Review Problems for Midterm 2 W_solutions 2014
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Transcript of Review Problems for Midterm 2 W_solutions 2014
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8/10/2019 Review Problems for Midterm 2 W_solutions 2014
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ReviewProblems:
1. Assumetheexcessreturnonstockjcanbewrittenas with
Cov(e,)=0. Alsoassumethatstockjhasabetaof1.5,astandarddeviationofexcess
returnsof.4,andthatthemarketexcessreturnhasastandarddeviationof.1.
a.
FindE[
b. FindVar[]
c. Whatproportionoftheassetjsvarianceissystematic?
d. Whatdeterminessystematicrisk? Toanswerthisquestionusetheformulasfor
thesystematicrisk.
e. Whatproportionofthevarianceisunsystematic?
f. What isthetotalsystematicandunsystematicvarianceinstockj?
g. WhatisRSquared? Toanswerthisquestionfirstanswerwithwordsandthenusinga
formulaandnotation.
h.
Inthesimpleregressionabove(i.e.noteithasonlyoneexplanatoryvariable)whatistherelationbetweenthecorrelationbetweenRjandRMandtheRsquare?
i. Whatistheformulaforthebetashowninthemodelabove?
j. Whatistheformulaforthealphashownabove?
k. Whatistheformulaforthesecuritycharacteristiclineforassetj?
E[Rj]=j+jE[Rm] notethatE(e)=0.
Var[Rj]=Var[j+jRm+ej]=Var[jRm]+Var[ej]
=2(jRm) +2(ej)
=j22m +2(ej)
=systematicrisk+firmspecificrisk
Thismeansthatthesystematicvarianceinstockjsexcessreturnsisequalto
j22m andthatthefirmspecificvarianceis2(ej).
Thismeansthatthetotalriskinvolvedis j22m +2(ej)
Thisalsomeansthatratioofassetjssystematicrisktototalriskis
j22m /[j22m +2(ej) ]
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Puttingitalltogether:
Systematicriskinjsreturns=j22m=1.5*1.5*.1*.1=.0225
Totalrisk(variance)injsreturns= 2j=.4*.4=.16
Proportionofvariancethatissystematic=j22m /[j22m +2(ej) ]
=.0225/.16=.1406
Rsquareistheratiooftheexplainedvariancetototalvariance. Inotherwordsit
representstheproportionofthevariationinthedependentvariableexplainedby
theexplanatoryvariable. Usingnotationfromaboveitwouldbej22m /[j22m
+2(ej) ]
=.0225/.16=.1406
IfpisthecorrelationbetweenRjandRM thenp2istheRsquarevalue
TheformulaforthebetaisCov(Rj,RM )/Var(RM).
TheformulafortheinterceptisIntercept=Average(Rj)average(RM). Notethatif
Igaveyoutheaveragevaluesusedinthisformulathatyoucouldsolveforthe
intercept.
ThesecuritycharacteristiclineforassetjwouldbeRj=intercept+beta(Rm)
2. Drawthe3figures(withlabeledaxis)thatshowcasethefollowingterms: CAL,CML,
efficientfrontier,SML,alpha,minimumvarianceportfolio,tangentportfolio,security
characteristicline,intercept,regressionline,beta,Rsquare,sharperatio,riskfreerate,standarddeviation.
See3bookfiguresandrelateddiscussioninthetext (6.10+6.11+7.1),(6.12+7.4),and
(7.2)
3.
Whatfeestructurewascommonathedgefundsafewyearsago?
20%ofprofits+aflatmanagementfeeof1.52%ofassets.
4. Assumetheriskfreerateis4%.Assumethehistoricalmarketriskpremiumis9%. Assume
investorsanticipatethatstockXwithabetaof0.9toofferarateofreturnof12percent.
A) WhatwilltheCAPMexpectedreturnofthestockbe,assumingthebetaofthestockand
theoverallexpectedmarketreturndonotchange?
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CAPMsuggeststhattheexpectedreturnshouldbe %1.12)09.0(*9.004.0
12.1%
B) Giventhisinformation,whatwouldhappeninthemarket? Howwouldtheprice
change?
CAPMsuggeststhattheexpectedreturnshouldbe %1.12)09.0(*9.004.0 so
themarketsexpectationislessthanthatspecifiedbytheCAPM. Thismeansthat
investorswouldbebetteroffinvestinginotherassetsthatatleastprovidethe
appropriateexpectedreturnfortheamountofriskinvolved.
