Chapter 12
Bond Prices and the
Importance of Duration
1
We cannot gamble with anything
so sacred as money.
- William McKinley
2
Outline
 Introduction
 Review
of bond principles
 Bond pricing and returns
 Bond risk
3
Introduction
 The
investment characteristics of bonds
range completely across the risk/return
spectrum
 As
part of a portfolio, bonds provide both
stability and income
• Capital appreciation is not usually a motive for
acquiring bonds
4
Review of Bond Principles
 Identification
of bonds
 Classification of bonds
 Terms of repayment
 Bond cash flows
 Convertible bonds
 Registration
5
Identification of Bonds
 A bond
is identified by:
• The issuer
• The coupon
• The maturity
 For
example, five IBM “eights of 10”
means $5,000 par IBM bonds with an 8%
coupon rate and maturing in 2010
6
Classification of Bonds
 Introduction
 Issuer
 Security
 Term
7
Introduction
 The
bond indenture describes the details of
a bond issue:
•
•
•
•
•
Description of the loan
Terms of repayment
Collateral
Protective covenants
Default provisions
8
Issuer
 Bonds
can be classified by the nature of the
organizations initially selling them:
•
•
•
•
Corporation
Federal, state, and local governments
Government agencies
Foreign corporations or governments
9
Security
 Definition
 Unsecured
debt
 Secured debt
10
Definition
 The
security of a bond refers to what backs
the bond (what collateral reduces the risk of
the loan)
11
Unsecured Debt
 Governments:
• Full faith and credit issues (general obligation
issues) is government debt without specific
assets pledged against it
– E.g., U.S. Treasury bills, notes, and bonds
12
Unsecured Debt (cont’d)
 Corporations:
• Debentures are signature loans backed by the
good name of the company
• Subordinated debentures are paid off after
original debentures
13
Secured Debt
 Municipalities
issue:
• Revenue bonds
– Interest and principal are repaid from revenue
generated by the project financed by the bond
• Assessment bonds
– Benefit a specific group of people, who pay an
assessment to help pay principal and interest
14
Secured Debt (cont’d)
 Corporations
issue:
• Mortgages
– Well-known securities that use land and buildings as
collateral
• Collateral trust bonds
– Backed by other securities
• Equipment trust certificates
– Backed by physical assets
15
Term
 The
term is the original life of the debt
security
• Short-term securities have a term of one year or
less
• Intermediate-term securities have terms ranging
from one year to ten years
• Long-term securities have terms longer than ten
years
16
Terms of Repayment
 Interest
only
 Sinking fund
 Balloon
 Income bonds
17
Interest Only
 Periodic
payments are entirely interest
 The
principal amount of the loan is repaid at
maturity
18
Sinking Fund
 A sinking
fund requires the establishment
of a cash reserve for the ultimate repayment
of the bond principal
• The borrower can:
– Set aside a potion of the principal amount of the
debt each year
– Call a certain number of bonds each year
19
Balloon
 Balloon
loans partially amortize the debt
with each payment but repay the bulk of the
principal at the end of the life of the debt
 Most
balloon loans are not marketable
20
Income Bonds
 Income
bonds pay interest only if the firm
earns it
 For
example, an income bond may be
issued to finance an income-producing
project
21
Bond Cash Flows
 Annuities
 Zero
coupon bonds
 Variable rate bonds
 Consols
22
Annuities
 An
annuity promises a fixed amount on a
regular periodic schedule for a finite length
of time
 Most
bonds are annuities plus an ultimate
repayment of principal
23
Zero Coupon Bonds
 A zero
coupon bond has a specific maturity
date when it returns the bond principal
 A zero
coupon bond pays no periodic
income
• The only cash inflow is the par value at
maturity
24
Variable Rate Bonds
 Variable
rate bonds allow the rate to
fluctuate in accordance with a market index
 For
example, U.S. Series EE savings bonds
25
Consols
 Consols
pay a level rate of interest
perpetually:
• The bond never matures
• The income stream lasts forever
 Consols
are not very prevalent in the U.S.
26
Convertible Bonds
 Definition
 Security-backed
bonds
 Commodity-backed bonds
27
Definition
 A convertible
bond gives the bondholder
the right to exchange them for another
security or for some physical asset
 Once
conversion occurs, the holder cannot
elect to reconvert and regain the original
debt security
28
Security-Backed Bonds
 Security-backed
convertible bonds are
convertible into other securities
• Typically common stock of the company that
issued the bonds
• Occasionally preferred stock of the issuing
firm, common stock of another firm, or shares
in a subsidiary company
29
Commodity-Backed Bonds
 Commodity-backed
bonds are convertible
into a tangible asset
 For
example, silver or gold
30
Registration
 Bearer
bonds
 Registered bonds
 Book entry bonds
31
Bearer Bonds
 Bearer
bonds:
• Do not have the name of the bondholder printed
on them
• Belong to whoever legally holds them
• Are also called coupon bonds
– The bond contains coupons that must be clipped
• Are no longer issued in the U.S.
32
Registered Bonds
 Registered
bonds show the bondholder’s
name
 Registered
bondholders receive interest
checks in the mail from the issuer
33
Book Entry Bonds
 The
U.S. Treasury and some corporation
issue bonds in book entry form only
• Holders do not take actual delivery of the bond
• Potential holders can:
– Open an account through the Treasury Direct
System at a Federal Reserve Bank
– Purchase a bond through a broker
34
Bond Pricing and Returns
 Introduction
 Valuation
equations
 Yield to maturity
 Realized compound yield
 Current yield
 Term structure of interest rates
 Spot rates
35
Bond Pricing and Returns
(cont’d)
 The
conversion feature
 The matter of accrued interest
36
Introduction
 The
current price of a bond is the market’s
estimation of what the expected cash flows
are worth in today’s dollars
 There
is a relationship between:
• The current bond price
• The bond’s promised future cash flows
• The riskiness of the cash flows
37
Valuation equations
 Annuities
 Zero
coupon bonds
 Variable rate bonds
 Consols
38
Annuities
 For
a semiannual bond:
2N
P0  
t 1
Ct
1  ( R / 2)
t
where N  term of the bond in years
Ct  cash flow at time t
R  annual yield to maturity
P0  current price of the bond
39
Annuities (cont’d)
 Separating
interest and principal
components:
2N
P0  
t 1
C

