Portfolio Diversification and Supporting Financial InstitutionsWord文档下载推荐.docx
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Portfolio Diversification and Supporting Financial InstitutionsWord文档下载推荐.docx
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sabadoutcomeforanyone,that'
stheoutcomeofsomerandomdraw.Whenpeoplegetintorealtroubleintheirlives,it'
sbecauseofasequenceofbadeventsthatpushthemintounfortunatepositionsand,veryoften,financialriskmanagementispartofthethingthatpreventsthatfromhappening.
Thefirst--letmego--Iwanttostartthislecturewithsomemathematics.It'
sacontinuationofthesecondlecture,whereItalkedabouttheprincipleofdispersalofrisk.Iwantnowtocarrythatforwardintosomethingalittlebitmorefocusedontheportfolioproblem.I'
mgoingtostartthislecturewithadiscussionofhowoneconstructsaportfolioandwhatarethemathematicsofit.Thatwillleadusintothecapitalassetpricingmodel,whichisthecornerstoneofalotofthinkinginfinance.I'
mgoingtogothroughthisratherquicklybecausethereareothercoursesatYalethatwillcoverthismorethoroughly,notably,JohnGeanakoplos'
sEcon251.Ithinkwecangetthebasicpointshere.
Let'
sstartwiththebasicidea.Iwanttojustsayitinthesimplestpossibleterms.Whatisitthat--Firstofall,aportfolio,let'
sdefinethat.Aportfolioisthecollectionofassetsthatyouhave--financialassets,tangibleassets--it'
syourwealth.Thefirstandfundamentalprincipleis:
youcareonlyaboutthetotalportfolio.Youdon'
twanttobesomeonelikethefishermanwhoboastsaboutonebigfishthathecaughtbecauseit'
snot--we'
retalkingaboutlivelihoods.It'
sallthefishthatyoucaught,sothere'
snothingtobeproudofifyouhadonebigsuccess.That'
sthefirstverybasicprinciple.Doyouagreewithmeonthat?
So,whenwesayportfoliomanagement,wemeanmanagingeverythingthatgivesyoueconomicbenefit.
Now,underlyingourtheoryistheideathatwemeasuretheoutcomeofyourinvestmentinyourportfoliobythemeanofthereturnontheportfolioandthevarianceofthereturnontheportfolio.Thereturn,ofcourse,inanygiventimeperiodisthepercentageincreaseintheportfolio;
or,itcouldbeanegativenumber,itcouldbeadecrease.Theprincipleisthatyouwanttheexpectedvalueofthereturntobeashighaspossiblegivenitsvarianceandyouwantthevarianceofthereturnontheportfoliotobeaslowaspossiblegiventhereturn,becausehighexpectedreturnisagoodthing.Youcouldsay,Ithinkmyportfoliohasanexpectedreturnof12%--thatwouldbebetterthanifithadanexpectedreturnof10%.But,ontheotherhand,youdon'
twanthighvariancebecausethat'
srisk;
so,bothofthosematter.Infact,differentpeoplemightmakedifferentchoicesabouthowmuchriskthey'
rewillingtobeartogetahigherexpectedreturn.Butultimately,everyoneagreesI--that'
sthepremisehere,thatforthe--ifyou'
recomparingtwoportfolioswiththesamevariance,thenyouwanttheonewiththehigherexpectedreturn.Ifyou'
recomparingtwoportfolioswiththesameexpectedreturn,thenyouwanttheonewiththelowervariance.Allrightisthatclearand--okay.
Solet'
stalkabout--whydon'
tIjustgiveitinaveryintuitiveterm.Supposewehadalotofdifferentstocksthatwecouldputintoaportfolio,andsupposethey'
reallindependentofeachother--thatmeansthere'
snocorrelation.WetalkedaboutthatinLecture2.There'
snocorrelationbetweenthemandthatmeansthatthevariance--andIwanttotalkaboutequally-weightedportfolio.So,we'
regoingtohavenindependentassets;
theycouldbestocks.Eachonehasastandarddeviationofreturn,callthatσ.Let'
ssupposethatallofthemarethesame--theyallhavethesamestandarddeviation.We'
regoingtocallrtheexpectedreturnoftheseassets.Then,wehavesomethingcalledthesquarerootrule,whichsaysthatthestandarddeviationoftheportfolioequalsthestandarddeviationofoneoftheassets,dividedbythesquarerootofn.Canyoureadthisintheback?
AmImakingthatbigenough?
Justbarely,okay.
