Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock...

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Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock i has expected revenue rate ; with variance of for i=1,..., n, and the covariance of the revenues of stocks i and j is given by oi for i j,i, j = 1,..., n. The proportion of stock i in the portfolio is denoted by . a) Showing your working carefully, show that the expected revenue from the port- folio isipi, and find an expression for the variance of the portfolio revenue, again showing your working carefully. b) Still for a general number of n stocks, formulate this as an optimization problem using Lagrange multipliers, and find a set of linear equations for the optimal values of the xs. You do not need to solve the problem at this stage. c) Now consider a problem with three stocks, where the means, variances and co- variances are as follows: -0.1, 013 = 0.1 Find the optimal portfolio (i.e. the one with the minimum variance) for the expected rate of return of 0.08. = 0.06, 2 = 0.08, 3 = 0.09, o = 0.1, 0 = 0.3, o = 0.6, 012 and 023 = 0.2. d) Now suppose that the target rate of return is reduced to 0.07. Find the optimal portfolio in this case. Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock i has expected revenue rate ; with variance of for i=1,..., n, and the covariance of the revenues of stocks i and j is given by oi for i j,i, j = 1,..., n. The proportion of stock i in the portfolio is denoted by . a) Showing your working carefully, show that the expected revenue from the port- folio isipi, and find an expression for the variance of the portfolio revenue, again showing your working carefully. b) Still for a general number of n stocks, formulate this as an optimization problem using Lagrange multipliers, and find a set of linear equations for the optimal values of the xs. You do not need to solve the problem at this stage. c) Now consider a problem with three stocks, where the means, variances and co- variances are as follows: -0.1, 013 = 0.1 Find the optimal portfolio (i.e. the one with the minimum variance) for the expected rate of return of 0.08. = 0.06, 2 = 0.08, 3 = 0.09, o = 0.1, 0 = 0.3, o = 0.6, 012 and 023 = 0.2. d) Now suppose that the target rate of return is reduced to 0.07. Find the optimal portfolio in this case.