Bodie Essentials Ch. 10 Solutions

Chapter 10 - Bond Prices and Yields

CHAPTER 10

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BOND PRICES AND YIELDS

1! a!

Catastrophe bond( "ypically issed by an insrance co#pany! "hey are si#ilar to an insrance policy in that the in)estor recei)es copons and par )ale' ) ale' bt ta*es a loss in part or all o$ o $ the principal i$ a #a+or insrance clai# is $iled a,ainst the isser! "his is pro)ided in echan,e $or hi,her h i,her than nor#al copons! b! Erobond( "hey are bonds issed in the crrency o$ one contry bt sold in other national #ar*ets! c! .ero-copon .ero-copon bond( bond( .ero-co .ero-copon pon bonds bonds are are bonds that pay pay no copons copons bt do pay a par )ale at #atrity! d! /a#rai /a#rai bond( Yen Yen-deno# -deno#inated inated bonds bonds sold sold in apan by non-ap non-apanese anese issers are called /a#rai bonds! e! n* bond( bond( "hose "hose rated rated BBB or or abo)e /P /P'' itch4 itch4 or Baa and abo)e Moody5s4 Moody5s4 are considered in)est#ent ,rade bonds' while lower-rated bonds are classi$ied as speclati)e ,rade or +n* bonds! $! Con)er Con)ertib tible le bond( bond( Con)er Con)ertib tible le bonds bonds #ay be be echan,e echan,ed' d' at the the bondholder5s discretion' $or a speci$ied n#ber o$ shares o$ stoc*! Con)ertible bondholders 6pay7 $or this option by acceptin, a lower copon rate on the secrity! ,! /erial bond( 8 serial bond is an isse isse in which which the $ir# $ir# sells sells bonds with sta,,ered #atrity dates! 8s bonds #atre se9entially se9en tially'' the principal repay#ent brden $or the $ir# is spread o)er ti#e +st as it is with a sin*in, $nd! /erial bonds do not inclde call pro)isions! h! E9ip#ent E9ip#ent obli,ation obli,ation bond( bond( 8 bond that is issed issed with with speci$ic speci$ic e9ip#ent e9ip#ent pled,ed as collateral a,ainst the bond! i!

:ri,inal :ri,inal isse isse discont discont bonds( bonds( :ri,inal :ri,inal isse isse discont discont bonds are are less less co##on than copon bonds b onds issed at par! "hese are bonds that are issed intentionally with low copon rates that case the bond to sell at a discont $ro# par )ale!

+!

;ndeed bond( ;ndeed bonds #a*e pay#ents that are tied to a ,eneral price inde or the price o$ a particlar co##odity!



2!

Callab Callable le bonds bonds ,i)e ,i)e the iss isser er the the optio option n to ete etend nd or retir retiree the bond bond at at the call call date' while the etendable or pttable bond ,i)es this option to the bondholder!

3! a! Y"M will will drop drop since since the co#pany has #ore #ore #oney #oney to pay pay the interest interest on its bonds! b! Y"M will increase since the co#pany has #ore debt and the ris* to the eistin, bondholders is now increased! c! Y"M will will decrease decrease since the the $ir# has has either either $ewer crrent crrent liabiliti liabilities es or an increase in )arios crrent assets!
/e# /e#i-ann -anna all cop copon on = >1'0 >1'000 00 × ?@ × 0!A = >30!

Annual Coupon Payment 2 Day s since Last Coupon Coupon Payment Payment Days Separating Coupon Payment 8ccred ;nterest =

×

= >30 × 30124 = >1'10 F >1'1
sin, sin , a $ina $inanci ncial al cal calcl clat ator or'' P = I
?!

8 bond5 bond5ss copon copon inte interes restt pay#en pay#ents ts and and princi principal pal repay repay#ent #ent are are not not a$$ec a$$ected ted by chan,es in #ar*et rates! Conse9ently' i$ #ar*et rates increase' bond in)estors in the secondary #ar*ets are not willin, to pay pa y as #ch $or a clai# on a ,i)en bond5s $ied $ied interest and principal pay#ents as they wold i$ #ar*et rates were lower! "his "his relationship is apparent $ro# the in)erse relationship between interest rates and present )ale! 8n increase in the discont rate i!e!' the #ar*et rate4 decreases the present )ale o$ the $tre cash $lows!

