9.3 BOND YIELDS

We have noted that the current yield of a bond measures only the cash income provided by the

bond as a percentage of bond price and ignores any prospective capital gains or losses. We

would like a measure of rate of return that accounts for both current income as well as the

price increase or decrease over the bond™s life. The yield to maturity is the standard measure

of the total rate of return of the bond over its life. However, it is far from perfect, and we will

explore several variations of this measure.

yield to maturity

(YTM)

Yield to Maturity

The discount rate that

makes the present

In practice, an investor considering the purchase of a bond is not quoted a promised rate of

value of a bond™s

return. Instead, the investor must use the bond price, maturity date, and coupon payments to

payments equal

infer the return offered by the bond over its life. The yield to maturity (YTM) is defined as

to its price.

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Essentials of Investments, Companies, 2003

Fifth Edition

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9 Bond Prices and Yields

the discount rate that makes the present value of a bond™s payments equal to its price. This rate

is often viewed as a measure of the average rate of return that will be earned on a bond if it is

bought now and held until maturity. To calculate the yield to maturity, we solve the bond price

equation for the interest rate given the bond™s price.

For example, suppose an 8% coupon, 30-year bond is selling at $1,276.76. What average

rate of return would be earned by an investor purchasing the bond at this price? To answer this

question, we find the interest rate at which the present value of the remaining 60 semiannual

bond payments equals the bond price. This is the rate that is consistent with the observed price

of the bond. Therefore, we solve for r in the following equation

60

$40 $1,000

$1,276.76

(1 r)t (1 r)60

t 1

or, equivalently,

1,276.76 40 Annuity factor(r, 60) 1,000 PV factor(r, 60)

These equations have only one unknown variable, the interest rate, r. You can use a financial

calculator to confirm that the solution to the equation is r .03, or 3% per half-year.6 This is

considered the bond™s yield to maturity, as the bond would be fairly priced at $1,276.76 if the

fair market rate of return on the bond over its entire life were 3% per half-year.

The financial press reports yields on an annualized basis, however, and annualizes the

bond™s semiannual yield using simple interest techniques, resulting in an annual percentage

rate or APR. Yields annualized using simple interest are also called bond equivalent yields.

Therefore, the semiannual yield would be doubled and reported in the newspaper as a bond

equivalent yield of 6%. The effective annual yield of the bond, however, accounts for com-

pound interest. If one earns 3% interest every six months, then after one year, each dollar in-

vested grows with interest to $1 (1.03)2 1.0609, and the effective annual interest rate on

the bond is 6.09%.

The bond™s yield to maturity is the internal rate of return on an investment in the bond. The

yield to maturity can be interpreted as the compound rate of return over the life of the bond

under the assumption that all bond coupons can be reinvested at an interest rate equal to the

bond™s yield to maturity.7 Yield to maturity therefore is widely accepted as a proxy for aver-

age return.

Yield to maturity can be difficult to calculate without a financial calculator. However, it is

easy to calculate with one. Financial calculators are designed with present value and future

value formulas already programmed. The basic financial calculator uses five keys that corre-

spond to the inputs for time value of money problems such as bond pricing:

n i PV FV PMT

• n is the number of time periods. In the case of a bond, n equals the number of periods until

the bond matures. If the bond makes semiannual payments, n is the number of half-year

periods or, equivalently, the number of semiannual coupon payments. For example, if the

bond has 10 years until maturity, you would enter 20 for n, since each payment period is

one-half year.

6

Without a financial calculator, you still could solve the equation, but you would need to use a trial-and-error

approach.

7

If the reinvestment rate does not equal the bond™s yield to maturity, the compound rate of return will differ from

YTM. This is demonstrated in Examples 9.5 and 9.6.

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

308 Part THREE Debt Securities

• i is the interest rate per period, expressed as a percentage (not a decimal). For example, if

the interest rate is 6%, you would enter 6, not 0.06.

• PV is the present value. Many calculators will require that PV be entered as a negative

number, in recognition of the fact that purchase of the bond is a cash outflow, while the

receipt of coupon payments and face value are cash inflows.

• FV is the future value or face value of the bond. In general, FV is interpreted as a one-time

future payment of a cash flow, which, for bonds, is the face (i.e., par) value.

• PMT is the amount of any recurring payment. For coupon bonds, PMT is the coupon

payment; for zero-coupon bonds, PMT will be zero.

Given any four of these inputs, the calculator will solve for the fifth. We can illustrate with

some examples.

Consider the yield to maturity problem that we just solved. We would enter the following in-

puts (in any order):

9.3 EXAMPLE

n 60 The bond has a maturity of 30 years, so it makes 60 semiannual

Bond Valuation payments.

Using a

PMT 40 Each semiannual coupon payment is $40.

Financial

PV ( )1,276.76 The bond can be purchased for $1,276.76, which on some

Calculator

calculators must be entered as a negative number as it is a cash

outflow.

FV 1,000 The bond will provide a one-time cash flow of $1,000 when it

matures.

Given these inputs, you now use the calculator to find the interest rate at which $1,276.76

actually equals the present value of the 60 payments of $40 each plus the one-time payment

of $1,000 at maturity. On most calculators, you first punch the “compute” key (labeled

COMP or CPT) and then enter i to have the interest rate computed. If you do so, you will find

that i 3, or 3% semiannually, as we claimed. (Notice that just as the cash flows are paid

semiannually, the computed interest rate is a rate per semiannual time period.)

