Question

Stock Y has a beta of 0.7 and an expected return of 9.55 percent. Stock Z has a beta of 1.7 and an expected return of 14.39 percent. What would the risk-free rate (in percent) have to be for the two stocks to be correctly priced relative to each other? Answer to two decimals.

Answer 1

Let the risk free rate be R

Let the market portfolio return be M

Expected return = risk free rate + (beta)*(market portfolio return - risk free rate)

Stock Y :

9.55 = R + (0.7)*(M - R)

9.55 = R + 0.7M - 0.7R

9.55 = 0.3R + 0.7M

Stock Z :

14.39 = R + (1.7)*(M - R)

14.39 = R + 1.7M - 1.7R

14.39 = -0.7R + 1.7M

The two equations to solve are :

0.3R + 0.7M = 9.55

-0.7R + 1.7M = 14.39

To solve this, we multiply the first equation by 0.7 and the second by 0.3

(0.7)*(0.3R + 0.7M = 9.55) gives 0.21R + 0.49M = 6.685

(0.3)*(-0.7R + 1.7M = 14.39) gives -0.21R + 0.51M = 4.317

the two equations we have now are :

0.21R + 0.49M = 6.685

-0.21R + 0.51M = 4.317

Adding these together, we get :

0.21R + (-0.21R) + 0.49M + 0.51M = 6.685 + 4.317

M = 11.002

Plugging this value of M into the equation :

0.3R + 0.7(11.002) = 9.55

0.3R = 1.8486

R = 6.16%

the risk free rate is 6.16%