金融市場的風(fēng)險評估管理英文版)金融市場的風(fēng)險評估管理英文版)金融市場的風(fēng)險評估管理英文版)Lecture2-TheUniversalPrincipleofRiskManagement:PoolingandtheHedgingofRisks

Overview:

Statisticsandmathematicsunderliethetheoriesoffinance.ProbabilityTheoryandvariousdistributiontypesareimportanttounderstandingfinance.Riskmanagement,forinstance,dependsontoolssuchasvariance,standarddeviation,correlation,andregressionanalysis.Financialanalysismethodssuchaspresentvaluesandvaluingstreamsofpaymentsarefundamentaltounderstandingthetimevalueofmoneyandhavebeeninpracticeforcenturies.

Readingassignment:

JeremySiegel,StocksfortheLongRun,chapter1andAppendix2,p.12FinancialMarkets:Lecture2Transcript

January16,2008

ProfessorRobertShiller:TodayIwanttospend--Thetitleoftoday'slectureis:TheUniversalPrincipleofRiskManagement,PoolingandtheHedgingofRisk.WhatI'mreallyreferringtoiswhatIthinkistheveryoriginal,thedeepconceptthatunderliestheoreticalfinance--Iwantedtogetthatfirst.Itreallyisprobabilitytheoryandtheideaofspreadingriskthroughriskpooling.So,thisideaisanintellectualconstructthatappearedatacertainpointinhistoryandithashadanamazingnumberofapplicationsandfinanceisoneofthese.Someofyou--Thisincidentallywillbeamoretechnicalofmylecturesandit'salittlebitunfortunatethatitcomesearlyinthesemester.Forthoseofyouwhohavehadacourseinprobabilityandstatistics,therewillbenothingnewhere.Well,nothingintermsofthemath.Theprobabilitytheoryisnew.Othersthough,Iwanttotellyouthatitdoesn't--ifyou'reshopping--Ihadastudentcomebyyesterdayandask--he'salittlerustyinhismathskills--ifheshouldtakethiscourse.Isaid,"Wellifyoucanunderstandtomorrow'slecture--that'stoday'slecture--thenyoushouldhavenoproblem."

Iwanttostartwiththeconceptofprobability.Doyouknowwhataprobabilityis?Weattachaprobabilitytoanevent.Whatistheprobabilitythatthestockmarketwillgoupthisyear?Iwouldsay--mypersonalprobabilityis.45.That'sbecauseI'mabearbut--Doyouknowwhatthatmeans?That45timesoutof100thestockmarketwillgoupandtheother55timesoutof100itwillstaythesameorgodown.That'saprobability.Now,you'refamiliarwiththatconcept,right?Ifsomeonesaystheprobabilityis.55or.45,wellyouknowwhatthatmeans.Iwanttoemphasizethatithasn'talwaysbeenthatwayandthatprobabilityisreallyaconceptthataroseinthe1600s.Beforethat,nobodyeversaidthat.

IanHacking,whowroteahistoryofprobabilitytheory,searchedthroughworldliteratureforanyreferencetoaprobabilityandcouldfindnoneanywherebefore1600.Therewasanintellectualleapthatoccurredintheseventeenthcenturyanditbecameveryfashionabletotalkintermsofprobabilities.Itspreadthroughouttheworld--theideaofquotingprobabilities.Butitwas--It'sfunnythatsuchasimpleideahadn'tbeenusedbefore.Hackingpointsoutthatthewordprobability--orprobable--wasalreadyintheEnglishlanguage.Infact,Shakespeareusedit,butwhatdoyouthinkitmeant?Hegivesanexampleofayoungwoman,whowasdescribingamanthatsheliked,andshesaid,Ilikehimverymuch,Ifindhimveryprobable.Whatdoyouthinkshemeans?Cansomeoneanswerthat?DoesanyoneknowElizabethanEnglishwellenoughtotellme?Whatisaprobableyoungman?I'maskingforananswer.Itsoundslikepeoplehavenoidea.Cananyoneventureaguess?Noonewantstoventureaguess?

Student:fertile?

