Problem 1: You have mean-variance preferences of the form U = µ − ασ 2 . Moreover, if you have $20, you are indifferent between a lottery that pays you $80 with probability 0.2, and $10 otherwise, and a lottery that pays $1000 with probability 0.1 and zero otherwise.

Today, you receive your first paycheck of $35,000, and you consider saving roughly half of it, or $12,000. Your bank offers Certificate of Deposits (CDs) that pay 5%, and are insured by the FDIC (that is, they are risk free).  Otherwise, Avanguard offers you an Exchange Traded Fund (ETF) that tracks with minimal error (assume zero) the SP500 index.  The average return of the SP 500 index over the last 20 years has been 7%, and its volatility has been 20%.

Answer the following questions:

1. What is your coefficient of risk-aversion α?

2. Sketch the assets and the Capital Allocation Line (CAL) in the standard- deviation/expected return space.

3. What is your optimal portfolio? Describe both the weights of the optimal portfolio, as well as their interpretation in terms of securities trading.

4. Suppose that you also consider a third option, which is a Bond ETF with an expected return of 6% and volatility of 10%.  Assume that the stock- bond correlation is −0.5.

(a) What is the Minimum Variance Portfolio of risky assets (that is, the MVP between the two ETFs)?

(b) What is the correlation between the MVP and the S&P500?

(c) Sketch the three assets and draw the Capital Allocation Line (CAL) in the standard-deviation/expected return space.

(d) At the tangency portfolio, what security are you shorting, if any?

Problem 2:  Go to Ken French’s Data Library and download the monthly returns of the Fama-French 3 factors, which are long-short portfolios of stocks. Answer the following questions.

1. Using the full time series, estimate the average (expected) returns, volatil- ity and correlations for each of the three factors. What do you observe?

2. First, consider only the MKT factor.  Using the additional information on the latest risk-free rate which the dataset provides, sketch the Capital Allocation Line (CAL) in the standard-deviation/expected return space. What are the weights of the optimal portfolio when investors have mean- variance preferences and their risk-aversion is α = 2?

3. What is the correlation between the MVP and the MKJT factor?

4. Using excel, calculate the Minimum Variance Portfolio (MVP). Describe both the weights and their interpretation.

5. Using the additional information on the latest risk-free rate, sketch the Capital Allocation Line (CAL) for the 3-risky assets case in the standard- deviation/expected return space.

Problem 3:  Suppose that the SP 500 index has an expected return of 7% and a standard deviation of returns of 15%.   The risk-free rate is 5%.   The portfolio on the Capital Allocation Line (CAL) that has an expected return of 20% invests weight 65% in the SP 500.  Answer if true or false, and show all your calculations.

Problem 4: Oil prices and airlines stock prices tend to have a strong nega- tive correlation (planes burn a lot of oil). Assume this correlation is − 1. More- over, assume that the return to holding oil is 10%, with a volatility of 25%, while the return of airline stocks is 9% with a volatility of 15%.  Answer the following questions.

1. What is the MVP between these two risky assets?

2. Suppose that the risk-free rate is 3%. Is there an arbitrage opportunity? If so, design a trading strategy that profits from it.

3. Remove now the risk-fre asset, and sketch the MVP and the two risky as- sets in the standard-deviation/expected return space. Without any calcu- lation, find a tangency portfolio and characterize its weights. Is it unique?

4. Suppose that you are mean-variance optimizer and your risk aversion is α = 1. What is your optimal portfolio?

Problem  5:  Hotel stocks and airlines stock prices tend to have a strong positive correlation.  Assume this correlation is 1.  Moreover, assume that the return to hotel stocks is 8%, with a volatility of 10%, while the return of airline stocks is 9% with a volatility of 15%. Answer the following questions.

1. What is the MVP between these two risky assets?