Bonds and Their Valuation
After reading this chapter, students should be able to:

List
the
four
main
classifications
differentiate among them.

Identify the key characteristics common to all bonds.

Calculate the value of a bond with annual or semiannual
interest payments.

Explain why the market value of an outstanding fixed-rate
bond will fall when interest rates rise on new bonds of
equal risk, or vice versa.

Calculate the current yield, the yield to maturity, and/or
the yield to call on a bond.

Differentiate between interest rate risk, reinvestment rate
risk, and default risk.

List major types of corporate bonds and distinguish among
them.

Explain the importance of bond ratings and list some of the
criteria used to rate bonds.

Differentiate
among
the
following
liquidation, and reorganization.

Read and understand the information provided on the bond
market page of your newspaper
of
terms:
bonds
and
Insolvent,
Characteristics of Bonds
A bond is a long-term contract under which a borrower (the
issuer) agrees to make payments of interest and principal, on
specific dates, to the holders (creditors) of the bond.
Bearer bond - Bonds that are not registered on the books of the
issuer. Such bonds are held in physical form by the owner, who
receives interest payments by physically detaching coupons from
the bond certificate and delivering them to the paying agent.
Registered bond - A bond whose issuer records ownership and
interest payments. Differs from a bearer bond, which is traded
without record of ownership and whose possession is the only
evidence of ownership.
1
Par value - Also called the maturity value or face value; the
amount that an issuer agrees to pay at the maturity date.
Coupon interest rate - With bonds, notes, or other fixed income
securities, the stated percentage rate of interest, usually paid
twice a year (semiannually). Coupon payments - A bond's dollar
interest payments.
Maturity date - Date on which the principal amount of a bond or
other debt instrument becomes due and payable.
Call provision - An embedded option granting a bond issuer the
right to buy back all or part of an issue prior to maturity.
Bond indenture - Contract that sets forth the promises of a
corporate bond issuer and the rights of investors.
Sinking fund - A fund to which money is added on a regular basis
that is used to ensure investor confidence that promised
payments will be made and that is used to redeem debt securities
or preferred stock issues.
Sinking fund provision - A condition included in some corporate
bond indentures that requires the issuer to retire a specified
portion of debt each year.
Convertible bond - General debt obligation of a corporation
that can be exchanged for a set number of common shares of the
issuing corporation at a prestated conversion price.
Warrant - A security entitling the holder to buy a proportionate
amount of stock at some specified future date at a specified
price, usually one higher than current market price. Warrants
are traded as securities whose price reflects the value of the
underlying stock. Corporations often bundle warrants with
another class of security to enhance the marketability of the
other class. Warrants are like call options, but with much
longer time spans-sometimes years. And, warrants are offered by
corporations, while exchange-traded call options are not issued
by firms.
Bond Valuation
The value of any financial asset -– a stock, a bond, etc., or
even a physical asset -– is simply the present value of the cash
flows (discounted at the asset’s required rate of return) which
the asset is expected to generate over its lifetime.
2
VALUE = PV =
CF3
CF1
CF2
CFn


 ... 

(1  k)1
(1  k)2
(1  k)3
(1  k)n
n
CFt
 (1  k) .
t 1
t
Bond terminology:
kd = the bond’s market rate of interest or required rate of
return; also called the yield to maturity (YTM); (can
change many times over the bond’s life).
N = the number of years before the bond matures.
M = the par, or maturity, value of the bond (usually $1,000).
CIR = the coupon interest rate (does not change over the bond’s
life).
INT = the dollar amount of interest the bond pays per year
(INT = CIR x M).
Note:
Always discount the bond’s cash flows with kd.
Bond valuation model with annual coupons:
VB =  (1 INT
 k)
N
t
t 1
d

M
(1  kd)N
=
INT(PVIFA kd,n) + M(PVIF kd,n)
Bond valuation model with semi-annual coupons:
VB =  (1 INT/2
+
 k /2)
2N
t1
t
d
M
(1  kd /2)2N
=
INT/2(PVIFA kd/2,2n) + M(PVIF kd/2,2n)
Zero Coupon bond valuation model:
VZero = M/(1 + kd)n = M(PVIFkd,n)
Bond Valuation:

Important Relationships
A decrease in interest rates (required rates of return)
will cause the value of a bond to increase; an interest
rate increase will cause a decrease in value.
The change
in value caused by changing interest rates is called
interest rate risk. A bondholder owning a long-term bond is
3
exposed to greater interest rate risk than when owning a
short-term bonds.

