Call Able Bonds 3

8/12/2019 Call Able Bonds 3

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AdvancedFixedIncome CallableBonds ProfessorAnhLe

1Whatarecallablebonds?

Whenyoutakeoutafixedratemortgagetobuyahouse,youusuallyhavetheoptionofprepayingthe

mortgage.Thecommontermtouseistorefinance.Andpeoplewouldrefinancetheirmortgages

whenitischeaptodosowheninterestratesarelow.

Callablebondsareverysimilarexceptthatnowcompaniesaretheborrowers.Theyissuecallablebonds

toborrowmoneyforwhateverreason(notnecessarilytobuyhouses).Beingcallable,suchbondsgive

themtherighttocallhomethebondsprepaytheirborrowingswhentheyseefit,whichusually

meanswheninterestratesarelow.

Topayoffthebonds,theissuersusuallyhavetopaytheholderthefacevalueofthebonds.Formany

callablebonds,however,theissuerswillneedtopaysomepremiumontopofthefacevalue.This

premiumactsassomecompensationforthelenderswhouponbeingprepaid,havetofindnew

borrowersatgenerallylowerinterestrates.Thepricethattheissuershavetopayisthecallprice.

Sincecallablebondsareattractivetoborrowers,theyaredislikedbylenders.Althoughlendersgetcompensatedthroughhighercouponrateswithcallablebonds,totonedowncallriskswithcallable

bonds,manyissuersintroduceacallprotectionperiodduringwhichacallablebondcannotbecalled.A

typicalcallablebondstructurewilllooklike10NC5:whichmeansthebondhas10yearstillmaturity

andonlycallableafteryear5.

2Whyarecallablebonds?

Itisobviousthatcallablebondsgiveborrowerstheoptiontorefinancewheninterestratesarelow.In

otherwords,itisonewaycompanieshedgeagainstpossibledecreasesinfutureinterestrates.Forthis

reason,callablebondsareverypopularbefore1990.Infact,before1970almostallcorporatebonds

wereissuedwithcallfeatures.Between1970and1990,about80%offixedratecorporatebondswere

callable.Duetothedevelopmentoftheinterestratederivativesmarketsinthelateeightiesandearly

nineties,therehasbeenabigdropincallablebondsissuancenowaccountingforonly30%ofthe

total.Thisisunderstandablesincewithderivatives,itbecomesevereasiertohedgeagainstinterestrate

risks.Withcallablebonds,providersofcapital(lenders)alsoactasinsuranceproviders.Thismaynotbe

necessarilyoptimal thesamewayapersonmaynotbegoodatbothtennisandfinance.Duetocost

savingsfromspecialization,companiesmayfinditmorecosteffectivetoborrowbyissuingstraight

bondsandbuyinsuranceagainstinterestraterisksfromspecializedinsuranceproviders.

However,anotherreasonwhyfirmsmaystillfindcallablebondsdesirableisthatbyissuingcallable

bondstheycansendastrongpositivesignaltothemarketsaboutthequalityoftheirbusiness.The

reasoninggoesasfollows:Ifafirmisconfidentabouttheirbusinessandbelievesthattheircreditquality

willimproveinthefuture(whichwilllowertheirborrowingcosts),itmakessenseforthemtoissuea

callablebond.Assoonasthemarketrealizestheirbettervalues,theycansimplycalltheoldexpensive

bondandreplaceitwithabondwhichpayslowercoupons.Ontheotherhand,ifafirmknowsthatthey

arenotdoingparticularlywellandtheircreditqualityisverylikelytodeteriorate,itmakessensefor

themtoissueanoncallablebondtolockinaborrowingrate.


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3Yieldstocallandyieldstoworst

Tradersliketothinkintermsofyieldtomaturitysimplybecauseitisseeminglyeasiertounderstand.A

bondistradingatayieldof5%seemsmorestraightforwardascomparedtoabondtradingat95.24%

offacevalue.Forthisreason,marketshavecomeupwithyieldsmeasuresforcallablebondsaswell.We

willtalkaboutthesemeasuresinthissection.

Strictlyspeaking,yieldtomaturityisoutofquestionforcallablebonds.Thesimplereasonisthatcallable

bondsdonthavefixedmaturities.Takeforanexample,the10NC5bond(10yearstatedmaturity,only

callableafter5years).Iftheissuer,forsomereasons,decidestocallthebondatendofyear

5/beginningofyear6,thematurityofthebondis5years.However,itisalsopossiblethattheissuer

mayletthebondliveuntilitsusualmaturityof10years.Withoutafixedmaturity,wearenotcertain

aboutthecashflowseitherandassuchayieldtomaturitycannotbecomputed.

However,tradersareinlovewithyieldsmeasuresandthustheyhavecomeupwithatleast2waysof

computingyieldsforcallablebonds.

First,theyassumethatthebond,thoughcallable,willnotbecalledatallduringitsentirelife.Inour10NC5bondexample,thismeansthebondsmaturitywillbe10years.Yieldcomputed

withthisassumptionisstillcalledyieldtomaturity.

Second,theyassumethatthebondwillbecalledwithcertainty.Inour10NC5bondexample,thismeansthatthebondwillmatureatyear5.Yieldcomputingwiththisassumptionisyieldto

call.Manycallablebondshoweverhavemultiplecalldates.Forexample,our10NC5bondcan

becalledanytimeafteryear5untilyear10.Inthiscase,weneedtobeveryspecificaboutthe

callassumption.Ifweassumethatthebondwillbecalledattheendofyear5withcertainty,

strictlyspeaking,theresultingyieldwillbecalledyieldtofirstcall.

Toavoidpossibleconfusion,letmegiveasimpleexampleforustoquicklygrasptheconcept.Tomakeit

simple,letsworkwitha2yearbondthatcanonlybecalledatendofyear1foracallpriceof$100.This

bond,currentlysellingfor$99,hasafacevalueof$100andispayingasemiannualcouponrateof8%

p.a.

Yieldtomaturity

Tocomputeyieldtomaturityofthiscallablebond,wewillmaketheassumptionthatthebondwillbe

heldtomaturityregardless.Therefore,thecashflowsfromthebondwillsimplybe:

Attime0.5: $4 Attime1.0: $4 Attime1.5: $4 Attime2.0: $104

Theyieldtomaturityofthebondwillthenbeysuchthat:


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99 41 2

41 2

4

1 2

1041 2

Solvethisfory,wehave:y=8.55%.

Yieldtocall

Tocomputeyieldtocallofthiscallablebond,wewillmaketheassumptionthatthebondwillbecalled

withcertainty.Therefore,thecashflowsfromthebondwillbe:

Attime0.5: $4 Attime1.0: $4+$100=$104

Theyieldtocallofthebondwillthenbeysuchthat:

99 4

1 2 104

1 2

Solvethisfory,wehave:y=9.07%.

Yieldtoworst

Foracallablebond,yieldtoworstissimplytheminimumbetweentheyieldtomaturityandtheyieldto

call.Intheaboveexample,yieldtoworstissimplyminimumof(8.55%,9.07%)=8.55%.

