Part 1: Preferences and the Equity Premium Puzzle

Assume that you use a quadratic utility function to make your fifinancial decisions. The

average return for large US stocks is 11.26% with a standard deviation of 19.13%. Use these

for your estimates of E(r) and .

1. Suppose that in choosing a portfolio consisting of a risk-free asset (where rf

= 3%)

and large US stocks, you invest 60% of your money in large US stocks (and the rest in

the risk-free asset). What does this imply about your risk aversion coeffiffifficient, A?

2. If your preferences are consistent (i.e. you use the same utility function, including the

same A as above), which would you prefer?

(a) an asset which has E(r) = 5% and = 0

(b) an asset with E(r) = 10% and = 20%

Assume that you are only investing in (a) or (b) and not mixing the two assets into a

portfolio.

3. Suppose that we interview a group of investors who chose to invest 60% of their portfolio

in large US stocks and 40% in the risk-free asset. We then ask them which asset from

(2) that they prefer. Most answer that they prefer (b). If we believe that the investors

in the group are consistent in their choices, what does this imply about the quadratic

utility function? If we believe that the quadratic utility function is the correct utility

function, what does this imply about the consistency of investors' preferences?

1Part 2: Markowitz Model and Optimal Portfolios

For this part of the assignment, use the data in assignment2 data.xlsx.

1. Generating some summary statistics:

(a) Report the mean and standard deviation of monthly returns for each of the six

stocks.

(b) Report the covariance matrix of returns (6 by 6).1

Are the covariances between

returns on these stocks generally positive or negative? Are the signs of the co

variances surprising?

2. Solving the Markowitz Problem: Take the means above as the expected returns and

the estimated covariance matrix above as your best estimate of the covariance between

the returns of the six stocks.

(a) Suppose that the target return is 0.8%. What are the portfolio weights for a

portfolio with this return and the minimum possible variance?

(b) Repeat for target returns of 0.9%, 1%, 1.1%, all the way to 1.8%. Report a table of

portfolio weights, expected returns, and volatilities. Plot a graph of the expected

return (y-axis) versus the volatility (x-axis) of the optimal portfolios.

(c) Would an investor who likes higher expected returns and dislikes volatility ever

invest in the portfolio constructed in (a)? Why or why not?

3. Now suppose that the risk-free rate is 0.4167%.

(a) What is the optimal risky portfolio?

(b) What is the expected return if we invest 50% in the optimal risky portfolio and

50% in the risk-free asset? What about 150% in the optimal risky portfolio and

-50% in the risk-free asset?

(c) For an investor with a utility function of E(r) ) 1

2A2r and a coeffiffifficient of risk

aversion of 4, what is the optimal asset allocation?

Part 3: Diversifification

Start with asset A which has an expected return of 10% and a volatility of 30%.

1. Suppose that we introduce asset B with an expected return of 10% and a volatility

of 30%. The correlation between the two asset returns is 0.9. What is the optimal

combination of A and B? What is the volatility of this portfolio? [Note: The choice of

the risk-free rate is not important. You can use rf = 3%.]

2. Now suppose that instead of introducing B, we had introduced asset C with an expected

return of 10% and a volatility of 30%. The returns of asset C are uncorrelated with

both the returns of asset A. What is the optimal combination of A and C? What is

the volatility of this portfolio?

3. Did the introduction of B or C have a greater effffect in decreasing the portfolio volatility?

Why is this the case?

Use the COVARIANCE.S function in Excel.