Chapter 6
Interest Rates
And Bond Valuation

Principles of Managerial Finance
Sixteenth Edition
Chapter 6
Interest Rates and Bond
Valuation
Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.

Learning Goals (1 of 2)
LG 1 Describe interest rate fundamentals, the term
structure of interest rates, and risk premiums.
LG 2 Review the legal aspects of bond financing and
bond cost.
LG 3 Discuss the general features, yields, prices,
ratings, popular types, and international issues
of corporate bonds.
LG 4 Understand the key inputs and basic model
used in the bond valuation process.

Learning Goals (2 of 2)
LG 5 Apply the basic valuation model to bonds,
and describe the impact of required return
and time to maturity on bond prices.
LG 6 Explain yield to maturity (YTM), its
calculation, and the procedure used to
value bonds that pay interest semiannually.

6.1 Interest Rates and Required Returns (1
of 14)
• Interest Rate Fundamentals
– Interest Rate
• Usually applied to debt instruments such as bank
loans or bonds
• the compensation paid by the borrower of funds to
the lender
• from the borrower’s point of view, the cost of
borrowing funds
• The interaction of supply and demand determines
interest rates
– Required Return
• The return that the supplier of funds requires the
demander to pay
• Applies to almost any kind of investment

Figure 6.1 Supply-Demand
Relationship

6.1 Interest Rates and Required Returns (2 of 14)
• Interest Rate Fundamentals
– Inflation
• A rising trend in the prices of most goods and
services
– Liquidity Preference
• A general tendency for investors to prefer short-
term (i.e., more liquid) securities

6.1 Interest Rates and Required Returns (3 of 14)
• Interest Rate Fundamentals
– Negative Interest Rates
• When a loan carries an interest rate below
zero, the lender essentially pays interest to the
borrower rather than the other way around
• A natural question is, why would anyone buy an
investment if it paid an interest rate below zero?
– The answer is that there is no good, safe
alternative offering a better return

Matter of Fact (1 of 2)
Fear Turns T-Bill Rates Negative
When the coronavirus outbreak reached the United States
in early 2020, financial markets reacted in dramatic
fashion. On March 25, interest rates on one-month and
three-month Treasury bills briefly turned negative,
meaning that investors paid more to the Treasury than the
Treasury promised to pay back. Why would people put
their money into an investment they know will lose money?
Fear about the effects of COVID-19 on the economy sent
stock prices tumbling, and investors moved their money
into safer Treasury securities. The demand for those
securities was so high, temporarily, that investors
effectively paid the U.S. Treasury to hold their funds for a
short time.

6.1 Interest Rates and Required
Returns (4 of 14)
• Interest Rate Fundamentals
– Nominal and Real Interest Rates
• Nominal Rate of Interest
– The actual rate of interest charged by the
supplier of funds and paid by the demander
• Real Rate of Interest
– The rate of return on an investment
measured not in dollars but in the increase in
purchasing power that the investment
provides
– The real rate of interest measures the rate of
increase in purchasing power

6.1 Interest Rates and Required
Returns (5 of 14)
• Interest Rate Fundamentals
– Nominal and Real Interest Rates
    
*
1+ = 1+ 1+ (6.1)
r r i
*
(6.1a)
r r i
 
• r = nominal interest rate
• r* = real interest rate
• i = expected inflation rate

Personal Finance Example 6.1 (1 of 3)
Burt Gummer is a survivalist who constantly worries
that the apocalypse may happen any day now. To be
prepared, Burt stores food with a long shelf life in his
basement. Burt has $100 to add to his food stores,
and he is considering the purchase of 100 cans of
Spam for $1 each. Burt expects the inflation rate
over the coming year to be 9%, so a can of Spam will
cost $1.09 each in a year. Burt’s wife, Heather, has
heard of an investment that will pay a 21% nominal
return over the next year, so she thinks Burt should
invest the money rather than use it to buy Spam.

Personal Finance Example 6.1 (2 of 3)
Equation 6.1a says that the approximate real return
on Burt’s potential investment is 12%:
12% ≈ 21% − 9%
On the basis of this calculation, Burt might expect
that if he invests $100, he could buy 112 cans of
Spam next year rather than 100 cans this year (a
12% increase in purchasing power). Suppose that
Burt invests the money, earns a 21% rate of return,
and one year later has $121. By that time, one can of
Spam costs $1.09, so Burt is just barely able to
purchase 111 cans (111 cans × $1.09 = $120.99).

Personal Finance Example 6.1 (3 of 3)
Burt’s purchasing power has increased by 11%, not by the
12% that he expected. Equation 6.1 reveals that the exact
real return on Burt’s investment is 11%:
(1 0.21) (1 *)(1 0.09)
1.21
1 *
1.09
1.110 1 0.11 11% *
r
r
r
   
 
   

6.1 Interest Rates and Required Returns (6 of 14)
• Interest Rate Fundamentals
– Nominal Interest Rates, Inflation, and Risk
• RF = Risk-free rate
*
(6.2)
F
R r i
 
• rj = Nominal return on investment j
• RPj = Risk premium above the risk-free rate for
investment j
(6.3)
j F j
r R RP
 

Figure 6.2 Inflation and the Risk-free Interest Rates
a Interest rate on a one-year Treasury bill issued in January of each calendar year. Data from
selected Federal Reserve Bulletins.
b Change in consumer price index in each calendar year. Data from U.S. Department of Labor
Bureau of Labor Statistics.