Awordofcaution

LetsassumethatwegoouttoaBloombergterminaltocheckoutpricesofbondsofcomparablecredit

qualitytothecallablebondaboveandfindoutthefollowing:

A2yearnoncallablebondistradingatayieldof8.5%(orapriceof$99.10) A1yearnoncallablebondistradingatayieldof8.4%(orapriceof$99.62)

Comparingthepricinginformationheretothatofthecallablebond,itseemsreallyweird.Fromour

calculations,

thecallablebondoffersayieldof8.55%ifitisheldtomaturity.Inthiscase,itscashflowsareexactlythesameasa2yearnoncallablebondwhichoffersayieldofonly8.5%.

thecallablebondoffersayieldof9.07%ifitiscalledregardless.Inthiscase,itscashflowsareexactlythesameasa1yearnoncallablebondwhichoffersayieldofonly8.4%.

inotherwords,worstcomestoworst,thebondearnsayieldtoworstof8.55%whichisstillbetterthaneitheroftheyieldsofferedbythe1yearor2yearnoncallablebond.

Itseemsthatthecallableoffershigher(thannecessary)yieldswhencomparedtothenoncallable.Putit

differently,thecallableissellingfor$99whichischeaperthanbothofthe1yearand2yearnon

callable.Whatisgoingon?Isthemarketnotfunctioningwell?Orarewemissingsomething?


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Itturnsoutthatifthemarketisfunctioningwell,thecallableoughttobecheaperthanboththe1year

aswellasthe2yearnoncallable.Toseewhythecallableshouldbecheaperthanthenoncallable

letscomparetheircashflows:

s,

time 1yearNon

callable

2yearNon

callable

callable

0.5 $4 $4 $4

1.0 $104 $4+Marketprice =$4+minimumof($100,market

priceattime 2yearnon

callable)

attime1.0 1.0ofthe

Thesecondcolumncontainscashflowsoft nca .The

thirdcolumncontainsthecashflowsofthe2yearnoncallableuptotime cashflowisof

oursethe$4couponattime0.5.Tocomeupwithacashflowforthe2yearnoncallableattime1.0,I

assumewecollectthecouponof$4andsellthisbondattime1.0.Thecashflowattime1forthisbond

ehowtheissuerofthecallablemakeshis/herdecisionattime

1.0.Itturnsouttobequitesimple.Sincethecallgivestheissuertherighttobuybackthe2yearnon

d

callable,whichislessthanthe

cashflowofthe1yearnoncallable.

.

eenthoseofthe1yearnoncallableand2yearnon

callable.Inotherwords,comparedtoeitherofthenoncallable,thecallableentailsstrictlylessthanor

he1yearno llablewhichisquitestraightforward

1.0.Thefirst

c

willbe$4+itsmarketpriceattime1.0.

Thelastcolumnofthetablecontainscashflowstothecallable.Nothingisspecialaboutthefirstcash

flowsimplyacouponof$4.Thecashflowattime1.0,however,iscrucialsincethisiswherethebond

issuercanexercisetheircallright.Letsse

callablebondattime1.0forapriceof$100,itonlymakessensefortheissuertobuythe2yearnon

callablebackifitissellingformorethan$100attime1.0.Therefore:

Ifthemarketvalueofthe2yearnoncallableislessthan$100,theissuershallnotcallthebonthecallablewillbethesameasthe2yearnoncallable.Inthiscase,thecashflowofthe

callableattime1.0willbethesameasthatofthe2yearnon

Ifthemarketvalueofthe2yearnoncallableisgreaterthan$100,theissuerwillcallthebondthecallablewillbethesameasthe1yearnoncallable.Inthiscase,thecashflowofthe

callableattime1.0willbethesameasthatofthe1yearnoncallable,whichislessthanthe

cashflowofthe2yearnoncallable.

Ascanbeseen,theissuersobjectiveistominimizehis/hercashflowobligationsofthecallablebond

Therefore,byexercisingthecallfeatureoptimally,theissuermakessurethatthecashflowofthe

callableattime1.0willbetheminimumbetw

equalcashflows.Assuchitisobviousthatthecallablehastobecheaperthanboththe1yearand2

yearnoncallable.

4Valuationofacallablebond

Youagreethatthecallablebondaboveshouldsellforlessthanthe1yearnoncallableaswellasthe2

yearnoncallable,butexactlyhowmuchless?Ofcourseitiseasyifyouknowthemarketprice.Butwhat


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ifthebonddoesnttradethatfrequent?Sothatyouknow,80%ofbondsnevertrademorethanoncea

year.Insuchinstances,tovalueacallablebond,ourmodelingknowledgebecomeshandy.Thisis

becauseitisquitestraightforwardtovalueacallablebondifwehaveaninterestratetree.Letsassume

wehavethefollowingtreeofsemiannualinterestrates.

0 0.5 1 1.515.44%

11.91%

9.19% 11.44%

7.09% 8.83%

6.81% 8.48%

6.54%

6.28%

Asyoumaynotice,therearesomecrazyinterestrates( 44%)inthetree,butthatsallright,every

tree,ifextendedlongenough,wouldgivethat.Importantly,proba forextremeoccurrencesare

uitesmall.Andalso,ourcurrentfocusisonhowtousethetreeforpricing,nothowreasonablethe

treeis.

like15.

bilities

q

Pricingofthe2yearnoncallable

Letsstarttoseehowwecanusethistreetopricethe2yearnoncallable.AndIpromisethatitisa

smoothtransitionfrompricingnoncallablebondstopricingcallablebonds.Rememberthatthecash

:

fcoursedotheusualthingbywalkingbackwardsalongthetree,startingat

Attime1.5,wearenotsurewhatthepriceofthebondwillbe,butweknowthatthereare4thebondwillsimplybe

.%

flowsfromthisbondareasfollows

Attime0.5: $4 Attime1.0: $4 Attime1.5: $4 Attime2.0: $104

Topricethisbond,wewillo

time1.5.

scenarios.Ifthe6monthinterestrateis15.44%,theprice(excludingthe$4couponattime1.5)

of 96.55.Similarly,ifthe6monthinterestrateattime1.5is

1.At

neutralpricingequation.Forexample,ifweareinthehighestnodeattime1,thepriceofthe

.%

11.44%or8.48%or6.28%,thevalueofthebondattime1.5wouldbe$98.37,$99.77,$100.83

respectively.

Letstakeastepbacktotime time1,thepriceofthebondcanbecomputedusingtheriskbondwillbe

... $95.75.The$4inthenumeratoris,ofcourse,thecouponthat

wewillreceiveattime1.5regardlessofwhereweare(goingupordown).Ifweareinthe


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middlenode(orthelowestnode),bysimilarcalculations,thepriceofthebondwouldbe98.72

(or101.00).

Nowletstake e0.5.Ifyouareintheuppernode(orlowernode),byvery Finally,takeastepbacktothecurrenttimetime0.Applyingtheriskneutralpricingequa

onelasttime,thepriceofthebondis...

astepbacktotim

similarcalculations,thepriceofthebondwouldbe$96.79(or$100.44).

tion

.% $99.10.