The market value of the bond will always approach its par
value as its maturity date approaches, provided the firm
does not go bankrupt.

If the bondholder's required rate of return (current
interest rate) equals the coupon interest rate, the bond
will sell at "par," or maturity value.

If the current interest rate exceeds the bond's coupon
rate, the bond will sell below par value or at a
"discount."

If the current interest rate is less than the bond's coupon
rate, the bond will sell above par value or at a "premium."
Bond Yields
Yield to maturity - The percentage rate of return paid on a
bond, note, or other fixed income security if the investor buys
and holds it to its maturity date. The calculation for YTM is
based on the coupon rate, length of time to maturity, and market
price. It assumes that coupon interest paid over the life of the
bond will be reinvested at the same rate.
Estimated YTM =
INT + [(M – VB)/n]
-------------------------(M + 2VB)/3
Current yield - The annual interest payment on a bond divided by
the bond's current price.
Current yield = INT/VB
Capital gains (loss) yield - The price change portion of a
bond's return.
Capital gains (loss) yield = (VB+1 - VB)/VB
Therefore, the total return on a bond is equal to the bond's
YTM, and the YTM = Current yield + Capital gains (loss) yield.
YTM = INT/VB + (VB+1 - VB)/VB
4
Yield to call - The percentage rate of return on a bond or note
if the investor buys and holds the security until the call date.
This yield is valid only if the security is called prior to
maturity. Generally bonds are callable over several years and
normally are called at a slight premium. The calculation of
yield to call is based on coupon rate, length of time to call,
call price and market price.
Estimated YTC =
INT + [(Call Price – VB)/Years to Call]
----------------------------------------(Call Price + 2VB)/3
Yield to maturity on a zero coupon bond:
YTMZero = (M/VZero)1/n
-
1.0
Types of bonds




