完整版投资学第10版课后习题答案Chap007Word文件下载.docx
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完整版投资学第10版课后习题答案Chap007Word文件下载.docx
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E(rMin)=(0.1739×
.20)+(0.8261×
.12)=.1339=13.39%
σMin=
=[(0.17392900)+(0.82612225)+(20.17390.826145)]1/2
=13.92%
5.
Proportion
inStockFund
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
16.80
19.53
18.40
24.48
100.00
0.00
20.00
30.00
Graphshownbelow.
6.Theabovegraphindicatesthattheoptimalportfolioisthetangencyportfoliowithexpectedreturnapproximately15.6%andstandarddeviationapproximately16.5%.
7.Theproportionoftheoptimalriskyportfolioinvestedinthestockfundisgivenby:
Themeanandstandarddeviationoftheoptimalriskyportfolioare:
E(rP)=(0.4516×
.20)+(0.5484×
.12)=.1561
=15.61%
σp=[(0.45162900)+(0.54842225)+(20.45160.5484×
45)]1/2
=16.54%
8.Thereward-to-volatilityratiooftheoptimalCALis:
9.a.Ifyourequirethatyourportfolioyieldanexpectedreturnof14%,thenyoucanfindthecorrespondingstandarddeviationfromtheoptimalCAL.TheequationforthisCALis:
IfE(rC)isequalto14%,thenthestandarddeviationoftheportfoliois13.04%.
b.TofindtheproportioninvestedintheT-billfund,rememberthatthemeanofthecompleteportfolio(i.e.,14%)isanaverageoftheT-billrateandtheoptimalcombinationofstocksandbonds(P).LetybetheproportioninvestedintheportfolioP.ThemeanofanyportfolioalongtheoptimalCALis:
SettingE(rC)=14%wefind:
y=0.7884and(1−y)=0.2119(theproportioninvestedintheT-billfund).
Tofindtheproportionsinvestedineachofthefunds,multiply0.7884timestherespectiveproportionsofstocksandbondsintheoptimalriskyportfolio:
Proportionofstocksincompleteportfolio=0.78840.4516=0.3560
Proportionofbondsincompleteportfolio=0.78840.5484=0.4323
10.Usingonlythestockandbondfundstoachieveaportfolioexpectedreturnof14%,wemustfindtheappropriateproportioninthestockfund(wS)andtheappropriateproportioninthebondfund(wB=1−wS)asfollows:
0.14=0.20×
wS+0.12×
(1−wS)=0.12+0.08×
wSwS=0.25
Sotheproportionsare25%investedinthestockfundand75%inthebondfund.Thestandarddeviationofthisportfoliowillbe:
σP=[(0.252900)+(0.752225)+(20.250.7545)]1/2=14.13%
Thisisconsiderablygreaterthanthestandarddeviationof13.04%achievedusingT-billsandtheoptimalportfolio.
11.a.
Eventhoughitseemsthatgoldisdominatedbystocks,goldmightstillbeanattractiveassettoholdasapartofaportfolio.Ifthecorrelationbetweengoldandstocksissufficientlylow,goldwillbeheldasacomponentinaportfolio,specifically,theoptimaltangencyportfolio.
b.
Ifthecorrelationbetweengoldandstocksequals+1,thennoonewouldholdgold.TheoptimalCALwouldbecomposedofbillsandstocksonly.Sincethesetofrisk/returncombinationsofstocksandgoldwouldplotasastraightlinewithanegativeslope(seethefollowinggraph),thesecombinationswouldbedominatedbythestockportfolio.Ofcourse,thissituationcouldnotpersist.Ifnoonedesiredgold,itspricewouldfallanditsexpectedrateofreturnwouldincreaseuntilitbecamesufficientlyattractivetoincludeinaportfolio.
12.SinceStockAandStockBareperfectlynegativelycorrelated,arisk-freeportfoliocanbecreatedandtherateofreturnforthisportfolio,inequilibrium,willbetherisk-freerate.Tofindtheproportionsofthisportfolio[withtheproportionwAinvestedinStockAandwB=(1–wA)investedinStockB],setthestandarddeviationequaltozero.Withperfectnegativecorrelation,theportfoliostandarddeviationis:
σP=Absolutevalue[wAσAwBσB]
0=5×
wA−[10(1–wA)]wA=0.6667
Theexpectedrateofreturnforthisrisk-freeportfoliois:
E(r)=(0.6667×
10)+(0.3333×
15)=11.667%
Therefore,therisk-freerateis:
11.667%
13.False.Iftheborrowingandlendingratesarenotidentical,then,dependingonthetastesoftheindividuals(thatis,theshapeoftheirindifferencecurves),borrowersandlenderscouldhavedifferentoptimalriskyportfolios.