Puttingthebondsvaluesateverynodeofthetreetogether,wewillhavethefollowingpricetree.This

wouldbeattime0.5. owthatitcouldonlybe

either96.79or100.43withequalriskneutralprobabilitiesof50%.The96.79(100.43)pricecorresponds

pricetreecorrespondstotheinteresttreethatwestartwith.Thewayweinterpretthistreeisthesame

ashowweinterprettheinterestratetree.Forexample,weknowthepriceofthebondis$99.10now,

butwearenotsurewhatthepriceofthebond Butwekn

tothescenariowheninterestrateis9.19%(6.81%)attime0.5.Andsoon,eachoftheprice/valuewe

seeherecorrespondstooneinterestratenodeweseeontheinterestratetree.

0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

99.10046 98.716

100.4394 99.7719

101.0012

100.8344

OK,youunderstandwhywen erestratetree becauseitallowsustopricethebondabove.Butonceyouhavetheprice(attim that erputtingalltheprices

togethertobuildtheabovepricetree?Whatpurpose rv soutthat,fromtheabove

ee,thepriceofthe2yearcallablebondisonlyafewcalculations nowturntohowwecan

usethetreetopricethe2yearcallablebondwithacallpriceof$100.

eedanint simplye0),isnt theend?

doesitse

Whyboth

e?Itturn

away.Letstr

Pricingofthe2yearcallable

Inpricingthe2yearcallable,thekeyisjusttorememberthatattime1theissuerofthecallablewill

optimallyusehis/hercallright.

Attime1.5,ifthecallablehasnotbeencalled,itwillbethesameasa2yearnoncallablebond.thecallableandthenoncallablemustbeidenticalateverynodeofthe

aycallthebond.However,theissuerwillcallthebondonlyifthevalue

ofthebondishigherthanwhatheneedstopayincallingit:$100.Checkingthe3scenariosat

time1,itonlymakessensefortheissuertobuybackthebondifthevalueofthebondis

$101.00.Paying$100forthebond,effectivelytheissuernets$1.00thankstothecallfeatureof

Assuch,thevaluesof

treeattime1.5.

Attime1.0,theissuerm


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thebond.Onthecontrary,itwillnotmakeanysensefortheissuertocallbackthebondifits

totalvalueiseither95.75or98.72.Insuchinstances,itisbetterfortheissuertoleavethebon

uncalled.Toprice

d

thecallable,therefore,requiresonemodificationinthebondvalueattime1

onthelowestnode.Insteadof101.00,sincetheissuerwouldoptimallycallthebondhere,the

valueofthecallableshouldreallybe$100atthisnode.Thismodification,inturn,willlowerthe

valueofthecallableattime0.5(lowernodeonly)andultimatelythepriceofthecallableat

time0.

Steppingbacktotime0.5,thetotalvalueofthebondattheuppernoderemainsunchanged.Thetotalvalueofthebondatthelowernodehoweverwillchangeto

...%

$99.96. Finally,letstakethestepbacktotime0.Thepriceofthecallablewouldbe:

....%

$98.87.

Puttingallthevaluesthatwejustcalculatedaboveinatree,wehave:

0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

98.86668 98.716

99.9552 99.7719

100

100.8344

Notethatthedifferencestothetreeofnoncallable sarehighlightedwiththebluecolor.

Thesearethenodesthatareaffectedbyt beingcalledat thatthesenodes

correspondtothelowestbran whereinte sarelow.Thismakessensebecause,as

weknow,bondsarebestcalled/refinanced terestratesar

Pricingofthe2yearcallableWhatifthebondiscallableattim ll?

bondprice

hebonds time1.Note

chofthetree restrate

whenin elow.

e1.5aswe

Ialwaysliketostartthingsoutsimple.ThatswhyIveillustratedhowtopricethecallablebond

e

ent

tnowcallableeitherat

time1withacallpriceof$102orattime1.5withacallpriceof$100.Thekeytopricingthisbondis

it

assumingthatwecanonlycallthebondattime1.Youcanaskthequestionofwhatifthebondcanb

calledattime1.5aswell.Infactmanybondsallowformultiplecalldates.Further,whatifatdiffer

calldates,wehavedifferentcallprices?Fortunately,thoughmorecomplicated,alltheseconcernscan

beaddressedusingthesameframeworkthatwevejustgonethrough.

Letsconsiderthesamecallablebond:face$100,semiannualcouponof8%,bu

simplytostartwiththepricetreeofthenoncallable,andthencheckattime1.5and1,whether

makessensetocallthebondatanyofthenode.


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First,letscheckifitsoptimaltocallthebondattime1.5.Rememberthatthecallpriceattime1.5is

$100.Ifhe/shedoesntcallthebondanddecidestoletthebondliveuntilmaturity,thebondsvaluewill

bethesameasthatofitsnoncallablecounterpart.Assuch,startingwiththepricetreeforthenon

callablebondandfocusonitsvalueattime1.5,wewillbeabletotellwhentheissuerwouldcallthe

bond:onlywhenthevalueofthebondexceeds100.

0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

99.10046 98.716

100.4394 99.7719

101.0012

100.8344

0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

99.10046 98.716

100.4394 99.7719

101.0012

100

Thetreeontheleftisth ncallablebondprice.Ifyouarewonderingwhe tree

from,Ijustco stedfroma nwewerepricingthenon bond.Ifyoua

f wegotthesenu pleasegoback seem calculationsup now,I

onlypaintor odesattime1 lightthefactthatweare whether

callableattime1.5.

xaminingthe4possiblescenariosattime1.5,itiseasytoseethattheissuerwillonlycallthebondat

thelowestnodewherethevalueofthebondifletalive(untilmaturity)is$100.8344.Assuchforthe

itis

ndanywhereattime1.Youmaybethinking:easystuffweddothesamething

again,checkingforallpossiblescenariosattime1andcomparingthevalueofthenoncallablebondto

tthelowestnodeofthetreeattime1?Itis

simplytheriskneutralexpectedcashflowsdiscountedattheriskfreerateof6.54%.Attime1.0,ifyou

etreeofno reIgotthe

piedandpa bovewhe callable lready

orgothow mbers, and ydetailed there.For

angethen .5tohigh onlychecking itis

E

treeontheright,Ireplacethevalueofthebondatthelowestnodeby100andpaintthenodeblueto

showthatthebondwouldbecalledifwegettothisnode.

Sothatisdoneattime1.5.Thesecondthingwewouldliketodoistogobacktotime1andcheckif

optimaltocallthebo

thecallprice.Ifyouthinkso,thatwouldbetoofast!Beforewegettothatstage,weneedtoadjustthe

bondvaluesattime1toreflectthechange(s)wehavemadetothetreeattime1.5.

Specifically,rememberhowwegetthevalueof101.0012a

areinthelowestnode,weknowthatthevalueofthebondwouldbeeither99.7719or100.8344.As

such,thevalueofthebondincludingthecouponwouldbe:

....% $101.0012.Thatis,

forthenoncallablebond.Nowforthecallable,wealreadyworkoutthatifthevalueofthebondis

100.83attime1.5,theissuerwillcallthebond.Assuch,thevalueofthebondthebondonthelowest..

nodeattime1.0wouldbe:.%

$100.60.