ProfessorRobertShiller:Thathecanfatherchildren?Idon'tthinkthat'swhatshemeantbutmaybe.No,whatapparentlyshemeantistrustworthy.That'saveryimportantqualityinapersonIsuppose.

So,ifsomethingisprobableyoumeanthatyoucantrustitandsoprobabilitymeanstrustworthiness.Youcanseehowtheymovedfromthatdefinitionofprobabilitytothecurrentdefinition.ButIanHacking,beingagoodhistorian,thoughtthatsomeonemusthavehadsomeconceptofprobabilitygoingbefore,eveniftheydidn'tquoteitasanumbertheway--itmusthavebeenintheirheadorintheiridea.Hesearchedthroughworldliteraturetotrytofindsomeuseofthetermthatprecededthe1600sandheconcludedthattherewereprobablyanumberofpeoplewhohadtheidea,buttheydidn'tpublishit,anditneverbecamepartoftheestablishedliteraturepartlybecause,hesaid,throughouthumanhistory,therehasbeenaloveofgamblingandprobabilitytheoryisextremelyusefulifyouareagambler.Hackingbelievesthatthereweremanygamblingtheoristswhoinventedprobabilitytheoryatvarioustimesinhistorybutneverwroteitdownandkeptitasasecret.

Hegivesanexample--Iliketo--hegivesanexamplefromabookthat--orit'sacollection--Ithink,acollectionofepicpoemswritteninSanskritthatgoesback--itwasactuallywrittenoveracourseof1,000yearsanditwascompletedinthefourthcentury.Well,there'sastory--there'salongstoryintheMahabarahtaaboutanemperorcalledNalaandhehadawifenamedDamayantiandhewasaverypureandverygoodperson.TherewasanevildemoncalledKaliwhohatedNalaandwantedtobringhisdownfall,sohehadtofindaweaknessofNala.Hefoundfinallysome,eventhoughNalawassopureandsoperfect--hefoundoneweaknessandthatwasgambling.Nalacouldn'tresisttheopportunitytogamble;sotheevildemonseducedhimintogamblingaggressively.Youknowsometimeswhenyou'relosingandyouredoubleandyoukeephopingtowinbackwhatyou'velost?Inafitofgambling,Nalafinallygambledhisentirekingdomandlost--it'saterriblestory--andNalathenhadtoleavethekingdomandhiswife.Theywanderedforyears.Heseparatedfromherbecauseofdirenecessity.

Theywerewanderingintheforestsandhewasindespair,havinglosteverything.Butthenhemeetssomeonebythenameof--wehaveNalaandhemeetsthisman,Rituparna,andthisiswhereaprobabilitytheoryapparentlycomesin.RituparnatellsNalathatheknowsthescienceofgamblingandhewillteachittoNala,butthatithastobedonebywhisperingitinhisearbecauseit'sadeepandextremesecret.Nalaisskeptical.HowdoesRituparnaknowhowtogamble?SoRituparnatriestoprovetohimhisabilitiesandhesays,seethattreethere,Icanestimatehowmanyleavesthereareonthattreebycountingleavesononebranch.Rituparnalookedatonebranchandestimatedthenumberofleavesonthetree,butNalawasskeptical.HestayedupallnightandcountedeveryleafonthetreeanditcameveryclosetowhatRituparnasaid;sohe--thenextmorning--believedRituparna.Nowthisisinteresting,Hackingsays,becauseitshowsthatsamplingtheorywaspartofNala'stheory.Youdon'thavetocountalltheleavesonthetree,youcantakeasampleandyoucountthatandthenyoumultiply.

Anyway,thestoryendsandNalagoesbackandisnowarmedwithprobabilitytheory,weassume.Hegoesbackandgamblesagain,buthehasnothinglefttowagerexcepthiswife;soheputsherandgamblesher.Butremember,nowheknowswhathe'sdoingandsohereallywasn'tgamblinghiswife--hewasreallyaverypureandhonorableman.Sohewonbacktheentirekingdomandthat'stheending.

