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Chapter 12
Bond Prices and the Importance of Duration
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Outline Introduction Review of bond principles Bond pricing and returns Bond risk The meaning of bond diversification Choosing bonds Example: monthly retirement income
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
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Definition The security of a bond refers to what backs
the bond (what collateral reduces the risk of the loan)
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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
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
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
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
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
15
Terms of Repayment Interest only Sinking fund Balloon Income bonds
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Interest Only Periodic payments are entirely interest
The principal amount of the loan is repaid at maturity
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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
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
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
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Bond Cash Flows Annuities Zero coupon bonds Variable rate bonds Consols
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
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
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
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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.
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
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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
27
Commodity-Backed Bonds Commodity-backed bonds are convertible
into a tangible asset
For example, silver or gold
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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.
29
Registered Bonds Registered bonds show the bondholder’s
name
Registered bondholders receive interest checks in the mail from the issuer
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
31
Bond Pricing and Returns Introduction Valuation equations Yield to maturity Realized compound yield Current yield Term structure of interest rates Spot rates
32
Bond Pricing and Returns (cont’d)
The conversion feature The matter of accrued interest
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
34
Valuation equations Annuities Zero coupon bonds Variable rate bonds Consols
35
Annuities For a semiannual bond:
2
01
0
1 ( / 2)
where term of the bond in years
cash flow at time
annual yield to maturity
current price of the bond
Nt
tt
t
CP
R
N
C t
R
P
36
Annuities (cont’d) Separating interest and principal
components:
2
0 21 1 ( / 2) 1 ( / 2)
where coupon payment
N
t Nt
C ParP
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?
38
Annuities (cont’d)Example (cont’d)
Solution: Using a financial calculator and the following input provides the solution:
N = 20PV = $870PMT = $35FV = $1,000CPT I = 4.50
This bond’s yield to maturity is 4.50% x 2 = 9.00%.
39
Zero Coupon Bonds For a zero-coupon bond (annual and
semiannual compounding):
0
0 2
(1 )
(1 / 2)
t
t
ParP
R
ParP
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)?
41
Zero Coupon Bonds (cont’d)Example (cont’d)
Solution:
0 2
40
(1 / 2)
$1,000$400
(1 / 2)
4.63%
t
ParP
R
R
R
42
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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 returnDaily interest rate 0.0139% <-- =(B3/B2)^(1/B4)-1YTM--the annualized rate 5.2086% <-- =(1+B7)^365-1
Method 2: Calculate the continuously compounded returnContinuously compounded 5.0775% <-- =LN(B3/B2)*(365/B4)
Future value in one year using each methodMethod 1 10,257.84 <-- =B2*(1+B8)Method 2 10,257.84 <-- =B2*EXP(B11)
COMPUTING THE YIELD TO MATURITY (YTM) ON TREASURY BILLS
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:
2
01 (1 )
where interest rate at time
Nt
tt t
t
CP
I
I t
44
Consols Consols are perpetuities:
0
CP
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?
46
Consols (cont’d)Example (cont’d)
Solution:
0
$606.67%
$900
CP
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
48
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A B C D E F G H I
Market price of bond 1000.00Data table
Year Bondcash flow Bond price Bondvalue0 -1,000.00 7.00% <-- =B14 , data table header1 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
YT
M
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A B C
Market price of bond 1050.00
Year Bond cash flow15-May-01 -1,050.0015-Dec-01 70.0015-Dec-02 70.0015-Dec-03 70.0015-Dec-04 70.0015-Dec-05 70.0015-Dec-06 70.0015-Dec-07 1,070.00
YTM of bond 6.58% <-- =XIRR(B5:B12,A5:A12)
YIELD TO MATURITYFor uneven date spacing
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 ( / ) 1
where yield to maturity
number of payment periods per year
xR 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?
52
Realized Compound Yield (cont’d)
Example (cont’d)
Solution:
2
Effective annual rate 1 ( / ) 1
1 (.009 / 2) 1
9.20%
xR x
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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
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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
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
58
Information Used to Build A Yield Curve
59
Theories of Interest Rate Structure
Expectations theory Liquidity preference theory Inflation premium theory
60
Expectations Theory According to the expectations theory of
interest rates, investment opportunities with different time horizons should yield the same return:
22 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?
62
Expectations Theory (cont’d)Example (cont’d)
Solution:2
2 1 1 2
21 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
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
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
66
Spot Rates (cont’d)
Spot Rate Curve
Yield to Maturity
Time Until the Cash Flow
Inte
rest
Rat
e
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.
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:
22
22
22
2
21.00 1,0211,020
(1 .03/ 2) (1 / 2)
1,021999.31
(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
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
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
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
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?