7.

The bond bond call callabl ablee at 105 105 shoul should d sell sell at a lowe lowerr price price beca because use the the call call prov provisi ision on is more valuable to the firm. Therefore, its yield to maturity should be higher.

8.

The bond bond pric pricee will will be be lower lower.. As time time passes passes,, the the bond bond price price,, which which is is now now above par value, will approach par.

9.

Current yield =

Annual Coupon Bond Price

=

$1,000 $"#0

4.!

= 4.95%



10.

a. The purchase purchase of a credit credit default default swap. swap. The The investor investor believes believes the the bond may may increase in credit risk, which raises the prices of the credit default swaps because of the widened swap spread.

11.

c. When credit risk increases, increases, the swap swap premium premium increases increases because because of higher higher chances of default on the firm. When the interest rate risk increases, the price of the CDS decreases because the cash flows are discounted at a higher rate for bearing more risk.

12!

"he crrent crrent yield and the the annal copon rate rate o$ ?@ i#ply i#ply that that the the bond price price was at par a year a,o! sin, a $inancial calclator'  = 1'000' n =' PM" = ?0' and i = ,i)es s a sellin, price o$ >D
－ $1,000  $"4%.11  $%0 $1,000

= 0!00?1 = 0!?1@ 13! .ero copon bonds pro)ide no copons to be rein)ested! rein)ested! "here$ore' the $inal )ale )ale o$ the in)estorJs proceeds co#es entirely $ro# the principal o$ the bond and is independent o$ the rate at which copons cold be rein)ested i$ they were paid4! "here is no rein)est#ent rate ncertainty with %eros! 1
(

$100, $100,000 000 $"#,%4&

)

4

I 1 = 1!02<124< I 1 = 0!1000 = 10@

b! E$$ecti)e annal interest rate on copon bond payin, A@ se#iannally( 1 F 0!0A42 I 1 = 0!102A = 10!2A@ "here$ore' the copon bond has the hi,her e$$ecti)e annal interest rate!

15.

The effec effectiv tivee annual annual yield yield on the the semian semiannual nual coup coupon on bonds bonds is (1.04) (1.04)2 = 8.16%. If the annual coupon bonds bo nds are to sell at par they must offer the same yield, which requires an annual coupon of 8.16%.

16. a. The bond pays pays $50 $50 ever every y six six months months.. Current price: [$50 × Annuity factor(4%, 6)] + [$1000 × PV factor(4%, 6)] = $1,052.42 Assuming the market interest rate remains 4% per half year, price six months from now: 10-< © 2013 by McGraw-Hill Edcation! "his is proprietary #aterial solely $or athori%ed instrctor se! &ot athori%ed $or sale or distribtion in any #anner! "his doc#ent #ay not be copied' scanned' dplicated' $orwarded' distribted' or posted on a website' in whole or part!


[$50 × Annuity factor(4%, 5)] + [$1000 × PV factor(4%, 5)] = $1,044.52 b! Kate o$ Ketrn =

$&0  '$1,044.&2 － $1,0&2.42( $1,0&2.42

=

$ & 0 － $#."0 $1,0&2.42 = 0!0<00 =
E$$ecti)e annal yield to #atrity = 1!0<2?4 I 1 = 0!00 = !0@

b. Since the the bond is is selling selling at par, par, the yield yield to to maturity maturity on a semi-annual semi-annual basis is the same as the semi-annual coupon, 4%. The bond equivalent yield to maturity is 8%. 2

Effective annual yield to maturity = (1.04) – 1 = 0.0816 = 8.16% c! eepin, eepin, other inpts nchan,ed nchan,ed bt settin, settin, P = I1'0A0' I1'0A0' we $ind $ind a bond e9i)alent yield to #atrity o$ !A2@' or 3!?@ on a se#i-annal basis! 2

E$$ecti)e annal yield to #atrity = 1!03?4 I 1 = 0!0?? = !??@ 1! /ince the bond pay#ents are now #ade annally instead o$ se#i-annally' se#i-annally' the bond e9i)alent yield to #atrity is the sa#e as the e$$ecti)e annal yield to #atrity! "he inpts are( n = 20'  = 1000' P = Iprice' PM" = 0! "he resltin, yields $or the three bonds are(

