Bond prices are determined by evaluating the expected cash flows of the bond discounted at the yield to maturity. The yield to maturity incorporates the interest payments and expected capital gains or losses. Bonds with higher yields have higher risks. Duration is an important measure of bond risk as it indicates how sensitive the bond price will be to changes in interest rates. Investors must consider factors like bond type, yield, and duration when choosing bonds for their portfolio.

1. Bond Prices and the
Importance of Duration
1

2. Outline
• Introduction
• Review of bond principles
• Bond pricing and returns
• Bond risk
• The meaning of bond diversification
• Choosing bonds
• Example: monthly retirement income
2

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
3

4. Review of Bond Principles
• Identification of bonds
• Classification of bonds
• Terms of repayment
• Bond cash flows
• Convertible bonds
• Registration
4

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
5

6. Classification of Bonds
• Introduction
• Issuer
• Security
• Term
6

7. Introduction
• The bond indenture describes the details of a
bond issue:
– Description of the loan
– Terms of repayment
– Collateral
– Protective covenants
– Default provisions
7

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
8

9. Definition
• The security of a bond refers to what backs
the bond (what collateral reduces the risk of
the loan)
9

10. 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

11. 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

12. 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

13. 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

14. 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

15. Terms of Repayment
• Interest only
• Sinking fund
• Balloon
• Income bonds
15

16. Interest Only
• Periodic payments are entirely interest
• The principal amount of the loan is repaid at
maturity
16

17. 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

18. 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

19. 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

20. Bond Cash Flows
• Annuities
• Zero coupon bonds
• Variable rate bonds
• Consols
20

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. Commodity-Backed Bonds
• Commodity-backed bonds are convertible into
a tangible asset
• For example, silver or gold
27

28. 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

29. Registered Bonds
• Registered bonds show the bondholder’s
name
• Registered bondholders receive interest
checks in the mail from the issuer
29

30. 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

31. Bond Pricing and Returns
• Introduction
• Valuation equations
• Yield to maturity
• Realized compound yield
• Current yield
• Term structure of interest rates
• Spot rates
31

32. Bond Pricing and Returns (cont’d)
• The conversion feature
• The matter of accrued interest
32

33. 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

34. Valuation equations
• Annuities
• Zero coupon bonds
• Variable rate bonds
• Consols
34

35. Annuities
• For a semiannual bond:
35
 
2
0
1
0
1 ( / 2)
where term of the bond in years
cash flow at time
annual yield to maturity
current price of the bond
N
t
t
t
t
C
P
R
N
C t
R
P









36. Annuities (cont’d)
• Separating interest and principal components:
36
   
2
0 2
1 1 ( / 2) 1 ( / 2)
where coupon payment
N
t N
t
C Par
P
R R
C

 
 



37. 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

38. Annuities (cont’d)
Example (cont’d)
Solution: Using a financial calculator and the following input provides the
solution:
N = 20
PV = $870
PMT = $35
FV = $1,000
CPT I = 4.50
This bond’s yield to maturity is 4.50% x 2 = 9.00%.
38

39. Zero Coupon Bonds
• For a zero-coupon bond (annual and
semiannual compounding):
39
0
0 2
(1 )
(1 / 2)
t
t
Par
P
R
Par
P
R





40. 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

41. Zero Coupon Bonds (cont’d)
Example (cont’d)
Solution:
41
0 2
40
(1 / 2)
$1,000
$400
(1 / 2)
4.63%
t
Par
P
R
R
R






42. 1
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15
A B C
Purchase price 9,750.00
Face value 10,000.00
Time to maturity (days) 182 <-- =26*7
Method 1: Compound the daily return
Daily interest rate 0.0139% <-- =(B3/B2)^(1/B4)-1
YTM--the annualized rate 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 10,257.84 <-- =B2*(1+B8)
Method 2 10,257.84 <-- =B2*EXP(B11)
COMPUTING THE YIELD TO MATURITY (YTM)
ON TREASURY BILLS
42

43. 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:
43
2
0
1 (1 )
where interest rate at time
N
t
t
t t
t
C
P
I
I t






44. Consols
• Consols are perpetuities:
44
0
C
P
R


45. 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

46. Consols (cont’d)
Example (cont’d)
Solution:
46
0
$60
6.67%
$900
C
P
R
R

 

