Chapter 7 Bonds and Their Valuation SOLUTIONS TO END-OF-CHAPTER PROBLEMS 7-1 71
With Wit h your your fin finan ancia cial l calc calcula ulato tor, r, ente enter r the the foll follow owing ing: : N = 10; I = YTM = 9%; PMT = 0.08 PV = $935.82.
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×
1,000 = 80; FV = 1000; PV = VB = ?
With Wit h your your finan financia cial l calcul calculat ator, or, ente enter r the foll follow owing ing to to find find YTM: YTM: N = 10 × 2 = 20; PV = -1100; PMT = 0.08/2 = ? YTM = 3.31% × 2 = 6.62%.
×
1,000 = 40; FV = 1000; I = YTM
With your financial calculator, enter the following to find YTC: N = 5 × 2 = 10; PV = -1100; PMT = 0.08/2 YTC = 3.24% × 2 = 6.49%.
7-3 73
×
1,000 = 40; FV = 1050; I = YTC = ?
The probl The problem em asks asks you you to find find the pr pric ice e of a bo bond nd, , given given the the follo followi wing ng facts: N = 16; I = 8.5/2 = 4.25; PMT = 45; FV = 1000. With a financial calculator, solve for PV = $1,028.60.
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VB = $985; M = $1,000; Int = 0.07
×
$1,000 = $70.
a. Cur Current rent yield yield = Annual Annual interes interest/Cu t/Curren rrent t price price of bond = $70/$985.00 = 7.11%. b. N = 10; PV = -985; -985; PMT PMT = 70; 70; FV = 1000; 1000; YTM = ? Solve for I = YTM = 7.2157% ≈ 7.22%. c. N = 7; I = 7.215 7.2157; 7; PMT PMT = 70; FV FV = 1000; 1000; PV = ? Solve for VB = PV = $988.46.
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a. 1. 5 %:
B ond L : Bond Bon d S:
2. 8% 8%: :
Bond Bo nd L: Bond Bon d S:
Input N = 15 15, I = 5, 5, P PM MT = 100, FV FV = 1000, PV PV = ?, PV = $1,518.98. Chang Cha nge e N = 1, PV = ? PV = $1, $1,04 047.6 7.62. 2. From Fr om Bo Bond nd S inp input uts, s, ch chan ange ge N = 15 15 and and I = 8, 8, PV PV = ?, ?, PV = $1,171.19. Chang Cha nge e N = 1, PV = ? PV = $1, $1,01 018.5 8.52. 2.
Answers and Solutions:
7-1
3. 12 12% %: Bo Bon nd L:
From Bon Fro Bond S inpu put ts, ch chan ang ge N = 15 and and I = ?, PV = $863.78. Chang Cha nge e N = 1, PV = ? PV = $982. $982.14. 14.
Bond Bon d S:
= 12, 12, PV
b. Think Think about about a bond bond that that matur matures es in one one month month. . Its prese present nt value value is is influenced primarily by the maturity value, which will be received in only one one month. month. Even if interest interest rates rates double, double, the price price of the the bond bon d will still still be close close to $1,00 $1,000. 0. A 1-year 1-year bond’s bond’s value value would would fluc fl uctu tuat ate e mo more re th than an th the e on onee-mo mont nth h bo bond nd’s ’s va valu lue e be beca caus use e of th the e diffe di ffere rence nce in the timing timing of rec receip eipts ts. . Howev Ho wever er, , its value value would would still be be fairly close to $1,000 even even if interest interest rates doubled. doubled. A long-term long-te rm bond paying semiannual coupons, on the other hand, will be dominated by distant receipts, receipts that are multiplied by 1/(1 + kd/2)t, and if kd incr increase eases, s, the these se mult multipli ipliers ers will decr decreas ease e sign si gnif ific ican antl tly. y. Anot An othe her r wa way y to view view th this is probl problem em is fr from om an opportunity opportu nity point point of view. view. A 1-month 1-month bond can can be reinvested reinvested at the new rate very quickly, and hence the opportunity to invest at this new rate is not lost; however, the long-term bond locks in subnormal returns for a long period of time.
N
7-6
a. VB =
INT
∑ (1 + t=1
+
t
kd)
M = $1,000. 1. VB = $829:
M N
(1 + kd)
I = 0.09($1,000) = $90. Input N = 4, PV = -829, PMT = 90, FV = 1000, I = ? I =
14.99%. 2. VB = $1,104:
Change PV = -1104, I = ?