"he yields co#pted in this case are lower than the yields calclated with se#iannal copon pay#ents! 8ll else e9al' bonds with annal pay#ents are less attracti)e to in)estors becase #ore ti#e elapses be$ore pay#ents are recei)ed! ;$ the bond price is the sa#e with annal pay#ents' then the bondJs yield to #atrity is lower! 1D! &o#inal Ketrn =

)nterest )nterest  Price Appre Appr e ciation ciation )nitial Price 10-A

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Keal Ketrn =

1  *omin *ominal al +eturn +eturn 1  )n latio lation n +ate +ate

I 1

"he second year &o#inal Ketrn = Keal Ketrn =

$42. $42.02  $-0.%0 $1,020.00

1 $ 0.0#11"% 1 $ 0.0-

I 1=

= 0!011D? = !12@

1.0#11"% 1.0-

I 1= 1!0<00 I 1 =
"he third year &o#inal Ketrn = Keal Ketrn =

$42.4 $42.44 4  $ 10. &1 = 0!0A0<00@ $1,0&0.%0

1 $ 0.0&0400 1 $ 0.01

I 1 =

1.0&0400 1.01

I 1= 1!0<00I 1 =
"he real rate o$ retrn in each year is precisely the <@ real yield on the bond! 20!

Ke#e#ber that the con)ention is to se se#i-annal periods( Price o$ a .ero-Copon Bond =

ace /alue '1$ Semiannu Semiannual al   ( 

Bond E9i)alent Y"M = /e#i-annal Y"M × 2

21!

sin, a $inancial calclator' inpt P = I00'  = 1'000' n = 10' PM" = 0! "he Y"M is 11!11!0! 22! "he reported bond price is( 100 232 percent o$ par = >1'000!?2A0 1A days ha)e passed since the last se#iannal copon was paid' so there is an accred interest' which can be calclated as( 10-? © 2013 by McGraw-Hill Edcation! "his is proprietary #aterial solely $or athori%ed instrctor se! &ot athori%ed $or sale or distribtion in any #anner! "his doc#ent #ay not be copied' scanned' dplicated' $orwarded' distribted' or posted on a website' in whole or part!


Annual Coupon Payment 2 Day s since Last Coupon Coupon Payment Payment Days Separating Coupon Payment

8ccred ;nterest =

×

= >3A × 1A124 = >2!1'003!A0D? ≒ 1'003!A1 23! ;$ the yield to #atrity #atrity is ,reater than than crrent yield' then the bond o$$ers o$$ers the prospect o$ price appreciation as it approaches its #atrity date! "here$ore' "here$ore' the bond is sellin, below par )ale! 2
2?! "he soltion soltion is obtained obtained sin, Ecel(

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2! sin, $inancial $inancial calclator calclator'' n = 10N P = ID00N  = 1'000N PM" = 1<0 "he stated yield to #atrity e9als 1?!0A@! Based on expected copon copon pay#ents o$ >0 annally' the epected yield to #atrity is( !A2A@! 2! "he bond is sellin, at par )ale! ;ts yield to to #atrity e9als the copon rate' 10@! ;$ the $irst-year copon is rein)ested at an interest rate o$ r percent' percent' then total proceeds at the end o$ the second year will be( O100 × 1 F r 4 F 1100! "here$ore' reali%ed co#pond yield to #atrity will be a $nction o$ r as as ,i)en in the $ollowin, table( r

@

"otal "o tal proceed proceedss Keali% Keali%ed ed Y"M Y"M = √ Proceeds Proceeds / 1,000 = 1 1,20 / 1,000 I 1 = 0!0DD1 = D!D1@ √ 1,20 >1'20

10 @

>1'210

1,210 / 1,000 I 1 = 0!1000 = 10!00@ √ 1,210

12 @

>1'212

1,210 / 1,000 I 1 = 0!100D = 10!0D@ √ 1,210

2D! 8pril 1A is #idway #idway thro,h the se#i-annal copon period! "here$ore' "here$ore' the in)oice price will be hi,her than the stated as* price by an a#ont e9al to one-hal$ o$ the se#iannal copon! "he as* price is 101!12A 101 !12A percent o$ par' so the in)oice price is( >1'011!2A F 12 × >A04 = >1'03?!2A

30. Factors that might make the the ABC debt more attractive to to investors, therefore justifying a lower coupon rate and yield to maturity, are:

•

The ABC debt is a larger issue and therefore may sell with greater liquidity.

