Unformatted text preview: WEB APPENDIX 7A
Zero Coupon Bonds
To understand how zeros are used and analyzed, consider the zeros that are going
to be issued by Vandenberg Corporation, a shopping center developer. Vandenberg is developing a new shopping center in San Diego, California; and it needs
$50 million. The company does not anticipate major cash flows from the project for
about 5 years. However, Pieter Vandenberg, the president, plans to sell the center
once it is fully developed and rented, which should take about 5 years. Therefore,
Vandenberg wants to use a financing vehicle that will not require cash outflows
for 5 years; and he has decided on a 5year zero coupon bond issue, each bond
having a maturity value of $1,000.
Vandenberg Corporation is an Arated company, and Arated zeros with 5year
maturities yield 6% at this time (5year coupon bonds also yield 6%). The company
is in the 40% federalplusstate tax bracket. Pieter Vandenberg wants to know the
firm’s aftertax cost of debt if it uses 6%, 5year maturity zeros; and he wants to
know what the bond’s cash flows will be. Table 7A1 provides an analysis of the
situation, and the following numbered paragraphs explain the table.
1. The information in the “Basic Data” section, except the issue price, was given
in the preceding paragraph; and the information in the “Analysis” section was
calculated using the known data. The maturity value of the bond is always set
at $1,000 or some multiple thereof.
2. The issue price is the PV of $1,000, discounted back 5 years at the rate rd ¼ 6%,
annual compounding. Using a financial calculator, we input N ¼ 5, I/YR ¼ 6,
PMT ¼ 0, and FV ¼ 1000, then press the PV key to find PV ¼ $747.26. Note
that $747.26 compounded annually for 5 years at 6% will grow to $1,000 as
shown in Table 7A1’s time line on Line 1.
3. The accrued values as shown on Line 1 in the analysis section represent the
compounded value of the bond at the end of each year. The accrued value for
Year 0 is the issue price; the accrued value for Year 1 is found as $747.26(1.06) ¼ Table 7A1
Basic Data
Maturity value
rd
M a tu r i ty
C o r p o r a te ta x r a te
I s s ue p ric e Analysis of a Zero Coupon Bond from Issuer’s Perspective
$1, 000
6.00%, annual compounding
5 y e a rs
40. 00%
$747. 26 Analysis
0 6% 1 2 3 4 5 Years (1) Yearend accrued value
$747.26
$792.10
$839.62
$890.00
$943.40
$1,000.00
(2) Interest deduction
44.84
47.52
50.38
53.40
56.60
(3) Tax savings (40%)
17.94
19.01
20.15
21.36
22.64
(4) Cash flow to Vandenberg
747.26
17.94
19.01
20.15
21.36
977.36
Aftertax cost of debt
3.60%
Face value of bonds the company must issue to raise $50 million = Amount needed/Issue price as % of par
= $50,000,000/0.74726
≈ $66,911,000 7A1 7A2 Web Appendix 7A $792.10; the accrued value at the end of Year 2 is $747.26(1.06)2 ¼ $839.62; and in
general, the value at the end of any Year N is as follows:
7A1 4. Accrued value at the end of year N ¼ Issue price Â ð1 þ rd ÞN The interest deduction as shown on Line 2 represents the increase in accrued
value during the year. Thus, interest in Year 1 ¼ $792.10 À $747.26 ¼ $44.84.
The general equation is as follows:
Interest in Year N ¼ Accrued valueN À Accrued valueNÀ1 7A2 5. This method of calculating taxable interest is specified in the Tax Code.
The company can deduct interest each year even though the payment is not
made in cash. This deduction lowers the taxes that would otherwise be paid,
producing the following tax savings:
Tax savings ¼ ðInterest deductionÞðTÞ
¼ $44:84ð0:4Þ 7A3 ¼ $17:94 in Year 1 6. 7. Line 4 represents cash flows on a time line; it shows the cash flow at the end of
Years 0 through 5. At Year 0, the company receives the $747.26 issue price. The
company also has positive cash inflows equal to the tax savings during Years
1 through 4. Finally, in Year 5, it must pay the $1,000 maturity value; but
it receives one more year of interest tax savings. Therefore, the net cash flow in
Year 5 is –$1,000 þ $22.64 ¼ À$977.36.