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0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

99.10046 98.716

100.4394 99.7719

101.0012

100

0 0.5 1 51.

96.5451

95.75492

96.7878 98.3727

99.10046 98.716

100.4394 99.7719

100.6

100

Onlyafteradjustingthenodesofthe 1asshownabove,wecanproceedandcheck

itisoptimalfortheissue bondattime1.Rememberthatthecallpricea nt.

It up tree time1, value bond $95.75.

is not pay for d node. ertwo s

1,itturnsout for tocallthebondeither. that is

duetothehighcallprice time1. ecallpriceattime1werestill$100,the wouldptimallycallthebondatthelowestnode he/shewouldpay$100forthebondthatisworth

$100.6.

treeattime whether

rtocallthe ttime1isdiffere

is$102.Examining

therefore

the mostnod

$102

eofthe at

atthat

thetotal

Similarly,

ofthe isonly

node

It

worthto

thatitisnot

thebon

theissuer

fortheoth

Youcansee

attime

simplyoptimal

($102)at

this

issuerIfthwhere

o

Now,goingbacktotime0.5,weneedtoadjustthevalueofthebondatthelowestnodeattime0.5as

well.Thismodificationisnecessaryduetothechangeswemadetothevaluesofthebondsattime1.

Thetotalvalueofthebondatthelowestnodeattime0.5shouldbe:...

.% $100.24.

Similarly,goingbacktotime0,thevalueofthebondshouldbe:....%

$99

0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

99.10046 98.716

100.4394 99.7719

100.6

104

0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

99 98.716

100.24 99.7719

100.6

104

5Spreadsd onalityuetoopti

Lets s dthink we done:seemingly,all c

usingtreesofinterestrates! f calculationsmakeusmissdearlythesimple

thatweusedtodo:allweneedisjustay curveandthenwewilljustdiscount1year flows

the1yeardiscountrate,2yearcash usingthe2yeardiscountrateandsoon,allweneed

takea tepbackan aboutallthat have ofasudden,discou

wepri ebonds

cisesAloto ntingexer

cashield

flowsusing to


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careaboutistheconsistencybetweenthetimingofthecashflowsandthehorizonoftheinterestrates.

Allweknowis:allofasudden,lifegetssocomplicated!Canwegetbacktothesimplediscounting

alculations?

reethatwestartwith(whichIputbelowtosaveyoutimeflippingback

c

Alright,letsdothat.Fromthet

thepages),Icancomputetheinterestratesfordifferenthorizons.

0 0.5 1 1.5

15.44%

11.91%

9.19% 11.44%

7.09%

Horizon Interest

0.5 7.09%

1 7.54%8.83%

6.81% 8.48%

6.54%

6.28%

1.5 8.03%

Iassumethat youkn howt interestratesofdifferenthorizonsfromaninterestrate

tree.Andassuch,Iwontshowth ofmy Howeve r inghowI

gottheabove rates, should backto erestRate els.

Pri noncallab

2 8.55%

allof ow ocalculate

edetails calculationshere.

mynotesonInt

r,ifyoua ewonder

interest you go Mod

cingofthe lebond:

Giventheaboveinterest ordertopricethe callablebond, wene oisto

discountitscashflowsusingtheappropriateinterestrates:

4

rates,in 2yearnon all edtod

1 7.09%2 1 7.54%2

4

1 8.03%2

4

1 8.55%2

104

$99.10

Youcanseethattheprice,$99.10,matcheswhatwegotbeforefromtheinteresttree.

Pricingofthecallablebond:

However,whenitcomestopricingthecallablebond,withoutthetree,wearestuck!Fromourearlier

calculations,weunderstandthatthepriceofthecallableislower

thanthepriceofthenoncallableand

shouldbe$98.87.Butitseemsthat,withouthavingthetreetodeterminewhenitisoptimaltocallthe

bond,wewontbeabletoarriveatthisprice.

Sometraders,however,dontliketocarryabulkytreearound.Theypreferthe ofthefamiliar

ble(duetoits

callability)willbecheaperthanitsnoncallablecounterpart,inordertopricethecallablebond,they

ratesusedtopricethenoncallable.Letssaythespreadis13basis

easiness

discountingexercises.Assuch,theydecidetodothefollowing:sincetheyknowthecalla

willaddaspreadtothediscount

points.Thepriceofthecallablebondwillbe:


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41 7.09%0.13%2

4

1 7.54%0.13%2

4

1 8.03%0.13%2

104

1 8.55%0.13%

2

$98.87

whichexactlymatchesthepriceofthecallablebondwehaveabove.IamsureyouknowhowIcome

withsuchaspreadthatgivestheexactpriceof

up

$98.87Solver,whatelse?

Oncewehavethisspread,itisseeminglyconvenientbecausewecanthencarrythespreadaroundand

priceothercallablebondsbyaddingthesamespreadtotheirdiscountrates.Thispracticeisdangerous,

however,sincethevalueofacalloptionisdifferentfrombondtobonddependingontheircoupon

rates,theircallpriceetc.Assuch,pleasebecarefulifyoueverdothisatwork.Ifyoutreasuresafety,I

wouldrecommendusinganinterestratetree.

6Zerovolatilityspread(orZspreadorstaticspread)

Wehavebeenusingtheriskfreeinterestratetreetopricethesetwobondswiththeimplicit

assumptionthattheycomewithoutdefaultrisk.Thisisnotreasonable.Infact,accountingfordefault

risk,liquidityrisketc.,pricesofthebondswouldbelowerthanwhatwehadpreviously.Letsassume

thatbecauseofthesefactors(defaultrisk,liquidityrisk),thecallableisonlysellingfor$97.33insteadof

$98.87.Tolookforaspreadforthisbond,weagainchooseanumbersthatwhenaddedtotheriskfree

discountrateswillrecoverthemarketpriceof$97.33.

$97.33 41 7.09%

2

4

1 7.54%

2

4

1 8.03%

2

104

1 8.55%

2

ofthebondbutalsothecreditandliquidityrisks

associatedwithit.

Thisspreadiscalledthezerovolatilityspreadorthezerospreadorthezspreadorthestaticspreadof

thebond.Nowzspreadorzerospreadisjustashortformforzerovolatilityspread.Whyi itcalled

is

erestrates(likewhatwehavehere)asopposedtoonethatcomesoffatreelacks

thevolatilityelement,henceiscalledzerovolatilityspread.

UsingSolver,Ihaves=100basispoints.Notsurprising!Fromthelastsection,evenwithoutcreditand

liquidityrisk,andjustduetooptionality,wealreadyhaveapositivespreadof13basispoints.Nowthe

bondhasmorerisksattachedtoit,thepriceisreducedtoreflecttheextrarisks,assuchthespread

shouldbelargertoaccountfornotonlytheoptionality

s

zerovolatilityorstaticspread?Well,itisstaticrelativetootherspreadsthatcomeofftheinterestrate

treethatwewouldconsiderinthenextsection.Looselyspeaking,aspreadthatcomesoffabulkytreeseemsmoredynamic.Likewise,withatree,we,sortof,seethevolatilityofinterestrates.Ifthetree

fat,interestratesarevolatile,ifitisthin,interestratesarestable.Assuch,aspreadthatcomesfrom

justtheriskfreeint

Namingbusinessaside,twothingsareimportantaboutstaticspreads:


12/24


Ittakestheshapeoftheyieldcurveintoaccount(sinceitisaconstantspreadaddedtoeachofthediscountrateforeachmaturityweneedayieldcurvetocomputethespread)

Itissomesortofatotalspreadsinceitincludeseverything:someelementofoptionality,someelementofcreditrisk,someelementofliquidityrisketc.

