HullOFOD8eSolutionsCh05.doc

HullOFOD8eSolutionsCh05.doc

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CHAPTER5

DeterminationofForwardandFuturesPrices

PracticeQuestions

Problem5.1.

Explainwhathappenswhenaninvestorshortsacertainshare.

Theinvestor’sbrokerborrowsthesharesfromanotherclient’saccountandsellsthemintheusualway.Tocloseouttheposition,theinvestormustpurchasetheshares.Thebrokerthenreplacesthemintheaccountoftheclientfromwhomtheywereborrowed.Thepartywiththeshortpositionmustremittothebrokerdividendsandotherincomepaidontheshares.Thebrokertransfersthesefundstotheaccountoftheclientfromwhomtheshareswereborrowed.Occasionallythebrokerrunsoutofplacesfromwhichtoborrowtheshares.Theinvestoristhenshortsqueezedandhastocloseoutthepositionimmediately.

Problem5.2.

Whatisthedifferencebetweentheforwardpriceandthevalueofaforwardcontract?

Theforwardpriceofanassettodayisthepriceatwhichyouwouldagreetobuyorselltheassetatafuturetime.Thevalueofaforwardcontractiszerowhenyoufirstenterintoit.Astimepassestheunderlyingassetpricechangesandthevalueofthecontractmaybecomepositiveornegative.

Problem5.3.

Supposethatyouenterintoasix-monthforwardcontractonanon-dividend-payingstockwhenthestockpriceis$30andtherisk-freeinterestrate（withcontinuouscompounding）is12%perannum.Whatistheforwardprice?

Theforwardpriceis

Problem5.4.

Astockindexcurrentlystandsat350.Therisk-freeinterestrateis8%perannum（withcontinuouscompounding）andthedividendyieldontheindexis4%perannum.Whatshouldthefuturespriceforafour-monthcontractbe?

Thefuturespriceis

Problem5.5.

Explaincarefullywhythefuturespriceofgoldcanbecalculatedfromitsspotpriceandotherobservablevariableswhereasthefuturespriceofcoppercannot.

Goldisaninvestmentasset.Ifthefuturespriceistoohigh,investorswillfinditprofitabletoincreasetheirholdingsofgoldandshortfuturescontracts.Ifthefuturespriceistoolow,theywillfinditprofitabletodecreasetheirholdingsofgoldandgolonginthefuturesmarket.Copperisaconsumptionasset.Ifthefuturespriceistoohigh,astrategyofbuycopperandshortfuturesworks.However,becauseinvestorsdonotingeneralholdtheasset,thestrategyofsellcopperandbuyfuturesisnotavailabletothem.Thereisthereforeanupperbound,butnolowerbound,tothefuturesprice.

Problem5.6.

Explaincarefullythemeaningofthetermsconvenienceyieldandcostofcarry.Whatistherelationshipbetweenfuturesprice,spotprice,convenienceyield,andcostofcarry?

Convenienceyieldmeasurestheextenttowhichtherearebenefitsobtainedfromownershipofthephysicalassetthatarenotobtainedbyownersoflongfuturescontracts.Thecostofcarryistheinterestcostplusstoragecostlesstheincomeearned.Thefuturesprice,,andspotprice,,arerelatedby

whereisthecostofcarry,istheconvenienceyield,andisthetimetomaturityofthefuturescontract.

Problem5.7.

Explainwhyaforeigncurrencycanbetreatedasanassetprovidingaknownyield.

Aforeigncurrencyprovidesaknowninterestrate,buttheinterestisreceivedintheforeigncurrency.Thevalueinthedomesticcurrencyoftheincomeprovidedbytheforeigncurrencyisthereforeknownasapercentageofthevalueoftheforeigncurrency.Thismeansthattheincomehasthepropertiesofaknownyield.

Problem5.8.

Isthefuturespriceofastockindexgreaterthanorlessthantheexpectedfuturevalueoftheindex?

Explainyouranswer.

Thefuturespriceofastockindexisalwayslessthantheexpectedfuturevalueoftheindex.ThisfollowsfromSection5.14andthefactthattheindexhaspositivesystematicrisk.Foranalternativeargument,letbetheexpectedreturnrequiredbyinvestorsontheindexsothat.Becauseand,itfollowsthat.

Problem5.9.

Aone-yearlongforwardcontractonanon-dividend-payingstockisenteredintowhenthestockpriceis$40andtherisk-freerateofinterestis10%perannumwithcontinuouscompounding.

a）Whataretheforwardpriceandtheinitialvalueoftheforwardcontract?

b）Sixmonthslater,thepriceofthestockis$45andtherisk-freeinterestrateisstill10%.Whataretheforwardpriceandthevalueoftheforwardcontract?

a）Theforwardprice,,isgivenbyequation（5.1）as:

or$44.21.Theinitialvalueoftheforwardcontractiszero.

b）Thedeliverypriceinthecontractis$44.21.Thevalueofthecontract,,aftersixmonthsisgivenbyequation（5.5）as:

i.e.,itis$2.95.Theforwardpriceis:

or$47.31.

Problem5.10.

Therisk-freerateofinterestis7%perannumwithcontinuouscompounding,andthedividendyieldonastockindexis3.2%perannum.Thecurrentvalueoftheindexis150.Whatisthesix-monthfuturesprice?

