CHAPTER 7
CHAPTER 7
Bonds Valuation
CHAPTER ORIENTATION
This chapter introduces the concepts that underlie asset
valuation. We are specifically concerned with bonds. We also look
at the concept of the bondholder's expected rate of return on an
investment.CHAPTER OUTLINE
I.Types of bonds
A.Debentures: unsecured long-term debt.
B.Subordinated debentures: bonds that have a lower claim on
assets in the event of liquidation than do other senior
debtholders.
C.Mortgage bonds: bonds secured by a lien on specific assets of
the firm, such as real estate.
D.Eurobonds: bonds issued in a country different from the one in
whose currency the bond is denominated; for instance, a bond issued
in Europe or Asia that pays interest and principal in U.S.
dollars.
E.Zero and low coupon bonds allow the issuing firm to issue
bonds at a substantial discount from their $1,000 face value with a
zero or very low coupon.
1.The disadvantages are, when the bond matures, the issuing firm
will face an extremely large nondeductible cash outflow much
greater than the cash inflow they experienced when the bonds were
first issued.
2.Zero and low coupon bonds are not callable and can be retired
only at maturity.
3.On the other hand, annual cash outflows associated with
interest payments do not occur with zero coupon bonds.
F.Junk bonds: bonds rated BB or below.
II.Terminology and characteristics of bonds
A.A bond is a long-term promissory note that promises to pay the
bondholder a predetermined, fixed amount of interest each year
until maturity. At maturity, the principal will be paid to the
bondholder.
B.In the case of a firm's insolvency, a bondholder has a
priority of claim to the firm's assets before the preferred and
common stockholders. Also, bondholders must be paid interest due
them before dividends can be distributed to the stockholders.
C.A bond's par value is the amount that will be repaid by the
firm when the bond matures, usually $1,000.
D.The contractual agreement of the bond specifies a coupon
interest rate that is expressed either as a percent of the par
value or as a flat amount of interest which the borrowing firm
promises to pay the bondholder each year. For example: A $1,000 par
value bond specifying a coupon interest rate of 9 percent is
equivalent to an annual interest payment of $90.
E.The bond has a maturity date, at which time the borrowing firm
is committed to repay the loan principal.
F.An indenture (or trust deed) is the legal agreement between
the firm issuing the bonds and the bond trustee who represents the
bondholders. It provides the specific terms of the bond agreement
such as the rights and responsibilities of both parties.
G.The current yield on a bond refers to the ratio of annual
interest payment to the bonds market price.
H.Bond ratings
1. Three primary rating agencies existMoodys, Standard &
Poors, and Fitch Investor Services.
2. Bond ratings are simply judgments about the future risk
potential of the bond in question. Bond ratings are extremely
important in that a firms bond rating tells much about the cost of
funds and the firms access to the debt market.
3.The different ratings and their implications are
described.
III.Definitions of value
A.Book value is the value of an asset shown on a firm's balance
sheet which is determined by its historical cost rather than its
current worth.
B.Liquidation value is the amount that could be realized if an
asset is sold individually and not as part of a going concern.
C.Market value is the observed value of an asset in the
marketplace where buyers and sellers negotiate an acceptable price
for the asset.
D.Intrinsic value is the value based upon the expected cash
flows from the investment, the riskiness of the asset, and the
investor's required rate of return. It is the value in the eyes of
the investor and is the same as the present value of expected
future cash flows to be received from the investment.
IV.Valuation: An Overview
A.The value of an asset is a function of three elements:
1.The amount and timing of the asset's expected cash flows
2.The riskiness of these cash flows
3.The investors' required rate of return for undertaking the
investment
B.Expected cash flows are used in measuring the returns from an
investment.
V.Valuation: The Basic Process
The value of an asset is found by computing the present value of
all the future cash flows expected to be received from the asset.
