You work for an asset management firm, and your job is to put together portfolios for different clients. You construct portfolios that contain a money market risk-free asset that pays rf per year and an "optimal" risky portfolio.

While clients may invest in both, your firm lends clients, who do not wish to invest in the risk-free asset, at the fixed rate of rb, so they can invest more heavily (and only) in the "optimal" risky portfolio.

Your need to offer the "correct" asset allocation within the risky portfolio. You have narrowed down the list of assets into two funds: a debt fund and an equity fund.

The annual return of the risky portfolio is the weighted average of the returns on these two risky assets: rp = w1r1 + w2r2 with w1 + w2 = 1 (short-selling is allowed). Tables I and II summarize the notations on the properties of these two risky assets.

Table I: Major Characteristics Table II: Covariance matrix

Expected return S.D Debt Equity    

Debt E(r1)   s1 Debt s²₁   s₁₂

Equity E(r2) s₂ Equity s21   s ²₂   

(a) Set up the Sharpe ratio maximization problem that you would solve to determine the optimal asset allocation in the risky portfolio (define Sharpe ratio using rf ).

(b) Derive the first order conditions for Sharpe ratio maximization problem, and solve for the optimal wights of debt and equity funds in the risky portfolio (w * 1 and w * 2 ). [Hint: see lecture notes to find out ways to simplify the maximization problem.]

(c) Suppose you have the following information: rf=2% and

Table II: Major Characteristics Table IV: Covariance matrix

Expected return S.D Debt Equity

Debt 0.08 0.12 Debt 0.0144 0.0072

Equity 0.13 0.2 Equity 0.0072 0.04

What are w * 1 , w * 2 , and expected return and volatility of the optimal portfolio?

(d) Redo part (c) assuming that the client would borrow at the rate rb = 5%. Compare results with those from part (c), and interpret your findings.

(e) Mr. Rango is your new client, and you have concluded that his utility of investment is given by U(E(r), s) = E(r) - 1 2 5s 2 , where E(r) and s are the expected return and volatility of the complete portfolio you offer him. Recall that 1-w denotes weight of investment in the risk-free asset, and that (1 - w) + w(w1 + w2) = 1.

Given the information in parts (c) and (d), what is the optimal capital/asset allocation for him? Will he be willing to invest in the money market fund, or will he opt to only invest in the risky portfolio by borrowing at the 5% rate? To answer this question, you need to compare the utility levels under each scenario. Show your results using graphs.

(f) How will the optimal capital allocation change if Mr. Rango would like for his standard deviation of complete portfolio to be at most 13%?