7

Option

adjusted

spread

Optionadjustedspreadisanimportant(thoughpotentiallyconfusing)conceptoftenusedincontextsof

callablebond,mortgage,mortgagebacksecuritiespricing.

wing

enticalbondissuedbythesame

issuerexceptthatitisnoncallable.Imaketheprevioussentenceboldtoshowthatitisimportantto

tterunderstandingtheconceptofoptionadjustedspread.

thenoncallablebecausethezspread

ofthecallablebondincludeseverything.Itincludesnotonlycreditrisk,liquidityriskbutalsothe

ty

o

me

Naturally,therefore,wewouldliketotakeawaythepartthatisduetooptionalityofthecallableand

the

Staticspread=optionadjustedspread +spreadduetooptionalityofthebond

appropriate!Thattreewasdefaultfree.

Nowthatourbondsaresubjecttodefaultrisksandliquidityrisks,weneedtodiscounttheircashflows

heavier.An spreads

toeachoftheinterestratesinthebinomialtree.Thatway,wewoulddiscountthebondscashflows

Tounderstandoptionadjustedspreadaswellaswhyithassuchaname,thinkaboutthefollo

situation:Weobservethepriceofthecallablebondtobe$97.33andwewouldliketousethis

informationtomakesomeinferencesregardingthepriceofanid

bearthiscontextinmindinbe

Alright,fromthecalculationsintheprevioussection,weknowthatthezspreadofthecallablebondis100basispoints.But,ofcourse,wecantusethisspreadtoprice

optionalityofthecallablebond.Whilethepartofthespreadthataccountsforcreditriskandliquidi

riskshouldbethesameforboththecallableandnoncallablebonds,thenoncallablebondhasn

optioninit.Assuch,itwouldbeunfairtopricethenoncallablebondusingaspreadthatincludesso

optionalitycomponentinit.

usetheremainingparttopricethenoncallablebond.Thismakessensebecauseifyoutakeawaythe

optionalitycomponent

from

the

callables

zspread,

the

remaining

spread

must

be

due

to

credit

risks

andliquidityriskswhicharethesameforboththecallableandnoncallable.Thisspreadiscalled

optionadjustedspread.Thenamederivesfromthefactthatwestartfromthestaticspreadofa

callableandinordertopricethenoncallable,weneedtoadjustthespreadfortheoptionality

componentinit.Totieeverythinginanequation,wehave:

Buthowwouldwedothat?Howcouldwedisentangletheoptionandnonoptioncomponentsofthe

staticspreadofthecallablebond?Theanswer:Weneedaninterestratetree.Youmayhaveawhya

treequestionrightnow,butletmedeferansweringthatquestionlater,letmeshowyouhowwefind

theoptionadjustedspreadfromatreefirstandthenexplainwhylater.

Firstofall,theinteresttreethatweusedbeforeisnolonger

dweneedtodothatateverynodeofthetree.Thissuggeststhatweneedtoadda

heavierateverynode.


13/24


15.44%+s15.44%

11.91% 11.91%+s

9.19%+s 11.44%+s9.19% 11.44%

7.09% 8.83%

6.81% 8.48%

6.54%

6.28%

7.09%+s 8.83%+s

6.81%+s 8.48%+s

6.54%+s6.28%+s

Inlookingfortheoptionadjusted thecallablebondwhichissellingatt$97.33,Iwi

spreadsinawaythatw theresultingtree(ontherightabove)toprice ,itwould

recoverthem lueoftheca n 7.33).Asusua sscanonlybe l

and canbeautomatedbythe ct .

IwillleavetheSolverpar .Fornow,tofurtherillustratehowtheprocess strya

randomvalueofs=99basispoints. 99basispoints,wewillhaveanewtreeofinteres

ccountsfordefault/liquidityrisksofthebond.Thetreewillbeasfollows:

spreadof llchoosea

henIuse thecallable

arketva llablebo d(at$9

Solverfun

l,thisproce donebytria

errorwhich ioninExcel

ttoyou works,let

Ifits= tratesthat

a

0 0.5 1 1.5

96.103

0 0.5 1 1.5

16.43%

12.90%

10.18% 12.43%

4

94.8869

95.48639 97.9142

97.33178 97.807

99.0416 99.3003

100

8.08% 9.82%

7.80% 9.47%

7.53%

7.27% 100.3528

Uponhavingthetree,wecanuse topriceourcallablebondfollowingtheusualproces

spaceandtime,Iwillno detailsofthepricingprocess.Rather,Ijustinclu efinaltree

ofbondvalu like,youcan e calculationsyourse kyouranswe

aga Ifyouaren howtoprice bon interestratet sereferto

section4ofth .Inpricingthe rememberthatthisbondhas eof$100,p

semiannualcouponrate canbecalledforacallpriceof$100attime1.0and 1.0only.

NotethatIpaintblueallthenodes edadjustmentsduetothecallfeatureofthecallab

Amazing

enough,

with

a

spread

of

99

basis

points,

we

indeed

recover

the

market

price

of

the

callable

n

thetree s.Tosave

tshowthe dehereth

es.Ifyou doallth pricing

acallable

lfandchec rs

instmine. otsure dusingan ree,plea

isnode bond, afacevalu aying

of8%and time

thatne lebond.

bondwhichis$97.33.Ok,soIcheated.Isaidletstryarandomvalueofs=99basispoints.Thevalue

of99basispointsIchosetotrywasnotrandom.IusedSolverinExcelandworkedoutthats=99basis

pointswouldgivemethepriceofthecallablebondthatIwant($97.33).

Hopefully,bynowyouunderstandatleastinatechnical,mechanicalsensehowtocomputetheoptio

adjustedspreadfromtheinterestratetree.(AndasImentionearlier,comparedtothestaticspread,

thisoptionadjustedspreadseemsmoredynamic,lessstaticflavorsinceitcomesoffatree.)Still,you


14/24


maybewonderingwhysuchaprocedurewouldgiveustheexactspreadthatwewanttheoption

adjustedspreadthespreadthathasnooptioncomponentinit.

Fairenoughletmeexplainit.

