Chapter 11
Bond Valuation

Outline
Learning Goals
I.
The Behavior of Market Interest Rates
A) Keeping Tabs on Market Interest Rates
B) What Causes Interest Rates to Move?
C) The Term Structure of Interest Rates and Yield Curves
1. Types of Yield Curves
2. Plotting Your Own Curves
3. Explanations of the Term Structure of Interest Rates
a. Expectations Hypothesis
b. Liquidity Preference Theory
c. Market Segmentation Theory
d. Which theory is right?
4. Using the Yield Curve in Investment Decisions
Concepts in Review
II.
The Pricing of Bonds
A) Annual Compounding
B) Semiannual Compounding
Concepts in Review
III. Measures of Yield and Return
A) Current Yield
B) Yield-to-Maturity
1. Using Semi-Annual Compounding
2. Yield Properties
3. Finding the Yield on a Zero
C) Yield-to-Call
D) Expected Return
E) Valuing a Bond
Concepts in Review
Chapter 11
Bond Valuation
197
IV. Duration and Immunization
A) The Concept of Duration
B) Measuring Duration
1. Duration for a Single Bond
2. Duration for a Portfolio of Bonds
C) Bond Duration and Price Volatility
D) Uses of Bond Duration Measures
1. Bond Immunization
Concepts in Review
V.
Bond Investment Strategies
A) Passive Strategies
B) Trading on Forecasted Interest Rate Behavior
C) Bond Swaps
Concepts in Review
Summary
Putting Your Investment Know-How to the Test
Discussion Questions
Problems
Case Problems
11.1. The Bond Investment Decisions of Kelley and Erin Coates
11.2. Grace Decides to Immunize Her Portfolio
Excel with Spreadsheets

Key Concepts
1.
The important role that interest rates play in the bond investment process and the basic determinants
of market rates.
2.
The term structure of interest rates and yield curves.
3.
Fundamentals of bond valuation, including basic measures of yield and return.
4.
The concept of duration and its measurement; how duration is applied in immunizing bond portfolios.
5.
Various types of bond investment programs and the ways debt securities can be used by investors.
Employment of bond ladders is a passive strategy, whereas buying high duration bonds prior to
interest rate drops would be a more active and risky strategy.
198
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition

Overview
1.
Interest rates are an integral component of the bond valuation process. Some class time should be
spent discussing the economics of interest rates. The various forces that drive interest rates should be
covered next. In this context, the instructor can introduce the term structure of interest rates.
2.
The text then presents three different explanations of the term structure of interest rates: the
Expectations Hypothesis, the Liquidity Preference Theory, and the Market Segmentation Theory. The
discussion of this important topic should include yield curves, how they are plotted, and their use in
making investment decisions.
3.
The next section discusses the bond valuation process. It shows how, given the market rate of interest
and other details regarding the bond (such as the maturity, coupon, and face value), it is possible to
compute the “correct” price of the bond. An example showing this computation should be worked out
in class, including how fluctuations in the market interest rates induce changes in the price of the
bond. The magnitude of price changes depends on the amount of change in the market interest rate, as
well as on the maturity and coupon of the bond.
4.
The concepts of bond yields and returns, along with the computation and use of current yield,
promised yield, yield-to-call, and expected yield, are discussed next. The instructor may wish to
demonstrate the trial-and-error procedure, to calculate the yield-to-maturity using tables. It is also
important to emphasize that what matters to investors is the return from the bond, not its yield.
5.
Bond duration is one of the most important concepts in bond valuation and investing. After
demonstrating the shortcomings of yield-to-maturity, the concept and measurement of duration can
be illustrated. In this regard, the instructor can work out an example to illustrate how duration and
modified duration aid investors in gauging a bond’s price volatility.
6.
Bond immunization is presented next. This technique preserves the value of a bond portfolio. Bond
immunization involves constructing a bond portfolio with a weighted average duration that matches
the investor’s investment horizon.
7.
Bond investment strategies can be either active or passive. Passive investment strategies include buyand-hold and bond ladders. Trading on interest rate swings and bond swaps are considered active
strategies. The instructor might point out the advantages and disadvantages (risks) of each technique.
8.
One interesting teaching strategy is to start out with a bond priced at par and show the decine/rise
from an interest rate decrease/increase of the same magnitude. Due to convexity, bond prices will rise
faster than they decline!

