The risk-free rate (Rf) in this scenario can be found using the following logic and the APT (Arbitrage Pricing Theory) one-factor model:

1. APT Model: The APT model suggests a linear relationship between an asset’s expected return (E(Ri)) and the risk-free rate (Rf) along with a factor (often market risk) represented by beta (βi). This can be expressed as:

   E(Ri) = Rf + βi * (Rm – Rf)

   where:

   • Ri: Expected return on asset i
   • Rf: Risk-free rate
   • βi: Sensitivity of asset i to the market factor (beta)
   • Rm: Market return (which is not given in this case)

2. No Arbitrage Opportunity: Since we are assuming there are no arbitrage opportunities, it implies that both portfolios A and B are expected to earn the market return (Rm) based on their respective betas.

3. Setting Up Equations: We can set up a system of equations based on the information provided for portfolios A and B:

   • Equation 1 (for Portfolio A): 0.12 (E(RA)) = Rf + 0.5 (βA) * (Rm – Rf)
   • Equation 2 (for Portfolio B): 0.24 (E(RB)) = Rf + 1.5 (βB) * (Rm – Rf)

4. Solving for Rf: We can solve for the risk-free rate (Rf) by eliminating Rm from the equation. Since both equations have Rm, subtracting equation 1 from equation 2 eliminates Rm and leaves us with an equation to solve for Rf.

   • (0.24 – 0.12) = (1.5 – 0.5) * (Rf)
   • 0.12 = 1 * (Rf)

5. Risk-Free Rate: Therefore, the risk-free rate (Rf) in this scenario is 0.12 or 12%.

In conclusion, given the information about the expected returns and betas of portfolios A and B, and assuming no arbitrage opportunities exist, the risk-free rate in this one-factor APT model is 12%.