Suppose there are 3 securities S, S2, S3 with return vector r= (r1, 72, r3), and...

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Suppose there are 3 securities S₁, S2, S3 with return vector r= (r1, 72, r3), and [0.0427] [0.01 0.002 0.001] μ = E[r]= 0.0015;= Cov(r) = 0.002 0.011 0.003 [0.0285 0.001 0.003 0.02 Let x = (1, 22, 23) be your weight of the portfolio, then the expected portfolio return is = μr, and the variance of the portfolio return is o= rr. Suppose you want to find the optimal portfolio weight z such that is minimized, subject to μp 20.05 and 2₁+ ₂+ 3 ≤1.¹ So the problem becomes 1 min f(x) = ¹ Ex = (0.01x² +0.002x1x2 +0.001*1*3 Er +0.0022₁ +0.011x2 +0.003x2x3 +...) subject to: Tr= 0.04271 +0.0015x2+0.0285x3 20.05 2₁+₂+3 ≤1 • Write this problem to find f, ui and ci. Then use function constrOptim() to find the optimal portfolio weights and the optimal value, you can choose starting point as (2,-2, 0). (Hint: you can use x%*%Sigma%*%x in R to calculate z Er) Write this problem to find D, d, A and b. Then use function solve.QP() to find the optimal portfolio weights and the optimal value. Suppose there are 3 securities S₁, S2, S3 with return vector r= (r1, 72, r3), and [0.0427] [0.01 0.002 0.001] μ = E[r]= 0.0015;= Cov(r) = 0.002 0.011 0.003 [0.0285 0.001 0.003 0.02 Let x = (1, 22, 23) be your weight of the portfolio, then the expected portfolio return is = μr, and the variance of the portfolio return is o= rr. Suppose you want to find the optimal portfolio weight z such that is minimized, subject to μp 20.05 and 2₁+ ₂+ 3 ≤1.¹ So the problem becomes 1 min f(x) = ¹ Ex = (0.01x² +0.002x1x2 +0.001*1*3 Er +0.0022₁ +0.011x2 +0.003x2x3 +...) subject to: Tr= 0.04271 +0.0015x2+0.0285x3 20.05 2₁+₂+3 ≤1 • Write this problem to find f, ui and ci. Then use function constrOptim() to find the optimal portfolio weights and the optimal value, you can choose starting point as (2,-2, 0). (Hint: you can use x%*%Sigma%*%x in R to calculate z Er) Write this problem to find D, d, A and b. Then use function solve.QP() to find the optimal portfolio weights and the optimal value.

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Step 1 Write the problem as a minimization problem The problem is to find the portfolio weights z th... View the full answer