券和组合课后习题答案：sm09

CHAPTER9

MULTIFACTORMODELSOFRISKANDRETURN

AnswerstoOuestions

1.BoththeCapitalAssetPricingModelandtheArbitragePricingModelrestonthe

assumptionthatinvestorsarerewardwithnon-zeroreturnforundertakingtwoactivities:

(1)committingcapital(non-zeroinvestment);and(2)takingrisk.Ifaninvestorcould

earnapositivereturnfornoinvestmentandnorisk,thenitshouldbepossibleforall

investorstodothesame.Thiswouldeliminatethesourceofthe"somethingfornothing,,

return.

Ineithermodel,superiorperformancerelativetoabenchmarkwouldbefoundbypositive

excessreturnsasmeasuredbyastatisticallysignificantpositiveconstantterm,oralpha.

Thiswouldbethereturnnotexplainedbythevariablesinthemodel.

2.CFAExaminationIII(1989)

2a.TheCapitalAssetpricingModel(CAPM)isanequilibriumassetpricingtheoryshowing

thatequilibriumratesofexpectedreturnonallriskyassetsareafunctionoftheir

covariancewiththemarketportfolio.TheCAPMisasingle-indexmodelthatdefines

systematicriskinrelationtoabroad-basedmarketportfolio(i.e.,themarketindex).This

singlefactor("beta")isunchanging:

Rj=Rf+Bj(Rm-Rf)

where

Rj=expectedreturnonanassetorportfolio

Rf=risk-freerateofreturn

Rm=expectedreturnonthemarket

Bj=volatilityoftheassetorportfoliotothatofthemarketm.

ArbitragePricingTheory(APT)isanequilibriumassetpricingtheoryderivedfroma

factormodelbyusingdiversificationandarbitrage.TheAPTshowsthattheexpected

returnonanyriskyassetisalinearcombinationofvariousfactors.Thatis,theAPT

assertsthatanassefsriskinessand,hence,itsaveragelong-termreturn,isdirectlyrelated

toitssensitivitiestocertainfactors.Thus,theAPTisamulti-factormodelwhichallows

forasmanyfactorsasareimportantinthepricingofassets.However,themodelitself

doesnotdefinethesevariables.UnliketheCAPM,whichrecognizesonlyone

unchangingfactor,thekeyfactorsinAPTcanchangeovertime.

Rj=Rf+Bj)(RFi-Rf)+...+Bjk(Rpk-Rf)

where

Ri=returnonanasset

Rf=risk-freerateofreturn

Bi=sensitivityofanassettoaparticularfactor

RF=expectedreturnonaportfoliowithanaverage(1.0)sensitivitytoafactork

J.

k=anasset

=afactor

Researchsuggeststhatseveralmacroeconomicfactorsmaybesignificantinexplaining

expectedstockreturns(i.e.,thesefactorsaresystematicallypriced):

z1x

(>

\2zInflation;

z

f\

t7

\3\Industrialproduction;

7»Riskpremiaasmeasuredbythespreadbetweenlowandhighgradebonds;

z4\

(1

x/Yieldcurve,(i.e.,slopeofthetermstructureofinterestrates.

Otherresearchershaveidentifiedadditionalfactorswhichmayinfluenceanassetsreturn.

z5X

(

\7RealGNPgrowth;

z6\

(

\7Rateofgrowthofrealoilprices(i.e.,anenergyfactor);

z7\

(!

x/Realdefensespending;

8\

>

/Marketindex.

2b.BecauseofAPTsmoregeneralfbnnulation,itismorerobustandintuitivelyappealing

thantheCAPM.Manyfactors,notjustthemarketportfolio,mayexplainassetreturns.

Thispermitsthestockselectionprocesstotakeintoaccountasmanyeconomicvariables

asarebelievedtobesignificantinavaluationcontext-notonlyastoindividualissues,

butalsoastogroups,sectorsoreventhemarketasawhole.Forexample,givenaforecast

ofasuddenspurtintheinflationrate,andoftheresultingeffectoninterestrates,the

analystorportfoliomanagercan,viaAPT,arriveatanestimateofvaluationchanges

acrosshis/herselectionuniverseandadjustportfolioexposuresaccordingly.Alternatively,

stockscouldbeselectedandportfoliosformedbasedonspecificfactor-sensitivitycriteria

setupinadvance.