47. 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

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A B C D E F G H I
Market price of bond 1000.00
Data table
Year Bond cash flow Bond price Bond value
0 -1,000.00 7.00% <-- =B14 , data table header
1 70.00 950 7.96% 7%
2 70.00 960 7.76% 7%
3 70.00 970 7.57% 7%
4 70.00 980 7.38% 7%
5 70.00 990 7.19% 7%
6 70.00 1,000 7.00% 7%
7 1,070.00 1,010 6.82% 7%
1,020 6.63% 7%
YTM of bond 7.00% <-- =IRR(B5:B12) 1,030 6.45% 7%
1,040 6.28% 7%
1,050 6.10% 7%
1,060 5.93% 7%
1,070 5.76% 7%
1,080 5.59% 7%
1,090 5.42% 7%
YIELD TO MATURITY
Yield to Maturity (YTM) of XYZ Bond
5.00%
5.50%
6.00%
6.50%
7.00%
7.50%
8.00%
8.50%
950 1,000 1,050 1,100
Market price
YTM
48

49. 1
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8
9
10
11
12
13
14
A B C
Market price of bond 1050.00
Year Bond cash flow
15-May-01 -1,050.00
15-Dec-01 70.00
15-Dec-02 70.00
15-Dec-03 70.00
15-Dec-04 70.00
15-Dec-05 70.00
15-Dec-06 70.00
15-Dec-07 1,070.00
YTM of bond 6.58% <-- =XIRR(B5:B12,A5:A12)
YIELD TO MATURITY
For uneven date spacing
49

50. Realized Compound Yield
• The effective annual yield is useful to compare
bonds to investments generating income on a
different time schedule
50
 
Effective annual rate 1 ( / ) 1
where yield to maturity
number of payment periods per year
x
R x
R
x
  



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?
51

52. Realized Compound
Yield (cont’d)
Example (cont’d)
Solution:
52
 
 
2
Effective annual rate 1 ( / ) 1
1 (.009/ 2) 1
9.20%
x
R x
  
  


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
53

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
54

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?
55

56. Current
Yield (cont’d)
Example (cont’d)
Solution:
Current yield = $70/$870 = 8.17%
56

57. 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

58. Information Used to
Build A Yield Curve
58

59. Theories of
Interest Rate Structure
• Expectations theory
• Liquidity preference theory
• Inflation premium theory
59

60. Expectations Theory
• According to the expectations theory of
interest rates, investment opportunities with
different time horizons should yield the same
return:
60
2
2 1 1 2
1 2
(1 ) (1 )(1 )
where the forward rate from time 1 to time 2
R R f
f
   


61. 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

62. Expectations Theory (cont’d)
Example (cont’d)
Solution:
62
2
2 1 1 2
2
1 2
2
1 2
1 2
(1 ) (1 )(1 )
(1.05) (1.045)(1 )
(1.05)
(1 )
(1.045)
5.50%
R R f
f
f
f
   
 
 


63. 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

64. 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

65. 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

66. Spot Rates (cont’d)
66
Spot Rate Curve
Yield to Maturity
Time Until the Cash Flow
Interest
Rate

67. Spot Rates (cont’d)
Example
A six-month T-bill currently has a yield of 3.00%. A one-year T-
note with a 4.20% coupon sells for 102.
Use bootstrapping to find the spot rate six months from now.
67

68. 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:
68
2
2
2
2
2
2
2
21.00 1,021
1,020
(1 .03/ 2) (1 / 2)
1,021
999.31
(1 / 2)
(1 / 2) 1.022
2.16%
r
r
r
r
 
 


 