Answers to Concepts in Review
1.
There is no single market rate of interest applicable to all segments of the bond market. Instead, a
series of market yields exists for a variety of market instruments. In general, the interest rate on a
particular bond issue will depend on a variety of issue characteristics, including the type of issuer, the
amount of tax exposure, its call feature, coupon, and time to maturity. The investment implications of
such a market are simple: Investors can pick the segments which have the return, risk, and other
characteristics that best meet their investment needs. For example, they can move from agency bonds
with a fairly low return (and risk) to corporate bonds and receive a higher return. In short, it opens up
the investment alternatives and investment opportunities for investors.
Chapter 11
Bond Valuation
199
2.
The behavior of interest rates is perhaps the single most important element in determining the level of
return from a bond investment program. Interest rates affect the level of current income earned by
conservative investors, as well as the amount of capital gains generated by aggressive bond traders.
Whereas conservative investors are primarily concerned with the level of interest rates, aggressive
investors are interested chiefly in movements in interest rates (the amount of interest rate volatility).
Some of the major determinants of interest rates include: inflation, the money supply, the demand for
loanable funds, the amount of deficit spending by the Federal Government, and actions of the Federal
Reserve (like changes in the discount rate). Individual investors can monitor interest rates and
formulate interest rate expectations on an informal basis through the use of reports obtained from
their brokers, from investor services (e.g., S&P’s Creditweek), and/or by following columns/articles
in such business and financial publications as The Wall Street Journal or Business Week.
3.
The term structure of interest rates is the relationship between the interest rate or yield and the time to
maturity for any class of similar risk securities. The yield curve is just a graphic representation of the
term structure of interest rates at a given point in time. To plot a yield curve, you need to know the yield
to maturity for different maturities of similar risk bonds. As market conditions change, the yield curve’s
shape and location also change.
The upward-sloping yield curve indicates that yields tend to increase with longer maturities. The
longer a bond has to go to maturity, the greater the potential for price volatility and the risk of loss.
Thus, investors require higher yields on longer maturity bonds. Flat yield curves indicate that yields
will be the same across maturities. Given that longer-term bonds have more default and maturity rate
risk, a flat yield curve implies that inflation rates are expected to decline.
4.
Analyzing the changes in yield curves over time provides investors with information about future
interest rate movements and how they can affect price behavior and comparative returns. For
example, if over a specific time period, the yield curve begins to rise sharply, it usually means that
inflation is increasing. Investors can expect that interest rates, too, will rise. Under these conditions,
most seasoned bond investors would turn to short or intermediate (three to five year maturities). A
downward-sloping yield curve would signal that rates have peaked and are about to fall.
Differences in yields on different maturities at a particular point in time, or the “steepness” of the
curve, is an indication that long-term rates are likely to fall somewhat to narrow the spread, providing
an incentive to invest in longer-term securities. Steep yield curves are generally viewed as a signs that
long-term rates are near their peak.
Even among longer-term maturities, the spread between different longer-term maturities should be
considered before making a decision to invest. For example, if the spread between 10 and 30-year
maturities is not large enough (say, less than 20 basis points), then the investor should favor the
10-year bond, because he would not gain enough to compensate for investing in the much riskier
30-year maturity. In any case, the investor would have to consider his or her own risk tolerance to
determine whether the risk premium was sufficient for the additional risk of buying longer-term
securities.
Bond prices are driven by market yields. In the marketplace, the appropriate yield at which the bond
should sell is determined first, and then that yield is used to find the price of the bond. The yield is a
function of certain market and economic forces, such as the risk-free rate and inflation, as well as key
issue and issuer characteristics, such as the maturity of the issue and agency rating assigned to the
bond.
You cannot value a bond without knowing its market yield, which functions as the discount rate in the
bond valuation process.
5.
200
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
6.
Bonds are usually priced using semiannual compounding because in practice, most bonds pay interest
every six months. Semiannual compounding makes discounting of semiannual coupon payments
more precise, resulting in more accurate valuation. However, using annual compounding simplifies
the valuation process a bit and does not make very much difference in value. With higher coupons
and longer maturities, the difference increases more. Bonds offering semiannual payments will be
priced higher.
7.
Current yield is a measure of a bond’s current income. It is the amount of current income a bond
provides relative to its prevailing market price. Yield-to-maturity is a more complete measure and
evaluates both interest income and price appreciation. Yield-to-maturity indicates the fullycompounded rate of return earned by an investor, given that the bond is held to maturity and all
principal and interest payments are made in a prompt and timely fashion.
Promised yield is the same as yield-to-maturity. Promised yield is computed assuming the bond is
held to maturity and the coupon cash flows are reinvested at the bond’s computed promised yield.
Realized yield is the rate of return an investor can expect to earn by holding a bond over a period of
time that is less than the life of the issue. Realized yield is used by bond traders who trade in and out
of bonds over short holding periods.
8.
When we are dealing with semi-annual cash flows, to be technically correct, we should find the
bond’s “effective” annual yield. But the market convention for finding the annual yield is to double
the semiannual yield. This practice produces what the market refers to as the bond-equivalent yield.
Thus, given a semi-annual yield of 4%, according to the bond-equivalent yield convention, the annual
rate of return of this bond if held to maturity is 8%. This is also the same as the bond’s promised yield
or yield-to-maturity.
9.
The reinvestment of interest income is an important consideration, because it is this rate that an
investor must earn on each of the interim cash throw-offs in order to realize a return equal to or
greater than the promised yield on a bond. As cash is received from interest income, the equation for
promised yield implicitly assumes this cash payment will be reinvested at a rate of return equal to the
issue’s promised yield; failure to do so means the investor will generate a realized yield that is less
than promised.
10. Duration is a measure of bond price volatility. It captures both price and reinvestment risks in a
single measure and indicates how a bond’s price will react to different interest rate environments. It is
the effective maturity of a fixed-income security. On the other hand, the bond’s actual maturity does
not consider all of the bond’s cash flows nor does it consider the time value of money. Duration is a
far superior measure of the effective timing of a bond’s cash flows, because it explicitly considers
both the time value of money and the bond’s coupon and principal payments.
When the market undergoes a big change in yield, duration will understate price appreciation when
rates fall and overstate the price decline when rates increase. Modified duration is used to overcome
this problem by linking interest rate changes to changes in bond price. First, you can compute the
modified duration using the bond’s computed duration and the computed yield-to-maturity. Then, the
change in bond price based upon a change in interest rates can be computed as follows:
Percent change in bond price  –1  Modified duration  Change in interest rates
Chapter 11
Bond Valuation
201
11. Market interest rate changes have two effects: the price effect and the reinvestment effect, which
occur in opposite directions. When a bond portfolio is immunized, these two effects exactly offset
each other and leave the value of the portfolio unchanged. This happens when the weighted average
duration of the bond portfolio is exactly equal to the desired investment horizon. If a portfolio is
constructed and continuously rebalanced such that the weighted average duration is equal to the
desired investment horizon at any particular point in time, then the portfolio is said to be immunized
from the effects of interest rate changes.
Bond immunization allows an investor to derive a specified rate of return from bond investments
regardless of what happens to market interest rates over the course of the holding period. That is, the
investor’s bond portfolio is “immunized” from the effects of changes in market interest rates over a
given investment horizon.
Portfolio immunization is not a passive investment strategy; it requires continual portfolio
rebalancing on the part of the investor in order to maintain a fully-immunized portfolio. The
composition of the portfolio should change every time interest rates change, and also with the passage
of time.
12. Bond ladders are a passive investment strategy whereby an equal amount of money is invested in a
series of bonds with staggered maturities. Suppose an investor wants to confine her investing to fixed
income securities with maturities of 10 years or less. She could set up the ladder by investing in
roughly equal amounts of 3-, 5-, 7-, and 10-year issues. When the 3-year issue matures, the proceeds
would be reinvested in a new 10-year note. Similar rollovers would occur whenever a bond matures.
Eventually, the investor would hold a full ladder of staggered 10-year notes. Rolling into new 10-year
issues every two or three years allows the investor to do a kind of dollar cost averaging and thereby
lessen the impact of swings in market rates.
Tax swaps involve replacement of a bond with a capital loss with a similar security. By selling the
bond with the capital loss, an investor can offset a capital gain generated in another part of the
portfolio and thereby reduce the overall tax liability. Identical issues cannot be used for this kind of
swap; the IRS will rule such swaps as “wash sales” and therefore disallow the capital loss.
13. An aggressive bond investor would employ the highly risky forecasted interest rate behavior strategy.
The intent of this strategy is to take advantage of interest rate swings by timing the market. Usually
these swings are short-lived, so aggressive bond traders will try to magnify their returns by trading on
margin. These investors try to generate capital gains when interest rates are expected to decline and to
preserve capital when an increase in interest rates is expected.
14. The interest sensitivity of a bond determines how much the bond’s price will fluctuate for a given
change in interest rates. Obviously, when rates drop, bond traders want to capitalize on this and as
such, require issues that will respond to these interest rate changes. Bonds with longer maturities
and/or lower coupons respond more vigorously to changes in market rates; therefore, they undergo
greater price swings. High-grade issues are widely used by active bond traders since these issues are
generally more interest-sensitive than lower-rated bonds—for example, market behavior is such that a
triple A corporate will generally be far more responsive to interest rates than a triple B issue. A
deteriorating economy will result in a decline in the demand for money and hence interest rates, but it
might cause more default risk to the holder of the triple B bond issue.
202

Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
Suggested Answers to Investing in Action Questions
There’s More to Bond Returns than Yield Alone (p. 472)
What is the difference between bond yield and its return?
Answer:
The total return on a bond consists of the initial yield, interest on reinvested interest, and bond price
changes. Only in the case of short-term investments is yield a good measure of return. For longer-term
bonds, the most important factor is the interest earned on one’s investment. Over a single year, a key to
return is the change in price of one’s bond. Bond price changes can swamp coupon rates and bond yields.
Bonds with high coupon payments are normally priced at a premium, which reduces the yield on these
investments. Furthermore, one has to be cognizant of changes in bond price arising from shift in interest
rates, firm quality, and other factors.
Getting Started in Bond Investment (p. 486)
(a) What are the advantages and disadvantages of buying bonds directly?
(b) What are the advantages and disadvantages of buying bond mutual funds?
Answers:
(a) Individual bond investment advantages include knowledge of the interest rate and par value payments
and timing. (Zero coupon bonds dates can be purchased at a discount and can be chosen so that their
maturity matches your investment horizon.) However, it is difficult to find information on individual
bonds. An individual bond might be called, while bond mutual funds simply reinvest the proceeds in
other bonds without the need for investor activity. Research has shown that it takes $50,000 invested
in bonds to be well diversified.
(b) Purchase of a bond mutual fund results in investment in a diversified set of bond meeting an
investment objective. Bankruptcy on the part of some issuers is unlikely to wipe out an investor’s
position. Bond fund managers also have more information resources at their fingertips in order to
assess the issuer ability to meet payment obligations. However, there are load fees, management fees,
and transaction fees. In addition one is unable to benefit from positive results of a single issuer (e.g.,
improved quality resulting in a lower required rate of return and hence a higher bond price). Since
mutual funds are continually buying new bonds, you cannot predict a specific cash flow on a future
date. As with individual bonds, those funds with higher durations will be more sensitive to interest rate
changes. While the duration of an individual bond declines with time as a bond approaches maturity,
the duration of a bond fund is more constant.

Suggested Answers to Discussion Questions
1.
Expectations hypothesis: The yield curve reflects investor expectations above all else. Future
behavior of interest rates with respect to the present is affected most by expectations regarding
inflation. Higher expected inflation requires higher interest rates today. The result is an upwardsloping yield curve. To produce a downward-sloping yield curve under this hypothesis, the expected
future inflation would be lower, but the current rates would remain higher.
Chapter 11
Bond Valuation
203
Liquidity preference theory: Long-term bond rates should be higher than shorter-term due to the
condition there are more liquid market rates in the short-term. Uncertainty increases over time
causing the demand for a higher risk premium (bond interest rate). This theory expects upwardsloping yield curves. Downward-sloping curves would not occur in this theory since it would
contradict the basic notion that uncertainty increases with time and the risk premium adjusts
accordingly.
Market Segmentation theory: The debt market is segmented according to length of maturity and
preferences. An equilibrium exists in the short-term between suppliers and demanders of funds. There
are different inhabitants in each segment with different motivations. In the short-term, banks
predominate, but in the long-term, life insurance and real estate firms determine the equilibriums. In
this theory yield curves may be either upward- or downward-sloping, as determined by the general
relationship between rates in each market segment.
2.
Answers will vary with each student and the conditions prevailing in the markets at the time the
assignment is made.
3.
(a)
(b)
(c)
(d)
4.
(a) An aggressive investor would be more concerned with capital gains, that is, price appreciation.
In this process, we would advise her to invest in new companies that may be found on the OTC
markets. These companies have higher risk but higher returns. We might also suggest a more
speculative strategy such as margin buying or short selling dependent on the current conditions in
the markets.
(b) A very conservative investor might include only investment grade corporate, government, and
municipal bonds in her portfolio. This would produce a minimum of market losses and risk.
(c) (1) an insurance company that must rely on predictable income streams
(2) an investor who wants to maximize yield
(3) an individual or institution that is not as conservative
(4) an investor who has a buy-and-hold investment strategy and a specific date at which the
funds will be needed
5.
Answers will vary with each student.