3.Thesmallfirmeffectreferstothetendencyofsmallcapitalizationstockstooutperform

largecapitalizationstocks.Inandofitself,suchevidencewouldnotnecessarilyconstitute

evidenceofmarketinefficiency,becausemosttestsofsuch“anomalies“areinfactjoint

testsoftwoelements:(1)anassetpricingmodel(e.g.,theCAPMorAPT)whichreflects

therisk-returntradeoff;and(2)marketefficiency,inthesenseofpricesreflectingsome

setofinformation.Ifatestfailstoaccountforsuchanomalies,itcouldbeduetoeither(1)

amis-specifiedassetpricingmodel;or(2)marketinefficiency,orboth.

4.Studiesoftheefficientmarketshypothesissuggestthatadditionalfactorsaffecting

estimatesofexpectedreturnsincludefirmsize,theprice-earningsratio,andfinancial

leverage.Thesevariableshavebeenshowntohavepredictiveabilitywithrespectto

securityreturns.

5.CFAExamination11(1998)

5(a).APTvs.theCAPM

Arbitragepricingtheorydoesnotincludeanyofthefollowingthreeassumptions

incorporatedinthecapitalassetpricingmodel(CAPM):notedaspartsi,ii,andiii:

i.Investorutilityfunctionorquadraticutilityfunction.Capitalmarkettheoryassumes

investorswanttomaximizeutilityintermsofriskandreturnpreferences;maximum

returnperunitofriskorminimumriskperunitofreturn.FromtheMarkowitzmodel

forward,relevantriskhasbeenmeasuredbyvarianceofreturnsorstandarddeviation.

APTmakesnoassumptionsregardinginvestorpreferences;themultifactormodel

commonlyusedinAPTdoesnotincludeanyexponentshigherthan1.

ii.Normallydistributedreturns.Theprobabilitydistributionofexpectedreturnsofan

investmentandtheassociateddispersionorvariabilityofthosereturnsformthebasis

ofMarkowitzportfoliotheoryandtheCAPM.Normalorsymmetricallydistributed

securityreturnsenableestimationofavarianceterm.IntheCAPM,allinvestorshave

identicalestimatesfortheprobabilitydistributionsoffuturereturns(homogeneous

expectations).

APTdoesnotdescribeorspecifyorrequireanassumptionaboutsecurityreturn

distributionsofanykind.

Hi.Themarketportfolio.TheCAPMassumesthatpricing,valuation,risk,andreturnare

solelyfunctionsofanassefsrelationshiptoamarketportfolioofallriskyassets.In

practice,themarketportfolioisdifficulttospecify,soamistakenlyspecifiedmarket

portfoliomightresult.Rollcalledthismisspecification"benchmarkerror.^^

APTdoesnotconsiderorincludeanassumptionofamarketportfolio.APTis

predicatedonacommonsetofseveral(macroeconomic)factors.

5(b).ConceptualDifferencebetweenAPTandtheCAPM

Conceptually,inAPT,returnisafunctionofasetofcommonfactors.IntheCAPM,

returnisafunctionofamarketportfolioofallriskyassets.Thus,onedifferencebetween

APTandtheCAPMcanbedescribedbythefactthatAPTisamultifactormodelthat

attemptstocaptureseveralnon-marketinfluencesthatcausesecuritiesorassetstochange

inpricewhereastheCAPMisasingle-indexmodelthatassumessecuritiesorassets

changeinpricebecauseofacommonco-movementwithonemarketportfolioofallrisky

assets.

AnotherconceptualdifferencebetweenAPTandtheCAPMisthat,inapplicationofthe

theory,themarketportfolio(or"factor")requiredbytheCAPMisspecified.InAPT,the

commonfactorsarenotidentified,butthecommonfactorsinAPTareoftendescribedor

acceptedasincludinginflationorunanticipateddeflation;defaultrisk,government

corporatesecurityspread,riskpremiumsorinterestratespreads;changesintheterm

structureofinterestrates;changesinrealfinalsales,GDPgrowthorasimilarproxyfor

long-runprofitsonaneconomy-widebasis;majorpoliticalupheavals;andexchangerates.