Inexplainingit,Ifindithelpstolookbackandseehowthingsmovealong.First,wepricethenon

callabledefaultfreebondtobe$99.10.Second,weshowthatifthebondbecomescallable,thecallab

defaultfreebondshouldbepricedatalowervalueof$98.87,areductionof$0.23.Finally,ifweallow

forthefactthatthebondisdefaultable,thepriceofthecallabledefaultablebondshouldbeevenlow

at$97.33,anadditionalreductionof$1.54.

le

er

reductionsinbondpriceoccurindifferentItisimportanttorecognizethat,inusingthetree,thetwo

manners.

bond.Itisimportanttounderstandthatallwedohereisto

adjustthecashflowsdownwards.Weneverhavetomodifyourinterestratesateachnodeof

accountfordefault/liquidityrisks,unlikehowweallowforcallability,wedontforcibly

modifythecashflows.Instead,wesimplydiscountthecashflowsheavier.Thisinvolvespushing

Toaccountforthecallabilityofthebond,weadjustdownwardsthevaluesofthebondsattime1.0atnodeswhereitisoptimalfortheissuertocallthebond.Thisisbecausetheissuerofthe

callablebondwilloptimallycallthebondwheneverthevalueofthebondisgreaterthanthecallpricehe/shehastopayincallingthe

thetreetoaccountforthecallabilityofthebondbecauseitisunnecessary.

Toupourinterestratesateachnodeofthetreebyapositivespread.

Ifyoucanthinkofpricingasgenerallydividingexpectedfuturecashflowsby(1+thediscountrate),or

,thelooselyspeaking,thefirstpricereduction(toaccountforthebondscallability)occursthroughareductionoffuturecashflows(areductioninthenumerator).Onthotherhand,thesecondpricereduction(toaccountforthebondsdefault/liquidityris

e

ks)occursthrough

wealreadyaccountforthecallabilityofthebondbyadjustingthecashflowsdownwheneverthe

ondiscalled,thespread(99basispointsintheaboveexample)weaddtotheriskfreeinterestrate

anincreaseinthediscountrate(anincreaseinthedenominator).

Since

b

CF

Pushingthe

wholetre

eup

Howtoadjustforcallability

Adjustingcashflowsdown

whenthebondiscalled

Howtoadjustforcreditrisks/liquidityrisks


15/24


treehasnothingtodowiththecallabilityofthebond.Inotherwords,suchaspreadbywhichwepush

thewholetreeupinpricingthecallableonlyaccountsforthecreditrisksandliquidityrisksofthe

allablebond.Therefore,thespreadof99basispointsthatwefoundaboveistheoptionadjusted

preadthatweneedaspreadwithouttheoptionalitycomponent!

c

s

Oncewefindtheoptionadjustedspread,wecanuseittopricethenoncallablebondsincewewouldhaveanewinterestratetreethatallowsforthecredit/liquidityrisksofthebondissuer.

0 0.5 1 1.5

16.43%

12.90%

10.18% 12.43%

0 0.5 1 1.5

96.1034

94.8869

95.48639 97.9142

8.08% 9.82%

7.80% 9.47%

7.53%

7.27%

97.34568 97.807

99.0705 99.3003

100.0601

100.3528

Usingthetree th lla shouldbestraightforward s

thecallablebon for optimal the call

bond. ,Iwont thr detailsof ingprocess Rath

resultingprice yo calculationsto.

Accordingtomycalculations,thefin ofthenoncallablebondis$97.35,justslightlyab

riceofthecallablebondat$97.33.Thismeansthatthevalueofthecallablefeatureisonly$0.02?Orin

ticspreadis100basispoints,thespread

duetooptionalityisreallysmall:10099=1basispoint!Thisiscrazy!Duetoourcalculationsearlier,the

?

Ifthefirmhascreditrisks/liquidityrisks,itsbondshouldgenerallysellforlesscomparedtothecase

whenithasnocred ebecauseto

ofthedefaultablebondisalwayslessthanits

defaultfreecounter part.

toprice

dbecause

enonca blebond

evenhave

andeven

for

implerthan pricing

thewedont

oughthe

tocheck

thepric

whenitis

here.

issuerto

puthereAgain go

treefor

er,Iwould the

utocompareyour

alprice ovethe

p

otherwords,ifwegobacktoourequation:

Staticspread=optionadjustedspread+spreadduetooptionality

Sincetheoptionadjustedspreadis99basispointsandthesta

spreadduetooptionalityis13basispoints(remember?).Whathappenstoitthatreducesitby13fold

Answer:theextracredit/liquidityrisks.Butwhy?

itorliquidityproblems.Thisshouldbetrueateverynodeofthetre

accountforcredit/liquidityrisks,weneedtousehigherdiscountratesateverynodeofthetree.To

betterillustratethis,Iwillputthepricetreeofthenoncallablebondwithandwithoutcredit/liquidityriskstogetherandhopefullyyoucanhaveasenseofwhatImean.Justcomparinganypairof

correspondingnodes,youwouldseethatthevalue


16/24


Defaultfreenoncallablebond Defaultablenoncallablebond

0 0.5 1 1.5

96.5451

95.75492

96.7878 98.3727

99.10046 98.716

100.4394 99.7719

101.0012

100.8344

1.50 0.5 1

96.1034

94.8869

95.48639 97.9142

97.34568 97.807

99.0705 99.3003

100.0601

100.3528

Asaconsequence,thevalueoftheca theissuerofthedefaultablecallablebondwill

smaller.Thisshould cle nodeattime1where two

will of d greaterthan the the

each Th the ltfr pocketsthe of betwe the

thebond(ifle calling($100).Ontheother ,the the

defaultablebondearnsonly

notherwayofthinkingaboutthisis:relativetothedefaultfreecase,ifyouhavetheextra

call

is

ksto1basispointwhenweallowforcredit/liquidityrisks.

model

aswellasmodelrisks.

del

lloptionto be

be

thevalue

arbylookingatthelowest

hereis

theissuer

comparing

ofthe bonds

calltocall(because

issuer.

thebon

defau

$100)and

difference

valueof

eneissuerof

talive)and

eebond

topayin

$1.00012

hand

valueof

whathehas

$0.0601.

issuerof

A

credit/liquidityrisksandthushavetofacerelativelyhigherborrowingcosts,youwillbelesslikelyto

thebondthesamewayyouwillbelesslikelytorefinanceyourmortgageifthecurrentinterestrates

arehigh.Ifyouarelesslikelytocallthebond,itsvalueshouldbesmaller.

Andifthevalueofthecallablefeaturebecomessmaller,naturallythespreadduetooptionality

becomessmalleraswell.Thisexplainswhythespreadduetooptionalityhasdecreasedfrom13bas

pointsincaseofnodefault/liquidityris

Assomefinalwordsofcaution,theoptionadjustedspreadmeasuresthatwelearnsofarare

dependent.Inotherwords,weneedsomeinterestratetreesomemodelofinterestratetocalculate

thismeasure.Wheneverwetalkaboutmodel,thereisoneextradimensionofrisk,namelymodelrisk.

Assuchtobeprecise,theoptionadjustedspreadcontainscredit/liquidityrisks

Thepartsrelatedtocredit/liquidityrisksshouldbepositive!However,thepartrelatedtothemo

risks,itcouldbepositiveornegativedependingonhowweconstructthemodel.

8Callablebondpricesandinterestrates

AsIalreadymentionedintheintroclass,bondpricesandinterestratesareliketwoendsofaseesaw

Wheninterestratesgoup,bondpricesgodownandviceversa.Thesameanalogyappliestothe

relationshipbetweencallablebondpricesandinterestrates.Ifyouplotthepriceofthe2yearcallable

bondconsideredearlieragainstinterestrates,youwillhaveanegativelyslopedgraphasyouwouldfor

otherbonds.