Anyway,thatshowsthatIthinkprobabilitytheorydoeshavealonghistory,but--itnotbeinganintellectualdiscipline--itdidn'treallyinformagenerationoffinancetheory.Whenyoudon'thaveatheory,thenyoudon'thaveawaytoberigorous.So,itwasinthe1600sthatprobabilitytheorystartedtogetwrittendownasatheoryandmanythingsthenhappenedinthatcenturythat,Ithink,areprecursorsbothtofinanceandinsurance.

Onewasinthe1600swhenpeoplestartedconstructinglifetables.Whatisalifetable?It'satableshowingtheprobabilityofdyingateachage,foreachageandsex.That'swhatyouneedtoknowifyou'regoingtodolifeinsurance.So,theystartedtodocollectingofdataonmortalityandtheydevelopedsomethingcalledactuarialscience,whichisestimatingtheprobabilityofpeopleliving.Thatthenbecamethebasisforinsurance.Actually,insurancegoesbacktoancientRomeinsomeform.InancientRometheyhadsomethingcalledburialinsurance.Youcouldbuyapolicythatprotectedyouagainstyourfamilynothavingthemoneytoburyyouifyoudied.Inancientculturepeopleworriedagreatdealaboutbeingproperlyburied,sothat'saninterestingconcept.TheyweresellingthatinancientRome;butyoumightthink,butwhyjustforburial?Whydon'tyoumakeitintofull-blownlifeinsurance?Youkindofwonderwhytheydidn't.Ithinkmaybeit'sbecausetheydidn'thavetheconceptsdown.InRenaissanceItalytheystartedwritinginsurancepolicies--Ireadoneoftheinsurancepolicies,it'sintheJournalofRiskandInsurance--andtheytranslateaRenaissanceinsurancepolicyandit'sveryhardtounderstandwhatthispolicywassaying.Iguesstheydidn'thaveourlanguage,theydidn't--theywereintuitivelyhalfwaytherebuttheycouldn'texpressit,soIthinktheindustrydidn'tgetreallystarted.Ithinkitwastheinventionofprobabilitytheorythatreallystarteditandthat'swhyIthinktheoryisveryimportantinfinance.

SomepeopledatefireinsurancewiththefireofLondonin1666.Thewholecityburneddown,practically,inaterriblefireandfireinsurancestartedtoproliferaterightafterthatinLondon.Butyouknow,youkindofwonderifthat'sagoodexampleforfireinsurancebecauseifthewholecityburnsdown,theninsurancecompanieswouldgobankruptanyway,right?Londoninsurancecompanieswouldbecausethewholeconceptofinsuranceispoolingofindependentprobabilities.Nonetheless,thatwasthebeginning.

We'realsogoingtorecognize,however,thatinsurancegotaslowstartbecause--Ibelieveitisbecause--peoplecouldnotunderstandtheconceptofprobability.Theydidn'thavetheconceptfirmlyinmind.Therearelotsofaspectstoit.Inordertounderstandprobability,youhavetotakethingsascomingfromarandomeventandpeopledon'tclearlyhavethatintheirmindfromanintuitivestandpoint.TheyhavemaybeasensethatIcaninfluenceeventsbywillingorwishingandifIthinkthat--ifIhavekindofamysticalsidetome,thenprobabilitiesdon'thaveaclearmeaning.Ithasbeenshownthateventodaypeopleseemtothinkthat.Theydon'treallytake,atanintuitivelevel,probabilitiesasobjective.Forexample,ifyouaskpeoplehowmuchtheywouldbewillingtobetonacointoss,theywilltypicallybetmoreiftheycantossthecoinortheywillbetmoreifthecoinhasn'tbeentossedyet.Itcouldhavebeenalreadytossedandconcealed.Whywouldthatbe?Itmightbethatthere'sjustsomeintuitivesensethatIcan--Idon'tknow--IhavesomemagicalforcesinmeandIcanchangethings.

Theideaofprobabilitytheoryisthatno,youcan'tchangethings,therearealltheseobjectivelawsofprobabilityouttherethatguideeverything.Mostlanguagesaroundtheworldhaveadifferentwordforluckandrisk--orluckandfortune.Luckseemstomeansomethingaboutyou:likeI'maluckyperson.Idon'tknowwhatthatmeans--likeGodorthegodsfavormeandsoI'mluckyorthisismyluckyday.Probabilitytheoryisreallyamovementawayfromthat.Wethenhaveamathematicallyrigorousdiscipline.