69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

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A B C D E F
Face value of bonds bought 1,000.00 Accrued interest calculation
Coupon rate 6.00%
Today's date 12-Feb-01
Market price 1,059.51 Last coupon date 15-Aug-00
Accrued interest 29.51 <-- =E12 Next coupon date 15-Feb-01
Actual price paid 1,089.02 Days since last coupon 181 <-- =E4-E5
Days between coupons 184 <-- =E6-E5
Cash flows to GI at bond issue
Date Cash flow Semi-annual coupon 30 <-- =B3/2*B2
12-Feb-01 -1,089.02 <-- =-B8 Accrued interest 29.51 <-- =E8/E9*E11
15-Feb-01 30.00 <-- =$B$3*$B$2/2
15-Aug-01 30.00
15-Feb-02 30.00
15-Aug-02 30.00
15-Feb-03 30.00
15-Aug-03 30.00
15-Feb-04 30.00
15-Aug-04 30.00
15-Feb-05 30.00 ACCRUED INTEREST IN EXCEL
15-Aug-05 30.00
15-Feb-06 30.00
15-Aug-06 30.00
15-Feb-07 30.00
15-Aug-07 30.00
15-Feb-08 30.00
15-Aug-08 30.00
15-Feb-09 30.00
15-Aug-09 1,030.00 <-- =$B$3*$B$2/2+B2
XIRR (annualized IRR) 5.193% <-- =XIRR(B12:B30,A12:A30)
Excel's Yield function 5.128% <-- =YIELD(A12,A30,B3,B5/10,100,2,3)
Excel's Yield annualized 5.193% <-- =(1+B33/2)^2-1
UNITED STATES TREASURY BOND, 6%, MATURING 8 AUGUST 2009
75

76. Bond Risk
• Price risks
• Convenience risks
• Malkiel’s interest rate theories
• Duration as a measure of interest rate risk
76

77. Price Risks
• Interest rate risk
• Default risk
77

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
78

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
79

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
80

81. Convenience Risks
• Definition
• Call risk
• Reinvestment rate risk
• Marketability risk
81

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
82

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
83

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
84

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
85

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
86

87. Malkiel’s
Interest Rate Theorems
• Definition
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
87

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
88

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
89

90. Theorem 2
• Bonds with longer maturities will fluctuate
more if interest rates change
• Long-term bonds have more interest rate risk
90

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30
A B C D E F G
Market interest rate 6.50%
Year
Bond
cash flow
Market
interest rate
Bond
value
1 70 0.00% 1,490.00 <-- =NPV(E5,$B$5:$B$11)
2 70 1.00% 1,403.69 <-- =NPV(E6,$B$5:$B$11)
3 70 2.00% 1,323.60 <-- =NPV(E7,$B$5:$B$11)
4 70 3.00% 1,249.21 <-- =NPV(E8,$B$5:$B$11)
5 70 4.00% 1,180.06
6 70 5.00% 1,115.73
7 1,070 6.00% 1,055.82
7.00% 1,000.00
Value of the bond 1,027.42 <-- =NPV(B2,B5:B11) 8.00% 947.94
9.00% 899.34
10.00% 853.95
11.00% 811.51
12.00% 771.81
13.00% 734.64
14.00% 699.82
VALUING THE XYZ CORPORATION BONDS
XYZ Bond Value
650
750
850
950
1,050
1,150
1,250
1,350
1,450
0% 2% 4% 5% 7% 9% 11% 12% 14%
Market interest rate
Bond
value
91

92. 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

93. 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

94. Theorem 5
• Capital gains from an interest rate decline
exceed the capital loss from an equivalent
interest rate increase
94

95. Duration as A Measure of Interest
Rate Risk
• The concept of duration
• Calculating duration
95

96. 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

97. 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

98. Calculating Duration
• The traditional duration calculation:
98
1 (1 )
where duration
cash flow at time
yield to maturity
current price of the bond
years until bond maturity
time at which a cash flow is received
N
t
t
t
o
t
o
C
t
R
D
P
D
C t
R
P
N
t












99. 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
A B C D E F G H
BASIC DURATION CALCULATION
YTM 7%
Year Ct,A t*Ct,A /PA*(1+YTM)t
Ct,B t*Ct,B /PB*(1+YTM)t
1 70 0.0654 130 0.0855
2 70 0.1223 130 0.1598
3 70 0.1714 130 0.2240
4 70 0.2136 130 0.2791
5 70 0.2495 130 0.3260
6 70 0.2799 130 0.3657
7 70 0.3051 130 0.3987
8 70 0.3259 130 0.4258
9 70 0.3427 130 0.4477
10 1070 5.4393 1130 4.0413
Bond price Duration Bond price Duration
1,000
$ 7.5152 1,421
$ 6.7535
=NPV(B3,B6:B15) =SUM(F6:F15)
Excel formula 7.5152 <-- =DURATION(DATE(1996,12,3),DATE(2006,12,3),7%,B3,1)
(need to have the tool "Analysis ToolPak" added in Excel)
99

