Portfolio Diversification and Supporting Financial Institutions.docx

Portfolio Diversification and Supporting Financial Institutions.docx

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PortfolioDiversificationandSupportingFinancialInstitutions

FinancialMarkets:

Lecture4Transcript

January28,2008

ProfessorRobertShiller:

Today'slectureisaboutportfoliodiversificationandaboutsupportingfinancialinstitutions,notablymutualfunds.It'sactuallykindofacrusadeofmine--Ibelievethattheworldneedsmoreportfoliodiversification.Thatmightsoundtoyoualittlebitodd,butIthinkit'sabsolutelytruethatthesamekindofcausethatEmmettThompsongoesthrough,whichistohelpthepoorpeopleoftheworld,canbeadvancedthroughportfoliodiversification--Iseriouslymeanthat.Therearealotofhumanhardshipsthatcanbesolvedbydiversifyingportfolios.WhatI'mgoingtotalkabouttodayappliesnotjusttocomfortablewealthypeople,butitappliestoeveryone.It'sreallyaboutrisk.Whenthere'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=x1²σ1²+x2²σ2²+2x1x2σ12;that'sjusttheformulaforthevarianceoftheportfolioasafunctionof--Now,sincetheyhavetosumto1,Icanwritethisasx1²σ1²+（1-x1）²σ2²+2x1（1-x1）σ12andsothattogethertracesout--Icanchooseanyvalueofx1Iwant,itcanbenumberfromm