Q2 (Essential to cover - a, b, c, d)

Suppose that the risk-free rate is 1% and that the corresponding optimal risky portfolio has an

expected return of 15% and a standard deviation of 20%. Consider an investor with preference

represented by the utility function U = E(r) - 0.5A² , where A = 3.

(a) What fraction of her wealth should she invest in the risky portfolio?

(b) Should she invest more or less fraction of her wealth in the optimal risky portfolio if she is more risk-averse? Why?

(c) Should she invest more or less fraction of her wealth in the optimal risky portfolio if the optimal risky portfolio offers a higher expected return (but the same standard deviation)? Why?

(d) If the risk-free rate is higher, what would you expect the optimal risky portfolio to differ from the current one (for 1% risk-free rate) in terms of expected return and standard deviation?

(e) Now suppose that the investor faces a higher risk-free rate when borrowing (because you are facing more borrowing constraints than the government, or people won't allow you to borrow at the same rate as the government). Specifically, the investor may lend at a risk-free rate of 1% but has to borrow at a risk-free rate of 5%. Assume for simplicity that the optimal risky portfolio is the same for either 1% risk-free rate or 5% risk-free rate, i.e., with an expected return of 15% and a standard deviation of 20% (this should not be the case in reality, as you will find out in question (d)). What fraction of her wealth should the investor now put in the risky portfolio?

Hint: The risk-free rate of 1% will imply a fraction₁of wealth invested in the optimal risky portfolio. The risk-free rate of 5% will imply another fraction₂of wealth invested in the optimal risky portfolio. However, 1% is only available for the investor to lend (so she cannot borrow to invest more than her wealth to invest in the optimal risky portfolio). If she wants to borrow to invest more than her wealth to invest in the optimal risky portfolio, she will have to go with 5% risk-free rate (but she won't be able to lend at 5% rate).

The key here is: you would need to understand what weights in the optimal risky portfolio mean she will lend, and what weights mean she will borrow.

(f) What is the expected return and standard deviation of the portfolio that you found in (e)?