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