Expressed as a general present value equation, the value of an
asset is found as follows:
V=
where Ct
=the cash flow to be received at time t
V =the intrinsic value or present value of an asset producing
expected future cash flows, Ct, in years 1 through N
k=the investor's required rate of return
N=the number of periods
VI.Bond Valuation
A.The value of a bond is simply the present value of the future
interest payments and maturity value discounted at the bondholder's
required rate of return. This may be expressed as:
Vb=
where It =the dollar interest to be received in each payment
M=the par value of the bond at maturity
kb=the required rate of return for the bondholder
N=the number of periods to maturity
In other words, we are discounting the expected future cash
flows to the present at the appropriate discount rate (required
rate of return).
B.If interest payments are received semiannually (as with most
bonds), the valuation equation becomes:
Vb=
VII.Bondholder's Expected Rate of Return (Yield to Maturity)
A. We compute the bondholder's expected rate of return by
finding the discount rate that gets the present value of the future
interest payments and principal payment just equal to the bond's
current market price.
B. The bondholder's expected rate of return is also the rate the
investor will earn if the bond is held to maturity, provided, of
course, that the company issuing the bond does not default on the
payments.
VIII.Bond Value: Five Important Relationships
A.First relationship
A decrease in interest rates (required rates of return) will
cause the value of a bond to increase; an interest rate increase
will cause a decrease in value. The change in value caused by
changing interest rates is called interest rate risk.
B.Second relationship
1.If the bondholder's required rate of return (current interest
rate) equals the coupon interest rate, the bond will sell at par,
or maturity value.
2.If the bondholder's required rate of return exceeds the bond's
coupon rate, the bond will sell below par value or at a
"discount."
3. If the bondholder's required rate of return is less than the
bond's coupon rate, the bond will sell above par value or at a
"premium."
C.Third relationship
As the maturity date approaches, the market value of a bond
approaches its par value.
1.The premium bond sells for less as maturity approaches.
2.The discount bond sells for more as maturity approaches.
D.Fourth relationship
A bondholder owning a long-term bond is exposed to greater
interest rate risk than when owning a short-term bond.
E.Fifth relationship
The sensitivity of a bond's value to changing interest rates
depends not only on the length of time to maturity, but also on the
pattern of cash flows provided by the bond.
1.The duration of a bond is simply a measure of the
responsiveness of its price to a change in interest rates. The
greater the relative percentage change in a bond price in response
to a given percentage change in the interest rate, the longer the
duration.
2.Calculating duration
duration=
where t=the year the cash flow is to be received
N=the number of years to maturity
Ct=the cash flow to be received in year t
kb=the bondholder's required rate of return
P0=the bond's present value
ANSWERS TO
END-OF-CHAPTER QUESTIONS
7-1.Book value is the asset's historical value and is
represented on the balance sheet as cost minus accumulated
depreciation. Liquidation value is the dollar sum that could be
realized if the assets were sold individually and not as part of a
going concern. Market value is the observed value for an asset in
the marketplace where buyers and sellers negotiate a mutually
acceptable price. Intrinsic value is the present value of the
asset's expected future cash flows discounted at an appropriate
discount rate.
7-2.The value of a security is equal to the present value of
cash flows to be received by the investor. Hence, the terms value
and present value are synonymous.
7-3.The first two factors affecting asset value (the asset
characteristics) are the asset's expected cash flows and the
riskiness of these cash flows. The third consideration is the
investor's required rate of return. The required rate of return
reflects the investor's risk-return preference.
7-4.The relationship is inverse. As the required rate of return
increases, the value of the security decreases, and a decrease in
the required rate of return results in a price increase.
7-5.(a)The par value is the amount stated on the face of the
bond. This value does not change and, therefore, is completely
independent of the market value. However, the market value may
change with changing economic conditions and changes within the
firm.
(b)The coupon interest rate is the rate of interest that is
contractually specified in the bond indenture. As such, this rate
is constant throughout the life of the bond. The coupon interest
rate indicates to the investor the amount of interest to be
received in each payment period. On the other hand, the investor's
required rate of return is equivalent to the bonds current yield to
maturity, which changes with the changing bond's market price. This
rate may be altered as economic conditions change and/or the
investor's attitude toward the risk-return trade-off is
altered.
