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BondMathema,cs
Akshat Shankar
Outlineo TimeValueofMoneyo Introduc,ontoBondso SpotRateso BondPricingo ForwardRates
TimeValueofMoney:Idea
o Moneygrowswith,meifitisinvested.o Ifmoneyisdepositedinabank,itearnsmoremoneyalong
withit,whichiscalled‘interest’.o Rs100todaycanbeRs110oneyearfromtoday.o Rs91oneyearbackmayhavebecomeRs100today.
Present
ALerSomeTime
TimeValueofMoney:Idea
o Ques,on:o Youhavejoboffersfromtwocompaniescalled‘A’and‘B’.
‘A’paysasalaryofRs10,000atthebeginningofthemonth.
o ‘B’paysRs11,000attheendofthemonth.o Whichoneyoushouldchoose?
o Solu,on:o YoucaninvestRs10,000inbankatthestartofthemonth.o Itmaybethecasethattheymayvalueless/morethanRs
11,000inonemonth.o Youshouldchoosethecompanywhichgivesmaximum
moneyattheendofthemonth.
Present&FutureValue
o ValueofMoneycannotbeevaluatedwithoutknowingwhenitisactuallyreceived.
o Rs100today=Rs110oneyearlater=Rs91oneyearback.o ValueoftheMoneyonehastoday,atsomefuturepointis
called‘FutureValue’o ValueoftheMoneyinfuture,atpresentiscalled‘Present
Value’
PresentValue
Discoun,ng
FutureValue
Accruing
Computa,onofInteresto L(Lender)lendstoB(Borrower)amount‘P’for‘T’,me.o Howmuchinterest‘I’shouldbepaidtoAat‘T’years?
o ‘I’shouldbemoreif‘P’ismore.o ‘I’shouldbemoreif‘T’ismore.
P
P+I
Lender Borrower
Computa,onofInterest
o Interestisdirectlypropor,onaltoo Amountinvested‘P’o Numberofyears‘T’
TPRITPI⋅⋅=
⋅∝
o ‘R’isthepropor,onalityconstantthatisknownas‘interestrate’.
o ‘R’istherateatwhichlenderwantsmoneytogrow.o Theequa,oncanbere-formulatedas
TRPI ⋅⋅=
SimpleInteresto Thepreviouscomputa,onofinterestiscalledSimpleInterest.o Interest=Principal(P).Rate(R).Time(T)
P
P+PRT
Lender Borrower
SimpleInterest:Problemo LgivesBRs.100for2yearsatarateof10%.o BisintelligentandlendsRs100toCforoneyear.(at10%)o ALeroneyearCreturnsRs110toB.o BlendsRs110toDforoneyear.(at10%)o Oneyearlater,DreturnsRs121toB.
100,t=0
120,t=2
L
B
100,t=0
110,t=1
121,t=2
110,t=1
Keeps1inhispocketatt=2!
BC
D
CompoundInterest
o Arbitragecanbeearnedifinterestratesaremen,onedas‘SimpleInterest’.
o CompoundInterestassumesthattheinterestisre-investedatsomedefinedfrequencies.
o Example:Rs100isinvestedfor2yearsatarateof10%compoundedannually
A=P(1+RT) A=P(1+RT)110100R=10%T=1 121
R=10%T=1
t=0 t=1 t=2
A=P(1+R)T 121100R=10%T=2
t=0 t=2
CompoundInterest:Example
o CompoundInterestcanbeexpressedintermsofdifferentcompoundingfrequencies.
o Example:Rs100isinvestedfor2yearsatarateof10%compoundedsemi-annually.
A=P(1+RT) A=P(1+RT)105100 R=10%T=1/2 110.25R=10%
T=1/2t=0 t=1/2
t=1
A=P(1+R/2)2T 121.55100R=10%T=2
t=0 t=2
A=P(1+RT) A=P(1+RT)115.76121.55R=10%T=1/2
t=2 t=3/2
R=10%T=1/2
R=10%T=1/2
CompoundInterest
o CompoundInterestcanbedefinedfordifferentcompoundingfrequencies.
mT
mRPA ⎟⎠
⎞⎜⎝
⎛ +⋅= 1
o IfTisnotamul,pleof1/m,compoundinterestiscalculatedusingtheaboveformula,llthehighestmul,pleof1/mlessthanT.