Par
1  ( R / 2) 1  ( R / 2)
t
2N
where C  coupon payment
40
Annuities (cont’d)
Example
A bond currently sells for $870, pays $70 per year (Paid
semiannually), and has a par value of $1,000. The bond
has a term to maturity of ten years.
What is the yield to maturity?
41
Annuities (cont’d)
Example (cont’d)
Solution: Using a financial calculator and the following input provides
the solution:
N
PV
PMT
FV
CPT I
= 20
= $870
= $35
= $1,000
= 4.50
This bond’s yield to maturity is 4.50% x 2 = 9.00%.
42
Zero Coupon Bonds
 For
a zero-coupon bond (annual and
semiannual compounding):
Par
P0 
(1  R )t
Par
P0 
(1  R / 2) 2t
43
Zero Coupon Bonds (cont’d)
Example
A zero coupon bond has a par value of $1,000 and
currently sells for $400. The term to maturity is twenty
years.
What is the yield to maturity (assume semiannual
compounding)?
44
Zero Coupon Bonds (cont’d)
Example (cont’d)
Solution:
Par
P0 
(1  R / 2) 2t
$1, 000
$400 
(1  R / 2) 40
R  4.63%
45
Variable Rate Bonds
 The
valuation equation must allow for
variable cash flows
 You cannot determine the precise present
value of the cash flows because they are
unknown:
2N
Ct
P0  
t
(1

