Review Problems for Midterm 2 W_solutions 2014

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%

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Review Problems for Midterm 2 W_solutions 2014