Chapter
Seven
Interest Rates and
Bond Valuation
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
7.1
Key Concepts and Skills





Know the important bond features and bond types
Understand bond values and why they fluctuate
Understand bond ratings and what they mean
Understand the impact of inflation on interest rates
Understand the term structure of interest rates and the
determinants of bond yields
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.2
Chapter Outline







Bonds and Bond Valuation
More on Bond Features
Bond Ratings
Some Different Types of Bonds
Bond Markets
Inflation and Interest Rates
Determinants of Bond Yields
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.3
Bond Definitions 7.1
 Bond – a long term debt obligation issued by a corporation
 Par value (face value) – the amount of money that will be
repaid at the maturity date of the bond
 Coupon rate – the percentage of the bond’s face value that
will be paid in interest every year
 Coupon payment – the dollar value of the interest that is paid
 Maturity date – the point in time when the bond will be
redeemed by the issuer. The investor will receive cash equal
to the face value of the bond.
 Yield or Yield to maturity – also known as the Internal Rate of
Return on the bond. It is the return to the investor that is
derived from both the coupons received plus any capital gain
or loss.
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.4
Treasury Bills
 Treasury Bills (T Bills) are issued by both provincial & federal
governments
 Federal government T Bills are considered the risk-free security
 T Bills are bought at a discount and mature at their face value. The
difference between the purchase price and face value is the interest
earned on the T Bill
 To price a T Bill, use the following formula
PVTreasury 
Bill
 Where:
Face Value

 n 
1  BEY   
B

BEY = the bond equivalent yield
n = the number of days until maturity
B = the annual basis (365 days in Canada)
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.5
Treasury Bills – Example
 Assume that you want to purchase a 91 day Treasury Bill with a face
value of $1,000,000. If the BEY is 6%, what is the purchase price?
Face Value

 n 
1  BEY   
B

1,000,000


 91  
1  .06 


365



PVTreasury 
Bill
 $985.261.57
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.6
Present Value of Cash Flows as Rates Change
 The market price of a bond is simply the present value of the
bond’s future cash flows
 Bonds have two types of future cash flows
 Interest annuity (the stream of coupons)
 Face value at maturity
 When interest rates go up, the market value of the bond will
go down
 When interest rates go down, the market value of the bond
will go up
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.7
The Bond-Pricing Equation
 1-1  r  t  FaceValue
Bond Value  $Coupon 

t
r
(1

r)


Where:
$ Coupon = the periodic interest paid by the bond
r = the yield-to-maturity for the bond
t = the number of time periods to maturity
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.8
Valuing a Bond with Annual Coupons
 Consider a bond with a $1,000 face value, a coupon rate of
10%, paid annually and 5 years to maturity. The yield to
maturity is 11%. What is the market value of the bond?
Formula Approach
 1-1  r  t  FaceValue
Bond Value  $Coupon 

r
(1  r)t


 1-1  .115 
1,000
 100 


5
.11

 1  .11
 $963.04
Calculator Approach
1,000
FV
100
PMT
5
N
11
I
PV
$963.04
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.9
Valuing a Bond with Annual Coupons
 Now consider a second bond, also with a $1,000 face value
and a 10% coupon, paid annually, but with 20 years to
maturity and a yield to maturity of 8%. What is the price of
this bond?
Formula Approach
 1-1  r  t  FaceValue
Bond Value  $Coupon 

