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http://www.smartmoney.com/bonds

Related Websites

This site has several tools that apply to concepts

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http://www.bondresources.com in the analysis of bond swaps, as well as a portfolio

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n this chapter, we turn to various strategies that bond managers can pursue, mak-

I ing a distinction between passive and active strategies. A passive investment strat-

egy takes market prices of securities as fairly set. Rather than attempting to beat

the market by exploiting superior information or insight, passive managers act to

maintain an appropriate risk-return balance given market opportunities. One special

case of passive management is an immunization strategy that attempts to insulate

the portfolio from interest rate risk.

An active investment strategy attempts to achieve returns that are more than

commensurate with the risk borne. In the context of bond portfolios, this style of man-

agement can take two forms. Active managers either use interest rate forecasts

to predict movements in the entire bond market, or they employ some form of intra-

market analysis to identify particular sectors of the market (or particular securities)

that are relatively mispriced.

Because interest rate risk is crucial to formulating both active and passive strate-

gies, we begin our discussion with an analysis of the sensitivity of bond prices to in-

terest rate fluctuations. This sensitivity is measured by the duration of the bond, and

we devote considerable attention to what determines bond duration. We discuss sev-

eral passive investment strategies, and show how duration-matching techniques can

be used to immunize the holding-period return of a portfolio from interest rate risk.

After examining the broad range of applications of the duration measure, we consider

refinements in the way that interest rate sensitivity is measured, focusing on the con-

cept of bond convexity. Duration is important in formulating active investment strate-

gies as well, and we next explore several of these strategies. We consider

strategies based on intramarket analysis as well as on interest rate fore-

casting. We also show how interest rate swaps may be used in bond port-

folio management.

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340 Part THREE Debt Securities

10.1 INTEREST RATE RISK

You know already that there is an inverse relationship between bond prices and yields and that

interest rates can fluctuate substantially. As interest rates rise and fall, bondholders experience

capital losses and gains. It is these gains or losses that make fixed-income investments risky,

even if the coupon and principal payments are guaranteed, as in the case of Treasury

obligations.

Why do bond prices respond to interest rate fluctuations? In a competitive market, all secu-

rities must offer investors fair expected rates of return. If a bond is issued with an 8% coupon

when competitive yields are 8%, then it will sell at par value. If the market rate rises to 9%,

however, who would purchase an 8% coupon bond at par value? The bond price must fall until

its expected return increases to the competitive level of 9%. Conversely, if the market rate falls

to 7%, the 8% coupon on the bond is attractive compared to yields on alternative investments.

Investors eager for that return will respond by bidding the bond price above its par value until

the total rate of return falls to the market rate.

Interest Rate Sensitivity

The sensitivity of bond prices to changes in market interest rates is obviously of great concern

to investors. To gain some insight into the determinants of interest rate risk, turn to Figure

10.1, which presents the percentage changes in price corresponding to changes in yield to

maturity for four bonds that differ according to coupon rate, initial yield to maturity, and time

200

Percentage change in bond price

150

100

50

Initial

0 Bond Coupon Maturity YTM

“5 “4 “3 “2 “1 0 1 2 3 4 5 A 12% 5 years 10%

B 12% 30 years 10%

C 3% 30 years 10%

“50 D 3% 30 years 6%

Change in yield to maturity (percentage points)

F I G U R E 10.1

Change in bond price as a function of change in yield to maturity

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10 Managing Bond Portfolios

to maturity. All four bonds illustrate that bond prices decrease when yields rise, and that the

price curve is convex, meaning that decreases in yields have bigger impacts on price than

increases in yields of equal magnitude. We summarize these observations in the following two

propositions:

1. Bond prices and yields are inversely related: as yields increase, bond prices fall; as

yields fall, bond prices rise.

2. An increase in a bond™s yield to maturity results in a smaller price change than a decrease

in yield of equal magnitude.

Now compare the interest rate sensitivity of bonds A and B, which are identical except for

maturity. Figure 10.1 shows that bond B, which has a longer maturity than bond A, exhibits

greater sensitivity to interest rate changes. This illustrates another general property:

3. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of

short-term bonds.

Although bond B has six times the maturity of bond A, it has less than six times the interest

rate sensitivity. Although interest rate sensitivity seems to increase with maturity, it does so

less than proportionally as bond maturity increases. Therefore, our fourth property is that:

4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as

maturity increases. In other words, interest rate risk is less than proportional to bond

maturity.

Bonds B and C, which are alike in all respects except for coupon rate, illustrate another point.