7-6.In the case of insolvency, claims of debt holders in
general, including bonds, are honored before those of both common
stock and preferred stock. However, different types of debt may
also have a hierarchy among themselves as to the order of their
claim on assets.
Bonds also have a claim on income that comes ahead of common and
preferred stock. If interest on bonds is not paid, the bond
trustees can classify the firm insolvent and force it into
bankruptcy. Thus, the bondholder's claim on income is more likely
to be honored than that of common and preferred stockholders, whose
dividends are paid at the discretion of the firm's management.
7-7.Ratings involve a judgment about the future risk potential
of the bond. Although they deal with expectations, several
historical factors seem to play a significant role in their
determination. Bond ratings are favorably affected by (1) a greater
reliance on equity, and not debt, in financing the firm, (2)
profitable operations, (3) a low variability in past earnings, (4)
large firm size, and (5) little use of subordinated debt. In turn,
the rating a bond receives affects the rate of return demanded on
the bond by the investors. The poorer the bond rating, the higher
the rate of return demanded in the capital markets.
For the financial manager, bond ratings are extremely important.
They provide an indicator of default risk that in turn affects the
rate of return that must be paid on borrowed funds.
7-8.The term debentures applies to any unsecured long-term debt.
Because these bonds are unsecured, the earning ability of the
issuing corporation is of great concern to the bondholder. They are
also viewed as being more risky than secured bonds and as a result
must provide investors with a higher yield than secured bonds
provide. Often the issuing firm attempts to provide some protection
to the holder through the prohibition of any additional encumbrance
of assets. This prohibits the future issuance of secured long-term
debt that would further tie up the firm's assets and leave the
bondholders less protected. To the issuing firm, the major
advantage of debentures is that no property has to be secured by
them. This allows the firm to issue debt and still preserve some
future borrowing power.
A mortgage bond is a bond secured by a lien on real property.
Typically, the value of the real property is greater than that of
the mortgage bonds issued. This provides the mortgage bondholders
with a margin of safety in the event the market value of the
secured property declines. In the case of foreclosure, the trustees
have the power to sell the secured property and use the proceeds to
pay the bondholders. In the event that the proceeds from this sale
do not cover the bonds, the bondholders become general creditors,
similar to debenture bondholders, for the unpaid portion of the
debt.
7-9.(a)Eurobonds are not so much a different type of security as
they are securities, in this case bonds, issued in a country
different from the one in whose currency the bond is denominated.
For example, a bond that is issued in Europe or in Asia by an
American company and that pays interest and principal to the lender
in U.S. dollars would be considered a Eurobond. Thus, even if the
bond is not issued in Europe, it merely needs to be sold in a
country different from the one in whose currency it is denominated
to be considered a Eurobond.
(b)Zero and very low coupon bonds allow the issuing firm to
issue bonds at a substantial discount from their $1,000 face value
with a zero or very low coupon. The investor receives a large part
(or all on the zero coupon bond) of the return from the
appreciation of the bond at maturity.
(c)Junk bonds refer to any bond with a rating of BB or below.
The major participants in this market are new firms that do not
have an established record of performance. Many junk bonds have
been issued to finance corporate buyouts.
7-10.The expected rate of return is the rate of return that may
be expected from purchasing a security at the prevailing market
price. Thus, the expected rate of return is the rate that equates
the present value of future cash flows with the actual selling
price of the security in the market.
7-11.When the coupon interest rate does not equal the
bondholder's required rate of return, the bond will be issued at
either a premium or discount. If the investor's required rate of
return is higher than the coupon interest rate, the bond will be
issued at a discount to the investor. If the coupon rate is higher
that the investor's required rate, the bond will be issued at a
premium.
7-12.A premium bond is issued when the coupon rate is higher
than the bondholder's required rate of return. The premium is the
excess of the market value over the face value of the bond. A
discount bond is issued when the bondholder's required rate of
return is higher than the coupon rate. The discount is the excess
of the face value of the bond over the market value.
Over time, the premium or discount on a bond is amortized. This
amortization allows the bondholder to realize an effective yield
equal to their required rate of return.