Usingequation（5.3）thesixmonthfuturespriceis

or$152.88.

Problem5.11.

Assumethattherisk-freeinterestrateis9%perannumwithcontinuouscompoundingandthatthedividendyieldonastockindexvariesthroughouttheyear.InFebruary,May,August,andNovember,dividendsarepaidatarateof5%perannum.Inothermonths,dividendsarepaidatarateof2%perannum.SupposethatthevalueoftheindexonJuly31is1,300.WhatisthefuturespriceforacontractdeliverableonDecember31ofthesameyear?

Thefuturescontractlastsforfivemonths.Thedividendyieldis2%forthreeofthemonthsand5%fortwoofthemonths.Theaveragedividendyieldistherefore

Thefuturespriceistherefore

or$1331.80.

Problem5.12.

Supposethattherisk-freeinterestrateis10%perannumwithcontinuouscompoundingandthatthedividendyieldonastockindexis4%perannum.Theindexisstandingat400,andthefuturespriceforacontractdeliverableinfourmonthsis405.Whatarbitrageopportunitiesdoesthiscreate?

Thetheoreticalfuturespriceis

Theactualfuturespriceisonly405.Thisshowsthattheindexfuturespriceistoolowrelativetotheindex.Thecorrectarbitragestrategyis

1.Buyfuturescontracts

2.Shortthesharesunderlyingtheindex.

Problem5.13.

Estimatethedifferencebetweenshort-terminterestratesinMexicoandtheUnitedStatesonMay26,2010fromtheinformationinTable5.4.

Thesettlementpricesforthefuturescontractsareto

Sept:

0.76375

Dec:

0.75625

TheDecember2009priceisabout0.98%belowtheSeptember2009price.Thissuggeststhattheshort-terminterestrateintheMexicoexceededshort-terminterestrateintheUnitedStatesbyabout0.98%perthreemonthsorabout3.92%peryear.

Problem5.14.

Thetwo-monthinterestratesinSwitzerlandandtheUnitedStatesare2%and5%perannum,respectively,withcontinuouscompounding.ThespotpriceoftheSwissfrancis$0.8000.Thefuturespriceforacontractdeliverableintwomonthsis$0.8100.Whatarbitrageopportunitiesdoesthiscreate?

Thetheoreticalfuturespriceis

Theactualfuturespriceistoohigh.ThissuggeststhatanarbitrageurshouldbuySwissfrancsandshortSwissfrancsfutures.

Problem5.15.

Thespotpriceofsilveris$15perounce.Thestoragecostsare$0.24perounceperyearpayablequarterlyinadvance.Assumingthatinterestratesare10%perannumforallmaturities,calculatethefuturespriceofsilverfordeliveryinninemonths.

Thepresentvalueofthestoragecostsforninemonthsare

or$0.176.Thefuturespriceisfromequation（5.11）givenbywhere

i.e.,itis$16.36perounce.

Problem5.16.

Supposethatandaretwofuturescontractsonthesamecommoditywithtimestomaturity,and,where.Provethat

whereistheinterestrate（assumedconstant）andtherearenostoragecosts.Forthepurposesofthisproblem,assumethatafuturescontractisthesameasaforwardcontract.

If

aninvestorcouldmakearisklessprofitby

1.Takingalongpositioninafuturescontractwhichmaturesattime

2.Takingashortpositioninafuturescontractwhichmaturesattime

Whenthefirstfuturescontractmatures,theassetispurchasedforusingfundsborrowedatrater.Itisthenhelduntiltimeatwhichpointitisexchangedforunderthesecondcontract.Thecostsofthefundsborrowedandaccumulatedinterestattimeis.Apositiveprofitof

isthenrealizedattime.Thistypeofarbitrageopportunitycannotexistforlong.Hence:

Problem5.17.

Whenaknownfuturecashoutflowinaforeigncurrencyishedgedbyacompanyusingaforwardcontract,thereisnoforeignexchangerisk.Whenitishedgedusingfuturescontracts,thedailysettlementprocessdoesleavethecompanyexposedtosomerisk.Explainthenatureofthisrisk.Inparticular,considerwhetherthecompanyisbetteroffusingafuturescontractoraforwardcontractwhen

a）Thevalueoftheforeigncurrencyfallsrapidlyduringthelifeofthecontract

b）Thevalueoftheforeigncurrencyrisesrapidlyduringthelifeofthecontract

c）Thevalueoftheforeigncurrencyfirstrisesandthenfallsbacktoitsinitialvalue

d）Thevalueoftheforeigncurrencyfirstfallsandthenrisesbacktoitsinitialvalue

Assumethattheforwardpriceequalsthefuturesprice.

Intotalthegainorlossunderafuturescontractisequaltothegainorlossunderthecorrespondingforwardcontract.Howeverthetimingofthecashflowsisdifferent.Whenthetimevalueofmoneyistakenintoaccountafuturescontractmayprovetobemorevaluableorlessvaluablethanaforwardcontract.Ofcoursethecompanydoesnotknowinadvancewhichwillworkoutbetter.Thelongforwardcontractprovidesaperfecthedge