Chapter 12
Bond Prices and the
Importance of Duration
1
Outline
 Introduction
 Review
of bond principles
 Bond pricing and returns
 Bond risk
 The meaning of bond diversification
 Choosing bonds
 Example: monthly retirement income
2
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
3
Review of Bond Principles
 Identification
of bonds
 Classification of bonds
 Terms of repayment
 Bond cash flows
 Convertible bonds
 Registration
4
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
5
Classification of Bonds
 Introduction
 Issuer
 Security
 Term
6
Introduction
 The
bond indenture describes the details of
a bond issue:
•
•
•
•
•
Description of the loan
Terms of repayment
Collateral
Protective covenants
Default provisions
7
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
8
Definition
 The
security of a bond refers to what backs
the bond (what collateral reduces the risk of
the loan)
9
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
10
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
11
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
12
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
13
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
14
Terms of Repayment
 Interest
only
 Sinking fund
 Balloon
 Income bonds
15
Interest Only
 Periodic
payments are entirely interest
 The
principal amount of the loan is repaid at
maturity
16
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
17
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
18
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
19
Bond Cash Flows
 Annuities
 Zero
coupon bonds
 Variable rate bonds
 Consols
20
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
21
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
22
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
23
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.
24
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
25
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
26
Commodity-Backed Bonds
 Commodity-backed
bonds are convertible
into a tangible asset
 For
example, silver or gold
27
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.
28
Registered Bonds
 Registered
bonds show the bondholder’s
name
 Registered
bondholders receive interest
checks in the mail from the issuer
29
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
30
Bond Pricing and Returns
 Introduction
 Valuation
equations
 Yield to maturity
 Realized compound yield
 Current yield
 Term structure of interest rates
 Spot rates
31
Bond Pricing and Returns
(cont’d)
 The
conversion feature
 The matter of accrued interest
32
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
33
Valuation equations
 Annuities
 Zero
coupon bonds
 Variable rate bonds
 Consols
34
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
35
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
36
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?
37
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%.
38
Zero Coupon Bonds
 For
a zero-coupon bond (annual and
semiannual compounding):
Par
P0 
(1  R )t
Par
P0 
(1  R / 2) 2t
39
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)?
40
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%
41
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B
C
COMPUTING THE YIELD TO MATURITY (YTM)
ON TREASURY BILLS
Purchase price
Face value
Time to maturity (days)
9,750.00
10,000.00
182 <-- =26*7
Method 1: Compound the daily return
Daily interest rate
YTM--the annualized rate
0.0139% <-- =(B3/B2)^(1/B4)-1
5.2086% <-- =(1+B7)^365-1
Method 2: Calculate the continuously compounded return
Continuously compounded
5.0775% <-- =LN(B3/B2)*(365/B4)
Future value in one year using each method
Method 1
Method 2
10,257.84 <-- =B2*(1+B8)
10,257.84 <-- =B2*EXP(B11)
42
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
43
Consols
 Consols
are perpetuities:
C
P0 
R
44
Consols (cont’d)
Example
A consol is selling for $900 and pays $60 annually in
perpetuity.
What is this consol’s rate of return?
45
Consols (cont’d)
Example (cont’d)
Solution:
C
P0 
R
$60
R
 6.67%
$900
46
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
47
A
B
C
E
F
Data table
Bond price
950
960
970
980
990
1,000
1,010
1,020
1,030
1,040
1,050
1,060
1,070
1,080
1,090
G
H
Bond value
7.00% <-- =B14 , data table header
7.96%
7.76%
7.57%
7.38%
7.19%
7.00%
6.82%
6.63%
6.45%
6.28%
6.10%
5.93%
5.76%
5.59%
5.42%
I
7%
7%
7%
7%
7%
7%
7%
7%
7%
7%
7%
7%
7%
7%
7%
YTM
1
YIELD TO MATURITY
2 Market price of bond
1000.00
3
4
Year
Bond cash flow
5
0
-1,000.00
6
1
70.00
7
2
70.00
8
3
70.00
9
4
70.00
10
5
70.00
11
6
70.00
12
7
1,070.00
13
14 YTM of bond
7.00% <-- =IRR(B5:B12)
15
16
17
Yield to Maturity (YTM) of XYZ Bond
18
19
8.50%
20
21
8.00%
22
7.50%
23
7.00%
24
6.50%
25
26
6.00%
27
5.50%
28
5.00%
29
950
1,000
1,050
30
31
Market price
32
D
1,100
48
A
B
C
YIELD TO MATURITY
For uneven date spacing
1
2 Market price of bond
3
4
Year
5
15-May-01
6
15-Dec-01
7
15-Dec-02
8
15-Dec-03
9
15-Dec-04
10
15-Dec-05
11
15-Dec-06
12
15-Dec-07
13
14 YTM of bond
1050.00
Bond cash flow
-1,050.00
70.00
70.00
70.00
70.00
70.00
70.00
1,070.00
6.58% <-- =XIRR(B5:B12,A5:A12)
49
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
50
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?