Now,I'mgoingtogothroughsomeofthetermsofprobabilityand--thiswillbereviewformanyofyou,butitwillbesomethingthatwe'regoingtouseinthe--SoI'llusethesymbolPorIcansometimeswriteitoutasprobtorepresentaprobability.Itisalwaysanumberthatliesbetweenzeroandone,orbetween0%and100%."Percent"meansdividedby100inLatin,so100%isone.Iftheprobabilityiszerothatmeanstheeventcan'thappen.Iftheprobabilityisone,itmeansthatit'scertaintohappen.Iftheprobabilityis--Caneveryoneseethisfromoverthere?Icanprobablymovethisorcan'tI?Yes,Ican.Now,canyounow--you'rethemostdisadvantagedpersonandyoucanseeit,right?Sothat'sthebasicidea.

Oneofthefirstprinciplesofprobabilityistheideaofindependence.Theideaisthatprobabilitymeasuresthelikelihoodofsomeoutcome.Let'ssaytheoutcomeofanexperiment,liketossingacoin.Youmightsaytheprobabilitythatyoutossacoinanditcomesupheadsisahalf,becauseit'sequallylikelytobeheadsandtails.Independentexperimentsareexperimentsthatoccurwithoutrelationtoeachother.Ifyoutossacointwiceandthefirstexperimentdoesn'tinfluencethesecond,wesaythey'reindependentandthere'snorelationbetweenthetwo.

Oneofthefirstprinciplesofprobabilitytheoryiscalledthemultiplicationrule.Thatsaysthatifyouhaveindependentprobabilities,thentheprobabilityoftwoeventsisequaltotheproductoftheirprobabilities.So,theProb(AandB)=Prob(A)*Prob(B).Thatwouldn'tholdifthey'renotindependent.Thetheoryofinsuranceisthatideallyaninsurancecompanywantstoinsureindependentevents.Ideally,lifeinsuranceisinsuringpeople--orfireinsuranceisinsuringpeople--againstindependentevents;soit'snotthefireofLondon.It'stheproblemthatsometimespeopleknockoveranoillampintheirhomeandtheyburntheirownhousedown.It'snotgoingtoburnanyotherhousesdownsinceit'sjustcompletelyindependentofanythingelse.So,theprobabilitythatthewholecityburnsdownisinfinitesimallysmall,right?ThiswillgeneralizetoprobabilityofAandBandCequalstheprobabilityofAtimestheprobabilityofBtimestheprobabilityofCandsoon.Iftheprobabilityis1in1,000thatahouseburnsdownandthereare1,000houses,thentheprobabilitythattheyallburndownis1/1000tothe1000thpower,whichisvirtuallyzero.Soinsurancecompaniesthen–Basically,iftheywritealotofpolicies,thentheyhavevirtuallynorisk.Thatisthefundamentalideathatmayseemsimpleandobvioustoyou,butitcertainlywasn'tbackwhentheideafirstcameup.

Incidentally,wehaveaproblemset,whichIwantyoutostarttodayanditwillbeduenotinaweekthistime,becausewehaveMartinLutherKingDaycomingup,butitwillbeduetheMondayfollowingthat.

Ifyoufollowthroughfromtheindependenttheory,there'soneofthebasicrelationsinprobabilitytheory--it'scalledthebinomialdistribution.I'mnotgoingtospendawholelotoftimeonthisbutitgivestheprobabilityofxsuccessesinntrialsor,inthecaseofinsurancex,ifyou'reinsuringagainstanaccident,thentheprobabilitythatyou'llgetxaccidentsandntrials.Thebinomialdistributiongivestheprobabilityasafunctionofxandit'sgivenbytheformulawherePistheprobabilityoftheaccident:.Thatistheformulathatinsurancecompaniesusewhentheyhaveindependentprobabilities,toestimatethelikelihoodofhavingacertainnumberofaccidents.They'reconcernedwithhavingtoomanyaccidents,whichmightexhausttheirreserves.Aninsurancecompanyhasreservesandithasenoughreservestocoverthemforacertainnumberofaccidents.Itusesthebinomialdistributiontocalculatetheprobabilityofgettinganyspecificnumberofaccidents.So,thatisthebinomialdistribution.I'mnotgoingtoexpandonthisbecauseIcan'tgetinto--ThisisnotacourseinprobabilitytheorybutI'mhopefulthatyoucanseetheformulaandyoucanapplyit.Anyquestions?Isthisclearenough?Canyoureadmyhandwriting?