Next, we can determine the aftertax cost (or aftertax yield to maturity) of
issuing the bonds. Since the cash flow stream is uneven, the aftertax yield to
maturity is found by entering the aftertax cash flows, shown on Line 4 of Table
7A1, into the cash flow register and pressing the IRR key on the financial
calculator. The IRR is the aftertax cost of zero coupon debt to the company.
Conceptually, here is the situation:
N
X $747:26
ð1 þ rdðATÞ Þ0 8. þ $17:94
ð1 þ rdðATÞ Þ1 þ CFN t¼ 1 7A4 ð1 þ rdðATÞ ÞN $19:01
ð1 þ rdðATÞ Þ2 þ ¼0 $20:15
ð1 þ rdðATÞ Þ3 þ $21:36
ð1 þ rdðATÞ Þ4 þ À $977:36
ð1 þ rdðATÞ Þ5 ¼0 The value of rd(AT) ¼ 0.036 ¼ 3.6%, found with a financial calculator, produces
the equality; and it is the cost of this debt. (Input in the cash flow register
CF0 ¼ 747.26, CF1 ¼ 17.94, and so forth, out to CF5 ¼ –977.36. Then press the
IRR key to find rd(AT) ¼ 3.6%.)
Note that rd(1 À T) ¼ 6%(0.6) ¼ 3.6%. As we will see in Chapter 10, the cost of
capital for regular coupon debt is found using the formula rd(1 – T). Thus,
there is symmetrical treatment for tax purposes for zero coupon and regular
coupon debt; that is, both types of debt use the same aftertax cost formula.
This was Congress’s intent, which is why the Tax Code specifies the treatment
set forth in Table 7A1.1 The purchaser of a zero coupon bond must calculate interest income on the bond
in the same manner as the issuer calculates the interest deduction.
Table 7A2 shows the resulting tax payments for an investor in the 28% tax bracket
who purchases the Vandenberg bond. Given this tax treatment, investors pay
1 Note too that we have analyzed the bond as if the cash flows accrued annually. Generally, to facilitate
comparisons with semiannual payment coupon bonds, the analysis is conducted on a semiannual basis. Web Appendix 7A Table 7A2
Basic Data
M a tu r i ty va l u e
rd
M a tu r i ty
P e r s o na l ta x r a te
I s s ue p ric e Analysis of a Zero Coupon Bond from an Investor’s Perspective
$1, 000
6.00%, annual compounding
5 y e a rs
28. 00%
$747. 26 Analysis
0
(1) Yearend accrued value
(2) Interest income
(3) Tax payment (28%)
(4) Cash flow to investor
Aftertax return $747.26 747.26
4.32% 1 2 3 4 5 $792.10
44.84
12.56
12.56 6% $839.62
47.52
13.31
13.31 $890.00
50.38
14.11
14.11 $943.40
53.40
14.95
14.95 $1,000.00
56.60
15.85
984.15 taxes in each year even though they don’t receive any cash flows until the bond is
sold or matures, a situation that many investors find unattractive. Consequently,
because of the tax situation, pension funds and other taxexempt entities buy most
zero coupon bonds. Individuals do, however, buy taxable zeros for their individual retirement accounts (IRAs). Also, state and local governments issue “taxexempt muni zeros” that are purchased by individuals in high tax brackets.
Not all original issue discount bonds (OIDs) have zero coupons. For example,
Vandenberg might have sold an issue of fiveyear bonds with a 5% coupon at a
time when other bonds with similar ratings and maturities were yielding 6%. Such
bonds would have had a value of $957.88:
Bond value ¼ 5
X $50
$1,000
¼ $957:88
tþ
ð1:06Þ5
t¼ 1 ð1:06Þ If an investor had purchased these bonds at a price of $957.88, the yield to
maturity would have been 6%. The discount of $1,000 – $957.88 ¼ $42.12 would
have been amortized over the bond’s fiveyear life, and it would have been
handled by both Vandenberg and the bondholders exactly as the discount on the
zeros was handled.