r
(1  r)t


 1-1  .08 20 
1,000
 100 


20
0.08

 1  .08
 $1,196.36
Calculator Approach
1,000
FV
100
PMT
20
N
8
I
PV
$1,196.36
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.10
Graphical Relationship Between Price and
Yield-to-Maturity
The important concept
to note is the inverse
relationship between
price & YTM
1500
1400
1300
1200
Based on a 10 year,
$1,000 bond with an
8% coupon
1100
1000
900
800
700
600
0%
2%
4%
6%
8%
10%
12%
14%
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.11
Bond Prices: Relationship Between Coupon
and Yield
 If YTM = coupon rate, then face value = market price
 If YTM > coupon rate, then face value > market price
 The bond is selling at a discount
 Why?
 If YTM < coupon rate, then face value < market price
 The bond is selling at a premium
 Why?
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.12
Example – Semiannual Coupons
 Most bonds in Canada make coupon payments semi-annually.
 Suppose you have an 8% semi-annual pay bond with a face value of
$1,000 that matures in 7 years. If the yield is 10%, what is the price of
this bond?
 Formula Approach
 1-1  r t  FaceValue
Bond Value  $Coupon 

r
(1  r)t


  0.10  7 x 2 

 1- 1 

80  
1,000
2 


0.10
  0.10  7 x 2
2 


 1 
2
2 

 
Calculator Approach
1,000
FV
80 ÷ 2 =
PMT
7x2=
N
10 ÷ 2 =
I
PV
$901.01
 $901.01
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.13
Interest Rate Risk
 Price Risk
 Change in price due to a change in interest rates
 Long-term bonds have more price risk than short-term bonds
 Bonds with low coupons have more price risk than bonds with high
coupons
 Reinvestment Rate Risk
 Uncertainty concerning the interest rate at which future cash flows can
be reinvested
 Long-term bonds have more reinvestment rate risk than short-term
bonds
 Bonds with high coupons have more reinvestment rate risk than bonds
with low coupons
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.14
Figure 7.2 – Interest Rate Risk and Time to Maturity
This graph shows the impact on price
as the YTM changes for both a oneyear bond and a thirty-year bond.
Both bonds have a $1,000 face value
and a 10% coupon. Note how much
more sensitive the long bond is to a
change in the YTM.
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.15
Computing Yield-to-Maturity
 Yield-to-maturity is the discount rate implied by the current bond price
 For example, assume that a ten year, $1,000 bond with a 6% coupon, paid
annually, is currently trading at $950 in the market. What is the bond’s
yield-to-maturity (YTM)?
Formula Approach
 1-1  r t  FaceValue
Bond Value  $Coupon 

r
(1  r)t


 1-1  r 10  1,000
$950  60

10
r

 1  r 
r  6.7021%
Calculator Approach
1,000
FV
60
PMT
950 +/-
PV
10
I
N
6.7021%
 The YTM is the discount rate that makes the equality true in the bond
pricing formula.
 Solve for it using the function keys on the calculator or trial & error if
using algebra
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.16
Example – Finding the YTM
 Consider another bond with a 10% annual coupon rate, 15
years to maturity and a face value of $1000. The current price
is $928.09.
 Will the yield-to-maturity be more or less than 10%?
Calculator Approach
1,000
FV
100
PMT
928.09 +/-
PV
15
I
N
11%
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.17
Understanding YTM
 The YTM can always be decomposed into its two component parts.
These are:
 Coupon Yield
 Capital gain or loss
 Assume that you buy a 10 year, $1,000 bond with an 8% coupon,
priced to yield 6%.
 Step #1: First calculate the price of the bond.
 1-1  r  t  FaceValue
Bond Value  $Coupon 

t
r
(1

r)


 1-1.06 10  1,000
 80 

10
0
.
06


1
.
06


 $1,147.20
Calculator Approach
1,000
FV
80
PMT
10
N
6
I
PV
$1,147.20
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.18
Understanding YTM
 In Step #2, we calculate the new price after one year has passed, assuming
all else remains equal.
Calculator Approach
t
 1-1  r   FaceValue
1,000
FV
Bond Value  $Coupon 

t
r
(1  r)


80
PMT
9
 1-1.06   1,000
9
N
 80 

9
 0.06  1.06 
6
I
 $1,136.03
PV
$1,136.03
 Step #3: Calculate the coupon yield by dividing the annual coupon
payment by the beginning price
Coupon Payment
Initial Price
80