I
)
t 1
t
where I t  interest rate at time t
46
Consols
 Consols
are perpetuities:
C
P0 
R
47
Consols (cont’d)
Example
A consol is selling for $900 and pays $60 annually in
perpetuity.
What is this consol’s rate of return?
48
Consols (cont’d)
Example (cont’d)
Solution:
C
P0 
R
$60
R
 6.67%
$900
49
Yield to Maturity
 Yield
to maturity captures the total return
from an investment
• Includes income
• Includes capital gains/losses
 The
yield to maturity is equivalent to the
internal rate of return in corporate finance
50
Realized Compound Yield
 The
effective annual yield is useful to
compare bonds to investments generating
income on a different time schedule
Effective annual rate  1  ( R / x)  1
x
where R  yield to maturity
x  number of payment periods per year
51
Realized Compound
Yield (cont’d)
Example
A bond has a yield to maturity of 9.00% and pays interest
semiannually.
What is this bond’s effective annual rate?
52
Realized Compound
Yield (cont’d)
Example (cont’d)
Solution:
Effective annual rate  1  ( R / x)   1
x
 1  (.009 / 2)   1
2
 9.20%
53
Current Yield
 The
current yield:
• Measures only the return associated with the
interest payments
• Does not include the anticipated capital gain or
loss resulting from the difference between par
value and the purchase price
54
Current Yield (cont’d)
 For
a discount bond, the yield to maturity is
greater than the current yield
 For
a premium bond, the yield to maturity is
less than the current yield
55
Current
Yield (cont’d)
Example
A bond pays annual interest of $70 and has a current
price of $870.
What is this bond’s current yield?
56
Current
Yield (cont’d)
Example (cont’d)
Solution:
Current yield = $70/$870 = 8.17%
57
Term Structure of
Interest Rates
 Yield
curve
 Theories of interest rate structure
58
Yield Curve
 The
yield curve:
• Is a graphical representation of the term
structure of interest rates
• Relates years until maturity to the yield to
maturity
• Is typically upward sloping and gets flatter for
longer terms to maturity
59
Information Used to
Build A Yield Curve
60
Theories of
Interest Rate Structure
 Expectations
theory
 Liquidity preference theory
 Inflation premium theory
61
Expectations Theory
 According
to the expectations theory of
interest rates, investment opportunities with
different time horizons should yield the
same return:
(1  R2 ) 2  (1  R1 )(1  1 f 2 )
where 1 f 2  the forward rate from time 1 to time 2
62
Expectations Theory (cont’d)
Example
An investor can purchase a two-year CD at a rate of 5
percent. Alternatively, the investor can purchase two
consecutive one-year CDs. The current rate on a one-year
CD is 4.75 percent.
According to the expectations theory, what is the expected
one-year CD rate one year from now?
63
Expectations Theory (cont’d)
Example (cont’d)
Solution:
(1  R2 )  (1  R1 )(1  1 f 2 )
2
(1.05) 2  (1.045)(1  1 f 2 )
(1.05) 2
(1  1 f 2 ) 
(1.045)
1 f 2  5.50%
64
Liquidity Preference Theory
 Proponents
of the liquidity preference
theory believe that, in general:
• Investors prefer to invest short term rather than
long term
• Borrowers must entice lenders to lengthen their
investment horizon by paying a premium for
long-term money (the liquidity premium)
 Under
this theory, forward rates are higher
than the expected interest rate in a year
65
Inflation Premium Theory
 The
inflation premium theory states that
risk comes from the uncertainty associated
with future inflation rates
 Investors who commit funds for long
periods are bearing more purchasing power
risk than short-term investors
• More inflation risk means longer-term
investment will carry a higher yield
66
Spot Rates
 Spot
rates:
• Are the yields to maturity of a zero coupon
security
• Are used by the market to value bonds
– The yield to maturity is calculated only after
learning the bond price
– The yield to maturity is an average of the various
spot rates over a security’s life
67
Spot Rates (cont’d)
Interest Rate
Spot Rate Curve
Yield to Maturity
Time Until the Cash Flow
68
Spot Rates (cont’d)
Example
A six-month T-bill currently has a yield of 3.00%. A oneyear T-note with a 4.20% coupon sells for 102.
Use bootstrapping to find the spot rate six months from
now.
69
Spot Rates (cont’d)
Example (cont’d)
Solution: Use the T-bill rate as the spot rate for the first
six months in the valuation equation for the T-note:
1, 020 
21.00
1, 021

(1  .03 / 2) (1  r2 / 2) 2
1, 021
999.31 
(1  r2 / 2) 2
(1  r2 / 2) 2  1.022
r2  2.16%
70
The Conversion Feature
Convertible bonds give their owners the right to
exchange the bonds for a pre-specified amount or
shares of stock
 The conversion ratio measures the number of
shares the bondholder receives when the bond is
converted

• The par value divided by the conversion ratio is the
conversion price
• The current stock price multiplied by the conversion
ratio is the conversion value
71
The Conversion
Feature (cont’d)
The market price of a bond can never be less than
its conversion value
 The difference between the bond price and the
conversion value is the premium over conversion
value

