call date for issues quoted above par and to the maturity date for issues below

par. *-When issued.

Source: Dow Jones/Cantor Fitzgerald.

U.S. Treasury strips as of 3 p.m. Eastern time, also based on transactions of

$1 million or more. Colons in bid-and-asked quotes represent 32nds; 99:01

means 99 1/32. Net changes in 32nds. Yields calculated on the asked quotation.

ci-stripped coupon interest. bp-Treasury bond, stripped prinicipal. np-Treasury

note, stripped principal. For bonds callable prior to maturity, yields are computed

to the earliest call date for issues quoted above par and to the maturity date for

issues below par.

Source: Bear, Stearns & Co. via Street Software Technology Inc.

Maturity

GOVT. BONDS & NOTES Ask

Maturity Ask Rate Mo/Yr Bid Asked Chg. Yld.

Rate Mo/Yr Bid Asked Chg. Yld. 57/8 Nov 01n 100:22 100:24 + 2 5.53

57/8 Jul 99n 99:31 100:01 . . . . 4.98 61/8 Dec 01n 101:07 101:09 + 1 5.56

67/8 Jul 99n 100:00 100:02 . . . . 5.20 61/4 Jan 02n 101:17 101:19 + 1 5.57

6 Aug 99n 100:01 100:03 . . . . 4.75 141/4 Feb 02 120:16 120:22 + 1 5.55

8 Aug 99n 100:07 100:09 . . . . 4.45 61/4 Feb 02n 101:18 101:20 + 1 5.57

57/8 Aug 99n 100:03 100:05 . . . . 4.52 65/8 Mar 02n 102:16 102:18 + 1 5.59

67/8 Aug 99n 100:07 100:09 . . . . 4.50 65/8 Apr 02n 102:18 102:20 + 1 5.59

5 Apr 01n 99:05 99:07 + 1 5.46 71/2 May 02n 104:27 104:29 + 1 5.60

61/4 Apr 01n 101:07 101:09 . . . . 5.48 61/2 May 02n 102:10 102:12 + 2 5.59

55/8 May 01n 100:05 100:07 + 1 5.49 61/4 Jun 02n 101:22 101:24 + 1 5.60

8 May 01n 104:07 104:09 . . . . 5.50 35/8 Jul 02i 99:01 99:02 -1 3.96

31/8 May 01 112:31 113:03 . . . . 5.50 6 Jul 02n 101:00 101:02 + 1 5.61

51/4 May 01n 99:17 99:18 + 1 5.49 63/8 Aug 02n 102:00 102:02 + 1 5.64

61/2 May 01n 101:22 101:24 . . . . 5.50 61/4 Aug 02n 101:21 101:23 + 1 5.64

53/4 Jun 01n 100:13 100:14 + 1 5.51 57/8 Sep 02n 100:21 100:23 + 2 5.62

65/8 Jun 01n 101:30 102:00 + 1 5.53 53/4 Oct 02n 100:10 100:12 + 2 5.62

65/8 Jul 01n 102:02 102:04 + 1 5.51 115/8 Nov 02 117.18 117:22 + 2 5.71

77/8 Aug 01n 104:15 104:17 . . . . 5.54 77/8 Nov 02-07 105:31 106:01 + 2 5.85

33/8 Aug 01 115:05 115:09 . . . . 5.51 3 5/8 Jan 08i 97:05 97:06 -1 4.02

61/2 Aug 01n 101:28 101:30 + 1 5.52 51/2 Feb 08n 97:26 97:26 + 4 5.82

63/8 Sep 01n 101:21 101:23 + 2 5.53 55/8 May 08n 98:15 98:17 + 4 5.84

61/4 Oct 01n 101:14 101:16 + 1 5.54 83/8 Aug 03-08 108:25 108:27 + 3 5.90

71/2 Nov 01n 104:05 104:07 + 1 5.54 4 3/4 Nov 08n 92:12 92:13 + 4 5.81

153/4 Nov 01 121:30 122:04 -2 5.50 83/4 Nov 03-08 110:19 110:23 . . . . 5.91

Source: Reprinted by permission of Dow Jones, from The Wall Street Journal, July 16, 1999. Permission

conveyed through Copyright Clearance Center, Inc.

tice that the spread for the 6 percent bonds is only 2„32, or about .06 percent of the bond™s

value. Don™t you wish that used-car dealers charged similar spreads?