1,147.20
 6.97%
Coupon Yield 
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7.19
Understanding YTM
 Step #4: Calculate the capital gain or loss
Capital Gain / Loss 
P1  P0
P0
1136.03  1147.20
1,147.20
 0.9737%

 Step #5: Calculate the Yield to Maturity
YTM  Coupon Yield  Capital Gain / Loss
 6.9735  0.9737
 6.0%
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.20
YTM with Semiannual Coupons
 Suppose a 20 year, $1,000 bond with a 10% coupon, paid
semi-annually, is selling for $1197.93.
 Is the YTM more or less than 10%?
 What is the semiannual coupon payment?
 How many periods are there?
Calculator Approach
1,000
FV
100 ÷ 2 =
PMT
1,197.93 +/-
PV
20 x 2 =
I
N
4 x 2 = 8%
Since the coupon is
paid twice a year, the
rate you calculate is a
semi-annual yield.
Double it to obtain
the quoted YTM.
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.21
Bond Pricing Theorems
 Bonds of similar risk (and maturity) will be priced to yield
about the same return, regardless of the coupon rate
 If you know the price of one bond, you can estimate its YTM
and use that to find the price of the second bond
 This is a useful concept that can be transferred to valuing
assets other than bonds
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.22
Bond Prices with a Spreadsheet
 There is a specific formula for finding bond prices on a
spreadsheet




PRICE(Settlement,Maturity,Rate,Yld,Redemption,Frequency,Basis)
YIELD(Settlement,Maturity,Rate,Pr,Redemption, Frequency,Basis)
Settlement and maturity need to be actual dates
The redemption and Pr need to given as % of par value
 Click on the Excel icon for an example
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.23
Differences Between Debt and Equity 7.2
 Debt
 Not an ownership interest
 Bondholders do not have
voting rights
 Interest is considered a cost
of doing business and is tax
deductible
 Bondholders have legal
recourse if interest or
principal payments are
missed
 Excess debt can lead to
financial distress and
bankruptcy
 Equity
 Ownership interest
 Common shareholders vote
for the board of directors
and other issues
 Dividends are not
considered a cost of doing
business and are not tax
deductible
 Dividends are not a liability
of the firm and shareholders
have no legal recourse if
dividends are not paid
 An all equity firm can not go
bankrupt
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7.24
The Bond Indenture
 Contract between the company (the issuer of the bond) and the
bondholders and includes






The basic terms of the bonds
The total amount of bonds issued
A description of property used as security, if applicable
Sinking fund provisions
Call provisions
Details of protective covenants
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7.25
Bond Classifications
 Registered vs. Bearer Bonds
 Registered – can only be sold by the registered owner
 Bearer – similar to cash; possession implies ownership
 Security
 Collateral – assets pledged as a secondary source of repayment
 Mortgage – secured by real property, normally land or buildings
 Debentures – unsecured debt with original maturity of 10 years or
more
 Notes – unsecured debt with original maturity less than 10 years
 Seniority
 Senior versus Junior debt – refers to preference in position with
respect to other lenders (Senior debt is paid first; junior debt paid last)
 Subordinated debt – indicates that it has a lower priority than other,
more senior debt
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7.26
Bond Classifications Continued
 Repayment
 Sinking Fund – Account managed by the bond trustee for early bond
redemption. Reduces default risk & improves marketability.
 Call Provision – allows the company to repurchase all or a
portion of the issue prior to the original maturity
 Call premium – amount above the face value the borrower agrees to
pay, should they call the bond before its original maturity
 Deferred call - The issuer usually cannot call the bond during
 Call protected - the years immediately after the issue date.
 Canada plus call – Protects the investor against a call by providing
compensation equal to the foregone interest, should a call occur
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7.27
Covenants
 Protective Covenants
 Negative covenants – things the borrower agrees to not do
 Agrees to limit the amount of dividends paid
 Agree to not pledge assets to other lenders
 Agree to not merge with, sell to or acquire another firm
 Agree to not buy new capital assets above $x in value
 Agree to not issue new debt
 Positive covenants – things the borrower agrees to do
 Maintain a minimum current ratio
 Provide audited financial statements
 Maintain collateral in good condition
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7.28
Bond Characteristics and Required Returns
 The coupon rate depends on the risk characteristics of the
bond when issued
 Which bonds will have the higher coupon, all else equal?