.

However,asIshowedabove,sincethe2yearcallableisboundedfromabovebypricesofthe1year

le,thepricingfunctionofthe2yearcallablebondwilltakeanoncallableaswellasthe2yearnoncallab


17/24


18/24


9Duration/dollardurationandconvexity/dollarconvexityofcallablebonds

Beforetalkingabouteitherdurationordollardurationofcallablebonds,Iwouldliketoremindyouof

howIshowed,inourintroclass,thatthedollardurationofbondsissimplytheslopeofthetangentline

fthepricingfunction.Afterall,(dollar)durationmeasuresthesensitivityofbondpricewithrespectto

changesininterestrates.Iftheslopeofthetangentlineissteep,agivenchangeininterestratewillleadalargechangeinbondprice.Ontheotherhand,whenthetangentlineisratherflat,agivenchange

.

o

to

ininterestrateswillonlycauseasmalldeviationinbondprices.

Itturnsoutthatdollardurationaswellasdurationofregularbondsdecreaseasinterestratesincrease

Whyisthat?Justthinkofhowyoucomputedurationofazerocouponbond:

whereTismaturity

andyisthesemiannualinterestrate.Obviously,asinterestrates(y)increase,durationgoesdown.

Graphicallyspeaking,aswemovealongthepricingfunctionofregularbonds(likewhatwehavebelow

onthelefthandside),theslopeofthetangentlinegraduallydecreasesthebondbecomeslessand

lesssensitivetochangesininterestrates.

Positiveconvexity Negativeconvexity

Itispreciselythisdecreaseindurationasinterestratesincreasethatgivesrisetopositiveconvexityfor

theregularbonds.TocontrastbetweenpositiveconvexityandnegativeconvexityIalsoincludeonthe

righthandsideanexampleofnegativeconvexity.Thereyouseethatasinterestratesincreasethe

tangentlinesbecomesteeperandsteeper.

Anotherinterestingobservationisthat:

Withpositiveconvexity,thebluecurvealwaysliesabovethetangentlines.Ifyouremember

ethe

curveinsteadalwaysliesbelowthetangentlines.Thistime,

thedifferencebetweenthebluecurveandtheredlineisalwaysnegative.Assuchtheconvexity

adjustmentforthiscasewillalwaysbenegativehencethenamenegativeconvexity.

wellthedurationandconvexityapproximation,thedifferencebetweenthebluecurveandthe

redtangentlineispreciselywhatourconvexityadjustmentsgoafter.Ifthebluecurvealways

liesabovetheredline,thisexplainswhyourconvexityadjustmentisalwayspositivehenc

namepositiveconvexity.

Withnegativeconvexity,theblue


19/24


Letsnowgetbacktoour2yearcallablebondandthinkaboutitsdollardurationandhowitwillchan

asinterestratesincrease.Forillustrationpurposes,Iplotonagraphherethe$durationofthe2year

noncallable(inblue)andthe$durationofthe1yearnoncallable(inred)togetherwiththe$duration

ofthecallablebond(inpurple).Youcanseethatthe$durationofthe2yearnoncallableand1year

noncallabledecreaseasinterestrates

ge

increaseasIexplainedabove.Inaddition,$durationofthe2year

uration

ofthecallablechangesasinterestrateschange.

Forreallylowinterestrates,borrowersarelikelytorefinancetheirborrowingsequivalently,therearehighchancesthe2yearcallablebondwillbecalled.Therefore,forverylowinterest

rates,the2yearcallablebondbehavesverysimilarlytothe1yearnoncallable.Assuch,forthelowrangeofinterestrates,theduration/dollardurationofthe2yearcallablewilllookvery

muchlikethatofthe1yearnoncallable.Inotherwords,thepurplegraphshouldcomereally

closetotheredlinewheninterestratesarereallylow.

Forreallyhighinterestrates,borrowersarealmostcertainnottorefinanceorinotherwords,therearehighchancesthatthe2yearcallablewithliveuntilmaturity.Assuch, thehigh

rangeofinterestrates,theduration/dollardurationofthe2yearcallablewilllookverymuch

.

,meansthat$durationofthecallable

noncallableisalwayshigherthan$durationofthe1yearnoncallable.Thismakessensesincefora

longermaturity,the2yearnoncallableshouldbemoresensitivetointerestratechangesthanthe1

yearnoncallable.Letmeknowexplaintheshapeofthepurplegraphwhichtellsushowthe$d

for

likethatofthe2yearnoncallable.Inotherwords,thepurplegraphshouldcomereallycloseto

thebluelinewheninterestratesarereallyhigh.

Fortheintermediaterangeofinterestrates,wedontknowexactlyhowthepurplegraphwillturnout.However,onethingweknowforsureisthatthepurplegraphshouldbecontinuous

Togetfromwhereitiswheninterestratesarelow(closetotheredline)towhereitiswhen

interestratesarehigh(closetotheblueline),therefore

bondwillincreaseasinterestratesincreaseatleastforsomeintermediaterangeofinterest

rates.


20/24


Asexplainedabove,thisbehaviorofduration(increasewheninterestratesincrease)creates

negativeconvexity.ThisisexactlywhatweseefromthefigureIincludeinsection8,whichIwillput

Onewaytothinkabouthowthisnegativeconvexitycomesaboutistoimaginethatyouaredriving

alongtheredcurveandapproachingwhetherthebluecurveandtheredcurveinterests.Thenyou

wanttomakearightturnintothebluecurve.Whenyouaremakingarightturn,aslongasyouare

notgoingdeadlyslow,youwillmakeabendingshapesimilartothepurplegraphthatwehave,

whichisnegativeconvexity.

Afterall,whatisthedealaboutnegativeconvexity?Whyisitsoimportantthatwehavewasted

quitesome

time

talking

about

it?

Itturns

out

that

ithas

quite

important

implications

in

hedging,

especiallyforthosecompaniesthatinvestinfixedratemortgagesthat,asyouwillsee teron,also

s.

10ComputationofDuration/dollardurationandconvexity/dollarconvexityofcallablebonds

hereagaintosaveyoutimeflippingback:

la

displaynegativeconvexity.Inminimizingtheirexposuretointerestraterisks,naturallythesefirms

wouldliketobalance/matchthedurationsandconvexitiesoftheirassetsandliabilities.Therefore,if

theyhavenegativeconvexityassets,theywouldliketohavenegativeconvexityliabilitiesthatgive

themanaturalhedge.Andonewaytohavenegativeconvexityliabilitiesistoissuecallablebond

Asy nt

from

dur

you

sho

con

themfromtheothermeasuresthatwehavelearnt.Forthesakeofbrevity,inwhatfollows, Iwillomit

tand

thatIrefertoeffective(dollar)durationsand(dollar)convexity.

