Lecture 14 Portfolio Theory - MIT OpenCourseWare

1 Portfolio TheoryPortfolio TheoryMIT KempthorneFall 2013 MIT TheoryLecture 14: 1 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresOutline1 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMIT Theory2 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance

2 Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMarkowitz Mean-Variance Analysis (MVA)Single-Period Analyisismrisky assets:i= 1,2,..,mSingle-Period Returns:m variate random vectorR= [R1,R2,..,Rm] Mean and Variance/Covariance of Returns: 1 1,1 1, [R] = = .. ,Cov[R] = =m .. m,1 m,m Portfolio :m vector of weights indicating the fraction ofportfolio wealth held in each assetwm= (w1,..,wm) :i=1wi= Return:Rw R=i=1wiRia with w=E[Rw]= w 2w=var[Rw] =w wMIT Theory3 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMarkowitz Mean Variance AnalysisEvaluate different portfolioswusing the mean-variance pair of theportfolio: ( w, 2w) with preferences forHigher expected returns wLower variancevarwProblem I: Risk Minimization.

3 For a given choice of target meanreturn 0,choose the portfoliowtoMinimize:1w2 wSubject to:w = 0w 1m= 1 Solution:Apply the method of Lagrange multipliers to the convexoptimization (minimization) problem subject to linear constraints:MIT Theory4 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresRisk Minimization ProblemDefine the LagrangianL(w, 1, 2) =1w2 w+ 1( 0 w ) + 2(1 w 1m)Derive the first-order conditions L=0 wm= w 1 21m L 1= 0= 0 w L= 0= 1 2 w 1mSolve forwin terms of 1, 2:w0= 1 1 + 2 11mSolve for 1, 2by substituting forw.

4 =w = ( 1 )+ ( 100121m)1=w 1= ( 110m11m) + 2(1 m 1m) a b = 01=with1b c 2a= ( 11 [),b=]( [ ][ ]1 a b 1m), andc= (1 m 1m) 1 a bMIT Theory5 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresRisk Minimization ProblemVariance of Optimal Portfolio with Return 0 With the given values of 1and 2, the solution portfoliow0= 1 1 + 2 11mhas minimum variance equal to 20=w 0 w0= 2[1( ] 1 ) + 2 1 2( 1[ ][ ]1m) + 2(1 m 121m)

5 1 a b 1= 2 Substituting= [ 1 2 ]a[b]c[ 21a b 0]gives 20=[01] [b c1bb c] 1[ 01]=1ac b2(c 20 2b 0+a)Optimal Portfolio has variance 20: parabolic in the meanreturn Theory6 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresEquivalent Optimization ProblemsProblem II: Expected Return maximization : For a given choiceof target return variance 20, choose the portfoliowtoMaximize:E(Rw) =w Subject to:w w= 20w 1m= 1 Problem III: Risk Aversion Optimization: Let 0 denote theArrow-Prattrisk aversion index gauging the trade-ff between riskand return.

6 Choose the portfoliowtoMaximize:[E(Rw) 12 var(Rw)]=w 12 w wSubject to:w 1m= I,II, and III solved by equivalent LagrangiansEfficient Frontier:{( 0, 2) = (E(Rw00),var(Rw0))|w0optimal}Efficient Frontier: traces of 0(I), 2(II), or (III)0 MIT Theory7 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresOutline1 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMIT Theory8 Portfolio TheoryMarkowitz Mean-Variance

7 OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMean-Variance Optimization with Risk-Free AssetRisk-Free Asset: In addition to the risky assets (i= 1,..,m)assume there is a risk-free asset (i= 0) for whichR0 r0, ,E(R0) =r0,andvar(R0) = With Investment in Risk-Free AssetSuppose the investor can invest in themrisky investment aswell as in the risk-free 1mm= i=1wiis invested in risky assets and1 w1mis invested in the risk-free borrowing allowed, (1 w1m) can be :Rw=w R+ (1 w 1m)R0, whereR= (R1.)

8 ,Rm), has expected return and variance: w=w + (1 w 1m)r0 2w=w wNote:R0has zero variance and is uncorrelated withRMIT Theory9 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMean-Variance Optimization with Risk-Free AssetProblem I : Risk Minimization with Risk-Free AssetFor a given choice of target mean return 0,choose the portfoliowtoMinimize:1w2 wSubject to:w + (1 w 1m)r0= 0 Solution:Apply the method of Lagrange multipliers to the convexoptimization (minimization):Define the LagrangianL(w, 1) =12w w+ 1[( 0 r0) w ( 1mr0)]Derive the first-order conditions L w=0m= w 1[ 1mr0] L= 0= ( 0 1 r0) w ( 1mr0)Solve forwin terms of 1.

9 W0= 1 1[ 1mr0]and 1= ( 0 r0)/[( 1mr0) 1( 1mr0)]MIT Theory10 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMean-Variance Optimization with Risk-Free AssetAvailable Assets for Investment:Risky Assets (i= 1,..,m) with returns:R= (R1,..,Rm)withE[R] = andCov[R] = Risk-Free Asset with returnR0:R0 r0,a PortfolioP: Target Return= 0 Invests in risky assets according to fractional weights vector:w0= 1 1[ 1mr0], where( ) 1= 1P) =0 r(0( 1mr0) 1( 1mr0)Invests in the risk-free asset with weight (1 w 01m) Portfolio return.

10 RP=w 0R+ (1 w 01m)r0 MIT Theory11 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMean-Variance Optimization with Risk-Free AssetPortfolio return:RP=w 0R+ (1 w 01m)r0 Portfolio variance:Var(RP) =Var(w 0R+ (1 w 01m)r0) =Var(w 0R)=w 0 w0= ( 0 r0)2/[( 1mr0) 1( 1mr0)]MIT Theory12 Portfolio TheoryMarkowitz Mean-Variance OptimizationMean-Variance Optimization with Risk-Free AssetVon Neumann-Morgenstern Utility TheoryPortfolio Optimization ConstraintsEstimating Return Expectations and CovarianceAlternative Risk MeasuresMarket PortfolioMThe fully-invested optimal Portfolio withwM:w M1m= 1 1[ ( 1mr0], where 1= 1(M) =1 1 1m[ 1mr0]) Market Portfolio Return.