Straight Coupon Bond Suppose you are considering investing in a straight coupon bond that:

Straight Coupon Bond Suppose you are considering investing in a straight coupon bond that:

■ Promises interest of $100, paid at the end of each year. ■ Promises to pay the principal amount of $1,000 at the end of 12 years. ■ Investors require an annual yield of 5%.

What is this bond worth today? We are given the following:

Interest, C = $100 every year

Number of periods, N = 12 years Maturity value, M

= $1,000 Yield, r d = 5% per year

---------------------------- $100 + ---------------------------- $1,000 = $886.32 + $556.84 = ∑ $1,443.16

t = 1 ( 1 + 0.05 ) ( 1 + 0.05 )

Using a financial calculator,

Hewlett-Packard Hewlett-Packard Hewlett-Packard Texas Instruments 10B

12C

17B

BA-II Plus

12 N 1000 FV

12 n

12 N

1000 FV 5 I/YR

5 I%YR

CPT PV This bond has a present value greater than its maturity value, so we

PV

PV

say that the bond is selling at a premium from its maturity value. Does this make sense? Yes: The bond pays interest of 10% of its face value every year. But what investors require on their investment—the capitali- zation rate considering the time value of money and the uncertainty of the future cash flows—is 5%. So what happens? The bond paying 10% is attractive— so attractive that its price is bid upward to a price that gives investors the going rate, 5%. In other words, an investor who buys the bond for $1,443.16 will get a 5% return on it if it is held until matu- rity. We say that at $1,443.16, the bond is priced to yield 5% per year.

Valuation of Securities and Options

Suppose, instead, the interest on the bond is $50 every year—a 5% coupon rate—instead of $100 every year. Then,

Interest, C = $50 every year

Number of periods, N = 12 years Maturity value, M

= $1,000 Yield, r d = 5% per year

---------------------------- $50 + ---------------------------- $1,000 = $443.16 + $556.84 = ∑ $1,000.00

t = 1 ( 1 + 0.05 ) ( 1 + 0.05 )

The bond’s present value is equal to its maturity value and we say that the bond is selling “at par.” Investors will pay the maturity value for a bond that pays the going rate for bonds of similar risk. In other words, if an investor buys the 5% coupon bond for $1,000.00, the investor will earn a 5% annual return on the investment if the bond is held until maturity. 3

Suppose, instead, the interest on the bond is $20 every year—a 2% coupon rate. Then,

Interest, C = $20 every year

Number of periods, N = 12 years Maturity value, M

= $1,000 Yield, r d = 5% per year

---------------------------- $20 + ---------------------------- $1,000 = $177.26 + $556.84 = ∑ $734.10

1 + 0.05 t t 1 ( ) ( ) The bond sells below its maturity value and is said to be trading at a dis-

1 + 0.05 t

count from its maturity value. Why? Because investors are not going to pay the maturity value for a bond that pays less than the going rate for bonds of similar risk. If an investor can buy other bonds that yield 5%, why pay the maturity value—$1,000 in this case—for a bond that pays only 2%? They wouldn’t. Instead, the price of this bond would fall to a price that provides an investor a yield of 5%. In other words, if an investor buys the 2% coupon bond for $734.10, the investor will earn a 5% annual return on the investment if the bond is held until maturity.

3 This statement will be qualified later when we discuss assumptions inherent in a yield-to-maturity calculation.

THE FUNDAMENTALS OF VALUATION

So when we look at the value of a bond, we see that its present value is dependent on the relation between the coupon rate and the yield. We can see this relation in our example:

If a bond has a yield of 5% so we say it and a coupon rate of ...

it will sell for ... is selling at ...

a premium 5%

a discount As another example for valuing a straight coupon bond, suppose we

have a $1,000 face value bond with a 10% coupon rate, that pays inter- est at the end of each year and matures in five years. If the required yield is 5%, the value of the bond is:

V = ∑ ---------------------------- $100 + ----------------------------- $1,000 = $432.95 + $783.53 = $1,216.48

5 t = 1 ( 1 + 0.05 ) ( 1 + 0.05 )

If the yield is 10%, the same as the coupon rate, the bond sells at matu- rity value:

---------------------------- $100 + ----------------------------- ∑ $1,000 = $379.08 + $620.92 = $1,000.00

If the yield is 15%, the bond’s value is less than its maturity value: 5 $100

---------------------------- + ----------------------------- ∑ $1,000 = $335.21 + $497.18 = $832.39

When we hold the coupon rate constant and vary the required yield, we see that:

If a bond has a coupon rate so we say it of 10% and a yield of ...

it will sell for ... is selling at ...

a premium 10%

a discount

Valuation of Securities and Options

We see a relation developing between the coupon rate, the yield, and the value of a debt security:

■ If the coupon rate is more than the yield, the security is worth more than its maturity value—it sells at a premium. ■ If the coupon rate is less than the yield, the security is less than its maturity value—it sells at a discount. ■ If the coupon rate is equal to the yield, the security is valued at its maturity value.

We can extend the valuation of debt to securities that pay interest every six months. But before we do this, we must grapple with a bit of semantics. In Wall Street parlance, the term yield-to-maturity is used to describe an annualized yield on a security if the security is held to matu- rity. For example, if a bond has a return of 5% over a six-month period, the annualized yield-to-maturity for a year is 2 times 5% or 10%.

Yield-to-maturity = r d ×2

If a debt security promises interest every six months, there are a couple of things to watch out for in calculating the security’s value. First, the r d we use to discount cash flows is the six-month yield, not an annual yield. Second, the number of periods is the number of six-month periods until maturity, not the number of years to maturity.

Suppose we are interested in valuing a bond with a maturity value of $1,000 that matures in five years and promises a coupon of 4% per year, with interest paid semiannually. This 4% coupon rate tells us that 2%, or $20, is paid every six months. What is the bond’s value if the yield-to-maturity is 6%? From the bond’s description we know that:

Interest, C = $20 every six months

Number of periods, N=5 × 2 = 10 six-month periods Maturity value, M

= $1,000 Yield, r d = 6%/2 = 3% for six-month period

The value of the bond is:

---------------------------- + ------------------------------- = $170.60 + $744.09 = ∑ $914.70

t = 1 ( 1 + 0.03 ) t ( 1 + 0.03 ) 10

If the yield-to-maturity is 8%, then:

THE FUNDAMENTALS OF VALUATION

Interest, C = $20 every six months

Number of periods, N=5 × 2 = 10 six-month periods Maturity value, M

= $1,000 Yield, r d = 8%/2 = 4% for six-month period

and the value of the bond is:

---------------------------- $20 + ------------------------------- $1,000 = $162.22 + $675.56 = ∑ $837.78

t = 1 ( 1 + 0.04 ) ( 1 + 0.04 )

We can see the relation between the yield-to-maturity and the value of the 4% coupon bond in Exhibit 9.3. The greater the required yield, the lower the present value of the bond. This makes sense since a higher yield-to-maturity required by the market means that the future cash flows are discounted at higher rates.

EXHIBIT 9.3 Value of a 4% Coupon Bond with Five Years to Maturity and

Semiannual Interest

Valuation of Securities and Options