Inotherwords,E[r_X]
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relationship(i.e.therealbeta)is0whenusingoursampleweestimatedittobethe
valuereported.
e. Whatrangeofnumbersaregenerallyusedtodescribetheaveragehistoricalrisk
premium?
Thetypicalrangeofvaluesis59%. InclassIsaidthatIexpectthistobecloseto~5%
goingforward.
6. Assumeyouhaveaportfoliothatis$35,000inTBills,$50,000intheS&P500(i.e.$50,000in
themarket),and$15,000inIBM. AssumethatIBMhasabetaof.73,theriskfreerateis.04
andtheaveragehistoricalmarketriskpremiumis6%.
a.
Whatistheportfoliobeta?
Theportfoliobetaisaweightedaverageofthebetasofthesecuritiesintheportfolio.
IBMhasabetaof.73,Tbillshaveabetaof0,andtheS&P500hasabetaof1.
Portfoliobeta=(35/100)(0)+(50/100)(1)+(15/100)(.73)=.705
b. Whatisthelongtermexpectedreturnontheportfolio?
E[r]=.04+.705(.06)=8.23%
c.
Inpartawesawthattheportfoliobetaistheweightedaverageofthesecuritiesinthe
portfolio. Istheriskontheportfoliotheweightedaverageoftheriskintheindividual
securities?
Ifbyriskyouaretalkingaboutbeta(i.e.nondiversifiablerisk)then thebetaof
theportfolioisjusttheweightedaverageoftheindividualbetas. Ifbyriskyou
aretalkingabouttotalriskthendiversificationmakestheriskintheportfolio
notequaltotheweightedaverageofthetotalriskinthesecurities.
7.
Trueorfalse? Explainorqualifyeachasnecessary.
a.
Investorsdemandhigherexpectedratesofreturnonstockswithmorevariableratesof
return.
b. TheCAPMpredictsthatasecuritywithabetaof0willofferazeroexpectedreturn.
c. Aninvestorwhoputs$10,000inTreasurybillsand$20,000intheaveragemarket
portfoliowillhaveabetaof2.0.
d. Investorsdemandhigherexpectedratesofreturnfromstockswithreturnsthatarevery
sensitivetofluctuationsinthestockmarket.
e.
Investorsdemandhigherexpectedratesofreturnfromstocksofsmallcompaniesthan
stocksoflargecompanies.
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Solutions
FALSE TheCAPMindicatesthattheexpectedreturnisafunctionofafirm'sbeta. Thebetaincreasesin
thecovarianceofthereturnswiththemarket,notnecessarilyinthevariabilityofthereturnsofthe
stock.
FALSE TheCAPMpredictsanexpectedreturnequaltotheriskfreeratewhenthebetaiszero.
FALSE AssumeTreasuryBillstobelikeariskfreeassetwithbeta=0. Themarketportfoliohasabeta
equalto1. Sointhisportfolioyouwouldexpect(1/3)(0)+(2/3)(1)=(2/3)tobetheportfolio
beta
TRUE Ifby"verysensitive"youmeanhighlypositivelycorrelatedwiththemarketfluctuationsthenyes
thehigherthepositivecorrelationthehigherthebetawhichmeansthehighertheexpected
return. Onecouldarguethat"verysensitive"couldalsomeanhighlynegativelycorrelatedwith
themarketmovements. IfthisisthecasethentheanswerwouldbeFALSEbecausethebeta
wouldbenegative.
FALSE Theexpectedreturnisafunctionofthebetaofthestockindependentofsize. Itmaybetrue
thatsmallfirm'sstockstendtohavehigherbetasbutthisisnot trueforallsmallfirms.
8.
Compareclosedendandopenendfunds. Howaretheysimilar? Howdotheydiffer?
a. Openend
i. AlwaysredeemableatNAVsopriceandNAVdontdiverge
ii.
Whenopenendfundholdersliquidatetheirholdingsthesharesaresoldbacktothefund
iii.
Donttradeonexchange
iv. Boardofdirectors
v. Hiredmanagementcompany
b. Closedend
i.
Whenclosedendfundholdersliquidatetheirholderstheydontsellbackto
thefund,insteadtheyfindsomeoneelsetobuytheirholdings.
ii. Sharestradedonexchanges.
iii. PriceisoftenlowerthantheNAV.
iv.