The lower-coupon bond exhibits greater sensitivity to changes in interest rates. This turns out

to be a general property of bond prices:

5. Interest rate risk is inversely related to the bond™s coupon rate. Prices of high-coupon

bonds are less sensitive to changes in interest rates than prices of low-coupon bonds.

Finally, bonds C and D are identical except for the yield to maturity at which the bonds cur-

rently sell. Yet bond C, with a higher yield to maturity, is less sensitive to changes in yields.

This illustrates our final property:

6. The sensitivity of a bond™s price to a change in its yield is inversely related to the yield to

maturity at which the bond currently is selling.

The first five of these general properties were described by Malkiel (1962) and are sometimes

known as Malkiel™s bond-pricing relationships. The last property was demonstrated by Homer

and Liebowitz (1972).

These six propositions confirm that maturity is a major determinant of interest rate risk.

However, they also show that maturity alone is not sufficient to measure interest rate sensitiv-

ity. For example, bonds B and C in Figure 10.1 have the same maturity, but the higher coupon

bond has less price sensitivity to interest rate changes. Obviously, we need to know more than

a bond™s maturity to quantify its interest rate risk.

To see why bond characteristics such as coupon rate or yield to maturity affect interest rate

sensitivity, let™s start with a simple numerical example.

Table 10.1 gives bond prices for 8% annual coupon bonds at different yields to maturity

and times to maturity. (For simplicity, we assume coupons are paid once a year rather than

semiannually.) The shortest term bond falls in value by less than 1% when the interest rate

increases from 8% to 9%. The 10-year bond falls by 6.4% and the 20-year bond by more

than 9%.

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342 Part THREE Debt Securities

Bond™s Yield to Maturity T 1 Year T 10 Years T 20 Years

TA B L E 10.1

Prices of 8% 1,000 1,000 1,000

8% annual 9% 990.83 935.82 908.71

coupon

Percent change in price* 0.92% 6.42% 9.13%

bonds

*Equals value of bond at a 9% yield to maturity minus value of bond at (the original) 8% yield, divided by the value at 8% yield.

Bond™s Yield to Maturity T 1 Year T 10 Years T 20 Years

TA B L E 10.2

Prices 8% 925.93 463.19 214.55

of zero- 9% 917.43 422.41 178.43

coupon

Percent change in price* 0.92% 8.80% 16.84%

bonds

*Equals value of bond at a 9% yield to maturity minus value of bond at (the original) 8% yield, divided by the value at 8% yield.

Let us now look at a similar computation using a zero-coupon bond rather than the 8%

coupon bond. The results are shown in Table 10.2.

For both maturities beyond one year, the price of the zero-coupon bond falls by a greater

proportional amount than the price of the 8% coupon bond. The observation that long-term

bonds are more sensitive to interest rate movements than short-term bonds suggests that in

some sense a zero-coupon bond represents a longer term investment than an equal-time-to-

maturity coupon bond. In fact, this insight about effective maturity is a useful one that we can

make mathematically precise.

To start, note that the times to maturity of the two bonds in this example are not perfect

measures of the long- or short-term nature of the bonds. The 8% bond makes many coupon

payments, most of which come years before the bond™s maturity date. Each payment may be

considered to have its own “maturity date,” which suggests that the effective maturity of the

bond should be measured as some sort of average of the maturities of all the cash flows paid

out by the bond. The zero-coupon bond, by contrast, makes only one payment at maturity. Its

time to maturity is a well-defined concept.

Duration

To deal with the concept of the “maturity” of a bond that makes many payments, we need a

measure of the average maturity of the bond™s promised cash flows to serve as a summary sta-

tistic of the effective maturity of the bond. This measure also should give us some information

on the sensitivity of a bond to interest rate changes because we have noted that price sensitiv-

ity tends to increase with time to maturity.

Macaulay™s

Frederick Macaulay (1938) called the effective maturity concept the duration of the bond.

duration

Macaulay™s duration is computed as the weighted average of the times to each coupon or

A measure of the

principal payment made by the bond. The weight applied to each time to payment clearly

effective maturity of a

should be related to the “importance” of that payment to the value of the bond. Therefore,

bond, defined as the

weighted average of the weight for each payment time is the proportion of the total value of the bond accounted

the times until each for by that payment. This proportion is just the present value of the payment divided by the

payment, with weights

bond price.

proportional to the

Figure 10.2 can help us interpret Macaulay™s duration by showing the cash flows made by

present value of the

an eight-year maturity bond with a coupon of 9%, selling at a yield to maturity of 10%. In the

payment.

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343

10 Managing Bond Portfolios

1,200

1,100

1,000