Consider the one-factor APT

Sample Solution

No arbitrage opportunities in a one-factor APT model with the given information allows us to solve for the risk-free rate (Rf) using the Capital Asset Pricing Model (CAPM) equation.

Here's why:

• APT vs CAPM: While APT uses multiple factors, a one-factor APT can be analogous to CAPM if we assume the single factor driving returns is the market return.

• CAPM Equation: E(ri) = Rf + βi * (E(rm) - Rf), where:

  • E(ri) is the expected return on investment i
  • Rf is the risk-free rate
  • βi is the beta of investment i relative to the market
  • E(rm) is the expected market return

• Given Information:

  • E(rA) = 12% (expected return of portfolio A)
  • βA = 0.5 (beta of portfolio A)
  • E(rB) = 24% (expected return of portfolio B)
  • βB = 1.5 (beta of portfolio B)

• Limited Information: We don't have the explicit value of E(rm) (expected market return). However, we can leverage the relationship between portfolios and the market return in a one-factor APT context.

• Assumption: Assuming portfolio B with a higher beta (1.5) captures more of the market return compared to portfolio A (beta 0.5), we can rewrite the CAPM equation to solve for Rf.

Full Answer Section

Solving for Risk-Free Rate (Rf):

1. Rewrite CAPM for portfolio B: E(rB) = Rf + βB * (E(rm) - Rf)
2. Substitute known values: 0.24 = Rf + 1.5 * (E(rm) - Rf)
3. Since we're looking for Rf, rearrange the equation: Rf = E(rB) - 1.5 * (E(rm) - Rf)

Notice: We can't solve for an exact value of Rf because E(rm) is unknown. However, we can see that the risk-free rate (Rf) is dependent on the expected return of portfolio B (E(rB)) and the difference between the market return (E(rm)) and the risk-free rate itself (Rf).

Interpretation:

• A higher expected return for portfolio B (E(rB)) would imply a higher risk-free rate (Rf).
• A larger gap between the expected market return (E(rm)) and the risk-free rate (Rf) would also lead to a higher risk-free rate (Rf) to compensate for the additional market risk.

In conclusion, while we cannot determine the exact risk-free rate due to the missing information about the expected market return, we can derive the relationship between the risk-free rate and the other given variables within the one-factor APT framework using the CAPM equation.