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•

8n option to etend the ter# $ro# 10 years to 20 years is $a)orable i$ interest rates ten years $ro# now are lower than today5s today5s interest rates! ;n contrast' i$ interest rates are risin,' the in)estor can present the bond $or pay#ent and rein)est the #oney $or better retrns!

•

;n the e)ent o$ troble' the 8BC debt is a #ore senior clai#! ;t has #ore nderlyin, secrity in the $or# o$ a $irst clai# a,ainst real property!

•

"he call $eatre on the QY. bonds #a*es the 8BC bonds relati)ely #ore attracti)e since 8BC bonds cannot be called $ro# the in)estor!

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•

"he QY. bond has a sin*in, $nd re9irin, QY. to retire part o$ the isse each year! /ince #ost sin*in, $nds ,i)e the $ir# the option option to retire this a#ont at the lower o$ par or #ar*et )ale' the sin*in, $nd can wor* to the detri#ent o$ bondholders!

31! a! "he $loatin $loatin,-rat ,-ratee note pays a copon that ad+sts ad+sts to #ar*et le)els! le)els! "here$ore' it will not eperience dra#atic price chan,es as #ar*et yields $lctate! "he $ied rate note there$ore will ha)e a ,reater price ran,e! b! loatin, rate notes #ay not sell at par $or any o$ these reasons( "he yield spread between one-year "reasry bills and other #oney #ar*et instr#ents o$ co#parable #atrity cold be wider than it was wh en the bond was issed! "he credit standin, o$ the $ir# #ay ha)e eroded relati)e to "reasry secrities that ha)e no credit ris*! "here$ore' the 2@ pre#i# wold beco#e ins$$icient to sstain the isse isse at par! "he copon increases are i#ple#ented with a la,' i!e!' once e)ery year! Rrin, a period o$ risin, interest rates' e)en this brie$ la, will be re$lected in the price o$ the secrity! c! "he ris* ris* o$ call is is low! low! Becase Becase the bond bond will al#ost al#ost srely srely not not sell $or $or #ch abo)e par )ale ) ale ,i)en its ad+stable copon rate4' it is nli*ely that the bond will e)er be called! d! "he $ied-ra $ied-rate te note crren crrently tly sells sells at only only D3@ o$ the call call price' price' so that that yield to #atrity is abo)e the copon rate! Call ris* is crrently low' since yields wold ha)e to $all sbstantially $or the $ir# to se its option to call the bond! e! "he D@ copon copon notes notes crrently crrently ha)e a re#aini re#ainin, n, #atrity #atrity o$ o$ $i$teen $i$teen years years and sell at a yield to #atrity o$ D!D@! "his is the copon rate that wold be needed $or a newly issed $i$teen-year #atrity bond to sell at par! par! $! Becase Becase the $loati $loatin, n, rate rate note note pays pays a variable stream o$ interest pay#ents to #atrity' its yield-to-#atrity yield-to-#atrity is not a well-de$ined concept! "he cash $lows one #i,ht want to se to calclate yield to #atrity are not yet *nown! "he e$$ecti)e #atrity $or co#parin, interest rate ris* o$ $loatin, rate debt secrities with other debt secrities is better tho,ht o$ as the net copon reset date rather than the $inal #atrity date! "here$ore' 6yield-to-recopon date7 is a #ore #eanin,$l #easre o$ retrn!



32! a! "he bond bond sells sells $or >1'121'121'0A0' we wold set  = 1'0A0 and redo part a4 to $ind that yield to call is 2!D?3@ se#i-annally' A!DA2A@ annally! Lith Lith a lower call price' the yield to call is lower! c! Yield Yield to call is is 3!0312@ 3!0312@ se#iann se#iannally ally'' ?!0?2A@ ?!0?2A@ annally annally(( On =
300! "here$ore' its its yield to #atrity #atrity is ?!211 11< !?0! "otal "otal taable inco#e is( ><0 F >1A10A0' o$$erin, a #ai## possible ,ain o$ only A@! "he disad)anta,e o$ the !A@ copon bond in ter#s o$ )lnerability to a call shows p in its hi,her promised yield yield to #atrity!

b. If an investor investor expects expects rates rates to fall fall substant substantially ially,, the 4% bond offers offers a greater expected return.



c! ;#plicit ;#plicit call call protection protection is o$$ered o$$ered in the sense sense that that any li*ely li*ely $all in in yields wold not be nearly eno,h to #a*e the $ir# consider callin, the bond! ;n this sense' the call $eatre is al#ost irrele)ant! 3?! "re! nder the the epectations hypothesis' there are no ris* pre#ia bilt into bond prices! "he only reason $or lon,-ter# yields to eceed short-ter# yields yields is an epectation o$ hi,her short-ter# rates in the $tre!

