Assume that both portfolios A andB are well diversified, that E (rA) = 12%, and E (rB) = 9%. Assume the economy has only one risk factor, thebeta of A = 1.2 and the beta of B = 0.8. Using the expected return-beta relationship, what must be the riskfreerate?

The risk free rate (Rf) is 28,2%

Explanation:

We will substituting the portfolio expected return (Er) and the betas of the portfolio in the expected return & beta relationship, that is:

E[r] = Rf + Beta * (Risk Premium)

On doing this we get 2 equations in which the risk free rate (Rf) and the risk premium [P] are not known to use:

12% = Rf + 1 * (P - Rf)

9% = Rf + 1.2 * (P - Rf)

On solving first equation (of Portfolio A) for P (risk premium), we get:

12% = Rf + 1 * (P - Rf)

12% = Rf + P - Rf

(Rf and Rf cancels each other)

P = 12%

Now, on using the value of P in second equation (of Portfolio B), and solving for Rf (risk free rate), we get:

9% = Rf + 1.2 * (12.2% - Rf)

9% = Rf + 14.64% - 1.2Rf

1.2Rf - Rf = 14.64% - 9%

0.2Rf = 5,64%

Rf = 5.64% / 0.2

Rf = 28,2%

So, the risk free rate (Rf) is 28,2%