M
O
D
U
L
E
Valuation &
Analysis of
Fixed-Income
Investments
D:\116097501.doc
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© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
This publication may not be duplicated in any way without the express written consent of the
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incorporated in any commercial programs, other books, databases, or any kind of software or any
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for any purpose other than your own is a violation of United States copyright laws.
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education requirement such as this CFP Board-Registered Program, have met its ethics,
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Printed in the United States of America.
Table of Contents
How to Study this Material ............................................ i
Study Plan/Syllabus ....................................................... 1
Learning Activities ........................................................... 3
1 Bond Yield Curves ..................................................... 7
What is a yield curve? ...................................................... 7
How to Construct a Yield Curve .................................. 10
Using Yield Curves to Make Investment Decisions... 10
2 Valuation, Risk & Return........................................ 13
Prices and Yields ............................................................. 13
Duration ........................................................................... 16
Bond Calculations ........................................................... 19
3 Convertible Bonds ................................................... 36
Conversion Value ........................................................... 36
4 Summary ................................................................... 42
5 Module Review ........................................................ 44
Questions ......................................................................... 44
Answers.......................................................................... 101
6 References .............................................................. 143
7 Exhibits ................................................................... 145
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How to Study this Material
Plan to invest from 100 to 150 hours of study time for this second course
in the CFP® Professional Education Program. If you study at least 10
hours each week, you should be able to work through the materials in
about 12 weeks. With an additional two weeks for review, you should be
ready to sit for the first exam in about 14 weeks. This means that, on a
self-study basis, you should be able to complete this course within four to
six months.
A number of study plans will work, but the steps outlined below have
proven to be effective.
1. Read the Learning Activities section in the
Study Plan/Syllabus to know what readings in
the Mayo book and in the College for Financial
Planning sections are required for each
learning objective.
2. Read the Mayo book chapters first for each
learning objective.
3. Read the College’s section readings next for
each learning objective.
4. Write out the answers to the review questions
for each learning objective. If you just read the
Mayo and College readings, you will retain
only about 10% of what you read–hardly
sufficient to pass the end-of-course test. If you
physically write in the answers to all the
i
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
questions, you will increase your retention by a
multiple of four to six times.
5. Read the answers to the review questions and
compare your answers to the approved
solutions. If your answers are sufficiently
close, move on; if not, rewrite the correct
answer so that you will better remember the
correct answer.
Required Textbooks and Readings
Mayo, Herbert B., Investments: An Introduction, 7th edition. Mason,
OH: South-Western, 2003.
Supplemental Resources
Reading Barron’s and The Wall Street Journal while studying the
Investment Planning course can help you better understand and apply
what you learn. If you are not currently a subscriber to either publication,
you can subscribe at half the normal subscription rate while you are a
College student. To subscribe, call Jolene Idler at 303-220-4996, or e-mail
Jolene.Idler@apollogrp.edu.
Recommended Software and Connectivity
Investment Analysis Calculator included in Mayo textbook
word processing software
spreadsheet software
Web browser that provides access to graphics
e-mail capability
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© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, a ll rights reserved.
Exam Formula Sheet
See formula sheet in Module 3
Study Tips
In the Study Plan/Syllabus section of this module, you’ll see a
smaller version of the learning pyramid identifying the level of each
learning objective. As you master each learning objective, you’ll know
where it fits in the hierarchy of learning.
Each learning objective is individually numbered (corresponding with the
module number) for review purposes. In addition, learning objectives are
boxed to make them stand out from the surrounding text. Look for the
boxes throughout each module to guide your studies.
The study materials for this course are designed to maximize your
capacity to assimilate important financial planning concepts.
iii
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
iv
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, a ll rights reserved.
Study Plan/Syllabus
Understanding how bonds are valued is a key to understanding how
bond prices change as economic conditions and interest rates change.
This module helps you learn how to value bonds, how to determine the
expected price volatility of bonds, and how to use the computations to
make decisions about buying and selling bonds.
The sections in this module are:
Bond Yield Curves
Valuation, Risk & Return
Convertible Bonds
The material in this module provides focus on bond valuation and
volatility and explains how to use the valuation tools to make fixedincome investment decisions.
Upon completion of this module, you should be able to use bond valuation
and duration formulas, calculate bond yields, interpret bond yield curves,
and make bond portfolio decisions for clients.
Study Plan/Syllabus
1
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
LO 7–1 is important because you must be able to apply the information
you learn to make judgments and decisions about appropriate times to
use long-term bonds or short-term bonds. Being able to interpret yield
curves aids bond decision making.
LOs 7–2 through 7–6 are very important. You must know how to define
and calculate bond intrinsic values, various types of yields, and duration.
Even more important is that you know how to interpret the information
contained in each calculation, how to assess the effect when one or more
of the assumptions changes, and how to compare bonds to help clients
make decisions about which bonds to purchase.
You should expand on the exercises given in the Module Review
Questions to practice more “what if” scenarios until you are confident
that you can intuitively understand how intrinsic value, yield-tomaturity, duration, and so forth are affected by changes in inputs. Those
who use the Mayo software will find this task greatly simplified.
Convertible bonds are especially complex. You must know how to use the
conversion value formula and—more importantly—know the
relationships among conversion value, investment value, conversion
premium, conversion ratio, and other convertible bond and convertible
preferred stock factors. Knowing how to calculate these values (LO 7–7) is
important, but knowing what the computations mean (LO 7–8) is even
more important.
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Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
Learning Activities
Learning Activities
Learning Objective
7–1
Evaluate the
investment
implications of yield
curves.
Readings
Investments:
An
Introduction
pages 490–
491 and
581–586
Module
Review
Questions
1–5
Module 7:
Bond Yield
Curves
7–2
Explain factors that
affect the price, yield,
or duration of fixedincome securities.
Investments:
An
Introduction
Ch. 16
Applications
Application A
Research the yield curve
in the Credit Markets
section of the Wall Street
Journal or in the
interactive version of The
Wall Street Journal
(www.wsj.com). Decide
what actions you would
take on the day that you
review the chart if you
had $1 million to invest in
bonds on that day.
6–17
Module 7:
Valuation,
Risk &
Return
Study Plan/Syllabus
3
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
Learning Activities
Learning Objective
7–3
Calculate the price,
compound return,
yield-to-maturity,
yield-to-call, taxableequivalent yield, or
duration of fixedincome securities.
Readings
Investments:
An
Introduction
Ch. 16 and
pages 563–
564 and
598–600
Module
Review
Questions
Applications
18–32
Module 7:
Valuation,
Risk &
Return
7–4
4
Analyze the
relationships among
bond ratings, yields,
maturities, and
durations to
determine
comparative price
volatility.
Investments:
An
Introduction
Ch. 16
33–37
Module 7:
Valuation,
Risk &
Return
Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
Application B
Go to the Web site for
Bonds Online
(www.bondsonline.com),
familiarize yourself with
the site, and find the
Capital Markets
Commentary and The
Outlook sections of the
site. Read the
commentary and
examine charts on the
yield curve and on sector
comparisons so that you
understand the
relationships among
various bond
characteristics and risk.
Learning Activities
Learning Objective
Readings
Module
Review
Questions
7–5
Assess how changes
in variables affect
bond risk and price
volatility.
38–43
7–6
Evaluate investor
profiles to
recommend
appropriate fixedincome securities for
purchase.
44–48
7–7
Calculate the
conversion value,
investment value,
investment premium,
conversion premium,
and downside risk of
convertible securities.
7–8
Analyze the
relationships among
conversion value,
investment value,
and market value of
convertible securities.
Investments:
An
Introduction
Ch. 18
Applications
49–52
Module 7:
Convertible
Bonds
53–55
Application C
Use The Wall Street
Journal or Barron’s to
find a corporate bond
that is convertible
(identified by “cv” in the
current yield column).
Then go to that
company’s Web site,
click on its most recent
annual report, and look
for the details of the
convertible issue in the
long-term debt footnote
to the financial
statements. If you have
trouble finding a
company with a
convertible bond, try
Hilton Hotels.
Study Plan/Syllabus
5
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
Look for the boxed objectives throughout this module to guide your
studies.
6
Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
1
Bond Yield Curves
Reading this section will enable you to:
7–1
Evaluate the investment implications of yield curves.
What is a yield curve?
A picture is worth a thousand words. A yield curve shows graphically
how much return a bond investor can achieve for his or her willingness to
hold the bond for a specified number of years. In theory, the greater the
number of years until the bond’s maturity, the greater the return an
investor should expect. Graphically, a yield curve based on this principle
should look like Figure 1 (the return is yield-to-maturity).
Figure 1: Hypothetical Yield Curve
15%
Return
Rf
0
Years Until Maturity
30
Bond Yield Curves
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© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
In Figure 1, the point at which the line intersects the vertical axis, Rf, is the
risk-free rate of return. In other words, if the risk-free rate of return
currently is 5%, then an investor who buys a three-month Treasury bill
would expect to earn a 5% annualized return over the next three months.
In a perfectly rational world, for each additional year that an investor
agrees to own a bond, he or she should earn an incrementally higher
return, which is reflected by the straight line sloping up and to the right
from the risk-free rate.
The investment world, however, is neither perfect nor rational. Therefore,
one seldom sees a yield curve in reality that looks like the one shown in
Figure 1. More often, the yield curve looks like the one shown in Figure 2.
Figure 2: Positive Yield Curve
15%
Return
Rf
0
Years Until Maturity
30
When the curve slopes upward to the right, it is known as a normal, or
positive, yield curve. The term normal is appropriate. Investors expect to
be paid a higher rate of interest for each additional year they agree to
hold a bond. However, as the curve indicates, sometimes investors
agreeing to hold a bond for 30 years are paid a lower rate of interest than
investors who are willing to hold the bond only 15 years. In other words,
the marginal utility declines after a certain point, instead of increasing, as
one would rationally expect.
Sometimes the yield curve looks like the one shown in Figure 3.
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Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
Figure 3: Negative Yield Curve
15%
Return
Rf
0
Years Until Maturity
30
When the curve slopes downward and to the right, it is known as a
negative yield curve. Short-term interest rates are higher than long-term
rates. A negative yield curve is seen infrequently. A negative yield curve
existed in the early 1980s and in 2000. Negative yield curves generally
occur when inflation is high; the Fed may increase short-term interest
rates to decrease money supply growth—all in an effort to break the back
of inflation. This is what happened in the 1980s. The negative yield curve
in 2000 was a consequence of a government announcement that longterm bonds would be repurchased and retired. This caused a strong
demand for long-term bonds, causing their prices to rise and their yields
to fall.
Short-term rates are higher than long-term rates in this instance. The Fed
raises short-term rates by raising the discount rate. It only has direct
control over short-term rates; long-term rates are a function of the
marketplace. When the Fed raises short-term rates to a high level,
investors begin to have confidence that the Fed’s actions may soon bear
fruit. This causes long-term rates to slow their increase and causes
investors to reenter the market, believing that all interest rates may soon
begin to decline.
Bond Yield Curves
9
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
How to Construct a Yield Curve
A yield curve for U.S. Treasury securities is easy to construct. Each day’s
issue of The Wall Street Journal provides a table listing the previous day’s
prices and yields-to-maturity of all Treasury bonds. If one wants to
construct a yield curve manually, the yield-to-maturity for about a dozen
representative maturities can be selected and plotted on graph paper. The
Wall Street Journal makes this easy by printing a Treasury yield curve in
the Credit Markets column each day. Yield curves are shown in The Wall
Street Journal for the previous day, for one week earlier, and for four
weeks earlier. This enables investors to see how the curve has changed
over the past four weeks. If you follow this section of the Journal until you
finish this course, you will be better prepared to answer test questions
that ask you to interpret a yield curve.
Yield curves for bonds other than Treasury securities are more difficult to
construct. U.S. Treasury bonds are available for almost all maturity
periods. U.S. Treasury securities are all of AAA quality. No single
company or municipality has such a wide variety of bonds available at a
consistent quality level across so many maturity periods. To construct a
yield curve for municipal and corporate bonds requires some ingenuity
and creativity. As a consequence, the only yield curve most investors see
is the U.S. Treasury yield curve. However, that curve alone often can give
bond investors a good idea of the overall shape of yield curves for all
types of bonds.
Using Yield Curves to Make Investment Decisions
Yield curves can provide investors with a wealth of information for
decision making. Some general principles are discussed here, but it
should be noted that the principles are generalizations, and they may not
always apply.
Flat Yield Curve
A yield curve is flat when its shape is normal but the incremental increase
in return over time is minimal. It does not have to be absolutely flat to be
10
Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
considered a flat yield curve. It may rise sharply for the first five years
and then flatten out. It may be relatively flat from the first year through
the 30th year. Many other possibilities exist.
A flat yield curve communicates that investors are not being paid for the
additional risk that they are taking by extending maturities beyond the
point at which the curve turns relatively flat.
Therefore, income investors, whose goal is to maximize income, should
not buy bonds with maturities that are beyond the point at which the
curve flattens. Speculators should buy long-term bonds only if they are
highly confident that interest rates will fall in the near future. For
speculators, the risk of a relatively flat curve is twofold. First, long-term
rates could rise to bring about a more rational relationship between
return and risk; second, long-term rates could rise as a consequence of an
increase in rates over the entire yield curve.
Inverted Yield Curve
An inverted yield curve communicates that actual or anticipated inflation
is a concern and that the Fed has increased short-term interest rates
(federal funds and/or the discount rate) to try to break the back of
inflation.
For some time thereafter, as the Fed tightens the money supply, rates
throughout the entire yield curve spectrum will rise, causing losses in
bond portfolios. At some point, however, rates will stabilize, inflation will
begin to fall, and a bond market rally will begin. At that time, an inverted
yield curve can be most favorable to both income investors and
speculators. Income investors can obtain bonds with high current yields
(if there is call protection) that also have tremendous price appreciation
potential. Speculators have the opportunity to leverage their positions
and generate large capital gains as interest rates fall.
The last time this inflation-induced scenario occurred was in the early
1980s, when interest rates were in the mid-teens. The subsequent fall in
interest rates for more than 15 years created the greatest bond market
rally of the 20th century. These opportunities may present themselves
only once in a person’s lifetime. However, when they occur, many
Bond Yield Curves
11
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investors fail to act out of a fear that the economy is ready to collapse
completely or that virulent inflation may shortly recur.
Steeply Sloped Yield Curve
Sometimes a yield curve slopes upward sharply over a period of years.
For example, a curve may slope upward sharply for the first 5 years and
then turn relatively flat or resume a more normal slope over the next 25
years. Both income investors and speculators may benefit from this
unusually steep curve. Income investors may be able to generate a much
higher coupon in return for their willingness to invest for just three or
four more years. Speculators may have an opportunity to generate
significant capital gains if the steep slope later returns to a more rational
shape.
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Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
2
Valuation, Risk & Return
Reading the first part of this section will enable you to:
7–2
Explain factors that affect the price, yield, or duration of fixedincome securities.
