Ms. Mary Kelley has initial wealth W0 = $1200 and faces an uncertain future that she partitions into two states, s = 1 and s = 2. She can invest in two securities, j and k, with initial prices of pi = $10 and pk = $12, and the following payoff table:
a) If she buys only security j, how many shares can she buy? If she buys only security k, how many can she buy? What would her final wealth, Ws, be in both cases and each state?
b) Suppose Ms. Kelley can issue as well as buy securities; however, she must be able to meet all claims under the occurrence of either state. What is the maximum number of shares of security j she could sell to buy security k? What is the maximum number of shares of security k she could sell to buy security j? What would her final wealth be in both cases and in each state?
c) What are the prices of the pure securities implicit in the payoff table?
d) What is the initial price of a third security i for which Q„ = $5 and Qi2 = $12?
e) Summarize the results of (a) through (d) on a graph with axes W1 and W2.
f) Suppose Ms. Kelley has a utility function of the form U Wt. Find the optimal portfolio, assuming the issuance of securities is possible, if she restricts herself to a portfolio consisting only of j and k. How do you interpret your results?