F I G U R E 10.3

Duration Zero-coupon bond

Duration as a

30

function of maturity

25

20

15% coupon YTM = 6%

15

3% coupon YTM = 15%

10

15% coupon YTM = 15%

5

0 Maturity

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

We have already established:

Rule 1: The duration of a zero-coupon bond equals its time to maturity.

We also have seen that the three-year coupon bond has a lower duration than the three-year

zero because coupons early in the bond™s life reduce the bond™s weighted average time until

payments. This illustrates another general property:

Rule 2: Holding time to maturity and yield to maturity constant, a bond™s duration and

interest rate sensitivity are higher when the coupon rate is lower.

This property corresponds to Malkiel™s fifth bond-pricing relationship, and is attributable to

the impact of early coupons on the average maturity of a bond™s payments. The lower these

coupons, the less weight these early payments have on the weighted average maturity of all

the bond™s payments. Compare the plots in Figure 10.3 of the durations of the 3% coupon and

15% coupon bonds, each with identical yields of 15%. The plot of the duration of the 15%

coupon bond lies below the corresponding plot for the 3% coupon bond.

Rule 3: Holding the coupon rate constant, a bond™s duration and interest rate sensitivity

generally increase with time to maturity. Duration always increases with maturity

for bonds selling at par or at a premium to par.

This property of duration corresponds to Malkiel™s third relationship and is fairly intui-

tive. What is surprising is that duration need not always increase with time to maturity. For

some deep discount bonds, such as the 3% coupon bond selling to yield 15% in Figure 10.3,

duration may eventually fall with increases in maturity. For virtually all traded bonds, how-

ever, it is safe to assume that duration increases with maturity.

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

348 Part THREE Debt Securities

Notice in Figure 10.3 that for the zero-coupon bond, maturity and duration are equal. For

all the coupon bonds, however, duration increases by less than a year for each year™s increase

in maturity. The slope of the duration graph is less than 1.0, and duration is always less than

maturity for positive-coupon bonds.

While long-maturity bonds generally will be high-duration bonds, duration is a better

measure of the long-term nature of the bond because it also accounts for coupon payments.

Only when the bond pays no coupons is time to maturity an adequate measure; then maturity

and duration are equal.

Notice also in Figure 10.3 that the two 15% coupon bonds have different durations when

they sell at different yields to maturity. The lower yield bond has longer duration. This makes

sense, because at lower yields the more distant payments have relatively greater present

values and thereby account for a greater share of the bond™s total value. Thus, in the weighted-

average calculation of duration, the distant payments receive greater weights, which results in

a higher duration measure. This establishes

Rule 4: Holding other factors constant, the duration and interest rate sensitivity of a coupon

bond are higher when the bond™s yield to maturity is lower.

Rule 4, which is the sixth bond-pricing relationship noted above, applies to coupon bonds. For

zeros, duration equals time to maturity, regardless of the yield to maturity.

Finally, we present an algebraic rule for the duration of a perpetuity. This rule is derived

from and is consistent with the formula for duration given in Equation 10.1, but it is far easier

to use for infinitely lived bonds.

Rule 5: The duration of a level perpetuity is (1 y)/y. For example, at a 15% yield, the

duration of a perpetuity that pays $100 once a year forever will equal 1.15/.15

7.67 years, while at an 8% yield, it will equal 1.08/.08 13.5 years.

Rule 5 makes it obvious that maturity and duration can differ substantially. The maturity of the

perpetuity is infinite, while the duration of the instrument at a 15% yield is only 7.67 years.

The present-value-weighted cash flows early on in the life of the perpetuity dominate the com-

putation of duration. Notice from Figure 10.3 that as their maturities become ever longer, the

durations of the two coupon bonds with yields of 15% both converge to the duration of the

perpetuity with the same yield, 7.67 years.

>

3. Show that the duration of a perpetuity increases as the interest rate decreases, in

Concept

accordance with Rule 4.

CHECK

Durations can vary widely among traded bonds. Table 10.3 presents durations for several

bonds all assumed to pay annual coupons and to yield 8% per year. Duration decreases as

Coupon Rates (% per year)

TA B L E 10.3

Years to Maturity 6 8 10 12

Durations of annual

coupon bonds (initial

1 1.000 1.000 1.000 1.000

bond yield 8%)

5 4.439 4.312 4.204 4.110

10 7.615 7.247 6.996 6.744

20 11.231 10.604 10.182 9.880

Infinite (perpetuity) 13.500 13.500 13.500 13.500

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

EXCE L Applications www.mhhe.com/bkm

> Duration

The Excel duration model analyzes the duration of bonds using the same methodology employed

in Spreadsheet 10.1. The model allows you to analyze durations of bonds for varying coupon rates

and yields. It contains a sensitivity analysis for price and duration of bonds relative to coupon rate

and yield to maturities. The concepts discussed in Section 10.1 can be investigated using the

spreadsheet.

You can learn more about this spreadsheet model by using the interactive version available on

our website at www.mhhe.com/bkm.