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 = $935.56
75
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2122232425262728293031323334
A B C D E F
Face value of bonds bought 1,000.00 Accrued interest calculationCoupon rate 6.00%
Today's date 12-Feb-01Market price 1,059.51 Last coupon date 15-Aug-00Accrued interest 29.51 <-- =E12 Next coupon date 15-Feb-01
Actual price paid 1,089.02 Days since last coupon 181 <-- =E4-E5Days between coupons 184 <-- =E6-E5
Cash flows to GI at bond issueDate Cash flow Semi-annual coupon 30 <-- =B3/2*B2
12-Feb-01 -1,089.02 <-- =-B8 Accrued interest 29.51 <-- =E8/E9*E1115-Feb-01 30.00 <-- =$B$3*$B$2/215-Aug-01 30.0015-Feb-02 30.0015-Aug-02 30.0015-Feb-03 30.0015-Aug-03 30.0015-Feb-04 30.0015-Aug-04 30.00
15-Feb-05 30.00 ACCRUED INTEREST IN EXCEL15-Aug-05 30.0015-Feb-06 30.0015-Aug-06 30.0015-Feb-07 30.0015-Aug-07 30.0015-Feb-08 30.0015-Aug-08 30.0015-Feb-09 30.0015-Aug-09 1,030.00 <-- =$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
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
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A B C D E F G
Market interest rate 6.50%
YearBond
cash flowMarket
interest rateBondvalue
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% 1,180.066 70 5.00% 1,115.737 1,070 6.00% 1,055.82
7.00% 1,000.00Value of the bond 1,027.42 <-- =NPV(B2,B5:B11) 8.00% 947.94
9.00% 899.3410.00% 853.9511.00% 811.5112.00% 771.8113.00% 734.6414.00% 699.82
VALUING THE XYZ CORPORATION BONDS
XYZ Bond Value
650
750850
9501,0501,1501,250
1,3501,450
0% 2% 4% 5% 7% 9% 11% 12% 14%
Market interest rate
Bo
nd
val
ue
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
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
94
Theorem 5 Capital gains from an interest rate decline
exceed the capital loss from an equivalent interest rate increase
95
Duration as A Measure of Interest Rate Risk
The concept of duration Calculating duration
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
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
98
Calculating Duration The traditional duration calculation:
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
Nt
tt
o
t
o
Ct
RD
P
D
C t
R
P
N
t
99
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10111213141516171819202122
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 0.08552 70 0.1223 130 0.15983 70 0.1714 130 0.22404 70 0.2136 130 0.27915 70 0.2495 130 0.32606 70 0.2799 130 0.36577 70 0.3051 130 0.39878 70 0.3259 130 0.42589 70 0.3427 130 0.4477
10 1070 5.4393 1130 4.0413Bond 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)
100
Calculating Duration (cont’d) The closed-end formula for duration:
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 NC
R R RD
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?
102
Calculating Duration (cont’d)Example (cont’d)
Solution: Using the closed-form formula for duration:
1
2
31
2 30 30
(1 ) (1 ) ( )(1 ) (1 )
(1.052) (1.052) (0.052 30) 1,000 3050
0.052 (1.052) (1.052)
98515.69 years
N
N N
o
R R R N F NC
R R RD
P
103
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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 21Maturity 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 duration9.0110 <-- =B10
5 4.516310 7.482715 8.814820 9.039825 8.788130 8.446135 8.163340 7.966945 7.842150 7.766855 7.722860 7.697765 7.683770 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 80Maturity
Du
rati
on
104
31323334353637383940414243444546
A B C D E F G HData table: Effect of coupon on duration
9.0110 <-- =B100% 21.00001% 13.12042% 10.78653% 9.66774% 9.01105% 8.57926% 8.27367% 8.04599% 7.7294
13% 7.370715% 7.259317% 7.1729
Effect of Coupon on Duration Maturity = 21, YTM = 15.00%
5.07.0
9.011.013.0
15.017.019.0
21.0
23.0
0% 5% 10% 15%Coupon rate
Dur
atio
n
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
The Meaning of Bond Diversification
Introduction Default risk Dealing with the yield curve Bond betas
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
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
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
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
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
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
Choosing Bonds Client psychology and bonds selling at a
premium Call risk Constraints
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
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
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
Constraints Specifying return Specifying grade Specifying average maturity Periodic income Maturity timing Socially responsible investing
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
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
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
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
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
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
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
Example: Monthly Retirement Income
The problem Unspecified constraints Using S&P’s Bond Guide Solving the problem
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
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
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
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
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
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
Using S&P’s Bond Guide (cont’d)
133
Solving the Problem Setup Dealing with accrued interest and
commissions Choosing the bonds Overspending What about convertible bonds?
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
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
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
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
BondsSecurity 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
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
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
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
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
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 )
Mt
B tt
PV
r
144
Suppose now that interest rates change from r to r+r. The new values of the future obligation and of the bond are:
00 0 0 0 1
11
(1 )
(1 )
N
MtB
B B B B tt
dV NQV V V r V r
dr r
tPdVV V V r V r
dr r
145
Rearranging terms and recalling that V0=VB
yields the following expression:
1
1
(1 )
Mt
ttB
tPN
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
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
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
123456789
10111213141516
A B C D
BASIC IMMUNIZATION EXAMPLE WITH 3 BONDS
Yield to maturity 6%
Bond 1 Bond 2 Bond 3Coupon rate 6.70% 6.988% 5.90%Maturity 10 15 30Face value 1,000 1,000 1,000
Bond price $1,051.52 $1,095.96 $986.24Face 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
When the interest rate increases:
When the interest rate decreases:
THE IMMUNIZATION PROBLEMIllustrated 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
Values 10 years later, assuming interest rates do not change
192021222324252627
A B C D E F G H INew yield to maturity, 10 years later 6%
Bond 1 Bond 2 Bond 3Bond 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
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)
192021222324252627
A B C D E F G H INew yield to maturity, 10 years later 5%
Bond 1 Bond 2 Bond 3Bond 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
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
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
Te
rmin
al
va
lue
Bond 1
Bond 2
Bond 3
