Ms. Lynch has a choice of two assets: The first is a risk-free asset that offers a rate of return of rf, and the second is a risky asset (a china shop that caters to large mammals) that has an expected rate of return of rm and a standard deviation of σm.
(a) If x is the percent of wealth Ms. Lynch invests in the risky asset, what is the equation for the expected rate of return on the portfolio? ______. What is the equation for the standard deviation of the portfolio? ______
(b) By solving the second equation above for x and substituting the result into the first equation, derive an expression for the rate of return on the portfolio in terms of the portfolio’s riskiness. rx = rm−rf / σm σx+rf.
(c) Suppose that Ms. Lynch can borrow money at the interest rate rf and invest it in the risky asset. If rm = 20, rf = 10, and σm = 10, what will be Ms. Lynch’s expected return if she borrows an amount equal to 100% of her initial wealth and invests it in the risky asset? (Hint: This is just like investing 200% of her wealth in the risky asset.)
(d) Suppose that Ms. Lynch can borrow or lend at the risk-free rate. If rf is 10%, rm is 20%, and σm is 10%, what is the formula for the “budget line” Ms. Lynch faces? ______. Plot this budget line in the graph below.
(e) Which of the following risky assets would Ms. Lynch prefer to her present risky asset, assuming she can only invest in one risky asset at a time and that she can invest a fraction of her wealth in whichever risky asset she chooses? Write the word “better,” “worse,” or “same” after each of the assets.
Asset A with ra =17% and σa = 5%. ______.
Asset B with rb =30% and σb = 25%.______.
Asset C with rc =11% and σc = 1%. ______.
Asset D with rd =25% and σd = 14%. ______.
(f) Suppose Ms. Lynch’s utility function has the form u (rx, σx) = rx−2σx. How much of her portfolio will she invest in the original risky asset? (You might want to graph a few of Ms. Lynch’s indifference curves before answering; e.g., graph the combinations of rx and σx that imply u(rx, σx) = 0, 1, . . .)