Boardofdirectorsv.
Hiredmanagementcompany
9. Dohedgefundshavethesamefeestructureasmutualfunds?
Nowhereasmutualfundshavethefrontend,annual,andbackendfeesasdiscussed
inclass,thehedgefundmanagerstendtoclaimaround20%oftheprofitseachyearin
additionto12%ofassetvalueundermanagement.
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10.Whatroledoescompetitionplayintheefficientmarketshypothesis?
a.
Itisthroughcompetitionforprofitthatinformationheldbyindividualsgetsimpounded
intheprices.
11.DescribewhatalphaisontheSMLplot,intheDardencase,andinaregression.
a. TheverticaldistancefromtheSMListhealphaontheplot.
b.
IntheDardencasewetalkedaboutriskadjustedreturnsasbeingconceptuallylike
alpha.
c. Inaregressionofstockjsexcessreturnonthemarketexcessreturns,theestimated
interceptisthealpha. Ifitissignificantitindicatesthatthereissomethingsystematic
thatexplainsthereturnsthatisnotcapturedbythebeta.
12.Whymightsomeonesaythattheyareseekingpositivealphas?
a. Ifalphacapturestheportionofthereturnnotexplainedbybetaanditispositiveit
representspositivereturnsaboveandbeyondwhattheriskwouldjustify.
13.
CAPM:
a. Whatassumptionsareusedtoderivethemodel?
i.
Seelistinslides,homework,orassignedreading.
b. Dotheassumptionsholdintherealworld? Ifnot,whatdoesthisimplyaboutCAPMs
applicabilitytoourworld?
i.
Nowehavedifferenttaxrates,differenttransactioncosts,different
informationsets,andaccesstodifferentassets. Theseviolationsofthetheorys
assumptionsimplythatweeachwouldcalculateaslightlydifferentefficient
frontierandtangencyportfolio. Despitetheassumptionviolations,themain
intuitionoftheCAPMstillholds: thereisariskreturnrelation,onlynon
diversifiableriskiscompensatedinexpectation,thereisaprice
ofrisk.
c.
Dobetasexplainthevariationweobserveintheactualdata? DoesCAPMdoagoodjob
inexplainingreturns?
i.
Betasexplainedvariationinreturnspriorto1980butarenotgoodatpredicting
returnstoday. SotheCAPMdoesapoorjobofexplainingreturns.
d. Whatotherfactorshavebeenshowntoexplainreturns? Aretheseconsistentwith
CAPM? Aretheyconsistentwithmarketefficiency?
i.
Seediscussionofeffectsandanomalies(i.e.B/M,size,momentum,etc.)
e.
DescribethenatureofsomeofthetestspeoplehaveusedtotesttheCAPM?
i. Arehistoricalreturnslinearlyrelatedtobetas?
ii. Doportfoliosortsformedbysortingonnonbetacharacteristics(i.e.,size,b/m,
etc)explainvariationinreturns? I.e.,isitvariationinthesortingcharacteristic
orinbetathatappearstobeexplainingvariationinreturns?
f. IfCAPMisflawed,whyisitsowidelyused?
i. Itiseasytouse
ii. Theintuitionbehindthemodeliscompelling
iii. Itprovidesabenchmark. Evenifweallunderstandthatitisfarfromperfect,it
providesastandardizedwaytocalculateandtalkaboutexpectedreturns.
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g. HowisCAPMusedinpractice?
i.
Costofequity/hurdlerate
ii. Expectedreturncalculations
iii. Benchmark
14.Whatistheintuitionbehindusingthedifferenceinreturnsbetweensmallandbig,andhigh
M/BandlowM/Bfirmsasfactorsinamultifactormodel?
a. Therearetwogeneralwaystoidentifyadditionalriskfactorsforfactormodels:
(1) identifymajorsystematicrisksfacinginvestors. Eachsourceofrisk
wouldcarryitsownriskpremium.
(2) Searchforcharacteristicsthatseemonempiricalgroundstoproxyfor
exposuretosystematicrisk.
b. FortheFamaFrenchmodeltheytookthesecondapproach. Differentsizedfirmsand
firmswithdifferentM/Bratiosappearedtohavesystematicallydifferentreturn
movements. Sobytakingthedifferenceinreturnsbetweenfirmswithopposite
characteristicsalongthesedimensionscreatesaproxymeasureoftheunderlyingrisk
factor.
