# 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?"

© BrainMass Inc. brainmass.com October 25, 2018, 3:28 am ad1c9bdddfhttps://brainmass.com/statistics/variance/variance-portfolio-proportions-of-stock-price-of-the-stock-341016

#### 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

Finance Problems

4. In a world in which your investment choices consist of a risky portfolio and a risk-free asset, the expected return on the former is 15%, and the return on the latter is 10% (which is also your borrowing rate). The standard deviation of the return on the risky portfolio is 20%. If your complete portfolio has a standard deviation of 25%, what are the proportions in which you have allocated your wealth between the risky portfolio and the risk-free asset?

5. Consider two perfectly negatively correlated risky securities, A and B. Security A has an expected rate of return of 16% and a standard deviation of return of 20%. B has an expected rate of return 10% and a standard deviation of return of 30%. What would the weights of these two securities be in the minimum-variance portfolio consisting of just these two assets?

7. You have just checked the newspaper and learned that zero-coupon government

bonds with maturities ranging from 1 to 5 years currently have the following yields:

6.00%, 7.50%, 7.99%, 8.49%, and 10.70%. If you believed that the expectations theory of the term structure were correct, what would you say that the market expects the oneyear zero rate to be one year from now?

12. You have purchased one Texas Instruments August 75 call at $8.50 and have written one Texas Instruments August 80 call at $6.00. If at expiration, shares of TI were trading at $79, what would be your profit or loss on this position?

13. How could you create an investment position involving a put, a call, and riskless

lending or borrowing that would have the same payoff structure at expiration as a long position in the common stock?

14. The stock of a company currently trades at $100. In the simple world in which we are investing, over the next year, the stock will either rise in price to $160 or decline to $60, with equal probabilities. The risk-free interest rate is 6%. In this world, what would be the value today of a put option that expires in one year with an exercise price of $135?

23. One year ago, you purchased a bond with a par value of $1,000 that had 5 years to maturity. The coupon on the bond is 6%, paid annually, and at the time that you bought the bond, its YTM was 4%. You have just sold the bond after receiving one coupon payment, by which time its YTM had declined to 3%. What was your holding period yield on this investment?

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