51
Realized Compound
Yield (cont’d)
Example (cont’d)
Solution:
Effective annual rate  1  ( R / x)   1
x
 1  (.009 / 2)   1
2
 9.20%
52
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
53
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
54
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?
55
Current
Yield (cont’d)
Example (cont’d)
Solution:
Current yield = $70/$870 = 8.17%
56
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
57
Information Used to
Build A Yield Curve
58
Theories of
Interest Rate Structure
 Expectations
theory
 Liquidity preference theory
 Inflation premium theory
59
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
60
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?
61
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%
62
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
63
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
64
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
65
Spot Rates (cont’d)
Interest Rate
Spot Rate Curve
Yield to Maturity
Time Until the Cash Flow
66
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.
67
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%
68
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
69
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
70
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
71
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
72
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?
73
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 = $935.56
74
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
B
C
D
E
F
UNITED STATES TREASURY BOND, 6%, MATURING 8 AUGUST 2009
Face value of bonds bought
Coupon rate
1,000.00
6.00%
Market price
Accrued interest
1,059.51
29.51 <-- =E12
Actual price paid
1,089.02
Cash flows to GI at bond issue
Date
Cash flow
12-Feb-01
-1,089.02 <-- =-B8
15-Feb-01
30.00 <-- =$B$3*$B$2/2
15-Aug-01
30.00
15-Feb-02
30.00
15-Aug-02
30.00
15-Feb-03
30.00
15-Aug-03
30.00
15-Feb-04
30.00
15-Aug-04
30.00
21
15-Feb-05
22
15-Aug-05
23
15-Feb-06
24
15-Aug-06
25
15-Feb-07
26
15-Aug-07
27
15-Feb-08
28
15-Aug-08
29
15-Feb-09
30
15-Aug-09
31
32 XIRR (annualized IRR)
33 Excel's Yield function
34 Excel's Yield annualized
30.00
30.00
30.00
30.00
30.00
30.00
30.00
30.00
30.00
1,030.00 <-- =$B$3*$B$2/2+B2
5.193% <-- =XIRR(B12:B30,A12:A30)
5.128% <-- =YIELD(A12,A30,B3,B5/10,100,2,3)
5.193% <-- =(1+B33/2)^2-1
Accrued interest calculation
Today's date
Last coupon date
Next coupon date
Days since last coupon
Days between coupons
Semi-annual coupon
Accrued interest
12-Feb-01
15-Aug-00
15-Feb-01
181 <-- =E4-E5
184 <-- =E6-E5
30 <-- =B3/2*B2
29.51 <-- =E8/E9*E11
ACCRUED INTEREST IN EXCEL
75
Bond Risk
 Price
risks
 Convenience risks
 Malkiel’s interest rate theories
 Duration as a measure of interest rate risk
76
Price Risks
 Interest
rate risk
 Default risk
77
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
78
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
79
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
80
Convenience Risks
 Definition
 Call
risk
 Reinvestment rate risk
 Marketability risk
81
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
82
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
83
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
84
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
85
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
86
Malkiel’s
Interest Rate Theorems
 Definition
 Theorem
1
 Theorem 2
 Theorem 3
 Theorem 4
 Theorem 5
87
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
88
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
89
Theorem 2
 Bonds
with longer maturities will fluctuate
more if interest rates change
 Long-term
bonds have more interest rate
risk
90
A
B
D
E
F
G
VALUING THE XYZ CORPORATION BONDS
1
2 Market interest rate
3
6.50%
1
2
3
4
5
6
7
Bond
cash flow
70
70
70
70
70
70
1,070
Market
