投资学chap0077thed.docx

投资学chap0077thed.docx

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投资学chap0077thed

CHAPTER7:

OPTIMALRISKYPORTFOLIOS

1.Thecorrectchoiceisc.Intuitively,wenotethatsinceallstockshavethesameexpectedrateofreturnandstandarddeviation,wechoosethestockthatwillresultinlowestrisk.ThisisthestockthathasthelowestcorrelationwithStockA.

Moreformally,wenotethatwhenallstockshavethesameexpectedrateofreturn,theoptimalportfolioforanyrisk-averseinvestoristheglobalminimumvarianceportfolio（G）.WhentheportfolioisrestrictedtoStockAandoneadditionalstock,theobjectiveistofindGforanypairthatincludesStockA,andthenselectthecombinationwiththelowestvariance.Withtwostocks,IandJ,theformulafortheweightsinGis:

Sinceallstandarddeviationsareequalto20%:

Cov（rI,rJ）=ρσIσJ=400ρandwMin（I）=wMin（J）=0.5

Thisintuitiveresultisanimplicationofapropertyofanyefficientfrontier,namely,thatthecovariancesoftheglobalminimumvarianceportfoliowithallotherassetsonthefrontierareidenticalandequaltoitsownvariance.（Otherwise,additionaldiversificationwouldfurtherreducethevariance.）Inthiscase,thestandarddeviationofG（I,J）reducesto:

σMin（G）=[200（1+ρIJ）]1/2

ThisleadstotheintuitiveresultthatthedesiredadditionwouldbethestockwiththelowestcorrelationwithStockA,whichisStockD.TheoptimalportfolioisequallyinvestedinStockAandStockD,andthestandarddeviationis17.03%.

2.No,theanswertoProblem1wouldnotchange,atleastaslongasinvestorsarenotrisklovers.Riskneutralinvestorswouldnotcarewhichportfoliotheyheldsinceallportfolioshaveanexpectedreturnof8%.

3.No,theanswerstoProblems1and2wouldnotchange.Theefficientfrontierofriskyassetsishorizontalat8%,sotheoptimalCALrunsfromtherisk-freeratethroughG.ThebestPortfolioGis,again,theonewiththelowestvariance.Theoptimalcompleteportfoliodependsonriskaversion.

4.b.

5.False.Iftheborrowingandlendingratesarenotidentical,then,dependingonthetastesoftheindividuals（thatis,theshapeoftheirindifferencecurves）,borrowersandlenderscouldhavedifferentoptimalriskyportfolios.

6.c.

7.σP=30=yσ=40yy=0.75

E（rP）=12+0.75（30-12）=25.5%

8.d.PortfolioYcannotbeefficientbecauseitisdominatedbyanotherportfolio.Forexample,PortfolioXhasbothhigherexpectedreturnandlowerstandarddeviation.

9.Theparametersoftheopportunitysetare:

E（rS）=20%,E（rB）=12%,σS=30%,σB=15%,ρ=0.10

Fromthestandarddeviationsandthecorrelationcoefficientwegeneratethecovariancematrix[notethatCov（rS,rB）=ρσSσB]:

Bonds

Stocks

Bonds

225

45

Stocks

45

900

Theminimum-varianceportfolioiscomputedasfollows:

wMin（S）=

wMin（B）=10.1739=0.8261

Theminimumvarianceportfoliomeanandstandarddeviationare:

E（rMin）=（0.1739⨯20）+（0.8261⨯12）=13.39%

σMin=

=[（0.17392900）+（0.82612225）+（20.17390.826145）]1/2

=13.92%

10.

Proportion

instockfund

Proportion

inbondfund

Expected

return

Standard

Deviation

0.00%

100.00%

12.00%

15.00%

17.39%

82.61%

13.39%

13.92%

minimumvariance

20.00%

80.00%

13.60%

13.94%

40.00%

60.00%

15.20%

15.70%

45.16%

54.84%

15.61%

16.54%

tangencyportfolio

60.00%

40.00%

16.80%

19.53%

80.00%

20.00%

18.40%

24.48%

100.00%

0.00%

20.00%

30.00%

Graphshownbelow.

11.

Thegraphindicatesthattheoptimalportfolioisthetangencyportfoliowithexpectedreturnapproximately15.6%andstandarddeviationapproximately16.5%.

12.Theproportionoftheoptimalriskyportfolioinvestedinthestockfundisgivenby:

wB=10.4516=0.5484

Themeanandstandarddeviationoftheoptimalriskyportfolioare:

E（rP）=（0.4516⨯20）+（0.5484⨯12）=15.61%

σp=[（0.45162900）+（0.54842225）+（20.45160.5484⨯45）]1/2

=16.54%

13.Thereward-to-variabilityratiooftheoptimalCALis:

14.a.Ifyourequirethatyourportfolioyieldanexpectedreturnof14%,thenyoucanfindthecorrespondingstandarddeviationfromtheoptimalCAL.TheequationforthisCALis:

SettingE（rC）equalto14%,wefindthatthestandarddeviationoftheoptimalportfoliois13.04%.

b.TofindtheproportioninvestedintheT-billfund,rememberthatthemeanofthecompleteportfolio（i.e.,14%）isanaverageoftheT-billrateandtheoptimalcombinationofstocksandbonds（P）.LetybetheproportioninvestedintheportfolioP.ThemeanofanyportfolioalongtheoptimalCALis:

E（rC）=（l-y）rf+yE（rP）=rf+y[E（rP）-rf]=8+y（15.61-8）

SettingE（rC）=14%wefind:

y=0.7884and（1-y）=0.2116（theproportioninvestedintheT-billfund）.