Treasury bonds - Debt obligations of the US Treasury that
have maturities of 10 years or more.
Municipal bond - State or local governments offer muni
bonds or municipals, as they are called, to pay for special
projects such as highways or sewers. The interest that
investors receive is exempt from some income taxes.
Foreign bond - A bond issued on the domestic capital market
of another country.
Corporate bonds - Debt obligations issued by corporations.
1.
2.
3.
Mortgage bond - A bond in which the issuer has
granted the owner a lien against the pledged assets.
Debenture - Any debt obligation backed strictly by
the borrower's integrity, e.g. an unsecured bond. A
debenture is documented in an indenture.
Subordinated debenture bond - An unsecured bond that
ranks after secured debt, after debenture bonds, and
often after some general creditors in its claim on
assets and earnings.
Bond Ratings
Bond ratings are critical to a company's ability to issue debt
at an acceptable interest rate.
5
1. What is the purpose of rating debt?
Answer: Unrated debt is extremely difficult, if not impossible
to sell. Corporations desiring to sell bonds must submit their
proposals to an independent rating company like Moody's Investor
Service or Standard & Poor’s Corporation for the debt to be
assigned a rating. This rating, coupled with market rates of
interest and the special features of the debt, will determine
how much the company will have to pay in interest.
2. Why would a company's debt rating be changed subsequent to
issue?
Answer: Any time new information relevant to the company's
ability to repay the debt (i.e. changes in the company's
financial health) is discovered and determined to be of
sufficient impact, that company's debt rating might be changed.
Obviously, good information should result in a better rating or
upgrade and bad information should result in a downgrade.
3. What impact would a change in debt rating have on the value
of an outstanding bond?
Answer: Bonds of companies that suffered downgrades would
decline in value to equate the yield to the new level of risk.
Conversely, bonds that were upgraded should increase in price.
Bond ratings are important for firms raising capital via a debt
offering. Companies that are financially sound enough to get a
higher rating will be able to sell debt with lower interest
payments than their riskier counterparts. Bond rating agencies
also actively monitor outstanding debt they have rated for
changes in status. Companies placed on watchlists are companies
who are facing potential changes in the rating of their debt.
Junk bond - A bond with a speculative credit rating of BB (S&P)
or Ba (Moody's) or lower. Junk or high-yield bonds offer
investors higher yields than bonds of financially sound
companies.
6
Bond Quotations
1
Bonds
Chiquita 10 1/2 04
K Mart 6.2s97
Disney zr05
2
Cur Yld
10.7
cv
…
3
Vol
144
50
414
4
Close
98 1/4
91
45 3/4
5
Net Change
+ 3/8
+ 1/4
+ 3/4
Column 1: Bond, Coupon Rate, Date of Maturity
A bond issued by Chiquita which matures in 2004 has a coupon
rate of 10 1/2. This stated interest rate represents the 10.5
percent paid on the bond's $1,000 face value. The holder of this
bond will receive $105 annually.
The "s" in the K Mart quotation separates the 6.2 percent rate
from the 1997 maturity rate. Note this bond is listed in
fractions of 10s instead of 8s.
The Disney bonds are zero coupon bonds as indicated by the "zr."
They do not pay annual interest.
Column 2: Current Yield
At this day's price, the holder of a Chiquita bond annually will
receive 10.7 percent or $10.70 for every $100 invested. The
current yield is calculated by dividing the annual interest by
the closing price.
"cv" indicates the K Mart bond is convertible and can be
exchanged for K Mart stock.
Column 3: Volume
On this day, 500,000 K Mart bonds were sold. The number 50 has
been multiplied by 10,000.
Columns 4 & 5: Close, Net Change
The final price for Chiquita bonds was $982.50 which was $3.75
more than the final price on the day before.
7
1
Bonds
USAir 16 1/4 09
IBM zr13
CBS 9.8s20
2
Cur Yld
14.5
…
cv
3
Vol
15
20
32
4
Close
111 7/8
42 1/8
109 1/2
5
Net Change
…
- 3/4
+2
1. How many transactions of USAir bonds were made? _________
2. What year are each of these bonds due?


USAir __________
CBS
__________
3. What is the stated coupon interest paid to the bondholder
for each of these bonds?



USAir _______________
IBM
_______________
CBS
_______________
4. If you bought the following bonds at these prices, what
would your yield to maturity be?


USAir _________________
IBM
_________________
5. What was the closing price for these bonds on the previous
day?