Anotherimportantconceptinprobabilitytheorythatwewillusealotisexpectedvalue,themean,oraverage--thoseareallroughlyinterchangeableconcepts.Wehaveexpectedvalue,meanoraverage.Wecandefineitinacoupleofdifferentwaysdependingonwhetherwe'retalkingaboutsamplemeanorpopulationmean.Thebasicdefinition--theexpectedvalueofsomerandomvariablex--E(x)--IguessIshouldhavesaidthatarandomvariableisaquantitythattakesonvalue.Ifyouhaveanexperimentandtheoutcomeoftheexperimentisanumber,thenarandomvariableisthenumberthatcomesfromtheexperiment.Forexample,theexperimentcouldbetossingacoin;Iwillcalltheoutcomeheadsthenumberone,andI'llcalltheoutcometailsthenumberzero,soI'vejustdefinedarandomvariable.Youhavediscreterandomvariables,liketheoneIjustdefined,ortherearealso--whichtakeononlyafinitenumberofvalues--andwehavecontinuousrandomvariablesthatcantakeonanynumberofvaluesalongacontinuum.Anotherexperimentwouldbetomixtwochemicalstogetherandputathermometerinandmeasurethetemperature.That'sanotherinventionofthe1600s,bytheway--thethermometer.Andtheylearnedthatconcept--perfectlynaturaltous--temperature.Butitwasanewideainthe1600s.Soanyway,that'scontinuous,right?Whenyoumixtwochemicalstogether,itcouldbeanynumber,there'saninfinitenumberofpossiblenumbersandthatwouldbecontinuous.

Fordiscreterandomvariables,wecandefinetheexpectedvalue,orµx--that'stheGreeklettermu--asthesummationi=1toinfinityof.[P(x=xi)times(xi)].Ihaveitdownthattheremightbeaninfinitenumberofpossiblevaluesfortherandomvariablex.Inthecaseofthecointoss,thereareonlytwo,butI'msayingingeneraltherecouldbeaninfinitenumber.Butthey'reaccountableandwecanlistallpossiblevalueswhenthey'rediscreteandformaprobabilityweightedaverageoftheoutcomes.That'scalledtheexpectedvalue.Peoplealsocallthatthemeanortheaverage.But,notethatthisisbasedontheory.Theseareprobabilities.Inordertocomputeusingthisformulayouhavetoknowthetrueprobabilities.There'sanotherformulathatappliesforacontinuousrandomvariablesandit'sthesameideaexceptthat--I'llalsocallitµx,exceptthatit'sanintegral.WehavetheintegralfromminusinfinitytoplusinfinityofF(x)*x*dx,andthat'sreally--youseeit'sthesamethingbecauseanintegralisanalogoustoasummation.

Thosearethetwopopulationdefinitions.F(x)isthecontinuousprobabilitydistributionforx.That'sdifferentwhenyouhavecontinuousvalues--youdon'thaveP(x=xi)becauseit'salwayszero.Theprobabilitythatthetemperatureisexactly100°iszerobecauseitcouldbe100.0001°orsomethingelseandthere'saninfinitenumberofpossibilities.Wehaveinsteadwhat'scalledaprobabilitydensitywhenwehavecontinuousrandomvariables.You'renotgoingtoneedtoknowalotaboutthisforthiscourse,butthisis--Iwantedtogetthebasicideasdown.Thesearecalledpopulationmeasuresbecausetheyrefertothewholepopulationofpossibleoutcomesandtheymeasuretheprobabilities.It'sthetruth,buttherearealsosamplemeans.Whenyouget--thisisRituparna,countingtheleavesonatree--youcanestimate,fromasample,thepopulationexpectedvalues.Thepopulationmeanisoftenwritten"x-bar."Ifyouhaveasamplewithnobservations,it'sthesummationi=1tonofxi/n--that'stheaverage.Youknowthatformula,right?Youcountnleaves--youcountthenumberofleaves.Youhavenbranchesonthetreeandyoucountthenumberofleavesandsumthemup.Onewouldbe--I'mhavingalittletroubleputtingthisintotheRituparnastory,butyouseetheidea.Youknowtheaverage,Iassume.That'sthemostelementaryconceptandyoucoulduseittoestimateeitheradiscreetorcontinuousexpectedvalue.