interest rate
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
10.00%
11.00%
12.00%
13.00%
14.00%
1,027.42 <-- =NPV(B2,B5:B11)
XYZ Bond Value
Bond value
4
Year
5
6
7
8
9
10
11
12
13 Value of the bond
14
15
16
17
1,450
18
1,350
19
1,250
20
1,150
21
1,050
22
950
23
850
24
25
750
26
650
27
0%
28
29
30
C
2%
4%
5%
7%
9%
11%
12%
Bond
value
1,490.00
1,403.69
1,323.60
1,249.21
1,180.06
1,115.73
1,055.82
1,000.00
947.94
899.34
853.95
811.51
771.81
734.64
699.82
<-- =NPV(E5,$B$5:$B$11)
<-- =NPV(E6,$B$5:$B$11)
<-- =NPV(E7,$B$5:$B$11)
<-- =NPV(E8,$B$5:$B$11)
14%
Market interest rate
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
A
B
C
D
E
F
G
H
1
BASIC DURATION CALCULATION
2
3 YTM
7%
4
5
Year
Ct,A
t*Ct,A /PA*(1+YTM)t
Ct,B
t*Ct,B /PB*(1+YTM)t
6
1
70
0.0654
130
0.0855
7
2
70
0.1223
130
0.1598
8
3
70
0.1714
130
0.2240
9
4
70
0.2136
130
0.2791
10
5
70
0.2495
130
0.3260
11
6
70
0.2799
130
0.3657
12
7
70
0.3051
130
0.3987
13
8
70
0.3259
130
0.4258
14
9
70
0.3427
130
0.4477
15
10
1070
5.4393
1130
4.0413
16
Bond price
Duration
Bond price
Duration
17
$ 1,000
7.5152
$ 1,421
6.7535
18
19
=NPV(B3,B6:B15)
=SUM(F6:F15)
20
21 Excel formula
7.5152
<-- =DURATION(DATE(1996,12,3),DATE(2006,12,3),7%,B3,1)
22
(need to have the tool "Analysis ToolPak" added in Excel)
99
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
100
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?
101
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
102
A
C
D
E
F
G
H
EFFECTS OF COUPON AND MATURITY ON DURATION
Current date
Maturity, in years
Maturity date
YTM
Coupon
Face value
Duration
5/21/1996 <-- =DATE(1996,5,21)
21
5/21/2017 <-- =DATE(1996+B4,5,21)
15% Yield to maturity (i.e., discount rate)
4%
1,000
9.0110 <-- =DURATION(B3,B5,B7,B6,1)
Data table: Effect of maturity on duration
9.0110 <-- =B10
5
4.5163
Effect of Maturity on Duration
10
7.4827
Coupon rate = 4.00%, YTM = 15.00%
15
8.8148
10.0
20
9.0398
9.0
25
8.7881
30
8.4461
8.0
35
8.1633
7.0
40
7.9669
6.0
45
7.8421
50
7.7668
5.0
55
7.7228
4.0
60
7.6977
0
20
40
60
80
65
7.6837
Maturity
70
7.6759
Duration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
B
103
Duration
A
B
C
D
E
F
G
H
31 Data table: Effect of coupon on duration
32
9.0110 <-- =B10
33
0% 21.0000
34
1% 13.1204
Effect of Coupon on Duration
35
2% 10.7865
Maturity = 21, YTM = 15.00%
23.0
36
3%
9.6677
21.0
37
4%
9.0110
19.0
38
5%
8.5792
17.0
15.0
39
6%
8.2736
13.0
40
7%
8.0459
11.0
41
9%
7.7294
9.0
42
13%
7.3707
7.0
43
15%
7.2593
5.0
44
17%
7.1729
0%
5%
10%
15%
Coupon rate
45
46
104
Bond Selection - Introduction
 In
most respects selecting the fixed-income
components of a portfolio is easier than
selecting equity securities
 There
are ways to make mistakes with bond
selection
105
The Meaning of
Bond Diversification
 Introduction
 Default
risk
 Dealing with the yield curve
 Bond betas
106
Introduction
 It
is important to diversify a bond portfolio
 Diversification of a bond portfolio is
different from diversification of an equity
portfolio
 Two types of risk are important:
• Default risk
• Interest rate risk
107
Default Risk
 Default
risk refers to the likelihood that a
firm will be unable to repay the principal
and interest of a loan as agreed in the bond
indenture
• Equivalent to credit risk for consumers
• Rating agencies such as S&P and Moody’s
function as credit bureaus for credit issuers
108
Default Risk (cont’d)
 To
diversify default risk:
• Purchase bonds from a number of different
issuers
• Do not purchase various bond issues from a
single issuer
– E.g., Enron had 20 bond issues when it went
bankrupt
109
Dealing With the Yield Curve
 The
yield curve is typically upward sloping
• The longer a fixed-income security has until
maturity, the higher the return it will have to
compensate investors
• The longer the average duration of a fund, the
higher its expected return and the higher its
interest rate risk
110
Dealing With the