The next column in the table shows the change in price since the previous day. The

price of the 6 percent bonds has increased by 1„32. Finally, the column “Ask Yld” stands

for ask yield to maturity, which measures the return that investors will receive if they

buy the bond at the asked price and hold it to maturity in 2002. You can see that the 6

percent Treasury bonds offer investors a return of 5.61 percent. We will explain shortly

how this figure was calculated.

Find the 6 1/4 August 02 Treasury bond in Figure 3.2.

Self-Test 1

a. How much does it cost to buy the bond?

b. If you already own the bond, how much would a bond dealer pay you for it?

c. By how much did the price change from the previous day?

d. What annual interest payment does the bond make?

e. What is the bond™s yield to maturity?

Valuing Bonds 259

Bond Prices and Yields

In Figure 3.1, we examined the cash flows that an investor in 6 percent Treasury bonds

would receive. How much would you be willing to pay for this stream of cash flows?

To find out, you need to look at the interest rate that investors could earn on similar se-

curities. In 1999, Treasury bonds with 3-year maturities offered a return of about 5.6

percent. Therefore, to value the 6s of 2002, we need to discount the prospective stream

of cash flows at 5.6 percent:

$60 $60 $1,060

PV = + +

(1 + r) (1 + r)2 (1 + r)3

$60 $60 $1,060

= + + = $1,010.77

(1.056) (1.056) (1.056)3

2

Bond prices are usually expressed as a percentage of their face value. Thus we can

say that our 6 percent Treasury bond is worth 101.077 percent of face value, and its

price would usually be quoted as 101.077, or about 101 2„32.

Did you notice that the coupon payments on the bond are an annuity? In other words,

the holder of our 6 percent Treasury bond receives a level stream of coupon payments

of $60 a year for each of 3 years. At maturity the bondholder gets an additional payment

of $1,000. Therefore, you can use the annuity formula to value the coupon payments

and then add on the present value of the final payment of face value:

PV = PV (coupons) + PV (face value)

= (coupon annuity factor) + (face value discount factor)

[ ]

1 1 1

= $60 — + 1,000 —

“

.056 .056(1.056)3 1.0563

= $161.57 + $849.20 = $1,010.77

If you need to value a bond with many years to run before maturity, it is usually easiest

to value the coupon payments as an annuity and then add on the present value of the

final payment.

Calculate the present value of a 6-year bond with a 9 percent coupon. The interest rate

Self-Test 2

is 12 percent.

Bond Prices and Semiannual Coupon Payments

EXAMPLE 1

Thus far we™ve assumed that interest payments occur annually. This is the case for

bonds in many European countries, but in the United States most bonds make coupon

payments semiannually. So when you hear that a bond in the United States has a coupon

rate of 6 percent, you can generally assume that the bond makes a payment of $60/2 =

$30 every 6 months. Similarly, when investors in the United States refer to the bond™s

interest rate, they usually mean the semiannually compounded interest rate. Thus an

interest rate quoted at 5.6 percent really means that the 6-month rate is 5.6/2 = 2.8

260 SECTION THREE

FIGURE 3.3

Cash flows to an investor in $1,030

the 6 percent coupon bond

maturing in 2002. The bond

$1,000

pays semiannual coupons, so

$30 $30 $30 $30 $30

there are two payments of

$30 each year. July $30

1999

Jan July Jan July Jan July

2000 2000 2001 2001 2002 2002

percent.2 The actual cash flows on the Treasury bond are illustrated in Figure 3.3. To

value the bond a bit more precisely, we should have discounted the series of semiannual

payments by the semiannual rate of interest as follows:

$30 $30 $30 $30 $30 $1,030

PV = + + + + +

(1.028) (1.028)2 (1.028)3 (1.028)4 (1.028)5 (1.028)6

= $1,010.91

which is slightly more than the value of $1,010.77 that we obtained when we treated the

coupon payments as annual rather than semiannual.3 Since semiannual coupon pay-

ments just add to the arithmetic, we will stick to our approximation for the rest of the

material and assume annual interest payments.

