call able bonds 3
TRANSCRIPT
-
8/12/2019 Call Able Bonds 3
1/24
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.
-
8/12/2019 Call Able Bonds 3
2/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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:
-
8/12/2019 Call Able Bonds 3
3/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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?
-
8/12/2019 Call Able Bonds 3
4/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
5/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
6/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
7/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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.
-
8/12/2019 Call Able Bonds 3
8/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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.
-
8/12/2019 Call Able Bonds 3
9/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
10/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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:
-
8/12/2019 Call Able Bonds 3
11/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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:
-
8/12/2019 Call Able Bonds 3
12/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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.
-
8/12/2019 Call Able Bonds 3
13/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
14/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
15/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
16/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
17/24
-
8/12/2019 Call Able Bonds 3
18/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
19/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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.
-
8/12/2019 Call Able Bonds 3
20/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
21/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
(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
-
8/12/2019 Call Able Bonds 3
22/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
-
8/12/2019 Call Able Bonds 3
23/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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.
-
8/12/2019 Call Able Bonds 3
24/24
AdvancedFixedIncome CallableBonds ProfessorAnhLe
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
.