Tofindtheproportionsinvestedineachofthefunds,multiply0.7884timestherespectiveproportionsofstocksandbondsintheoptimalriskyportfolio:

Proportionofstocksincompleteportfolio=0.78840.4516=0.3560

Proportionofbondsincompleteportfolio=0.78840.5484=0.4324

15.Usingonlythestockandbondfundstoachieveaportfolioexpectedreturnof14%,wemustfindtheappropriateproportioninthestockfund（wS）andtheappropriateproportioninthebondfund（wB=1-wS）asfollows:

14=20wS+12（1-wS）=12+8wSwS=0.25

Sotheproportionsare25%investedinthestockfundand75%inthebondfund.Thestandarddeviationofthisportfoliowillbe:

σP=[（0.252900）+（0.752225）+（20.250.7545）]1/2=14.13%

Thisisconsiderablygreaterthanthestandarddeviationof13.04%achievedusingT-billsandtheoptimalportfolio.

16.False.Theportfoliostandarddeviationequalstheweightedaverageofthecomponent-assetstandarddeviationsonlyinthespecialcasethatallassetsareperfectlypositivelycorrelated.Otherwise,astheformulaforportfoliostandarddeviationshows,theportfoliostandarddeviationislessthantheweightedaverageofthecomponent-assetstandarddeviations.Theportfoliovarianceisaweightedsumoftheelementsinthecovariancematrix,withtheproductsoftheportfolioproportionsasweights.

17.d.

18.SinceStockAandStockBareperfectlynegativelycorrelated,arisk-freeportfoliocanbecreatedandtherateofreturnforthisportfolio,inequilibrium,willbetherisk-freerate.Tofindtheproportionsofthisportfolio[withtheproportionwAinvestedinStockAandwB=（1–wA）investedinStockB],setthestandarddeviationequaltozero.Withperfectnegativecorrelation,theportfoliostandarddeviationis:

σP=Absolutevalue[wAσAwBσB]

0=5wA-[10（1–wA）]wA=0.6667

Theexpectedrateofreturnforthisrisk-freeportfoliois:

E（r）=（0.6667⨯10）+（0.3333⨯15）=11.667%

Therefore,therisk-freerateis11.667%.

19.a.

Eventhoughitseemsthatgoldisdominatedbystocks,goldmightstillbeanattractiveassettoholdasapartofaportfolio.Ifthecorrelationbetweengoldandstocksissufficientlylow,goldwillbeheldasacomponentinaportfolio,specifically,theoptimaltangencyportfolio.

b.Ifthecorrelationbetweengoldandstocksequals+1,thennoonewouldholdgold.TheoptimalCALwouldbecomprisedofbillsandstocksonly.Sincethesetofrisk/returncombinationsofstocksandgoldwouldplotasastraightlinewithanegativeslope（seethefollowinggraph）,thesecombinationswouldbedominatedbythestockportfolio.Ofcourse,thissituationcouldnotpersist.Ifnoonedesiredgold,itspricewouldfallanditsexpectedrateofreturnwouldincreaseuntilitbecamesufficientlyattractivetoincludeinaportfolio.

20.a.Restrictingtheportfolioto20stocks,ratherthan40to50stocks,willincreasetheriskoftheportfolio,butitispossiblethattheincreaseinriskwillbeminimal.Supposethat,forinstance,the50stocksinauniversehavethesamestandarddeviation（）andthecorrelationsbetweeneachpairareidentical,withcorrelationcoefficientρ.Then,thecovariancebetweeneachpairofstockswouldbeρσ2,andthevarianceofanequallyweightedportfoliowouldbe:

Theeffectofthereductioninnonthesecondtermontheright-handsidewouldberelativelysmall（since49/50iscloseto19/20andρσ2issmallerthanσ2）,butthedenominatorofthefirsttermwouldbe20insteadof50.Forexample,if

σ=45%andρ=0.2,thenthestandarddeviationwith50stockswouldbe20.91%,andwouldriseto22.05%whenonly20stocksareheld.Suchanincreasemightbeacceptableiftheexpectedreturnisincreasedsufficiently.

b.Hennessycouldcontaintheincreaseinriskbymakingsurethathemaintainsreasonablediversificationamongthe20stocksthatremaininhisportfolio.Thisentailsmaintainingalowcorrelationamongtheremainingstocks.Forexample,inpart（a）,withρ=0.2,theincreaseinportfolioriskwasminimal.Asapracticalmatter,thismeansthatHennessywouldhavetospreadhisportfolioamongmanyindustries;concentratingonjustafewindustrieswouldresultinhighercorrelationsamongtheincludedstocks.

21.Riskreductionbenefitsfromdiversificationarenotalinearfunctionofthenumberofissuesintheportfolio.Rather,theincrementalbenefitsfromadditionaldiversificationaremostimportantwhenyouareleastdiversified.RestrictingHennessyto10insteadof20issueswouldincreasetheriskofhisportfoliobyagreateramountthanwouldareductioninthesizeoftheportfoliofrom30to20stocks.Inourexample,restrictingthenumberofstocksto10willincreasethestandarddeviationto23.81%.The1.76%increaseinstandarddeviationresultingfromgivingup10of20stocksisgreaterthanthe1.14%increasethatresultsfromgivingup30of50stocks.

22.Thepointiswelltaken