37. If the yield curve is upward sloping, we cannot conclude that investors expect shortterm interest rates to rise because the rising slope could be due to either expectations of future increases in rates or the demand of investors for a risk premium on longterm bonds. In fact the yield curve can be upward sloping even in the absence of expectations of future increases in rates.

3!

39. Uncertain. Lower inflation usually usually leads to lower nominal interest rates. Nevertheless, if the liquidity premium is sufficiently great, long-term yields can exceed short-term yields despite expectations of falling short rates. <0! a! Le s##ari%e s##ari%e the $orward $orward rates rates and crrent crrent prices prices in the the $ollowin, $ollowin, table( table( Matrity years4 1 2 3

Y"M @ 11!0@ 12!0@

orward rate

Price $or part c4

12!01@ 1
>D 0D !0D >11!?2 >11!

Year 1 Price( 1'0001 F 10@4 = D0D!0D Year 2 Price( 1'0001 F 11@42 = 11!?2 orward Kate( 1 F 11@421 F 10@4 I 1 = 0!1201 = 12!01@ 10-12 © 2013 by McGraw-Hill Edcation! "his is proprietary #aterial solely $or athori%ed instrctor se! &ot athori%ed $or sale or distribtion in any #anner! "his doc#ent #ay not be copied' scanned' dplicated' $orwarded' distribted' or posted on a website' in whole or part!


Year 3 Price( 10001 F 12@43 = 11! orward Kate( 1F12@431F11@42 I 1 = 0!1<03 = 1
b. We obtain obtain next year’s prices and yields yields by discount discounting ing each zero’s face value at the forward rates derived in part (a): Matrity years4 1 2

Price ! >   2 !D 3

Y"M O = 1'0001!1201 12!01@ O = 1'0001!1201 × 1!1<034 13!02@

&ote that this year5s pward slopin, yield yield cr)e i#plies' accordin, to the epectations hypothesis' a shi$t pward in net year5s cr)e!

c. Next year, year, the two-yea two-yearr zero will will be a one-year one-year zero, zero, and it will therefo therefore re sell at: $1000/1.1201 = $892.78 Similarly, the current three-year zero will be a two-year ze ro, and it will sell for: $782.93 Epected total rate o$ retrn( "wo-Year Bond(

$"2.# $11.%2

I 1 = 0!1000 = 10!00@

"hree-Year Bond(

$#2."$#11.#

I 1 = 0!1000 = 10!00@

<1! a! "he "he $or $orwa ward rd rate rate $ 24 is the rate that #a*es the retrn $ro# rollin, o)er one-year bonds the sa#e as the retrn $ro# in)estin, in the two-year #atrity bond and holdin, to #atrity( 1 F @4 × 1 F $ 24 = 1 F D@4 2 ⇒ $ 2 = 0!1001 = 10!01@ b! 8ccordin, to the epectations hypothesis' the $orward rate e9als the epected )ale o$ the short-ter# interest rate net year' so the best ,ess wold be 10!01@! c! 8ccordin, 8ccordin, to the the li9idity li9idity pre$erence pre$erence hypothes hypothesis' is' the the $orward $orward rate rate eceeds the epected short-ter# interest rate net year' so the best ,ess wold be less than 10!01@!