Prices and Yields
The current price of a bond is the discounted present value of the bond’s
future cash flow stream. A financial calculator can be used to compute a
bond’s current price (its present value) because the four inputs needed
(semiannual payment, par value, number of periods until maturity, and
current market interest rate for comparable bonds) are readily available.
For bond problems in this course, assume that all bonds, including zerocoupon bonds, accrue interest semiannually unless you are told
otherwise.
Since the coupon and par value are fixed at the time a bond is issued and
are not changed during the life of the bond, a bond’s present value
changes as current market interest rates change. Current market interest
rates are the discount rates used to compute the present value of a bond.
As the discount rate rises, the present value of a bond decreases. As the
discount rate declines, the present value of a bond increases. When a
bond sells above its par value (par value is generally $1,000), it is said to
be selling at a premium; when it sells below its par value, it is said to be
Valuation, Risk & Return
13
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
selling at a discount. The inverse relationship between market interest
rates and bond prices can be represented by the following seesaw
illustrations.
A bond at par might look like this.
If interest rates increase, the seesaw might look like this.
If interest rates decrease, the seesaw might look like this.
The current yield of a bond is the annual coupon rate divided by the
current price of the bond. When a bond is originally issued, the current
yield and the coupon rate are the same. If the price of a bond declines
because market interest rates have risen, the coupon is divided by a lower
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Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
price; therefore, the current yield is greater than the coupon yield. If the
price of a bond rises, then the coupon is divided by a higher price and the
current yield is less than the coupon yield.
For example, assume that a new bond is issued with a 6% coupon; it pays
$60 of interest per year, in semiannual payments of $30. Assume market
rates have risen and the bond now sells for $900; the current yield is $60
divided by $900, or 6.67%. In bond market terminology, the bond yield is
now 67 basis points higher. Assume market rates have declined, and the
bond now sells for $1,100; the current yield is $60 divided by $1,100, or
5.45%. In bond market terminology, the bond yield is now 55 basis points
lower.
A bond’s yield-to-maturity (YTM) is the sum of the current yield and the
appreciation or depreciation the bond will experience between the
current date and its maturity date. In the first example in the previous
paragraph, assume that the bond has 20 years until its maturity date (40
semiannual periods). The YTM is 6.93%, consisting of a current yield of
6.67% and a compound semiannual return over the 20 years of 0.26%
($100 of appreciation compounded over 40 periods). (After you learn the
keystrokes for computing YTM in the section on Bond Calculations,
confirm this calculation and the YTC calculation below.) Note that the
YTM is greater than the current yield because the YTM includes
appreciation; in the second case in the previous paragraph, in which the
YTM includes depreciation of the value of the asset from $1,100 to $1,000,
the YTM will be less than the current yield.
A bond’s yield-to-call (YTC) is similar to the YTM, except that the number
of periods until the call date is always less than the number of periods
until the maturity date. The YTC on a bond selling at a discount will
always be higher than the YTM because the dollar amount of
appreciation will be returned faster. However, discount bonds are seldom
called because the issuing corporation could buy the bond on the market
at a lower price than it would have to pay if it called the bond. The YTC
on a bond selling at a premium will always be lower than its YTM
because the dollar amount of depreciation will be incurred faster.
Valuation, Risk & Return
15
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Duration
Duration is the weighted-average amount of time (measured in years)
that it takes to collect a bond's principal and interest payments. Duration
is used to calculate the expected change in bond price when interest rates
change. Interest rate sensitivity and interest rate risk are directly related
to duration. Duration for a bond is similar to beta for a stock, in that both
duration and beta are volatility measures that are multiplied by the
expected change in interest rates (bonds) or the expected market risk
premium (stocks) to arrive at an expected change in the market value of
the subject bond or the expected risk premium of the subject stock. High
durations, like high betas, indicate high risk and high volatility; low
durations indicate low risk and low volatility. Treasury bills have low
durations and 30-year zero-coupon bonds have high durations.
Bonds have different characteristics and features. One bond may have a
20-year maturity, a 7% coupon, and an AAA rating. A second bond may
have a 12-year maturity, an 8% coupon, and a BB rating. The market
interest rate for the AAA bond may be 6%, and the market rate for the BB
bond may be 7.5%. Investors may have a difficult time applying this
information to analyze which of the two bonds will be the most volatile
when interest rates change. Duration is a relative measure of the data that
allows investors to determine which of the two bonds is likely to be the
most volatile.
Formulas are used to compute duration. The best way to understand how
the formulas work is to recognize that duration is a computation of the
time-weighted average term-to-maturity of a bond’s cash flow (Downes
and Goodman 1995). The time weighting means that cash flows that are
received later receive a proportionately higher weight than cash flows
that are received sooner. Therefore, the large $1,000 payment of principal
at a bond’s maturity tilts the scale to the right.
A simple way to think of duration is viewing it as a seesaw. The fulcrum
point of the seesaw is at the duration point. In other words, the timeweighted average of the bond’s cash flows is at the point where the
seesaw balances. Consider Figure 4, which follows.
16
Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
Figure 4: A Graphical Representation of Duration
Each column on top of the seesaw in Figure 4 represents the present value
of the cash flows to the investor. The small columns are the present
values of the semiannual interest payments, and the larger column at the
right end is the present value of the $1,000 par value of the bond plus the
final semiannual coupon payment.
Note that the present values of the semiannual interest payments
decrease over time (the columns are not drawn to scale). Thus, the
present value of a coupon payment received 10 years from today is less
valuable than the present value of a coupon payment received 1 year
from today. The declining present values are offset by the weighting,
which becomes heavier with each succeeding cash flow.
The present value of the large $1,000 payment received when the bond
matures is weighted heavily because it is received many years from
today. This means that proportionally more weight is on the right side of
the seesaw, even though the present value of the $1,000 is not very large.
Because of the weighting of the present values of cash flows, the fulcrum
will be closer to the right end of the seesaw than to the left end.
To see how duration might change as coupon rates, market interest rates,
and time to maturity change, consider how the fulcrum point moves as
these factors change.
Valuation, Risk & Return
17
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If we have several bonds that are equal in all respects except that their
coupon rates are different, then the price of each bond will also be
different because of the inverse relationship between interest rates and
bond prices. Bonds with higher coupons (and, therefore, greater cash
flows), discounted at the current market interest rate, will have higher
present values for each coupon payment than bonds with lower coupons
(and cash flows). Thus, the seesaw tilts downward on the left side, and
the center of gravity moves to the left. (The weighted present value of the
$1,000 principal payment is the same for all bonds of the same maturity).
In other words, bonds with higher coupon rates have lower durations
and are less volatile to interest rate changes than bonds with lower
coupon rates. Note that the duration of a zero-coupon bond is equal to its
maturity, since the only cash flow from a zero-coupon bond is the $1,000
principal payment at maturity.
If market interest rates are higher, but the coupon rate and maturity of a
bond stay constant, then the present value of each coupon payment and
of the par value will decrease. The fact that the cash flows are time
weighted means that the present value of the $1,000 par value payment
decreases proportionately more. Therefore, the right side of the seesaw
will rise, and the center of gravity will shift to the left. So, an increase in
market interest rates decreases duration, assuming that all other factors
are equal.
If the maturity of a bond increases, but the coupon and market interest
rates stay constant, then the right side of the seesaw becomes longer, and
the center of gravity shifts to the right. Therefore, an increase in maturity
increases duration, assuming that all other factors are equal.
These principles can be summarized as follows. Duration is inversely
related to changes in market and coupon interest rates, and it is directly
related to changes in maturity. The following matrix may help:
18
Coupon
Current Market
Interest Rates
Maturity
Increases Duration
Decreases
Decreases
Increases
Decreases Duration
Increases
Increases
Decreases
Valuation & Analysis of Fixed-Income Investments
© 1983, 1986, 1989, 1996, 2002, 2003, College for Financial Planning, all rights reserved.
Reading the next part of this section will enable you to:
7–3
Calculate the price, compound return, yield-to-maturity, yield-tocall, taxable-equivalent yield, or duration of fixed-income securities.
Bond Calculations
The keystrokes for computing the price, yield-to-maturity, and yield-tocall for bonds are the same as those used for single sums combined with
annuities. The single sums are the present value of the bond (the
purchase price or current market price of the bond) and the future value
of the bond (generally $1,000). The annuities are the semiannual coupon
payments. On the HP-10B/10BII financial calculator, use the top row of
keys for bond problems. The top row contains five variables (n, i, PV,
PMT, and FV). Input four of the variables and solve for the unknown fifth
variable.
When performing these types of bond calculations, make the following
assumptions unless the problem specifically states otherwise.
1. The face value is $1,000. This is input as a positive number in FV since
it is money that is paid to the client when the bond matures.
2. Coupon interest is given as an annual percentage rate based on the
face value ($1,000 unless stated otherwise). Coupon interest is paid
twice a year, so a payment is received every six months by the
investor. Coupon payments are a positive input into the calculator.
The amount of each payment is found by dividing the annual coupon
interest earned by two.
Semiannual coupon payment (PMT)
$1,000 Annual coupon rate
2
3. Since payments are received twice a year, the number of
compounding periods (n) is twice the number of years left to
maturity.
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4. If a return on “comparable bonds of the same maturity and grade” (i)
is given as an input for a bond problem, it will be given as an average
annual yield-to-maturity. If you are calculating the price of a bond,
this annual rate is a necessary input.
5. There are six months until the next semiannual coupon interest
payment will be paid to the investor. This means that bond problems
should be calculated as if each payment occurs at the end of each
period of n. This is an ordinary annuity type of problem.
6. The present value of the bond, PV, is entered as a negative number
because this is considered to be a cash outflow. Any time an investor
spends money, or purchases an investment, the amount is entered as
a negative number.
Calculating the Yield-to-Maturity for a Bond Investment
What is the YTM (IRR) on an investment in a bond with a $1,000 face
value, a current market price of $966, a 10% coupon, and 3 years to
maturity?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
2
3, gold,
xP/YR
?
(966)
50
1,000
Answer: 11.37%
Calculating the Yield-to-Call for a Bond Investment
What is the YTC on an investment in a bond with a call price of $1,050, a
current market price of $926, a 9% coupon, and 8 years until call?
Set the calculator to “end.”
20
P/YR
N
I/YR
PV
PMT
FV
2
8, gold,
xP/YR
?
(926)
45
1,050
Valuation & Analysis of Fixed-Income Investments
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Answer: 10.81%
Calculating the Price of a Bond
What is the price (or intrinsic value) of a bond with a $1,000 face value, a
10% coupon, and 3 years to maturity, if comparable bonds of the same
maturity and grade are yielding 11.5%?
Set the calculator to “end.”
P/YR
2
N
3, gold,
xP/YR
I/YR
11.5
PV
?
PMT
50
FV
1,000
Answer: $962.83
Calculating the Return on a Zero-Coupon Bond
What is the YTM on an investment in a zero-coupon bond with a $1,000
face value, a current market price of $746, and 3 years to maturity?
Note: Zero-coupon bonds have no coupon interest payments. However,
semiannual compounding is still used.
Set the calculator to “end.”
P/YR
2
N
3, gold,
xP/YR
I/YR
?
PV
(746)
PMT
0
FV
1,000
Answer: 10.01%
Calculating the Price of a Zero-Coupon Bond
What is the intrinsic value (or price) of a zero-coupon bond with a $1,000
face value, a YTM of 10.01%, and three years to maturity?
Set the calculator to “end.”
P/YR
2
N
3, gold,
I/YR
10.01
PV
?
PMT
0
FV
1,000
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xP/YR
Answer: $746.00
Duration Computations
Calculating duration for bonds is not as simple as computing the price or
YTM. A rather complex-looking formula is required. The formula for
computing a bond’s duration is as follows.
Duration
1 y ( 1 y ) n( c y )
y
c[(1 y)n 1] y
where
y = Yield-to-maturity per period
c = Coupon rate per period
n = Number of periods until maturity
If the compounding period is annual, then all numbers reflect annual
rates; if the compounding period is semiannual, then the number of
periods is twice the number of years, and the coupon rate and YTM are
one-half of the annual rates.
Annual compounding. What is the duration of a bond that has 20 years to
maturity and a coupon of 8% when the current market interest rate is 6%?
Assume annual compounding.
Duration
Duration
1 .06 (1 .06) 20(.08 .06)
.06
.08[(1 .06) 20 1] .06
1.06
1.06 .4
1.46
17.67
11.59 periods
.06 .08[ 2.21] .06
.24
Answer: 11.59 years
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Since the compounding period is annual, the 11.59 periods is also the
number of years.
Semiannual compounding. What is the duration of a bond that has 20
years to maturity and a coupon of 8% when the current market interest
rate is 6%? Assume semiannual compounding.
Duration
Duration
34.33
1 .03 (1 .03) 40(.04 .03)
.03
.04[(1 .03) 40 1] .03
1 .03
1.03 .40
.03
.04[ 2.26] .03
1.43
22.41 periods 2 11.21 years
.12
Answer: 11.21 years
Since the compounding period is semiannual, the duration in periods
must be divided by two to get the duration in years. The semiannual
computation should result in a lower duration because compounding
takes place more frequently than with annual compounding.
Change in bond price. Duration is a useful tool to help investors
determine the expected change in the price of a bond for a given change
in interest rates. A rule of thumb approach is to multiply the duration by
the expected change in rates. Using the data from the examples above, we
could say that, if interest rates are expected to change 1%, the
approximate percentage change in the price of the bond is 11.21% (when
semiannual compounding is used). If rates are expected to change onehalf of 1%, then the expected percentage change in bond price is 5.61%
(11.21% 2).
For a more precise answer, the following general formula is used.
P D
y
PB
1 y
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where
y
D
PB
y
=
=
=
=
Current yield-to-maturity
Duration
Price of the bond
Expected change in yield
Using this general formula can be confusing. Refer to the two examples
above for annual and semiannual compounding. If you have computed
the duration of one bond using annual compounding and the duration
for a different identical bond using semiannual compounding, you might
conclude that the second bond is less risky than the first bond because the
duration of the first bond (11.59) is greater than that of the second bond
(11.21).
Modified duration. The problem is that the durations for the bonds were
computed using different assumptions (annual versus semiannual
compounding). Therefore, the bonds’ durations must be adjusted to
account for this difference so that we are comparing apples to apples
when using duration to determine the price sensitivity of two or more
bonds. The method used to do this is called modified duration.
Modified duration is calculated for each bond by using part of the
preceding formula. Modified duration is then multiplied by the expected
annual percentage change in market yield to obtain the percentage
change in price. If the formula above were rewritten in this manner, it
would look like the following formula.
