B-242 SOLUTIONS
b. We need to find the portfolio weights that result in a portfolio with a of 0.95. We know the of
the risk-free asset is zero. We also know the weight of the risk-free asset is one minus the weight
of the stock since the portfolio weights must sum to one, or 100 percent. So:
p = 0.95 = wS(1.35) + (1 – wS)(0)
0.95 = 1.35wS + 0 – 0wS
w
S = 0.95/1.35
w
S = .7037
And, the weight of the risk-free asset is:
w
Rf = 1 – .7037 = .2963
c. We need to find the portfolio weights that result in a portfolio with an expected return of 8 percent.
We also know the weight of the risk-free asset is one minus the weight of the stock since the
portfolio weights must sum to one, or 100 percent. So:
E(Rp) = .08 = .16wS + .048(1 – wS)
.08 = .16wS + .048 – .048wS
.032 = .112wS
w
S = .2857
So, the
of the portfolio will be:
p = .2857(1.35) + (1 – .2857)(0) = 0.386
d. Solving for the of the portfolio as we did in part a, we find:
p = 2.70 = wS(1.35) + (1 – wS)(0)
w
S = 2.70/1.35 = 2
w
Rf = 1 – 2 = –1
The portfolio is invested 200% in the stock and –100% in the risk-free asset. This represents
borrowing at the risk-free rate to buy more of the stock.
18. First, we need to find the of the portfolio. The of the risk-free asset is zero, and the weight of the
risk-free asset is one minus the weight of the stock, the of the portfolio is:
ß
p = wW(1.25) + (1 – wW)(0) = 1.25wW
So, to find the of the portfolio for any weight of the stock, we simply multiply the weight of the stock
times its .