Solutions to Problems
1.
Bond A: $1,000 par value, 5% coupon, 15-year life, priced to yield 8%
Bond B: $1,000 par value, 7.5% coupon, 20-year life, priced to yield 6%
Bond A, with a 5% coupon and an 8% yield, must sell at a discount; it will be priced below $1,000.
Bond B, on the other hand, is a premium bond (its coupon is greater than its yield) and it will sell at a
much higher price than Bond A:
Price of Bond A  $50  PVIFA8%,15 yrs.  $1,000  PVIF8%,15 yrs.
 $50  8.560  $1,000  0.315
 $428  315  $743
Price of Bond B  $75  PVIFA6%,20 yrs.  $1,000  PVIF6%,20 yrs.
 $75  11.470  $1,000  0.312
Higher yields lead to shorter durations, lower yields lead to longer durations.
Longer maturities mean longer durations, shorter maturities mean shorter durations.
Higher coupons result in shorter durations, lower coupons result in longer durations.
Yield-to-maturity has increased.
204
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
 $860.25  312  $1,172.25
Chapter 11
2.
Bond Valuation
205
Bond prices using semi-annual compounding:
(a) 10½%, 15 years, YTM of 8%
Price  $52.50  PVIFA4%,30 yrs.  $1,000  PVIF4%,30 yrs.
 $52.50  17.292  $1,000(0.308)
 $907.83  $308  $1,215.83
(b) 7%, 10 years, YTM of 8%:
Price  $35  PVIFA4%,20 yrs.  $1,000  PVIF4%,20 yrs.
 $35  13.590  $1,000(0.456)
 $475.65  $456  $931.65
(c) 12%, 20 years, YTM of 10%:
Price  $60  PVIFA5%,40 yrs.  $1,000  PVIF5%,40 yrs.
 $60  17.159  $1,000(0.142)
 $1,029.54  $142  $1,171.54
Bond prices using annual compounding:
(a) 10½%, 15 years, YTM of 8%:
Price  $105  PVIFA8%,15 yrs.  $1,000  PVIF8%,15 yrs.
 $105  8.559  $1,000(0.315)
 $898.70  $315  $1,213.70
(b) 7%, 10 years, YTM of 8%:
Price  $70  PVIFA8%,10 yrs.  $1,000  PVIF8%,10 yrs.
 $70  6.710  $1,000(0.463)
 $469.70  $463  $932.70
(c) 12%, 20 years, YTM of 10%:
Price  $120  PVIFA10%,20 yrs.  $1,000  PVIF10%,20 yrs.
 $120  8.514  $1,000(0.149)
 $1,021.68  $149  $1,170.68
The price difference using the two compounding methods are:
Semi-annual
Annual
Difference
(a)
Premium
$1,215.83
1,213.70
$2.13
(b)
Discount
$931.65
932.70
–$1.05
(c)
Premium
$1,171.54
1,170.68
$0.86
Overall, the difference between bond prices computed using either method are very small, ranging in
absolute value from $0.86 to $2.13. As the above comparison demonstrates, if a bond sells at a
premium its value is higher with semi-annual compounding. When it sells at a discount, its value is
greater with annual compounding.
3.
PVIFA9%, 15 periods  8.061
PVIF9%, 15 periods  0.275
Bond price  ($758.061)  ($1,0000.275)  $604.58  $275  $879.57
206
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
4.
PVIFA4%, 40 periods  19.793
PVIF4%, 40 periods  0.208
Bond price  ($5019.793)  ($1,0000.208)  $989.65  $208  $1,197.65
5.
Return  interest income plus price appreciation
Return  $100  $50  $150
Holding period return  $150/$900  0.1667 or 16.67%.
6.
Current yield is equal to annual income divided by current price
 $80/$1,150  0.0696 or 6.9%.
7.
Price of bond today (8%, 18 years, 10% yield):
Price  $80  PVIFA10%,18 yrs.  $1,000  PVIF10%,18 yrs.
 $80  8.201  $1,000(0.180)
 $656.08  $180  $836.08
Price of bond in one year (8%, 17 years, 9% yield):
Price  $80  PVIFA9%,17 yrs.  $1,000  PVIF9%,17 yrs.
 $80  8.544  $1,000(0.231)
 $683.52  $231  $914.52
If the investor’s expectations are accurate, the price of the bond should go up by $78.44 ($914.52) –
$836.08) over the next year. The holding period return will be:
HPR  Annual interest income  Capital gains
Purchase price
$80  $78.44
 18.95%



$836.08
8.
$1,170.68  (1,000PVIFx%, 20 periods)  ($120PVIFAx%, 20 periods)
Using 20 years and 10%
($1,0000.149)  ($1208.514)  $149  $1,021.68  $1,170.68
Calculator solution:
9.
20N, –1170.58 PV, 120 PMT, 1000FV
CPT I/Y  10.0 %
$1,098.62  (1,000PVIFx%, 20 periods)  ($90PVIFAx%, 20 periods)
Using 20 years and 8%
($1,0000.215)  ($1209.818)  $215  $883.62  $1,098.62
Calculator solution:
20N, –1098.62 PV, 90 PMT, 1000FV
CPT I/Y  8.0 %
Chapter 11
10. Current yield 

Bond Valuation
207
Annual interest income
Current market price of bond
$100 *
 8.33%
$1, 200
*Annual interest income  0.10  $1,000 (assumed face value).
Using annual compounding, the promised yield (YTM) on the bond can be calculated as follows:
Current price: $1,200
Coupon Payment: $100
Holding period  25 years
Future Price  $1,000
Let r% be the promised yield. We have the following:
1,200  100  PVIFAr%,25 period  $1,000  PVIFr%,25 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the promised
yield is: 8.1%
Using Semi-Annual Compounding, the promised yield (YTM) on the bond can be calculated as
follows:
Current price: $1,200
Coupon Payment: $100 ÷ 2  50
Holding period  25 years  2  50 periods
Future Price  $1,000
Let r% be the promised Yield. We have the following:
1,200  50  PVIFAr%,50 period  $1,000  PVIFr%,50 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the promised
semi-annual yield is: 4.06%
Hence the annual promised yield  4.06%  2  8.12%
11. (a) Current yield  Annual interest income/Current market price of the bond
$100
$1200
 8.33%

Yield-to-Maturity (using interpolation) for a market price of $1200
PV @ 4%  ($50PVIFA4%,50periods )  ($1000PVIF4%,50periods)
$1215  ($50  21.482)  ($1000  21.482)
PV @ 5%  ($50  PVIFA5%,50periods )  ($1000  PVIF5%,50periods)
$999  ($50  18.256)  ($1000  0.087)
Interest Rate
4%
I
5%
Bond Price
$1215
1200
$999
$15 difference

 $216 difference

YTM  4%  (15/216) (1%)  4%  0.07%  4.07%  2  8.14%
208
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
Yield-to-Call
PV @ 3%
$1226.30
PV @ 4%
$1132.25
 ($50PVIFA3%,10periods)  ($1075  PVIF3%,10periods)
 ($508.530)  (10750.744)  426.50  799.80
 ($50PVIFA4%,10periods)  ($1075PVIF4%,10periods)
 ($508.111)  ($10750.676)  405.55  726.70
Interest Rate
3%
I
4%
Bond Price
$1226
1200
$1132
$26 difference

 $94 difference

YTC  3%  (26/94) (1%)  4%  0.277%  3.287%2  6.56%
Thus, the YTC on this bond is 6.56% while the YTM is 8.14%. Bond traders would compare
both rates. Since the convention is to use the lower more conservative measure of yield as the
appropriate indicator of value, we would use the YTC of 6.56%. Calculator Solution:
Calculator Solution:
YTM : 50N, –1200 PV, 50PMT, 1000 FV; CPT I/Y  4.062  8.12%
YTM : 10N, –1200 PV, 50PMT, 1075 FV; CPT I/Y  3.272  6.54%
$100
$850
 11.8%
(b) Current Yield 
Yield-to-Maturity (using interpolation) for a market price of $850
PV @ 5%  ($50  PVIFA5%,50periods )  ($1000  PVIF5%,50periods)
$999.80  ($50  18.256)  ($1000  0.087)
PV @ 6%  ($50  PVIFA6%,50periods )  ($1000XPVIF6%,50periods)
$ 842.10  ($50  15.762)  ($1000  0.054)
Interest Rate
5%
I
6%
Bond Price
$999
$149 difference
850
$842