7-13.A change in current interest rates (required rate of
return) causes a change in the market value of a bond. However, the
impact on value is greater for long-term bonds than it is for
short-term bonds. The reason long-term bond prices fluctuate more
than short-term bond prices in response to interest rate changes is
simple. Assume an investor bought a 10-year bond yielding a 12
percent interest rate. If the current interest rate for bonds of
similar risk increased to 15 percent, the investor would be locked
into the lower rate for 10 years. If, on the other hand, a
shorter-term bond had been purchased, say one maturing in 2 years,
the investor would have to accept the lower return for only 2 years
and not the full 10 years. At the end of year 2, the investor would
receive the maturity value of $1,000 and could buy a bond offering
the higher 15 percent rate for the remaining 8 years. Thus,
interest rate risk is determined, at least in part, by the length
of time an investor is required to commit to an investment.
7-14.The duration of a bond is simply a measure of the
responsiveness of its price to a change in interest rates. The
greater the relative percentage change in a bond price in response
to a given percentage change in the interest rate, the longer the
duration.
SOLUTIONS TO
END-OF-CHAPTER PROBLEMS7-1A.Value (Vb)=
12 EQ \x(N)
12 EQ \x(I/Y)
80 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-752.23
7-2A.If the interest is paid semiannually:
Value (Vb)=
16 EQ \x(N)
4 EQ \x(I/Y)
45 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,058.26
If interest is paid annually:
Value (Vb)=
8 EQ \x(N)
8 EQ \x(I/Y)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,057.47
7-3A.$900 =
20 EQ \x(N)
900 EQ \x(+/-) EQ \x(PV)
40 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER4.79%semiannual rate
The rate is equivalent to 9.6 percent annual rate compounded
semiannually, or 9.8 percent (1.0482 - 1) compounded annually.
7-4A.$945 =
20 EQ \x(N)
945 EQ \x(+/-) EQ \x(PV)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER9.63%
7-5A.$1,150 =
12 EQ \x(N)
1150 EQ \x(+/-) EQ \x(PV)
70 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER5.28%
7-6A.a.$1,085 =
15 EQ \x(N)
1085 EQ \x(+/-) EQ \x(PV)
80 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER7.06%
b.Vb =
15 EQ \x(N)
10 EQ \x(I/Y)
80 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-847.88
c.Since the expected rate of return, 7.06 percent, is less than
your required rate of return of 10 percent, the bond is not an
acceptable investment. This fact is also evident because the market
price, $1,085, exceeds the value of the security to the investor of
$847.88.
7-7A.a.Value
Par Value$1,000.00
Coupon$ 100.00
Required Rate of Return0.12
Years to Maturity15
Market Value$ 863.78
b.Value at Alternative Rates of Return
Required Rate of Return0.15
Market Value$ 707.63
Required Rate of Return0.08
Market Value$1,171.19
c.As required rates of return change, the price of the bond
changes, which is the result of "interest-rate risk." Thus, the
greater the investor's required rate of return, the greater will be
his/her discount on the bond. Conversely, the less his/her required
rate of return below that of the coupon rate, the greater the
premium will be.
d.Value at Alternative Maturity Dates
Years to Maturity5
Required Rate of Return0.15
Market Value$ 832.39
Required Rate of Return0.08
Market Value$1,079.85
e.The longer the maturity of the bond, the greater the interest
rate risk the investor is exposed to, resulting in greater premiums
and discounts.
7-8A.$1,250=
15 EQ \x(N)
1250 EQ \x(+/-) EQ \x(PV)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER6.36%
7-9A.(a)Vb =
20 EQ \x(N)
9 EQ \x(I/Y)
110 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,182.57
(b)(i)Vb=
20 EQ \x(N)
12 EQ \x(I/Y)
110 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-925.31
(b)(ii)Vb=
20 EQ \x(N)
6 EQ \x(I/Y)
110 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,573.50
(c)We see that value is inversely related to the investor's
required rate of return.
7-10A.