6.1 Interest Rates and Required Returns (7 of 14)
• Term Structure of Interest Rates
– Yield Curves
• Term Structure of Interest Rates
– The relationship between the maturity and rate
of return for bonds with similar levels of risk
• Yield Curve
– A graphic depiction of the term structure of
interest rates
• Yield to Maturity (YTM)
– Compound annual rate of return earned on a
debt security purchased on a given day and held
to maturity
– An estimate of the market’s required return on a
particular bond

6.1 Interest Rates and Required Returns (8 of 14)
• Term Structure of Interest Rates
– Yield Curves
• Normal Yield Curve
– An upward-sloping yield curve indicates that
long-term interest rates are generally higher
than short-term interest rates
• Inverted Yield Curve
– A downward-sloping yield curve indicates that
short-term interest rates are generally higher
than long-term interest rates
• Flat Yield Curve
– A yield curve that indicates that interest rates do
not vary much at different maturities

Figure 6.3 Treasury Yield Curves
Source: Data from Daily Treasury Yield Curve Rates.
https://www.treasury.gov/resource-center/datachart- center/interest-
rates/Pages/ TextView.aspx?data=yield

6.1 Interest Rates and Required Returns (9 of 14)
• Term Structure of Interest Rates
– Yield Curves
• Deflation
– A general trend of falling prices
• The shape of the yield curve may affect the firm’s
financing decisions
– A financial manager who faces a downward-
sloping yield curve may be tempted to rely
more heavily on cheaper, long-term financing
– When the yield curve is upward sloping, the
manager may believe it wise to use cheaper,
short-term financing

Figure 6.4 The Slope of the Yield Curve and
the Business Cycle

Matter of Fact (2 of 2)
Bond Yields Hit Record Lows
On March 9, 2020, the 10-year Treasury note yield reached
an all-time low of 0.318% as fears about the spreading
coronavirus and its impact on the economy prompted
investors to pour money into safe Treasury securities.
Lower interest rates are usually good news for the housing
market. Many mortgage rates are linked to rates on
Treasury securities. For example, the traditional 30-year
mortgage rate is typically linked to the yield on 10-year
Treasury notes. Mortgage rates did reach new lows, but the
slowing economy discouraged home buyers (and sellers)
from taking advantage, though many homeowners did
benefit from the falling rates when they refinanced their
mortgages.

6.1 Interest Rates and Required
Returns (10 of 14)
• Term Structure of Interest Rates
– Theories of the Term Structure
• Expectations Theory
– The theory that the yield curve reflects
investor expectations about future interest
rates; an expectation of rising interest rates
results in an upward-sloping yield curve, and
an expectation of declining rates results in a
downward-sloping yield curve

Example 6.2 (1 of 3)
Suppose that a one-year T-bill currently offers a 3.5%
return and a two-year Treasury note offers a 3.0% annual
return. Thus, the short-term rate is higher than the long-
term rate and the yield curve slopes down. According to
the expectations theory, what belief must investors hold
about the rate of return that a one-year T-bill will offer next
year? First, recognize that by purchasing the two-year
note, investors can earn a return of 6.09% over two years:
 
2
1 0.030 1.0609
 

Example 6.2 (2 of 3)
If the market is in equilibrium, then the expected return on
a strategy of purchasing a sequence of two one-year T-bills
must offer the same return, so we have
 
1 0.035 1 E r 1.0609
1 E r 1.0609 1.035
E r 0.025 2.5%
( ( ))
( )
( )
  
  
 

Example 6.2 (3 of 3)
Investors believe the T-bill will offer a 2.5% return next year,
which is lower than the 3.0% return currently offered by
one-year T-bills. In other words, investors are indifferent
between earning 3.0% for two consecutive years on the
Treasury note or earning 3.5% this year and 2.5% next year
on a sequence of two one-year T-bills. Thus, today’s
downward-sloping yield curve implies that investors expect
falling rates.

6.1 Interest Rates and Required
Returns (11 of 14)
• Term Structure of Interest Rates
– Theories of the Term Structure
• Liquidity Preference Theory
– Theory suggesting that long-term rates are
generally higher than short-term rates
(hence, the yield curve is upward sloping)
because investors perceive short-term
investments as more liquid and less risky
than long-term investments
– Borrowers must offer higher rates on long-
term bonds to entice investors away from
their preferred short-term securities

6.1 Interest Rates and Required
Returns (12 of 14)
• Term Structure of Interest Rates
– Theories of the Term Structure
• Market Segmentation Theory
– Theory suggesting that the market for loans
is segmented on the basis of maturity and
that the supply of and demand for loans
within each segment determine its prevailing
interest rate; the slope of the yield curve is
determined by the general relationship
between the prevailing rates in each market
segment

6.1 Interest Rates and Required
Returns (13 of 14)
• Risk Premiums: Issuer and Issue
Characteristics
– The nominal rate of interest for security j is equal to the risk-
free rate, consisting of the real rate of interest plus the
expected inflation rate, plus the risk premium
– The risk premium varies with characteristics of the company
that issued the security as well as characteristics of the
security itself

Example 6.3 (1 of 3)
The nominal interest rates on a number of classes of
long-term securities in April 2020 were as follows:
Security Nominal interest rate
One-Year U.S. Treasury bills 0.2%
Corporate bonds:
Investment grade 3.0%
High-yield 8.0%

Example 6.3 (2 of 3)
Using the one-year Treasury bill rate as our benchmark risk-free rate,
we can calculate the risk premium of the other securities by subtracting
the risk-free rate, 0.2%, from rates offered by each of the other
corporate securities:
Security Risk premium
Corporate bonds:
Investment grade 3.0% − 0.2% = 2.8%
High-yield 8.0% − 0.2% = 7.8%

Example 6.3 (3 of 3)
Investment grade bonds pay a risk premium over Treasury
bills in part because of their term (investment grade bonds
have a longer term than Treasury bills), but also because
they are issued by corporations and are not backed by the
full faith and credit of the U.S. government as are Treasury
securities. High-yield bonds pay an even larger risk
premium because they are issued by less creditworthy
corporations.