15.Doyouexpectwellmanagedfirmstoprovidehighratesofreturn?Whyorwhynot?
Weexpectthemtohaverelativelyhighreturnoninvestmentsonthefinancialstatementsbut
wedonotexpectthemtonecessarilyhavehighreturnsontheirstockvalue. Thisisbecause
investorsexpectationforfuturecashflowsaffectstodaysprice. Henceinvestorswillexpect
relativelyhighcashflowsinthefuturefromawellruncompanywhichwillmakethecurrent
pricemoveup. Investorsthatbuyintothestockafterthepricehasalreadyriseninexpectation
willnotgainabnormalreturnsgoingforwardunlessthefirmproducesevenhigherfuturecash
flowsthanalreadyexpected.
16.YouwanttocalculatetheexpectedreturnatCocaColausingthe3factormodel. Youusedata
overthepast5yearstoestimatethefactorbetasofCocaColausingtheFamaFrench3factor
model. Specificallyyouregressthemonthlyexcessreturn(therealizedreturnineachmonth
minustheriskfreerate)ofCocaColasstockontheexcessreturnforeachofthe3factor
portfolios(rmrf,SmB,HmL). ByyourestimatestheFamaFrenchmodelbetasare:
Marketbeta=.15
SmBbeta=.30
HmLbeta=.49
Thecurrentriskfreemonthlyrateis.42%(i.e.,5%/12). Determinethecostofequitycapital
usingthese3factors. Todothisyouwillneedtheriskpremiumsforeachofthesefactors.
Assumetheaveragemonthlyhistoricalriskpremiumonthemarketis.64%,theaverage
monthlyhistoricalpremiumontheSmBis.17%,andtheaveragemonthlyhistoricalpremiumon
theHmLis.53%.
E[r]=.0042+.15*.0064+.30*.0017+.49*.0053
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17.Twoinvestmentadvisersarecomparingperformance. Oneaverageda19%returnandthe
othera16%return. However,thebetaofthefirstadviserwas1.5whilethatofthesecondwas
1. Assumethemarketpremiumis8%.
a. Whichadviserwaslikelyabetterselectorofindividualstocks?
Inordertodeterminewhichinvestorwasabetterselectorofindividualstockswelookattheabnormalreturn,whichistheexpostalpha;thatis,theabnormalreturnisthe
differencebetweentheactualreturnandthatpredictedbytheSML.Withoutadditional
information(i.e.,theriskfreerate)wecannotdeterminewhichinvestmentadviseris
thebetterselectorofindividualstocks.
Inpartsbandcwewillinvestigatewhatthesolutionwillbefordifferentriskfreerates
andhistoricalmarketpremiums.
b. Iftheriskreeratewere6%, whichadviserwouldbethesuperiorstockselector?
Ifrf=6%andusingalphafortheabnormalreturn:
1=19%[6%+1.5(8%)]=19%18%=1%
2=16%[6%+1.0(8%)]=16%14%=2%
Here,thesecondinvestmentadviserhasthelargerabnormalreturnandthusappearsto
bethebetterselectorofindividualstocks. Bymakingbetterpredictions,thesecond
adviserappearstohavetiltedhisportfoliotowardunderpricedstocks.
c.
WhatiftheTbillratewere3%andthemarketriskpremiumwere12%(12%wouldbe
anexcessivelyhighassumption)?
Ifrf=3%andrM=15%,then:
1=19%[3%+1.5(12%)]=19%21%=2%
2=16%[3%+1.0(12%)]=16%15%=1%
Here,notonlydoesthesecondinvestmentadviserappeartobeabetterstockselector,
butthefirstadviser'sselectionsappearvalueless(orworse).
18.
Doestheefficientmarketshypothesisimplythatyoucantmakeaprofit?
Notheefficientmarketshypothesissuggeststhatpricesreflectcurrentinformation.
Hencetheexpectedfuturecashflows(basedonexistinginformation)arealready
anticipatedanddiscountedattheriskappropriateratetoobtaincurrentprices.
(Rememberthatcurrentpricescanbethoughtoftheasthepresentvalueofallfuture
expectedcashflows.) Ifthefuturecashflowsoccuracrosstimeascurrentlyexpected
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thenyouwillearntheratethatwasappliedthefuturecashflowstodiscountthem. I.e.,
youwillearnafairrateofreturngiventherisk.
