十五 Asset Valuation Debt Investments Analysis and Valuation.docx
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十五 Asset Valuation Debt Investments Analysis and Valuation.docx
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十五AssetValuationDebtInvestmentsAnalysisandValuation
十五AssetValuation:
DebtInvestments:
AnalysisandValuation
1.A:
IntroductiontotheValuationofFixedIncomeSecurities
a:
Describethefundamentalprinciplesofbondvaluation.
Bondinvestorsarebasicallyentitledtotwodistincttypesofcashflows:
1)theperiodicreceiptofcouponincomeoverthelifeofthebond,and2)therecoveryofprincipal(parvalue)attheendofthebond'slife.Thus,invaluingabond,you'redealingwithanannuityofcouponpayments,plusalargesinglecashflow,asrepresentedbytherecoveryofprincipalatmaturity,orwhenthebondisretired.Thesecashflows,alongwiththerequiredrateofreturnontheinvestment,arethenusedinapresentvaluebasedbondmodeltofindthedollarpriceofabond.
b:
Explainthethreestepsinthevaluationprocess.
Thevalueofanyfinancialassetcanbedeterminedasthesumoftheasset’sdiscountedcashflows.Therearethreesteps:
∙Estimatethecashflows.
∙Determinetheappropriatediscountrate.
∙Calculatethesumofpresentvaluesoftheestimatedcashflows.
c:
Explainwhatismeantbyabond'scashflow.
ThisLOSisverystraightforward.Abond'scashflowisthecouponorprincipalvalue.Foranoption-freebond(meaningthatthebondisnotcallable,putable,convertible,etc.),theexpectedcashflowstructureisshownonthetimelinebelow.
Wherem=maturity,par,orfacevalue(usually$1,000,£1,000,etcetera),CPN=(maturityvalue*statedcouponrate)/#couponsperyear,andN=#ofyearstomaturity*#couponsperyear.So,foranarbitrarydiscountratei,thebond’svalueis:
Bondvalue=
CPN1
+
CPN2
+...+
CPNn*m+M
(l+i/m)1
(1+i/m)2
(l+i/m)n*m
Where:
i=interestrateperannum(yieldtomaturityorYTM),m=numberofcouponsperyear,andn=numberofyearstomaturity.
d:
Discussthediffultiesofestimatingtheexpectedcashflowsforsometypesofbondsandidentifythebondsforwhichestimatingtheexpectedcashflowsisdifficult.
Normally,estimatingthecashflowstreamofahigh-qualityoption-freebondisrelativelystraightforward,astheamountandtimingofthecouponsandprincipalpaymentsareknownwithahighdegreeofcertainty.Removethatcertainty,anddifficultieswillariseinestimatingthecashflowstreamofabond.Asidefromnormalcreditrisks,thefollowingthreeconditionscouldleadtodifficultiesinforecastingthefuturecashflowstreamofevenhigh-qualityissues:
∙Thepresenceofembeddedoptions,suchascallfeaturesandsinkingfundprovisions-inwhichcase,thelengthofthecashflowstream(lifeofthebond)cannotbedeterminedwithcertainty.
∙Theuseofavariable,ratherthanafixed,couponrate-inwhichcase,thefutureannualorsemi-annualcouponpaymentscannotbedeterminedwithcertainty.
∙Thepresenceofaconversionorexchangeprivilege,soyou'redealingwithaconvertible(orexchangeable)bond,ratherthanastraightbond-inwhichcase,it'sdifficulttoknowhowlongitwillbebeforethebondisconvertedintostock.
e:
Computethevalueofabond,giventheexpectedcashflowsandtheappropriatediscountrates.
Example:
Annualcoupons.Supposethatwehavea10-year,$1,000parvalue,6%annualcouponbond.Thecashvalueofeachcouponis:
CPN=($1,000*0.06)/1=$60.Thevalueofthebondwithayieldtomaturity(interestrate)of8%appearsbelow.Onyourfinancialcalculator,N=10,PMT=60,FV=1000,I/Y=8;CPTPV=865.80.Thisvaluewouldtypicallybequotedas86.58,meaning86.58%ofparvalue,or$865.80.
Bondvalue=[60/(1.08)1]+[60/(1.08)2]+[60+100/(1.08)3]=$865.80
Example:
Semiannualcoupons.Supposethatwehavea10-year,$1,000parvalue,6%semiannualcouponbond.Thecashvalueofeachcouponis:
CPN=($1,000*0.06)/2=$30.Thevalueofthebondwithayieldtomaturity(interestrate)of8%appearsbelow.Onyourfinancialcalculator,N=20,PMT=30,FV=1000,I/Y=4;CPTPV=864.10.Notethatthecouponsconstituteanannuity.
BondValue=
n*m
∑
t=1
30
(1+0.08/2)t
+
1000
(1+0.08/2)n*m
=864.10
f:
Explainhowthevalueofabondchangesifthediscountrateincreasesordecreasesandcomputethechangeinvaluethatisattributabletotheratechange.
Therequiredyieldtomaturitycanchangedramaticallyduringthelifeofabond.Thesechangescanbemarketwide(i.e.,thegenerallevelofinterestratesintheeconomy)orspecifictotheissue(e.g.,achangeincreditquality).However,forastandard,option-freebondthecashflowswillnotchangeduringthelifeofthebond.Changesinrequiredyieldarereflectedinthebond’sprice.
Example:
changesinrequiredyield.Usingyourcalculator,computethevalueofa$1,000parvaluebond,withathreeyearlife,paying6%semiannualcouponstoaninvestorwitharequiredrateofreturnof:
3%,6%,and12%.