You can also find bond prices given a yield to maturity. For example, we saw in Example

9.2 that if the yield to maturity is 5% semiannually, the bond price will be $810.71. You can

confirm this with the following inputs on your calculator:

n 60; i 5; FV 1,000; PMT 40

and then computing PV to find that PV 810.71. Once again, your calculator may report

the result as 810.71.

Yield to maturity is different from the current yield of a bond, which is the bond™s annual

current yield

coupon payment divided by the bond price. For example, for the 8%, 30-year bond currently

Annual coupon

selling at $1,276.76, the current yield would be $80/$1,276.76 0.0627, or 6.27% per year.

divided by bond price.

In contrast, recall that the effective annual yield to maturity is 6.09%. For this bond, which is

selling at a premium over par value ($1,276 rather than $1,000), the coupon rate (8%) exceeds

the current yield (6.27%), which exceeds the yield to maturity (6.09%). The coupon rate ex-

ceeds current yield because the coupon rate divides the coupon payments by par value

($1,000) rather than by the bond price ($1,276). In turn, the current yield exceeds yield to ma-

turity because the yield to maturity accounts for the built-in capital loss on the bond; the bond

premium bonds bought today for $1,276 will eventually fall in value to $1,000 at maturity.

This example illustrates a general role: for premium bonds (bonds selling above par

Bonds selling above

value), coupon rate is greater than current yield, which in turn is greater than yield to maturity.

par value.

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

309

9 Bond Prices and Yields

For discount bonds (bonds selling below par value), these relationships are reversed (see discount bonds

Concept Check 3). Bonds selling below

It is common to hear people talking loosely about the yield on a bond. In these cases, they par value.

almost always are referring to the yield to maturity.

<

3. What will be the relationship among coupon rate, current yield, and yield to matu- Concept

rity for bonds selling at discounts from par? Illustrate using the 8% (semiannual

CHECK

payment) coupon bond assuming it is selling at a yield to maturity of 10%.

Yield to Call

Yield to maturity is calculated on the assumption that the bond will be held until maturity.

What if the bond is callable, however, and may be retired prior to the maturity date? How

should we measure average rate of return for bonds subject to a call provision?

Figure 9.4 illustrates the risk of call to the bondholder. The colored line is the value at var-

ious market interest rates of a “straight” (that is, noncallable) bond with par value of $1,000,

an 8% coupon rate, and a 30-year time to maturity. If interest rates fall, the bond price, which

equals the present value of the promised payments, can rise substantially. Now consider a

bond that has the same coupon rate and maturity date but is callable at 110% of par value, or

$1,100. When interest rates fall, the present value of the bond™s scheduled payments rises, but

the call provision allows the issuer to repurchase the bond at the call price. If the call price is

less than the present value of the scheduled payments, the issuer can call the bond at the ex-

pense of the bondholder.

The dark line in Figure 9.4 is the value of the callable bond. At high interest rates, the risk

of call is negligible, and the values of the straight and callable bonds converge. At lower rates,

however, the values of the bonds begin to diverge, with the difference reflecting the value of

the firm™s option to reclaim the callable bond at the call price. At very low rates, the bond is

called, and its value is simply the call price, $1,100.

This analysis suggests that bond market analysts might be more interested in a bond™s yield

to call rather than its yield to maturity if the bond is especially vulnerable to being called. The

yield to call is calculated just like the yield to maturity, except that the time until call replaces

time until maturity and the call price replaces the par value. This computation is sometimes

called “yield to first call,” as it assumes the bond will be called as soon as it is first callable.

F I G U R E 9.4

Prices ($)

2,000

Bond prices: Callable

1,800 and straight debt.

1,600 Coupon 8%;

maturity 30 years;

1,400 Straight bond

semiannual payments

1,200

1,100

1,000

Callable

800 bond

600

400

200 Interest

0 rate

3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13%

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

310 Part THREE Debt Securities

Suppose the 8% coupon, 30-year maturity bond sells for $1,150 and is callable in 10 years

at a call price of $1,100. Its yield to maturity and yield to call would be calculated using the

9.4 EXAMPLE following inputs:

Yield to Call

Yield to Call Yield to Maturity

Coupon payment $40 $40

Number of semiannual periods 20 periods 60 periods

Final payment $1,100 $1,000

Price $1,150 $1,150

Yield to call is then 6.64% [to confirm this on your calculator, input n 20; PV

( )1,150; FV 1,100; PMT 40; compute i as 3.32%, or 6.64% bond equivalent yield],

while yield to maturity is 6.82% [to confirm, input n 60; PV ( )1,150; FV 1,000; PMT

40; compute i as 3.41%, or 6.82% bond equivalent yield].

We have noted that most callable bonds are issued with an initial period of call protection.

In addition, an implicit form of call protection operates for bonds selling at deep discounts

from their call prices. Even if interest rates fall a bit, deep-discount bonds still will sell below

the call price and thus will not be subject to a call.

Premium bonds that might be selling near their call prices, however, are especially apt to

be called if rates fall further. If interest rates fall, a callable premium bond is likely to provide

a lower return than could be earned on a discount bond whose potential price appreciation is

not limited by the likelihood of a call. Investors in premium bonds often are more interested

in the bond™s yield to call rather than yield to maturity as a consequence, because it may ap-

pear to them that the bond will be retired at the call date.

In fact, the yield reported for callable Treasury bonds in the financial pages of the newspa-

per (see Figure 9.1) is the yield to call for premium bonds and the yield to maturity for dis-

count bonds. This is because the call price on Treasury issues is simply par value. If the bond

is selling at a premium, it is more likely that the Treasury will find it advantageous to call the

bond when it enters the call period. If the bond is selling at a discount from par, the Treasury

will not find it advantageous to exercise its option to call.