100. Calculating Duration (cont’d)
• The closed-end formula for duration:
100
1
2
(1 ) (1 ) ( )
(1 ) (1 )
where par value of the bond
number of periods until maturity
yield to maturity of the bond per period
N
N N
o
R R R N F N
C
R R R
D
P
F
N
R

 
     

 
 
 





101. 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

102. Calculating Duration (cont’d)
Example (cont’d)
Solution: Using the closed-form formula for duration:
102
1
2
31
2 30 30
(1 ) (1 ) ( )
(1 ) (1 )
(1.052) (1.052) (0.052 30) 1,000 30
50
0.052 (1.052) (1.052)
985
15.69 years
N
N N
o
R R R N F N
C
R R R
D
P

 
     

 
 
 

 
   

 
 



103. 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
A B C D E F G H
EFFECTS OF COUPON AND MATURITY ON DURATION
Current date 5/21/1996 <-- =DATE(1996,5,21)
Maturity, in years 21
Maturity date 5/21/2017 <-- =DATE(1996+B4,5,21)
YTM 15% Yield to maturity (i.e., discount rate)
Coupon 4%
Face value 1,000
Duration 9.0110 <-- =DURATION(B3,B5,B7,B6,1)
Data table: Effect of maturity on duration
9.0110 <-- =B10
5 4.5163
10 7.4827
15 8.8148
20 9.0398
25 8.7881
30 8.4461
35 8.1633
40 7.9669
45 7.8421
50 7.7668
55 7.7228
60 7.6977
65 7.6837
70 7.6759
Effect of Maturity on Duration
Coupon rate = 4.00%, YTM = 15.00%
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0 20 40 60 80
Maturity
Duration
103

104. 31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
A B C D E F G H
Data table: Effect of coupon on duration
9.0110 <-- =B10
0% 21.0000
1% 13.1204
2% 10.7865
3% 9.6677
4% 9.0110
5% 8.5792
6% 8.2736
7% 8.0459
9% 7.7294
13% 7.3707
15% 7.2593
17% 7.1729
Effect of Coupon on Duration
Maturity = 21, YTM = 15.00%
5.0
7.0
9.0
11.0
13.0
15.0
17.0
19.0
21.0
23.0
0% 5% 10% 15%
Coupon rate
Duration
104

105. 105
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

106. 106
The Meaning of
Bond Diversification
Introduction
Default risk
Dealing with the yield curve
Bond betas

107. 107
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

108. 108
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

109. 109
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

110. 110
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

111. 111
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

112. 112
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

113. 113
Choosing Bonds
Client psychology and bonds selling at a
premium
Call risk
Constraints

114. 114
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

115. 115
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

116. 116
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

117. 117
Constraints
Specifying return
Specifying grade
Specifying average maturity
Periodic income
Maturity timing
Socially responsible investing

118. 118
Specifying Return
To increase the expected return on a bond
portfolio:
• Choose bonds with lower ratings
• Choose bonds with longer maturities
• Or both

119. 119
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

120. 120
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

121. 121
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

122. 122
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

123. 123
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

124. 124
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

125. 125
Example: Monthly
Retirement Income
The problem
Unspecified constraints
Using S&P’s Bond Guide
Solving the problem

126. 126
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

127. 127
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

128. 128
The Problem (cont’d)
Stock Value Qrtl Div. Payment Month
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.
Total $494,000 $3,010

129. 129
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%

130. 130
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

131. 131
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

132. 132
Using S&P’s
Bond Guide (cont’d)

133. 133
Solving the Problem
Setup
Dealing with accrued interest and
commissions
Choosing the bonds
Overspending
What about convertible bonds?

134. 134
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)

135. 135
Setup (cont’d)
Security Price Jan. Feb. March April May June
3,000 AAC $51,000 $380 $380
1,000 BBL 50,000 370 370
2,000 XXQ 49,000 $400 $400
5,000 XZ 52,000 $270 $270
7,000 MCDL 53,000
1,000 ME 49,000 370 370
2,000 LN 51,000 500 500
4,000 STU 47,000 260 260
3,000 LLZ 49,000 290 290
6,000 MZN 43,000 170 170
Equities $494,000 $1,420 $1,060 $530 $1,420 $1,060 $530

136. 136
Dealing With Accrued
Interest and Commissions
Bond prices are typically quoted on a net
basis (already include commissions)
Calculate accrued interest using the mid-
term heuristic
• Assume every bond’s accrued interest is half of
one interest check

137. 137
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?