Secured versus unsecured debt?
Subordinated debenture versus senior debt?
A bond with a sinking fund versus one without?
A callable bond versus a non-callable bond?
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.29
Credit Ratings: Investment Grade
Credit Risk
Moody’s
Standard &
Poors
Fitch
Duff &
Phelps
Highest
quality
Aaa
AAA
AAA
AAA
High quality
(very strong)
Aa
AA
AA
AA
Upper
Medium
(strong)
A
A
A
A
Medium
grade
Baa
BBB
BBB
BBB
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7.30
Credit Ratings: Speculative or Junk Debt
Credit Risk
Moody’s
Standard &
Poors
Fitch
Duff &
Phelps
Lower
Medium
Ba
BB
BB
BB
Low grade
(Speculative)
B
B
B
B
Poor Quality
Caa
CCC
CCC
CCC
Most
speculative
Ca
CC
CC
CC
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
Standard & Poor's Bond Rating Scale
7.31












AAA - An obligor rated 'AAA' has EXTREMELY STRONG capacity to meet its financial commitments.
AA - An obligor rated 'AA' has VERY STRONG capacity to meet its financial commitments. It differs from the highest rated
obligors only in small degree.
A - An obligor rated 'A' has STRONG capacity to meet its financial commitments but is somewhat more susceptible to the
adverse effects of changes in circumstances and economic conditions than obligors in higher-rated categories.
BBB - An obligor rated 'BBB' has ADEQUATE capacity to meet its financial commitments. However, adverse economic
conditions or changing circumstances are more likely to lead to a weakened capacity of the obligor to meet its financial
commitments.
BB - An obligor rated 'BB' is LESS VULNERABLE in the near term than other lower-rated obligors. However, it faces
major ongoing uncertainties and exposure to adverse business, financial, or economic conditions which could lead to the
obligor's inadequate capacity to meet its financial commitments.
B - An obligor rated 'B' is MORE VULNERABLE than the obligors rated 'BB', but the obligor currently has the capacity to
meet its financial commitments. Adverse business, financial, or economic conditions will likely impair the obligor's capacity
or willingness to meet its financial commitments.
CCC - An obligor rated 'CCC' is CURRENTLY VULNERABLE, and is dependent upon favorable business, financial, and
economic conditions to meet its financial commitments.
CC - An obligor rated 'CC' is CURRENTLY HIGHLY VULNERABLE.
R - An obligor rated 'R' is under regulatory supervision owing to its financial condition. During the pendency of the
regulatory supervision the regulators may have the power to favor one class of obligations over others or pay some
obligations and not others. Please see Standard & Poor's issue credit ratings for a more detailed description of the effects of
regulatory supervision on specific issues or classes of obligations.
SD and D - An obligor rated 'SD' (Selective Default) or 'D' has failed to pay one or more of its financial obligations (rated or
unrated) when it came due. A 'D' rating is assigned when Standard & Poor's believes that the default will be a general default
and that the obligor will fail to pay all or substantially all of its obligations as they come due. An 'SD' rating is assigned when
Standard & Poor's believes that the obligor has selectively defaulted on a specific issue or class of obligations but it will
continue to meet its payment obligations on other issues or classes of obligations in a timely manner. Please see Standard &
Poor's issue credit ratings for a more detailed description of the effects of a default on specific issues or classes of
obligations.
Note: Obligors rated 'BB', 'B', 'CCC', and 'CC' are regarded as having significant speculative characteristics. 'BB' indicates
the least degree of speculation and 'CC' the highest. While such obligors will likely have some quality and protective
characteristics, these may be outweighed by large uncertainties or major exposures to adverse conditions. Plus (+) or minus
(?): Ratings from 'AA' to 'CCC' may be modified by the addition of a plus or minus sign to show relative standing within the
major rating categories.
Source: Standard & Poors
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7.32
Stripped or Zero-Coupon Bonds 7.4




