econ4510 1 finance theory candidate 17773 · econ4510 1 finance theory candidate 17773 9/10 knytte...

ECON45101FinanceTheory Candidate17773

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Candidateinstructions

ECON4510-FinanceTheory

ThisissomeimportantinformationaboutthewrittenexaminECON4510.Pleasereadthiscarefullybeforeyoustartansweringtheexam.Dateofexam:Monday,May13,2019Timeforexam:09.00-12.00(3hours)Theproblemset:Theproblemsetconsistsof3problems.Theywillbegivenweightasindicated.Someadvice:Startbyreadingthroughthewholeexam,andmakesurethatyouallocatetimetoansweringproblemssyoufindeasy.Youcangetagoodgradeeveniftherearepartsofproblemsthatyoudonothavetimetosolve.Itisbettertotrytodosomethingoneachproblemthantogetboggeddownwithoneproblem.Ifyoufindyouarespendingtoomuchtimeononeproblem,stopworkingonitandplantogetbacktoitifyouhavetimeattheend.Makesureyoustateanyassumptionsyoumake.Sketches:Youmayusesketchesonallquestions.Youaretousethesketchingsheetshandedtoyou.Youcanusemorethanonesketchingsheetperquestion.Seeinstructionsforfillingoutsketchingsheetsatthebottom.Itisveryimportantthatyoumakesuretoallocatetimetofillintheheadings(thecodeforeachproblem,candidatenumber,coursecode,dateetc.)onthesheetsthatyouwillusetoaddtoyouranswer.Youwillfindthecodeforeachproblemundertheproblemtext.YouwillNOTbegivenextratimetofilloutthe"generalinformation"onthesketching.Access:Youwillnothaveaccesstoyourexamrightaftersubmission.Thereasonisthatthesketcheswithequationsandgraphsmustbescannedintoyourexam.Youwillgetaccesstoyourexamwithin2-3days.Resourcesallowed:Nowrittenorprintedresources-orcalculator-isallowed(exceptifyouhavebeengranteduseofadictionaryfromtheFacultyofSocialSciences).Grading:Thegradesgiven:A-F,withAasthebestandEastheweakestpassinggrade.Fisfail.Gradesaregiven:Monday,June3,2019

1 Problem1-35%

Assumethatallinvestorshavelinear-quadraticpreferencesandsupposetheycaninvesttheirwealthinasetofriskyassets.Moreover,thereturnsontheassetsareimperfectlycorrelatedandtheexpectedreturnandvariancediffersacrossassets.(A)[5%]Supposethereisnoriskfreeasset.Illustrate,usingadiagram,thesetofportfoliosthatinvestorswillchooseinequilibrium.(B)[5%]Explainhowtheoptimalportfolioschangewhenariskfreeassetisintroduced(C)[10%]Supposeariskyassetihasanexpectedreturnequaltotheriskfreerate andthesamestandarddeviationasthereturnonthemarketportfolio.i.Explainwhyallindividualswanttoholdthisassetinequilibriumeventhoughitoffersalowequilibriumreturn.ii.Whatmustthecovariancebebetweentherateofreturnontheasset, ,andthereturnonthemarketportfolioinequilibrium?

(D)[15%]EmpiricaltestingofCAPM:i.ExplainhowCAPMcanbetestedempirically.ii.ReviewsomeempiricalevidencethatCAPMshouldberejected.iii.Howcouldadie-hardCAPMbelieverholdontoCAPMdespitetheevidenceabove?

Fillinyouranswerhereand/oronsketchingpaper

ECON45101FinanceTheory Candidate17773

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Fillinyouranswerhereand/oronsketchingpaper

(C)i.Alloftheindividualswillwanttoholdaportfoliomadeoutofonly2assets,theriskfreeassetandthemarketportfolio.Sincethemarketportfolioiscomprisedofallassetsinthemarket(aslongastheexpectedreturnoftheassetisnotsmallerthantheexpectedreturnoftheriskfreeasset)allindividualswillholdtheriskyassetithroughthemarketportfolio.ii.WeusetheCAPMequation:Sincetheriskyassethasthesamereturnastheriskfreeassetthelefthandsideoftheequationequals0.Wethenhave:Sinceinequilibriumthereturnofthemarketportfoliocannotbethesameasthereturnoftheriskfreeassettheonlywaythattheequationholdsisif .

