72金融学概论讲义(北大光华管理学院)lecture04

1PrinciplesofFinanceLecture04PortfolioTheory2ProbabilityDistributionofReturnsReturnsoninvestmentareuncertain(risky)WemodeluncertaintyoffuturereturnsusingExpectedreturn:thereturnyouexpecttoreceiveonaverageVolatility(standarddeviation):degreeofdispersionoffuturereturnsThelargerastock’svolatility,thewidertherangeofpossibleoutcomesandthelargertheprobabilitiesofthosereturnsattheextremeoftherange3DistributionofReturnsonTwoStocks0.00.51.01.52.02.53.03.5-100%-50%0%50%100%ReturnProbabilityDensityNORMCOVOLCO4ExpectedReturnnnniiiRRRRRE22111~RE~:Expectedrateofreturnforinvestmenti:ProbabilityofoccurrenceofithstateiR:Estimatedrateofreturnforthatstaten:Numberofpossiblestates5ExampleofCalculatingExpectedReturnStateofEconomyProbabilityReturnonRiscoReturnonGencoStrong0.2050%30%Normal0.6010%10%Weak0.2030%10%%1030.020.010.060.050.020.0~RiscoRE%1010.020.010.060.030.020.0~GencoRE6VarianceandStandardDeviationniiiiRERRERER1222~~~VarianceforRisco:StateofEconomyProbabilityReturnDeviationfrommeanSquaredDeviationProbabilitySquaredDeviationStrong0.2050%40%0.160.032Normal0.6010%000Weak0.2030%40%0.160.032064.0~2RiscoRand%3.25253.0~RiscoR7RiskandReturnRevisitedIknowsomestatistics…Ifmyinvestmenthorizonislongenough,doesitmeanthatIshouldchoosetheonewithhigherexpectedreturn,regardlessofitshighervolatility?(PerhapsintheendIwillprobablyhitahighend-of-periodwealth)No.Imagineascenariolikethis:a1%chanceofearning$1billion,anda99%chanceoflosingeverything.Youhaveahighexpectedreturn,butyouwillbebankruptwithalmostcertainty!8PortfolioReturnandRiskStateofEconomyProbabilityReturnonAReturnonB10.205%19%20.6010%10%30.2035%4%Portfolioweight:6.0Aand4.0B9PortfolioReturnandRiskStateofEconomyProb.ReturnonAReturnonBPortfolioReturn10.205%19%4.6%20.6010%10%10%30.2035%4%19.4%RE~12%9%10.8%%8.10%4.192.0%106.0%6.42.0~PREBBAARERE~~0.6*12%+0.4*9%=10.8%Portfolioreturnisaweightedaverageofsecurityreturns10PortfolioReturnandRiskStateofEconomyProbabilityDeviationfrommeanSquaredDeviationProbabilitySquaredDeviation10.206.2%0.0038440.000768820.600.8%0.0000640.000038430.208.6%0.0073960.00147920022864.0~2PRand%78.4~PR11PortfolioReturnandRisk0166.0~2ARand%88.12~AR00544.0~2BRand%38.7~BRBBAARR~~0.6*12.88%+0.4*7.38%=10.68%Thestandarddeviationofreturnsonaportfolioislessthanaweightedaverageofconstituentsecuritystandarddeviation!(IfBARR~and~arelessthanperfectlypositivelycorrelated)12CovarianceandCorrelationCoefficientCovariance:Ameasureofhowtwosecurities’returnsmovetogetherandthesizeofthoseco-movementsBBiAiABARERRERERR~~~,~covniBBiAiAiRERRER1~~StateofEconomyProb.Deviationfrommean:ADeviationfrommean:BCovarianceterm10.2017%10%0.0034020.602%1%0.0001230.2023%13%0.00598BARR~,~cov0.0095013CovarianceandCorrelationCoefficientThecovariancemeasureisaffectedbyboththedirectionthatassetsmovetogetherandthesizeofthosemovementsAsaresult,themagnitudeofBARR~,~covissometimesdifficulttointerpretForthisreasonwealsocalculatethecorrelationcoefficientCorrelationCoefficient:Astandardizedmeasureofthewayinwhichthereturnsfromtwoassetsmovetogether14CovarianceandCorrelationCoefficientBABABARRRR~~~,~cov,119994.00738.01288.000950.0,BASpecialcase:0,1,1.15ExpectedReturn,Variance,CovariancennniiiRRRRRE22111~niiiiRERRERER1222~~~BBiAiABARERRERE~~,niBBiAiAiRERRER1~~BABABARRRR~~~,~cov,Weuser,,andforsimplicity.16PortfolioofTwoRiskySecuritiesAssumethatweinvestwproportionofthewealthinsecurity1andproportionof1wofthewealthinsecurity2Theexpectedreturnofsecurity1is1r,andtheexpectedreturnofsecurity2is2rThestandarddeviationofsecurity1is1,andthestandarddeviationofsecurity2is217PortfolioofTwoSecuritiesTheexpectedreturnoftheportfolioistheweightedaverageofthecomponentreturns211rwwrrPThevolatilityoftheportfolioisnotquitesimple211wwP(wrong!)2222112212211218PortfoliowiththeRisklessAssetandaSingleRiskyAssetRisklessasset:futurereturniscertainAssumeaworldwithasingleriskyassetandtherisklessassetTheriskyassetis,intherealworld,aportfolioofriskyassetsAssumethatyouinvestwproportionofyourwealthintheriskyasset(portfolio)1;1-winvestedintherisklessasset2Riskyasset:%20and%1411rRisklessasset:%62rand0219PortfoliowiththeRisklessAssetandaSingleRiskyAssetPortfolioreturnandstandarddeviation211rwwrrP212222211221221121wPIf25.0wthen%5and%8PPrIf75.0wthen%15and%12PPr20PortfolioofaRiskyandaRisklessSecurity00.050.10.150.20.2500.10.20.30.4StandardDeviationExpectedReturn21PortfolioofTwoRiskySecurities12isthecorrelationcoefficientSpecialcorrelationcases:Perfectly(positively)correlated(112)12(1)pwwPerfectlynegativelycorrelated(112)12(1)pwwIngeneral:1112211wwp22ExampleofPortfoliowithTwoRiskyAssetsSecurity1:%141rand%201Security2:%82rand%15201225.0w%5.9%875.0%