14.False.Theportfoliostandarddeviationequalstheweightedaverageofthecomponent-assetstandarddeviationsonlyinthespecialcasethatallassetsareperfectlypositivelycorrelated.Otherwise,astheformulaforportfoliostandarddeviationshows,theportfoliostandarddeviationislessthantheweightedaverageofthecomponent-assetstandarddeviations.Theportfoliovarianceisaweightedsumoftheelementsinthecovariancematrix,withtheproductsoftheportfolioproportionsasweights.
15.Theprobabilitydistributionis:
Probability
RateofReturn
0.7
100%
0.3
−50
Mean=[0.7×
100%]+[0.3×
(-50%)]=55%
Variance=[0.7×
(100−55)2]+[0.3×
(-50−55)2]=4725
Standarddeviation=47251/2=68.74%
16.σP=30=y×
σ=40×
yy=0.75
E(rP)=12+0.75(30−12)=25.5%
17.Thecorrectchoiceis(c).Intuitively,wenotethatsinceallstockshavethesameexpectedrateofreturnandstandarddeviation,wechoosethestockthatwillresultinlowestrisk.ThisisthestockthathasthelowestcorrelationwithStockA.
Moreformally,wenotethatwhenallstockshavethesameexpectedrateofreturn,theoptimalportfolioforanyrisk-averseinvestoristheglobalminimumvarianceportfolio(G).WhentheportfolioisrestrictedtoStockAandoneadditionalstock,theobjectiveistofindGforanypairthatincludesStockA,andthenselectthecombinationwiththelowestvariance.Withtwostocks,IandJ,theformulafortheweightsinGis:
Sinceallstandarddeviationsareequalto20%:
Thisintuitiveresultisanimplicationofapropertyofanyefficientfrontier,namely,thatthecovariancesoftheglobalminimumvarianceportfoliowithallotherassetsonthefrontierareidenticalandequaltoitsownvariance.(Otherwise,additionaldiversificationwouldfurtherreducethevariance.)Inthiscase,thestandarddeviationofG(I,J)reducesto:
ThisleadstotheintuitiveresultthatthedesiredadditionwouldbethestockwiththelowestcorrelationwithStockA,whichisStockD.TheoptimalportfolioisequallyinvestedinStockAandStockD,andthestandarddeviationis17.03%.
18.No,theanswertoProblem17wouldnotchange,atleastaslongasinvestorsarenotrisklovers.Riskneutralinvestorswouldnotcarewhichportfoliotheyheldsinceallportfolioshaveanexpectedreturnof8%.
19.Yes,theanswerstoProblems17and18wouldchange.Theefficientfrontierofriskyassetsishorizontalat8%,sotheoptimalCALrunsfromtherisk-freeratethroughG.Thisimpliesrisk-averseinvestorswilljustholdTreasurybills.
20.Rearrangethetable(convertingrowstocolumns)andcomputeserialcorrelationresultsinthefollowingtable:
NominalRates
Small
Company
Large
Long-Term
Government
Intermed-TermGovernment
Treasury
Bills
Inflation
1920s
-3.72
18.36
3.98
3.77
3.56
-1.00
1930s
7.28
-1.25
4.60
3.91
0.30
-2.04
1940s
20.63
9.11
3.59
1.70
0.37
5.36
1950s
19.01
19.41
0.25
1.11
1.87
2.22
1960s
13.72
7.84
1.14
3.41
3.89
2.52
1970s
8.75
5.90
6.63
6.11
6.29
7.36
1980s
12.46
17.60
11.50
12.01
9.00
5.10
1990s
13.84
18.20
8.60
7.74
5.02
2.93
SerialCorrelation
0.46
-0.22
0.60
0.59
0.63
0.23
Forexample:
tocomputeserialcorrelationindecadenominalreturnsforlarge-companystocks,wesetupthefollowingtwocolumnsinanExcelspreadsheet.Then,usetheExcelfunction“CORREL”tocalculatethecorrelationforthedata.
Decade
Previous
-1.25%
18.36%
9.11%
19.41%
7.84%
5.90%
17.60%
18.20%
Notethateachcorrelationisbasedononlysevenobservations,sowecannotarriveatanystatisticallysignificantconclusions.Lookingattheresults,however,itap
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