I = 6.00%.
b. Yes. Yes. At a pric price e of $829, $829, the the yiel yield d to matur maturit ity, y, 15 perc percen ent, t, is greater gre ater than than your requir required ed rate of return return of 12 percen percent. t. If your required rate of return were 12 percent, you should be willing to buy the bond at any price below $908.88.
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The rate of ret etu urn is appro rox xima mat tely calculator using the following inputs:
15.03
N = 6; PV = -1000; PMT = 140; FV = 1090; I = ?
7-8 78
perce cen nt,
fou fo und
with
a
Solve for I = 15.03%.
a. Us Usin ing g a fi fina nanc ncia ial l cal calcu cula lato tor, r, in inpu put t the the fo foll llow owin ing: g: N = 20, PV = -1100, PMT = 60, FV = 1000, and solve for I = 5.1849%. Howeve Howe ver, r, this this is a pe peri riod odic ic rate. rate. 5.1849%(2) = 10.3699% ≈ 10.37%.
The Th e
nomi no mina nal l
annu an nual al
rate rate
=
b. The current current yield = $120/$1, $120/$1,100 100 = 10.91%. c.
YTM = Current Yield + Capital Gains (Loss) Yield Answers and Solutions:
7-2
10.37% = 10.91% + Capital Loss Yield -0.54% = Capital Loss Yield. d. Using a financia financial l calculator, calculator, input input the followin following: g: N = 8, PV = , PMT = 60, FV = 1060, and solve for I = 5.0748%. Howeve Howe ver, r, this this is a pe peri riod odic ic rate. rate. 5.0748%(2) = 10.1495% ≈ 10.15%.
7-9
The Th e
nomi no mina nal l
annu an nual al
rate rate
=
The prob problem lem ask asks s you you to solv solve e for for the the YTM, YTM, give given n the the follow following ing fac facts: ts: N = 5, PMT = 80, and FV = 1000.
In order to solve for I we need PV.
However, you are also give However, given n that the curr current ent yield is equa equal l to 8.21%. Given this information, we can find PV. Current yield = Annual interest/Current price 0.0821 = $80/PV PV = $80/0.0821 = $974.42. Now, solve for the YTM with a financial calculator: N = 5, PV = -974. -974.42, 42, PMT PMT = 80, 80, and FV FV = 1000. 1000. 8.65%.
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Solve Sol ve for for I = YTM =
The proble lem m asks you to so sol lve for the curren ent t yiel eld d, giv ive en the following facts: N = 14, I = 10.5883/ following 10.5883/2 2 = 5.29415, 5.29415, PV = -1020, -1020, and and FV = 1000. 1000 . In order order to solve solve for the the current current yield yield we need need to find find PMT. PMT. With a financi financial al calcula calculator, tor, we find find PMT = $55.00. $55.00. However How ever, , because because the bond is a semiann semiannual ual coupon bond this amount needs to be multipli multiplied ed by 2 to obtain obtain the annua annual l in inter teres est t pa payme yment nt: : $55.0 $5 5.00( 0(2) 2) = $11 $110.0 0.00. 0. Finally, find the current yield as follows: Current yield = Annual interest/Current price = $110/$1,020 = 10.78%.
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The Th e bo bond nd is se sell llin ing g at a la larg rge e pr prem emiu ium, m, which which means means that that it its s co coup upon on rate is much much higher higher than the the going rate rate of interest interest. . Therefo Ther efore, re, the bond is likely to be called--it is more likely to be called than to remain outstandin outstanding g until it it matures. matures. Thus, it will probably probably provide provide a return equal equal to the YTC rather rather than the YTM. So, there is no point point in calculating the YTM--just calculate the YTC. Enter these values: N = 10, PV = -1353.54, PMT = 70, FV = 1050, and then solve for I. The periodic rate is 3.2366 percent, so the nominal YTC is 2 × 3.2366% = 6.4733% ≈ 6.47 6.47%. %. This would would be close close to the the going going rate, rate, and it is about what the firm would have to pay on new bonds.
