INVESTMENTS(LL)W/CONNECT
11th Edition
ISBN: 9781260433920
Author: Bodie
Publisher: McGraw-Hill Publishing Co.
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Chapter 10, Problem 17PS
Summary Introduction
To define: Various weighting schemes which will generate well-diversified portfolios.
Introduction:
Well-diversified portfolios: Well-diversified portfolios are portfolios that are constructed with a combination of various securities such as stocks, fixed income, and commodities. When the weight of any of such securities is calculated, it will always be small. Systematic risk is the type of risk found in well-diversified portfolios.
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2. Suppose that you have a riskfree asset and N risky assets for investment. The rate of return
on the riskfree asset is r,, while the (Nx1) vector of the rate of return on the N risky
assets is r, which is multivariate normal, i.e., r N(u, E). Your utility function for a
portfolio that consists of the riskfree asset and the N risky asset is u(r,)=r,-=o,
2
Suppose that the sum of investment proportions on the riskfree and risky assets is one.
Answer the following question.
A. What is your optimal investment proportion in the risky assets? How is your investment
on the riskfree asset affected by different values of 2?
B. Suppose that there is only one risky asset i. Show the effects of the Sharpe ratio
(4,/0, ) on the investment proportion in the risky asset.
In order to create an efficient set of portfolios thru optimization using concepts from Markowitz portfolio theory, you would need to forecast only 2 variables including expected return and standard deviation or variance for the asset classes or securities in focus.
True or false
Consider the expected return and standard deviation of the following two assets:
Asset 1: E[r1]=0.1 and σ1=0.2
Asset 2: E[r2]=0.3 and σ2=0.4
(a) Draw (e.g. with Excel) the set of achievable portfolios in mean-standard deviation
space for the cases: (i) ρ12= -1, (ii) ρ12=0.
(b) Suppose ρ12=-1. Which portfolio has the minimal variance? What is the variance and
expected return of that portfolio?
(c) Derive the formula for the variance of a portfolio with four assets.
Chapter 10 Solutions
INVESTMENTS(LL)W/CONNECT
Ch. 10 - Prob. 1PSCh. 10 - Prob. 2PSCh. 10 - Prob. 3PSCh. 10 - Prob. 4PSCh. 10 - Prob. 5PSCh. 10 - Prob. 6PSCh. 10 - Prob. 7PSCh. 10 - Prob. 8PSCh. 10 - Prob. 9PSCh. 10 - Prob. 10PS
Ch. 10 - Prob. 11PSCh. 10 - Prob. 12PSCh. 10 - Prob. 13PSCh. 10 - Prob. 14PSCh. 10 - Prob. 15PSCh. 10 - Prob. 16PSCh. 10 - Prob. 17PSCh. 10 - Prob. 18PSCh. 10 - Prob. 19PSCh. 10 - Prob. 1CPCh. 10 - Prob. 2CPCh. 10 - Prob. 3CPCh. 10 - Prob. 4CPCh. 10 - Prob. 5CPCh. 10 - Prob. 6CPCh. 10 - Prob. 7CPCh. 10 - Prob. 8CP
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- The market has three risky assets. The variance-covariance matrix of the risky assets are as follows: r1 r2 r3 r1 0.25 0 -0.2 r2 0 4 0.1 r3 -0.2 0.1 1 Assume the market portfolio is M = 0.2 ◦ r1 + 0.5 ◦ r2 + 0.3 ◦ r3. Further assume E(rM) = 0.08. (1) What is the variance of M?(2) What is the covariance of r2 and M?(3) What is β2?(4) If the rate of return of the risk-free asset is 0.02. Then what is the fair expected rate of return of security 2?(5) An investor wants to invest in a portfolio P = 0.4◦r1+0.6◦r3. What is its “fair” expected rate of return?arrow_forwardAssume we beleive a 1 factor APT model describes securities returns. Consider 2 assets with the following data Security A B Suppose the relevant variances are: Component Systematic Factor Expected Return 5.65% 9.06% € A EB Variance 10.0365 0.0387 0.039 Beta 0.5 1.6 1. The beta of an equally weighted portfolio is: Number 2. The the variance of an equally weighted portfolio is (answer exactly): Number 3. Compute the risk free rate : Numberarrow_forwardAssume a Portfolio of two assets A and B whose standard deviations of their returns are 8.6% and 10.8% respectively, while their correlation coefficient of returns is Pas= - 0.61. You are given the right to do portfolio optimization without restrictions. What proportions would you choose and why?arrow_forward
- The optimal proportion of the risky asset in the complete portfolio is given by the equation below y*= E(Rp− Rf) A0² For each of the variables on the right side of the equation, discuss the impact of the variable's effect on y* and why the nature of the relationship makes sense intuitively. Assume the investor is risk aversearrow_forwardAssume that security returns are generated by the single-index model, = : αj + BiRM + ei Ri where Ri is the excess return for security i and RM is the market's excess return. The risk-free rate is 4%. Suppose also that there are three securities A, B, and C, characterized by the following data: Security Bi E(Ri) o(ej) A 1.1 11% 24% B 1.3 13 C 1.5 15 a. If om = Security A Security B Security C 10 19 22%, calculate the variance of returns of securities A, B, and C. Variancearrow_forwardf. Assume a Portfolio of two assets A and B whose standard deviations of their returns are 8.6% and 10.8% respectively, while their correlation coefficient of returns is Pas = - 0.61. You are given the right to do portfolio optimization without restrictions. What proportions would you choose and why?arrow_forward
- assume that every asset has the same expected return and variance. furthermore, all assets have the same covariance with each other. as number of assets in a portfolio grows, which becomes more important: variance or covariance? clarify your answer using words, diagrams, formulae or a practical example.arrow_forwardExpected retun and standard deviation. Use the following information to answer the questions: a. What is the expected return of each asset? b. What is the variance and the standard deviation c. What is the expected return of a portfolio with 1 1 Data Table d. What is the portfolio's variance and standard de - X Hint Make sure to round all intermediate calculatio Swers yo (Click on the following icon D in order to copy its contents into a spreadsheet.) a. What is the expected return of asset J? (Round to four decimal places.) Return on Return on Return on Probability of State State of Asset J in Asset Kin State 0.200 0.140 0.040 Asset L in Economy State State Вoom 0.28 0.070 0.260 0.180 Growth 0.37 0.25 0.070 Stagnant 0.070 0.060 -0.210 Recession 0.10 0.070 -0.100 Print Donearrow_forward3.1 Show that for any portfolio w = equal to (wi,..., wn) the Beta of the portfolio is Bw = wiB1 + + wn Bn, !! ... where B; for i = 1,...,n are Betas of individual assets.arrow_forward
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