as a function of the change in its yield would plot as a straight line, with slope equal to D*.

Yet we know from Figure 10.1, and more generally from Malkiel™s five bond-pricing rela-

tionships (specifically relationship 2), that the relationship between bond prices and yields is

not linear. The duration rule is a good approximation for small changes in bond yield, but it is

less accurate for larger changes.

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10 Managing Bond Portfolios

F I G U R E 10.6

100

Bond price convexity.

The percentage

80

Percentage change in bond price

change in bond price

Actual price

is a convex function

60 change

of the change in

yield to maturity.

40

Duration

20 approximation

0

“5 “4 “3 “2 “1 0 1 2 3 4 5

“20

“40

“60

Change in yield to maturity (percentage points)

Figure 10.6 illustrates this point. Like Figure 10.1, this figure presents the percentage

change in bond price in response to a change in the bond™s yield to maturity. The curved line

is the percentage price change for a 30-year maturity, 8% coupon bond, selling at an initial

yield to maturity of 8%. The straight line is the percentage price change predicted by the du-

ration rule: The modified duration of the bond at its initial yield is 11.26 years, so the straight

line is a plot of D* y 11.26 y. Notice that the two plots are tangent at the initial

yield. Thus, for small changes in the bond™s yield to maturity, the duration rule is quite accu-

rate. However, for larger changes in yield, there is progressively more “daylight” between the

two plots, demonstrating that the duration rule becomes progressively less accurate.

Notice from Figure 10.6 that the duration approximation (the straight line) always under-

states the value of the bond; it underestimates the increase in bond price when the yield falls,

and it overestimates the decline in price when the yield rises. This is due to the curvature of

the true price-yield relationship. Curves with shapes such as that of the price-yield relation-

ship are said to be convex, and the curvature of the price-yield curve is called the convexity convexity

of the bond. The curvature of

We can quantify convexity as the rate of change of the slope of the price-yield curve, ex- the price-yield

pressed as a fraction of the bond price.2 As a practical rule, you can view bonds with higher relationship

of a bond.

convexity as exhibiting higher curvature in the price-yield relationship. The convexity of non-

callable bonds, such as that in Figure 10.6, is positive: The slope increases (i.e., becomes less

negative) at higher yields.

2

If you have taken a calculus class, you will recognize that Equation 10.3 for modified duration can be written as

dP/P D* dy. Thus, D* 1/P dP/dy is the slope of the price-yield curve expressed as a fraction of the bond

price. Similarly, the convexity of a bond equals the second derivative (the rate of change of the slope) of the price-

yield curve divided by bond price: 1/P d2P/dy2. The formula for the convexity of a bond with a maturity of n years

making annual coupon payments is:

n

1 CFt

s (t2

Convexity t)t

2

(1 y)t

P (1 y) t 1

where CFt is the cash flow paid to the bondholder at date t; CFt represents either a coupon payment before maturity

or final coupon plus par value at the maturity date.

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360 Part THREE Debt Securities

Convexity allows us to improve the duration approximation for bond price changes.

Accounting for convexity, Equation 10.3 can be modified as follows:3

1

P

( y)2

D* y Convexity (10.4)

2

P

The first term on the right-hand side is the same as the duration rule, Equation 10.3. The sec-

ond term is the modification for convexity. Notice that for a bond with positive convexity, the

second term is positive, regardless of whether the yield rises or falls. This insight corresponds

to the fact noted just above that the duration rule always underestimates the new value of a

bond following a change in its yield. The more accurate Equation 10.4, which accounts for

convexity, always predicts a higher bond price than Equation 10.3. Of course, if the change in

yield is small, the convexity term, which is multiplied by ( y)2 in Equation 10.4, will be ex-

tremely small and will add little to the approximation. In this case, the linear approximation

given by the duration rule will be sufficiently accurate. Thus, convexity is more important as

a practical matter when potential interest rate changes are large.

Convexity is the reason that the immunization examples we considered above resulted in

small errors. For example, if you turn back to Table 10.5 and Figure 10.5, you will see that the

single payment obligation that was funded with a coupon bond of the same duration was well

immunized for small changes in yields. However, for larger yield changes, the two pricing

curves diverged a bit, implying that such changes in yields would result in small surpluses.

This is due to the greater convexity of the coupon bond.

The bond in Figure 10.6 has a 30-year maturity, an 8% coupon, and sells at an initial yield to

maturity of 8%. Because the coupon rate equals yield to maturity, the bond sells at par value,

10.4 EXAMPLE or $1,000. The modified duration of the bond at its initial yield is 11.26 years, and its convex-

ity is 212.4. (Convexity can be calculated using the formula in footnote 2.) If the bond™s yield

Convexity

increases from 8% to 10%, the bond price will fall to $811.46, a decline of 18.85%. The dura-

tion rule, Equation 10.3, would predict a price decline of

P

D* y 11.26 0.02 0.2252 22.52%

P

which is considerably more than the bond price actually falls. The duration-with-convexity rule,

Equation 10.4, is more accurate:

P 1

D* y Convexity ( y)2

P 2

1

(0.02)2

11.26 0.02 212.4 0.1827 18.27%

2

which is far closer to the exact change in bond price.

