more dispersed cash flows give more convexity.
Positive convexity means bigger gains than losses for equal yield moves.
Weight each bond by its market value. Par-value and equal weights give the wrong answer.