138. 138
Bonds
Security Price Jan. Feb. March April May June
$80,000 Empire
71/2s02
$86,400 $3,000
$80,000 Energen
8s07
82,900 $3,200
$100,000 Enhance
61/4s03
105,500 $3,370
$80,000 Enron
65/8s03
84,500 $2,650
$90,000 Enron
6.7s06
97,200 $3,010
$100,000
Englehard 6.95s28
100,630 $3,470
Bonds subtotal $557,130 $3,000 $3,200 $3,370 $2,650 $3,010 $3,470
Total income $4,420 $4,260 $3,900 $4,070 $4,070 $4,000

139. 139
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

140. 140
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

141. 141
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.

142. 142
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.

143. 143
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:
1 (1 )
M
t
B t
t
P
V
r





144. 144
Suppose now that interest rates change from
r to r+Dr. The new values of the future
obligation and of the bond are:
0
0 0 0 0 1
1
1
(1 )
(1 )
N
M
t
B
B B B B t
t
dV NQ
V V V r V r
dr r
tP
dV
V V V r V r
dr r



 

 D   D   D  

 

 D   D   D



145. 145
Rearranging terms and recalling that V0=VB
yields the following expression:
1
1
(1 )
M
t
t
t
B
tP
N
V 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).

146. 146
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.

147. 147
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.

148. 148
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A B C D
BASIC IMMUNIZATION EXAMPLE WITH 3 BONDS
Yield to maturity 6%
Bond 1 Bond 2 Bond 3
Coupon rate 6.70% 6.988% 5.90%
Maturity 10 15 30
Face value 1,000 1,000 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)

149. 149
When the interest rate increases:
When the interest rate decreases:
THE IMMUNIZATION PROBLEM
Illustrated for the 30-year bond.
0
Year 10:
Future obligation of
$1,790.85 due.
30
Buy $1,014
face value
of 30-year
bond.
Reinvest
coupons
from bond
during years
1-10.
Sell bond for PV of
remaining coupons
and redemption in
year 30.
Value of
reinvested
coupons
increases.
Value of bond in
year 10 decreases.
Value of
reinvested
coupons
decreases.
Value of bond in
year 10
increases.

150. 150
Values 10 years later, assuming
interest rates do not change
19
20
21
22
23
24
25
26
27
A B C D E F G H I
New yield to maturity, 10 years later 6%
Bond 1 Bond 2 Bond 3
Bond price $1,000.00 $1,041.62 $988.53 <-- =-PV($B$19,D7-10,D6*D8)+D8/(1+$B$19)^(D7-10)
Reinvested coupons $883.11 $921.07 $777.67 <-- =-FV($B$19,10,D6*D8)
Total $1,883.11 $1,962.69 $1,766.20
Multiply by percent of face value bought 95.10% 91.24% 101.40%
Product 1,790.85
$ 1,790.85
$ 1,790.85
$
(The goal of getting $1,790.85 is still met)

151. 151
Values 10 years later, assuming
interest rates change to 5% right
after we buy the bonds
(The goal of getting $1,790.85 is not met by Bond 1 anymore)
19
20
21
22
23
24
25
26
27
A B C D E F G H I
New yield to maturity, 10 years later 5%
Bond 1 Bond 2 Bond 3
Bond price $1,000.00 $1,086.07 $1,112.16 <-- =-PV($B$19,D7-10,D6*D8)+D8/(1+$B$19)^(D7-10)
Reinvested coupons $842.72 $878.94 $742.10 <-- =-FV($B$19,10,D6*D8)
Total $1,842.72 $1,965.01 $1,854.26
Multiply by percent of face value bought 95.10% 91.24% 101.40%
Product 1,752.43
$ 1,792.97
$ 1,880.14
$

152. 152
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.

153. 153
We can compute and plot the bonds’
terminal values in year 10
Immunization Properties of the Three Bonds
$1,550
$1,750
$1,950
$2,150
$2,350
$2,550
$2,750
$2,950
0% 2% 4% 6% 8% 10% 12% 14% 16%
New interest rate
Terminal
value
Bond 1
Bond 2
Bond 3