6.1 Interest Rates and Required Returns (14 of 14)
• Risk Premiums: Issuer and Issue Characteristics
– Several factors influence a bond’s risk premium
• Default risk or credit risk
– The likelihood that a corporation will fail to make
interest payments or principle repayments on its
bonds
• Bond characteristics
– e.g., Time to maturity
• Interest rate risk
– When interest rates in the economy change,
bond prices move in the opposite direction
– Some bonds are more sensitive to interest rate
risk and therefore pay higher risk premiums to
compensate investors

6.2 Government and Corporate Bonds
(1 of 15)
• Municipal Bond
– A bond issued by a state or local government body
• Corporate Bond
– A long-term debt instrument indicating that a
corporation has borrowed a certain amount of money
and promises to repay it in the future under clearly
defined terms

6.2 Government and Corporate Bonds
(2 of 15)
• Par Value, Face Value, Principal
– The amount of money the borrower must repay at
maturity, and the value on which periodic interest
payments are based
• Coupon Rate
– The percentage of a bond’s par value that will be paid
annually, typically in two equal semiannual payments,
as interest

6.2 Government and Corporate Bonds
(3 of 15)
• Legal Aspects of Corporate Bonds
– Bond Indenture
• A legal document that specifies both the rights of the
bondholders and the duties of the issuing corporation
– Standard Debt Provisions
• Provisions in a bond indenture specifying certain record-
keeping and general business practices that the bond
issuer must follow; normally, they do not place a burden
on a financially sound business
– Restrictive Covenants
• Provisions in a bond indenture that place operating and
financial constraints on the borrower

6.2 Government and Corporate Bonds
(4 of 15)
• Legal Aspects of Corporate
Bonds
– Subordination
• In a bond indenture, the stipulation that
subsequent creditors agree to wait until all claims
of the senior debt are satisfied
– Sinking-Fund Requirement
• A restrictive provision often included in a bond
indenture, providing for the systematic retirement
of bonds prior to their maturity

6.2 Government and Corporate Bonds
(5 of 15)
• Legal Aspects of Corporate
Bonds
– Collateral
• A specific asset against which bondholders have a
claim in the event that a borrower defaults on a
bond
– Secured Bond
• A bond backed by some form of collateral
– Unsecured Bond
• A bond backed only by the borrower’s ability to
repay the debt

6.2 Government and Corporate Bonds
(6 of 15)
• Legal Aspects of Corporate
Bonds
– Trustee
• A paid individual, corporation, or commercial bank
trust department that acts as the third party to a
bond indenture and can take specified actions on
behalf of the bondholders if the terms of the
indenture are violated

6.2 Government and Corporate Bonds
(7 of 15)
• Cost of Bonds to the Issuer
– Impact of Bond Maturity
• Usually, long-term debt pays higher interest rates
than short-term debt
– Impact of Offering Size
• Bond flotation and administration costs per dollar
borrowed are likely to decrease as offering size
increases
• However, the risk to the bondholders may
increase, because larger offerings result in greater
risk of default, all other factors held constant
– Impact of the Issuer’s Risk
• The greater the issuer’s default risk, the higher the
interest rate

6.2 Government and Corporate Bonds
(8 of 15)
• Cost of Bonds to the Issuer
– Impact of the Risk-free Rate
• The risk-free rate in the capital market determines
a floor on the cost of a bond offering
• Generally, the rate on U.S. Treasury securities of
equal maturity is the lowest cost of borrowing
money
– To that basic rate is added a risk premium that
reflects the factors mentioned prior (maturity,
offering size and issuer’s risk)

6.2 Government and Corporate Bonds
(9 of 15)
• General Features of a Bond Issue
– Conversion Feature
• A feature of convertible bonds that allows bondholders to
change each bond into a stated number of shares of
common stock
– Call Feature
• A feature included in nearly all corporate bond issues that
gives the issuer the opportunity to repurchase bonds at a
stated call price prior to maturity
– Call Price
• The stated price at which a bond may be repurchased, by
use of a call feature, prior to maturity

6.2 Government and Corporate Bonds
(10 of 15)
• General Features of a Bond
Issue
– Call Premium
• The amount by which a bond’s call price exceeds
its par value
– Stock Purchase Warrants
• Instruments that give their holders the right to
purchase a certain number of shares of the
issuer’s common stock at a specified price over a
certain period of time

6.2 Government and Corporate Bonds (11 of 15)
• Bond Yields
– Current Yield
• A measure of a bond’s cash return for the year
• Calculated by dividing the bond’s annual
interest payment by its current price
– Yield to Maturity (YTM)
– Yield to Call (YTC)

6.2 Government and Corporate Bonds (12 of 15)
• Bond Prices
– Though corporate bonds are held mostly by
institutional investors and are not as actively traded
as stocks, it is still important to understand market
conventions for quoting bond prices and yields.
– Basis points
• A way of quoting an interest rate such that 100
basis points equals one percentage point

Table 6.1 Data on
Selected Bonds
Company Coupon Maturity Price Yield (YTM)
Verizon 5.050% Mar. 15, 2034 123.88 2.915%
Apple 3.850 Aug. 4, 2046 124.36 2.554
Tesla 5.300 Aug. 15, 2025 94.50 6.542
Safeway 7.450 Sep. 15, 2027 105.10 6.566
Amazon 4.250 Aug. 22, 2057 99.98 4.251
Bond data from http://finra-markets.morningstar.com/BondCenter/ActiveUSCorpBond.jsp, accessed on April 23, 2020.

6.2 Government and Corporate Bonds (13 of 15)
• Bond Ratings
– Independent agencies such as Moody’s, Fitch, and
Standard & Poor’s assess the riskiness of publicly
traded bond issues
– These agencies derive their ratings by using
financial ratio and cash flow analyses to assess the
likely payment of bond interest and principal
– Normally an inverse relationship exists between the
quality of a bond and the rate of return that it must
provide bondholders

Table 6.2 Moody’s and Standard &
Poor’s Bond Ratings
Note: Some ratings may be modified to show relative standing within a major rating category; for
example, Moody’s uses numerical modifiers (1, 2, 3), whereas Standard & Poor’s uses plus (+) and
minus (−) signs.
Sources: Based on Moody’s Investors Service, Inc., and based on Standard & Poor’s Corporation.