o Fortheremainingperiod,useSimpleInterest.o CompoundingFrequenciesusedoLen:
o m=1:AnnualCompoundingo m=2:Semi-annualCompoundingo m=365:DailyCompounding
OnlyifTisamul,pleof1/m
Con,nuousCompounding
o Whatifinterestisre-investedeverymoment.(momentcanbeassmallaspossible)
o Mathema,cally,itmeansΔt→0andm→infinity.mT
m mRPLimA ⎟⎠
⎞⎜⎝
⎛ +⋅=∞→
1
RTePA ⋅=
o FromLimitRules
Con,nuousCompounding:Intui,on
o A(t)denotesthevalueoftheinvestmentat,met.o A(0)=P(principalinvestmentini,ally)o Amountisreinvestedeveryinstantandduringeveryinstant,
amountgrowsbysimpleinterest.
t t+dt
A(t)
A(t+dt)
A(t).R.dtInterestaccruesintheinfinitesimalperiodaspersimpleinterest
Con,nuousCompounding:Deriva,on
RT
TA
A
T
eATARTATA
RRTATA
RdttAAdgIntegratin
RdttAAd
RdttAdARdttAtAdttARdttAtAdttARdttAdttA
)0()())0()(ln(
0.))0(ln())(ln(
)()(
)()(
)()()()()()()()1)(()(
)(
)0( 0
=⇒=
−=−
=⇒
=⇒
=⇒=−+⇒
+=+⇒+=+
∫ ∫
TRePTA ⋅⋅=)(
Inter-conversion
o DifferentRatescanbecomparedonlyiftheyaremen,onedinthesamefrequency.
o EquivalentRatescanbecomputedusingthefollowingmethodology
m-Compounding
Con,nuousCompounding
TrceP ⋅⋅mT
m
mrP ⎟⎠
⎞⎜⎝
⎛ +⋅ 1
P P
⇒⎟⎠
⎞⎜⎝
⎛ +⋅=mT
mTr
mrPeP c 1. ⎟
⎠
⎞⎜⎝
⎛ +=mrmr m
c 1ln
Inter-conversion:Example
o Example:Considertwoop,onsforFixedDeposito ICICIBank:1yearmaturity,9.9%quarterlycompoundedo HDFCBank:1yearmaturity,10%semi-annualcompounded.
o Ques:Whichoneisbemer?o Ans:FindtheICICIrateintermsofsemi-annualcompounding.
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−⎟
⎠
⎞⎜⎝
⎛ +⋅=⇒⎟⎠
⎞⎜⎝
⎛ +⋅=⎟⎠
⎞⎜⎝
⎛ +⋅⋅⋅
14
124
12
14
42
144
122 rrrPrP
%0225.1014099.012
2/4
2 =⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠
⎞⎜⎝
⎛ +⋅=r
o FixedDepositofICICIisbemer!
Discoun,ng
o PresentValueofafutureinvestmentcanbefoundbyaninverseprocedurewhichiscalledDiscoun,ng.
Discoun,ng
Accruingx
rtxe
F?x
rtrt eFxFxe −=⇒= . rteFVPV −= .
Outlineo TimeValueofMoneyo Introduc,ontoBondso SpotRateso BondPricingo ForwardRates
Bonds:Introduc,on
o Abondisaformalcontracttorepayborrowedmoneywithinterestatfixedintervals
o Bondbuyergetscashflowsatdifferent,mesfromtheseller.o Cashflowsarenotfixedbutareclearlydefined.
Bonds:Terminology
o FaceValue:Amountofmoneyreceivedatmaturity.(apartfromcoupon)
o Maturity:Timeforwhichthebondisac,ve.o CouponRate:CouponRate*FaceValueisthecoupon
receivedeveryyear.o CouponSchedule:Datesatwhichcouponisreceived.
Sellsthebond
Givesthemoney
Issuer Buyer
Borrower Lender
Bonds:Structure
o Bondcanbevisualizedasaseriesofcashflows.
o BondwithanabovestructureiscalledaBulletBond.o Cashflowsofabondcanbevariableaswell.o Cashflowsofabondcanalsobeprobabilis,caldepending
uponcertainevents.