Value Bond P
Par Value$1,000.00
Coupon$ 100.00
Required Rate of Return8%
Years to Maturity5
Market Value$ 1,079.85
Value Bond Q
Par Value$1,000.00
Coupon$ 70.00
Required Rate of Return8%
Years to Maturity5
Market Value$ 960.07
Value Bond R
Par Value$1,000.00
Coupon$ 120.00
Required Rate of Return8%
Years to Maturity10
Market Value$ 1,268.40
Value Bond S
Par Value$1,000.00
Coupon$ 80.00
Required Rate of Return8%
Years to Maturity10
Market Value$ 1,000.00
Value Bond T
Par Value$1,000.00
Coupon$ 65.00
Required Rate of Return8%
Years to Maturity15
Market Value$ 871.61
BondPQRST
EQ \a(Bond,value) $1,079.85$960.07$1,268.40$1,000.00$871.61
YearsCtt*PV(Ct)Ctt*PV(Ct)Ctt*PV(Ct)Ctt*PV(Ct)Ctt*PV(Ct)
1$100$93$70$65$120$111$80$74$65$60
2100171701201202068013765111
3100238701671202868019165155
4100294702061203538023565191
51,1003,7431,0703,6411204088027265221
6
1204548030265246
7
1204908032765265
8
1205198034665281
9
1205408036065293
10
1,1205,1881,0805,00265301
11
65307
12
65310
13
65311
14
65310
15
1,0655,036
4,539
4,198
8,554
7,247
8,398
Duration
4.20
4.37
6.74
7.25
9.63
7-11A.a.$1,100 =
7 EQ \x(N)
1100 EQ \x(+/-) EQ \x(PV)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER7.14%
b.Vb =
7 EQ \x(N)
7 EQ \x(I/Y)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,107.79
c.Since the expected rate of return, 7.14 percent, is more than
your required rate of return of 7 percent, the bond is an
acceptable investment. This fact is also evident because the market
price, $1,100, is less than the value of the security to the
investor of $1,107.79.
7-12A.a.$915=
12 EQ \x(N)
915 EQ \x(+/-) EQ \x(PV)
50 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER6.01%
b.Since the required rate of return(9%) is greater than the
expected rate of return(6%), you should not purchase the bond.
7-13A.
Value Bond I
Par Value$1,000.00
Coupon$ 130.00
Required Rate of Return7%
Years to Maturity 7
Market Value$ 1,323.36
Value Bond II
Par Value$1,000.00
Coupon$ 90.00
Required Rate of Return7%
Years to Maturity6
Market Value$1,095.33
Value Bond III
Par Value$1,000.00
Coupon$ 110.00
Required Rate of Return7%
Years to Maturity12
Market Value$1,317.71
Value Bond IV
Par Value$1,000.00
Coupon$ 125.00
Required Rate of Return7%
Years to Maturity5
Market Value$1,225.51
Value Bond V
Par Value$1,000.00
Coupon$ 80.00
Required Rate of Return7%
Years to Maturity10
Market Value$1,070.24
BondIIIIIIIVV
Bond Value$1,323.36$1,095.33$1,317.71$1,225.51$1,070.24
YearsCttPV(Ct)CttPV(Ct)CttPV(Ct)CttPV(Ct)CttPV(Ct)
1$130$121$90$84$110$103$125$117$80$75
2$130$227$90$157$110$192$125$218$80$140
3$130$318$90$220$110$269$125$306$80$196
4$130$397$90$275$110$336$125$381$80$244
5$130$463$90$321$110$392$1,125$4,011$80$285
6$130$5201,090$4,358$110$440$80$320
71,130$4,926$110$480$80$349
8$110$512$80$372
9$110$538$80$392
10$110$559$1,080$5,490
111,110$5,801
12
Sum of
t*PV(Ct)$6,973$5,415$9,622$5,033$7,863
Duration5.27 4.94 7.30 4.11 7.35
7-14A.(a)Vb =
15 EQ \x(N)
9 EQ \x(I/Y)
85 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-959.70
(b)(i)Vb=
15 EQ \x(N)
11 EQ \x(I/Y)
85 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-820.23
(b)(ii)Vb=
15 EQ \x(N)
7 EQ \x(I/Y)
85 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,136.62
(c)As long as the required rate of return is less than the
expected rate of return of 9%, you should purchase the bond Thus,
if your required rate of return decreases to 7%, you should
purchase the bond.