USAir _________________
CBS
_________________
8
Interest Rate Price Risk for 10 percent Coupon
Bonds with Different Maturities
Bond Value
($)
1,800
1,400
1,000
1-Year
5-Year
10-Year
20-Year
30-Year
600
5
6
7
8
9
10
11
12
13
14
15
Interest Rate (%)
An Investor's Guide to the Many Meanings of Yield
Let us assume that you are reading the financial pages of your
favorite newspaper. You read that even though stock returns have
been dismal for the last two years, bond returns have been very
good. In fact, you read that over the past two years, many bond
funds returned well over 15%. While the returns look pretty darn
good relative to stocks, you may wonder: Does that mean I can
expect to earn 15% next year if I buy bonds? If the answer is
not obvious, read on.
The direction of interest rates is one of the chief determinants
of bond prices. A strong market for bonds is one in which
interest rates are declining. That causes bond prices to go up.
A weak market is one in which interest rates are going up. That
causes bond prices to decline. Clearly, then, since changes in
interest rate levels affect bond prices, they also affect what
you earn from investments in bonds.
But that is only the beginning. In order to understand what you
actually earn from bonds, you need to understand two different
concepts: yield and total return.
When you buy an individual bond, you can expect to receive
coupon payments (usually every six months) for most bonds. When
you buy a bond fund, you can expect a monthly payout of the
income earned by the bond fund. That stream of income is
variously described as the bond’s “yield.” But you also have to
9
bear in mind that when you sell or redeem your bond (or bond
fund), you may sell at a higher or at a lower price than the
price you paid. That difference can be an additional source of
earnings, or it may result in a loss. That change in price is
one of the main factors that determines a bond’s total return.
What may be confusing, however, is that the term yield has a
number of different meanings. Even more confusing is the fact
that these meanings are not directly comparable for individual
bonds and for bond funds. Moreover, individual bonds are usually
sold to investors and are discussed primarily in terms of yield,
not returns. But discussions of bond funds often focus on total
return.
Yield
When you buy an individual bond, you derive income from three
different sources: Simple interest, Interest on interest, and
Return of principal at maturity, or proceeds from the sale of
the bond at an earlier date.
Simple interest consists of the bond’s coupons, which are
usually paid twice a year. Let us say you invest $10,000 in a
four-year bond, paying 8% a year, semiannually. In return, you
will receive two coupon (or interest) payments of $400 each, at
six-month intervals every year. If you hold the bond until it
matures, you will receive eight coupons that total $3,200. Those
eight coupons are the simple interest.
If the coupon payments are spent, only the simple interest is
earned. But if the coupons are reinvested, they produce
additional interest; subsequently, if those earnings are
reinvested, you earn interest on that interest, and so on. That
entire income stream is called, logically enough, interest-oninterest, or compounded interest. Both interest income, and
interest-on-interest, in different combinations, lie behind
the different meanings of yield.
Yield appears in a number of phrases: coupon yield, current
yield and yield to maturity. Each has a very precise meaning.
Let’s look at each in turn.
Coupon Yield
Coupon yield is set when a bond is issued. It is the interest
rate paid by the bond (for example, 5½%,7¾%), and it is listed
as a percentage of par, or face value, which is the principal
10
amount that will be owed at maturity.
The coupon yield designates a fixed dollar amount that never
changes through the life of the bond. If a $1,000 par value bond
is described as having a 10% coupon, that coupon will always be
$100 for each bond, paid out in two $50 increments for the
entire life of the bond—no matter what happens to the price of
the bond, or to interest rates. That is the reason bonds are
called fixed-income securities.
Current Yield
Almost as soon as a bond starts trading in the secondary market,
it ceases to trade at par due to changing interest rates. A
bond’s current yield is its coupon divided by its market price.
To illustrate, let us assume you purchased three bonds: the
first you bought at par, for $1,000; the second you bought at a
premium to par, and paid $1,200; the third you bought at a
discount to par, for $800. Each bond has a 10% coupon, and so
each pays $100 in annual coupons. Dividing the coupon ($100) by
the price results in a current yield of 10% for the par bond;
8.33% for the premium bond; and 12.5% for the discount bond.
Thus, the current yield is equal to coupon yield for the par
bond; the current yield is lower than the coupon yield for the
premium bond; and the current yield is higher than the coupon
yield for the discount bond. Current yield is quoted for fixedincome securities of any maturity, whether short or long.
Yield to Maturity
You can see from the above description that current yield is
based only on the coupon and the current market price. Current
yield, therefore, fails to measure two important sources of
income that investors earn from bonds: interest-on-interest and
capital gains or losses.
Yield to maturity (YTM) is a more comprehensive measure of
potential earnings than “current yield.” It estimates the total
amount that a bond will earn over the entire life of an
individual bond, from all possible sources of income — coupon
income, interest-on-interest, and capital gains or losses due to
the difference between the price paid when the bond was
purchased and par, the return of principal at maturity—based on
a number of assumptions regarding the holding period, reinvested
income and interest rates over the life of the bond.
11
Yield to maturity calculations are not easily made using paper
and pencil, but they can easily be determined using either a
financial calculator, or by using the various calculators
available on the Internet.
YTM is the measure most widely quoted by brokers when selling
individual bonds. However, it is not a prediction of what you
will actually earn on a bond. Your actual return is likely to
differ from the YTM, perhaps considerably, because the YTM will
only be realized under certain conditions, which are:



That you hold the bond to maturity;
That the coupons are reinvested (rather than spent); and
That coupons are reinvested at an interest rate equal to
the yield-to-maturity.
Let’s look briefly at each assumption:
-Holding to Maturity: The YTM quote is based on the redemption
price of par (that is, $1,000). If you sell a bond before it
matures at a price other than par, then the capital gain or loss
will considerably alter what you actually earn.
-Reinvesting Coupons: YTM calculations are based on the
assumption that coupons are never spent; they are always
reinvested. If you spend coupons, then you do not earn the
interest-on-interest, and your return would be less than the
anticipated YTM. How much less depends both on how many coupons
you spend and on the maturity of the bonds.
-Coupons are Reinvested at an Interest Rate Equal to the YTM:
This may sound like double talk, but it means that if a bond has
a YTM of 7%, it is assumed that each and every coupon is
reinvested at a rate of 7%. However, if in reality you reinvest
coupons at a higher rate than 7%, you will earn more than the
bond’s stated YTM, while if you reinvest coupons at lower rates
than 7%, you will earn less.
Reinvestment Risk
Reinvestment risk is the risk that coupons may have to be
reinvested at a lower interest rate, in which case an investor’s
actual return would then be lower than the YTM quoted at the
time of purchase. On the other hand, the reinvestment risk may
work in your favor if coupons are reinvested at a higher rate,
and that would increase the actual return above the YTM quoted
at the time of purchase.
12
If YTM does not predict your actual return, what does it tell
you? The chief usefulness of YTM quotes is that they allow you
to compare different kinds of bonds—those with dissimilar
coupons, different market prices relative to par (for instance,
bonds selling at premiums or discounts), and different
maturities.
Default Risk
The risk that an issuer of a bond may be unable to make timely
principal and interest payments.
Zero Coupon Bonds