AthirddifferencebetweenAPTandtheCAPMliesintheincorporationofsensitivity

coefficientstomeasureordescribetheriskofassetsorsecurities.APTincorporatesa

numberofsensitivitycoefficients.Thesecoefficientsdeterminehoweachindependent

variableormacroeconomicfactoraffectseachasset.Differentassetsareaffectedto

differentdegreesorextentsbythecommonfactors.IntheCAPM,theonlysensitivity

factorisbeta,anasset'ssensitivitytothechangesinthemarketportfolio(oftencalledan

asset's"systematicrisk").

6.Amarketfactorof1.2meansthemutualfundis1.2timesassensitiveasthemarket

portfolio,allotherfactorsheldequal.TheSMB("smallminusbig")factoristhereturnof

aportfolioofsmallcapitalizationstocksminusthereturntoaportfoliooflarge

capitalizationstocks.Avalueof-0.3indicatesthefundtendstoreactnegativelytosmall

capfactors;thusthefundisprimarilyinvestedinlargecapstocks.TheHML("high

minuslow")factoristhereturntoaportfolioofstockswithhighratiosofbook-to-market

valuelessthereturntoaportfoliooflowbook-to-marketreturns.Avalueof1.4indicates

thefundreactspositivelytothisfactor;thusthefundisweightedtowardhighvalue

stocks.

7.CFAExaminationffl(1990)

Thevalueofstockandbondscanbeviewedasthepresentvalueofexpectedfuturecash

flowsdiscountedatsomediscountratereflectingrisk.Anticipatedeconomicconditions

arealreadyincorporatedinreturns.Unanticipatedeconomicconditionsaffectreturns.

Industrialproduction.Industrialproductionisrelatedtocashflowsinthetraditional

discountedcashflowformula.Therelativeperformanceofaportfoliosensitiveto

unanticipatedchangesinindustrialproductionshouldmoveinthesamedirectionasthe

changeinthisfactor.Whenindustrialproductionturnsupordown,sotoosharesinthe

returnontheportfolio.Portfoliossensitivetounanticipatedchangesinindustrial

productionshouldbecompensatedfortheexposuretothiseconomicfactor.

InflaliorLInflationisreflectedinthediscounirate,whichgenerallyisasfollows:K=real

return+hedgeforinflation+riskpremium.Unexpectedinflationwillquicklybefactored

intokbythemarketasinvestorsattempttohedgethelossofpurchasingpower.Thus,the

discountratewouldmoveinthesamedirectionasthechangeininflation.

HighinflationcouldalsoaffectcashflowsintheDCFformula,buttheeffectwill

probablybemorethanoffsetbyhigherKsnominalcashflowgrowthratesdonotalways

matchexpectedinflationrates.Ifthegrowthinnominalcashflowslagstheinflationrate,

therelativeperformanceofaportfoliosensitivetorisinginflationshoulddeclineover

time.Investmentsinbondsaresubjecttosignificantadverseinflationeffects.Hence,

higherunanticipatedinflationwillnegativeaffectportfoliovalues.

RiskDremiaorqualityspreads.RiskpremiaaffectthemagnitudeoftheDCFdiscount

rate.Theriskpremiummeasurerepresentsinvestorattitudestowardrisk-bearingand

perceptionsaboutthegenerallevelofuncertainty.Whenthereturninlowversushigh

qualitybondswidens,thereislikelytobeanegativeimpactonthevaluesofstockand

bonds,particularlyforlowerqualitycompanies.Riskiercashflowsrequirehigher

discountrates,andwideningqualityspreadsoftensignalgreateruncertaintyaboutprofit

levelsanddebtservicerequirement.

Yieldcurveshifts.Yieldcurveshiftsaffectthediscounlidle.Ifaparallelupwardshift

occurs,itwouldmeaninvestorsarerequiringhigherreturntoholdallassets.Hence,with

highdiscountrates,portfoliovalueswouldfall.Ifthecurvebecomessteeper,longer

durationassets,suchaslong-termbondsandgrowthstocks,wouldbenegativelyaffected

morethanshorter-termassets.

8.Themacroeconomicapproachtoidentifyingthefactorsinamulti-factorassetpricing

modeltriestofindvariablesthatexplain,theunderlyingreasonsforvariationsinthecash

flowsandinvestmentreturnsovertime.Themicroeconomicapproachconcentratesonthe

characteristicsofthesecuritiesthemselves.