SOLUTION TO INTEGRATIVE PROBLEM
1.Young Corp. Bond Value (Vb)=
10 EQ \x(N)
6 EQ \x(I/Y)
78 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-$1,132.48
Thomas Resorts Bond Value (Vb)=
17 EQ \x(N)
9 EQ \x(I/Y)
75.00 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-$871.85
Entertainment, Inc. Bond Value (Vb)=
4 EQ \x(N)
8 EQ \x(I/Y)
79.75 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-$999.17
2.Young Corporation: $1,030=
10 EQ \x(N)
1,030 EQ \x(+/-) EQ \x(PV)
78 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER7.37%
Thomas Resorts: $973=
17 EQ \x(N)
973 EQ \x(+/-) EQ \x(PV)
75.00 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER7.79%
Entertainment, Inc.: $1,035=
4 EQ \x(N)
1,035 EQ \x(+/-) EQ \x(PV)
79.75 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER 6.94%
3.i.Young Corp. Bond Value (Vb)=
10 EQ \x(N)
9 EQ \x(I/Y)
78 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER- $922.99
(Vb)=
17 EQ \x(N)
12 EQ \x(I/Y)
75.00 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER- $679.62
Entertainment, Inc. Bond Value (Vb)=
4 EQ \x(N)
11 EQ \x(I/Y)
79.75 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER- $906.15
3.iiYoung Corp. Bond Value (Vb)=
10 EQ \x(N)
3 EQ \x(I/Y)
78 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER- $1,409.45
(Vb)=
17 EQ \x(N)
6 EQ \x(I/Y)
75.00 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER- $1,157.16
Entertainment, Inc. Bond Value (Vb)=
4 EQ \x(N)
5 EQ \x(I/Y)
79.75 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER- $1,105.49
4.As the interest rates rise and fall, we see the different
effects on the bond prices depending on the length of time to
maturity and whether the investor's required rate of return is
above or below the coupon interest rate. If the investors required
rate of return is above the coupon interest rate, the bond will
sell at a discount (below par value), but if the investors required
rate of return is below the coupon interest rate, the bond will
sell at a price above its par value (premium).
5.Duration of bonds
Young Corp.Thomas ResortsEntertainment, Inc.
Bond Value$1,030.00$ 973.00$1,035.00
Required rate of return6%9%8%
YearCtt* PV(Ct)Ctt* PV(Ct)Ctt* PV(Ct)
1$ 78.00$ 73.58$ 75.00$ 68.81$ 79.75$ 73.84
278.00138.8475.00126.2579.75136.75
378.00196.4775.00173.7479.75189.92
478.00247.1375.00212.531,079.753,174.59
578.00291.4375.00243.72
678.00329.9275.00268.32
778.00363.1275.00287.19
878.00391.5175.00301.12
978.00415.5175.00310.79
101,078.006,019.5075.00316.81
1175.00319.71
1275.00319.98
1375.00318.02
1475.00314.21
1575.00308.86
1675.00302.24
171,075.004222.86
Sum of t*PV(Ct)8,467.028,415.173,575.11
Duration8.228.653.45
The value of the Entertainment, Inc. bonds will be less
sensitive to interest rate changes than will Young Corporation and
Thomas Resorts bonds.
6.Although the Young Corporation bonds and the Thomas Resorts
bonds have different terms to maturity, the duration of the two
bonds is very similar. These two bonds will likely have similar
sensitivity to changes in interest rates as evidenced by their
duration values.
7.The Entertainment, Inc. and Thomas Resorts bonds have lower
expected rates of return than your required rate of return. Young
Corporations expected rate of return is greater than your required
rate of return. So we would buy Young Corporation and not
Entertainment, Inc. or Thomas Resorts.