Wefindthatinequilibrium andsince thecovariancebetweentheriskyassetandthemarketportfolioisequalto0.(D)i.TheCAPMcanbetestedusingtheßofalltheportfolios.TheCAPMmodelverifiestheequation

whereriisthereturnofportfolioiandrmthereturnofthemarketportfolio.Usingempiricaldatawecanevaluatetheportfolios,ifsomeofthemdonotverifytheequationtherecouldbeanarbitrageopportunity.ii.CAPMshouldberejectedconsideringthatitisveryhardtofindthetruemarketportfolio.Toreplace/rejectthemodelwecouldusethefactormodelwherewecancreate0investmentportfoliosusingprice/bookratio,small/bigcompaniesorlow/highbeta.Forexamplewewouldbuystockswithlowprice/bookratioandshortstockswithhighprice/bookratio.Samefortheothervariables,wecancreateanother0investmentwherewegolongonsmallcapstocksandshortonbigcap.These0investmentportfolioshaveempiricallyanexcessreturnsotheyareasourceofarbitrageandagoodargumenttorejecttheCAPM.iii.HecouldarguethattheoreticallytheCAMPholdsbutempiricallyitisveryhardtofindthetruemarketportfolioandthusitisstillveryusefulinthetheoreticalcasebutnotempirically.

ß

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2 Problem2-45%

Consideranindividualwithpreferences ,wheretheutilityfunction is

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constantrelativeriskaversion(CRRA);

where .TheindividualhasaninitialwealthWandhasnootherincome.(A)[5%]Showthattheparameter istherelativeriskaversionforthisindividual.(B)[10%]Assumethatinperiod1theindividualcaninvestinstocksandbonds,wherestockspayariskyreturn inperiod2andbondshaveasafereturn inperiod2.i.Writedowntheoptimizationproblemfortheindividual.ii.Showthatthefirst-orderconditionsforthisindividualcanbeexpressedas

(C)[15%]Assumethat and areaggregateconsumptionandthatallhouseholdshavethesameCRRAutilityfunction above.Moreover,assumethattheaggregateconsumptiongrowth, ,andtheriskyreturn arejointlylog-normal.Showthattheequitypremiumcanbeexpressedas

(D)[7%]Giveaneconomicinterpretationoftheformulafortheequitypremiumimpliedbythemodel,i.e.,theequationin(2.C)above.(E)[5%]MehraandPrescott(1985)documentedthefollowingannualstatisticsforthe1889-1980periodforUSA: , ,andthefollowingcovariancematrix:

Variance-covariancematrix.USA1889-1978

0.0274 0.0022 0.0013

Explainwhytheseempiricalobservationsarepuzzlinginlightoftheequitypremiumimpliedbythemodel,i.e.,theequationin(2.C)above.(F)[3%]Assumethatanewasset isintroducedandthatthereturnonthisassetisnegativelycorrelatedwithconsumptiongrowth,i.e., .Willtheexpectedreturnonthisassetbelowerorhigherthan

thesafeinterestrate ?Givethoroughintuitionforyourargument.Fillinyouranswerhereand/oronsketchingpaper

(onpaper)

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3 Problem3-20%

Consideratwo-periodbinominalmodelforanasset.Thepriceoftheassetis inthefirstperiod.

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Assumethattheinterestrateiszero( ).Moreover,betweenperiod1and2thesharepriceoftheassetwilleitherincreasewith10%(whichhappenswithprobability0.6)orfallwith10%(whichhappenswithprobability0.4).Theassetdoesnotpayanydividends.Consideracalloptionwithastrikeprice inthesecondperiod.(A)[1%]When(i.e.,inwhichstates)shouldtheoptionbeexercised?(B)[9%]Calculatethereplicatingportfolio.Explainwhythereplicatingportfolio(foracalloption)inthebinominalmodelwillalwaysinvolveleveragedpositions,i.e.,debt( ).(C)[5%]Calculatethevalueofthecalloptioninthefirstperiod.(D)[5%]Whyistheprobabilityofapriceincreaseirrelevantforthepriceoftheoption?Fillinyouranswerhereand/oronsketchingpaper

(onpaper)

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