All Q4 please, i want to get a right answer, i am confusing thx

Finance Discipline Group UTS Business School 25503 Investment Analysis Tutorial 3 1. (a) Write down expressions for the following functions: \u0012 \u0013\u0012 \u0013 \u0001 1 1 x f (x, y) := x y 1 2 y and 1 1 1 x \u0001 1 2 2 y g(x, y, z) := x y z 1 2 3 z (b) Write down expressions for f (x, y) x and f (x, y). y (c) Find (x , y ) = minx,y f (x, y), and evaluate f (x , y ). (d) Use Lagrange multipliers to find (x , y , z ) = minx,y,z g(x, y, z), subject to the constraint x + y + z = 1, and evaluate g(x , y , z ). 2. (a) What criterion must a portfolio meet to be in the minimum variance set? (b) What criterion defines the global minimum variance portfolio (G)? (c) How does the efficient frontier differ from the minimum variance set? Would you prefer a portfolio of the same standard deviation (risk) on the inefficient or efficient frontier? 3. Consider a market containing three assets with the vector of expected returns = (0.4, 0.6, 0.8)> . Suppose that the variance-covariance matrix and its inverse are 2 1 0 0.75 0.5 0.25 1 0.5 . = 1 2 1 and 1 = 0.5 0 1 2 0.25 0.5 0.75 (a) Calculate the values of the scalars A, B, C and . (b) Write done the mean variance frontier equation. (c) Calculate the expected return and variance of returns for the global MVP, G. (d) Are any of the three assets efficient? (e) Draw a rough sketch of the MVS in mean-standard deviation space, indicating the coordinates of G as well as the coordinates of the three assets. 1 4. Consider a market containing three assets whose returns are mutually uncorrelated. The expected returns of the three assets are 1 = 10%, 2 = 20%, and 3 = 30%, and the variances of their returns are 12 = 22 = 32 = 0.2. (a) Suppose you wish to find the weights of the portfolio P with the minimum variance for a target portfolio return P = 25%. Formulate and solve the Markowitz problem using the method of Lagrange multipliers. What are the weights of P and what is P ? (b) Now calculate the scalars A, B, C and and verify your answers for x and P from part (a). Remember that a diagonal matrix can be inverted by inverting each element of the diagonal. (c) Calculate the expected return and standard deviation of returns for the global MVP, G. Is the portfolio P efficient? (d) Write down the equations for the asymptotes of the MVS. (e) Sketch the MVS and its asymptotes in mean-standard deviation space. Your diagram should indicate the positions of P , G, and the three underlying assets. You should also identify the efficient and inefficient components of the MVS. (f) Compare G with the three global MVP's that result when combining only two of the above assets at a time. Does adding a third asset improve things? 5. Consider a financial market consisting of three risky assets vector of expected returns and variance-covariance matrix: 0.05 0.2 0.1 = 0.10 and = 0.1 0.16 0.15 0.1 0.1 with the following 0.1 0.1 . 0.3 (a) Suppose an investor wishes to construct the portfolio with a minimum variance that has an expected return of 20%. Formulate the associated Markowitz problem and obtain the first order conditions by using the method of Lagrange multipliers. (b) Verify that 1 7.6 4.0 1.2 2 . = 4.0 10 1.2 2 4.4 (c) Compute the values of A, B, C and and write done the mean variance frontier equation. (d) Use the values of A, B, C and to determine x . (e) Sketch the MVS in mean-standard deviation space. Your diagram should indicate the positions of P , G, and the three underlying assets. You should also identify the efficient and inefficient components of the MVS. 2 6. Suppose the means, variances, and covariances of returns for six listed stocks are 0.0503 0.0778 0.1256 = 0.0513 0.1031 0.0148 and 0.3736 0.1792 0.1170 0.2561 0.0548 0.0081 0.1792 0.2068 0.0661 0.1385 0.1439 0.0422 0.1170 0.0661 0.4950 0.0033 0.1357 0.0977 . = 0.2561 0.1385 0.0033 0.5023 0.2191 0.0329 0.0548 0.1439 0.1357 0.2191 0.7729 0.1460 0.0081 0.0422 0.0977 0.0329 0.1460 0.1648 Enter this data into an Excel worksheet and perform the following tasks: (a) Calculate the parameters A, B, C and . (b) Calculate the weights, expected return, and standard deviation of returns for the global MVP, G. (c) Calculate the weights and standard deviation of returns for the three MVP's whose expected returns are 15%, 0%, and 15%, respectively. (d) Plot the MVS for expected returns between 20% and 20%. 3