<<

. 88
( 193 .)



>>

bond™s yield. If this were exactly so, however, a graph of the percentage change in bond price
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.
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition




359
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.
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition




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.
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition




361
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.
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition




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

<<

. 88
( 193 .)



>>