P
D
y PB (for annual compounding)
1 y
P
D
y PB (for semiannual compounding)
y
1 2
The first element of the equation, after the equal sign and before the first
multiplication sign, is the computation for modified duration. The
computation of modified duration for the two bonds in the preceding
examples is as follows:
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Valuation & Analysis of Fixed-Income Investments
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11.59
10.93 (for annual compounding)
1.06
11.21
10.88 (for semiannual compounding)
1.03
Modified duration adjusts for the fact that different assumptions were
used, and it standardizes both so that you are comparing apples to
apples. The durations now are virtually equivalent, with only a 5 basis
point difference, compared to the 38 basis point difference if the raw
figures (called the Macaulay duration) were used.
Computing modified duration is similar to computing risk-adjusted
returns for stocks. If two stocks have different standard deviations and
different returns, computing each stock’s risk-adjusted return
standardizes both stocks so that they can be compared with each other.
Computing modified duration accomplishes the same result for bonds.
Once the modified durations are computed, they can be multiplied by the
expected change in interest rates to compute the expected percentage
changes in the prices of the bonds.
Expected change in price. Assume that you expect market interest rates to
change from the current 6% to 6.25%. The expected change in “y” is .0625
– .0600, or .0025. Therefore, the expected percentage change in the price of
the bond (for which the semiannual compounding method was used) is
computed as follows.
Modified duration y = –10.88 .0025 = –.0272 or –2.72%
Note that the negative sign indicates that when interest rates rise, bond
prices fall. Next, multiply the percentage price change by the current
price of the bond to obtain the expected change in price (P).
The price of an 8% coupon bond maturing in 20 years with a current
market interest rate of 6% is $1,231.15 (computed with a financial
calculator). Therefore, the change in price is calculated as follows.
–.0272 $1,231.15 = –$33.49
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As originally stated, this can all be put into one equation, as follows.
P 11.21
.0025
1,231.15 $33.50
1 .03
The one-cent difference is due to calculator rounding. The price of the
bond is expected to decrease by $33.50 if interest rates rise from 6% to
6.25%.
The expected percentage change in price for an increase in market interest
rates of 25 basis points is 33.50 ÷ 1,231.15, or 2.72%.
When using the general formula, you must remember to adjust the
denominator by dividing the annual market interest rate by two if
semiannual compounding is used to compute the original duration.
Duration can be used to approximate the percentage change in price of a
bond only for small (100 basis points or less) changes in market interest
rates.
Convexity. Using duration to compute the expected price change given
an expected change in YTM assumes that a linear relationship applies to
the change in YTM and change in price. The linear relationship is
considered valid for relatively small changes in YTM, generally less than
1%. When the expected change in YTM is greater, then the linear
relationship does not apply.
Rather, a curvilinear relationship exists, as shown in the following
graphic.
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Bond
Price
Positive Convexity
Zero Convexity
Negative
Convexity
Interest Rate
In the figure, the straight line represents the linear relationship defined by
duration. Generally, straight bonds exhibit positive convexity,
represented by the upward-sloping line. As the curve shows, when
market interest rates decline, the actual price increase of the bond is
greater than would be computed using only duration; when market
interest rates increase, the actual price decrease of the bond is less than
would be computed using only duration. So, the general rule is that
duration understates the price increase when rates fall and duration
overstates the price decrease when rates rise. Convexity is a desirable
characteristic to have in bonds, especially during periods when interest
rates exhibit high volatility.
Callable bonds and mortgage-backed bonds are typical examples of
bonds with negative convexity. The graph helps explain why MBS and
callable bonds do not increase much in price when interest rates fall.
Convexity can be calculated; its calculation gives the mathematical
difference between the actual price-YTM curve and the zero-convexity
straight line that represents the price change expected solely due to
duration (the difference between the curved line and the straight line in
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the graph). The sum of the price change expected due to duration and the
price change expected due to convexity equals the total expected price
change of the bond. CFP students should not expect to have to make this
calculation on the CFP Board exam, however. Simply knowing the impact
that convexity has on the true expected price change due to a change in
interest rates is sufficient.
Taxable-Equivalent Yield
Investors in higher tax brackets (28% is often considered the lower
threshold) generally are advised to buy municipal bonds when bonds are
recommended for their portfolios. A key determinant of that decision is
the taxable-equivalent yield of the tax-free bonds.
If a tax-free bond has a yield of 5.5% and an investor is in the 28% tax
bracket, the taxable-equivalent yield is 7.64%. If the investor can find a
taxable bond with an equivalent credit rating and characteristics (but
with a yield greater than 7.64%), then the taxable bond will yield more,
after tax, than the tax-free bond; the taxable bond should probably be
purchased.
The taxable-equivalent yield (TEY) is computed as follows.
TEY
Tax- free yield
1 Marginal tax bracket
Problem: Carl Hudgins is in the 33% marginal tax bracket and is
considering investing in a municipal bond with a yield of 4.2%.
Equivalent-maturity Treasury bonds have a yield of 5.5%. What is the
TEY of the municipal bond?
TEY
Tax- free yield
1 Marginal tax bracket
TEY
4.2
6.27%
1 .33
Answer: The tax-free bond has a TEY of 6.27%, which is higher than the
yield of the Treasury Bonds.
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Valuation & Analysis of Fixed-Income Investments
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Reading the next part of this section will enable you to:
7–4
Analyze the relationships among bond ratings, yields, maturities,
and durations to determine comparative price volatility.
The reasons for a bond’s volatility are similar to the reasons for a stock’s
volatility. Bonds have both systematic and unsystematic risk.
Unsystematic risk is a function of the underlying company itself. A
bond’s unsystematic risk is reflected in the bond’s credit rating. The top
four credit ratings (AAA, AA, A, and BBB) generally indicate a company
with strong credit and, therefore, one with low unsystematic risk. Credit
ratings below BBB reflect companies with higher unsystematic risk. In
general, the bonds of companies with high credit ratings have less
business risk than the bonds of companies with lower credit ratings.
The financial uncertainty of companies with lower credit ratings makes
the repayment of principal for their bonds more unpredictable. In
general, when interest rates rise, the spread between high-quality and
low-quality debt widens; when interest rates fall, the spread narrows.
Investors assume that risk increases as rates rise and decreases as rates
fall.
Yields are also an indication of the credit risk of a company. To
compensate investors for a higher level of unsystematic risk, bonds with
lower credit ratings generally have higher coupons than bonds with
higher credit ratings. As discussed earlier, higher coupons help to lower
duration, thereby helping to lower the systematic risk of the bond. The
amount by which duration is lowered in high-coupon bonds is not
significant, however.
A direct relationship exists between a bond’s maturity and duration and
the bond’s volatility. Longer maturities and durations reflect higher
volatility. For a portfolio of bonds, the unsystematic risk associated with
credit ratings and yields becomes less important than the systematic risk
associated with maturity and duration.
Therefore, investors should pay the most attention to a bond’s (or a bond
fund’s) maturity and duration when judging the relative potential
Valuation, Risk & Return
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volatility of a single bond and of a portfolio of bonds. Investors who have
a low capacity for volatility should invest in short- to intermediate-term
bonds; investors with a higher capacity for volatility may invest in longterm bonds, zero-coupon bonds, and high-yield bonds when they are
confident about lower interest rates in the near future. When they are less
confident or when they expect higher interest rates in the near future,
they may sell their long-maturity, high-duration bonds and reinvest in
short-maturity, low-duration bonds.
Reading the next part of this section will enable you to:
7–5
Assess how changes in variables affect bond risk and price volatility.
Bond default risk is primarily a function of credit rating. Bonds with
lower credit ratings have a higher degree of risk of loss of principal. Loss
of principal is not an issue otherwise, since a bond will return its $1,000
principal at its stated maturity date.
Changes, or anticipated changes, in credit ratings can have an impact on a
bond’s price volatility. The prices of bonds for companies in financial
difficulty may decline sharply in anticipation of a possible downgrade in
a bond’s credit rating. Bonds that may be upgraded, especially from, say,
BB to BBB, might see a large increase in price. The reason for this is that
BBB is the lowest rating included in the larger category of investmentgrade bonds—meaning that the bonds are of sufficient quality to be
available for investment by many institutions, such as pension plans,
endowments, etc. Therefore, an upgrade to this level may result in a large
increase in demand for the bonds from these institutions. Professional
high-yield bond investors attempt to limit their credit risk by buying
seasoned issues with intermediate maturities instead of new issues with
long maturities.
The greatest changes in volatility are the result of changes in
creditworthiness and market interest rates. Therefore, bonds with high
durations are subject to the greatest degree of price volatility. Bond fund
managers constantly readjust the durations of their portfolios to minimize
volatility risk if they anticipate higher interest rates. Likewise, if they
anticipate lower interest rates, they will extend the durations in their
portfolios to the extent allowed in their charters.
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Individual investors can take similar actions with mutual funds, although
such actions can be offset by income taxes that flow from the transaction.
Investors can sell high-duration bond funds and buy low-duration funds
when they anticipate interest rate increases. They can move back into
high-duration funds when they anticipate interest rate decreases. In IRAs
and 401(k) plans, the tax consequences are not relevant, and such
switching may be profitable.
Immunization
When investors have a specific goal to fund at the end of a known time
horizon, they can take specific steps to “immunize” the goal against
interest rate and reinvestment rate risk. Immunization is practiced
primarily by institutional investors managing pension plans and
endowments, where future funding needs are targeted by year over a
long time horizon. Individual investors also can immunize, but on a more
limited basis, such as for ensuring that dollars are available to fund a
college education.
Immunization is the process of matching the duration (not maturity) of a
bond or a bond portfolio to the time horizon of a cash need. A single zerocoupon bond with a duration (and maturity in the case of a zero) equal to
the time until a child starts college immunizes against the cost of a college
education. A portfolio of bonds with a duration equal to the year pension
payments are required to be made to retirees immunizes the pension plan
against the liability due at that time.Technically, immunization offsets
interest rate risk and reinvestment rate risk. If interest rates rise after a
portfolio is immunized, the falling bond value is offset by the bond
coupon cash flows, which are assumed to be reinvested at increasingly
higher rates, thereby offsetting the bond’s price decline, and ensuring that
the cash needed to fund the goal is available. If interest rates fall after
immunization, the decline in interest earned on reinvested coupon
income is assumed to offset by the increase in the value of the bond.
Valuation, Risk & Return
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Institutional investors can use coupon bonds to immunize the multiple
liabilities typical of a pension or endowment plan, but individual
investors must rely on zero-coupon bonds. If individuals were to use
coupon bonds, they would have to sell one bond and purchase another
bond several times over the time horizon, since durations change as
market interest rates change. The trading costs on the odd lots typically
purchased by individual investors would quickly neutralize the benefits
of immunization.
Another approach investors use to minimize the impact of interest rate
risk when interest rates increase and reinvestment rate risk when interest
rates decrease is to construct laddered or barbell portfolios. Bond ladders
and barbells allow an investor with no opinion on the future direction of
interest rates to be hedged for either rising or falling rates.
In a ladder, bonds with maturities spread out over the time horizon are
used (e.g., buy 2-, 4-, 6-, 8-, and 10-year bonds). If interest rates increase
over the next two years, the 2-year bond is reinvested into a 10-year bond
(since the original 10-year bond now has an 8-year maturity) at a coupon
higher than the original 10-year bond. Although all bond prices have
declined, the reduction in time until maturity softens the impact. Because
all the bonds will be held until their maturity, the price decline will be
offset by future price increases until the par value is received at maturity.
In a barbell, the amount to be invested in bonds is divided between a
short-term issue and a long-term issue (e.g., a 5-year bond and a 25-year
bond). If rates increase, the large price decline of the 25-year bond is
softened by the small price decline of the 5-year bond; if rates decrease,
the large price increase of the 25-year bond is accompanied by a small
price increase of the 5-year bond.
In both ladders and barbells, the short-term bonds minimize losses if
rates rise, whereas the long-term bonds give the opportunity for
significant price appreciation if rates fall. Both allow investors to
minimize the regret that accompanies declines in bond values when
interest rates rise, and to experience the euphoria that accompanies
increases in bond values when interest rates fall.
Reading the next part of this section will enable you to:
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7–6
Evaluate investor profiles to recommend appropriate fixed-income
securities for purchase.
The two basic elements of a diversified investment portfolio are
allocations to financial assets (equities and fixed-income securities) and
real (or hard) assets (commodities, real estate, and natural resources).
Within the financial assets class, some proportion is allocated to equities
and, generally, a smaller proportion is allocated to fixed-income
investments.
After making a decision to allocate some percentage of assets to fixedincome investments, investors must decide which specific types of fixedincome investments to make. Investors who are more concerned with
stability of principal and income will focus on some types of bonds or
bond funds, such as Treasury bills, money market funds, and funds with
AAA-rated issues. Investors who want to focus on capital gains will select
other types of bonds or bond funds, such as zero-coupon bonds, highyield bond funds, or funds with long durations.
If an investor is in a 28% or higher marginal tax bracket, then tax-free
bonds may make more economic sense than taxable bonds. An investor in
a higher marginal tax bracket should always compute the taxableequivalent yield to determine if more after-tax income is possible in taxfree bonds than is possible in taxable bonds. No general obligation
municipal bond has ever defaulted; therefore, they offer the same sense of
security to investors as do taxable Treasury securities.
Investors who buy taxable bonds and who are concerned about default
risk should consider Treasury securities. If they live in a state that has a
state income tax, the income from Treasury securities is excluded from
the income reported on state tax returns. In states with high state and
local income taxes, the savings could be substantial.
U.S. agency bonds might be appropriate for investors who want a current
yield that is higher than those available on Treasury bills, notes, or bonds.
Agencies have the moral backing of the U.S. Treasury, even if they are not
fully guaranteed by the Treasury (although GNMA securities are
guaranteed). Some agencies are callable; the degree of call protection
should be determined prior to purchase.
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Duration is important for all bond investors. Risk-averse investors should
consider bonds with low durations. Aggressive investors should consider
bonds with high durations when they anticipate that interest rates will
decline, and they should consider bonds with low durations when they
anticipate that interest rates will rise.
An investor’s time horizon is more important than his or her age when
one is considering the duration and maturities of bonds in a portfolio.
Many investors decide that they should invest for the short term when in
retirement. However, these investors may have a 20-year life expectancy
at age 60 or 65. The joint life expectancy of a retired couple could exceed
20 years. If a bond investor’s time horizon could exceed 10 years, such an
investor would still need to invest in something other than Treasury bills.
Convertible bonds are an option for investors who like the higher income
stream that bonds provide and who want the opportunity for capital
gains from the same investment. As stock yields have dropped to record
lows in the 1990s, convertibles have become a more attractive option than
stocks for income-oriented investors.
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breakkk
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3
Convertible Bonds
Reading the first part of this section will enable you to:
7–7
Calculate the conversion value, investment value, investment
premium, conversion premium, and downside risk of convertible
securities.
Conversion Value
The formula for computing the conversion value of a convertible bond is
as follows.
Cs
FV
Ps
Pe
where
=
Conversion value
Pe
=
=
Face value of bond (generally $1,000)
Conversion price
Ps
=
Current market price of underlying stock
Cs
FV
The face value of the bond divided by the conversion price is known as
the conversion ratio. The conversion ratio is the number of shares of stock
into which the bond can be converted.