• Reflects the potential for future increases in the
common stock price

Mandatory convertibles convert automatically into
common stock after three or four years
72
The Matter of Accrued Interest
 Bondholders
earn interest each calendar day
they hold a bond
 Firms mail interest payment checks only
twice a year
 Accrued interest refers to interest that has
accumulated since the last interest payment
date but which has not yet been paid
73
The Matter of
Accrued Interest (cont’d)
 At
the end of a payment period, the issuer
sends one check for the entire interest to the
current bondholder
• The bond buyer pays the accrued interest to the
seller
• The bond sells receives accrued interest from
the bond buyer
74
The Matter of
Accrued Interest (cont’d)
Example
A bond with an 8% coupon rate pays interest on June 1
and December 1. The bond currently sells for $920.
What is the total purchase price, including accrued
interest, that the buyer of the bond must pay if he
purchases the bond on August 10?
75
The Matter of
Accrued Interest (cont’d)
Example (cont’d)
Solution: The accrued interest for 71 days is:
$80/365 x 71 = $15.56
Therefore, the total purchase price is:
$920 + $15.56 = $935.56
76
Bond Risk
 Price
risks
 Convenience risks
 Malkiel’s interest rate theories
 Duration as a measure of interest rate risk
77
Price Risks
 Interest
rate risk
 Default risk
78
Interest Rate Risk
 Interest
rate risk is the chance of loss
because of changing interest rates
 The
relationship between bond prices and
interest rates is inverse
• If market interest rates rise, the market price of
bonds will fall
79
Default Risk
 Default
risk measures the likelihood that a
firm will be unable to pay the principal and
interest on a bond
 Standard
& Poor’s Corporation and
Moody’s Investor Service are two leading
advisory services monitoring default risk
80
Default Risk (cont’d)
 Investment
grade bonds are bonds rated
BBB or above
 Junk
bonds are rated below BBB
 The
lower the grade of a bond, the higher its
yield to maturity
81
Convenience Risks
 Definition
 Call
risk
 Reinvestment rate risk
 Marketability risk
82
Definition
 Convenience
risk refers to added demands
on management time because of:
• Bond calls
• The need to reinvest coupon payments
• The difficulty in trading a bond at a reasonable
price because of low marketability
83
Call Risk
 If
a company calls its bonds, it retires its
debt early
 Call
risk refers to the inconvenience of
bondholders associated with a company
retiring a bond early
• Bonds are usually called when interest rates are
low
84
Call Risk (cont’d)
 Many
bond issues have:
• Call protection
– A period of time after the issuance of a bond when
the issuer cannot call it
• A call premium if the issuer calls the bond
– Typically begins with an amount equal to one year’s
interest and then gradually declining to zero as the
bond approaches maturity
85
Reinvestment Rate Risk
 Reinvestment
rate risk refers to the
uncertainty surrounding the rate at which
coupon proceeds can be invested
 The
higher the coupon rate on a bond, the
higher its reinvestment rate risk
86
Marketability Risk
 Marketability
risk refers to the difficulty of
trading a bond:
• Most bonds do not trade in an active secondary
market
• The majority of bond buyers hold bonds until
maturity
 Low
marketability bonds usually carry a
wider bid-ask spread
87
Malkiel’s
Interest Rate Theorems
 Definition
 Theorem
1
 Theorem 2
 Theorem 3
 Theorem 4
 Theorem 5
88
Definition
 Malkiel’s
interest rate theorems provide
information about how bond prices change
as interest rates change
 Any
good portfolio manager knows
Malkiel’s theorems
89
Theorem 1
 Bond
prices move inversely with yields:
• If interest rates rise, the price of an existing
bond declines
• If interest rates decline, the price of an existing
bond increases
90
Theorem 2
 Bonds
with longer maturities will fluctuate
more if interest rates change
 Long-term
bonds have more interest rate
risk
91
Theorem 3
 Higher
coupon bonds have less interest rate
risk
 Money
in hand is a sure thing while the
present value of an anticipated future
receipt is risky
92
Theorem 4
 When
comparing two bonds, the relative
importance of Theorem 2 diminishes as the
maturities of the two bonds increase
 A given
time difference in maturities is
more important with shorter-term bonds
93
Theorem 5
 Capital
gains from an interest rate decline
exceed the capital loss from an equivalent
interest rate increase
94
Duration as A Measure of
Interest Rate Risk
 The
concept of duration
 Calculating duration
95
The Concept of Duration
 For
a noncallable security:
• Duration is the weighted average number of
years necessary to recover the initial cost of the
bond
• Where the weights reflect the time value of
money
96
The Concept of
Duration (cont’d)
 Duration
is a direct measure of interest rate
risk:
• The higher the duration, the higher the interest
rate risk
97
Calculating Duration
 The
traditional duration calculation:
N
Ct
t

t
(1  R)
D  t 1
Po
where D  duration
Ct  cash flow at time t
R  yield to maturity
Po  current price of the bond
N  years until bond maturity
t  time at which a cash flow is received
98
Calculating Duration (cont’d)
 The
closed-end formula for duration:
 (1  R) N 1  (1  R)  ( R  N )  F  N
C


2
N
N
R
(1

R
)
(1

R
)

D 
Po
where F  par value of the bond
N  number of periods until maturity
R  yield to maturity of the bond per period
99
Calculating Duration (cont’d)
Example
Consider a bond that pays $100 annual interest and has a
remaining life of 15 years. The bond currently sells for
$985 and has a yield to maturity of 10.20%.
What is this bond’s duration?
100
Calculating Duration (cont’d)
Example (cont’d)
Solution: Using the closed-form formula for duration:
 (1  R) N 1  (1  R)  ( R  N )  F  N
C
  (1  R) N
2
N
R
(1

R
)

D 
Po
 (1.052)31  (1.052)  (0.052  30)  1, 000  30
50 


2
30
30
0.052
(1.052)
(1.052)

 
985
 15.69 years
101