Conceptually,itispossibleforthetwoapproachestoleadtothesameestimatefor

expectedreturns.Practicallyspeaking,itisnotlikelythattheywill.Thetwosetsof

factors,whileprobablysignificantlycorrelated,arenotlikelytobeperfectlycorrelated,

leadingtodifferencesinestimates.

9.ThethreefactorsusedbyFamaandFrenchinheirmicroeconomicmodelaretheexcess

marketreturn;SMB("smallminusbig")-hedifferencebetweenthereturnstoaportfolio

ofsmallcapitalizationstocksandaportfoliooflargecapitalizationstocks;andHML

("highminuslow")-thedifferenceinreturnstoaportfolioofhighbook-to-marketvalue

stocksandlowbook-to-marketstocks.Thistypeofapproachfocusesonthevarious

characteristicsoftheunderlyingsecurities.Themacroeconomicapproach,asitsname

implies,concentratesonmoregeneral,economy-widevariablesthatcausevariationinthe

cashflowsandinvestmentreturns.

10.Multifactormodeltypicallyincludeamarketportfolioreturnasoneofthefactors.Thus

thecoefficientsontheotherfactorsinthemodelindicatethemarginalsensitivityof

returnstothosespecificfactors.Highervaluesindicategreatersensitivityofreturnsto

thatfactor.

Inamacroeconomic-basedmodelsuchastheonedevelopedbyChen,Roll,andRoss,

returnsonaparticularsecuritymightbemuchmoresensitivetothecreditspreadthanto

oneoftheothermacrovariables.Theinvestorcouldthenassesswhetherthatsecurityfit

hisparticularinvestmentprofile.

Likewise,amicroeconomic-basedmodelwouldidentifysensitivitytocertain

characteristics,suchassmallvs.largefirm,etc.Again,theinvestorcouldthenselect

securitiesbasedonthesensitivitiestothevariouscharacteristics.

11.

11(a).OnestudythatdoesnotsupporttheAPTisthatofReinganumonthesmall-firmeffect.

ReinganumhypothesizedthatiftheAPTwereasuperiortheorytoCAPM,theAPT

wouldexplainthesmall-firmeffectwheretheCAPMcouldnot.However,hisresultsdid

notsupportthathypothesis.Thesmall-firmportfoliostillhadsignificantpositiveexcess

returns,whilethelargefirmportfoliohadsignificantnegativeexcessreturns.

AstudythatsupportstheAPTisthatofCho,Elton,andGruber.Theirstudytendedto

supportthebasiccontentionoftheAPTthatfactorsotherthanthemarketportfolioaffect

securityreturns.

11(b).Shanken'scontentionthattheAPTisnottestableisinessencethattheAPTisnot

falsifiable.Ifatestfindsthatreturnsarenotexplainedbyaparticularsetoffactors,thatis

nottakenasevidencethattheAPTisincorrect,butthattheparticularsetoffactorsis

wrong.Ontheotherhand,ifastudyfindsthatsomeothersetoffactorsdoesexplain

returns,thatistakenasevidenceiffavoroftheAPT.TheAPTgivesnoguidanceasto

thepropersetoffactors,sointhatsenseitcannotbefalsified.

DybvigandRosscounterthisargumentbystatingthatthePTistestableasanequality,

ratherthanthe"empiricalAPT"ofShanken.

12.CFAExamIII(1986)

12(a).ThebasicCapitalAssetPricingModel(CAPM)assumesthatinvestorscareonlyabout

portfolioriskandexpectedreturn;i.e.,theyareriskaverse.Fromthisassumptioncomes

theconclusionthataportfolio'sexpectedreturnwillberelatedtoonlyoneattribute-its

beta(sensitivity)relativetothebroadlybasedmarketportfolio.

ArbitragePriceTheory(APT)takesadifferentapproach:itisnotmuchconcernedabout

investorpreferences,anditassumesthatreturnsaregeneratedbyamulti-factormodel.

APTreflectsthefactthatseveralmajor(systematic)economicfactorsmayaffectagiven

assetinvaryingdegrees.Further,unliketheCAPM,whosesinglefactorisunchanging,

APTrecognizesthatthesekeyfactorscanchangeovertime(ascaninvestorpreferences).