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If the conversion price is $40 per share, then the conversion ratio is 25
shares, which is computed as follows.
$1,000
25
$40
This means that when the common stock is $40 per share, the investor
who converts a convertible bond into shares of stock will hold 25 shares
of stock with a market value that is equal to the face value of the bond.
When the stock sells below the conversion price of $40, the value of the
bond as stock is less than the face value of the bond. An investor
generally will not convert the bond if the stock is selling for less than $40
per share because he or she could hold the bond until its maturity and be
assured of receiving $1,000.
Bond Investment Value
A bond’s investment value is the same as its intrinsic value as a straight
bond. It can be calculated with a financial calculator as the present value
of cash flows from receipts of semiannual interest payments and from the
$1,000 face value received at maturity.
Assume that a convertible bond has a coupon rate of 6%, has 20 years to
maturity, and has a $1,000 face value, when current market interest rates
are 5%. The investment value (intrinsic value) of the bond is computed
with a financial calculator as follows.
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
2
20, gold,
xP/YR
5
?
30
1,000
Answer: $1,125.51
If this is the same bond as one that is convertible into 25 shares of stock,
then an investor will not convert the bond into stock if the stock is selling
at $30 per share. To do so, the investor would be giving $1,125 worth of
bond value to acquire $750 worth of value in the stock.
Convertible Bonds
37
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Investment Premium and Conversion Premium
Because a convertible bond is like a straight bond combined with an
option contract, an investor pays a premium whenever he or she buys a
convertible bond. Take the preceding bond with a current investment
value of $1,125. If this bond were not a convertible bond, the investor
would pay $1,125 to purchase the bond. Because it is a convertible bond
and because the investor has a call option to acquire 25 shares of the
company’s stock, the investor will have to pay more than the bond’s
investment value for this option.
Let’s say the current market price of the convertible bond is $1,250. The
investor, then, is paying an investment premium of $125 over the true
market value of the bond for the option. The investor is also paying a
conversion premium, the difference between the market price of the
convertible bond and the conversion value. If the market price of the
stock is currently $30 per share, then the conversion premium is $500,
which is the current market price of the convertible bond ($1,250) minus
the conversion value of the bond ($750).
Both of these premiums can be shown as percentages. The investment
premium is 11.1%, which is computed as follows.
$125
11.1%
$1,125
The conversion premium is 66.7%, which is computed as follows.
$500
66.7%
$750
In other words, the investor currently holds a convertible bond for which
he or she paid a premium that is 11.1% greater than the investment value
of the bond and 66.7% greater than the value of the bond as stock (if the
bond were converted to stock).
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Downside Risk
Because a convertible bond is purchased at a premium over its value as a
bond, the market value of the security could fall substantially if the
market price of the underlying stock falls. The point at which that fall is
cushioned is the investment value of the bond.
The downside risk of a convertible bond is the dollar or percentage
decline from the current market price of the convertible bond to the
investment value of the bond. In other words, the investment premium is
the measure of a bond’s downside risk. As computed previously, the
downside risk for the bond in the example is $125. However, the
percentage downside risk is not 11.1%; it is 10.0%, which is computed as
follows.
$125
10.0%
$1,250
Convertible Preferred Stock
The concepts for convertible preferred stock are similar to those for
convertible bonds. The conversion price is the number of shares of
common stock that will be received in exchange for the preferred stock
times the current market price of the common stock. The investment
value (intrinsic value) of the preferred stock is the dividend of the
preferred stock divided by the current market interest rate on comparable
convertible preferred stock. Investment value is computed as follows.
P
where
P =
D =
k =
D
k
Investment value
Annual preferred stock dividend
Comparable yield
Convertible Bonds
39
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Reading the next part of this section will enable you to:
7–8
Analyze the relationships among conversion value, investment
value, and market value of convertible securities.
Summary of Convertible Bond Relationships
Figure 5 summarizes the relationships among the values found in
convertible bonds.
Figure 5: Convertible Bond Relationships
The conversion value is directly proportional to the price of the
underlying stock. As long as the conversion value is less than the
investment value of the bond, the holder would be foolish to convert. He
or she would exchange a bond for stock that is worth less than what the
bond would be worth if it was a straight bond and not a convertible bond.
After the conversion value of the stock has reached the investment value
of the bond, then conversion might make sense. At that intersection point
(point A in Figure 5) and above, the investor would exchange a bond for
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stock that is worth more than what the bond would be worth if it were a
straight bond and not a convertible bond.
Even though the conversion value of the bond may be below the
investment value of the bond, the actual market price of the bond may
exceed the investment value. This is because, in effect, the investor holds
both a straight bond and an option to convert the bond into stock.
Therefore, the bond’s market value will not fall below the investment
value of the bond, and will, in fact, sell for a premium over the
investment value. This premium, which is paid by the investor, is shown
by the shaded area in Figure 5.
On the other hand, a benefit of convertible bonds occurs when the
conversion value is greater than the investment value of the bond (due to
strong upward movement in the stock price). Should the convertible
bond not be converted and the stock begin to fall in price, the market
value of the bond will not fall proportionally with the stock after the price
of the convertible nears the investment value. In effect, the investment
value of the bond acts as a floor; the option to convert becomes nearly
worthless, but the investment value of the bond remains intact.
Sometimes, when the market value of the underlying stock falls, the
company’s financial ratios deteriorate to the point that the company’s
ability to repay debt principal becomes impaired. In that case, the
investment value of the bond may fall, causing further losses in the
convertible bond, as the convertible price drops alongside the stock.
Investors should be made aware of this possibility, especially since the
companies that issue convertible bonds frequently are those with less
stable financial positions. For this reason, an investor should like the
underlying stock as a potential investment if he or she intends to buy the
company’s convertible bonds.
Convertible Bonds
41
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4
Summary
Yield curves can help bond investors understand more clearly the term
structure of interest rates. The shape of a yield curve provides clues about
investors’ inflationary expectations and about the additional return that is
possible for each incremental increase in risk. Yield curve analysis should
be an important element in bond portfolio decision making.
Knowing how to calculate bond yields, prices, and durations gives
investors a better sense of what these terms mean and how they are used
in bond portfolio management. Duration is especially important because
it gauges the volatility of individual bonds or bond portfolios.
Convertible bonds are a type of hybrid security that many investors find
difficult to understand. A convertible bond is a combination of a straight
bond and an option contract on the underlying stock. Investors who buy
convertibles must pay a premium for this option that exceeds the
investment value of the bond, and this fact causes many investors to shun
convertibles. However, if an investor understands convertible bonds, they
can be a valuable addition to his or her investment portfolio.
Having read the material in this module, you should be able to:
7–1
Evaluate the investment implications of yield curves.
7–2
Explain factors that affect the price, yield, or duration of fixedincome securities.
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7–3
Calculate the price, compound return, yield-to-maturity, yield-tocall, taxable-equivalent yield, or duration of fixed-income securities.
7–4
Analyze the relationships among bond ratings, yields, maturities,
and durations to determine comparative price volatility.
7–5
Assess how changes in variables affect bond risk and price volatility.
7–6
Evaluate investor profiles to recommend appropriate fixed-income
securities for purchase.
7–7
Calculate the conversion value, investment value, investment
premium, conversion premium, and downside risk of convertible
securities.
7–8
Analyze the relationships among conversion value, investment
value, and market value of convertible securities.
Summary
43
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5
Module Review
Questions
7–1
Evaluate the investment implications of yield curves.
1. Construct a yield curve and interpret the information it
communicates.
a. Construct a yield curve based on the following data.
% Yield
4.7
5.2
7.0
7.6
7.8
7.8
44
Years to Maturity
0.5
1.0
5.0
10.0
15.0
20.0
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b. What kind of slope does the yield curve exhibit?
c. At what point on the curve does the risk/return relationship
change?
d. What maturity range would be chosen by a risk-averse investor,
and what maturity range would be chosen by an aggressive
investor who believes that interest rates will decline in the near
future?
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2. Use the yield curves below to answer the questions that follow.
Assume YC1 changed over time and became YC2.
YC1
% Yield
YC2
Years to Maturity
a. Which yields, short-term or long-term, were higher at the time of
YC1? Which yields were higher at the time of YC2?
b. What happened to short-term yields and to long-term yields
between the time of YC1 and the time of YC2?
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c. Compare the degree of change in short-term rates to the degree of
change in long-term rates between the time of YC1 and the time of
YC2.
d. Is YC1 a positive, negative, or flat yield curve? What type of yield
curve is YC2?
e. At the time of YC1, if Investor 1 purchased 6-month Treasury bills
and Investor 2 purchased 30-year Treasury bonds, what would
have been the relative economic consequences?
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f.
Historically, a yield curve such as YC1 depicts a situation that will
change over time to a yield curve such as YC2. What should a
financial planner consider doing when a yield curve looks like
YC1?
3. Use the following yield curve to answer the questions that follow.
10
% Yield
8
6
1
5
10
Years to Maturity
20
a. Explain the change in the risk/return relationship in moving
between the following maturities.
(1) from 1 year to 5 years
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(2) from 5 years to 20 years
b. If interest rates are expected to increase during the next year and
you are investing for income, what action would you take?
c. If interest rates are expected to decrease during the next year and
you are investing for appreciation, what action would you take?
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4. Use the following yield curves to answer the questions that follow.
Assume that YC1 changed over time and became YC2.
a. If a client originally purchased bonds at par with 10-year
maturities (as shown at point A on YC1), approximately what
yield did the client receive?
b. If a client purchased bonds at par with five-year maturities (as
shown at point B on YC2), approximately what yield did the client
receive?
c. If a client sold the 10-year bonds depicted at point A and
reinvested in the 5-year bonds depicted at point B, what effect did
that change have on each of the following?
(1) interest rate risk
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(2) purchasing power risk (inflation risk)
(3) current yield (return)
(4) overall risk/return relationship
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5. Answer the following questions about the links among Federal
Reserve policies, stages of the economy, and the term structure of
interest rates.
a. If investors expect inflation to increase over the next several years,
how will the yield curve change, and why?
b. If the Fed also believes that inflation will increase, how will the
yield curve change, and why?
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c. When the economy appears to be close to entering a recession,
how will the yield curve change, and why?
d. What bond portfolio actions should an investor take if he or she
expects interest rates to increase? What such actions should an
investor take if he or she expects interest rates to decrease?
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Application A
Research the yield curve in the Credit Markets section of the Wall Street Journal or in the
interactive version of The Wall Street Journal (www.wsj.com). Decide what actions you would
take on the day that you review the chart if you had $1 million to invest in bonds on that day.
7–2
Explain factors that affect the price, yield, or duration of fixedincome securities.
6. On what four factors does the calculation of a bond’s price depend?
7. How is the price of each of the following determined?
a. a perpetual debt instrument
b. a bond with a maturity date
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8. Which contribute more to the present value of a bond, interest
payments received in the near future or those received in the distant
future? Explain your answer.
9. Explain why bond prices and interest rates are inversely related.
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10. What do the terms “discount” and “premium” mean in relation to the
pricing of a bond?
11. Describe what each of the following bond yields represents and
explain how each is determined.
a. current yield
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b. yield-to-maturity
c. yield-to-call
12. Describe the general circumstances under which each of the following
relationships exists.
a. The YTC is higher than the YTM.
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b. The YTC is lower than the YTM.
c. The current yield is higher than the YTM.
d. The current yield is lower than the YTM.
13. What factors determine the amount of price fluctuation in a bond?
14. Compare the price volatility of the following types of bonds.
a. bonds with long maturities compared to bonds with short
maturities, assuming both have the same coupon rate
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b. bonds with low coupon rates compared to bonds with high
coupon rates, assuming both have the same maturity
15. How can an investor minimize the uncertainty surrounding the
realized compound yield of a bond?
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16. What is duration, and how is it used?
17. Explain the following bond portfolio management strategies.
a. tax swap
b. substitution swap
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c. intermarket spread swap
d. pure yield pickup swap
e. rate anticipation swap
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f.
laddered portfolio
g. dumbbell (barbell) portfolio
h. immunization
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i.
7–3
dedication
Calculate the price, compound return, yield-to-maturity, yield-tocall, taxable-equivalent yield, or duration of fixed-income securities.
18. Your client asks what the market price of a particular bond should be.
The bond pays 12% coupon interest semiannually. The bond will
mature in 7 years and will pay a face value of $1,000. Comparable
bonds (bonds with similar maturities and of the same investment
grade) are yielding 14.9%. What should be the price of this bond?
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19. Your client asks what the market price of a particular zero-coupon
bond should be. The bond will mature in 7 years and will pay a face
value of $1,000. Comparable bonds (bonds with similar maturities
and of the same investment grade) are yielding 14.9%. What should
be the price of this bond?
20. Calculate the following bond values.
a. What is the intrinsic value (price) of a newly issued bond with a
12% coupon rate, 30 years to maturity, and a $1,000 maturity
value when current market rates for comparable bonds are at
12%?
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b. What will be the bond’s price one year after issue if market rates
drop to 9%?
c. What will be the bond’s price one year after issue if market rates
rise to 15%?
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21. A bond has a market price of $875. The bond pays 12% coupon
interest semiannually. The bond will mature in 7 years and will pay a
face value of $1,000.
a. What is the YTM (IRR) for this bond?
b. What is the YTM if the bond currently has a market price of
$1,200?
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22. Your client recently purchased a zero-coupon bond for $630. It has a
$1,000 face value and matures in 6 years. What is the YTM for this
bond?
23. Your client purchased a bond for $950. The bond has a coupon rate of
11%, it matures in 17 years, and it is callable in 5 years at $1,110. What
is the YTC for this bond?
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24. Calculate the duration and expected price change for each of the
following bonds.
a. Market rate greater than coupon rate. Assume that the coupon is 6%,
that the market interest rate is 7%, that there are 16 years until
maturity, and that compounding is annual. Also assume that
interest rates are subsequently expected to fall by 50 basis points.
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b. Coupon rate greater than market rate. Assume that the coupon is 8%,
that the market interest rate is 6%, that there are 22 years until
maturity, and that compounding is semiannual. Also assume that
interest rates are subsequently expected to rise by 60 basis points.
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c. Zero-coupon bond. Assume that the current market interest rate is
7%, that there are 18 years until maturity, and that compounding
is semiannual. Also assume that interest rates are subsequently
expected to fall by 30 basis points.
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25. IBM has a bond with a 7% coupon; the bond matures in 2025 for
$1,000. In 1998, the current price of the bond was 107 5/8 (107.625% of
par, or 1.07625 × $1,000 = $1,076.25). Assume that the bond had 26
years until maturity at that time. (Find the same bond in the
newspaper on the day you work this problem and rework the
problem based on current market prices).
a. What is the YTM of the IBM bond?
b. Using the YTM computed in part a. of this question (rounded to
the nearest tenth), what is the duration of the IBM bond?
