(L09)Markowitz3(f)

(L09)Markowitz3(f) - Lecture#9 Markowitz Portfolio Theory...

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Primbs/Investment Science 1 Lecture #9 Markowitz Portfolio Theory Reading: Luenberger Chapter 6, Sections 6 - 10
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Primbs/Investment Science 2 Markowitz Portfolio Theory The Markowitz Model The Two Fund Theorem Inclusion of a Risk Free Asset The One Fund Theorem Markowitz’s Message Solving the Optimization
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Primbs/Investment Science 3 Picture of Markowitz σ r For a given mean return, you would like to minimize your risk or the variance. x Minimum variance point for a given mean return
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Primbs/Investment Science 4 The Markowitz Model Markowitz formulated the problem of being on the efficient frontier as an optimization. Assume there are n risky assets with Mean returns: n r r r , , , 2 1 Covariances: ij σ for i,j=1,...,n
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Primbs/Investment Science 5 Markowitz Optimization Minimize: = n j i ij j i w w 1 , 2 1 σ = 1/2 (Variance) p n i i i r r w = = 1 Subject to: = Mean Return 1 1 = = n i i w = Weights sum to 1. Note: (1) We are allowing short selling! (2) We assume all assets are risky !
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Primbs/Investment Science 6 Markowitz Portfolio Theory The Markowitz Model The Two Fund Theorem Inclusion of a Risk Free Asset The One Fund Theorem Markowitz’s Message Solving the Optimization
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Primbs/Investment Science 7 Solving a General Optimization ) , ( u x f 1 1 ) , ( c u x g = Min: s.t.: Step 1: Convert all constraints to zero on the right hand side. 2 2 ) , ( c u x g = ) , ( u x f Min: s.t.: 0 ) , ( 1 1 = - c u x g 0 ) , ( 2 2 = - c u x g
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Primbs/Investment Science 8 Solving a General Optimization ) , ( u x f 1 1 ) , ( c u x g = Min: s.t.: Step 2: Associate a Lagrange multiplier with each constraint. ) , ( u x f Min: s.t.: 2 2 ) , ( c u x g = 0 ) , ( 1 1 = - c u x g 0 ) , ( 2 2 = - c u x g 1 λ 2 λ
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Primbs/Investment Science 9 Solving a General Optimization ) , ( u x f 1 1 ) , ( c u x g = Min: s.t.: Step 3: Form the Lagrangian by subtracting from the objective each constraint multiplied by its Lagrange multiplier. ) , ( u x f Min: s.t.: 2 2 ) , ( c u x g = 0 ) , ( 1 1 = - c u x g 0 ) , ( 2 2 = - c u x g 1 λ 2 λ ) ) , ( ( ) ) , ( ( ) , ( ) , , , ( 2 2 2 1 1 1 2 1 c u x g c u x g u x f u x L - - - - = λ λ λ λ
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Primbs/Investment Science 10 Solving a General Optimization ) , ( u x f 1 1 ) , ( c u x g = Min: s.t.: Step 4: Compute the partial derivatives of the Lagrangian with respect to all its variables and set equal to zero. 2 2 ) , ( c u x g = 0 2 2 1 1 = - - = x g x g x f x L λ λ 0 2 2 1 1 = - - = u g u g u f u L λ λ 0 ) , ( 1 1 1 = - = - c u x g L λ 0 ) , ( 2 2 2 = - = - c u x g L λ ) ) , ( ( ) ) , ( ( ) , ( ) , , , ( 2 2 2 1 1 1 2 1 c u x g c u x g u x f u x L - - - - = λ λ λ λ
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Primbs/Investment Science 11 Solving a General Optimization ) , ( u x f 1 1 ) , ( c u x g = Min: s.t.: Step 5: Solve these equations (for x, u, λ ) to find the optimal solution. 2 2 ) , ( c u x g = 0 2 2 1 1 = - - = x g x g x f x L λ λ 0 2 2 1 1 = - - = u g u g u f u L λ λ 0 ) , ( 1 1 1 = - = - c u x g L λ 0 ) , ( 2 2 2 = - = - c u x g L λ
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Primbs/Investment Science 12 Markowitz Optimization Minimize: = n j i ij j i w w 1 , 2 1 σ = 1/2 (Variance) p n i i i r r w = = 1 Subject to: = Mean Return 1 1 = = n i i w = Weights sum to 1.
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