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Top of Form
1 .
Consider, as in Lecture 7.1, a portfolio of two risky assets, with expected returns rˉ1,rˉ2, variances σ21,σ22 and covariance σ1,2. No other assets are available. You have to allocate $1 mln of investment in the portfolio of the two assets in order to minimize total portfolio variance. What is the optimal amount of investment in asset 1 (in mln dollars)? Assume expected returns are positive. radio button to select σ_22rˉ_2−σ_12rˉ_1σ_12+σ_22−2σ_1,2 as your response

σ22rˉ2−σ21rˉ1σ21+σ22−2σ1,2
radio button to select σ_12rˉ_1−σ_1,2rˉ_2σ_12+σ_22−2σ_1,2 as your response

σ21rˉ1−σ1,2rˉ2σ21+σ22−2σ1,2
radio button to select σ_22−σ_1,2σ_12+σ_22−2σ_1,2 as your response

σ22−σ1,2σ21+σ22−2σ1,2
radio button to select σ_22−σ_1,2σ_12+σ_22−σ_1,2 as your response

Instructor Explanation
Call a the proportion of total portfolio value invested in first stock. Then we have to minimize the total variance f(a)=a2σ21+(1−a)2σ22+2a(1−a)σ1,2
First-order conditions imply
0=∂f∂a=2aσ21−2(1−a)σ22+(2−4a)σ1,2
Solving this for a yields a=σ22−σ1,2σ21+σ22−2σ1,2
Second order conditions also hold, which is trivial to check.

2 .
Similarly to in-class problem, consider 3 assets with expected returns 1, 2 and 3, respectively. Assume all variances to be 1, σ1,2=0.5,σ1,3=σ2,3=0.5. Solve the Markowitz problem for this setup, and required portfolio return rˉ=2. What are the resulting portfolio weights?

radio button to select -3/2, 1, 3/2 as your response

-3/2, 1, 3/2
radio button to select -1/4, 1/2, 3/4 as your response

-1/4, 1/2, 3/4
radio button to select 3/8, 1/4, 3/8 as your response

3/8, 1/4, 3/8
radio button to select -5/8, 3/4, 9/8 as your response

-5/8, 3/4, 9/8
Correct Answer
3/8, 1/4, 3/8

Instructor Explanation
After writing down the Langrangian for the problem...

...John Doe
Fin 4980-01
Dr. Alex
2/18/2013
Project 1: “Foundations of PortfolioTheory” by. H.M. Markowitz (1991)
Foundations of PortfolioTheory by H.M. Markowitz is based on a two part lesson of microeconomics of capital markets. Part one being that taught by Markowitz, which is solely geared toward portfoliotheory and how an optimizing investor would behave, whereas part two focuses on the Capital Asset Pricing Model (CAPM) which is the work done by Sharpe and Lintner. In this article Markowitz speaks strictly on portfoliotheory.
He states that there are three major ways in which portfoliotheory differs from the theory of the firm and the theory of the consumer, which he was taught. The first way is concerned with investors; the second is concerned with economic factors that act under uncertainty. The third way is a theory where it can be directly used by large investors with adequate computer and database resources. With the first theory being pretty much self-explanatory Markowitz focuses more on expanding on the second and third theories.
When speaking about uncertainty he begins to relate it to his microeconomics course where the theory of the producer assumes that the competitive firm knows the price at which it will sell the goods...

...Mean-Variance Analysis
Mean-varianceportfoliotheory is based on the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return. Markowitz called this approach to portfolio formation mean-variance analysis. Mean-variance analysis is based on the following assumptions:
1. All investors are risk averse; they prefer less risk to more for the same level of expected return.
2. Expected returns for all assets are known.
3. The variances and covariances of all asset returns are known.
4. Investors need only know the expected returns, variances, and covariances of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes.
5. There are no transaction costs or taxes.
The Mean-Variance Approach
The mean-variancetheory postulated that in determining a strategic asset allocation an investor should choose from among the efficient portfolios consistent with that investor’s risk tolerance amongst other constraints and objectives. Efficient portfolios make efficient use of risk by offering the maximum expected return for specific level of...

...Harry W. Markowitz, the father of “Modern Portfoliotheory”, developed the mean-variance analysis, which focuses on creating portfolios of assets that minimizes the variance of returns i.e. risk, given a level of desired return, or maximizes the returns given a level of risk tolerance. This theory aids the process of portfolio construction by providing a quantitative take on it. It integrates the field of quantitative analysis with portfolio management. Meanvariance analysis has found wide applications both inside and outside financial economics. However it is based on certain assumptions which do not hold good in practice. Hence there have been certain revisions to it, so as to make it a more useful tool in portfolio management.
MeanVariance Analysis
Within the meanvariance approach of Markowitz, the basic assumption is that risk is measured by variance, and the investment decision is based on the trade-off between higher mean and lower variance of the returns. The locus of optimal mean-variance combinations is called the efficient frontier, on which all rational investors would desire to be positioned. Asset returns are assumed to be (jointly) normally distributed random...