Thus, zero coupon bonds are just one type of original issue discount bond.
Any nonconvertible bond whose coupon rate is set below the going market rate at
the time of its issue will sell at a discount, and it will be classified (for tax and other
purposes) as an OID bond.
Shortly after corporations began to issue zeros, investment bankers figured
out a way to create zeros from U.S. Treasury bonds, which at the time were issued
only in coupon form. In 1982, Salomon Brothers bought $1 billion of 12%, 30year
Treasuries. Each bond had 60 coupons worth $60 each, which represented the
interest payments due every six months. Salomon then in effect clipped the coupons and placed them in 60 piles. The last pile also contained the now “stripped”
bond, which represented a promise of $1,000 in the year 2012. These 60 piles of
U.S. Treasury promises were placed with the trust department of a bank and used
as collateral for “zero coupon U.S. Treasury Trust Certificates,” which are, in
essence, zero coupon Treasury bonds. Treasury zeros are, of course, safer than
corporate zeros; so they are very popular with pension fund managers. In response
to this demand, the Treasury has also created its own “Strips” program, which
allows investors to purchase zeros electronically.
Corporate (and municipal) zeros are generally callable at the option of the
issuer, just like coupon bonds, after some stated call protection period. The call
price is set at a premium over the accrued value at the time of the call. Stripped Years 7A3 7A4 Web Appendix 7A U.S. Treasury bonds (Treasury zeros) generally are not callable because the
Treasury normally sells noncallable bonds. Thus, Treasury zeros are completely
protected against reinvestment risk (the risk of having to invest cash flows from a
bond at a lower rate because of a decline in interest rates). QUESTIONS
7A1
7A2
7A3 Do all original issue discount (OID) bonds have zero coupon payments? Explain.
What is a Treasury strip? Are they callable? Explain.
Do Treasury zeros face any interest rate (price) or reinvestment rate risk? Explain. PROBLEMS
7A1
7A2
7A3 7A4
7A5 7A6 ZERO COUPON BONDS A company has just issued 4year zero coupon bonds with
a maturity value of $1,000 and a yield to maturity of 9%. Its tax rate is 40%. What is its
aftertax cost of debt?
ZERO COUPON BONDS An investor in the 35% tax bracket purchases the bond discussed
in Problem 7A1. What is the investor’s aftertax return?
STRIPPED U.S. TREASURY BOND McGwire Company’s pension fund projected that a
significant number of its employees would take advantage of an early retirement program
the company plans to offer in 5 years. Anticipating the need to fund these pensions, the
firm bought zero coupon U.S. Treasury Trust Certificates maturing in 5 years. When these
instruments were originally issued, they were 12% coupon, 30year U.S. Treasury bonds.
The stripped Treasuries are currently priced to yield 10%. Their total maturity value is
$6,000,000. What is their total cost (price) to McGwire today?
ZERO COUPON BOND At the beginning of the year, you purchased a 7year, zero coupon
bond with a yield to maturity of 6.8%. The bond has a face value of $1,000. Your tax rate
is 25%. What total tax will you have to pay on the bond during the first year?
ZEROS AND EXPECTATIONS THEORY A 2year, zero coupon Treasury bond with a
maturity value of $1,000 has a price of $873.4387. A 1year, zero coupon Treasury bond
with a maturity value of $1,000 has a price of $938.9671. If the pure expectations theory is
correct, for what price should 1year, zero coupon Treasury bonds sell 1 year from now?
ZERO COUPON BONDS AND EAR Assume that the city of Tampa sold taxexempt (muni)
zero coupon bonds 5 years ago. The bonds had a 25year maturity and a maturity value
of $1,000 when they were issued; and the interest rate built into the issue was a nominal
10%, but with semiannual compounding. The bonds are now callable at a premium of
10% over the accrued value. What effective annual rate of return would an investor
who bought the bonds when they were issued and who still owns them earn if they
are called today? ...
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 Fall '06
 Tapley
 Finance

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