ECON 1723 Fall 2013 Lecture 6 - Economics 1723 Capital...

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Economics 1723: Capital MarketsLecture 6John Y. CampbellEc1723September 19, 2013John Y. Campbell(Ec1723)Lecture 6September 19, 20131 / 51
RoadmapLast time:Started mean-variance analysis. Portfolio choicewith a risk-free and a risky asset.Today:Portfolio choice when there are various risky assets.Key concept isdiversi°cation.Plan:1Simple model to illustrate diversi°cation.2Portfolio choice with two risky assets.3Portfolio choice with many risky assets.4Portfolio choice with many risky assets and a risk-free asset.John Y. Campbell(Ec1723)Lecture 6September 19, 20132 / 51
Key questionsWhat is the bene°t of forming a diversi°ed portfolio?What are the minimum variance and e¢ cient frontiers?How would you form an optimal portfolio with one risk-free andmany risky assets?John Y. Campbell(Ec1723)Lecture 6September 19, 20133 / 51
Roadmap1Diversi°cation2Mean-variance analysis with two risky assets3Mean-variance analysis with many risky assets4Mean-variance analysis with one risk-free and many risky assetsJohn Y. Campbell(Ec1723)Lecture 6September 19, 20134 / 51
A simple model to illustrate diversi°cationSuppose there are two risky stocks, 1 and 2, with the sameexpected return:R1=R+"1andR2=R+"2:Suppose"1and"2(risks) arezero-meananduncorrelatedrandom variablesCov("1; "2) =0.Suppose"1and"2have the same variance,°2idio:What do these assumptions mean?
John Y. Campbell(Ec1723)Lecture 6September 19, 20135 / 51
A simple diversi°cation strategyIf you invest in either stock 1 or stock 2, you obtain expectedreturnRand variance°2idio.If you split your wealth equally between the two stocks, theportfolio return is:R1+R22=R+"1+"22:Mean return is stillR. Variance is14Var("1+"2)=140@Var("1)|{z}°2+2Cov("1; "2)|{z}0+Var("2)|{z}°21A=°2idio2Simple diversi°cation reduces the variance.John Y. Campbell(Ec1723)Lecture 6September 19, 20136 / 51
Consider the case with more than two assetsSuppose there are many such stocks:Ri=R+"ifori2 f1; ::;Ng.Idiosyncratic shocks,"i, are uncorrelated with one another, withvariance°2idio.If you split your wealth equally across stocks, you end up holding:NXi=1RiN=R+NXi=1"iN.Mean return isR. Variance is:VarNXi=1"iN!=NXi=1°2idioN2+cross-terms|{z}0 because uncorrelated=°2idioN.ForNlarge, diversi°cation eliminates idiosyncratic risks.John Y. Campbell(Ec1723)Lecture 6September 19, 20137 / 51

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