HOW BOND PRICES VARY WITH INTEREST RATES

As interest rates change, so do bond prices. For example, suppose that investors de-

manded an interest rate of 6 percent on 3-year Treasury bonds. What would be the price

of the Treasury 6s of 2002? Just repeat the last calculation with a discount rate of r =

.06:

$60 $60 $1,060

PV at 6% = + + = $1,000.00

(1.06) (1.06)2 (1.06)3

2 You may have noticed that the semiannually compounded interest rate on the bond is also the bond™s APR,

although this term is not generally used by bond investors. To find the effective rate, we can use a formula

that we presented earlier:

APR m

( )

Effective annual rate = 1 + “1

m

where m is the number of payments each year. In the case of our Treasury bond,

.056 2

( ) “ 1 = 1.028 “ 1 = .0568, or 5.68%

Effective annual rate = 1 + 2

2

3 Why is the present value a bit higher in this case? Because now we recognize that half the annual coupon

payment is received only 6 months into the year, rather than at year end. Because part of the coupon income

is received earlier, its present value is higher.

Valuing Bonds 261

Thus when the interest rate is the same as the coupon rate (6 percent in our example),

the bond sells for its face value.

We first valued the Treasury bond with an interest rate of 5.6 percent, which is lower

than the coupon rate. In that case the price of the bond was higher than its face value.

We then valued it using an interest rate that is equal to the coupon and found that bond

price equaled face value. You have probably already guessed that when the cash flows

are discounted at a rate that is higher than the bond™s coupon rate, the bond is worth less

than its face value. The following example confirms that this is the case.

Bond Prices and Interest Rates

EXAMPLE 2

Investors will pay $1,000 for a 6 percent, 3-year Treasury bond, when the interest rate

is 6 percent. Suppose that the interest rate is higher than the coupon rate at (say) 15 per-

cent. Now what is the value of the bond? Simple! We just repeat our initial calculation

but with r = .15:

$60 $60 $1,060

PV at 15% = + + = $794.51

(1.15) (1.15)2 (1.15)3

The bond sells for 79.45 percent of face value.

We conclude that when the market interest rate exceeds the coupon rate,

bonds sell for less than face value. When the market interest rate is below the

coupon rate, bonds sell for more than face value.

YIELD TO MATURITY VERSUS CURRENT YIELD

Suppose you are considering the purchase of a 3-year bond with a coupon rate of 10

percent. Your investment adviser quotes a price for the bond. How do you calculate the

rate of return the bond offers?

For bonds priced at face value the answer is easy. The rate of return is the coupon

rate. We can check this by setting out the cash flows on your investment:

Cash Paid to You in Year

You Pay 1 2 3 Rate of Return

$1,000 $100 $100 $1,100 10%

Notice that in each year you earn 10 percent on your money ($100/$1,000). In the final

year you also get back your original investment of $1,000. Therefore, your total return

is 10 percent, the same as the coupon rate.

Now suppose that the market price of the 3-year bond is $1,136.16. Your cash flows

are as follows:

Cash Paid to You in Year

You Pay 1 2 3 Rate of Return

$1,136.16 $100 $100 $1,100 ?

What™s the rate of return now? Notice that you are paying out $1,136.16 and receiving

an annual income of $100. So your income as a proportion of the initial outlay is

262 SECTION THREE

$100/$1,136.16 = .088, or 8.8 percent. This is sometimes called the bond™s current

CURRENT YIELD

Annual coupon payments yield.

divided by bond price. However, total return depends on both interest income and any capital gains or

losses. A current yield of 8.8 percent may sound attractive only until you realize that the

bond™s price must fall. The price today is $1,136.16, but when the bond matures 3 years

from now, the bond will sell for its face value, or $1,000. A price decline (i.e., a capi-

tal loss) of $136.16 is guaranteed, so the overall return over the next 3 years must be

less than the 8.8 percent current yield.

Let us generalize. A bond that is priced above its face value is said to sell at a pre-

mium. Investors who buy a bond at a premium face a capital loss over the life of the

bond, so the return on these bonds is always less than the bond™s current yield. A bond

priced below face value sells at a discount. Investors in discount bonds face a capital

gain over the life of the bond; the return on these bonds is greater than the current yield:

Because it focuses only on current income and ignores prospective price

increases or decreases, the current yield mismeasures the bond™s total rate of

return. It overstates the return of premium bonds and understates that of

discount bonds.

We need a measure of return that takes account of both current yield and the change

in a bond™s value over its life. The standard measure is called yield to maturity. The

YIELD TO MATURITY

yield to maturity is the answer to the following question: At what interest rate would the

Interest rate for which the

bond be correctly priced?

present value of the bond™s

payments equals the price.

The yield to maturity is defined as the discount rate that makes the present

value of the bond™s payments equal to its price.