42. The top row must be the the spot rates. The spot rates are (geometric) averages of the forward rates, and the top row is the average of the bottom row. For example, the spot rate on a two-year investment (12%) is the average of the two forward rates 10% and 14.0364%: 1 F 0!1242 = 1 F 0!104 × 1 F 0!1<03?<4 = 1!2A<< <3! sin, a $inancial calclator' P = 100' n = 3' PM"=0' i = ?!A! Price o$  = 120!DA! sin, a $inancial calclator' P = 100' n = <' PM"=0' i = !0! Price or  = 131!00! /ettin, P = I120!DA'  = 131!00' n = 1' PM"= 0! r = !A1@! <
$&0  '$#"-.2" － $#0 $#0&.4% ( $#0&.4%

= 0!1DA< = 1D!A<@

b! sin, :;R ta rles' the cost basis and i#pted interest nder the constant constant yield #ethod are obtained by discontin, bond pay#ents at the original @ @ yield to #atrity and si#ply redcin, #atrity by one year at a ti#e( P0 = >0A!11!D' so i#pted taable interest o)er the $irst year is( >11!D I >0A!?!<3 Copon recei)ed and i#pted taable interest in the year are taed as the ordinary inco#e( <0@ × >A0 F >?!<34 = >22!A Capital ,ain = 8ctal price at @ Y"M I Constant yield price = P1 I

'

P1

= >D3!2D I >11!D = >1!<0 "a on capital ,ain = 30@ × >1!<0 = >222!A F >2
$&0  '$#"-.2" － $#0&.4%( － $ 4%." 4%."" " $#0&.4% 10-1<

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= 0!12 = 12!@ d! ale ale o$ the bond bond a$ter a$ter two years e9als >D!2 Osin, Osin, n = 1N i =  "otal "o tal inco#e $ro# the two copons' incldin, rein)est#ent inco#e( >A0 × 1!034 F >A0 = >101!A0 "otal "o tal $nds a$ter two years( >D!2 F >101!A0 = >D00!32 "here$ore' the >0A!D00!32 a$ter two years! 0A!?!<34 &et cash $low in $irst year

>A0!00 I 20!00 I 2!A >2!<3

;$ yo in)est the year-1 cash $low at an a$ter-ta rate o$( 3@ × 1 I <0@4 = 1!@ By year 2' it will ,row to( >2!<3 × 1!01 = >2!D2 Yo Yo sell the bond in the second year $or( P2 = >1!<' so i#pted interest o)er the second year = >?!DA /ellin, price o$ the bond in the second year( "a on imputed interest in second year( Copon recei)ed in second year' net o$ ta( Capi Capittal ,ai ,ains ta on sales ales pri price sin, constant yield )ale( C $ro# $irst yearJs copon rein)ested4( ":"8T

>D!2 I 2!O<0@ × >?!DA F 30!00 O>A0 × 1 I <0@4 I23! I23!DD DD O30@ O30@ × >D!2 I >1!<4

F 2!D2 >2D!D

O$ro# abo)e

"hs' a$ter two years' the initial in)est#ent o$ >0A!2D!D( 0A!
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C8 1

8nswer( a! 34 "he "he yield yield on the callabl callablee bond #st #st co#pensat co#pensatee the in)estor in)estor $or the the ris* o$ call! Choice 14 is wron, becase' altho,h the owner o$ a callable bond recei)es principal pls a pre#i# in the e)ent e)en t o$ a call' the interest rate at which he can sbse9ently rein)est will will be low! "he low interest rate that #a*es it pro$itable $or the isser to call the bond #a*es it a bad deal $or the bond5s holder! Choice 24 is wron, becase a bond is #ore apt to be called when interest rates are low! "here will be an interest sa)in, $or the isser only i$ rates rates are low! b! 34 c! 24 d! 34

C8 2 8nswer( a! "he #atrity #atrity o$ each bond bond is 10 10 years' years' and we ass#e ass#e that that copons copons are are paid se#iannally! /ince /ince both bonds are sellin, at par )ale' the crrent yield to #atrity $or each bond is e9al to its copon rate! ;$ the yield declines by 1@ to A@ 2!A@ se#iannal yield4' the /entinal bond will increase in )ale to 10!D On=20N i = 2!AN  = 100N PM" = 3! "he price o$ the Colina bond bon d will increase' bt only to the call ca ll price o$ 102! "he present )ale o$ schedled pay#ents is ,reater than 102' bt the call price pts a ceilin, on the actal bond price! b! ;$ rates are epected to $all' the /entinal bond is #ore attracti)e( /ince it is not sb+ect to bein, called' its potential capital cap ital ,ains are hi,her! ;$ rates are epected to rise' Colina is a better in)est#ent! ;ts hi,her copon which pres#ably is co#pensation to in)estors $or the call $eatre $eatre o$ the bond4 will pro)ide a hi,her rate o$ retrn than that o$ the /entinal bond! c! 8n increase increase in the the )olatility )olatility o$ rates rates increase increasess the )ale )ale o$ the $ir#5 $ir#5ss option option to call bac* the Colina bond! ;$ rates ,o down' the $ir# can call the bond' which pts a cap on possible capital ,ains! /o' hi,her )olatility #a*es the option to call bac* the bond #ore )alable )alable to the isser! isser! "his #a*es the Colina bond less attracti)e to the in)estor!