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c. If the YTM is expected to fall 40 basis points in the next year, by
how much would the price of the IBM bond change?
26. What are the taxable-equivalent yields of municipal bonds with the
following tax-free yields for investors in the following marginal tax
brackets?
Tax-Free
Yield
TEY
28% Bracket
TEY
36% Bracket
4%
4.5%
5%
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TEY
40% Bracket
27. Jane Roberts owns a public purpose municipal bond that pays 5%.
Assuming she is in the 36% marginal tax bracket, what yield on
corporate bonds would be comparable to the yield on Jane’s current
investment?
28. Paulette Doyle’s marginal tax bracket is 40%. She is considering either
a corporate bond that pays 8% annually or a tax-exempt municipal
bond. What yield on the municipal bond would be comparable to the
yield on the taxable corporate bond?
29. If preferred stock does not have a required sinking fund or call
feature, it may be viewed as a perpetual debt instrument. How is the
intrinsic value of this type of preferred stock calculated?
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30. Explain how to determine the intrinsic value of preferred stock that
has a finite life.
31. If a preferred stock pays an annual dividend of $5 and investors can
earn 12% on alternative, comparable investments, what is the price
that should be paid for this stock?
32. If the preferred stock in the previous question had a call feature, and
if investors expected the stock to be called for $100 after 12 years,
what price would be paid for this stock?
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7–4
Analyze the relationships among bond ratings, yields, maturities,
and durations to determine comparative price volatility.
33. Consider the following three bonds and determine which bond is
most susceptible to price fluctuations.
Bond 1: A-rated, pays a coupon of 11%, matures in 12 years
Bond 2: AA-rated, pays a coupon of 12%, matures in 7 years
Bond 3: BBB-rated, pays a coupon of 9%, matures in 15 years
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34. Consider the following bonds.
Bond 1: BBB-rated, pays a coupon of 9%, matures in 6 years
Bond 2: BBB-rated, pays a coupon of 9%, matures in 11 years
Bond 3: BBB-rated, pays a coupon of 7%, matures in 11 years
Bond 4: BBB-rated, pays a coupon of 7%, matures in 6 years
a. Determine whether Bond 1 or Bond 2 has more potential for price
fluctuation and give a reason why.
b. Determine whether Bond 2 or Bond 3 has more potential for price
fluctuation and give a reason why.
c. Determine whether Bond 3 or Bond 4 has more potential for price
fluctuation and give a reason why.
d. Determine whether Bond 1 or Bond 4 has more potential for price
fluctuation and give a reason why.
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35. Consider the following bonds.
Bond 1: AA-rated, pays a coupon of 9%, matures in 7 years
Bond 2: BB-rated, pays a coupon of 9%, matures in 12 years
Bond 3: BB-rated, pays a coupon of 9%, matures in 7 years
Bond 4: AA-rated, pays a coupon of 9%, matures in 12 years
a. Determine whether Bond 1 or Bond 3 has more potential for price
fluctuation and give a reason why.
b. Determine whether Bond 2 or Bond 4 has more potential for price
fluctuation and give a reason why.
c. Determine whether Bond 1 or Bond 4 has more potential for price
fluctuation and give a reason why.
d. Determine whether Bond 2 or Bond 3 has more potential for price
fluctuation and give a reason why.
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36. Consider the following bonds.
Bond 1: BBB-rated, pays a coupon of 8%, matures in 5 years
Bond 2: AA-rated, pays a coupon of 12%, matures in 5 years
Bond 3: BBB-rated, pays a coupon of 12%, matures in 5 years
Bond 4: AA-rated, pays a coupon of 8%, matures in 5 years
a. Determine whether Bond 1 or Bond 3 has more potential for price
fluctuation and give a reason why.
b. Determine whether Bond 1 or Bond 4 has more potential for price
fluctuation and give a reason why.
c. Determine whether Bond 2 or Bond 4 has more potential for price
fluctuation and give a reason why.
d. Determine whether Bond 2 or Bond 3 has more potential for price
fluctuation and give a reason why.
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37. Review the Morningstar reports for the American Century-Benham
bond fund and the Alliance Bond Corporate bond fund.
a. Which of the two funds would you expect to be more volatile, and
why?
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b. What evidence is there in the Morningstar report of the greater
volatility of the fund you chose?
Application B
Go to the Web site for Bonds Online (www.bondsonline.com), familiarize yourself with the site,
and find the Capital Markets Commentary and The Outlook sections of the site. Read the
commentary and examine charts on the yield curve and on sector comparisons so that you
understand the relationships among various bond characteristics and risk.
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7–5
Assess how changes in variables affect bond risk and price volatility.
38. In Question 24, you computed the duration and expected price
change for several types of bonds. The characteristics and the
estimated percentage price changes of those bonds are summarized in
the following table (BP stands for basis points).
Bond
Coupon
Market Rate
Maturity
Duration
y
P (%)
A
6
7
16
10.42
50 BP
4.9%
B
8
6
22
11.80
60 BP
6.9%
C
0
7
18
18.00
30 BP
5.2%
What conclusions can you reach about bond risk and volatility
relative to different characteristics of these bonds and changes in
some of their variables?
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39. The following table shows characteristics of four bond funds. The
funds are listed in order of ascending credit quality. The first fund,
Merrill Lynch Corp Hi-Income A, is a high-yield bond fund. The last
three bonds are also ranked by decreasing average maturity.
Review the data in the table and explain how each fund’s risk and
volatility is affected by the differences in variables.
Fund
Average
Credit
Quality
Average
Weighted
Coupon
Average
Maturity
Average
Standard Effective
Deviation Duration
Merrill Lynch Corp Hi-Income A
B
7.8
NA
3.74
3.8
Alliance Bond Corp Bond A
BBB
8.1
23.1
7.84
9.3
Merrill Lynch Corp Invmt Gr A
A
7.3
12.5
3.92
5.9
Intermediate Bond Fd America
AA
7.9
4.4
2.1
3.3
Source: Morningstar, Inc., Morningstar Principia Pro Plus for Mutual Funds. Chicago: Morningstar, Inc., 1998.
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40. If you want to ensure that $40,000 is available in 13 years when your
child is about to enter college, would you select a zero-coupon bond
that matures in 13 years or a coupon bond that matures in 13 years?
Why did you select the one you did?
41. If you have a conservative client who is concerned about fluctuating
bond prices, but who wants to have relatively high income from a
bond portfolio, how would you construct a bond portfolio so that you
can help the client resolve both of these apparently conflicting
concerns?
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42. Referring to Figure 17.2 and Figure 17.3 in Chapter 17 of the Mayo
text, what conclusions can you draw concerning yields and prices of
state and local government bonds?
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43. Summarize all the relationships between price, coupon, maturity,
interest rates, and duration that you have discovered in this module.
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7–6
Evaluate investor profiles to recommend appropriate fixed-income
securities for purchase.
44. Robert Berens, age 65, is retiring and has $150,000 to invest. He is
interested in purchasing fixed-income securities to provide for his
income needs during retirement. Robert will not have any other
substantial income, and he will be in the 15% marginal income tax
bracket. He has invested in bonds in the past, and he plans to be
actively involved in this investment.
What kind of fixed-income security is appropriate for Robert, and
why? (Consider type, risk rating, marginal tax bracket, term, and
other relevant factors.)
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45. John Bloom, age 49, wants to take early retirement next year when he
turns 50. He wants to invest $200,000 in a fixed-income security to
provide him with additional income. He estimates that he will be in
the 31% marginal tax bracket. He has invested previously, and he is
willing to be aggressive with this investment to increase his return.
What kind of fixed-income security is appropriate for John, and why?
(Consider type, risk rating, marginal tax bracket, term, and other
relevant factors.)
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46. Kent Walters, age 32, has $40,000 to invest in a fixed-income security.
He has invested in various types of bonds for 10 years, he considers
himself to be an aggressive investor, and he is in the 28% marginal
income tax bracket. His primary goal is capital appreciation; income is
a secondary consideration. Kent’s financial planner has presented the
following securities and their before-tax yields.
a. 15-year, BB-rated, noncallable corporate bonds trading near par
with a yield of 11.8%
b. 20-year, A-rated, discount, public purpose, callable general
obligation municipal bonds with a taxable-equivalent yield of
12.2%
c. 10-year, A-rated, premium, callable, sinking fund, corporate
bonds with a yield of 9.5%
d. Treasury bills with a yield of 8.0%
Which one of these fixed-income securities would be an appropriate
choice for Kent, and why?
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47. Kathy Connelly, age 20, is just starting college and needs to invest
$25,000 in fixed-income securities. She is in the 15% tax bracket and
plans to use the interest income and principal as needed to pay her
college expenses for the next four years. She is looking for a low-risk
investment, and she knows she must receive principal periodically
from these securities. The following securities are available to Kathy
at the before-tax yields indicated.
a. BB-rated, public purpose, municipal revenue bonds with an aftertax yield of 7.0%
b. 12-year, B-rated, discount, callable corporate bonds with a beforetax yield of 8.8%
c. eight-year Treasury notes with a before-tax yield of 6.8%
d. AA-rated, noncallable, five-year corporate bonds with a before-tax
yield of 8.5%
Which one of these securities would be an appropriate choice for
Kathy, and why?
48. Answer the following questions about selecting bonds for client
portfolios.
a. What sort of characteristics would you look for in a bond chosen
for a client with a high risk tolerance?
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b. What sort of characteristics would you look for in a bond chosen
for a client with a moderate risk tolerance?
c. What sort of characteristics would you look for in a bond chosen
for a client with a low risk tolerance?
d. If you believe that interest rates will decline sharply in the future,
what bond characteristics would you search for?
e. If you believe that interest rates will rise sharply in the future,
what bond characteristics would you search for?
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7–7
Calculate the conversion value, investment value, investment
premium, conversion premium, and downside risk of convertible
securities.
49. Janice Carlysle owns a ZZT Corporation convertible bond. The bond
has a 9.5% coupon rate that is paid semiannually; the bond matures in
8 years. Comparable debt (with the same rating and maturity date) is
yielding 11%. Janice’s bond is convertible at $27 a share, the current
market price of ZZT common stock is $35, and the bond sells for
$1,400.
a. What is the conversion value of the bond?
b. What is the investment value of the bond?
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c. What is the bond’s investment premium?
d. What is the bond’s conversion premium?
e. What is the downside risk percentage of the bond?
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50. James Perry owns a QV, Inc., convertible bond. The bond has a
coupon rate of 10% that is paid semiannually; the bond matures in 12
years. Comparable debt yields 8% currently. His bond is convertible
into 24 shares of stock. The current market price of QV common stock
is $34, and the bond sells for $1,200.
a. What is the conversion value of the bond?
b. What is the investment value of the bond?
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c. What is the bond’s investment premium?
d. What is the bond’s conversion premium?
e. What is the downside risk percentage of the bond?
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51. Assume that a convertible bond has a face value of $1,000 and that it
is selling in the market for $890. Its conversion price is $50 per share.
The underlying common stock is selling for $38 per share. The bond
pays $40 semiannually in interest and matures in 20 years. The market
interest rate on comparable bonds is 12%.
a. What is the bond’s conversion ratio?
b. What is the conversion value?
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c. What is the investment value of the convertible bond?
d. Express the downside risk as a percentage.
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52. An investor can obtain 1.5 shares of common stock through
conversion of 1 share of preferred stock. The price of the common
stock is $35. The convertible preferred stock has no maturity date and
pays an annual dividend of $3. The yield on comparable
nonconvertible preferred stock is 12%.
a. What is the conversion value of this convertible preferred stock?
b. What is the investment value of this convertible preferred stock?
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7–8
Analyze the relationships among conversion value, investment
value, and market value of convertible securities.
53. In the following figure, what does the shaded area represent?
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54. Under what circumstances does a convertible bond become an
inferior investment?
55. A convertible bond with an 8% coupon has an investment value of
$900 and a conversion value of $1,150 when the market interest rate is
9%.
a. Would you expect the market value of the convertible bond to be
(1) less than the bond’s investment value, (2) between the
investment value and the conversion value, or (3) greater than the
conversion value? Explain your answer.
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b. Is the conversion price below or above the market price of the
common stock? Explain your answer.
c. Is the downside risk less than or greater than $100? Explain your
answer.
Application C
Use The Wall Street Journal or Barron’s to find a corporate bond that is convertible (identified by
“cv” in the current yield column). Then go to that company’s Web site, click on its most recent
annual report, and look for the details of the convertible issue in the long-term debt footnote to
the financial statements. If you have trouble finding a company with a convertible bond, try
Hilton Hotels.
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Answers
7–1
Evaluate the investment implications of yield curves.
1. Construct a yield curve and interpret the information it
communicates.
a. Construct a yield curve based on the following data.
% Yield
4.7
5.2
7.0
7.6
7.8
7.8
Years to Maturity
0.5
1.0
5.0
10.0
15.0
20.0
8.0
7.0
%
6.0
Yield
5.0
4.0
0
.5 1
5
10
Years to
Maturity
15
20
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b. What kind of slope does the yield curve exhibit?
The slope is normal, or positive.
c. At what point on the curve does the risk/return relationship
change?
The risk/return relationship changes at about 5 years to maturity and at
approximately 7% interest. As an investor increases the maturity from
zero to five years, a substantial increase in yield is achieved for each
incremental increase in maturity. After five years, the curve flattens and
little additional yield is achieved for each incremental increase in maturity
and interest rate risk.
d. What maturity range would be chosen by a risk-averse investor,
and what maturity range would be chosen by an aggressive
investor who believes that interest rates will decline in the near
future?
A risk-averse investor would choose maturities from 5 to 10 years to avoid
any significant interest rate risk. An aggressive investor would choose
longer maturities—probably longer than 10 years—so that he or she could
accumulate capital gains as interest rates decline and thus obtain an
attractive total return (coupon plus capital gain).
2. Use the yield curves below to answer the questions that follow.
Assume YC1 changed over time and became YC2.
YC1
% Yield
YC2
Years to Maturity
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a. Which yields, short-term or long-term, were higher at the time of
YC1? Which yields were higher at the time of YC2?
At the time of YC1, short-term yields were higher than long-term yields. At
the time of YC2, long-term yields were higher than short-term yields.
b. What happened to short-term yields and to long-term yields
between the time of YC1 and the time of YC2?
From the time of YC1 to the time of YC2, both short-term and long-term
yields decreased.
c. Compare the degree of change in short-term rates to the degree of
change in long-term rates between the time of YC1 and the time of
YC2.
From the time of YC1 to the time of YC2, short-term rates fell more than
long-term rates.
d. Is YC1 a positive, negative, or flat yield curve? What type of yield
curve is YC2?
YC1 is a negatively sloped (inverted) yield curve, and YC2 is a positively
sloped (normal) yield curve.
e. At the time of YC1, if Investor 1 purchased 6-month Treasury bills
and Investor 2 purchased 30-year Treasury bonds, what would
have been the relative economic consequences?