Maturity
Coupon
FaceValueCouponIssueDate
Bonds:Types
o CashFlowsinabondarewelldefined.o Basedonthecashflows,bondcanbeclassifiedas:
FixedCashflows
Bond
VariableCashflows
ZeroCouponBond
BulletBond
CallableBond
Outlineo TimeValueofMoneyo Introduc,ontoBondso SpotRateso BondPricingo ForwardRates
SpotRate
o SpotRateistheinterestrateonaninvestmentthatstartstodayandlasts,llmaturitywithalltheinterestandprincipalbeingrealizedattheend.
o Alltheinterestratesdefinedup,llnowareSpotRates.o AsSpotRateistheinterestrealizedonazerocouponbond
(nointermediatepayment),itiscalledzerorate.o Example:
o AFixedDepositatICICIispromising8%spotrate(compoundedquarterly)foramaturityofoneyear.
o WhatwouldthevalueofRs100investmentbeaLer1year?o Solu,on:
24.108
408.01100
4
Rs=⎟⎠
⎞⎜⎝
⎛ +
SpotRateCurve
o InterestRatewouldbedifferentfordifferentmaturi,es.o SpotRateCurveplotsSpotRateforeachmaturity.o SpotRateCurvewouldbedifferentfordifferentissuers.
SpotRateCu
rve
4.2%
5.3%
2years 10years
Ifonelendsfor2yearstherateis4.2%
$1todaywouldgrowto$1.e0.053*10in10years.
Maturity
Outlineo TimeValueofMoneyo Introduc,ontoBondso SpotRateso BondPricingo ForwardRates
BondPricing
o Priceofanyfinancialinstrumentissumofthepresentvalueoftheexpectedfuturecashflows.
o Asbondisacollec,onofcashflows,soitspriceshouldbethesumofthePresentValueofthesecashflows.
+ + + + + + +
FV
PV
=
TimelineofCashFlows
PRICE
BondPricing
o StepsforBondPricingo FindtheCashFlowsandCashFlow,mesforabond.o FindtheSpotRatecorrespondingtotheCashFlow,mes.o Discounteachcashflowusingthecorrespondingspotrate.o Sumallthediscountedcashflows.
SpotRateCurve
EachcashflowisdiscountedtothePresent
Cashflows
PriceoftheBond∑ −=i
tsi
iiecP
BondPricing:Example
o Example:Considerabondwitho Maturityof1.75yearso Couponof10%.(Paidsemi-annually)o SpotRatesaregivenasfollows.(con,nuouslycompounded)
Years 0.25 0.75 1.25 1.75 SpotRate
5% 6% 7% 6%
o CashFlows,mes:1.75,1.25,0.75,0.25o CashFlows:105,5,5,5o CorrespondingSpotRatesaregiven.
BondPricing:Example
o Followingtableshowsthecalcula,onforthebondprice.
Year SpotRate
CashFlow
PV
0.25 5% 5 4.94 0.75 6% 5 4.78 1.25 7% 5 4.58 1.75 6% 105 94.53
Dirty/FullPrice 108.83
o PriceofabondisthesumofallthePresentValues.o PriceobtainedaboveiscalledFullPrice/DirtyPrice.
78.4.5 75.006.0 =•−e
SampleCalcula,on
BondPricevsInterestRate
o Priceisthesumofdiscountedcashflows.o InterestRate↑,PresentValue↓,BondPrice↓
∑ −=i
tsi
iiecP
o BondPriceandInterestRateshaveaninverserela,onship.
BondPricing:CleanPrice
o PriceatwhichabondistradediscalledFullPrice/DirtyPrice.o FullPricevsTimegraphhasajumpateverycouponpayment
date.
Just(є)beforethecouponpayment
Just(є)aLerthecouponpayment
τ
τ
s
ss
eccPececP
−
∈+ʹ−∈−
+≈⇒∈→
+= ∈
54
)(54
0 τ
τ
s
s
ecPecP
−
∈−ʹ́−
≈⇒∈→
=
5
)(5
0
o PriceaLerthepaymentdecreasesbyanamountequaltocoupon.
Ifcoupon=$5.Pricebefore=$102PriceaLer=$97
BondPricing:CleanPrice
o IftheFullPriceofabonddecreases:o Op,onA:Qualityofthebondhasdeterioratedo Op,onB:Couponhasbeenrecentlypaid.
o Traderswantameasureofpricewhichis‘Clean’i.e.itdoesnotchangewithcouponpayments.
o Priceis‘cleaned’byremovingthe‘AccruedInterest’(AI).o AIistheinterestwhichhasbeenearnedbutnotpaid.
ti ti+1ti-1 tc
ttttAIii
i ⋅−
−=
+1
c
BondPricing:CleanPrice
o CleanPrice=FullPrice–AccruedInteresto TradersusethispriceandthisisquotedonBloomberg.