Solutions to Problem Set B7-1B.Value (Vb)
=
10 EQ \x(N)
15 EQ \x(I/Y)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-698.87
7-2B.If the interest is paid semiannually:
Value (Vb)=
22 EQ \x(N)
4.5 EQ \x(I/Y)
50 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1068.92
If interest is paid annually:
Value (Vb)=
11 EQ \x(N)
9 EQ \x(I/Y)
100 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1068.05
7-3B.$950 =
16 EQ \x(N)
950 EQ \x(+/-) EQ \x(PV)
45 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER4.96%
The rate is equivalent to 9.92 percent annual rate, compounded
semiannually or 10.17 percent (1.04962 - 1) compounded
annually.
7-4B.$975 =
20 EQ \x(N)
975 EQ \x(+/-) EQ \x(PV)
100 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER10.30%
7-5B.$1,175 =
15 EQ \x(N)
1175 EQ \x(+/-) EQ \x(PV)
80 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER6.18%
7-6B.a.$1,100 =
14 EQ \x(N)
1100 EQ \x(+/-) EQ \x(PV)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER7.80%
b.Vb =
14 EQ \x(N)
10 EQ \x(I/Y)
90 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-926.33
c.Since the expected rate of return, 7.80 percent, is less than
your required rate of return of 10 percent, the bond is not an
acceptable investment. This fact is also evident because the market
price, $1,100, exceeds the value of the security to the investor of
$926.33.
7-7B.a.Value
Par Value$1,000.00
Coupon$ 80.00
Required Rate of Return7%
Years to Maturity20
Market Value$ 1,105.94
b.Value at Alternative Rates of Return
Required Rate of Return10%
Market Value$ 829.73
Required Rate of Return6%
Market Value$1,229.40
c.As required rates of return change, the price of the bond
changes, which is the result of "interest-rate risk." Thus, the
greater the investor's required rate of return, the greater will be
his/her discount on the bond. Conversely, the less his/her required
rate of return is below that of the coupon rate, the greater the
premium will be.
d.Value at Alternative Maturity Dates
Years to Maturity10
Required Rate of Return10%
Market Value$ 877.11
Required Rate of Return6%
Market Value$1,147.20
e.The longer the maturity of the bond, the greater the
interest-rate risk the investor is exposed to, resulting in greater
premiums and discounts.
7-8B.$1,110 =
14 EQ \x(N)
1110 EQ \x(+/-) EQ \x(PV)
70 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER5.83%
7-9B.(a)Value (Vb)=
+
17 EQ \x(N)
8.5 EQ \x(I/Y)
70 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-867.62
(b)(i)Value (Vb)=
+
17 EQ \x(N)
11 EQ \x(I/Y)
70 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-698.05
(b)(ii)Value (Vb)=
+
17 EQ \x(N)
6 EQ \x(I/Y)
70 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,104.77
(c)We see that value is inversely related to the investor's
required rate of return.
7-10B.
Value Bond A
Par Value$1,000.00
Coupon$ 90.00
Required Rate of Return7%
Years to Maturity5
Market Value$ 1,082.00
Value Bond B
Par Value$1,000.00
Coupon$ 60.00
Required Rate of Return7%
Years to Maturity5
Market Value$ 959.00
Value Bond C
Par Value$1,000.00
Coupon$ 120.00
Required Rate of Return7%
Years to Maturity10
Market Value$ 1,351.18
Value Bond D
Par Value$1,000.00
Coupon$ 90.00
Required Rate of Return7%
Years to Maturity15
Market Value$ 1,182.16
Value Bond E
Par Value$1,000.00
Coupon$ 75.00
Required Rate of Return7%
Years to Maturity15
Market Value$ 1,045.54
BondABCDE
$1,082.00$959.00$1,351.18$1,182.16$1,045.54
YearsCtt*PV(Ct)Ctt*PV(Ct)Ctt*PV(Ct)Ctt*PV(Ct)Ctt*PV(Ct)1$90$84$60$56$120$112$90$84$75$70
290157601051202109015775131
390220601471202949022075184
490275601831203669027575229
51,0903,8861,0603,7791204289032175267
6
1204809036075300
7
1205239039275327
8
1205599041975349
9
1205879044175367
10
1,1205,6949045875381
11
9047075392
12
9048075400
13
9048675405
14
9048975407
15
1,0905,9261,0755,844
4,622
4,270
9,252
10,977
10,053
Duration
4.27
4.45
6.85
9.29
9.62
7-11B.a.$1,350 =
4 EQ \x(N)
1350 EQ \x(+/-) EQ \x(PV)
120 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER2.66%
b.Vb =
4 EQ \x(N)
9 EQ \x(I/Y)
120 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,097.19
c.Since the expected rate of return, 2.66 percent is much less
than your required rate of return of 9 percent, the bond is not an
acceptable investment. This fact is also evident because the market
price, $1,350, exceeds the value of the security to the investor of
$1,097.19.