Yield Curve (cont’d)
 The
client and portfolio manager need to
determine the appropriate level of interest
rate risk of a portfolio
111
Bond Betas
 The
concept of bond betas:
• States that the market prices a bond according
to its level of risk relative to the market average
• Has never become fully accepted
• Measures systematic risk, while default risk and
interest rate risk are more important
112
Choosing Bonds
 Client
psychology and bonds selling at a
premium
 Call risk
 Constraints
113
Client Psychology and
Bonds Selling at A Premium
 Premium
bonds held to maturity are
expected to pay higher coupon rates than
the market rate of interest
 Premium
bond held to maturity will decline
in value toward par value as the bond
moves towards its maturity date
114
Client Psychology & Bonds
Selling at A Premium (cont’d)
 Clients
may not want to buy something they
know will decline in value
 There
is nothing wrong with buying bonds
selling at a premium
115
Call Risk
 If
a bond is called:
• The funds must be reinvested
• The fund manager runs the risk of having to
make adjustments to many portfolios all at one
time
 There
is no reason to exclude callable bonds
categorically from a portfolio
• Avoid making extensive use of a single callable
bond issue
116
Constraints
 Specifying
return
 Specifying grade
 Specifying average maturity
 Periodic income
 Maturity timing
 Socially responsible investing
117
Specifying Return
 To
increase the expected return on a bond
portfolio:
• Choose bonds with lower ratings
• Choose bonds with longer maturities
• Or both
118
Specifying Grade
A
legal list specifies securities that are
eligible investments
• E.g., investment grade only
 Portfolio
managers take the added risk of
noninvestment grade bonds only if the yield
pickup is substantial
119
Specifying Grade (cont’d)
 Conservative
organizations will accept only
U.S. government or AAA-rated corporate
bonds
A
fund may be limited to no more than a
certain percentage of non-AAA bonds
120
Specifying Average Maturity
 Average
maturity is a common bond
portfolio constraint
• The motivation is concern about rising interest
rates
• Specifying average duration would be an
alternative approach
121
Periodic Income
 Some
funds have periodic income needs
that allow little or not flexibility
 Clients
will want to receive interest checks
frequently
• The portfolio manager should carefully select
the bonds in the portfolio
122
Maturity Timing
 Maturity
timing generates income as needed
• Sometimes a manager needs to construct a bond
portfolio that matches a particular investment
horizon
• E.g., assemble securities to fund a specific set
of payment obligations over the next ten years
– Assemble a portfolio that generates income and
principal repayments to satisfy the income needs
123
Socially Responsible Investing
 Some
clients will ask that certain types of
companies not be included in the portfolio
 Examples
are nuclear power, military
hardware, “vice” products
124
Example: Monthly
Retirement Income
 The
problem
 Unspecified constraints
 Using S&P’s Bond Guide
 Solving the problem
125
The Problem
A
client has:
• Primary objective: growth of income
• Secondary objective: income
• $1,100,000 to invest
• Inviolable income needs of $4,000 per month
126
The Problem (cont’d)
 You
decide:
• To invest the funds 50-50 between common
stocks and debt securities
• To invest in ten common stock in the equity
portion (see next slide)
– You incur $1,500 in brokerage commissions
127
The Problem (cont’d)
Stock
Value
Qrtl Div.
3,000 AAC
$51,000
$380
Jan./April/July/Oct.
1,000 BBL
50,000
370
Jan./April/July/Oct.
2,000 XXQ
49,000
400
Feb./May/Aug./Nov.
5,000 XZ
52,000
270
March/June/Sept./Dec.
7,000 MCDL
53,000
0
1,000 ME
49,000
370
Feb./May/Aug./Nov.
2,000 LN
51,000
500
Jan./April/July/Oct.
4,000 STU
47,000
260
March/June/Sept./Dec.
3,000 LLZ
49,000
290
Feb./May/Aug./Nov.
