# Variance portfolio; proportions of stock; price of the stock

See attached file for clarity.

Problem 1

Given:

E(RÌ?A) = 10% Ï?2A = 0.25

E(RÌ?B) = 12% Ï?2B = 0.35

PAB = 0.56 Target Return = 11.5%

Minimize: X2A(0.25) + X2B(0.35) + 2XAXB(0.56)(0.25)0.5(0.35)0.5

Subject to:

1 XA(10%) + XB(12%) = 11.5% E(RÌ?P) = 11.5%

2 XA + XB = 1.0 Investments total 100%

3 XA>0, XB>0 Proportions are positive

Solution:

XA = 25%, XB = 75%

Suppose there is a third security ©, with these characteristics: E(Rc) = 10%; Ï?2=0.20; Pac=0.78; and Pbc = 0.56. Construct the quadratic program that would minimize the risk of a three-security portfolio consisting of A, B, and C.

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Problem 2

A B C D E

98' 0.200 0.300 0.100 0.000 -0.100

99' -0.100 0.000 0.000 0.100 0.200

00' 0.400 0.500 0.100 0.400 0.300

01' 0.100 0.200 0.300 -0.100 0.000

02' 2.000 0.300 0.300 -0.200 0.200

03' -0.200 -0.200 -0.100 0.100 0.400

04' 0.500 0.500 0.000 0.300 0.300

05' -0.100 0.100 0.200 0.300 -0.100

06' 0.000 -0.100 0.200 0.100 -0.200

07' 0.300 0.400 0.300 0.100 0.000

E(RÌ?) 0.130 0.200 0.140 0.110 0.100

Ï? 0.219 0.232 0.136 0.176 0.195

A two-security portfolio contains Stocks B and C from above table. Using a spreadsheet package, do the following:

a. Prepare a plot showing the portfolio variance for various combinations of Stocks B and C.

b. Find the minimum variance portfolio.

c. Find the proportions of Stocks B and C that constitute a portfolio with the same risk as Stock C alone.

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Problem 3

"Consider the following information:

Stock price = $46.69

Current dividend = 1.98

Future dividend growth rate = 5.5%

Beta = 1.10

30-day T-bill rate = 2.55%

Equity risk premium = 8.2%

For this stock you want to set a buy limit at 90% of the intrinsic value of the stock as determined using the dividend discount model. What should that price be?"

#### Solution Preview

See the attached file.

Problem 1

Given: E(R̃A) = 10% σ2A = 0.25

E(R̃B) = 12% σ2B = 0.35

PAB = 0.56 Target Return = 11.5%

Minimize: X2A(0.25) + X2B(0.35) + 2XAXB(0.56)(0.25)0.5(0.35)0.5

Subject to:

1 XA(10%) + XB(12%) = 11.5% E(R̃P) = 11.5%

2 XA + XB = 1.0 Investments total 100%

3 XA>0, XB>0 Proportions are positive

Solution:

XA = 25%, XB = 75%

Suppose there is a third security ©, with these characteristics: E(Rc) = 10%; σ2=0.20; Pac=0.78; and Pbc = 0.56. Construct the quadratic program that would minimize the risk of a three-security portfolio consisting of A, B, and C.

E(R̃A) 10% σ2A 0.25

E(R̃B) 12% σ2B 0.35

E(R̃C) 10% σ2C ...

#### Solution Summary

This post shows how to calculate the minimum variance portfolio, proportions of stocks and the price of stock. It shows how to prepare a plot showing the portfolio variance for various combinations of stock