Thisisaspecialcase,though,becauseI'
veassumedthattheassetsareindependentofeachother,whichisn'
tusuallythecase.It'
slikeaninsurancewherepeopleimaginethey'
reinsuringpeople'
slivesandtheythinkthattheirdeathsareallindependent.I'
mtransferringthistotheportfoliomanagementproblemandyoucanseeit'
sthesameidea.I'
vemadeaveryspecialcasethatthisisthecaseofanequally-weightedportfolio.It'
saveryimportantpoint,ifyouseetheverysimplemaththatI'
mshowinguphere.Thereturnontheportfolioisr,butthestandarddeviationoftheportfolioisσ/√(n).So,theoptimalthingtodoifyouliveinaworldlikethisistogetnaslargepossibleandyoucanreducethestandarddeviationoftheportfolioverymuchandthere'
snocostintermsofexpectedreturn.Inthissimpleworld,you'
dwanttomaken100or1,000orwhateveryoucould.Supposeyoucouldfind10,000independentassets,thenyoucoulddrivetheuncertaintyabouttheportfoliopracticallyto0.Becausethesquarerootof10,000is100,whateverthestandarddeviationoftheportfoliois,youwoulddivideitby100anditwouldbecomereallysmall.Ifyoucanfindassetsthatallhave--thatareallindependentofeachother,youcanreducethevarianceoftheportfolioveryfar.That'
sthebasicprincipleofportfoliodiversification.That'
swhatportfoliomanagersaresupposedtobedoingallthetime.
Now,Iwanttobemoregeneralthanthisandtalkabouttherealcase.Intherealworldwedon'
thavetheproblemthatassetsareindependent.Thedifferentstockstendtomoveupanddowntogether.Wedon'
thavetheidealworldthatIjustdescribed,buttosomeextentwedo,sowewanttothinkaboutdiversifyinginthisworld.Now,Iwanttotalkaboutformingaportfoliowheretheassetsarenotindependentofeachother,butarecorrelatedwitheachother.WhatI'
mgoingtodonow--let'
sstartoutwiththecasewhere--nowit'
sgoingtogetalittlebitmorecomplicatedifwedroptheindependenceassumption.I'
mgoingtodropmorethantheindependenceassumption,I'
mgoingtoassumethattheassetsdon'
thavethesameexpectedreturnandtheydon'
thavethesameexpectedvariance.I'
mgoingto--let'
sdothetwo-assetcase.There'
sn=2,butnotindependentornotnecessarilyindependent.Asset1hasexpectedreturnr1.Thisisdifferent--Iwasassumingaminuteagothatthey'
reallthesame--ithasstandard--thisistheexpectationofthereturnofAsset1andr2istheexpectationofthereturn--I'
msorry,σ1isthestandarddeviationofthereturnonAsset1.WehavethesameforAsset2;
ithasanexpectedreturnofr2,ithasastandarddeviationofreturnofσ2.Thosearetheinputsintoouranalysis.Onemorething,Isaidthey'
renotindependent,sowehavetotalkaboutthecovariancebetweenthereturns.So,we'
regoingtohavethecovariancebetweenr1andr2,whichyoucanalsocallσ12andthosearetheinputstoouranalysis.
Whatwewanttodonowiscomputethemeanandvarianceoftheportfolio--orthemeanandstandarddeviation,sincestandarddeviationisthesquarerootofthevariance--fordifferentcombinationsoftheportfolios.I'
mgoingtogeneralizefromoursimplestoryevenmorebysayingthat,let'
snotassumethatwehaveequally-weighted.We'
regoingtoputx1dollars--let'
ssaywehave$1toinvest,wecanscaleitupanddown,itdoesn'
tmatter.Let'
ssayit'
s$1andwe'
regoingtoputx1inasset1andthatleavesbehind1-x1inasset2,becausewehave$1total.We'
renotgoingtorestrictx1tobeapositivenumberbecause,asyouknoworyoushouldknow,youcanholdnegativequantitiesofassets,that'
scalledshortingthem.Youcancallyourbrokerandsay,I'
dliketoshortstocknumberoneandwhatthebrokerwilldoisborrowthesharesonyourbehalfandsellthemandthenyouownnegativeshares.So,we'
renotgoingto--x1canbeanythingandx--thisisx2=1-x1,sox1+x2=1.
Now,wejustwanttocomputewhatisthemeanandvarianceoftheportfolioandthat'
ssimplearithmetic,basedonwhatwetalkedaboutbefore.I'
mgoingtoerasethis.Theportfoliomeanvariancewilldependonx1inthewaythatifyouput--ifyoumadex1=1,itwouldbeasset1andifyoumadex1=0,thenitwouldbethesameasasset2returns.But,inbetween,ifsomeothernumber,it'
llbesomeblendofthe--meanandvarianceof--theportfoliowillbesomeblendofthemeanandvarianceofthetwoassets.Theportfolioexpectedreturnisgoingtobegivenbythesummationi=1ton,ofxi*ri,.Inthiscase,sincen=2that'
sx1r1+x2r2,orthat'
sx1r1+(1-x1)r2;
that'
stheexpectedreturnontheportfolio.Thevarianceoftheportfolioσ²
--thisistheportfoliovariance--isσ²
=x1²
σ1²
+x2²
σ2²
+2x1x2σ12;
sjusttheformulaforthevarianceoftheportfolioasafunctionof--Now,sincetheyhavetosumto1,Icanwritethisasx1²
+(1-x1)²
+2x1(1-x1)σ12andsothattogethertracesout--Icanchooseanyvalueofx1Iwant,itcanbenumberfromm
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