Infinance,there'softenreferencetoanotherkindofaverage,whichIwanttoreferyoutoandwhich,intheJeremySiegelbook,alotismadeofthis.Theotherkindofaverageiscalledthegeometricaverage.We'llcallthat--I'llonlyshowthesampleversionofitG(x)=theproducti=1tonof(xi)^(1/n).Doeseveryone--Canyouseethat?InsteadofsummingthemanddividingbyM,Imultiplythemalltogetherandtakethenthrootofthem.Thisiscalledthegeometricaverageandit'susedonlyforpositivenumbers.So,ifyouhaveanynegativenumbersyou'dhaveaproblem,right?Ifyouhadonenegativenumberinit,thentheproductwouldbeanegativenumberand,ifyoutookarootofthat,thenyoumightgetanimaginarynumber.Wedon'twanttouseitinthatcase.

There'sanappendixtooneofthechaptersinJeremySiegel'sbookwherehesaysthatoneofthemostimportantapplicationsofthistheoryistomeasurehowsuccessfulaninvestoris.Supposesomeoneismanagingmoney.Havetheydonewell?Ifso,youwouldsay,"Well,they'vebeeninvestingmoneyoveranumberofdifferentyears.Let'staketheaverageoverallthedifferentyears."Supposesomeonehasbeeninvestingmoneyfornyearsandxiisthereturnontheinvestmentinagivenyear.Whatistheiraverageperformance?Thenaturalthingtodowouldbetoaveragethemup,right?ButJeremysaysthatmaybethat'snotaverygoodthingtodo.Whathesaysyoushoulddoinsteadistotakethegeometricaverageofgrossreturns.Thereturnonaninvestmentishowmuchyoumadefromtheinvestmentasapercentofthemoneyinvested.Thegrossreturnisthereturnplusone.Theworstyoucaneverdoinvestingisloseallofyourinvestment--lose100%.Ifweaddonetothereturn,thenyou'vegotanumberthat'snevernegativeandwecanthenusegeometricreturns.

JeremySiegelsaysthatinfinanceweshouldbeusinggeometricandnotarithmeticaverages.Whyisthat?WellI'lltellyouinverysimpleterms,Ithink.Supposesomeoneisinvestingyourmoneyandheannounces,Ihavehadverygoodreturns.IhaveinvestedandI'veproduced20%ayearfornineoutofthelasttenyears.Youthinkthat'sgreat,butwhataboutthelastyear.Theguysays,"OhIlost100%inthatyear."Youmightsay,"Alright,that'sgood."Iwouldaddup20%ayearfornineyearsandthanputinazero–no,120becauseit'sgrossreturnfornineyears--andputinazeroforoneyear.Maybethatdoesn'tlookbad,right?Butthinkaboutit,ifyouwereinvestingyourmoneywithsomeonelikethat,whatdidyouendupwith?Youendedupwithnothing.Iftheyhaveoneyearwhentheyloseeverything,itdoesn'tmatterhowmuchtheymadeintheotheryears.Jeremysaysinthetextthatthegeometricreturnisalwayslowerthanthearithmeticreturnunlessallthenumbersarethesame.It'salessoptimisticversion.So,weshouldusethat,butpeopleinfinanceresistusingthatbecauseit'salowernumberandwhenyou'readvertisingyourreturnyouwanttomakeitlookasbigaspossible.