6.2 Government and Corporate Bonds (14 of 15)
• Common Types of Bonds
– Debentures
– Subordinated Debentures
– Income Bonds
– Mortgage Bonds
– Collateral Trust Bonds
– Equipment Trust Certificates
– Zero- (or Low-) Coupon Bonds
– Junk (High-Yield) Bonds
– Floating-Rate Bonds
– Extendible Notes
– Putable Bonds

Table 6.3 Characteristics and Priority of
Lender’s Claim of Traditional Types of
Bonds (1 of 2)
Bond type Characteristics Priority of lender’s claim
Unsecured bonds
Debentures Unsecured bonds that only
creditworthy firms can issue.
Convertible bonds are normally
debentures.
Claims are the same as those of
any general creditor. May have
other unsecured bonds
subordinated to them.
Subordinated debentures Claims are not satisfied until
those of the creditors holding
certain (senior) debts have
been fully satisfied.
Claim is that of a general
creditor but not as good as a
senior debt claim.
Income bonds Payment of interest is required
only when earnings are
available. Commonly issued in
reorganization of a failing firm.
Claim is that of a general
creditor. Are not in default
when interest payments are
missed because they are
contingent only on earnings
being available.

Table 6.3 Characteristics and Priority of
Lender’s Claim of Traditional Types of
Bonds (2 of 2)
Bond type Characteristics Priority of lender’s claim
Secured Bonds
Mortgage bonds Secured by real estate or buildings. Claim is on proceeds from sale of mortgaged assets; if
not fully satisfied, the lender becomes a general
creditor. The first-mortgage claim must be fully
satisfied before distribution of proceeds to second-
mortgage holders and so on. A number of mortgages
can be issued against the same collateral.
Collateral trust
bonds
Secured by stock and (or) bonds that are owned by
the issuer. Collateral value is generally 25% to 35%
greater than bond value.
Claim is on proceeds from stock and/or bond
collateral; if not fully satisfied, the lender becomes a
general creditor.
Equipment trust
certificates
Used to finance “rolling stock,” such as airplanes,
trucks, boats, railroad cars. A trustee buys the asset
with funds raised through the sale of trust
certificates and then leases it to the firm; after
making the final scheduled lease payment, the firm
receives title to the asset. A type of leasing.
Claim is on proceeds from the sale of the asset; if
proceeds do not satisfy outstanding debt, trust
certificate lenders become general creditors.

Table 6.4 Characteristics of
Contemporary Types of Bonds (1
of 2)
Bond type Characteristicsa
Zero- (or low-)
coupon bonds
Issued with no (zero) or a very low coupon (stated interest) rate and sold at a large
discount from par. A significant portion (or all) of the investor’s return comes from
gain in value (i.e., par value minus purchase price). Generally callable at par value.
Junk (high-yield)
bonds
Debt rated Ba or lower by Moody’s or BB or lower by Standard & Poor’s. Commonly
used by rapidly growing firms to obtain growth capital, most often as a way to finance
mergers and takeovers. High-risk bonds with high yields, often yielding 2% to 3%
more than the best-quality corporate debt.
Floating-rate
bonds
Stated interest rate is adjusted periodically within stated limits in response to changes
in specified money market or capital market rates. Popular when future inflation and
interest rates are uncertain. Tend to sell at close to par because of the automatic
adjustment to changing market conditions. Some issues provide for annual
redemption at par at the option of the bondholder.
Extendible notes Short maturities, typically one to five years, that can be renewed for a similar period
at the option of holders. Similar to a floating-rate bond. An issue might be a series of
three-year renewable notes over a period of 15 years; every three years, the notes
could be extended for another three years, at a new rate competitive with market
interest rates at the time of renewal.

Table 6.4 Characteristics of
Contemporary Types of Bonds (2 of 2)
Bond type Characteristicsa
Putable bonds Bonds that can be redeemed at par (typically, $1,000) at the option of their holder
either at specific dates after the date of issue and every one to five years
thereafter or when and if the firm takes specified actions, such as being acquired,
acquiring another company, or issuing a large amount of additional debt. In return
for its conferring the right to “put the bond” at specified times or when the firm
takes certain actions, the bond’s yield is lower than that of a nonputable bond.
aThe claims of lenders (i.e., bondholders) against issuers of each of these types of bonds vary, depending on the bonds’ other
features. Each of these bonds can be unsecured or secured.

6.2 Government and Corporate Bonds (15 of 15)
• International Bond Issues
– Eurobond
• A bond issued by an international borrower and
sold to investors in countries with currencies other
than the currency in which the bond is
denominated
– Foreign Bond
• A bond that is issued by a foreign corporation or
government and is denominated in the investor’s
home currency and sold in the investor’s home
market
– Bulldog Bond
» Foreign bond issued in Britain
– Samurai Bond
» Foreign bond issued in Japan

6.3 Valuation Fundamentals (1 of 2)
• Valuation
– The process that links risk and return to determine the
worth of an asset
• Key Inputs
– Cash Flows
– Timing
– Risk and Required Return

Personal Finance Example 6.4 (1 of 2)
Celia Sargent wishes to estimate the value of three assets:
common stock in Michaels Enterprises, an interest in an oil
well, and an original painting by a well-known artist. Her
cash flow estimates for each are as follows:
Stock in Michaels Enterprises: Expects to receive cash
dividends of $300 per year indefinitely starting in one year.
Oil well: Expects to receive cash flows of $2,000 after one
year, $4,000 after two years, and $10,000 after four years,
when the well will run dry.

Personal Finance Example 6.4 (2 of 2)
Original painting: Expects to sell the painting in five
years for $85,000.
With these cash flow estimates, Celia has taken the
first step in valuation.

