OBSERVE

-    When the questions refer to T-bills, think of the risk-free asset (T-bills are short term zero coupon Treasury bonds).

-    The book also calls the Sharpe ratio the “reward-to-variability” ratio.

Question 1

Bob’s initial wealth was $1000. He now holds a portfolio of two assets: Dell and IBM. The weight on Dell is -45%.

a) What are the proceeds from short-selling Dell?

b) What is the weight on IBM?

c) What is the dollar amount invested in IBM?

Question 2

The stock of Gadget Corp sells for $40 a share. Its likely dividend payout and end-of-year price depends on the state of the economy by the end of the year as follows:

The interest rate on the risk-free asset is 4% per year.

a) Calculate the expected (one-year) holding-period return of Gadget Plus

b) Calculate the standard deviation of the (one-year) holding-period return of Gadget Plus.

c) Bob Bruno did not take FIN320 and holds a portfolio invested 60% in Gadget Plus, 40% in the risk-free asset. Calculate:

c.1) the expected return of Bob’s portfolio;

c.2) the standard deviation of Bob’s portfolio;

c.3) the Sharpe Ratio (aka reward-to-variability ratio) of Bob’s portfolio.

Question 3

Gloria Gomez has a portfolio worth $80,000. She subsequently inherits ABC Company common

stock worth $20,000. Her financial advisor provided her with the following forecasted information:

The correlation coefficient of ABC stock returns with the original portfolio returns is 0.40. The risk- free rate Gloria can earn by investing in government securities is 4% per year.

a) The inheritance changes Gloria’s overall portfolio and she is deciding whether to keep the ABC stock. Assuming Gloria keeps the ABC stock, calculate:

a.1) the expected return of Gloria’s new portfolio (which includes the ABC stock).

a.2) the standard deviation of Gloria’s new portfolio (which includes the ABC stock).

b) Based on conversations with her husband, Gloria is considering selling the $20,000 of ABC stock and acquiring $20,000 of XYZ Company common stock instead. XYZ stock has the same expected

return and standard deviation as ABC stock BUT DIFFERENT correlation with the original portfolio. Her husband comments, “It does not matter whether you keep all the ABC stock or replace it with

$20,000 of XYZ stock.” State whether her husband’s comment is correct or incorrect. Justify your response.

c) Assuming Gloria sells the ABC stock and invests all the proceeds from ABC stock with the risk- free asset, calculate:

c.1) the expected return of Gloria’s new portfolio (which includes the risk-free asset).

c.2) the standard deviation of Gloria’s new portfolio (which includes the risk-free asset).

d) Gloria has an utility function of the form U=E(rC) – ½ A (。C)2  with risk aversion A=3.

Is she better off keeping stock ABC (item a) or replacing it by the risk-free asset (item c)? Justify your answer.

e) Gloria has an utility function of the form U=E(rC) – ½ A (。C)2   with risk aversion A=3. Suppose she decides to allocate her total wealth (i.e., her original wealth plus the inheritance) optimally

between her original portfolio and the risk-free asset (so, having zero in ABC stock). How much money should she have in the risk-free asset?

Question 4

Bob considers investing in stock A and stock B. He estimated the following annual statistics: E[r_A] = 10%, sigma_A = 20%, E[r_B] = 15%, sigma_B = 30% and rho_A,B = 0.3. He also knows that r_f = 1%, and that his risk aversion parameter is 3.

a)   Set up the minimization problem to find the minimum variance portfolio for these two assets

b)  Write down the expression for the minimum variance weights, and plug-in the numbers

c)  What are the expected returns, standard deviation, and Sharpe ratio for the minimum variance risky portfolio?

d)  Set up Bob’s optimization problem for the optimal risky portfolio

e)  Write down the expression for the optimal weights, and plug-in the numbers

f)   What are the expected returns, standard deviation, and Sharpe ratio for the optimal portfolio?

g)  Set up Bob’s optimization problem for the complete portfolio

h)  Solve Bob’s optimization problem for the complete portfolio, and find the optimal weights

i)   What are the expected returns, standard deviation, and Sharpe ratio for the complete portfolio?

j)   What are the weights for individual assets in the complete portfolio?

k)  If Bob’s initial wealth was $100, what is the dollar amount invested in each asset (assets A, B and the risk-free asset)?