 $157 difference

YTM  5%  (149/157) (1%)  5%  0.95%  5.95%  2  11.9%
Chapter 11
Bond Valuation
209
Yield-to-Call (using interpolation) for a market price of $850
PV @ 7%  ($50PVIFA7%,10periods )  ($1075PVIF7%,10periods)
$897.30  ($507.024)  ($10750.508)
PV @ 8%  ($50PVIFA8%,10periods )  ($1000PVIF8%,10periods)
$ 833.23  ($506.710)  ($10750.463)
Interest Rate
7%
I
8%
Bond Price
$897
$47 difference
850
$ 833

 $64 difference

YTC  7%  (47/64)(1%)  7% 0.734%  7.73%2  15.46%
Calculator Solution:
YTM : 50N, –850 PV, 50PMT, 1000 FV; CPT I/Y  5.942  11.9%
YTM : 10N, –850 PV, 50PMT, 1075 FV; CPT I/Y  7.732  15.47%
12. Bond A:
Since 10.5% is an interest rate that does not appear in the tables, it is necessary to use a calculator to
price the bond.
N  202  40 semiannual periods
I/Y  10.5/2  5.25 percent per semiannual period
PMT  0.09 ($1,000)/2  $45 per semiannual period
FV  $1000
Compute PV  $875.59
Current yield
90
10.3% 
$876
Yield-to-Maturity
PV @ 5%  $45PVIFA5%,40period  $1000PVIF5%,40period
914
 (4517.159)  (10000.142)
PV @ 6%  $45PVIFA6%,40period  $1000PVIF6%,40period
774
 (4515.046)  (10000.097)
Interest Rate
5%
I
6%
Bond Price
$914
$38 difference
876
$774

 $140 difference

YTM  5%  (38/140) (1%)  5%  0.27%  5.27%2  10.54%
210
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
Yield-to-Call (using interpolation) for a market price of $876
PV @ 6%  ($45  PVIFA6%,40periods )  ($1000  PVIF6%,40periods)
$917
 ($457.360)  ($1050  0.558)  331.20  585.90
PV @ 7%  ($45PVIFA7%,10periods )  ($1050PVIF7%,10periods)
$ 849
 ($457.024)  ($10500.508) 316.08  533.40
Interest Rate
6%
I
7%
Bond Price
$917
876
$849
$41 difference

 $68 difference

YTC  6%  (41/68) (1%)  6% 0.60%  6.60%2  13.2%
Bond B:
Since 7.5% is an interest rate that does not appear in the tables, it is necessary to use a calculator to
price the bond.
N  20 annual periods
I/Y  7.5 percent per year
PMT  $80 per year
FV  $1000
Compute PV  $1,050.97
Current yield
7.6% 
$80
$1,051
Yield-to-Maturity(using interpolation) for a market price of $1051
PV @ 7%  $80  PVIFA7%,20period  $1000xPVIF7%,20period
1105.52  (8010.594)  (10000.258)
PV @ 8%  $80  PVIFA8%,20period  $1000  PVIF8%,20period
1000.44  (809.818)  (10000.215)
YTM
 7%  (54.52/105.08) 1%  7%  052%  7.52%
Yield-to-Call (using interpolation) for a market price of $1051
PV @ 7%  ($80  PVIFA7%,5periods )  ($1050PVIF7%,5periods)
$1076.65  ($804.100)  ($10500.713)  328  748.65
PV @ 8%  ($80PVIFA8%,5periods )  ($1050PVIF8%,5periods)
$1034.49  ($803.993)  ($1050  0.681)  319.44  715.05
YTC
 7%  (25.65/42.16) (1%)  7% 0.61%  7.61%
13. PVIF  Price/Par  0.209. PVIF of 0.209 for 15 years  11%.
Calculator Solution:
15N, –209PV, 1000FV; CPT I/Y  11.0%
Chapter 11
Bond Valuation
211
14. Price  ParPVIF  $1,0000.422  $422.00.
Calculator Solution
10N, 9I/Y, 1000FV; CPT PV  $422.41
15. Bond terms: 25 years, zero-coupon, priced at 11.625 (price of $116.25).
Current yield 
Annual interest income
Current market price of bond
$0
$116.25
The easiest (and most accurate) way to find the promised yield of a zero coupon issue without a
calculator that has the time value of money function is to use the table of present value interest factors
(Table B.3 in the appendix). First solve for the PVIF in the basic present value equation:
$116.25  $1,000  PVIF

PVIF

$116.25
 0.116
$1,000
The 25-year factor in Table B.3 that’s equal (or close) to 0.116 is 9%, which lies at the intersection of
25 years and 9%. (Note: Using the approximate yield equation results in a promised yield of only
6.33%, a figure that isn’t even close to the real promised yield—which illustrates why approximate
yield is not a very accurate measure of return for zero coupon bonds.)
Calculator Solution
25N, –116.25PV, 1000FV; CPT I/Y  8.99%