Make no periodic interest payments (coupon rate = 0%)
All cash flows occur on the maturity date
Sometimes called zeroes, or deep discount bonds
Bondholder must pay taxes on accrued interest every year,
even though no interest is received (thus are best held in a taxdeferred account, such as an RRSP)
 Market price is the PV of the face value at maturity
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.33
Zero Coupon or Stripped Bonds
 Assume that you want to purchase a 30 year stripped bond with a
$100,000 face value. If the appropriate YTM is 8%, how much will
you have to pay today to buy this bond?
Face Value
PVStripped 
Bond
1  YTM t
100,000

1.0830
 $9,937.73
Calculator Approach
100,000
FV
0
PMT
30
N
8
I
PV
$9,937.73
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7.34
Floating Rate Bonds
 Coupon rate floats depending on some index value
 There is less price risk with floating rate bonds
 The coupon floats, so it is less likely to differ substantially from the
yield-to-maturity
 Coupons may have a “collar” – the rate cannot go above a specified
“ceiling” or below a specified “floor”
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.35
Other Bond Types
 Disaster (CAT) bonds – payout to the investor is dependent on
the occurrence of some major catastrophic event
 Income bonds – coupons are tied to firm profitability
 Convertible bonds – bonds may be converted into common
stock
 Real Return bonds – the bond is adjusted for inflation
 Put bond (retractable bond) – the investor may sell the bond
back to the issuer at a fixed price
 LYON (Liquid Yield Option Note) - created by Merrill Lynch.
Bond is a callable, puttable, convertible, zero coupon,
subordinated note
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7.36
Bond Markets 7.5
 Primarily over-the-counter transactions with dealers connected
electronically
 Extremely large number of bond issues, but generally low
daily volume in single issues
 Makes getting up-to-date prices difficult, particularly on small
corporate issues. One alternative for the retail investor is
http://www.cbidmarkets.com
 Treasury securities are an exception
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.37
Bond Quotations
 From the Financial Post, October 20, 2005
Canada 10.000 Jun 01/08 115.94 3.55
Issuer
Coupon
Maturity
Price
YTM




The issuer is the Government of Canada
The coupon rate is 10% (assumed to be semiannual)
The maturity date is June 1, 2008
The quoted price can be interpreted as either the price per $100 of
face value or as a percentage of face value
 The yield to maturity is 3.55%
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7.38
Inflation and Interest Rates 7.6
 Nominal rate of interest – quoted rate of interest, includes
compensation for deferring consumption and expected
inflation
 Real rate of interest – compensation for deferring
consumption
 Expected Inflation – the expected fall in purchasing power of
the dollar, due to rising prices
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.39
The Fisher Effect
 The Fisher Effect defines the relationship between real rates, nominal
rates and inflation
 Exact relationship
1  RNominal   1  RReal 1  Expected Inflation 
 Where:
 RReal = the real interest rate
 RNominal = the nominal interest rate
 Expected Inflation = the expected future inflation rate
 Approximation of the above relationship is:
RNominal  RReal  Expected Inflation
Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved.
7.40
Example – Fisher Effect
 If we require a 4% real return and we expect inflation to be 6%, what is
the nominal rate?
1  RNominal   1  RReal 1  Expected Inflation 
 1  .041  .06
 1.1024
 Therefore, the nominal rate is 10.24%
 If both inflation and the real return are low, we can safely use the
approximation, which would give us a nominal rate of 10%
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7.41
Term Structure of Interest Rates 7.7
 Term structure is the relationship between time to maturity
and yield-to-maturity, all else equal
 The term structure is always derived using government bonds,
as they are the only issuer with sufficient bonds in all
maturities with the identical default risk (zero).
 The terms yield curve and term structure may be used
interchangeably. Both depict a graphical representation of the
relationship between term and yield
 The term structure typically has one of four shapes




