Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock i has expected revenue rate μ, with variance o for i = 1,..., n, and the covariance of the revenues of stocks i and j is given by oij for ij, i, j = 1,.. 1,..., n. The proportion of stock i in the portfolio is denoted by ₁. a) Showing your working carefully, show that the expected revenue from the port- folio is 1 ii, 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 fl₁ = 0.06,2 = 0.08,3 = 0.09, 0 = 0.1, 02 = 0.3, 03 = 0.6, 012 = -0.1, 013 and 023 = 0.2. Find the optimal portfolio (i.e. the one with the minimum variance) for the expected rate of return of 0.08. 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 μ, with variance o for i = 1,..., n, and the covariance of the revenues of stocks i and j is given by oij for ij, i, j = 1,.. 1,..., n. The proportion of stock i in the portfolio is denoted by ₁. a) Showing your working carefully, show that the expected revenue from the port- folio is 1 ii, 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 fl₁ = 0.06,2 = 0.08,3 = 0.09, 0 = 0.1, 02 = 0.3, 03 = 0.6, 012 = -0.1, 013 and 023 = 0.2. Find the optimal portfolio (i.e. the one with the minimum variance) for the expected rate of return of 0.08. d) Now suppose that the target rate of return is reduced to 0.07. Find the optimal portfolio in this case.
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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Please solve c part and get a thumbs up
Transcribed Image Text:Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock i
has expected revenue rate u; with variance o? for i = 1, ..., n, and the covariance of
the revenues of stocks i and j is given by ơij for i + j, i, j = 1,..., n. The proportion
of stock i in the portfolio is denoted by x;.
a) Showing your working carefully, show that the expected revenue from the port-
folio is E1 xili, 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 x;s. 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:
0.06, µ2 = 0.08, µz = 0.09, oỉ = 0.1, o = 0.3, ož = 0.6, 12
-0.1, 013 = 0.1
%3D
and 023 = 0.2.
Find the optimal portfolio (i.e. the one with the minimum variance) for the expected
rate of return of 0.08.
d) Now suppose that the target rate of return is reduced to 0.07. Find the optimal
portfolio in this case.
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