Wealsoneedsomemeasureof--We'vebeentalkinghereaboutmeasuresofcentraltendencyonlyandinfinanceweneed,aswell,measuresofdispersion,whichishowmuchsomethingvaries.Centraltendencyisameasureofthecenterofaprobabilitydistributionofthe--Centraltendencyisameasure--Varianceisameasureofhowmuchthingschangefromoneobservationtoanother.Wehavevarianceandit'softenrepresentedbyσ²,that'stheGreeklettersigma,lowercase,squared.Or,especiallywhentalkingaboutestimatesofthevariance,wesometimessayS²orwesaystandarddeviation².Thestandarddeviationisthesquarerootofthevariance.Forpopulationvariance,thevarianceofsomerandomvariablexisdefinedasthesummationi=1toinfinityoftheProb(x=xi)times(xi-µx)2.Somuisthemean--wejustdefineditofx--that'stheexpectationofxoralsoE(x),soit'stheprobabilityweightedaverageofthesquareddeviationsfromthemean.Ifitmovesalot--eitherwayfromthemean--thenthisnumbersquaredisabignumber.Themorexmoves,thebiggerthevarianceis.

There'salsoanothervariancemeasure,whichweuseinthesample--oralsoVarisusedsometimes--andthisis∑².There'salsoanothervariancemeasure,whichisforthesample.Whenwehavenobservationsit'sjustthesummationi=1tonof(x-xbar)²/n.Thatisthesamplevariance.Somepeoplewilldividebyn–1.IsupposeIwouldaccepteitheranswer.I'mjustkeepingitsimplehere.Theydividebyn-1tomakeitanunbiasedestimatorofthepopulationvariance;butI'mjustgoingtoshowitinasimplewayhere.Soyouseewhatitis--it'sameasureofhowmuchxdeviatesfromthemean;butit'ssquared.Itweightsbigdeviationsalotbecausethesquareofabignumberisreallybig.So,that'sthevariance.

So,thatcompletescentraltendencyanddispersion.We'regoingtobetalkingabouttheseinfinanceinregardstoreturnsbecause--generallytheideahereisthatwewanthighreturns.Wewantahighexpectedvalueofreturns,butwedon'tlikevariance.Expectedvalueisgoodandvarianceisbadbecausethat'srisk;that'suncertainty.That'swhatthiswholetheoryisabout:howtogetalotofexpectedreturnwithoutgettingalotofrisk.

Anotherconceptthat'sverybasichereiscovariance.Covarianceisameasureofhowmuchtwovariablesmovetogether.Covarianceis--we'llcallit--nowwehavetworandomvariables,soI'lljusttalkaboutitinasampleterm.It'sthesummationi=1tonof[(x–x-bar)times(y–y-bar)]/n.Soxisthedeviationforthei-subscript,meaningwehaveaseparatexiandyiforeachobservation.Sowe'retalkingaboutanexperimentwhenyougenerate--Eachexperimentgeneratesbothanxandayobservationandweknowwhenxishigh,yalsotendstobehigh,orwhetherit'stheotherwayaround.Iftheytendtomovetogether,whenxishighandyishightogetheratthesametime,thenthecovariancewilltendtobeapositivenumber.Ifwhenxislow,yalsotendstobelow,thenthiswillbenegativenumberandsowillthis,sotheirproductispositive.Apositivecovariancemeansthatthetwomovetogether.Anegativecovariancemeansthattheytendtomoveoppositeeachother.Ifxishighrelativetox-bar--thisispositive--thenytendstobelowrelativetoitsmeany-barandthisisnegative.Sotheproductwouldbenegative.Ifyougetalotofnegativeproducts,thatmakesthecovariancenegative.

ThenIwanttomovetocorrelation.Sothisisameasure--it'sascaledcovariance.WetendtousetheGreekletterrho.IfyouweretouseExcel,itwouldbecorrelorsometimesIsaycorr.That'sthecorrelation.Thisnumberalwaysliesbetween-1and+1.Itisdefinedasrho=[cov(xiyi)/SxSy]That'sthecorrelationcoefficient.ThathaskindofalmostenteredtheEnglishlanguageinthesensethatyou'llseeitquotedoccasionall