Notice that if the change in yield were smaller, say 0.1%, convexity would matter less. The

price of the bond actually would fall to $988.85, a decline of 1.115%. Without accounting for

convexity, we would predict a price decline of

P

D* y 11.26 0.001 0.01126 1.126%

P

Accounting for convexity, we get almost the precisely correct answer:

P 1

212.4 (0.001)2 0.01115 1.115%

11.26 0.001

P 2

Nevertheless, the duration rule is quite accurate in this case, even without accounting for

convexity.

3

To use the convexivity rule, you must express interest rates as decimals rather than percentages.

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10 Managing Bond Portfolios

Why Do Investors Like Convexity?

Convexity is generally considered a desirable trait. Bonds with greater curvature gain more in

price when yields fall than they lose when yields rise. For example, in Figure 10.7 bonds A

and B have the same duration at the initial yield. The plots of their proportional price changes

as a function of interest rate changes are tangent, meaning that their sensitivities to changes in

yields at that point are equal. However, bond A is more convex than bond B. It enjoys greater

price increases and smaller price decreases when interest rates fluctuate by larger amounts. If

interest rates are volatile, this is an attractive asymmetry that increases the expected return on

the bond, since bond A will benefit more from rate decreases and suffer less from rate in-

creases. Of course, if convexity is desirable, it will not be available for free: Investors will

have to pay more and accept lower yields on bonds with greater convexity.

10.4 ACTIVE BOND MANAGEMENT

Sources of Potential Profit

Broadly speaking, there are two sources of potential value in active bond management. The

first is interest rate forecasting; that is, anticipating movements across the entire spectrum of

the fixed-income market. If interest rate declines are forecast, managers will increase portfolio

duration; if increases seem likely, they will shorten duration. The second source of potential

profit is identification of relative mispricing within the fixed-income market. An analyst might

believe, for example, that the default premium on one bond is unnecessarily large and the

bond is underpriced.

These techniques will generate abnormal returns only if the analyst™s information or insight

is superior to that of the market. There is no way of profiting from knowledge that rates are

about to fall if everyone else in the market is onto this. In that case, the anticipated lower

future rates are built into bond prices in the sense that long-duration bonds are already selling

100

Percentage change in bond price

80

Bond A

60

40

Bond B

20

0

“5 “4 “3 “2 “1 0 1 2 3 4 5

“20

“40

“60

Change in yield to maturity (percentage points)

F I G U R E 10.7

Convexity of two bonds. Bond A has greater convexity than bond B.

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362 Part THREE Debt Securities

at higher prices that reflect the anticipated fall in future short rates. If the analyst does not have

information before the market does, it will be too late to act on that information”prices will

have responded already to the news. You know this from our discussion of market efficiency.

For now we simply repeat that valuable information is differential information. And it is

worth noting that interest rate forecasters have a notoriously poor track record.

Homer and Leibowitz have developed a popular taxonomy of active bond portfolio strate-

gies. They characterize portfolio rebalancing activities as one of four types of bond swaps. In

the first two swaps, the investor typically believes the yield relationship between bonds or sec-

tors is only temporarily out of alignment. Until the aberration is eliminated, gains can be real-

ized on the underpriced bond during a period of realignment called the workout period.

1. The substitution swap is an exchange of one bond for a nearly identical substitute. The

substitution swap

substituted bonds should be of essentially equal coupon, maturity, quality, call features,

Exchange of one

sinking fund provisions, and so on. A substitution swap would be motivated by a belief

bond for a bond with

that the market has temporarily mispriced the two bonds, with a discrepancy representing

similar attributes but

more attractively a profit opportunity.

priced. An example of a substitution swap would be a sale of a 20-year maturity, 9% coupon

Ford bond callable after five years at $1,050 that is priced to provide a yield to maturity

of 9.05% coupled with a purchase of a 9% coupon General Motors bond with the same

call provisions and time to maturity that yields 9.15%. If the bonds have about the same

credit rating, there is no apparent reason for the GM bonds to provide a higher yield.

Therefore, the higher yield actually available in the market makes the GM bond seem

relatively attractive. Of course, the equality of credit risk is an important condition. If the

GM bond is in fact riskier, then its higher yield does not represent a bargain.

2. The intermarket spread swap is an exchange of two bonds from different sectors of the

intermarket

bond market. It is pursued when an investor believes the yield spread between two