oualreadysee,unfortunately,durationandconvexitymeasuresofcallablebondsarequitediffere

thoseofthemoreregularbonds.Duetothis,allthetechniquesthatwelearnincomputing

ationandconvexityfortheregularnoncallablebondsarenolongerapplicablehere.Iwillfirstshow

theprocessbywhichwecancomputethe(dollar)durationofthe2yearcallablebond.Next,Iwill

wyouhowtheconvexitycanalsobecomputed,usingasimilarprocess.Thesedurationand

vexitymeasuresarecalledeffective(dollar)durationandeffective(dollar)convexitytodifferentiate

thewordeffectiveinfrontofdurationandconvexitywiththeimplicitassumptionthatyouunders


21/24


(Dollar)Duration:

First,Imentionedearlier,dollardurationistheslopeofthetangentlinetothepricingfunctionofany

bond,callableornoncallable.Fornoncallableregularbonds,fortunately,wehaveformulas.With

callablebonds,however,wedonthavethatluxury.Tocompute(dollar)durationforacallableb

needtogothroughasetofstepsthatturnouttobeapplicabletoeverybond:

1. First,wecomputethecurrentvalueofthebondatthecurrentlevelofinterestrates.LetssaythecurrentvalueofthebondisV0.

2. Second,weincreasetheinterestratelevelbyasmallam

ond,we

ounty.Howsmall?Somethingsmaller

ispointswouldbesmall.Wethencomputethevalueofthebondatthisnewlevelof

interestratesandcallitV+.

than10bas

3. Third,wedecreasetheinterestratelevelbythesamesmallamountyandcomputethevalueofthebondatthisnewlevelofinterestratesandcallitV.

4. Dollardurationofthebondwouldbe .5. Durationofthebondwouldbe .

se.Myexplanation(thoughshortand intuitive)onlyserves

anactuallyimplementthesesteps.

Basically,ifwehaveaniceandneatequationthatgivesustheslopeofthetangentlinetothepricing

functionordollarduration,thatwouldbenice.Otherwise,theslopeofthetangentlinewouldbesimilar

totheslopeofthebrowndottedlineonthegraphabove.Thisdottedlineconnects2pointsontheblue

curve:thefirstpointiswhenwedecreaseinterestratesbyasmallamountyandthesecondpointis

Iwillnowexplainwhythesestepsmakesen

thepurposeofsatisfyingthosecuriousaboutthereasoningbehindthesesteps.Itwontbeonthetest.

Assuch,thoseofyouwhothinkthatitistoomuchtoreadalready,youcanskipthissectionandgo

straighttotheexampleofhowwec


22/24


whenweincreaseinterestratesbythesamesmallamounty.Theslopeofthislineissimply

whichistheformulaweuseforthedollarduration.Sinceduration=dollarduration/price,theduration

ofthebondissimply

.

Importantly,none

of

the

steps

isparticular

to

callable

bonds,

which

means:

the

process

isapplicable

to

anykindofbondsorinterestratesensitivesecurities.

Toprovideaconcreteexampleofhowtocarryouttheabovesteps,letsconsidercomputingthedollar

durationanddurationofour2yearcallablebond.

Thegoodnewsisstep1isalreadydonebecausewealreadypricethecallablebondintheprecedingsections.Toremindyouofwhatwedid,Iincludeheretheinterestratetreeweused

aswellastheresultingpricetree.Again,nodespaintedbluearethoseaffectedbythebond

beingcalledattime1.

0 0.5 1 1.5 0 0.5 1 1.5

96.1034

94.8869

95.48639 97.9142

97.33178 97.807

99.0416 99.300

16.43%

12.90%

10.18% 12.43%

8.08% 9.82%

7.80% 9.47%

7.53%

7.27%

3

100

100.3528

SoourV0=97.33.

Instep2,weneedtoincreaseinterestratesbyasmallamounty.Letsassumey=10basisthetreetopricethecallablebondagain.Again,I

willnotgivethedetailsofthepricingcalculationsbutratherpostheretheresultingpricetree

points.Wewilltalkalittlebitaboutthislater,butfornowwewilladd10basispointstoeachof

thenodeoftheinterestratetreeandthenuse

foryoutocompareyourcalculationsagainst.

0 0.5 1 1.5

16.53%

13.00%

10.28% 12.53%

8.18% 9.92%

7.90% 9.57%

7.63%

7.37%

0 0.5 1 1.5

96.0591

94.79989

95.35627 97.8682

97.17068 97.7159

98.9337 99.253

99.9658

1

00.3044


23/24


Asyoucansee,asinterestr reaseby10basispoints,bondpricedecreas +=$97.17.

Itturnsoutthatatnowherealongthetreethatitisoptimaltocallthebondgiventhis

rthisreason,wedonthaveanybluenodesthistime.

atesinc etoV

increase

ininterestrates.Fo

Step3issimilartostep2,exceptthatwenowsubtract10basispointsfromtheoriginalinterestratetree.Theresultingratetreeandpricetreeareasfollows:

0 0.5 1 1.5

16.33%

12.80%

10.08% 12.33%

7.98% 9.72%

7.70% 9.37%

7.43%

7.17%

0 0.5 1 1.5

96.1479

94.97404

95.61675 97.9604

97.4853 97.8982

99.1332 99.3478

100

10

Asinterestratesdecrease,bondpriceincreasestoV=$97.49

GivenV

follows:

0.4012

+,V,V0andy=10basispoints,wecancomputethebondsdollardurationanddurationas

Dollarduration= ... 160.

Duration= . 1.6439.Thenegativesignsinfrontofdollardurationanddurationarejusttoindicatethatbondpricesand

interestratesmoveinoppositedirectio as estratesincr se,bond cesdecreaseandvice

versa.

(Dollar)Convexity:

ns: inter ea pri

Itt thatitisquit tforwardtocomputeconv dollarconvexi uhaveV+,

VandV0allre illfirstshowy formulastoperformthenee lations.Next,

curious,Ibrieflyandgrap explaintheideasbehindtheformulas.Again,thispartw tbeonthe

test.Therefore,ifyouarenotintere reasoning,youcouldsafelyskipit.

Tocomputedollarconvexity,weusethefollowingformula:

urnsout estraigh exityand tyonceyo

ady.Iw outhe dedcalcu forthose

hically illno

stedin

.

.

PluggingthevaluesforV+,V,V0intheformula,dollarconvexityofthecallablebondis:... 3800.


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convexity,wesimplydivi bytheprice,whichwillgive:

Tocompute dedollarconvexity. 39.04.

Letmenowbrieflyprovidetheintuitionbehindtheformula:

Asyoucansee,the(dollar)convexityforthecallableisnegative.

fordollarconvexity.

Asyoualreadyknow,thewholeideaofconvexityadjustmentistocorrectforthedeviationsbetween

thebluecurveanditstangentline.Givenanincreaseofyininterestrates,thedifferencebetweenthe

bluecurveandtheredtangentlineisthedistanceCC1.Givenadecreaseofyininterestrates,the

differencebetweenthebluecurveandtheredtangentlineisthedistanceAA1.Althoughwedontknow

exactlywhatCC1andAA1are,weknowtheiraveragewhichisthedistanceBB1.TheheightofBissimply

theaverageofV+andV.theheightofB1isV0.Therefore,thelengthofBB1issimply:

. Thisexplainswhytheconvexityformula,

Isproportionalto

.

Call Able Bonds 3

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