Personal Finance Example 6.5 (1 of 2)
Let’s return to Celia Sargent’s task of placing a value
on the original painting and consider two scenarios.
Scenario 1: Certainty A major art gallery has
contracted to buy the painting for $85,000 after five
years. Because this contract is already signed and
the art gallery is well established and reliable, Celia
views this asset as “money in the bank,” and uses
something close to the prevailing risk-free rate of 3%
as the required return when calculating the painting’s
value today.

Personal Finance Example 6.5 (2 of 2)
Scenario 2: High risk The values of original
paintings by this artist have fluctuated widely over the
past 10 years. Although Celia expects to sell the
painting for $85,000, she realizes that its sale price
could range between $30,000 and $140,000.
Because of the high uncertainty surrounding the
painting’s value, Celia believes that a 15% required
return is appropriate.
These two estimates illustrate how the valuation
process accounts for risk through the discount rate.
Although adjusting the discount rate for risk has a
subjective element, analysts use historical data and
forward-looking models to estimate the required
return with as much precision as possible.

6.3 Valuation Fundamentals (2 of 2)
• Basic Valuation Model
– The value of an asset is the present value of all the
future cash flows it is expected to provide.
     
1 2
0 1 2
(6.4)
1 1 1
n
n
CF
CF CF
V
r r r
   
  
• V0 = value of the asset at time zero
• CFt = cash flow expected in year t
• r = required return (discount rate)
• n = time period (investment’s life or investor’s
holding period)

Personal Finance Example 6.6 (1 of 3)
Celia Sargent values each asset by discounting its cash
flows using Equation 6.4. Because Michael’s stock pays a
perpetual stream of $300 dividends, Equation 6.4 reduces
to Equation 5.7, which says that the present value of a
perpetuity equals the dividend payment divided by the
required return. Celia decides that a 12% discount rate is
appropriate for this investment, so her estimate of the
value of Michael’s Enterprises stock is
$300 0.12 $2,500
 

Personal Finance Example 6.6 (2 of 3)
Next, Celia values the oil well investment, which she
believes is the most risky of the three investments.
Discounting the oil well’s cash flows using a 20%
required return, Celia estimates the well’s value to be
1 2 4
$2,000 $4,000 $10,000
$9,266.98
(1 0.20) (1 0.20) (1 0.20)
  
  

Personal Finance Example 6.6 (3 of 3)
Finally, Celia estimates the value of the painting by
discounting the expected $85,000 cash payment in
five years at 15%:
$85,000 ÷ (1 + 0.15)5 = $42,260.02
Note that, regardless of the pattern of the asset’s
expected cash flows, Celia can use the basic
valuation equation to determine the asset’s value.

6.4 Bond Valuation (1 of 11)
• Bond Fundamentals
– Bonds are long-term debt instruments used by
business and government to raise large sums of
money, typically from a diverse group of lenders
– Most corporate bonds
• pay interest semiannually (every six months) at a
stated coupon rate
• have an initial maturity of 10 to 30 years
• and have a par value, principal, or face value, of
$1,000 that the borrower must repay at maturity

Example 6.7
The Mills Company just issued a 6% coupon rate, 10-year
bond with a $1,000 par value that pays interest annually.
Investors who buy this bond receive the contractual right to
two types of cash flows: (1) $60 annual interest (6%
coupon rate × $1,000 par value) distributed at the end of
each year and (2) the $1,000 par value at the end of the
tenth year.

6.4 Bond Valuation (2 of 11)
• Bond Values for Annual Coupons
         
   
0 1 2 3
0
1
1 1 1 1 1
(6.5)
1 1
n n
n
t n
t
C C C C M
B
r r r r r
C M
B
r r

     
    
   
 
   
 
   
   

• B0 = value (or price) of the bond at time zero
• C = annual coupon interest payment in dollars
• n = number of years to maturity
• M = par value in dollars
• r = annual required return on the bond

6.4 Bond Valuation (3 of 11)
• Bond Values for Annual Coupons
   
0
1
1 (6.5a)
1 1
n n
C M
B
r r r
 
 
  
 
 
   
 
 
• B0 = value (or price) of the bond at time zero
• C = annual coupon interest payment in dollars
• n = number of years to maturity
• M = par value in dollars
• r = annual required return on the bond

Example 6.8 (1 of 4)
Tim Sanchez wishes to determine the current value of
the Mills Company bond. If the bond pays interest
annually and the required annual return on the bond
is 6% (equal to its coupon rate), then we can
calculate the bond’s value using Equation 6.5a:
0 10 10
0
$60 1 $1,000
1
0.06 (1 0.06) (1 0.06
$1,000 0.44161 $558.39 $441.61 $558
)
.39 $1,
[ 000 0
] . 0
B
B
 
 
  
  
 
 
 

  
The timeline on the next slide depicts the
computations involved in finding the bond value.

Example 6.8 (2 of 4)

Example 6.8 (3 of 4)
Calculator use Using the Mills Company’s
inputs shown at the left, you should find the
bond value to be exactly $1,000. If you
compare the calculator keystrokes to the
ones we displayed when we used a
calculator to find the present value of an
ordinary annuity earlier in this text, you will
see that there is an additional term here,
namely, the $1,000 future value (FV). We
must add that to our sequence of keystrokes
because the bond pays out an annuity plus a
$1,000 lump sum at the end. When we add
the FV keystroke, we are capturing the value
of that final lump sum payment when the
bond matures. Note that the calculated bond
value is equal to its par value, which will
always be the case when the required return
is equal to the coupon rate.

Example 6.8 (4 of 4)
Spreadsheet use We can also calculate the value of
the Mills Company bond using Excel’s PV function as
shown in the following Excel spreadsheet.