To find the price of this zero coupon bond, find the present value at 12% of $1,000 (par value) in 25
years:
Bond price  $1,000  PVIF12%,25 yrs.
 $1,000  0.059  $59
Calculator Solution:
25N, 12I/Y, 1000FV; CPT PV  $58.82
16. Using annual compounding, the realized yield on the bond can be calculated as follows:
Current price: $800
Coupon Payment: $80
Holding period  3 years
Future Price  $950
Let r% be the promised yield. We have the following:
950  80  PVIFAr%,3 period  $950  PVIFr%,3 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the realized
yield is: 15.38%
If this is a nine-month holding period, the hold period return is:
HPR 
$950  $800  $60
 26.25%
$800
The 15.38% is lower than the 26.25% holding period return. The latter is for nine months, while the
former is an annual yield. Dividing the nine-month holding period by 0.75 puts both rates on an
annual basis; that is, 26.25/0.75  35% annual rate of return.
212
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
17. (a) Bond terms: 9½%, 20 years, priced at $957.43
Current price: $957.43
Coupon Payment: $95
Holding period  20 years
Future Price  $1,000
Let r% be the Yield-To-Maturity. We have the following:
957.43  95  PVIFAr%,20 period  $  PVIFr%,20 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the YTM
is: 10%
(b) Bond terms: 16%, 15 years, priced at $1,684.76
Current price: $1,684.76
Coupon Payment: $160
Holding period  15 years
Future Price  $1,000
Let r% be the Yield-To-Maturity. We have the following:
1,684.76  160  PVIFAr%,15 period  $1,000  PVIFr%,15 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the YTM
is: 8%
(c) Bond terms: 5½%, 18 years, priced at $510.65
Current price: $510.65
Coupon Payment: $55
Holding period  18 years
Future Price  $1,000
Let r% be the Yield-To-Maturity. We have the following:
510.65  55  PVIFAr%,18 period  $1,000  PVIFr%,18 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the YTM
is: 12.42%
18. Modified duration  Macaulay Duration/(1  Yield)  9.5/1.075  8.84.
19. Percent change in bond price  –1modified durationchange in interest rates
Modified duration  Macaulay Duration/(1  Yield)  8.62/1.08  7.98.
Percent change in bond prices  –17.980.005  –0.0399 or –3.99%
20. Percent change in bond price  –1modified durationchange in interest rates
Modified duration  Macaulay Duration/(1  Yield)  8.62/1.08  7.98.
Percent change in bond prices  –17.98–0.005  0.0399 or 3.99%
21. To calculate the duration of the bond, first calculate the bond’s current market price:
Bond terms: 10% coupon, 20 years, 8% YTM
Price  $100  PVIFA8%,20 yrs.  $1,000  PVIF8%,20 yrs.

 $100  9.818  $1,000  0.215
 $981.80  $215  $1,196.80
Chapter 11
Bond Valuation
213
Duration analysis: 10% coupon, 20 years, 8% YTM
(1)
Year
(t)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
(2)
Weighted
Annual
Cash Flow
(C)
$100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
1,100
(3)
(4)
PVIF
(8%)
Present Value
of Cash Flows
(2)  (3)
$92.60
85.70
79.40
73.50
68.10
63.00
58.30
54.00
50.00
46.30
42.90
39.70
36.80
34.00
31.50
29.20
27.00
25.00
23.20
236.50
0.926
0.857
0.794
0.735
0.681
0.630
0.583
0.540
0.500
0.463
0.429
0.397
0.368
0.340
0.315
0.292
0.270
0.250
0.232
0.215
Modified duration 

(5)
PC (Ct)
Divided by
Current Price
of the Bond
4/$1,196.80
0.07737
0.07161
0.06634
0.06141
0.05690
0.05264
0.04871
0.04512
0.04178
0.03869
0.03585
0.03317
0.03075
0.02841
0.02632
0.02440
0.02256
0.02089
0.01939
0.19761
Duration
(6)
TimeRelative
Cash Flow
(1)  (5)
0.07737
0.14322
0.19902
0.02564
0.28450
0.31584
0.34097
0.36096
0.37602
0.38960
0.39435
0.39804
0.39975
0.39774
0.39480
0.39040
0.38352
0.37602
0.36841
3.95220
10.19 years
Duration in years
1  Yield to maturity
10.19
 9.44
1  0.08
% change in bond price  –1  Modified duration  change in interest rates
 –1  9.44  1%  –9.44%
If market yields rise 1%, the price of the bond will fall by 9.44%:
Price in one year  $100  PVIFA9%,19 yrs.  $1,000  PVIF9%,19 yrs.
 $100  8,950  $1,000  0.194
 $895  $194  $1,089
The change in bond price is –$107.80, or 9% of the purchase price. The change in price using the
modified duration method is 9.44%, overstating the actual price change by 0.44%. Duration is
therefore not a good predictor of price volatility if interest rates undergo a big swing. Since the priceyield relationship of a bond is convex in form—but duration is not—the duration measure will
overstate the price decline as the market experiences a big increase in rates. Here, although better, the
modified duration overstated the decline by almost 0.5%.
214
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
22. This question is about bond price volatility. We need to measure the responsiveness of a bond’s price
to a given change in market interest rates. To maximize capital gains, we need to select the bond that
has the maximum price volatility. To do this, first calculate the modified duration of each bond using
the following formula:
Modified duration 
Duration in years
1  Yield to maturity
Then calculate the price change with the following formula:
% change in bond price  –1  Modified duration  change in interest rates
(a) Bond with duration of 8.46 years with YTM of 7.5%:
Modified duration 
8.46
 7.87%
1  0.075
% change in bond price  –1  7.87  –0.5%  3.94%
(b) Bond with duration of 9.30 years with YTM of 10%:
9.30
 8.45%
1  0.10
% change in price  –1  8.45  –0.5% 4.23%
Modified duration 
(c) Bond with duration of 8.75 years with YTM of 5.75%:
8.75
 8.27%
1  0.0575
% change in price  –1  8.27  –0.5%  4.135%
Modified duration 
Bond (b) offers the potential for maximum capital appreciation. To maximize gains, this bond should
be selected over the others.
(Note: This question can be answered directly by looking at the modified duration. For a given
change in interest rates, the bond with the highest modified duration will offer maximum price
appreciation potential. Bond (b), with the highest modified duration, is the choice for the investor
who wishes to maximize capital gains.
23. Current price of the bonds at 9% market interest:
Zero-coupon bond:
Price  $1,000  PVIFA9%,25 yrs.  $1,000  0.116  $116
7½%, 20-year bond (assume annual payments):
Price  $75  PVIFA9%,20 yrs.  $1,000  PVIFA9%,20 yrs.
 $75  9.129  $1,000  0.178  $862.68
Prices based on 7% rate in 1 year:
Zero-coupon bond:
Price  $1,000  PVIFA7%,24 yrs.  $1,000  .197  $197
7½%, 19-year bond (assume annual payments):
Price  $75  PVIFA7%,19 yrs.  $1,000  PVIFA7%,19 yrs.
 $75  10.336  $1,000  .277  $1,052.20
Capital gains:
Zero-coupon bond: Gain  $197 – $116  $81
7½% bond:
Gain  $1,052.20 – 862.68  $189.52
Chapter 11
Bond Valuation
215
To maximize capital gains per bond, buy the 7½%, 20 year bond; but this doesn’t take into account
the big difference in the amount (cost) invested. To do that, we should compare holding period
returns:
HPR 
Zero-coupon bond:
7½% bond:
Interest  Capital gains
Purchase price
$81
 69.8%
$116
$75  $189.52
HPR 
 30.7%
$862.68
HPR 
The conclusion remains unchanged. Mary should purchase the zero-coupon bond.
We know from Chapter 9 that prices of bonds with lower coupons and/or longer maturities will
respond more vigorously to changes in market rates. This is exactly why the zero coupon bond
provided better capital gains than the 7½% bond as market rates went down; the zero coupon bond
pays no interest and, in this case, had a longer maturity than the other bond.
The duration of a zero-coupon bond is equal to its actual maturity, while the duration of a couponbearing bond is always less than its actual maturity. In this case, the zero-coupon bond’s duration is
longer (25 years) than that of the 7½% coupon bond. The zero-coupon bond, with its longer duration,
should be more price volatile than the other bond under consideration.
24. The duration and modified duration can be calculated using the IMD software. It gives the precise
duration measure because it avoids the rounding-off errors which are inevitable with manual
calculations. The following answers are computed using a Lotus 1-2-3 worksheet set up to mimic
manual calculations using present value factors from table B.3. The duration and modified duration
measures using IMD are provided for comparison.
(a) Duration and modified duration
T
[PV(Ct )  t ]
Duration
 