19.Whatempiricalevidenceexiststhatsupporttheideaofmarketefficiency?Whatempirical
evidenceexiststhatisinconsistentwiththeideaofmarketefficiency?
Seechapter8andtheassociatedlectureslides.
20.AnopenendfundhasaNAvof$10.70pershare. Itissoldwithafrontendloadof6%.Whatis
theofferingprice?
Theofferingpriceincludesa6%frontendload,orsalescommission,meaningthateverydollar
paidresultsinonly$0.94goingtowardpurchaseofshares. Therefore:
Offeringprice=06.01
70.10$
load1
NAV
==$11.38
21.
Aclosedendfundhasaportfoliocurrentlyworth$200million. Ithasliabilitiesof$3millionand
5millionsharesoutstanding.
a. WhatistheNAV?
b. Ifthefundsellsfor$36pershare,whatisthepremiumordiscountasapercentofNAV?
a. NAV=million5
million3$million200$ =$39.40
b. Premium(ordiscount)=NAV
NAVicePr =
40.39$
40.39$36$ =0.086=8.6%
Thefundsellsatan8.6%discountfromNAV
22.
AfundstartedtheyearwithaNAVof$12.50. ByyearenditsNAVequaled$12.10. Thefund
paidyearenddistributionsofincomeandcapitalgainsof$1.50.Whatwastherateofreturnto
aninvestorinthefund?
Rateofreturn=NAVyearofStart
onsDistributi)( NAV=
50.12$
50.1$40.0$ =0.0880=8.80%
23.LoadedUpFundchargesa12b1feeof1%andmaintainsanexpenseratioof.75%. Economy
fundchargesafrontendloadof2%buthasno12b1feeandanexpenseratioof.25%. Assume
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therateofreturnonbothfundsunderlyingportfolios(beforefees)is6%peryear. Howmuch
willaninvestmentineachfundgrowtoafter(a)1year,(b)3years,and(c)10years? Assumean
initialinvestmentof$100.
Assumeahypotheticalinvestmentof$100.
Loadedup
a.Year1=100x(1+.06.0175)=104.25
b.Year3=100x(1+.06.0175)^3=116.30
c.Year10=100x(1+.06.0175)^10=151.62
Economyfund
a.Year1=100x.98x(1+.06.0025)=103.64
b.Year3=100x.98x(1+.06.0025)^3=115.90
c.Year10=100x.98x(1+.06.0025)^10=171.41
24.Supposeyouobservetheinvestmentperformanceof350portfoliomanagersfor5yearsand
rankthembyinvestmentreturnsduringeachyear. After 5years,youfindthat11ofthefunds
haveinvestmentreturnsthatplacethefundinthetophalfofthesampleeachandeveryyearof
yoursample. Suchconsistencyofperformanceindicatestoyouthatthesefundsmustbethe
fundswhosemanagersareinfactskilled,andyouinvestyourmoneyinthesefunds. Isyour
conclusionwarranted?
Supposethatfinishinginthetophalfofallportfoliomanagersispurelyluck,andthatthe
probabilityofdoingsoinanyyearisexactly. Thentheprobabilitythatanyparticularmanager
wouldfinishinthetophalfofthesamplefiveyearsinarowis()5=1/32.Wewouldthen
expecttofindthat[350(1/32)]=11managersfinishinthetophalfforeachofthefive
consecutiveyears. Thisispreciselywhatwefound. Thus,weshouldnotconcludethatthe
consistentperformanceafterfiveyearsisproofofskill.Wewouldexpecttofindeleven
managersexhibitingpreciselythislevelof"consistency"evenifperformanceisduesolelytoluck.
25.Supposethateverytimeafundmanagertradesstock,transactioncostssuchascommissions
andbidaskspreadsamountto.4%ofthevalueofthetrade. Iftheportfolioturnoverrateof
50%,byhowmuchisthetotalreturnoftheportfolioreducedbytradingcosts?
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Theturnoverrateis50%. Thismeansthat,onaverage,50%oftheportfolioissoldandreplaced
withothersecuritieseachyear. Tradingcostsonthesellordersare0.4%;andthebuyordersto
replacethosesecuritiesentailanother0.4%intradingcosts. Totaltradingcostswillreduce
portfolioreturnsby:20.4%0.50=0.4%