6.4 Bond Valuation (4 of 11)
• Bond Values for Semiannual Coupons
– As a practical matter, most bonds make
semiannual rather than annual interest
payments
– Calculating the value for a bond paying
semiannual interest requires three
changes to the approach we’ve used so
far:
1. Convert the annual coupon payment, C, to a semiannual payment
by dividing C by 2
2. Recognize that if the bond has n years to maturity it will make 2n
coupon payments (i.e., in n years there are 2n semiannual
periods)
3. Discount each payment by using the semiannual required return
calculated by dividing the annual required return, r, by 2

6.4 Bond Valuation (5 of 11)
• Bond Values for Semiannual Coupons
0 1 2 3 2 2
2
0 2
1
2 2 2 2
1+ 1+ 1+ 1+ 1
2 2 2 2 2
2 (6.6)
1+ 1
2 2
n n
n
t n
t
C C C C
M
B
r r r r r
C
M
B
r r

     
         

         
         
   
   
   
 
   
   

   
   
   
   


6.4 Bond Valuation (6 of 11)
• Bond Values for Semiannual Coupons
0 2 2
/ 2 1
1 (6.6a)
/ 2
1 1
2 2
n n
C M
B
r r r
 
 
  
  
  
     
 
 
   
   
 
• B0 = value (or price) of the bond at time zero
• C = annual coupon interest payment in dollars
• n = number of years to maturity
• M = par value in dollars
• r = annual required return on the bond

Example 6.9 (1 of 3)
Assuming that the Mills Company bond pays interest
semiannually and that the required annual return, r, is 6%,
Equation 6.6a indicates that the bond’s value is
 
0 2(10) 2(10
0
)
$60/2 1 $1,000
1
0.06/2 0.06 0.06
1 1
$1,000 0.44632 $553.68 $446.32 $553.68 $1,0
2 2
00
B
B

 
 
  
  
  
     
 

   

   
   
 
As before, because the required return equals the coupon
rate, the bond sells at par value. We will soon see that
when the required return does not equal the coupon rate,
the bond may sell above or below par value.

Example 6.9 (2 of 3)
Calculator use When using a calculator
to find the price of a bond that pays
interest semiannually, N is the number of
semiannual periods until maturity and
PMT is the semiannual interest payment.
For the Mills Company bond, N equals
20 semiannual periods (2 × 10 years),
I/Y remains 6 for the annual required rate
of return, PMT equals an interest
payment of $30 ($60 ÷ 2), and FV is
1,000 for the par value. As shown on the
left, this calculation verifies that the
bond’s price is $1,000.
Spreadsheet use The value of the Mills Company bond paying
semiannual interest at an annual required return of 6% also can
be calculated as shown in the following Excel spreadsheet.

Example 6.9 (3 of 3)

6.4 Bond Valuation (7 of 11)
• Changes in Bond Prices
– When the required return rises, the
bond price falls, and when the
required return falls, the bond price
rises
– Required Returns and Bond Prices
• Discount
– The amount by which a bond sells below its par value
» When the required return is greater than the
coupon rate, the bond’s value will be less than its
par value
• Premium
– The amount by which a bond sells above its par value
» When the required return falls below the coupon
rate, the bond’s value will be greater than par

Example 6.10 (1 of 3)
Let’s reconsider the Mills Company bond paying a
6% coupon rate and maturing in 10 years (assume
annual interest payments for simplicity). Initially, we
assumed that the required return on this bond was
6%, so the bond’s value was equal to par value,
$1,000. Let’s see what happens to the bond’s value if
the required return is higher or lower than the coupon
rate. Table 6.5 shows that at an 8% required return,
the bond sells at a discount with a value of $865.80,
but if the required return is 4%, the bond sells at a
premium with a value of $1,162.22.

Example 6.10 (2 of 3)
Calculator use Using the inputs shown at
the left, you will find that when the required
annual return is 8%, the Mill’s annual
coupon bond sells for $865.80, which is a
discount of $134.20 below par value. At a
4% required annual return, the bond sells
for $1,162.22, a premium of $162.22.
Figure 6.5 illustrates the inverse
relationship between the required return
and the price of the Mills Company bond.

Example 6.10 (3 of 3)
Spreadsheet use The values for the Mills Company
bond at required returns of 8% and 4% also can be
calculated as shown in the following Excel
spreadsheet. Once this spreadsheet has been
configured, you can calculate the bond price for any
required return by simply changing the input values.

Table 6.5 Bond Values for Various Required
Returns (Mills Company’s 6% Annual
Coupon Bond with a 10-Year Maturity and
$1,000 Par Value)
Required return, r Bond value, B0 Status
8% $ 865.80 Discount
6 1,000.00 Par value
4 1,162.22 Premium

Figure 6.5 Bond Values and Required
Returns

6.4 Bond Valuation (8 of 11)
• Changes in Bond Prices
– Interest Rate Risk
• The chance that a bond’s required return will
change and thereby cause a change in the bond’s
price
• Other things being equal, Long-term bond prices
move more when rates change whereas short-
term bond prices are less sensitive to rate
changes
– Long-term bonds are associated with more
interest rate risk

6.4 Bond Valuation (9 of 11)
• Changes in Bond Prices
– Interest Rate Risk
• Other things being equal, higher coupon bonds
have less interest rate risk exposure compared to
lower coupon bonds
• Other things being equal, a bond with lower
required return has greater interest rate risk
compared with a bond that has a higher required
return

Figure 6.6 Interest Rate Risk and Bond
Characteristics

6.4 Bond Valuation (10 of 11)
• Changes in Bond Prices
– The Passage of Time and Bond Prices
• If a bond trades at a premium, and its required
return remains constant, as the bond gets
closer to maturity its price will fall, eventually
reaching par value at maturity
• If a bond sells at a discount and its required
return remains fixed, as time passes the bond’s
price will rise, eventually reaching par value at
the moment before the bond matures
• Bonds trading at par will remain at that value
until maturity as long as the required return
does not deviate from the coupon rate

Example 6.11
Figure 6.7 depicts the behavior of the bond values
calculated earlier and presented in Table 6.5 for Mills
Company’s 6% coupon rate bond paying annual
interest and having 10 years to maturity. Each of the
three required returns—8%, 6%, and 4%—remains
constant throughout the bond’s 10-year life. The
bond’s value at both 8% and 4% approaches and
ultimately equals the bond’s $1,000 par value at its
maturity, as the discount (at 8%) or premium (at 4%)
declines with the passage of time.