Pbond
t 1
Modified duration 
Duration in years
1  Yield to maturity
Bond 1: 13 years, 8¼, priced to yield 7.47%
Using Lotus 1-2-3, duration of this bond is 8.74. years
8.74
 8.13
1  0.0747
Using the software the duration is 8.58 years and the modified duration is 7.97%.
Modified duration 
Bond 2: 15 years, 77/8 , priced to yield 7.60%
Using Lotus 1-2-3, duration of this bond is 9.41. years
9.41
 8.75
1  0.0760
Using the software the duration is 9.37 years and the modified duration is 8.71%
Modified duration 
Bond 3: 20 years, zero coupon, priced to yield 8.22%
With a zero-coupon bond, the duration of this bond is the same as its maturity, 20 years.
20.00
Modified duration 
 18.48
1  0.0822
216
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
Bond 4: 24 years, 7½, priced to yield 7.90%
Using Lotus 1-2-3, duration of this bond is 11.59. years
11.59
Modified duration 
 10.70
1  0.0790
Using the software the duration is 11.56 years and the modified duration is 10.72%
(b) When Elliot invests $250,000 in each of the four bonds, the weighted average duration of the
portfolio is:
(1)
(2)
(3)
(4)
Bond 1
Bond 2
Bond 3
Bond 4
Bond
Particulars
13 years, 8.15%
15 years, 7.875%
20 years, 0%
24 years, 7.5%
Amount
Invested
$250,000
250,000
250,000
250,000
$1,000,000
Weight
0.25
0.25
0.25
0.25
1.00
(5)
Bond
Duration
8.74
9.41
20.00
11.59
(6)
Weighted
Duration
(4)  (5)
2.1850
2.3525
5.0000
2.8975
12.4350
The duration of the portfolio is 12.44 years.
(c) When Elliot invests $360,000 each into bonds 1 and 3, and $140,000 each into bonds 2 and 4, the
weighted average duration of the bond portfolio is:
(1)
(2)
Bond 1
Bond 2
Bond 3
Bond 4
Bond
Particulars
13 years, 8.25%
15 years, 7.875%
20 years, 0%
24 years, 7.5%
(3)
Amount
Invested
$360,000
140,000
360,000
140,000
$1,000,000
(4)
Weight
0.36
0.14
0.36
0.14
1.00
(5)
Bond
Duration
8.74
9.41
20.00
11.59
(6)
Weighted
Duration
(4)  (5)
3.1464
1.3174
7.2000
1.6226
13.2864
The duration of the portfolio is 13.29 years.
(d) Portfolio (c) has a higher duration than portfolio (b). If rates are about to rise, then it is safer to
invest in portfolio (b), because this would be less price volatile than the other portfolio.

Solutions to Case Problems
Case 11.1
The Bond Investment Decisions of Kelley and Erin Coates
In this case, the student is asked to evaluate two bond trading opportunities—one involves using bonds to
speculate on short-term interest rate movements, and the other deals with a bond swap.
(a) 1. The Coates are attempting to speculate on interest rates by seeking capital gains from an expected
drop in rates.
Chapter 11
Bond Valuation
217
2. The price of the bond in 2 years (when it has 23 years to maturity):
Price of bond  Coupon (PVIFA)  Maturity value (PVIF)
 $75  PVIFA8%,23 yrs.  $1,000  PVIF8%,23 yrs.
 $75  10.371  $1,000  0.170
 $778  $170  948
3. Using the formula for expected return
Current price: $852
Coupon Payment: $75
Holding period  2 years
Future Price  $948
Let r% be the Yield-To-Maturity. We have the following:
852  75  PVIFAr%,2 period  $948  PVIFr%,2 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the YTM
is 14%
4. Although this appears to be an attractive investment, one must compare the expected return with
other possible alternatives. Presuming the expected rate of return (of 14%) is commensurate with
the exposure to risk, Kelley & Erin should seriously consider this bond investment opportunity—
unless they feel strongly that they can do better elsewhere. Further, they should be well aware of
the fact that this high rate of return is due in large part to their ability to correctly forecast interest
rates (no easy task); they should fully appreciate the implications of this kind of risk exposure.
(b) 1. We will evaluate the current and promised yields using the text’s formulas.
Current yield  Annual interest/Current price
Beta Corporation
$70/$785
8.92%