...housing derivatives blew up, and other foundations were laid bare. Even the core of investing theories related to portfolios has come under pressure. Yet the belief in Modern PortfolioTheory has remained strong amongst the investors.
Modern PortfolioTheory (MPT) is a theory that tells investors how to minimise risks associated with investment and at the same time, maximise return on the investments by proper resource allocation and diversifying their portfolios – it is based on the theory that risk can be lessened by diversifying into uncorrelated asset classes. However, unless the correlations of the various asset classes are predictable, the reduction of risk may be lost.
Investors expect to be rewarded for the level of risk they are taking in a particular market. According to the theory, it's possible to construct an "efficient frontier" of optimal portfolios offering the maximum possible expected return for a given level of risk and there are four basic steps involved in portfolio construction: Security Valuation, Asset Allocation, Portfolio Optimization and Performance measurement.
This theory of portfolio selection was coined by Harry Markowitz in his paper ‘Portfolio Selection’ which was published in the Journal of Finance in March,...

...Post-Modern PortfolioTheory
PMPT Definition, Investment Strategy, and Differences With MPT
By Kent Thune
See More About
alternative investing
build a portfolio
mutual fund analysis
See More About
alternative investing
build a portfolio
mutual fund analysis
Definition: Post-Modern PortfolioTheory (PMPT) is an investing theory and strategic investment style that is a variation of Modern PortfolioTheory (MPT). Similar to MPT, PMPT is an investing method where the investor attempts to take minimal level of market risk, through diversification, to capture maximum-level returns for a given portfolio of investments.
PMPT History and Difference With MPT
PMPT is the culmination of research from many authors and has expanded over several decades as academics at universities in many countries tested these theories to determine whether or not they had merit. The term post-modern portfoliotheory was first used in 1991 to describe portfolio construction software created by engineers Brian M. Rom and Kathleen Ferguson. Rom and Ferguson first publicly described their ideas about PMPT in the 1993 Journal of Investing article, Post-Modern PortfolioTheory Comes of Age.
The difference between PMPT and MPT is the way they define risk and build...

...Case on Mean-Variance Frontiers
1. Ignoring the risk-free asset, draw the frontier in mean-std space.
We solve the problem by Matlab:
clear; clc;
% input data
temp = xlsread('30_Industry_Portfolios');
ret = temp(:,2:end)/100; （this step is to get all the returns from the file）
rf = 0.01/100; (The risk free rate is rf =0.01%= 0.0001 per month.)
% compute moments
er = (mean(ret))'; (the (30.1)vector of returns on the 30 industries)
V = cov(ret); (the covariance matrix of the returns)
% draw the frontier
reqrets = 0.00:0.001:0.02;
for i = 1:length(reqrets)
[trash, rfront(i), varfront(i)] = frontierp(reqrets(i),V,er);
end
figure (1);
plot (varfront.^.5,rfront);
title ('Mean-Std Frontier');
ylabel('E[ret]'); xlabel('\sigma(ret)');
2. Now also consider the risk-free rate. Draw the eﬃcient frontier (do it on the same ﬁgure as 1).
% now add frontier with rf
for i = 1:length(reqrets)
[trash, rfront(i), varfront(i)] = rffrontierp(reqrets(i),V,er,+rf);
end
hold on;
plot(varfront.^.5,rfront);
hold off
3. Compute the tangency portfolio and plot it in the ﬁgure. Write the expected return and standard deviation of the portfolio.
clear;
% input data
temp = xlsread('30_Industry_Portfolios');
ret = temp(:,2:end)/100;
rf = 0.01/100;
% compute moments
er = (mean(ret))';
V = cov(ret);
>> one = ones(T,1); % unit vector...

...they can invest? – The investment opportunity set
2. “Personal” part of asset allocation
– How should an individual investor choose the best risk-return combination from the set of feasible combinations?
3. Equilibrium
– When all investors optimize their portfolios, how are asset returns determined in equilibrium?
Agenda
• • • • • Risk, risk aversion, and utility Portfolio risk and return Diversification Allocation between one risky and a risk-free asset Optimal risky portfolios and the efficient frontier
“OCTOBER: This is one of the peculiarly dangerous months to speculate in stocks in. The other are July, January, September, April, November, May, March, June, December, August, and February.” (Mark Twain)
Key ideas of portfoliotheory
• Risk of a single investment vs. the new investment as part of one’s existing portfolio? • Diversification = “Don't put all your eggs in one basket” • Finding the least risky portfolio for any level of target return • Finding the portfolio with the highest expected return for any level of target risk • Assessing the risk-return relationship of various investments • Selecting an optimal portfolio
Practical value
Asset management • Asset allocation • Portfolio optimization • Performance measurement Risk management • Scenario analysis • Value-at-Risk (VaR) Banking • Credit risk...

...Sub: Finance Question:
Calculation of variance of portfolio.
Topic: Portfolio management
ClassOf1 provides expert guidance to College, Graduate, and High school students on homework and assignment problems in Math, Sciences, Finance, Marketing, Statistics, Economics, Engineering, and many other subjects.
Suppose there are three risky assets, A, B and C with the following expected returns, standard deviations of returns and correlation coefficients. E (rA)= 4% E (rB)=5% E (rC) =15% S.DEVA=5% S.DEVB=7% S.DEVC=10%
A, B=0.7 A, C=-0.2 B,C=0.3
QUESTION 1: Solving for the Global Minimum VariancePortfolio Consider a world where there are no risk free assets, and just these three risky assets. Suppose short sales are permitted. Solve for the weights and variance of the global minimum varianceportfolio. If short sales are not permitted is the solution affected?
Solution:
Global Minimum VariancePortfolio is that set of portfolios that will provide the minimum level of risk for a given level of expected return. Given a world with just the three given risk assets we can use the Solver function in Excel to ascertain the weights and variance of the global minimum varianceportfolio. We have to calculate the Variance Covariance matrix for the given set of...