Upward-sloping - long-term yields are higher than short-term yields
Downward-sloping - long-term yields are lower than short-term yields
Flat – yields of all maturities are the same
Humped – yields rise and then fall
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7.42
Theories of the Term Structure
 Unbiased Expectations Theory – the long rate is the geometric
average of expected future short rates
 Allows us to calculate an implied forward rate, which is the markets
consensus estimate of a future short rate
0
1
5%
The example shows a one year
spot rate of 5% and a two year
spot rate of 6%. What is the
market predicting for the one
year rate, one year from today?
2
?%
6%
Implied Forward Rate
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7.43
Unbiased Expectations Theory of the Term Structure
To solve for the implied forward rate, we solve the
following equation for R1,2
1  R   1  R 1  R 
1.06  1.051  R 
2
0, 2
0 ,1
1, 2
2
1, 2
R1, 2  7.01%
Where:
R0,2 = the rate starting at time period zero and ending at time period two
R0,1 = the rate starting at time period zero and ending at time period one
R1,2 = the implied forward rate, which is the rate starting at time period one
and ending at time period two
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7.44
Unbiased Expectations Theory of the Term Structure
 The implied forward rate is the market consensus estimate of
the short interest rate that will prevail at some future time
period
 As the market receives new information, the consensus
estimate will be updated in a continuous fashion, as market
participants buy and sell based on their expectations
 The unbiased expectations theory of the term structure tells us
that:
 An upward sloping term structure suggests future short rates will be
higher than current short rates
 A downward sloping term structure suggests that future short rates will
be lower than current short rates
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7.45
Theories of the Term Structure
 Liquidity Preference Theory
 The observed term structure includes a liquidity premium for less
liquid bonds
 Long bonds are less liquid than short bonds
 Investors prefer liquidity; therefore they are willing to accept a lower
yield for more liquid bonds
 Issuers are willing to pay a liquidity premium for longer bonds, since
they incur lower issue costs
 Segmented Markets/Preferred Habitat Theory
 Market participants operate in only one segment of the term structure,
due to institutional restrictions, asset/liability matching considerations,
etc.
 Probably has little validity in today’s world, given the existence of
derivatives
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7.46
Figure 7.4 – Upward-Sloping Yield Curve
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7.47
Figure 7.4 – Downward-Sloping Yield Curve
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7.48
Figure 7.5 – Government of Canada Yield Curve
November 29, 2002
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7.49
Factors Affecting Required Return
 Default risk – the probability that the issuer will not be able to
repay the bond as contractually obligated to do
 Liquidity premium – liquidity refers to the ability to:
 Convert to cash
 At or near face value
 Short bonds with high coupons that are more frequently traded have
greater liquidity & hence a lower required return
 Call features – since a call feature allows the bond to be
redeemed early, it increases risk to the bond investor
 Anything else that affects the risk of the cash flows to the
bondholders, will affect the required returns
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7.50
Quick Quiz
 How do you find the value of a bond and why do bond prices
change?
 What is a bond indenture and what are some of the important
features?
 What are bond ratings and why are they important?
 How does inflation affect interest rates?
 What is the term structure of interest rates?
 What factors determine the required return on bonds?
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7.51
Summary 7.8
 You should know:







How to price a bond or find the yield
Bond prices move inversely with interest rates
Bonds have a variety of features that are spelled out in the indenture
Bonds are rated based on their default risk
Most bonds trade OTC
Fisher effect links interest rates and inflation
Term structure of interest rates shows the relationship between interest
rates and maturity
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