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C8 3 8nswer Mar*et con)ersion )ale = ale i$ con)erted co n)erted into stoc* = 20!3 × >2 = >A3!2< Con)ersion pre#i# = Bond )ale I Mar*et con)ersion )ale = >A I >A3!2< = >1D1!? C8 < 8nswer( a! "he call call pro)ision pro)ision re9ires re9ires the the $ir# $ir# to o$$er o$$er a hi,her hi,her copon copon or or hi,her hi,her pro#ised yield to #atrity4 on the bond in order to co#pensate the in)estor $or the $ir#Js option to call bac* the bond at a speci$ied call price i$ interest rates $all s$$iciently! ;n)estors ;n)estors are willin, to ,rant this )alable option to the isser' bt only $or a price that re$lects the possibility that the bond will be called! "hat price is the hi,her pro#ised yield at which they are willin, to by the bond! b! "he call option redces the epected li$e o$ the bond! ;$ interest rates $all sbstantially so that the li*elihood o$ a call increases' in)estors will treat the bond as i$ it will U#atreU and be paid o$$ at the call date' not at the stated #atrity date! :n the other hand i$ rates rise' the bond #st be paid o$$ at the #atrity date' not later! "his asy##etry #eans that the epected li$e o$ the bond will be less than the stated #atrity! c! "he ad)anta,e ad)anta,e o$ a callabl callablee bond is the the hi,her hi,her copon copon and hi,her pro#ised yield to #atrity4 when the bond is issed! ;$ the bond is ne)er called' then an in)estor will earn a hi,her reali%ed co#pond yield on a callable bond issed at par than on a non-callable bond issed at par on the sa#e date! "he disad)anta,e o$ the callable bond is the ris* o$ call! ;$ rates $all and the bond is called' then the in)estor recei)es the call price and will ha)e to rein)est the proceeds at a t interest rates that are lower than the yield to #atrity at which the bond was ori,inally issed! ;n this e)ent' the $ir#Js sa)in,s in interest pay#ents are the in)estorJs loss! C8 A 8nswer( a! 14 Crrent Crrent yield = CoponPri CoponPrice ce = >0>D?0 = 0!02D = !2D@ !2D@ 24 Y"M = 3!DD3@ se#iannally se#iannally or !D?@ !D?@ annal bond e9i)alent e9i)alent yield yield On = 10N P = ID?0N  = 1000N PM" = 3A "hen co#pte the interest rate!

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34 Keali%ed Keali%ed co#pond yield is 3A each' rein)ested se#iannally at a per period rate o$ 3@( OP = 0N PM" = >3AN n = ?N i = 3 Co#pte  = >22?!3D "he bond will be sellin, at par )ale o$ >1'000 in three years' since copon is $orecast to e9al yield to #atrity! "here$ore' total proceeds in three years will be >1'22?!3D! "o $ind reali%ed co#pond yield on a se#iannal basis i!e!' $or si hal$-year periods4' we sol)e( ? >D?0 × 1 F r reali%ed reali%ed4 = >1'22?!3D ⇒ r reali%ed reali%ed =
b! /hortco#in,s o$ each #easre( 14 Crrent yield does not accont $or capital ,ains or losses on bonds bo,ht at prices other than par )ale! ;t also does not accont $or rein)est#ent inco#e on copon pay#ents! 24 Yield Yield to #atrity ass#es ass#es that the bond is held to to #atrity and that all copon inco#e can be rein)ested at a rate e9al to the yield to #atrity! 34 Keali%ed co#pond yield hori%on hori%on yield4 is a$$ected by the the $orecast o$ rein)est#ent rates' holdin, period' and yield o$ the bond at the end o$ o$ the in)estorJs holdin, period!

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Bodie Essentials Ch. 10 Solutions

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