Dental Floss, Inc
$75/$780
9.60%

Root Canal Products
$65/$885
7.35%

Kansas City Dental
$80/$950
8.42%

Insurance
Beta Corporation:
785  35  PVIFAr%,30 period  $1,000  PVIFr%,30 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the expected
return is: 4.87%  2  9.75%
Dental Floss, Inc:
780  37.50  PVIFAr%,30 period  $1,000  PVIFr%,30 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the expected
return is: 5.21%  2  10.42%
Root Canal Products:
885  33.50  PVIFAr%,26 period  $1,000  PVIFr%,26 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the expected
return is: 4%  2  8%
218
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
Kansas City Dental Insurance:
950  40  PVIFAr%,34 period  $1,000  PVIFr%,34 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the expected
return is: 4.28%  2  8.56%
2. Clearly, Dental Floss offers both higher current and promised yields than Beta Corporation.
3. The Coates should swap Beta for Dental Floss to obtain higher current income and promised
yield; this presumes the two have equal default risk, and that the Coates are sure the two are of
comparable quality.
Case 11.2
Grace Decides to Immunize Her Portfolio
(a) Current and Promised Yield Calculations
Current yield 
Annual interest income
Current market price of bond
Bond 1: 12 years, 7½% coupon; currently priced at $895
Current yield 
$75
 8.38%
$895
Yield to Maturity: 895  75  PVIFAr%,12 period  1,000  PVIFr%,12 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the expected
return is: 8.96%
Bond 2: 10 years, zero coupon; currently priced at $405
Current yield  0% for a zero coupon bond
Precise yield: Solve for PVIF:
$405  $1, 000  PVIF
$405
PVIF 
 0.405
$1, 000
The 10-year factor closest to 0.405 (from Table B.3) occur at 9% (0.422) and 10% (0.386). Because
0.405 is halfway between the two, the promised yield on this security should be 9.5%.
Using the software, the YTM is 9.45%.
Bond 3: 10 years, 10% coupon; currently priced at $1,080
Current yield 
$100
 9.26%
$1,080
Yield to Maturity: 1,080  100  PVIFAr%,10 period  1,000  PVIFr%,10 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the expected
return is: 8.77%
Chapter 11
Bond Valuation
Bond 4: 15 years, 9¾% coupon; currently priced at $980
Current yield 
$97.50
 9.95%
$980.00
Yield to Maturity: 980  97.50  PVIFAr%,15 period  1,000  PVIFr%,15 period
The r% can be calculated by trial and error using Tables. Using a financial calculator, the expected
return is: 10%
(b) Duration and Price Volatility
Bond 1: 12 years, 7½% coupon; currently priced at $895 to yield 9%
Using Lotus 1-2-3, duration of this bond is 8.07 years
8.07
 7.40
1  0.09
Percent change in bond price  –1  Modified duration  Change in interest rate
Modified duration 
 –1 7.40  0.75
 –5.55%
The price of the bond will fall by 5.55% if interest rate rise 0.75% and vice versa.
Bond 2: 10 years, zero coupon; currently priced at $405 to yield 9.5%
The duration of a zero coupon bond is the same as its maturity, or 10 years.
Modified duration 
10.00
 9.13
1  0.095
Percent change in bond price  –1  Modified duration  Change in interest rate
 –1 9.13  0.75
 –6.85%
The price of the bond will fall by 6.85% if interest rate rise 0.75% and vice versa.
Bond 3: 10 years, 10% coupon; currently priced at $1,080 to yield 8.75%
Using Lotus 1-2-3, duration of this bond is 6.89 years.
Modified duration 
6.89
 6.34%
1  0.0875
Percent change in bond price  –1  Modified duration  Change in interest rate
 –1 6.34  0.75
 –4.76%
The price of the bond will fall by 4.76% if interest rate rise 0.75% and vice versa.
219
220
Gitman/Joehnk • Fundamentals of Investing, Ninth Edition
Bond 4: 15 years, 9¾% coupon; currently priced at $980 to yield 10%
Using Lotus 1-2-3, duration of this bond is 8.41 years.
8.41
 7.65%
1  0.10
Percent change in bond price  –1  Modified duration  Change in interest rate
 –1 .65  0.75
 –5.74%
Modified duration 
The price of the bond will fall by 5.74% if interest rate rise 0.75% and vice versa.
(c) When Grace invests 450,000 in each of the four bonds, the weighted average duration of the bond
portfolio would be:
(1)
Bond 1
Bond 2
Bond 3
Bond 4
(2)
Bond
Particulars
12 years, 7.50%
10 years, zero
10 years, 10%
15 years, 9.75%
(3)
Amount
Invested
$50,000
50,000
50,000
50,000
$200,000
(4)
Weight
0.25
0.25
0.25
0.25
1.00
(5)
Bond
Duration
8.07
10.00
6.89
8.41
(6)
Weighted
Duration
(4)  (5)
2.0175
2.5000
1.7225
2.1025
8.3425
The duration of the portfolio is 8.34 years. Grace’s investment horizon is 7 years; therefore, the bond
portfolio is not immunized because the weighted average of the portfolio is greater than the investment
horizon.
(d) The bond with the highest duration is the zero-coupon bond (10 years). The bond with the lowest
duration is the 10%, 10-year bond. To lengthen the portfolio’s duration, Grace can invest in higher
duration bonds and shorten the duration by investing in lower duration bonds. By investing the entire
sum of $200,000 in the 10-year bond, she can achieve the shortest duration portfolio. Obviously,
investing the entire portfolio in the zero coupon bond results in the longest duration portfolio.
(e) Grace is planning to cash out of the bond portfolio in about 7 years and wants to immunize the
portfolio. To do so, we must find a portfolio with a weighted average duration of 7 years. The easiest
way to immunize her portfolio from interest rate risk is to invest all of the $200,000 in the 10-year,
10% bond, with its 6.89 year duration.
To achieve a fully immunized portfolio with a duration of exactly 7 years, we can consider the 12year, 7.50% bond with its 8.07 year duration and the 10-year, 10% bond with its 6.89 year duration.
The following portfolio has a 7.01 year duration and is therefore immunized from interest rate risk:
(1)
Bond 1
Bond 2
(2)
Bond
Particulars
12 years, 7.50%
10 years, 10%
(3)
Amount
Invested
$20,000
180,000
(4)
Weight
0.10
0.90
1.00
(5)
Bond
Duration
8.07
6.89
(6)
Weighted
Duration
(4  (5)
0.8070
6.2010
7.0080
Chapter 11
Bond Valuation
221
(f) Regardless of how Grace immunizes her bond portfolio, immunization is not meant to be a passive
strategy that she can “put away and forget about.” Immunization is a continued portfolio rebalancing
process that reflects changes in market interest rates.

Outside Project
Chapter 11
Realized Returns on Bonds vs. Their Promised Yields
What kind of returns have investors earned lately? How do last year’s realized returns stack up against the
yields (i.e., yields-to-maturity) promised at the time of purchase? Realized returns on bonds are of interest
to investors because past performance may give clues to the current trends and may suggest possible trend
shifts. The purpose of this project is to look at holding period returns for the past year on bonds.
Obtain a Wall Street Journal that’s approximately one year old, and select four corporate bonds that are
traded on the New York Stock Exchange (make sure they’re non-convertible). Select maturities of 5 years,
10 years, 15 years, and 20 years. Record the prices, coupons, and maturities of your four bonds; also
determine the promised yield for each issue. Now, look up the same bonds today. Calculate the holding
period return actually realized for each security over the past year. Note the effect of coupon and maturity
on each bond’s return. Contrast the promised yield of each bond with its realized return. How do you
explain the difference?