7-12
a. To fi find th the YT YTM: Answers and Solutions:
7-3
N = 10, PV = -1175, PMT = 110, FV = 1000 I = YTM = 8.35%. b. To find find the the YTC, YTC, if calle called d in Year Year 5: N = 5, PV = -1175, PMT = 110, FV = 1090 I = YTC = 8.13%. c. Th The e bond bonds s are sel selli ling ng at at a premium which indicates that interest rates have have fallen fallen since since the bonds bonds were were origin originall ally y issued issued. . Assuming Assumi ng that interest rates do not change from the present level, investors would expect to to earn the yield yield to call. call. (Note that that the YTC is is less than than the YTM.) d. Simila Similarly rly from above, above, YTC can be fou found, nd, if called called in eac each h sub subseq sequen uent t year. If called in Year 6: N = 6, PV = -1175, PMT = 110, FV = 1080 I = YTM = 8.27%. If called in Year 7: N = 7, PV = -1175, PMT = 110, FV = 1070 I = YTM = 8.37%. If called in Year 8: N = 8, PV = -1175, PMT = 110, FV = 1060 I = YTM = 8.46%. If called in Year 9: N = 9, PV = -1175, PMT = 110, FV = 1050 I = YTM = 8.53%. According to these calculations, According calculations, the latest investors might expect a call ca ll of the bond bonds s is in Year 6. This Th is is the last last year year that the the expected expecte d YTC will will be less than than the expected expected YTM. At this time, time, the firm fi rm sti still ll fin finds ds an adv advan antag tage e to ca calli lling ng the bo bonds nds, , ra rathe ther r th than an seeing them to maturity.
7-13 7-1 3
First, we must First, must find find the amount amount of of money money we can expec expect t to sell sell this bond bond for in 5 years. years. This is found found using using the fact that in five five years, there there will be 15 years remaining remaining until the bond matures and that the expecte expected d YTM for this bond at that time will be 8.5%. N = 15, I = 8.5, PMT = 90, FV = 1000 PV = -$1,041.52. VB = $1,041.52. This is the the value of the bond in in 5 years. Therefor Therefore, e, we can solve solve for the maximu maximum m pri price ce we wo would uld be wil willi ling ng to pay for th this is bon bond d tod today ay, , subject to our required rate of return of 10%. N = 5, I = 10, PMT = 90, FV = 1041.52 PV = -$987.87. VB = $987.87. We are willing to pay up to $987.87 for this bond today. Answers and Solutions:
7-4
7-14 714
Before Befo re yo you u ca can n so solv lve e fo for r th the e pr pric ice, e, we mu must st semiannual rate at which to evaluate this bond.
find fi nd
the th e
appr ap prop opri riat ate e
EAR = (1 + NOM/2)2 - 1 0.0816 = (1 + NOM/2)2 - 1 NOM = 0.08. Semiannual interest rate = 0.08/2 = 0.04 = 4%. Solving for price: N = 20, I = 4, PMT = 45, FV = 1000 PV = -$1,067.95. VB = $1,067.95.
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a. The The curre current nt yiel yield d is define defined d as the the annua annual l coupon coupon pay payme ment nt divid divided ed by the current price. CY = $80/$901.40 = 8.875%. b. Solvi Solving ng for YTM YTM: : N = 9, PV = -901.40, PMT = 80, FV = 1000 I = YTM = 9.6911%. c. Expecte Expected d capital capital gains yield yield can be fou found nd as the diff differe erence nce between between YTM and the current yield. CGY = YTM - CY = 9.691% - 8.875% = 0.816%. Alternatively, you can solve for the capital gains yield by first finding the expected price next year. N = 8, I = 9.6911, PMT = 80, FV = 1000 PV = -$908.76. VB = $908.76. Hence, the cap Hence, capital ital gains over the next year.
yield yiel d
is
the
percent perc ent
price pric e
appreci app reciatio ation n
CGY = (P1 - P0)/P0 = ($908.76 - $901.40)/$901.40 = 0.816%. 7-16 Us 7-16 Usin ing g income.
the th e
TIE TI E
rati ra tio, o, we
can ca n
solv so lve e
for fo r
the th e
firm fi rm's 's curre current nt opera operati ting ng
TIE = EBIT/Int Exp 3.2 = EBIT/$10,500,000 EBIT = $33,600,000. Using th Using the e sam same e me metho thodo dolog logy, y, you ca can n so solve lve fo for r the ma maxim ximum um expense the firm can bear without violating its covenant.
inter in teres est t
2.5 = $33,600,000/Int Exp Max Int Exp = $13,440,000. Therefor There fore, e, the fir firm m can rai raise se de debt bt to the poi point nt th that at it its s int intere erest st expense expe nse incr increas eases es by $2.9 $2.94 4 mill million ion ($1 ($13.44 3.44 − $10. $10.50). 50). The firm firm can
Answers and Solutions:
7-5
raise $25 million at 8%, which would increase the cost of debt by $25 × 0.08 0.0 8 = $2 millio million. n. Addit Add ition ional al debt debt will be issued issued at 10%, and and th the e amount of debt to be raised can be found, since we know that only an additional $0.94 million in interest expense can be incurred. Additional Int Exp = Additional Debt $0.94 million = Additional Debt Additional Debt = $9.40 million.
