Question

Suppose that the S&P 500, with a beta of 1.0, has an expected return of 10% and T-bills provide a risk-free return of 4%.

a. How would you construct a portfolio from these two assets with an expected return of 8%? Specifically, what will be the weights in the S&P 500 versus T-bills?

b. How would you construct a portfolio from these two assets with a beta of 0.4?

c. Find the risk premiums of the portfolios in (a) and (b), and show that they are proportional to their betas.

Answer 1

S&P 500 beta = 1.0

Expected return =10%

T-bills risk-free return =4%

Let Weight of S&P be W and T-bills (1-W)

(a) W*10% + (1-W) * 4% = 8%

=>W= 66.67%

Weightage of S&P 500 is 66.67 % and T-bills will be 33.33%.

(b) W*1.0% + (1-W) * 0.0 = 0.4

=>W= 40%

Weightage of S&P 500 is 40% and T-bills will be 60%.

(c) Let the risk premium be R, then

8%=4% + beta 1 *R

=>beta1 * R =4.0%

=40% * 10% + 60% * 4% = 4% + beta2 * R

=>beta2 * R =2.4%

beta1 * R / beta2 *R = 1.67

=>beta1 = 1.67 * 0.40

= 0.67

Hence R = 4% /0.67

Risk premium =6.0%