Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock i has expected revenue rate u, with variance of for i = 1,..., n, and the covariance of the revenues of stocks i and j is given by oij for i ‡j, i, j = 1,..., n. The proportion of stock i in the portfolio is denoted by x₁. 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 ris. You do not need to solve the problem at this stage.
Suppose that you have to choose an optimal portfolio from a list of n stocks. Stock i has expected revenue rate u, with variance of for i = 1,..., n, and the covariance of the revenues of stocks i and j is given by oij for i ‡j, i, j = 1,..., n. The proportion of stock i in the portfolio is denoted by x₁. 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 ris. You do not need to solve the problem at this stage.
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
Related questions
Question
Please solve this question in 2 hours to 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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, finance and related others by exploring similar questions and additional content below.Recommended textbooks for you
Essentials Of Investments
Finance
ISBN:
9781260013924
Author:
Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:
Mcgraw-hill Education,
Essentials Of Investments
Finance
ISBN:
9781260013924
Author:
Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:
Mcgraw-hill Education,
Foundations Of Finance
Finance
ISBN:
9780134897264
Author:
KEOWN, Arthur J., Martin, John D., PETTY, J. William
Publisher:
Pearson,
Fundamentals of Financial Management (MindTap Cou…
Finance
ISBN:
9781337395250
Author:
Eugene F. Brigham, Joel F. Houston
Publisher:
Cengage Learning
Corporate Finance (The Mcgraw-hill/Irwin Series i…
Finance
ISBN:
9780077861759
Author:
Stephen A. Ross Franco Modigliani Professor of Financial Economics Professor, Randolph W Westerfield Robert R. Dockson Deans Chair in Bus. Admin., Jeffrey Jaffe, Bradford D Jordan Professor
Publisher:
McGraw-Hill Education