AtI/Y=3%/2;n=3*2;FV=1000;PMT=60/2;computePV=-1,085.458
AtI/Y=6%/2;n=3*2;FV=1000;PMT=60/2;computePV=-1,000.000
AtI/Y=12%/2;n=3*2;FV=1000;PMT=60/2;computePV=-852.480
g:
Explainhowthepriceofabondchangesasthebondapproachesitsmaturitydateandcomputethechangeinvaluethatisattributabletothepassageoftime.
Abond’svaluecandiffersubstantiallyfromitsmaturityvaluepriortomaturity.However,regardlessofitsrequiredyield,thepricewillconvergetowardmaturityvalueasmaturityapproaches.Returningtoour$1,000parvaluebond,withathree-yearlife,paying6%semi-annualcoupons.Herewecalculatethebondvaluesusingrequiredyieldsof3,6,and12%asthebondapproachesmaturity.
TimetoMaturity
YTM=3%
YTM=6%
YTM=12%
3.0years
1,085.458
1,000.000
852.480
2.5
1,071.740
1,000.000
873.629
2.0
1,057.816
1,000.000
896.047
1.5
1,043.683
1,000.000
919.810
1.0
1,029.338
1,000.000
944.998
0.5
1,014.778
1,000.000
971.689
0.0
1,000.000
1,000.000
1,000.000
h:
Computethevalueofazero-couponbond.
Youfindthepriceormarketvalueofazerocouponbondjustlikeyoudoacoupon-bearingsecurity,except,ofcourse,youignorethecouponcomponentoftheequation.Theonlycashflowisrecoveryofparvalueatmaturity.Thusthepriceormarketvalueofazerocouponbondissimplythepresentvalueofthebond'sparvalue.
Bondvalue=M/(1+i/m)n*m
Example:
Azerocouponbond.Supposewehavea10-year,$1,000parvalue,zerocouponbond.Tofindthevalueofthisbondgivenitsbeingpricetoyield8%(compoundedsemiannually),you'ddothefollowing:
Bondvalue=1000/(1+.08/2)10*2=456.39
Onyourfinancingcalculator,N=10*2=20,FV=1000,I/Y=8/24;CPTPV=456.39(ignorethesign).
Thedifferencebetweenthe$456.39andtheparvalue($1000)istheamountofinterestthatwillbeearnedoverthe10-yearlifeoftheissue.
i:
Computethedirtypriceofabond,accruedinterest,andcleanpriceofabondthatisbetweencouponpayments.
Assumewearetryingtopricea3-year,$1,000parvalue,6%semiannualcouponbond,withYTM=12%,withamaturityofJanuary15,2005,andyouarevaluingthebondforsettlementonApril20,2002.ThenextcouponisdueJuly15,2002.Therefore,thereare85daysbetweensettlementandnextcoupon,and180daysinthecouponperiod.Thefractionalperiod(w)=85/180=0.4722.Thevalueofthebondcalculatesouttobe$879.105.
Notethatthisbondvalueincludestheaccruedinterest.Thisisoftenreferredtoasthedirtypriceorthefullprice.Unfortunately,whenusingafinancialcalculator,youcan'tjustinputNas5.4722,sincethecalculatorwillholdthefractionalperiodtotheendratherthanconsideritupfront,andyou'llendupwiththewronganswer($863.49).Theeasiestwaytocomputethedirtypriceonyourfinancialcalculatoristoaddupthepresentvaluesofeachcashflow.
∙N=0.4722,I/Y=6,FV=30,PMT=0;CPTPV=29.18
∙N=1.4722,I/Y=6,FV=30,PMT=0;CPTPV=27.53
∙N=2.4722,I/Y=6,FV=30,PMT=0;CPTPV=25.97
∙N=3.4722,I/Y=6,FV=30,PMT=0;CPTPV=24.50
∙N=4.4722,I/Y=6,FV=30,PMT=0;CPTPV=23.12
∙N=5.4722,I/Y=6,FV=1030,PMT=0;CPTPV=748.79
∙Addeachcashflowfora879.09(roundingerror)dirtyvaluation.
Bondpricesarequotedwithouttheaccruedinterest.Thisisoftenreferredtoasthecleanprice(orjusttheprice).Todeterminethecleanprice,wemustcomputetheaccruedinterestandsubtractthisfromthedirtyprice.Theaccruedinterestisafunctionoftheaccruedinterestperiod,thenumberofdaysinthecouponperiod,andthevalueofthecoupon.Theperiodduringwhichtheinterestisearnedbytheselleristheaccruedinterestperiod.Assumeafractionalperiodof0.4722andabondpriceof$879.105.Sincethe“w”previouslycalculatedisthenumberofdays’interestearnedbythebuyerdividedbythenumberofdaysinthecouponperiod,theAIperiodisthecomplementof“w”.Hence:
AI=(1-w)*CPN
=(1–0.4722)*30=$15.833
Therefore,thecleanpriceis:
CP=dirtyprice–AI=$879.105–15.833=863.272.Thebondwouldbequotedat86.3272%(orapproximately8610/32)ofpar.
j:
Explainthedeficiencyofthetraditionalapproachtovaluationinwhicheachcashflowisdiscountedatthesamediscountrate.
Theuseofasinglediscountfactor(i.e.,YTM)tovalueallbondcashflowsassumesthatinterestratesdonotvarywithtermtomaturityofthecashflow.Inpracticethisisusuallynotthecase—interestratesexhibitatermstructure,meaningthattheyvaryaccordingtotermtomaturity.Consequently,YTMisreallyanapproximationorweightedaverageofasetofspotrates(aninterestratetodayusedtodiscountasi
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