Summarizing,APT1)identifiesseveralkeysystematicmacroeconomicfactorsaspartof

theprocessthatgeneratessecurityreturnsvs.onlyonefactorrecognizedbytheCAPM,2)

recognizesthatthesekeyfactorscanchangeovertime,whereastheCAPM'ssinglefactor

isunchanging,3)makesfewerassumptionsaboutinvestorpreferencesthantheCAPM,

and4)recognizesthatthesepreferencescanchangeovertime.

12(b).ThefoursystematicfactorsidentifiedbyRollandRossareunanticipatedchangesin:1)

inflation,2)industrialproduction,3)riskpremiums,and4)theslopeofthetermstructure

ofinterestrates.

TheAPTassertsthatanasset9sriskinessand,hence,theaveragelong-termreturn,is

directlyrelatedtoitssensitivitiesinunanticipatedchangesinthesefoureconomic

variables.Thisrelationshipmaybeexpressedinequationformasfollows:

Ri=Ro+biiFi+卜2F2+...+binFn

Thismeansthereturn(Ri)thatacertainasset(i)willproduceisacombinationofsome

"base"return(Ro)plusreturnsoccasionedbytheinfluencesorsensitivities(bn)ofsome

systematicexternalfactors(Fn).

13.CFAExaminationII(2001)

StatewhetherIndicate,foreachincorrect

ArgumentArgumentisCorrectorargument,whytheargumentis

Incorrectincorrect

a.Boththe

CAPMandAPTCAPMrequiresthemean-variance

requireamean­IncorrectportfoliobutAPTdoesnot.

varianceefficient

marketportfolio.

b.Neitherthe

CAPMnorAPTCAPMassumesnormally

assumesnormallyIncorrectdistributedsecurityreturns,but

distributedAPTdoesnot.

returns.

c.TheCAPM

assumesthatone

specificfactorCorrect

explainssecurity

returns,butAPT

doesnot.

CHAPTER9

AnswerstoProblems

1.

1(a).IngeneralfortheAPT,E(Rq)=Xo+Xibqi+九2bq2

ForsecurityJ:

E(Rj)=0.05+0.02x0.80+0.04x1.40

=0.05+0.016+0.056

=.1220or12.20%

ForSecurityL:

E(RL)=0.05+0.02x1.60+0.04x2.25

=0.05+0.032+0.09

=.172or17.20%

1(b).Totalreturn=dividendyield+capitalgainyield

ForsecurityJ,thedividendyieldis$0.75/$22.50=().033or3.33%

ForsecurityL,thedividendyieldis$0.75/$15.00=().05or5%

ForsecurityJ,theexpectedcapitalgainistherefore12.20%-3.33%=8.87%

ForsecurityLtheexpectedcapitalgainistherefore17.20%-5.00%=12.20%

Therefore:

TheexpectedpriceforsecurityJis$22.50x(1.0887)=$25.50

TheexpectedpriceforsecurityLis$15.00x(1.1220)=$16.83

2.

2(a).For1981-2000:

RINTC=0.615x9.09-0.64x(-1.1)-1.476x4.48

=5.59+0.704-6.61

=-0.32%

RJPM=1.366x9.09-0.387x(-1.1)+0.577x4.48

=12.41+0.43+2.59

=15.43%

RWFMI=1.928x9.09+0.817x(-l.l)+1.684x4.48

=17.53-0.90+7.55

=24.18%

For1928-2000:

RINTC=0.615x7.02-0.64x3.09-1.476x4.39

=4.32-1.98-6.48

=-4.14%

RJPM=1.366x7.02-0.387x3.09+0.577x4.39

=9.59-1.20+2.53

=10.92%

RWFMI=1.928x7.02+0.817x3.09+1.684x4.39

=13.53+2.52+18.66

=34.71%

2(b).TheexcessreturnonIntel(INTC)inthe1981-2000periodseemsreasonable,inthatitis

veryclosetozero.INTC'sexcessreturninthe1928-2000periodisunusualinthatitis

negative.TheexcessreturnsofJ.P.Morgan(JPM)andWholeFoods(WFMI)appearto

beextremelylarge.Thisispartlyduetothefactthatweareusingout-of-samplenumbers

todotheestimating.Thatis,theregressionswereestimatedusing1996-2000data,but

theestimatesweremadeusingdatafrommuchlongerperiods.

