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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...

...“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...

...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...

...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...

...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...

...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...

...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...

...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...

...Sub: Finance Question:
Calculation of variance of portfolio.
Topic: Portfolio management
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Suppose there are three risky assets, A, B and C with the following expected returns, standard deviations of returns and correlation...