6,000 MZN
43,000
170
Jan./April/July/Oct.
$494,000
$3,010
Total
Payment Month
--
128
The Problem (cont’d)
 Characteristics
of the fund:
• Quarterly dividends total $3,001 ($12,004
annually)
• The dividend yield on the equity portfolio is
2.44%
• Total annual income required is $48,000 or
4.36% of fund
• Bonds need to have a current yield of at least
6.28%
129
Unspecified Constraints
 The
task is meeting the minimum required
expected return with the least possible risk
• You don’t want to choose CC-rated bonds
• You don’t want the longest maturity bonds you
can find
130
Using S&P’s Bond Guide
 Figure
11-4 is an excerpt from the Bond
Guide:
• Indicates interest payment dates, coupon rates,
and issuer
• Provides S&P ratings
• Provides current price, current yield
131
Using S&P’s
Bond Guide (cont’d)
132
Solving the Problem
 Setup
 Dealing
with accrued interest and
commissions
 Choosing the bonds
 Overspending
 What about convertible bonds?
133
Setup
 You
have two constraints:
• Include only bonds rated BBB or higher
• Keep the average maturities below fifteen years
 Set
up a worksheet that enables you to pick
bonds to generate exactly $4,000 per month
(see next slide)
134
Setup (cont’d)
Security
Price
Jan.
Feb.
March
April
3,000 AAC
$51,000
$380
$380
1,000 BBL
50,000
370
370
2,000 XXQ
49,000
5,000 XZ
52,000
7,000 MCDL
53,000
1,000 ME
49,000
2,000 LN
51,000
4,000 STU
47,000
3,000 LLZ
49,000
6,000 MZN
43,000
Equities
$400
May
$400
$270
$270
370
370
500
500
260
260
290
290
170
$494,000 $1,420
June
170
$1,060
$530
$1,420
$1,060
$530
135
Dealing With Accrued
Interest and Commissions
 Bond
prices are typically quoted on a net
basis (already include commissions)
 Calculate
accrued interest using the midterm heuristic
• Assume every bond’s accrued interest is half of
one interest check
136
Choosing the Bonds

The following slide shows one possible solution:
•
•
•
•

Stock cost: $494,000
Bond cost: $557,130
Accrued interest: $9,350
Stock commissions: $1,500
Do you think this solution could be improved?
137
Bonds
Security
Price
Jan.
Feb.
March April
May
June
$80,000 Empire
71/2s02
$86,400
$80,000 Energen
8s07
82,900
$100,000 Enhance
61/4s03
105,500
$80,000 Enron
65/8s03
84,500
$90,000 Enron
6.7s06
97,200
$100,000
Englehard 6.95s28
100,630
Bonds subtotal
$557,130 $3,000
$3,200
$3,370
$2,650 $3,010
$3,470
$4,420
$4,260
$3,900
$4,070 $4,070
$4,000
Total income
$3,000
$3,200
$3,370
$2,650
$3,010
$3,470
138
Overspending
 The
total of all costs associated with the
portfolio should not exceed the amount
given to you by the client to invest
 The
money the client gives you establishes
another constraint
139
What About
Convertible Bonds?
 Convertible
bonds can be included in a
portfolio
• Useful for a growth of income objective
• People buy convertible bonds in hopes of price
appreciation
• Useful if you otherwise meet your income
constraints
140
Immunization Strategies
A
portfolio of bonds is said to be
immunized (from interest rate risk) if its
payoff at some future date is independent of
the future levels of interest rates.
 Immunization
is closely related to the
concept of duration.
141
 Immunization
consists of matching the
duration of the portfolio’s assets and
liabilities (obligations).
 Suppose a firm has a future obligation Q.
The prevailing interest rate is r, and the
liability is N periods away.
 The
present value of this liability is denoted
by V0=Q/(1+r)N.
142
 Now
suppose that the firm is currently
hedging this liability with a bond whose
value VB = V0 and whose coupon payments
are denoted by P1,…,PM.
 We
thus have:
M
Pt
VB  
t
(1

r
)
t 1
143
 Suppose
now that interest rates change from
r to r+Dr. The new values of the future
obligation and of the bond are:
  NQ 
dV0
V0  DV0  V0 
Dr  V0  Dr 
N 1 
dr
(1

r
)


M
tPt
dVB
VB  DVB  VB 
Dr  VB  Dr 
t 1
dr
t 1 (1  r )
144
 Rearranging
terms and recalling that V0=VB
yields the following expression:
1
VB
M
tPt
N

t
t 1 (1  r )
 The
left-hand side represents the duration of
the bond, while the right-hand side
represents the duration of the obligation
(Since the obligation consisted of only one
payment, the duration is its maturity).
