HullFund8eCh05ProblemSolutions.docx

HullFund8eCh05ProblemSolutions.docx

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HullFund8eCh05ProblemSolutions

CHAPTER5

DeterminationofForwardandFuturesPrices

PracticeQuestions

Problem5.8.

Isthefuturespriceofastockindexgreaterthanorlessthantheexpectedfuturevalueoftheindex?

Explainyouranswer.

Thefuturespriceofastockindexisalwayslessthantheexpectedfuturevalueoftheindex.ThisfollowsfromSection5.14andthefactthattheindexhaspositivesystematicrisk.Foranalternativeargument,let

betheexpectedreturnrequiredbyinvestorsontheindexsothat

.Because

and

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）Thedeliveryprice

inthecontractis$44.21.Thevalueofthecontract,f,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$1,331.80.

Problem5.12.

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

Thetheoreticalfuturespriceis

Theactualfuturespriceisonly405.Thisshowsthattheindexfuturespriceistoolowrelativetotheindex.Thecorrectarbitragestrategyis

a）Buyfuturescontracts

b）Shortthesharesunderlyingtheindex.

Problem5.13.

Estimatethedifferencebetweenshort-terminterestratesinJapanandtheUnitedStatesonJuly13,2012fromtheinformationinTable5.4.

Thesettlementpricesforthefuturescontractsareto

Sept:

1.2619

Dec:

1.2635

TheDecember2012priceisabout0.13%abovetheSeptember2013price.Thissuggeststhattheshort-terminterestrateintheUnitedStatesexceededshort-terminterestrateinJapanbyabout0.13%perthreemonthsorabout0.52%peryear.

Problem5.14.

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

Thetheoreticalfuturespriceis

Theactualfuturespriceistoohigh.ThissuggeststhatanarbitrageurshouldbuySwissfrancsandshortSwissfrancsfutures.

Problem5.15.

Thecurrentpriceofsilveris$30perounce.Thestoragecostsare$0.48perounceperyearpayablequarterlyinadvance.Assumingthatinterestratesare10%perannumforallmaturities,calculatethefuturespriceofsilverfordeliveryinninemonths.

Thepresentvalueofthestoragecostsforninemonthsare

0.12+0.12e−0.10×0.25+0.12e−0.10×0.5=0.351

or$0.351.Thefuturespriceisfromequation（5.11）givenby

where

F0=（30+0.351）e0.1×0.75=32.72

i.e.,itis$32.72perounce.

Problem5.16.

Supposethat

and

aretwofuturespricesonthesamecommoditywherethetimestomaturityofthecontractsare

and

with

.Provethat

where

istheinterestrate（assumedconstant）andtherearenostoragecosts.Forthepurposesofthisproblem,assumethatafuturescontractisthesameasaforwardcontract.

If

aninvestorcouldmakearisklessprofitby

a）takingalongpositioninafuturescontractwhichmaturesattime

;and

b）takingashortpositioninafuturescontractwhichmaturesattime

Whenthefirstfuturescontractmatures,theassetispurchasedfor

usingfundsborrowedatrater.Itisthenhelduntiltime

atwhichpointitisexchangedfor

underthesecondcontract.Thecostsofthefundsborrowedandaccumulatedinterestattime

is

.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.Thelongfuturescontractprovidesaslightlyimperfecthedge.

a）Inthiscase,theforwardcontractwouldleadtoaslightlybetteroutcome.Thecompanywillmakealossonitshedge.Ifthehedgeiswithaforwardcontract,thewholeofthelosswillberealizedattheend.Ifitiswithafuturescontract,thelosswillberealizeddaybydaythroughoutthecontract.Onapresentvaluebasistheformerispreferable.

b）Inthiscase,thefuturescontractwouldleadtoaslightlybetteroutcome.Thecompanywillmakeagainonthehedge.Ifthehedgeiswithaforwardcontract,thegainwillberealizedattheend.Ifitiswithafuturescontract,thegainwillberealizeddaybydaythroughoutthelifeofthecontract.Onapresentvaluebasisthelatterispreferable.

c）Inthiscase,thefuturescontractwouldleadtoaslightlybetteroutcome.Thisisbecauseitwouldinvolvepositivecashflowsearlyandnegativecashflowslater.

d）Inthiscase,theforwardcontractwouldleadtoaslightlybetteroutcome.Thisisbecause,inthecaseofthefuturescontract,theearlycashflowswouldbenegativeandthelatercashflowwouldbepositive.

Problem5.18.

Itissometimesarguedthataforwardexchangerateisanunbiasedpredictoroffutureexchangerates.Underwhatcircumstancesisthisso?

FromthediscussioninSection5.14ofthetext,theforwardexchangerateisanunbiasedpredictorofthefutureexchangeratewhentheexchangeratehasnosystematicrisk.Tohavenosystematicrisktheexchangeratemustbeuncorrelatedwiththereturnonthemarket.

Problem5.19.

Showthatthegrowthrateinanindexfuturespriceequalstheexcessreturnoftheportfoliounderlyingtheindexovertherisk-freerate.Assumethattherisk-freeinterestrateandthedividendyieldareconstant.

Supposethat

isthefuturespriceattimezeroforacontractmaturingattime

and

isthefuturespriceforthesamecontractattime

.Itfollowsthat

where

and

arethespotpriceattimeszeroand

istherisk-freerate,and

isthedividendyield.Theseequationsimplythat

Definetheexcessreturnoftheportfoliounderlyingtheindexovertherisk-freerateas

.Thetotalreturnis

andthereturnrealizedintheformofcapitalgainsis

.Itfollowsthat

andtheaboveequationforreducesto

whichistherequiredresult.

Problem5.20.

Showthatequation（5.3）istruebyconsideringaninvestmentintheassetcombinedwithashortpositioninafuturescontract.Assumethatallincomefromtheassetisreinvestedintheasset.Useanargumentsimilartothatinfootnotes2and4andexplainindetailwhatanarbitrageurwoulddoifequation（5.3）didnothold.

Supposewebuy

unitsoftheassetandinvesttheincomefromtheassetintheasset.Theincomefromtheassetcausesourholdingintheassettogrowatacontinuouslycompoundedrateq.Bytime

ourholdinghasgrownto

unitsoftheasset.Analogouslytofootnotes2and4ofChapter5,wethereforebuy

unitsoftheassetattimezeroatacostof

perunitandenterintoaforwardcontracttosell

unitsfor

perunitattime

.Thisgeneratesthefollowingcashflows:

Time0:

Time1:

Becausethereisnouncertaintyaboutthesecashflows,thepresentvalueofthetime

inflowmustequalthe