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AswathDamodaran1ArbitrageAswathDamodaranAswathDamodaran2TheEssenceofArbitrageQInpurearbitrage,youinvestnomoney,takenoriskandwalkawaywithsureprofits.QYoucancategorizearbitrageintherealworldintothreegroups:•Purearbitrage,where,infact,yourisknothingandearnmorethantherisklessrate.•Neararbitrage,whereyouhaveassetsthathaveidenticaloralmostidenticalcashflows,tradingatdifferentprices,butthereisnoguaranteethatthepriceswillconvergeandthereexistsignificantconstraintsontheinvestorsforcingconvergence.•Speculativearbitrage,whichmaynotreallybearbitrageinthefirstplace.Here,investorstakeadvantageofwhattheyseeasmispricedandsimilar(thoughnotidentical)assets,buyingthecheaperoneandsellingthemoreexpensiveone.AswathDamodaran3PureArbitrageQForpurearbitrage,youhavetwoassetswithidenticalcashflowsanddifferentmarketpricesmakespurearbitragedifficulttofindinfinancialmarkets.QTherearetworeasonswhypurearbitragewillberare:•Identicalassetsarenotcommonintherealworld,especiallyifyouareanequityinvestor.•Assumingtwoidenticalassetsexist,youhavetowonderwhyfinancialmarketswouldallowpricingdifferencestopersist.•Ifinaddition,weaddtheconstraintthatthereisapointintimewherethemarketpricesconverge,itisnotsurprisingthatpurearbitrageismostlikelytooccurwithderivativeassets–optionsandfuturesandinfixedincomemarkets,especiallywithdefault-freegovernmentbonds.AswathDamodaran4FuturesArbitrageQAfuturescontractisacontracttobuy(andsell)aspecifiedassetatafixedpriceinafuturetimeperiod.QThebasicarbitragerelationshipcanbederivedfairlyeasilyforfuturescontractsonanyasset,byestimatingthecashflowsontwostrategiesthatdeliverthesameendresult–theownershipoftheassetatafixedpriceinthefuture.•Inthefirststrategy,youbuythefuturescontract,waituntiltheendofthecontractperiodandbuytheunderlyingassetatthefuturesprice.•Inthesecondstrategy,youborrowthemoneyandbuytheunderlyingassettodayandstoreitfortheperiodofthefuturescontract.•Inbothstrategies,youendupwiththeassetattheendoftheperiodandareexposedtonopriceriskduringtheperiod–inthefirst,becauseyouhavelockedinthefuturespriceandinthesecondbecauseyouboughttheassetatthestartoftheperiod.Consequently,youshouldexpectthecostofsettingupthetwostrategiestoexactlythesame.AswathDamodaran5a.StorableCommoditiesQStrategy1:Buythefuturescontract.Takedeliveryatexpiration.Pay$F.QStrategy2:Borrowthespotprice(S)ofthecommodityandbuythecommodity.Paytheadditionalcosts.(a)Interestcost(b)Costofstorage,netofconvenienceyield=SktQIfthetwostrategieshavethesamecosts,F*=+()()S1r-1t=+()()+=+()+()S1r-1SktS1rktttAswathDamodaran6AssumptionsunderlyingarbitrageQInvestorsareassumedtoborrowandlendatthesamerate,whichistherisklessrate.QWhenthefuturescontractisoverpriced,itisassumedthatthesellerofthefuturescontract(thearbitrageur)cansellshortonthecommodityandthathecanrecover,fromtheownerofthecommodity,thestoragecoststhataresavedasaconsequence.AswathDamodaran7Arbitragewithdifferentborrowingrateandnon-recoveryofstoragecosts…QAssume,forinstance,thattherateofborrowingisrbandtherateoflendingisra,andthatshortsellercannotrecoveranyofthesavedstoragecostsandhastopayatransactionscostofts.Thefuturespricewillthenfallwithinabound.QIfthefuturespricefallsoutsidethisbound,thereisapossibilityofarbitrageS-t1rF*S1rktsatbt()+()+()+()AswathDamodaran8b.StockIndexFuturesQStrategy1:Sellshortonthestocksintheindexforthedurationoftheindexfuturescontract.Investtheproceedsattherisklessrate.Thisstrategyrequiresthattheownersofthestocksthataresoldshortbecompensatedforthedividendstheywouldhavereceivedonthestocks.QStrategy2:Selltheindexfuturescontract.QTheArbitrage:Bothstrategiesrequirethesameinitialinvestment,havethesameriskandshouldprovidethesameproceeds.Again,ifSisthespotpriceoftheindex,Fisthefuturesprices,yistheannualizeddividendyieldonthestockandristherisklessrate,thearbitragerelationshipcanbewrittenasfollows:F*=S(1+r-y)tAswathDamodaran9AssumptionsunderlyingarbitrageQWeassumethatinvestorscanlendandborrowattherisklessrate.QWeignoretransactionscostsonbothbuyingstockandsellingshortonstocks.QWeassumethatthedividendspaidonthestocksintheindexareknownwithcertaintyatthestartoftheperiod.AswathDamodaran10ModifiedArbitrageQAssumethatinvestorscanborrowmoneyatrbandlendmoneyatraQAssumethatthetransactionscostsofbuyingstockistcandsellingshortists.Thebandwithinwhichthefuturespricemuststaycanbewrittenas:QArbitrageispossibleifthefuturespricestraysoutsidethisband.S-t1ra-yF*St1rb-ystct()+()+()+()AswathDamodaran11Adjustfortime-varyingdividends…Figure11.5:DividendYieldsbyMonthofYear-20000.0000%0.0200%0.0400%0.0600%0.0800%0.1000%0.1200%0.1400%0.1600%JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecemberMonthDividendsinmonth/IndexatstartofmonthDividendYieldAswathDamodaran12c.T.BondFuturesQThevaluationofatreasurybondfuturescontractfollowsthesamelinesasthevaluationofastockindexfuture,withthecouponsofthetreasurybondreplacingthedividendyieldofthestockindex.Thetheoreticalvalueofafuturescontractshouldbe–where,F*=TheoreticalfuturespriceforTreasuryBondfuturescontractS=SpotpriceofTreasurybondPVC=PresentValueofcouponsdurin