Zero coupon bonds result from the separation of coupons
from the body of a security.
Zeros sell at deep discounts from face value.
The difference between the purchase price of the zero and
its face value when redeemed is the investor's return.
Zeros can be purchased from private brokers and dealers,
but not from the Federal Reserve or any government agency.
Creating Zeros by Coupon Stripping - Coupon stripping is the act
of detaching the interest payment coupons from a note or bond
and treating the coupons and the body as separate securities.
Each coupon, or interest payment, entitles its owner to a
specified cash return on a specific date; the body of the
security calls for repayment of the principal amount at
maturity.
The body of the stripped securities and the separate coupons are
known as "zero coupons" or "zeros" because there are no periodic
interest payments on each instrument. After stripping, the body
and coupons are sold at a deep discount from their face values.
An owner benefits only from the difference between the purchase
price and the payment received upon sale or at maturity.
For example, a 20 year bond with a face value of $20,000 and a
10% interest rate could be stripped into its principal and its
40-semi-annual interest payments. The result would be 41
separate zero coupon instruments, each with its own maturity
date. The principal would be worth $20,000 upon maturity, and
each interest coupon $1,000, or one-half the annual interest of
10% on $20,000. Each of the 41 securities, now possessing a
distinct ID number, could be traded separately until its
13
maturity date at prices determined by the market.
Proliferation of Treasury STRIPS - Some Treasury securities were
traded in the secondary market without one or more of their
interest coupons in the late 1970s. Stripped securities offered
investors a financial instrument that had abundant supply, no
default risk, and low incidence of being "called," or paid off,
before their maturity date. However, their popularity raised
fears within the Treasury Department that zeros would result in
a sizable loss of tax revenues.
Detached coupons and the body of the security were sold at deep
discounts, $.05 or $.10 on the dollar. After purchase, an
investor claimed a capital loss on the difference between the
sale price of the security and its face value, thus reducing the
investor's overall tax liability.
The Tax Equity and Fiscal Responsibility Act (TEFRA) of 1982
adjusted the tax treatment of stripped securities to reduce
their tax advantage. The Treasury Department then withdrew its
objections to coupon stripping, prompting several securities
dealers to create new products incorporating receipts for
stripped debt securities.
TEFRA also required the Treasury to begin issuing all of its
securities in book-entry (electronic)form only, beginning on
January 1, 1983. This provision eliminated Treasury issues of
bearer notes and bonds with coupons attached. Physical stripping
would no longer be possible.
In response, bond dealers began to market receipts that
evidenced ownership of Treasury zeros held by a custodian. The
first of these "receipt products" were named Treasury Investment
Growth Receipts, or TIGRS. Similar products appeared in 1984,
such as Certificates of Accrual on Treasury Securities (CATS)
and Treasury Receipts (TRs). However, most of these securities
were not exchangeable with other stripped securities, and thus
lacked the liquidity customers had come to expect from "zero"
instruments.
In February 1985, the Treasury took a more active role by
introducing its own coupon stripping program called STRIPS, an
acronym for Separate Trading of Registered Interest and
Principal of Securities. The STRIPS program was intended
primarily to reduce the cost of financing the public debt "by
facilitating competitive private market initiatives."
14
Under the STRIPS program, U.S. government issues with maturities
of ten years or more became eligible for transfer over Fedwire.
The process involves wiring Treasury notes and bonds to the
Federal Reserve Bank of New York and receiving separated
components in return. This practice also reduced the legal and
insurance costs customarily associated with the process of
stripping a security. In May 1987, the Treasury began to allow
the reconstitution of stripped securities.
Part of a Balanced Portfolio - Stripped securities can be
purchased only from private dealers and brokers. Although the
Federal Reserve provides services to the zero coupon market, it
does not actually sell these securities for the Treasury.
Financial services companies decide when and what portion of an
eligible security are stripped and sold.
Because their increase in value is taxable yearly as it accrues,
zeros have become most popular for investments on which taxes
can be deferred, such as individual retirement accounts and
pension plans, or for nontaxable accounts. However, their known
cash value at specific future dates enables savers and investors
to tailor their use to a wide range of portfolio objectives.
15
Problems
1) The bonds of the Nordy Company have a coupon interest rate
of 9%. The interest on the bonds is paid semiannually, the
bonds mature in 8 years, and their par value is $1,000. If the
required rate of return, kd = 8%, what is the value of each bond?
What is the value of each bond if the interest is paid annually?
2) You own a bond that pays $100 in interest annually, has a
par value of $1000, and matures in 15 years. What is the value
of the bond if your required rate of return is 12%? What is the
value of the bond if your required rate of return (a) increases
to 15% or (b) decreases to 8%? Now, recompute all three answers
assuming that the bond matures in 5 years instead of 15 years.
3) Dullco Company bonds are selling in the market for $1,045
(104.50). These bonds will mature in 15 years and pay $70 in
interest annually. If the bonds are purchased at the market
price, what is the (a) coupon rate, (b) current yield,
(c) approximate yield to maturity and (d) capital gains yield?
4) Apex Company is planning to issue zero coupon bonds that
will mature at $1,000 in 20 years. If your required rate of
return on these bonds is 9.35%, what are you willing to pay for
the bonds? If these bonds are currently selling for $213.50,
what is their yield to maturity (YTM)?
Answers
1) $1,058.26; $1,057.47
2) $863.78; (a)$707.63; (b)$1,171.19
$927.90; (a)$832.39; (b)$1,079.85
3) (a)7%; (b)6.70%; (c)6.50%; Exact YTM = 6.5208%;
(d)-0.20%; Exact CGY = -0.1792%
4)
$167.35; 8.03%
16