7-12B.a.$915=
25 EQ \x(N)
915 EQ \x(+/-) EQ \x(PV)
80 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(I/Y) (ANSWER8.86%
b.Since the required rate of return(11%) is greater than the
expected rate of return(8.86%), you should not purchase the
bond.
7-13B.Value Bond J
Par Value$1,000.00
Coupon$ 95.00
Required Rate of Return10%
Years to Maturity4
Market Value$984.15
Value Bond P
Par Value$1,000.00
Coupon$115
Required Rate of Return10%
Years to Maturity12
Market Value$1,102.21
Value Bond Y
Par Value$1,000.00
Coupon$ 80
Required Rate of Return10%
Years to Maturity16
Market Value$843.53
Value Bond Q
Par Value$1,000.00
Coupon$ 70.00
Required Rate of Return10%
Years to Maturity20
Market Value$744.59
Value Bond Z
Par Value$1,000.00
Coupon$ 130.00
Required Rate of Return10%
Years to Maturity15
Market Value$1,228.18
BondJPYQZ
Bond Value$984.15$1,102.21$843.53$744.59$1,228.18
YearsCttPV(Ct)CttPV(Ct)CttPV(Ct)CttPV(Ct)CttPV(Ct)
1 $95 $86 $115 $105 $80 $73 $70 $64 $130 $118
2 $95 $157 $115 $190 $80 $132 $70 $116 $130 $215
3 $95 $214 $115 $259 $80 $180 $70 $158 $130 $293
4 $1,095 $2,992 $115 $314 $80 $219 $70 $191 $130 $355
5 $115 $357 $80 $248 $70 $217 $130 $404
6 $115 $389 $80 $271 $70 $237 $130 $440
7 $115 $413 $80 $287 $70 $251 $130 $467
8 $115 $429 $80 $299 $70 $261 $130 $485
9 $115 $439 $80 $305 $70 $267 $130 $496
10 $115 $443 $80 $308 $70 $270 $130 $501
11 $115 $443 $80 $308 $70 $270 $130 $501
12 1,115$4,263 $80 $306 $70 $268 $130 $497
13 $80 $301 $70 $264 $130 $490
14 $80 $295 $70 $258 $130 $479
15 $80 $287 $70 $251 $1,130 $4,058
16 $1,080 $3,761 $70 $244
17 $70 $235
18 $70 $227
19 $70 $217
20 1,070 $3,181
Sum of
t*PV(Ct)$3,449 $8,046 $7,581 $7,447 $9,799
Duration3.50 7.30 8.99 10.00 7.98
7-14B.(a)Vb =
20 EQ \x(N)
8 EQ \x(I/Y)
120 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,392.73
(b)(i)Vb=
20 EQ \x(N)
13 EQ \x(I/Y)
120 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-929.75
(b)(ii)Vb=
20 EQ \x(N)
6 EQ \x(I/Y)
120 EQ \x(PMT)
1000 EQ \x(FV)
EQ \x(PV) (ANSWER-1,688.20
(c)As long as the required rate of return is less than the
expected rate of return of 8%, you should purchase the bond Thus,
if your required rate of return decreases to 6%, you should
purchase the bond.
PAGE 203
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