# Expected return and Standard Deviation of a portfolio, CAPM

The following four questions need to be addressed with regards to each problem.

1. What financial concept or principle is the problem asking you to solve?

2. In the context of the problem, what are some business decisions that a manager would be able to make after solving the problem?

3. Is there any additional information missing from the problem that would enhance the decision making process?

4. Without showing mathematical equations, explain in writing how you would solve the problem.

PROBLEM 1

Suppose the expected returns and standard deviations of stocks A and B are E(RA)= 0.17, E(RB) = 0.27, StdDevA = 0.12, and StdDevB = 0.21, respectively.

a. Calculate the expected return and standard deviation of a portfolio that is composed of 35% A and 65% B when the correlation between the returns on A and B is 0.06.

b. Calculate the standard deviation of a portfolio that is composed of 35% A and 65% B when the correlation coefficient between the returns on A and B is -0.6.

c. How does the correlation between the returns on A and B affect the standard deviation?

PROBLEM 2

Suppose the expected return on the market portfolio is 14.7 percent and the risk-free rate is 4.9 percent. Morrow Inc. stock has a beta of 1.3. Assume the capital-asset-pricing model holds.

a. What is the expected return on Morrow's stock?

b. If the risk-free rate decreases to 4 percent, what is the expected return on Morrow's stock?

PROBLEM 3

A portfolio that combines the risk-free asset and the market portfolio has an expected return of 22 percent and a standard deviation of 5 percent. The risk-free rate is 4.9 percent, and the expected return on the market portfolio is 19 percent. Assume the capital-asset-pricing model holds.

What expected rate of return would a security earn if it had a 0.6 correlation with the market portfolio and a standard deviation of 3 percent?

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#### Solution Preview

Please see the attached file.

1. What financial concept or principle is the problem asking you to solve?

2. In the context of the problem, what are some business decisions that a manager would be able to make after solving the problem?

3. Is there any additional information missing from the problem that would enhance the decision making process?

4. Without showing mathematical equations, explain in writing how you would solve the problem.

PROBLEM 1

Suppose the expected returns and standard deviations of stocks A and B are E(RA)= 0.17, E(RB) = 0.27, StdDevA = 0.12, and StdDevB = 0.21, respectively.

a. Calculate the expected return and standard deviation of a portfolio that is composed of 35% A and 65% B when the correlation between the returns on A and B is 0.06.

b. Calculate the standard deviation of a portfolio that is composed of 35% A and 65% B when the correlation coefficient between the returns on A and B is -0.6.

c. How does the correlation between the returns on A and B affect the standard deviation?

1. What financial concept or principle is the problem asking you to solve?

The concept is the reduction of risk when a portfolio of stocks is formed as the stocks are not perfectly positively correlated with each other. The risks in holding individual stocks consist of unique risk and systematic risk. The unique risk of stocks can be diversified away if a portfolio of stocks is held rather than individual stock.

2. In the context of the problem, what are some business decisions that a manager would be able to make after solving the problem?

The business decisions can be

What is the optimal allocation of fund in the stocks A and B.

3. Is there any additional information missing from the problem that would enhance the decision making process?

The risk free rate is not mentioned. If this rate is mentioned a fund manager can invest part of the funds in a risk free security.

4. Without showing mathematical equations, explain in writing how you would solve the problem.

The expected return of a portfolio is the weighted average of the expected returns of individual stocks where. Here the weights and expected return of the stocks are given which can be used to calculate the expected return.

The standard deviation of a portfolio depends on the weights of the stocks in the portfolio, the standard deviation of individual stocks and the correlation between the stocks. Since these are mentioned we can use the data to calculate the standard deviation of the portfolio.

Though the problem is not asking for exact solution, the answers are given for completeness

Calculate the expected return and standard deviation of a portfolio that is composed of 35 percent A and 65 percent B when the correlation between the returns on A and B is 0.6 ...

#### Solution Summary

The solution answers 3 questions and calculates

1) The Expected return and Standard Deviation of a portfolio of 2 stocks given the expected returns and standard deviation of returns of the two stock, the coefficient of correlation of returns of the two stocks and the proportion of the two stocks in the portfolio.

2) Beta of a stock using CAPM given expected return on the market portfolio and the risk-free rate.

3) Expected rate of return of a security given its correlation with the market portfolio and a its standard deviation of returns

Expected Return, Standard Deviation, CAPM, Beta, CAL ( Capital Allocation Line)

1 Asset Expected Return Standard Deviation Weight

X 15% 22% 0.5

Y 10% 8% 0.4

Z 6% 3% 0.1

What is the expected return on this three -asset portfolio?

2) Karen Kay, a portfolio manager at Collins Asset Management, is using the capital asset pricing model for making recommendations to her clients. Her research department has developed the information shown in the following exhibit.

Forecasted returns, Standard deviations, and Betas

Forecasted return Standard deviation Beta

Stock X 14% 36% 0.8

Stock Y 17% 25% 1.5

Market Index 14% 15% 1

Risk-free rate 5%

a) Calculate the expected return on the stocks based on CAPM

b) Identify and justify which stock would be more appropriate for an investor who wants to

i) Add this stock to a well- diversified equity portfolio.

ii) Hold this stock as a single-stock portfolio

6. True or False and Explain

You can construct a portfolio with a Beta of .75 by investing .75 of the budget in T-Bills and the remainder in the market portfolio?

In problems 16-18 below, assume the risk-free rate is 8% and the expected rate of return on the market is 18%

16. A share of stock is now selling for $100. It will pay a dividend of $9 per share at the end of the year. Its beta is 1.0. What do the investors expect the stock to sell for at the end of the year.

18. A stock has an expected return of 6%. What is its beta?

For Problems 19-23, assume that you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7%.

19. a. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected return and standard deviation of your client's portfolio?

b. Suppose your risky portfolio includes the following investments in the given proportions:

Stock A 27%

Stock B 33%

Stock C 40%

What are the investment proportions of your client's overall portfolio, including the position in T-bills?

c. What is the reward-to-variability ratio (S) of your risky portfolio and your client's overall portfolio?

d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL.

Please see the attached file.

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