HullOFOD9eSolutionsCh05

CHAPTER5DeterminationofForwardandFuturesPricesPracticeQuestionsProblem5.1.Explainwhathappenswhenaninvestorshortsacertainshare.Theinvestor’sbrokerborrowsthesharesfromanotherclient’saccountandsellsthemintheusualway.Tocloseouttheposition,theinvestormustpurchasetheshares.Thebrokerthenreplacesthemintheaccountoftheclientfromwhomtheywereborrowed.Thepartywiththeshortpositionmustremittothebrokerdividendsandotherincomepaidontheshares.Thebrokertransfersthesefundstotheaccountoftheclientfromwhomtheshareswereborrowed.Occasionallythebrokerrunsoutofplacesfromwhichtoborrowtheshares.Theinvestoristhenshortsqueezedandhastocloseoutthepositionimmediately.Afeemaybechargedforborrowingshares.Problem5.2.Whatisthedifferencebetweentheforwardpriceandthevalueofaforwardcontract?Theforwardpriceofanassettodayisthepriceatwhichyouwouldagreetobuyorselltheassetatafuturetime.Thevalueofaforwardcontractiszerowhenyoufirstenterintoit.Astimepassestheunderlyingassetpricechangesandthevalueofthecontractmaybecomepositiveornegative.Problem5.3.Supposethatyouenterintoasix-monthforwardcontractonanon-dividend-payingstockwhenthestockpriceis$30andtherisk-freeinterestrate(withcontinuouscompounding)is12%perannum.Whatistheforwardprice?Theforwardpriceis0120530$3186eProblem5.4.Astockindexcurrentlystandsat350.Therisk-freeinterestrateis8%perannum(withcontinuouscompounding)andthedividendyieldontheindexis4%perannum.Whatshouldthefuturespriceforafour-monthcontractbe?Thefuturespriceis(008004)03333350$3547e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,0F,andspotprice,0S,arerelatedby()00cyTFSewherecisthecostofcarry,yistheconvenienceyield,andTisthetimetomaturityofthefuturescontract.Problem5.7.Explainwhyaforeigncurrencycanbetreatedasanassetprovidingaknownyield.Aforeigncurrencyprovidesaknowninterestrate,buttheinterestisreceivedintheforeigncurrency.Thevalueinthedomesticcurrencyoftheincomeprovidedbytheforeigncurrencyisthereforeknownasapercentageofthevalueoftheforeigncurrency.Thismeansthattheincomehasthepropertiesofaknownyield.Problem5.8.Isthefuturespriceofastockindexgreaterthanorlessthantheexpectedfuturevalueoftheindex?Explainyouranswer.Thefuturespriceofastockindexisalwayslessthantheexpectedfuturevalueoftheindex.ThisfollowsfromSection5.14andthefactthattheindexhaspositivesystematicrisk.Foranalternativeargument,letbetheexpectedreturnrequiredbyinvestorsontheindexsothat()0()qTTESSe.Becauserand()00rqTFSe,itfollowsthat0()TESF.Problem5.9.Aone-yearlongforwardcontractonanon-dividend-payingstockisenteredintowhenthestockpriceis$40andtherisk-freerateofinterestis10%perannumwithcontinuouscompounding.a)Whataretheforwardpriceandtheinitialvalueoftheforwardcontract?b)Sixmonthslater,thepriceofthestockis$45andtherisk-freeinterestrateisstill10%.Whataretheforwardpriceandthevalueoftheforwardcontract?a)Theforwardprice,0F,isgivenbyequation(5.1)as:0110404421Feor$44.21.Theinitialvalueoftheforwardcontractiszero.b)ThedeliverypriceKinthecontractis$44.21.Thevalueofthecontract,f,aftersixmonthsisgivenbyequation(5.5)as:0105454421fe295i.e.,itis$2.95.Theforwardpriceis:0105454731eor$47.31.Problem5.10.Therisk-freerateofinterestis7%perannumwithcontinuouscompounding,andthedividendyieldonastockindexis3.2%perannum.Thecurrentvalueoftheindexis150.Whatisthesix-monthfuturesprice?Usingequation(5.3)thesixmonthfuturespriceis(0070032)0515015288eor$152.88.Problem5.11.Assumethattherisk-freeinterestrateis9%perannumwithcontinuouscompoundingandthatthedividendyieldonastockindexvariesthroughouttheyear.InFebruary,May,August,andNovember,dividendsarepaidatarateof5%perannum.Inothermonths,dividendsarepaidatarateof2%perannum.SupposethatthevalueoftheindexonJuly31is1,300.WhatisthefuturespriceforacontractdeliverableonDecember31ofthesameyear?Thefuturescontractlastsforfivemonths.Thedividendyieldis2%forthreeofthemonthsand5%fortwoofthemonths.Theaveragedividendyieldistherefore1(3225)325%Thefuturespriceistherefore(0090032)041671300133180eor$1331.80.Problem5.12.Supposethattherisk-freeinterestrateis10%perannumwithcon