Subtract
AIFullPrice CleanPrice
AI=c AI=0
c
o FullPriceandAIhaveanequaljumpatthecouponpayment.o Hencetheirdifference(CleanPrice)remainsthesame.
Outlineo TimeValueofMoneyo Introduc,ontoBondso SpotRateso BondPricingo ForwardRates
ForwardRates:Idea
o ForwardRatesaretheinterestratesforafutureperiodwhichcanbelockedatpresent
o Businessmanexpectshugecashin2months.o Fearisthattheinterestmaygodownin2months.o Hewantstoimmediately‘lockin’arateforfutureperiod.o ThelockedinrateisknownasForwardRate.
t=0:forwardrate‘f’for(t1,t2)is‘lockedin’
BForwardRateBuyer
SForwardRateSeller
t=t1:BdepositsamountPwithSatrate‘f’
t=t2:SreturnsP+I(asperrate‘f’)toB
ForwardRates:Idea
o ForwardRatesis‘lockedin’forsomefutureperiodtoday.
FuturePeriodforwhichtheratehasbeenlockedin
InterestRateis
lockedhere
t0 t1 t2o Example:o Infosysexpectstoreceiveapaymentof$1,000,000two
monthsfromnow.o Itlocksinarateof10%(semi-annual)withRBSforaperiod
from2monthsto8months.(frompresent)o Infosyswouldbedeposi,ng$1,000,0002monthsfromnow.o RBSwouldbeobligedtopay$1,050,0008monthsfromnow.
ForwardRates:Formula
o ForwardRatescanbefoundfromtheSpotRates.
Case1o BuyazerocouponbondofFace
Value$1,maturityt2ataspotrates2
o Amountaccruedatt2=22.1$ tse
Case2o Buyaforwardfortheperiod
(t1,t2) ataratef1,2 o Buyazerocouponbondwith
$1,maturityt1ataspotrates1o Amountaccruedatt1=o Thisamountre-investedatthe
‘lockedin’ratef1,2 o Amountaccruedatt2=
11.1$ tse
)( 121211 ..1$ ttfts ee −
o Toavoidarbitrage,twoamountsshouldbeequal.
ForwardRates:Formula
)(11
.11
12121122
)(
)(
12121122
12121122
ttftstseeeee
ttftsts
ttftsts
−+=⇒
=⇒
=⇒−+
−
o Aspernoarbitragerela,onship,forwardratecanbeexpressedintermsofspotrates
12
112212 tt
tstsf−
−=⇒
ShortRate
o Ques,on:o SupposeLwantstolend$100toBforonesecond.o Whatwouldbetheinterest?o Whatwouldbetheinterestrate?
o Solu,on:o I=PRT,AsTisalmost0,Interestwouldbealmost0.o Thereisnothingwhichsuggeststhattheinterestratewouldbe0.Infactthisinterestratewouldbeanonzeroquan,ty.
o ShortRate:istheinterestrateatwhichonecanborrowmoneyforaninfinitesimallyshortperiodof,me.
InstantaneousForwardRates
o InstantaneousForwardRatefor,metistheforwardratethatcanbelockednowforaloanbetween,me‘t’and,me‘t+dt’.
o InstantaneousForwardRate‘locksin’theshortrateforsomefuture,met.
dtInterestRateislockedatt0foraperiodbetween(t,t+dt)
t0 t
o ShortRatesandInstantaneousForwardRatesareusedextensivelyinmodellingTermStructure.
o ThereisaonetoonemappingbetweenSpotRateCurveandInstantaneousForwardRateCurve.
SpotRate-ForwardRateParity
o InstantaneousForwardratecanbefoundasalimi,ngcaseofforwardratewitht1→t2.
⇒=−∈+
−∈+=
∈+==
−
−=
∈+
∈→)(
)(.).()(
,
0
21
12
112212
stdtd
tttstsLimtf
tttttttstsf
tt
∫=T
dfT
ts0
)()(1)( ττ
Ques,ons?