Investor 1 would have had a higher initial current yield but also would
have had substantial reinvestment rate risk; every six months a new sixmonth bill would have been purchased with a yield that was lower than the
yield was during the prior six months. Investor 2 would have settled for a
lower current yield initially. After short-term rates declined, Investor 2
would have had a higher current yield than Investor 1 had. Investor 2 also
would have had a capital gain when long-term rates fell, which would have
caused bond prices to increase in value—a positive outcome of interest
rate risk when rates fall.
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f.
Historically, a yield curve such as YC1 depicts a situation that will
change over time to a yield curve such as YC2. What should a
financial planner consider doing when a yield curve looks like
YC1?
When a yield curve is inverted, a financial planner should consider
purchasing long-term securities to lock in rates for a long period of time—
in anticipation of lower yields in the future. Such a move also might result
in a capital gain on the investment as bond prices rise.
3. Use the following yield curve to answer the questions that follow.
10
% Yield
8
6
1
5
10
Years to Maturity
20
a. Explain the change in the risk/return relationship in moving
between the following maturities.
(1) from 1 year to 5 years
Moving along the yield curve from maturities of one year to those of
five years, increased interest rate risk and increased returns would be
experienced. The risk/return trade-off appears to be beneficial
because the additional interest rate risk is minimal for a 33% increase
in yield.
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(2) from 5 years to 20 years
Moving along the yield curve from 5-year maturities to 20-year
maturities, increased interest rate risk with the same return would be
experienced. The risk/return trade-off does not appear to be beneficial
because no additional yield is obtained for a large increase in interest
rate risk. The investor would be assuming more interest rate risk,
inflation risk, reinvestment risk, and perhaps call risk, for no yield
increase.
b. If interest rates are expected to increase during the next year and
you are investing for income, what action would you take?
Securities with maturities of less than one year should be chosen to
minimize interest rate risk and to take advantage of the rising yields.
c. If interest rates are expected to decrease during the next year and
you are investing for appreciation, what action would you take?
Securities with maturities of 20 years should be chosen so that the high
current coupons could be achieved and so that the maximum opportunity
for capital appreciation would be available.
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4. Use the following yield curves to answer the questions that follow.
Assume that YC1 changed over time and became YC2.
a. If a client originally purchased bonds at par with 10-year
maturities (as shown at point A on YC1), approximately what
yield did the client receive?
8%
b. If a client purchased bonds at par with five-year maturities (as
shown at point B on YC2), approximately what yield did the client
receive?
8%
c. If a client sold the 10-year bonds depicted at point A and
reinvested in the 5-year bonds depicted at point B, what effect did
that change have on each of the following?
(1) interest rate risk
Interest rate risk decreased because the maturity decreased by five
years.
(2) purchasing power risk (inflation risk)
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Purchasing power risk (inflation risk) decreased because the
investment matured five years sooner.
(3) current yield (return)
There was no change in current yield.
(4) overall risk/return relationship
The client experienced less risk for the same return. A loss may have
been incurred, however, when the original bond was sold because
interest rates had increased and the price of the original bond had
decreased.
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5. Answer the following questions about the links among Federal
Reserve policies, stages of the economy, and the term structure of
interest rates.
a. If investors expect inflation to increase over the next several years,
how will the yield curve change, and why?
In general, all interest rates along the curve will rise. Initially, long-term
interest rates may increase more sharply than short-term rates; investors
who fear increased inflation will expect a higher current yield for bonds
with longer maturities, which will compensate them for loaning money for
an extended period of years. The yield curve should remain positively
sloped. If the perception among investors regarding inflation continues,
then short-term rates will also rise; at some point short-term rates may rise
faster than long-term rates, especially if the Fed intervenes.
b. If the Fed also believes that inflation will increase, how will the
yield curve change, and why?
The Fed can control short-term rates, not long-term rates. It will probably
raise the Fed funds rate and the discount rate, which would force up shortterm interest rates. This might cause short-term rates to rise more rapidly
than long-term rates, which would cause the yield curve to flatten and
possibly to become a negatively sloped curve.
c. When the economy appears to be close to entering a recession,
how will the yield curve change, and why?
To prevent a serious recession, the Fed will probably decrease the Fed
funds and discount rates so that short-term rates will fall rapidly, which will
turn the yield curve from a negatively sloped or relatively flat curve to a
positively sloped curve. Long-term rates will fall because the market will
know that the Fed is increasing liquidity and that inflation expectations are
probably lower. Long-term bonds will experience capital gains in addition
to their coupon yield, giving investors excellent total returns.
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d. What bond portfolio actions should an investor take if he or she
expects interest rates to increase? What such actions should an
investor take if he or she expects interest rates to decrease?
If interest rates are expected to increase, the yield curve will shift upward,
causing capital losses in bond portfolios. An investor who wants to
minimize capital losses should move from long-term bonds to short-term
bonds. This action would also allow the investor to reinvest at increasingly
higher yields as rates increase. Reinvestment rate risk would work in the
investor’s favor. If the investor expects rates to decrease, the yield curve
would shift downward, causing capital gains in bond portfolios. In this
case, an investor should move from short-term bonds to long-term bonds
to lock in higher coupons and to generate capital gains in the bond
portfolio.
Application A
Research the yield curve in the Credit Markets section of the Wall Street Journal or in the
interactive version of The Wall Street Journal (www.wsj.com). Decide what actions you would
take on the day that you review the chart if you had $1 million to invest in bonds on that day.
7–2
Explain factors that affect the price, yield, or duration of fixedincome securities.
6. On what four factors does the calculation of a bond’s price depend?
The price of a bond is related to (1) the interest paid by the bond, (2) the
interest rate available on comparable bonds of the same maturity and grade
(market interest rate), (3) the maturity date of the bond, and (4) the bond’s
principal or call amount.
7. How is the price of each of the following determined?
a. a perpetual debt instrument
The price of a perpetual debt instrument is equal to the present value of
an infinite stream of payments, which is determined as follows: annual
interest payment divided by the current market interest rate.
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b. a bond with a maturity date
The price of a bond with a maturity date is equal to the present value of
the interest payments plus the present value of the principal to be
received at maturity. (The present value of a bond, also known as its
intrinsic value, can be determined with a financial function calculator.)
8. Which contribute more to the present value of a bond, interest
payments received in the near future or those received in the distant
future? Explain your answer.
The interest payments received in the near future contribute more to the
present value of a bond because dollars received in the distant future have
less value today. The present value of $100 received in 3 years is greater than
the present value of $100 received in 20 years.
9. Explain why bond prices and interest rates are inversely related.
Because the dollar amount of interest paid by a bond is constant (i.e., there is
a fixed flow of income), the price (or intrinsic value) of the bond changes in the
opposite direction of a change in interest rates, which would encourage
investors to purchase it. For example, if the market interest rates of
comparable bonds increase, the value (price) of the bond declines, which
makes its flow of income attractive to investors (who could otherwise receive a
larger flow of income from other newly issued, higher-coupon bonds). When
market rates decrease, the price of the bond increases because its flow of
income is more valuable to investors (who would otherwise have to accept a
smaller flow of income from other newly issued, lower-coupon bonds).
10. What do the terms “discount” and “premium” mean in relation to the
pricing of a bond?
In relation to bonds, the discount is the amount by which a bond sells below its
maturity value to be competitive with bonds of comparable quality. The
premium is the amount by which a bond’s price exceeds its maturity value. If
the coupon rate of a bond is less than the market yield, the bond’s price is
below its maturity value (i.e., it is a discount bond). If the bond’s coupon rate is
greater than the market yield, the bond’s price exceeds its maturity value (it is
a premium bond).
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11. Describe what each of the following bond yields represents and
explain how each is determined.
a. current yield
The current yield of a bond is a measure of the return on the bond based
on the stated cash interest per year and the bond’s current market price.
Current yield is calculated by dividing the annual interest payment by the
market price. Current yield does not take into account the difference
between a bond’s purchase price and its redemption value.
b. yield-to-maturity
YTM is the compound yield earned on a bond from the time it is
purchased until its maturity date. (It includes both the periodic cash
income received and any capital gains or losses that arise because the
principal amount is greater or smaller than the current market price.) YTM
is the market rate of return, the interest rate that equates the stream of
interest payments and the par value at maturity to the bond’s current
price.
c. yield-to-call
YTC is a measure of the yield for bonds that are likely to be called. In
calculating YTC, the number of periods until the call date is used instead
of the number of periods until maturity, and the call price is used instead
of the face value.
12. Describe the general circumstances under which each of the following
relationships exists.
a. The YTC is higher than the YTM.
For a discount bond, the YTC is higher than the YTM if the bond is called
and the principal is redeemed early.
b. The YTC is lower than the YTM.
If a bond is selling at a premium and it is called by the issuing firm at par,
then the YTC would be lower than the YTM.
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c. The current yield is higher than the YTM.
If a bond sells at a premium, the current yield is higher than the YTM.
d. The current yield is lower than the YTM.
If a bond sells at a discount, the current yield is lower than the YTM.
13. What factors determine the amount of price fluctuation in a bond?
Price fluctuations are affected by a bond’s grade (credit/default risk), its
coupon rate, its length of time to maturity, its duration, and any changes in
market interest rates.
14. Compare the price volatility of the following types of bonds.
a. bonds with long maturities compared to bonds with short
maturities, assuming both have the same coupon rate
Bonds with long maturities are more volatile than bonds with short
maturities. The principal payment and coupon payments for longer-term
bonds occur further into the future, which raises the duration. See Exhibit
A at the end of this module for an example using data from Chapter 16 of
the Mayo text.
b. bonds with low coupon rates compared to bonds with high
coupon rates, assuming both have the same maturity
Bonds with low coupon rates are more volatile than bonds with high
coupon rates. Assuming everything else is equal, low-coupon bonds have
higher durations than high-coupon bonds because the present value of
their time-weighted cash flows is lower. See Exhibit B at the end of this
module for an example using data from Chapter 16 of the Mayo text.
15. How can an investor minimize the uncertainty surrounding the
realized compound yield of a bond?
An investor can reduce one source of risk by purchasing only noncallable
bonds, which are bonds that cannot be retired prior to maturity. (Noncallable
bonds tend to sell for lower yields, however.) The uncertainty associated with
changes in interest rates remains. A zero-coupon bond eliminates the
uncertainty about the reinvestment rate because there are no coupons to
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reinvest. Longer-term zero-coupon bonds have very volatile prices, however,
due to their 0% interest coupon.
16. What is duration, and how is it used?
Duration is the weighted-average amount of time it takes to collect a bond’s
interest and principal payments. Duration is used to compare the interest rate
risk of bonds that have different coupons and different maturities (i.e., to relate
bond price sensitivity to interest rate changes). Investors can reduce interest
rate risk by selecting bonds with shorter durations. They also can match the
duration of their portfolios with the timing of their cash flow needs. By
matching duration to the term of a goal, they optimize the trade-off between
interest rate risk and reinvestment rate risk.
17. Explain the following bond portfolio management strategies.
a. tax swap
A tax swap occurs when an investor sells a bond for a capital loss and
immediately reinvests the proceeds in a bond of similar characteristics
(yield, maturity, credit rating, etc.), but one that is from another issuer. The
investor does this to recognize the capital loss for tax purposes—and still
maintain his or her bond portfolio position.
b. substitution swap
A substitution swap occurs when an investor sells one bond and
purchases another bond with similar characteristics, but chooses one with
a higher yield-to-maturity.
c. intermarket spread swap
An intermarket spread swap is a variation of the substitution swap in
which the difference in yields (the spread) between two types of bonds
(e.g., corporate and government bonds) seems excessively high.
d. pure yield pickup swap
A pure yield pickup swap occurs when an investor sells short-term bonds
and purchases long-term bonds to increase the yield on the bond portfolio.
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e. rate anticipation swap
A rate anticipation swap occurs when an investor believes that interest
rates will change dramatically and adjusts the maturity of his or her
portfolio accordingly. The investor who anticipates that rates will rise will
shorten the average maturity and duration of his or her portfolio; the
investor who anticipates that rates will fall will lengthen the average
maturity and duration of his or her portfolio.
f.
laddered portfolio
An investor uses a laddered strategy to minimize interest rate risk. Instead
of trying to anticipate which way interest rates will change, the investor
spreads out money invested in bonds over some period of time (e.g., 1 to
10 years, every 5 years from 5 through 30 years, etc.). Regardless of
which way interest rates move, the investor will have some bonds that
benefit and some that suffer.
g. dumbbell (barbell) portfolio
A barbell approach (the preferred term is barbell rather than dumbbell,
which is used in the Mayo text) is a more dramatic variation of the
laddering strategy. Very short-term and very long-term bonds are
purchased so that the bond portfolio is heavily weighted in both long- and
short-maturity issues, with no bonds in the middle. The purpose is similar
to that of the laddering approach.
h. immunization
Immunization is an approach that attempts to match the duration of a
bond portfolio with the duration of cash needs. It is used frequently by
financial institutions and retirement plans that have cash obligations that
can be calculated with some degree of precision as to their time
requirements.
i.
dedication
A dedicated portfolio is an immunization strategy that is even more
precise. When the timing of specific cash flows is certain, bonds are
purchased that will mature precisely when needed.
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7–3
Calculate the price, compound return, yield-to-maturity, yield-tocall, taxable-equivalent yield, or duration of fixed-income securities.
18. Your client asks what the market price of a particular bond should be.
The bond pays 12% coupon interest semiannually. The bond will
mature in 7 years and will pay a face value of $1,000. Comparable
bonds (bonds with similar maturities and of the same investment
grade) are yielding 14.9%. What should be the price of this bond?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
2
7, gold,
xP/YR
14.9
?
60
1,000
Answer: $876.54
19. Your client asks what the market price of a particular zero-coupon
bond should be. The bond will mature in 7 years and will pay a face
value of $1,000. Comparable bonds (bonds with similar maturities
and of the same investment grade) are yielding 14.9%. What should
be the price of this bond?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
2
7, gold,
xP/YR
14.9
?
0
FV
1,000
Answer: $365.69
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20. Calculate the following bond values.
a. What is the intrinsic value (price) of a newly issued bond with a
12% coupon rate, 30 years to maturity, and a $1,000 maturity
value when current market rates for comparable bonds are at
12%?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
2
30, gold,
xP/YR
12
?
60
1,000
Answer: $1,000
b. What will be the bond’s price one year after issue if market rates
drop to 9%?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
2
29, gold,
xP/YR
9
?
60
1,000
Answer: $1,307.38
c. What will be the bond’s price one year after issue if market rates
rise to 15%?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
2
29, gold,
xP/YR
15
?
60
1,000
Answer: $803.02
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21. A bond has a market price of $875. The bond pays 12% coupon
interest semiannually. The bond will mature in 7 years and will pay a
face value of $1,000.
a. What is the YTM (IRR) for this bond?
Set the calculator to “end.”
P/YR
2
N
I/YR
7, gold,
xP/YR
?