Figure 6.7 Time to Maturity and Bond Prices

6.4 Bond Valuation (11 of 11)
• Yield to Maturity (YTM)
– The compound annual rate of return earned on a bond
purchased assuming that all payments arrive on time
and that the investor holds the bond to maturity
– Mathematically, a bond’s YTM is the discount rate that
equates the bond’s market price to the present value of
its cash flows

Personal Finance Example 6.12
(1 of 10)
Jaylen Washington wishes to find the YTM on Mills Company’s bond.
The bond currently sells for $929.76, has a 6% coupon rate and
$1,000 par value, pays interest annually, and has 10 years to maturity.
We can find the bond’s YTM using equation 6.5a and iteratively solving
for the discount rate, r, at which the present value of cash flows equals
the bond’s market price. Start by plugging the bond’s price and cash
flows into Equation 6.5a and substituting YTM for r.
10 10
$60 1 $1,000
$929.76 1
YTM (1 YTM) (1 YTM)
 
 
  
  
 
  

Personal Finance Example 6.12
(2 of 10)
Now make an initial guess for the YTM and solve for the present value
of the bond’s cash flows. Because the bond’s current price is less than
its par value of $1,000 (i.e., it’s a discount bond) its YTM must be
greater than its coupon rate. Make an initial guess for the YTM that is
greater than the bond’s 6% coupon rate, say 10%.
10 10
$60 1 $1,000
$929.76? 1
0.10 (1 0.10) (1 0.10)
$929.76 $600[0.61446] $385.54 $368.68 $385.54 $754.22
 
 
 
  
 
  
    

Personal Finance Example 6.12 (3
of 10)
If the discount rate is 10% the present value of cash flows
equals $754.22, which is less than the bond’s current
price, so the YTM must be less than 10%. Now try 8%.
10 10
$60 1 $1,000
$929.76 ? 1
0.08 (1 0.08) (1 0.08)
$929.76 $750 [0.53681] $463.19 $402.61 $463.19 $865.80
 
 
 
  
 
  
    

Personal Finance Example 6.12 (4 of 10)
Here again the present value of $865.80 is less than the $929.76
current price, so the YTM must be lower still. Trying 7% gives the
answer.
10 10
$60 1 $1,000
$929.76 ? 1
0.07 (1 0.07) (1 0.07)
$929.76 $857.14 [0.49165] $508.35 $421.41 $508.35 $929.76
 
 
 
  
 
  
    
At a 7% discount rate the present value of the bond’s
cash flows equals the current price, so 7% is the
bond’s YTM. In other words, the market’s required
annual rate of return for the Mills’ bond is 7%.

Personal Finance Example 6.12 (5 of 10)
Calculator use Financial
calculators require either the
present value (the bond
price in this case) or the
future cash flows (coupons
and principal in this case) to
be input as negative
numbers to calculate yield to
maturity. Using the inputs
shown at the left, you find
the YTM to be 7.0%.

Personal Finance Example 6.12 (6
of 10)
Spreadsheet use Excel makes it easier to calculate
the yield to maturity. One way to solve for the YTM is
to enter the bond’s cash flows and use Excel’s
internal rate of return (IRR) function. Start by entering
the bond’s price as a negative number, and then
enter subsequent coupon and principal payments as
positive values. The price is the cash outflow the
investor must pay to receive the subsequent cash
inflows that the bond provides.

Personal Finance Example 6.12 (7
of 10)
Spreadsheet use Excel’s IRR function simply finds
the discount rate at which the present value of cash
inflows (the coupon and principal payments) equals
the present value of cash outflows (the bond’s price).
But that is exactly the definition of the YTM—the
discount rate at which the present value of a bond’s
coupon and principal payments equals the bond’s
price. Therefore, the IRR function solves for a bond’s
YTM. Note also that the IRR function calculates the Y
TM per period. In this example, payments arrive
annually, so each period is one year, and the
resulting IRR equals the YTM per year.

Personal Finance Example 6.12
(8 of 10)

Personal Finance Example 6.12 (9
of 10)
Spreadsheet use Another way to calculate the YTM in
Excel recognizes that the typical bond’s interest payments
form an annuity, which allows you to solve for a bond’s
yield to maturity using Excel’s RATE function.

Personal Finance Example 6.12
(10 of 10)

Personal Finance Example 6.13 (1 of 5)
Now suppose Mills’ bond makes semiannual coupon
payments but as in the previous example, the bond sells
for $929.76. For that price investors receive 20 semiannual
coupon payments of $30 (6% coupon rate × $1,000 par
value ÷ 2 coupon payments per year)plus the $1,000 par
value at maturity. Before using the RATE function, let’s use
Equation 6.6a and iteratively solve for the YTM. Start by
plugging the bond’s given information into Equation 6.6a
and substituting YTM for r.
2(10) 2(10)
$60 / 2 1 $1,000
$929.76 1
YTM/2 YTM YTM
1 1
2 2
 
 
  
  
  
     
 
 
   
   
 

Personal Finance Example 6.13 (2 of 5)
The YTM will be higher than the coupon rate because the bond sells
below par. In the example with annual payments, the YTM was 7%,
and intuitively it might seem that the same answer will prevail here.
Plugging 7% into the equation for Y TM we find that the present
value is a bit too low.
2(10) 2(10)
$60 / 2 1 $1,000
$929.76 ? 1
0.07 / 2 0.070 0.070
1 1
2 2
$929.76 $857.14 [0.49743] $502.57 $928.94
 
 
  
 
  
     
 
 
   
   
 
  

Personal Finance Example 6.13 (3 of 5)
If the discount rate is 7%, the present value of cash flows
is $928.94 which is barely below the market price, so the Y
TM must be slightly lower than 7%. With more trial and
error, the correct YTM of 6.988% emerges.
2(10) 2(10)
$60 / 2 1 $1,000
$929.76 ? 1
0.06988 / 2 0.06988 0.06988
1 1
2 2
$929.76 $858.61 [0.49685] $503.15 $426.60 $503.15 $929.75
 
 
  
 
  
     
 
 
   
   
 
    
The discount rate of 6.988% results in a present
value of the bond’s cash flows that is within
$0.01 of the bond’s current price.

