1. You have estimated the returns of 2 companies for the next year depending on the general

economic condition.

Outcome Probability Company 1 Company 2

Boom 30% 15% 5%

Normal 50% 5% 0

Recession 20% 0% 10%

(a) Calculate the expected return and standard deviation of the returns of each company.

(b) Calculate the covariance and correlation between the returns of these 2 companies.

(c) Let?s combine these 2 companies 50%-50% in portfolio P. Calculate the expected return

and the standard deviation of the portfolio P.

(d) Using mean-variance criterion, does any of the 3 investments, company 1, company 2,

and portfolio P, dominate the other one?

2. Suppose that a fund that tracks the S&P has mean E(Rm) = 16% and standard deviation

?m = 10%, and suppose that the T-bill rate Rf = 8%. Answer the following questions:

(a) What is the expected return and standard deviation of a portfolio that is completely invested

in the risk-free asset?

(b) What is the expected return and standard deviation of a portfolio that has 50% of its wealth in

the risk-free asset and 50% in the S&P?

(c) What is the expected return and standard deviation of a portfolio that has 125% of its wealth

in the S&P, financed by borrowing 25% of its wealth at the risk-free rate?

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(d) What are the weights for investing in the risk-free asset and the S&P that produce a standard

deviation for the entire portfolio that is twice the standard deviation of the S&P? What is the

expected return on that portfolio?

(e) Assume an investor? preference is characterized by the utility function U = E[r]-0.5A(?)^2.

What is the optimal portfolio for an investor with A=4? (Hint: calculate the investor?s utility for

different portfolio combinations. Begin with 100% in risk free, 0% in S&P and go up to -50% in

risk free and 150% in S&P with 10% increments. Excel would be helpful here.)

3. Consider the following data:

Expected Return Standard Deviation

Russell Fund 16% 12%

Windsor Fund 14% 10%

S&P Fund 12% 8%

The correlation between the returns on the Russell Fund and the S&P Fund is 0.7. The rate on Tbills

is 6%. Which of the following portfolios would you prefer to hold in combination with Tbills

and why? (Hint: look at the sharp ratios)

(a) Russell Fund

(b) Windsor Fund

(c) S&P Fund

(d) A portfolio of 60% Russell Fund and 40% S&P Fund.

Principals of Investments Professor Saeid Hoseinzade Problem Set 1 1. You have estimated the returns of 2 companies for the next year depending on the general economic condition. Outcome Probability Company 1 Company 2 Boom 30% 15% 5% Normal 50% 5% 0 Recession 20% 0% 10% (a) Calculate the expected return and standard deviation of the returns of each company. (b) Calculate the covariance and correlation between the returns of these 2 companies. (c) Let's combine these 2 companies 50%-50% in portfolio P. Calculate the expected return and the standard deviation of the portfolio P. (d) Using mean-variance criterion, does any of the 3 investments, company 1, company 2, and portfolio P, dominate the other one? 2. Suppose that a fund that tracks the S&P has mean E(Rm) = 16% and standard deviation m = 10%, and suppose that the T-bill rate Rf = 8%. Answer the following questions: (a) What is the expected return and standard deviation of a portfolio that is completely invested in the risk-free asset? (b) What is the expected return and standard deviation of a portfolio that has 50% of its wealth in the risk-free asset and 50% in the S&P? (c) What is the expected return and standard deviation of a portfolio that has 125% of its wealth in the S&P, financed by borrowing 25% of its wealth at the risk-free rate? Page 1 of 2 (d) What are the weights for investing in the risk-free asset and the S&P that produce a standard deviation for the entire portfolio that is twice the standard deviation of the S&P? What is the expected return on that portfolio? (e) Assume an investor' preference is characterized by the utility function U = E[r]-0.5A()^2. What is the optimal portfolio for an investor with A=4? (Hint: calculate the investor's utility for different portfolio combinations. Begin with 100% in risk free, 0% in S&P and go up to -50% in risk free and 150% in S&P with 10% increments. Excel would be helpful here.) 3. Consider the following data: Expected Return Standard Deviation Russell Fund 16% 12% Windsor Fund 14% 10% S&P Fund 12% 8% The correlation between the returns on the Russell Fund and the S&P Fund is 0.7. The rate on Tbills is 6%. Which of the following portfolios would you prefer to hold in combination with Tbills and why? (Hint: look at the sharp ratios) (a) Russell Fund (b) Windsor Fund (c) S&P Fund (d) A portfolio of 60% Russell Fund and 40% S&P Fund. Page 2 of 2