PV
PMT
FV
(875)
60
1,000
Answer: 14.94%
b. What is the YTM if the bond currently has a market price of
$1,200?
Set the calculator to “end.”
P/YR
2
N
I/YR
7, gold,
xP/YR
?
PV
PMT
FV
60
1,000
(1,200)
Answer: 8.19%
22. Your client recently purchased a zero-coupon bond for $630. It has a
$1,000 face value and matures in 6 years. What is the YTM for this
bond?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
2
6, gold,
xP/YR
?
(630)
0
FV
1,000
Answer: 7.85%
23. Your client purchased a bond for $950. The bond has a coupon rate of
11%, it matures in 17 years, and it is callable in 5 years at $1,110. What
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is the YTC for this bond?
Set the calculator to “end.”
P/YR
N
I/YR
2
5, gold,
xP/YR
?
PV
PMT
FV
55
1,110
(950)
Answer: 14.02%
24. Calculate the duration and expected price change for each of the
following bonds.
a. Market rate greater than coupon rate. Assume that the coupon is 6%,
that the market interest rate is 7%, that there are 16 years until
maturity, and that compounding is annual. Also assume that
interest rates are subsequently expected to fall by 50 basis points.
Duration
1 y ( 1 y ) n( c y )
y
c[(1 y)n 1] y
Duration
1 .07 (1 .07) 16(.06 .07)
.07
.06[(1 .07)16 1] .07
15.29
1.07 .16
15.29 4.87 10.42
.06(1.95) .07
Using a financial calculator and assuming annual compounding, the
market price of the bond at current market rates is computed to be
$905.53.
P D
118
y
1 y
PB
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P 10.42
.0050
905.53 $44.09
1 .07
The bond’s price will increase by approximately $44 if interest rates fall by
50 basis points. Note that, due to positive convexity, the0 actual price
increase will be greater than the amount computed based on duration
alone.
b. Coupon rate greater than market rate. Assume that the coupon is 8%,
that the market interest rate is 6%, that there are 22 years until
maturity, and that compounding is semiannual. Also assume that
interest rates are subsequently expected to rise by 60 basis points.
Duration
1 y ( 1 y ) n( c y )
y
c[(1 y)n 1] y
Duration
1 .03 1 .03 44.04 .03
.03
.04[1 .0344 1] .03
34.33
1.03 .44
34.33 10.74 23.59 periods 11.80 years
.04( 2.67) .03
Using a financial calculator and assuming semiannual compounding, the
market price of the bond at current market rates is computed to be
$1,242.54.
P D
y
1 y
P 11.80
PB
.0060
1,242.54 $85.41
1 .03
The bond’s price will decrease by approximately $85 if interest rates rise
by 60 basis points. Note that, due to positive convexity, the actual price
decrease will be less than the amount computed based on duration alone.
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c. Zero-coupon bond. Assume that the current market interest rate is
7%, that there are 18 years until maturity, and that compounding
is semiannual. Also assume that interest rates are subsequently
expected to fall by 30 basis points.
Duration
1 y ( 1 y ) n( c y )
y
c[(1 y)n 1] y
Duration
1 .035 1 .035 36.00 .035
.035
.00[1 .03536 1] .035
29.57
1.035 1.26
29.57 6.43 36.00 periods 18.00 years
0 .035
Note that no calculation is necessary for a zero-coupon bond since the
duration of a zero-coupon bond is the remaining term (18 years in this
problem).
Using a financial calculator and assuming semiannual compounding, the
market price of the bond at current market rates is computed to be
$289.83.
P D
y
1 y
P 18.00
PB
.0030
289.83 $15.12
1 .035
The bond’s price will increase by approximately $15 if interest rates fall by
30 basis points. Note that, due to positive convexity, the actual price
increase will be greater than the amount computed based on duration
alone.
25. IBM has a bond with a 7% coupon; the bond matures in 2025 for
$1,000. In 1998, the current price of the bond was 107 5/8 (107.625% of
par, or 1.07625 × $1,000 = $1,076.25). Assume that the bond had 26
years until maturity at that time. (Find the same bond in the
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newspaper on the day you work this problem and rework the
problem based on current market prices).
a. What is the YTM of the IBM bond?
Set the calculator to “end.”
P/YR
2
N
I/YR
26, gold,
xP/YR
?
PV
(1,076.25)
PMT
FV
35
1,000
Answer: 6.4%
b. Using the YTM computed in part a. of this question (rounded to
the nearest tenth), what is the duration of the IBM bond?
Duration
1 y ( 1 y ) n( c y )
y
c[(1 y)n 1] y
Duration
1 .032 1 .032 52.035 .032
.032
.035[1 .03252 1] .032
32.25
1.032 .156
32.25 6.71 25.54 periods 12.77 years
.145 .032
c. If the YTM is expected to fall 40 basis points in the next year, by
how much would the price of the IBM bond change?
P D
y
1 y
P 12.77
PB
.0040
1,076.25 $53.27
1 .032
The bond’s price will increase by approximately $53 if interest rates fall by
40 basis points. Note that, due to positive convexity, the actual price
increase will be greater than the amount computed based on duration
alone.
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26. What are the taxable-equivalent yields of municipal bonds with the
following tax-free yields for investors in the following marginal tax
brackets?
Tax-Free
Yield
TEY
28% Bracket
TEY
36% Bracket
TEY
40% Bracket
4%
5.56%
6.25%
6.67%
4.5%
6.25%
7.03%
7.5%
5%
6.94%
7.81%
8.33%
27. Jane Roberts owns a public purpose municipal bond that pays 5%.
Assuming she is in the 36% marginal tax bracket, what yield on
corporate bonds would be comparable to the yield on Jane’s current
investment?
TEY
5.00
7.81%
1 .36
28. Paulette Doyle’s marginal tax bracket is 40%. She is considering either
a corporate bond that pays 8% annually or a tax-exempt municipal
bond. What yield on the municipal bond would be comparable to the
yield on the taxable corporate bond?
8.00
Tax- free yield
1 .40
Tax-free yield = 8.00 (.60) = 4.8%
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29. If preferred stock does not have a required sinking fund or call
feature, it may be viewed as a perpetual debt instrument. How is the
intrinsic value of this type of preferred stock calculated?
The fixed annual dividend (D) of this type of preferred stock is divided by the
yield (k) being earned on comparable preferred stock of a similar grade.
P
D
k
30. Explain how to determine the intrinsic value of preferred stock that
has a finite life.
The intrinsic value of preferred stock that has a finite life is equal to the
present value of the dividend payments plus the present value of the amount
that is returned to the stockholder when the preferred stock is retired. The
keystrokes for this are the same as those for a bond valuation problem.
31. If a preferred stock pays an annual dividend of $5 and investors can
earn 12% on alternative, comparable investments, what is the price
that should be paid for this stock?
The price paid should be $41.67.
P
5
$41.67
.12
32. If the preferred stock in the previous question had a call feature, and
if investors expected the stock to be called for $100 after 12 years,
what price would be paid for this stock?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
1
12
12.0
?
5
100
Answer: $56.64
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7–4
Analyze the relationships among bond ratings, yields, maturities,
and durations to determine comparative price volatility.
33. Consider the following three bonds and determine which bond is
most susceptible to price fluctuations.
Bond 1: A-rated, pays a coupon of 11%, matures in 12 years
Bond 2: AA-rated, pays a coupon of 12%, matures in 7 years
Bond 3: BBB-rated, pays a coupon of 9%, matures in 15 years
Bond 3 is most susceptible because it has the lowest rating, longest maturity,
and lowest coupon rate. (Bonds with lower coupon rates are subject to greater
price fluctuations than higher coupon bonds. If interest rates rise, for example,
the cash flows are discounted at the higher rate, and the present value falls
more than it would in a higher coupon bond, in which more cash is provided in
the form of interest payments.) See Exhibit C at the end of this module for an
example of the market interest rate changing from 6% to 15%.
34. Consider the following bonds.
Bond 1: BBB-rated, pays a coupon of 9%, matures in 6 years
Bond 2: BBB-rated, pays a coupon of 9%, matures in 11 years
Bond 3: BBB-rated, pays a coupon of 7%, matures in 11 years
Bond 4: BBB-rated, pays a coupon of 7%, matures in 6 years
a. Determine whether Bond 1 or Bond 2 has more potential for price
fluctuation and give a reason why.
Bond 2 has more potential for price fluctuation because it has a longer
maturity.
b. Determine whether Bond 2 or Bond 3 has more potential for price
fluctuation and give a reason why.
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Bond 3 has more potential for price fluctuation because it has a lower
coupon rate.
c. Determine whether Bond 3 or Bond 4 has more potential for price
fluctuation and give a reason why.
Bond 3 has more potential for price fluctuation because it has a longer
maturity.
d. Determine whether Bond 1 or Bond 4 has more potential for price
fluctuation and give a reason why.
Bond 4 has more potential for price fluctuation because it has a lower
coupon rate.
35. Consider the following bonds.
Bond 1: AA-rated, pays a coupon of 9%, matures in 7 years
Bond 2: BB-rated, pays a coupon of 9%, matures in 12 years
Bond 3: BB-rated, pays a coupon of 9%, matures in 7 years
Bond 4: AA-rated, pays a coupon of 9%, matures in 12 years
a. Determine whether Bond 1 or Bond 3 has more potential for price
fluctuation and give a reason why.
Bond 3 has more potential for price fluctuation because it has a lower
rating.
b. Determine whether Bond 2 or Bond 4 has more potential for price
fluctuation and give a reason why.
Bond 2 has more potential for price fluctuation because it has a lower
rating.
c. Determine whether Bond 1 or Bond 4 has more potential for price
fluctuation and give a reason why.
Bond 4 has more potential for price fluctuation because it has a longer
maturity.
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d. Determine whether Bond 2 or Bond 3 has more potential for price
fluctuation and give a reason why.
Bond 2 has more potential for price fluctuation because it has a longer
maturity.
36. Consider the following bonds.
Bond 1: BBB-rated, pays a coupon of 8%, matures in 5 years
Bond 2: AA-rated, pays a coupon of 12%, matures in 5 years
Bond 3: BBB-rated, pays a coupon of 12%, matures in 5 years
Bond 4: AA-rated, pays a coupon of 8%, matures in 5 years
a. Determine whether Bond 1 or Bond 3 has more potential for price
fluctuation and give a reason why.
Bond 1 has more potential for price fluctuation because it has a lower
coupon rate.
b. Determine whether Bond 1 or Bond 4 has more potential for price
fluctuation and give a reason why.
Bond 1 has more potential for price fluctuation because it has a lower
rating.
c. Determine whether Bond 2 or Bond 4 has more potential for price
fluctuation and give a reason why.
Bond 4 has more potential for price fluctuation because it has a lower
coupon rate.
d. Determine whether Bond 2 or Bond 3 has more potential for price
fluctuation and give a reason why.
Bond 3 has more potential for price fluctuation because it has a lower
rating.
37. Review the Morningstar reports for the American Century-Benham
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bond fund and the Alliance Bond Corporate bond fund.
a. Which of the two funds would you expect to be more volatile, and
why?
The Alliance bond fund should be more volatile because its duration is 9.3
years, compared to a duration of 5.2 years for the American Century bond
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fund. Also, the Alliance fund has bonds with an average credit quality of
BBB, compared to an average quality of A for the American Century fund.
b. What evidence is there in the Morningstar report of the greater
volatility of the fund you chose?
The standard deviation and beta of the Alliance fund are higher than those
of the American Century fund.
Application B
Go to the Web site for Bonds Online (www.bondsonline.com), familiarize yourself with the site,
and find the Capital Markets Commentary and The Outlook sections of the site. Read the
commentary and examine charts on the yield curve and on sector comparisons so that you
understand the relationships among various bond characteristics and risk.
7–5
Assess how changes in variables affect bond risk and price volatility.
38. In Question 24, you computed the duration and expected price
change for several types of bonds. The characteristics and the
estimated percentage price changes of those bonds are summarized in
the following table (BP stands for basis points).
Bond
Coupon
Market Rate
Maturity
Duration
y
P (%)
A
6
7
16
10.42
50 BP
4.9%
B
8
6
22
11.80
60 BP
6.9%
C
0
7
18
18.00
30 BP
5.2%
What conclusions can you reach about bond risk and volatility
relative to different characteristics of these bonds and changes in
some of their variables?
Bond C, the zero-coupon bond, has a maturity that is between the two coupon
bonds, yet the bond has a relatively large price change, considering the
relatively small change in the market interest rate. Zero-coupon bonds have a
large degree of price volatility because they have no coupon payments to
reduce duration.
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Bond B has a maturity that is six years longer than that of Bond A, which
should result in Bond B having a larger duration than Bond A. Although this is
the case, this effect is somewhat muted because Bond B also has a larger
coupon than Bond A. Larger coupons reduce duration, while longer maturities
increase duration. The decrease in Bond B’s duration due to its higher coupon
does not totally offset its longer maturity, so Bond B’s duration is, in fact,
higher than Bond A’s duration. The higher duration of Bond B increases the
bond’s price volatility over that of Bond A. Bond A’s duration is also helped
somewhat by the bond’s higher market interest rate. The higher market rate
could be due to the fact that Bond A may have a lower credit rating than Bond
B, making it even more risky than its duration alone indicates.
39. The following table shows characteristics of four bond funds. The
funds are listed in order of ascending credit quality. The first fund,
Merrill Lynch Corp Hi-Income A, is a high-yield bond fund. The last
three bonds are also ranked by decreasing average maturity.
Review the data in the table and explain how each fund’s risk and
volatility is affected by the differences in variables.
Fund
Average
Credit
Quality
Average
Weighted
Coupon
Average
Maturity
Average
Standard Effective
Deviation Duration
Merrill Lynch Corp Hi-Income A
B
7.8
NA
3.74
3.8
Alliance Bond Corp Bond A
BBB
8.1
23.1
7.84
9.3
Merrill Lynch Corp Invmt Gr A
A
7.3
12.5
3.92
5.9
Intermediate Bond Fd America
AA
7.9
4.4
2.1
3.3
Source: Morningstar, Inc., Morningstar Principia Pro Plus for Mutual Funds. Chicago: Morningstar, Inc., 1998.
The difference in credit quality seems to have little effect on the bonds’
coupons. All are within one percentage point of one another. The difference in
coupon rates does not appear to have any measurable effect on risk.
As the average maturity of the non-high-yield bond funds decreases, standard
deviation and duration also decrease. Therefore, both risk and volatility
decrease as average maturity decreases. Although the credit rating of the
high-yield bond fund is low, the standard deviation and duration of the fund is
relatively low (in the same range as the A and AA bond funds). Therefore,
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although this fund has some potential unsystematic risk, its volatility risk is not
significant. Assuming sufficient diversification of bonds within the fund, the
unsystematic risk may also be minimal.
40. If you want to ensure that $40,000 is available in 13 years when your
child is about to enter college, would you select a zero-coupon bond
that matures in 13 years or a coupon bond that matures in 13 years?