2(c).No,wewouldn'texpectthefactorbetastoremainconstantovertime.Thesensitivityofa

particularcompany'sreturntoaspecificfactorwillchangeasthecharacterofthefirm

changes.Forinstance,growthcompaniesdon'tremaingrowthcompaniesforever,but

tendtomature.Thustheirfactorbetaswouldchangetoreflecttheslowdowningrowth

butincreasedstabilityofearnings.

3(a).RQRS=4.5+7.5x1.24

=4.5+6.825

=13.8%

RTUV=4.5+7.5x0.91

=4.5+6.825

=11.325%

RWXY=4.5+7.5x1.03

=4.5+7.725

=12.225%

3(b).RQRS=4.5+7.5x1.24+(-0.3)x(-0.42)+0.6x().()0

=4.5+9.30+0.126+0.0()

=13.926%

RTUV=4.5+7.5x0.91+(0.3)x(0.54)+0.6x0.23

=4.5+6.825-0.162+0.138

=11.301%

RWXY=4.5+7.5x1.03+(-0.3)x(-0.09)+).6x0.00

=4.5+7.725+0.027+().0()

=12.252%

3(c).Assumingthatthefactorloadingsaresignificantthethreefactormodelshouldbemore

usefultotheextentthatthenon-marketfactorspickupmovementsinreturnsnot

capturedbythemarketreturn.

3(d).BecausethefactorloadingsonMACRO2arezerofortwoofthestocks,itappearsthat

MACRO2isnotasystematicfactor,i.e.,onethatgenerallyaffectsallstocks.Itmay

representindustry-orfirm-specificfactors.

4(a).E(RD)=5.0+1.2xXi+3.4x九2=13.1%

E(RE)=5.0+2.6xZi+2.6x九2=15.4%

SolvingthesecondequationforQintermsof九2,weget:

Q=[(10.4—2.6)/2.6]入2

Substitutingthatintothefirstequation:

1.2f(10.4-2.6)/2.6]x九2+3.4xX2=8.1

Solvingfor九2,wefind九2=1.5.Usingthisvaluewecandeterminefromeitherequation

thatXiisequalto2.5.

4(b).Becauseneitherstockpaysadividend,thetotalreturnisallduetopriceappreciation.

ThereforeforstockD:

Pox(1.131)=$55

Po=$55/1.131

=$48.63

AndforstockE:

Pox(1.154)=$36

Po=$36/1.154

=$31,20

4(c).Frompart(a),theriskpremiumforfactor1was2.5%.Thenewriskfactoristhus2.5%+

0.25%,or2.75%.Thenewexpectedreturnsare:

E(RD)=5.0+(1.2X2.75)+(3.4X1.5)

=5.0+33+5.1

=13.4%

E(RE)=5.0+(2.6X2.75)+(2.6xl.5)

=5.0+7.15+3.9

=16.05%

4(d).D:PDO(1+0.134)=$55

PDO=$55/1.134

PDO=$48.50

E:PEO(1+.1605)=$36

PEO=$36/(1.1605)

PEO=$31.02

5(a).Becausenostockpaysadividend,allreturnisduetopriceappreciation.

E(RA)=1.1x0.04+0.8x0.02

=0.044+0.016

=0.06or6%

E(PriceA)=$30(1.06)=$31.80

E(RB)=0.7x0.04+0.6x0.02

=0.28+.012

=0.04or4%

E(PriceB)=$30(1.04)=$31.20

E(Rc)=0.3x0.04+0.4x0.02

=0.12+0.008

=0.02or2%

E(PriceC)=$30(1.02)=$30.60

5(b).Inordertocreatearisklessarbitrageinvestment,aninvestorwouldshort1shareodA

andoneshareofC,andbuy2sharesofB.TheweightsofthisportfolioareWA=-0.5,

WB=+1.0,andWC=-0.5.Thenetinvestmentis:

Short1shareA=+$30

Buy2sharesB=-$60

Short1shareC=+$30

Netinvestment=$0

Theriskexposureis:

RiskExposureFactor1Factor2

A(-0.5)xl.l(-0.5)x().8

B(+1.0)x0.7(+1.0)x0.6

c(-().5)x().3(-0.5)x0.4

NetRiskExposure00

Attheendoftheperiodtheprofitisgivenby:

Profit=($30-$31.50)+2x($35-30)+($30-$30.50)

=-$1.50+$10-$0.50

=$8

6(a)