Personal Finance Example 6.13 (4 of 5)
Calculator use Using the
inputs shown at the left, you
will find a calculator solution
of 6.988% for the annual
YTM. Because your
calculator was set to two
periods per year (P/Y) it
accounts for the semiannual
payment frequency when
determining the annual rate
of return (I/Y). To find the
semiannual required rate of
return, simply divide the
annual YTM by two.

Personal Finance Example 6.13 (5 of 5)
Spreadsheet use The RATE function in Excel calculates
the YTM per period, or the YTM per half year in the case of
the Mills bond. To convert the 3.494% semiannual YTM
into an annual figure, simply multiply it by two.

Review of Learning Goals (1 of 10)
• LG 1
– Describe interest rate
fundamentals, the term structure of
interest rates, and risk premiums.
• Equilibrium in the flow of funds between savers and
borrowers produces the interest rate or required return
• Most interest rates are expressed in nominal terms
• The nominal interest rate represents the rate at which
money grows over time, whereas the real interest rate
represents the rate at which purchasing power grows
over time
• The difference between the nominal rate and the real
rate is (approximately) the inflation rate (or the expected
inflation rate)

Review of Learning Goals (2 of 10)
• LG 1 (Cont.)
– Describe interest rate
fundamentals, the term structure of
interest rates, and risk premiums.
• For risky assets, the nominal interest rate is the sum of
the risk-free rate and a risk premium reflecting issuer and
issue characteristics
• The risk-free rate is the real rate of interest plus an
inflation premium
• For any class of similar-risk bonds, the term structure of
interest rates is the relation between the rate of return
and the time to maturity
• Yield curves plot this relation on a graph and can be
downward sloping (inverted), upward sloping (normal), or
flat

Review of Learning Goals (3 of 10)
• LG 1 (Cont.)
– Describe interest rate fundamentals, the term structure
of interest rates, and risk premiums.
• The expectations theory, liquidity preference
theory, and market segmentation theory offer
different explanations of the shape of the yield
curve
• Risk premiums for non-Treasury debt issues result
from default or credit risk and other features such
as maturity and exposure to interest rate risk

Review of Learning Goals (4 of 10)
• LG 2
– Review the legal aspects of bond
financing and bond cost.
• Corporate bonds are long-term debt instruments that the
company must repay under clearly defined terms
• Most bonds are issued with maturities of 10 to 30 years
and a par value of $1,000
• The bond indenture, enforced by a trustee, states all
conditions of the bond issue
• It contains both standard debt provisions and restrictive
covenants, which may include a sinking-fund
requirement and/or a security interest
• The cost of a bond to an issuer depends on its maturity,
offering size, and issuer risk and on the risk-free rate

Review of Learning Goals (5 of 10)
• LG 3
– Discuss the general features, yields, prices, ratings,
popular types, and international issues of corporate
bonds.
• A bond issue may include a conversion feature, a
call feature, or stock purchase warrants
• The return on a bond can be measured by its
current yield, yield to maturity (YTM), or yield to
call (YTC)
• Bond prices are typically reported along with their
coupon, maturity date, and yield to maturity (YT
M).
• Bond ratings by independent agencies indicate
the risk of a bond issue

Review of Learning Goals (6 of 10)
• LG 3 (Cont.)
– Discuss the general features, yields, prices, ratings,
popular types, and international issues of corporate
bonds.
• Various types of traditional and contemporary
bonds are available
• Eurobonds and foreign bonds enable established
creditworthy companies and governments to
borrow large amounts internationally

Review of Learning Goals (7 of 10)
• LG 4
– Understand the key inputs and basic model used in
the bond valuation process.
• Key inputs to the valuation process include cash
flows, timing, risk, and the required return
• The value of any asset is equal to the present
value of all future cash flows it is expected to
provide over the relevant time period

Review of Learning Goals (8 of 10)
• LG 5
– Apply the basic valuation model to bonds, and
describe the impact of required return and time to
maturity on bond prices.
• The value of a bond is the present value of its
coupon payments plus the present value of its par
value
• The discount rate used to determine bond value is
the required return, which may differ from the
bond’s coupon rate
• A bond can sell at a discount, at par, or at a
premium, depending on whether the required
return is greater than, equal to, or less than its
coupon rate

Review of Learning Goals (9 of 10)
• LG 5 (Cont.)
– Apply the basic valuation model to bonds, and
describe the impact of required return and time to
maturity on bond values.
• The chance that interest rates will change and
thereby alter the required return and bond value is
called interest rate risk.
• The shorter the amount of time until a bond’s
maturity, the higher its coupon rate, or the higher
its required return, the less responsive is its
market value to a change in the required return.
• The passage of time affects bond prices. The
price of a bond will approach its par value as the
bond moves closer to maturity.

Review of Learning Goals (10 of 10)
• LG 6
– Explain yield to maturity (YTM), its calculation, and
the procedure used to value bonds that pay interest
semiannually.
• Yield to maturity is the rate of return investors earn if they
buy a bond at a specific price and hold it until maturity
• YTM can be calculated by using a financial calculator or
by using an Excel spreadsheet
• Bonds that pay interest semiannually are valued by using
the same procedure used to value bonds paying annual
interest except that the interest payments are one-half of
the annual interest payments, the number of periods is
twice the number of years to maturity, and the required
return is one-half of the stated annual required return on
similar-risk bonds

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