145
 In
conclusion, in order for a portfolio to be
immunized, you need to have:
 DURATIONASSETS
= DURATIONLIABILITIES
 Caveat:
this works only if the interest rates
of various maturities all change in the same
manner, i.e. if the yield curve shifts upward
or downward in a parallel shift.
146
Immunization Example
 You
need to immunize an obligation whose
present value V0 is $1,000. The payment is to be
made 10 years from now, and the current interest
rate is 6%. The payment is thus the future value of
1,000 at 6%, therefore it is:
1,000(1.06)10 = $1,790.85

The Excel spreadsheet on the next slide shows
three bonds that you have at your disposition to
immunize the liability.
147
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
D
BASIC IMMUNIZATION EXAMPLE WITH 3 BONDS
Yield to maturity
Coupon rate
Maturity
Face value
6%
Bond 1
6.70%
10
1,000
Bond 2
6.988%
15
1,000
Bond 3
5.90%
30
1,000
Bond price
$1,051.52
$1,095.96
$986.24
Face value equal to $1,000 of market value $
951.00 $
912.44 $ 1,013.96
Duration
7.6655
10.0000
14.6361
=dduration(B7,B6,$B$3,1)
148
THE IMMUNIZATION PROBLEM
Illustrated for the 30-year bond.
0
Buy $1,014
face value
of 30-year
bond.
Year 10:
Future obligation of
$1,790.85 due.
Reinvest
coupons
from bond
during years
1-10.
30
Sell bond for PV of
remaining coupons
and redemption in
year 30.
When the interest rate increases:
Value of
reinvested
coupons
increases.
Value of bond in
year 10 decreases.
When the interest rate decreases:
Value of
reinvested
coupons
decreases.
Value of bond in
year 10
increases.
149
Values 10 years later, assuming
interest rates do not change
A
19 New yield to maturity, 10 years later
20
21
22 Bond price
23 Reinvested coupons
24 Total
25
26 Multiply by percent of face value bought
27 Product
B
C
D
E
F
G
H
I
6%
Bond 1
$1,000.00
$883.11
$1,883.11
Bond 2
$1,041.62
$921.07
$1,962.69
Bond 3
$988.53 <-- =-PV($B$19,D7-10,D6*D8)+D8/(1+$B$19)^(D7-10)
$777.67 <-- =-FV($B$19,10,D6*D8)
$1,766.20
95.10%
91.24% 101.40%
$ 1,790.85 $ 1,790.85 $ 1,790.85
(The goal of getting $1,790.85 is still met)
150
Values 10 years later, assuming
interest rates change to 5% right
after we buy the bonds
A
19 New yield to maturity, 10 years later
20
21
22 Bond price
23 Reinvested coupons
24 Total
25
26 Multiply by percent of face value bought
27 Product
B
C
D
E
F
G
H
I
5%
Bond 1
$1,000.00
$842.72
$1,842.72
Bond 2
$1,086.07
$878.94
$1,965.01
Bond 3
$1,112.16 <-- =-PV($B$19,D7-10,D6*D8)+D8/(1+$B$19)^(D7-10)
$742.10 <-- =-FV($B$19,10,D6*D8)
$1,854.26
95.10%
91.24%
101.40%
$ 1,752.43 $ 1,792.97 $ 1,880.14
(The goal of getting $1,790.85 is not met by Bond 1 anymore)
151
Observations
 If
interest rates go down to 5%, Bond 1
does not meet the requirement anymore.
 Bond 3, on the other hand, exceeds the
payment that must be made in year 10.
 The ability of Bond 2 to meet the obligation
is barely affected. Why? Because its
duration is 10 years, exactly matching the
duration of the liability. Pick Bond 2.
152
We can compute and plot the bonds’
terminal values in year 10
Immunization Properties of the Three Bonds
$2,950
$2,750
Terminal value
$2,550
$2,350
$2,150
$1,950
Bond 1
Bond 2
Bond 3
$1,750
$1,550
0%
2%
4%
6%
8%
New interest rate
10%
12%
14%
16%
153