Introduction to Corporate Finance

4: Valuing Bonds

4-2b THe BASIC eQUATION (ASSUMING ANNUAL INTereST)

We can value ordinary bonds by developing a simplified version of Equation 4.1. With annual interest,

remember that a bond makes a fixed coupon payment each year. Assume that the bond makes annual

coupon payments of $C for n years, and at maturity the bond makes its final coupon payment and returns

the par value, $M, to investors. (We will deal with the more common occurrence of semiannual coupon

payments shortly.) Using these assumptions, we can replace Equation 4.1 with the following:

Eq. 4.2 P

C

r

CC

0 =(1 +) 12 +(1 +)rr++(1 +)nn+(1 +)

M

r

...

Equation 4.2 says that the bond’s price equals the present value of an n-year ordinary annuity plus

the present value of the lump-sum principal payment.

Price = PV of annuity + PV of lump sum

The annuity consists of a stream of coupon payments, and the lump sum is the bond’s principal or

face value. The bond’s price is simply the sum of the present values of these two components:

Price = PV of coupons + PV of principal

Next, we modify the bond pricing equation once more, borrowing from Equation 3.7 on page 90

to highlight that the price equals the sum of the present value of an annuity and the present value of a

lump-sum payment at maturity.

Eq. 4.2a P

C

rr

M

r

0 ×1

1

(1 )(1)

=−

+











nn+ +

example

On 1 January 2012, Platypus United had a bond outstanding with a coupon rate of 9.125% and a face value of
$1,000. At the end of each year, this bond pays investors $91.25 in interest (0.09125 × $1,000), and it matures
at the end of 2022. Figure 4.1 illustrates the sequence of cash flows that the bond promises investors over
time. Notice that we break up the bond’s cash payments into two separate components. The first component
is an 11-year annuity of $91.25 annual payments. The second component is a lump-sum payment of $1,000 at
maturity.
To calculate the price of this bond, we need to know what rate of return investors demand on bonds that
are as risky as Platypus’s bonds. Assume that the market currently requires an 8% return on these bonds.
Substituting the required return and the payments into Equation 4.2, we can express the current price of this
bond as follows:

Price= +++...++=

$91.25

(1.08)

$91.25

(1.08)

$91.25

(1.08)

$91.25

(1.08)

$1000

(1.08)

1231111 $1,080.31

Figure 4.1 shows that the present value of the 11-year coupon stream is $651.43, and the present value of
the principal repayment is $428.88. That gives a combined bond value of $1,080.31, as shown below.

=















×−

































=

==

=+=

ofcoupons

$91.25

0.08

1

1

1. 08

$651.4 3

ofprincipal

$1, 000

1. 08

$428.88

Priceofbond$ 651 .43 $428.88 $1,080.31

11

11

PV

PV

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CONCEPTS