Why did you select the one you did?
The appropriate bond is the one with duration close to the duration of the goal.
The goal’s duration is 13 years. The appropriate bond is the zero-coupon
bond, since a zero’s duration is equal to its maturity. The duration of a coupon
bond is less than its maturity. The coupon bond’s duration must be less than
13 years, since its maturity is 13 years.
41. If you have a conservative client who is concerned about fluctuating
bond prices, but who wants to have relatively high income from a
bond portfolio, how would you construct a bond portfolio so that you
can help the client resolve both of these apparently conflicting
concerns?
You would construct a laddered bond portfolio. Although the exact structure
could take on any number of formats, one structure might be to purchase
bonds with 3-, 6-, 9-, 12-, 15-, and 18-year maturities. The longer maturities
would have higher coupons providing a high income, but they would have
significant interest-rate risk. The shorter maturities would not provide much
income, but their price fluctuations would be small compared to the
fluctuations of the 15- and 18-year bonds. The overall portfolio would have an
above-average income and a below-average price volatility.
42. Referring to Figure 17.2 and Figure 17.3 in Chapter 17 of the Mayo
text, what conclusions can you draw concerning yields and prices of
state and local government bonds?
Yields on municipal bonds have experienced considerable fluctuation over the
last two decades. For example, the yields on Baa-rated municipal bonds have
been higher than those of Aaa-rated municipal bonds, but the yield spread
between the two has not been constant. During periods of higher interest
rates, the spread widened, reflecting the fact that investors perceived higher
risk on the lower-rated bonds as rates rose. The yields on U.S. government
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bonds exceeded the yields on municipal bonds during the 20-year period, but
the yield differential narrowed in 1986 due to tax reform and lower income tax
rates that diminished the attractiveness of municipal bonds. An increase in
yields means the prices of the bonds fall; a decrease in yields means the
prices rise.
43. Summarize all the relationships between price, coupon, maturity,
interest rates, and duration that you have discovered in this module.
Bond prices and interest rates are inversely related.
Long-term bonds are more affected by interest rate changes than are
short-term bonds (i.e., they have more price volatility).
Lower-coupon bonds are more affected by interest rate changes than are
higher-coupon bonds (i.e., they have more price volatility).
Lower-rated bonds have more price volatility than higher-rated bonds.
Bonds with longer durations are more volatile than bonds with shorter
durations.
There is a positive correlation between maturity and duration.
There is an inverse relationship between the market interest rate (YTM)
and duration.
There is an inverse relationship between coupon rate and duration.
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7–6
Evaluate investor profiles to recommend appropriate fixed-income
securities for purchase.
44. Robert Berens, age 65, is retiring and has $150,000 to invest. He is
interested in purchasing fixed-income securities to provide for his
income needs during retirement. Robert will not have any other
substantial income, and he will be in the 15% marginal income tax
bracket. He has invested in bonds in the past, and he plans to be
actively involved in this investment.
What kind of fixed-income security is appropriate for Robert, and
why? (Consider type, risk rating, marginal tax bracket, term, and
other relevant factors.)
A high-grade corporate bond (AA or AAA), a Treasury note or bond, or a
federal agency security like a Ginnie Mae would be appropriate. All of these
can be bought at par, pay periodic income, and have good marketability.
Because he is in a low marginal tax bracket, taxable securities would most
likely provide more after-tax income than municipal bonds. An intermediate
term of 7 to 15 years would give adequate yield with only moderate interest
rate risk.
45. John Bloom, age 49, wants to take early retirement next year when he
turns 50. He wants to invest $200,000 in a fixed-income security to
provide him with additional income. He estimates that he will be in
the 31% marginal tax bracket. He has invested previously, and he is
willing to be aggressive with this investment to increase his return.
What kind of fixed-income security is appropriate for John, and why?
(Consider type, risk rating, marginal tax bracket, term, and other
relevant factors.)
Because of his high tax bracket, municipal revenue bonds are appropriate,
assuming their equivalent yield exceeds the yield of corporate bonds.
Purchasing bonds with lower ratings (BB or BBB) would be consistent with his
aggressive attitude of attempting to increase his return while realizing
additional income from this investment. If rates fall, longer maturities may be
appropriate to provide capital gain potential.
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46. Kent Walters, age 32, has $40,000 to invest in a fixed-income security.
He has invested in various types of bonds for 10 years, he considers
himself to be an aggressive investor, and he is in the 28% marginal
income tax bracket. His primary goal is capital appreciation; income is
a secondary consideration. Kent’s financial planner has presented the
following securities and their before-tax yields.
a. 15-year, BB-rated, noncallable corporate bonds trading near par
with a yield of 11.8%
b. 20-year, A-rated, discount, public purpose, callable general
obligation municipal bonds with a taxable-equivalent yield of
12.2%
c. 10-year, A-rated, premium, callable, sinking fund, corporate
bonds with a yield of 9.5%
d. Treasury bills with a yield of 8.0%
Which one of these fixed-income securities would be an appropriate
choice for Kent, and why?
Investment “b.” is an appropriate choice. On an after-tax basis, it has the
highest return (8.80%), and when compared to the BB-rated bonds with an
8.50% after-tax return, the municipal bonds have a higher after-tax yield with a
better risk rating. Compared to the A-rated corporate bonds, the municipal
bonds are less likely to be called since they are trading at a discount. There is
no reason for the investor to seek the security of Treasuries (with a 5.76%
after-tax return), given his aggressive risk profile. Also, since his primary goal
is capital appreciation, the discounted, 20-year bond is most likely to provide
capital gains if interest rates decrease.
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47. Kathy Connelly, age 20, is just starting college and needs to invest
$25,000 in fixed-income securities. She is in the 15% tax bracket and
plans to use the interest income and principal as needed to pay her
college expenses for the next four years. She is looking for a low-risk
investment, and she knows she must receive principal periodically
from these securities. The following securities are available to Kathy
at the before-tax yields indicated.
a. BB-rated, public purpose, municipal revenue bonds with an aftertax yield of 7.0%
b. 12-year, B-rated, discount, callable corporate bonds with a beforetax yield of 8.8%
c. eight-year Treasury notes with a before-tax yield of 6.8%
d. AA-rated, noncallable, five-year corporate bonds with a before-tax
yield of 8.5%
Which one of these securities would be an appropriate choice for
Kathy, and why?
Investment “d.” is most appropriate. Because Kathy is in a marginal tax
bracket of 15%, the municipal bonds, which have a poor risk rating, result
in a taxable equivalent return of only 8.2%. The Treasury notes are too
long term, and they subject her to too much interest rate risk. The B-rated
bonds are too speculative, and they also have too long of a time frame.
48. Answer the following questions about selecting bonds for client
portfolios.
a. What sort of characteristics would you look for in a bond chosen
for a client with a high risk tolerance?
a bond with a high duration, a low or zero coupon, a long maturity, and a
relatively low credit rating
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b. What sort of characteristics would you look for in a bond chosen
for a client with a moderate risk tolerance?
a bond with a moderate duration (5 to 10 years) and an intermediate
maturity (7 to 15 years), a coupon that is near current market rates, and a
low credit rating (but one that is still considered investment grade, A or
BBB)
c. What sort of characteristics would you look for in a bond chosen
for a client with a low risk tolerance?
a bond with a low duration and a short maturity, a coupon that is at current
market rates, and a high investment-quality rating
d. If you believe that interest rates will decline sharply in the future,
what bond characteristics would you search for?
bonds that have long maturities and high durations and that have low (or
zero) coupons
e. If you believe that interest rates will rise sharply in the future,
what bond characteristics would you search for?
bonds that have short maturities and low durations and that have (if
available) high coupon rates
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7–7
Calculate the conversion value, investment value, investment
premium, conversion premium, and downside risk of convertible
securities.
49. Janice Carlysle owns a ZZT Corporation convertible bond. The bond
has a 9.5% coupon rate that is paid semiannually; the bond matures in
8 years. Comparable debt (with the same rating and maturity date) is
yielding 11%. Janice’s bond is convertible at $27 a share, the current
market price of ZZT common stock is $35, and the bond sells for
$1,400.
a. What is the conversion value of the bond?
The conversion value is $1,296.30, which is computed as follows.
Cs
FV
1,000
Ps
35 $1,296.30
Pe
27
b. What is the investment value of the bond?
Set the calculator to “end.”
P/YR
2
N
8, gold,
xP/YR
I/YR
PV
PMT
11
?
47.50
FV
1,000
Answer: $921.53
c. What is the bond’s investment premium?
The investment premium is $478.47, the difference between the bond’s
market price of $1,400 and the bond’s investment value of $921.53.
d. What is the bond’s conversion premium?
The conversion premium is $103.70, the difference between the bond’s
market price of $1,400 and the bond’s conversion value of $1,296.30.
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e. What is the downside risk percentage of the bond?
The downside risk is 34.2%, which is computed as follows.
1,400 921.53
1,400
34.2%
If the price of the underlying stock falls substantially, the maximum that
the price of the bond can fall is about 34%.
50. James Perry owns a QV, Inc., convertible bond. The bond has a
coupon rate of 10% that is paid semiannually; the bond matures in 12
years. Comparable debt yields 8% currently. His bond is convertible
into 24 shares of stock. The current market price of QV common stock
is $34, and the bond sells for $1,200.
a. What is the conversion value of the bond?
The conversion value is $816.00, which is computed as follows. Note that
the conversion ratio is given and does not have to be computed.
Cs
FV
P s 24 34 $816.00
Pe
b. What is the investment value of the bond?
Set the calculator to “end.”
P/YR
N
2
12, gold,
xP/YR
I/YR
PV
PMT
FV
8
?
50
1,000
Answer: $1,152.47
c. What is the bond’s investment premium?
The investment premium is $47.53 (the difference between the bond’s
market price of $1,200 and the bond’s investment value of $1,152.47).
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d. What is the bond’s conversion premium?
The conversion premium is $384.00 (the difference between the bond’s
market price of $1,200 and the bond’s conversion value of $816.00).
e. What is the downside risk percentage of the bond?
The downside risk is 4.0%, which is computed as follows.
1,200 1,152.47
1,200
4 .0 %
If the price of the underlying stock falls substantially, the maximum that
the price of the bond can fall is less than 4%.
51. Assume that a convertible bond has a face value of $1,000 and that it
is selling in the market for $890. Its conversion price is $50 per share.
The underlying common stock is selling for $38 per share. The bond
pays $40 semiannually in interest and matures in 20 years. The market
interest rate on comparable bonds is 12%.
a. What is the bond’s conversion ratio?
The conversion ratio is the face value divided by the conversion price.
CR
1,000
20 shares
50
b. What is the conversion value?
The conversion value is the conversion ratio times the market price of the
stock.
CV 20 38 $760
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c. What is the investment value of the convertible bond?
Set the calculator to “end.”
P/YR
N
I/YR
PV
PMT
FV
2
20, gold,
xP/YR
12
?
40
1,000
Answer: $699.07
d. Express the downside risk as a percentage.
The downside risk is 21.5%, which is computed as follows.
890 699
890
21.5%
52. An investor can obtain 1.5 shares of common stock through
conversion of 1 share of preferred stock. The price of the common
stock is $35. The convertible preferred stock has no maturity date and
pays an annual dividend of $3. The yield on comparable
nonconvertible preferred stock is 12%.
a. What is the conversion value of this convertible preferred stock?
The conversion value is $52.50, which is computed as follows.
1.5 $35 $52.50
b. What is the investment value of this convertible preferred stock?
The investment value is $25, which is computed as follows.
$3
$25
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7–8
Analyze the relationships among conversion value, investment
value, and market value of convertible securities.
53. In the following figure, what does the shaded area represent?
The shaded area represents the premium that an investor might pay to
purchase a convertible bond. Since a convertible bond is, in essence, a
straight bond plus an option contract, an investor usually pays more for such a
bond than its value as a straight bond. The shaded area also represents the
downside risk of the bond. The premium will be small until the value of the
underlying stock rises above the intersection of the conversion value and the
investment value. Above that point, the convertible bond will act more like a
stock than a bond.
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54. Under what circumstances does a convertible bond become an
inferior investment?
A convertible bond becomes an inferior investment (to a comparable
nonconvertible bond) when the price of the common stock does not rise (the
nonconvertible bond earns more interest). On the other hand, when compared
to the stock (when the stock’s price rises rapidly), the convertible bond seems
inferior to the stock in terms of the stock’s larger gain. In other words, the very
sources of a convertible bond’s attractiveness (i.e., potential capital growth
plus interest income) can also be the sources of its lack of appeal (i.e., inferior
growth and inferior interest).
55. A convertible bond with an 8% coupon has an investment value of
$900 and a conversion value of $1,150 when the market interest rate is
9%.
a. Would you expect the market value of the convertible bond to be
(1) less than the bond’s investment value, (2) between the
investment value and the conversion value, or (3) greater than the
conversion value? Explain your answer.
The market value should be greater than the conversion value. A
convertible bond usually will sell at a premium to the higher of investment
value or conversion value. Since this bond’s conversion value is higher
than its investment value, the market value will be greater than $1,150.
b. Is the conversion price below or above the market price of the
common stock? Explain your answer.
The conversion price is below the market price. If the market price were equal
to the conversion price, then the market price would equal $1,000. Since the
conversion value exceeds $1,000, the stock must be selling at a price that is
higher than the conversion price. Because interest rates would normally have
risen since the bond was issued, one would expect that the stock’s price has
fallen. Apparently, this company’s profits and earnings have increased, and the
company has profited in spite of higher interest rates.
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c. Is the downside risk less than or greater than $100? Explain your
answer.
The downside risk must be greater than $100 because the bond’s
investment value is $100 less than its maturity value. Downside risk is the
difference between the convertible bond’s current market price and its
investment value. Since the current market price exceeds $1,150, the
downside risk must be greater than $250.
Application C
Use The Wall Street Journal or Barron’s to find a corporate bond that is convertible (identified by
“cv” in the current yield column). Then go to that company’s Web site, click on its most recent
annual report, and look for the details of the convertible issue in the long-term debt footnote to
the financial statements. If you have trouble finding a company with a convertible bond, try
Hilton Hotels.
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6
References
BrainPower Technologies, Inc., <www.bondsonline.com> (July 2002).
Dow Jones & Company, Inc., <www.wsj.com> (July 2002).
Downes, John, and Jordan Goodman, Dictionary of Finance and Investment
Terms. Hauppauge, NY: Barron’s Educational Series, Inc., 1995.
Mayo, Herbert B., Investments: An Introduction, 7th edition. Mason, OH:
South-Western, 2003.
Morningstar, Inc., Morningstar Principia Pro Plus for Mutual Funds.
Chicago: Morningstar, Inc., 1998.
References
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Exhibits
See the following pages for exhibits.
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Exhibit A
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Exhibit B
Exhibits
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Exhibit C
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Exhibits
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