A B C D E F

1 Duration of Bonds

2

3 Bond

4 Coupon Rate 0.08

5 Par Value 1000

6 Years Mat 10

7 YTM 0.06 Present

8 Value of

9 Bond Price Years Cashflow Cashflow PVCF(t)

10 1 80 75.4717 75.4717

11 2 80 71.1997 142.3994

12 3 80 67.1695 201.5086

13 4 80 63.3675 253.4700

14 5 80 59.7807 298.9033

15 6 80 56.3968 338.3811

16 7 80 53.2046 372.4320

17 8 80 50.1930 401.5439

18 9 80 47.3519 426.1669

19 10 1080 603.0664 6030.6636

20

21 Sum 1147.2017 8540.9404

22

23 Price 1147.20

24 Duration 7.4450

25

26

27 YTM Price YTM Duration

28 One-Way Table 1147.20 7.4450

29 0.04 1324.44 0.04 7.6372

30 0.045 1276.95 0.045 7.5898

31 0.05 1231.65 0.05 7.5419

coupon rates increase and increases with time to maturity. According to Table 10.3 and Equa-

tion 10.2, if the interest rate were to increase from 8% to 8.1%, the 6% coupon, 20-year bond

would fall in value by about 1.04% ( 11.231 0.1%/1.08) while the 10% coupon, one-

year bond would fall by only 0.093% ( 1 0.1%/1.08). Notice also from Table 10.3 that

duration is independent of coupon rate only for perpetuities.

349

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

350 Part THREE Debt Securities

10.2 PASSIVE BOND MANAGEMENT

Passive managers take bond prices as fairly set and seek to control only the risk of their fixed-

income portfolios. Generally, there are two ways of viewing this risk, depending on the

investor™s circumstances. Some institutions, such as banks, are concerned with protecting the

portfolio™s current net worth or net market value against interest rate fluctuations. Risk-based

capital guidelines for commercial banks and thrift institutions require the setting aside of ad-

ditional capital as a buffer against potential losses in market value incurred from interest rate

fluctuations. The amount of capital required is directly related to the losses that may be in-

curred under various changes in market interest rates. Other investors, such as pension funds,

may have an investment goal to be reached after a given number of years. These investors are

more concerned with protecting the future values of their portfolios.

What is common to the bank and pension fund, however, is interest rate risk. The net worth

of the firm and its ability to meet future obligations fluctuate with interest rates. If they adjust

the maturity structure of their portfolios, these institutions can shed their interest rate risk.

Immunization and dedication techniques refer to strategies that investors use to shield their

immunization

net worth from exposure to interest rate fluctuations.

A strategy to shield

net worth from

interest rate

Immunization

movements.

Many banks and thrift institutions have a natural mismatch between the maturities of assets

and liabilities. For example, bank liabilities are primarily the deposits owed to customers;

these liabilities are short-term in nature and consequently of low duration. Assets largely com-

prise commercial and consumer loans or mortgages. These assets are of longer duration than

deposits, which means their values are correspondingly more sensitive than deposits to inter-

est rate fluctuations. When interest rates increase unexpectedly, banks can suffer serious de-

creases in net worth”their assets fall in value by more than their liabilities.

Similarly, a pension fund may have a mismatch between the interest rate sensitivity of the

assets held in the fund and the present value of its liabilities”the promise to make payments

to retirees. The nearby box illustrates the dangers that pension funds face when they neglect

the interest rate exposure of both assets and liabilities. The article points out that when inter-

est rates change, the present value of the fund™s liabilities change. For example, in some recent

years pension funds lost ground despite the fact that they enjoyed excellent investment returns.

As interest rates fell, the value of their liabilities grew even faster than the value of their as-

sets. The article concludes that funds should match the interest rate exposure of assets and li-

abilities so that the value of assets will track the value of liabilities whether rates rise or fall.

In other words, the financial manager might want to immunize the fund against interest rate

volatility.

Pension funds are not alone in this concern. Any institution with a future fixed obligation

might consider immunization a reasonable risk management policy. Insurance companies, for

example, also pursue immunization strategies. The notion of immunization was introduced by

F. M. Redington (1952), an actuary for a life insurance company. The idea behind immuniza-

tion is that duration-matched assets and liabilities let the asset portfolio meet the firm™s obli-

gations despite interest rate movements.

Consider, for example, an insurance company that issues a guaranteed investment contract,

or GIC, for $10,000. (GICs are essentially zero-coupon bonds issued by the insurance com-

pany to its customers. They are popular products for individuals™ retirement-savings accounts.)

If the GIC has a five-year maturity and a guaranteed interest rate of 8%, the insurance com-

pany is obligated to pay $10,000 (1.08)5 $14,693.28 in five years.

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

How Pension Funds Lost in Market Boom

By now, maybe you sense that pension liabilities

In one of the happiest reports to come out of Detroit

swing more, in either direction, than assets. How

lately, General Motors proclaimed Tuesday that its U.S.

come? In a phrase, most funds are “mismatched,”

pension funds are now “fully funded on an economic

meaning their liabilities are longer-lived than their in-

basis.” Less noticed was GM™s admission that, in ac-

vestments. The longer an obligation, the more its cur-

counting terms, it is still a few cents”well, $3 billion”

rent value reacts to changes in rates. And at a typical

shy of the mark.

pension fund, even though the average obligation is

Wait a minute. If GM™s pension plans were $9.3 bil-

15 years away, the average duration of its bond port-

lion in the hole when the year began, and if the com-

folio is roughly five years.

pany, to its credit, shoveled in $10.4 billion more during

If this seems to defy common sense, it does. No