7-17 7-1 7
× ×
Cost of debt 0.10
Hence, Henc e, th the e fi firm rm ma may y ra rais ise e up to $3 $34. 4.4 4 mi mill llio ion n in ad addi diti tion onal al without violating its bond covenants. First, Firs t, we must find the pric price e Baili Baili paid paid for for this this bond. bond.
debt de bt
N = 10, I = 9.79, PMT = 110, FV = 1000 PV = -$1,075.02. VB = $1,075.02. Then to find the one-period return, we must find the sum of the change in price and the coupon received divided by the starting price.
One-period return =
Ending price - Beginningprice + Coupon receive Beginningprice
One-period return = ($1,060.49 - $1,075.02 + $110)/$1,075.02 One-period return = 8.88%.
7-18 718
The answ The answer er dep depen ends ds on on when when one one wor works ks the the pro probl blem em. . Street Journal, February 3, 2003:
We use used d The Wall
a. AT&T’ AT&T’s s 8.625 8.625%, %, 2031 2031 bonds bonds had had an 8.6 8.6 perce percent nt curre current nt yiel yield. d. The Th e bonds sold at a premium premium, , 100.75% of par, so the coupon interest rate would have to be set lower than 8.625% for the bonds to sell at par. If we assume the bonds ar are en’t calla lab ble, we can do a ro rou ugh calculation calcula tion of their YTM. Using a financi financial al calculator, we input the following values:
N = 29
×
2 = 58, PV = 1.0075
×
-1,000 = -1007.50, PMT =
0.08 625 2
1,000 = 86.25/2 = 43.125, FV = 1000, and then solve for YTM = kd = 4.2773% × 2 = 8.5546%. ×
Thus, AT&T would have to set a rate of 8.55 percent on new long-term bonds. b. The return return on AT&T’s bonds bonds is the curr current ent yield of of 8.6 percent, percent, less a sm smal all l ca capi pita tal l lo loss ss in 20 2031 31. . The Th e to tota tal l re retu turn rn is ab abou out t 8. 8.55 55 percent.
7-19 719
a. Yi Yiel eld d to ma matu turi rity ty (Y (YTM TM): ): With a financial calculator, input N = 28, PV = -1165.75, PMT = 95, FV = 1000, I = ? I = kd = YTM = 8.00%.
Answers and Solutions:
7-6
Yield to call (YTC): With a calculato calculator, r, input N = 3, PV = -1165.75, PMT = 95, FV = 1090, I = ? I = kd = YTC = 6.11%. b. Knowle Knowledge dgeabl able e inv invest estors ors would would exp expect ect the return return to be clo closer ser to 6.1 percen per cent t than to 8 per percen cent. t. If interes interest t rates remain remain substa substanti ntiall ally y lower than 9.5 percent, the company can be expected to call the issue at the call date and to refund it with an issue having a coupon rate lower than 9.5 percent. c. If the bo bond nd had so sold ld at a di disc scou ount nt, , th this is would would im impl ply y th that at curren current t interest rates are above the coupon rate. Therefore, the company would not call the bonds, so the YTM would be more relevant than the YTC.
Answers and Solutions:
7-7
7-20 10-y ear, 10% annua l co upon 10-y ear z ero 5-y ear z ero 30-y ear z ero $100 perp etui ty
7-21
a.
t 0 1 2 3 4
Pri ce at 8% $1 ,134. 20 463. 19 680. 58 99. 38 1 ,250. 00
Pri ce o f Bon d C $1,0 12.79 1,0 10.02 1,0 06.98 1,0 03.65 1,0 00.00
Pr ice a t 7% $ 1,210 .71 508 .35 712 .99 131 .37 1,428 .57
Pe rcen tage cha nge 6. 75% 9. 75 4. 76 32. 19 14. 29
Pr ice o f Bo nd Z $ 693.0 4 759.5 7 832.4 9 912.4 1 1, 000.0 0
b.
7-8
