To find the beta for a portfolio with an expected return of **12.5%**, we use the Security Market Line (SML) from the Capital Asset Pricing Model (CAPM): \[ \text{Expected Return} = R_f + \beta (R_m - R_f) \] From the graph: - The risk-free rate (\( R_f \)) is **5%** (where Beta = 0). - The market return (\( R_m \)) is **10%** (where Beta = 1). Plug in the values: \[ 12.5\% = 5\% + \beta (10\% - 5\%) \] \[ 12.5\% = 5\% + 5\%\beta \] \[ 12.5\% - 5\% = 5\%\beta \] \[ 7.5\% = 5\%\beta \] \[ \beta = \frac{7.5\%}{5\%} = 1.5 \] **Answer:** The beta for a portfolio with an expected return of **12.5%** is **1.5**.

Let's determine the beta for a portfolio with an expected return of 12.5% step by step, using the Security Market Line (SML) from the Capital Asset Pricing Model (CAPM). **Step 1: Understand the given information** From the image and the data provided: - The risk-free rate (\( R_f \)) is **5%** (corresponds to Beta = 0). - The market return (\( R_m \)) is **10%** (corresponds to Beta = 1). - The portfolio's expected return is **12.5%**. **Step 2: Recall the CAPM formula (the Security Market Line equation):** \[ \text{Expected Return} = R_f + \beta (R_m - R_f) \] Where: - \( R_f \) = risk-free rate - \( R_m \) = market return - \( \beta \) = measure of systematic risk **Step 3: Plug in known values into the formula** \[ 12.5\% = 5\% + \beta (10\% - 5\%) \] Simplify the expression inside the parentheses: \[ 12.5\% = 5\% + \beta \times 5\% \] **Step 4: Isolate \( \beta \)** Subtract 5% from both sides: \[ 12.5\% - 5\% = \beta \times 5\% \] \[ 7.5\% = 5\% \times \beta \] **Step 5: Solve for \( \beta \)** Divide both sides by 5%: \[ \beta = \frac{7.5\%}{5\%} = \frac{7.5}{5} = 1.5 \] **Final answer:** The beta for the portfolio with an expected return of 12.5% is **1.5**. --- **Summary:** - The expected return of 12.5% lies above the risk-free rate and the market return, indicating higher risk. - The calculated beta of 1.5 confirms this higher systematic risk. - The approach involves plugging known values into the CAPM formula and solving for beta.

Let's carefully recheck the calculation step by step: **Given Data:** - Risk-free rate (\( R_f \)) = 5% - Market return (\( R_m \)) = 10% - Portfolio expected return (\( R_p \)) = 12.5% **Step 1: Write the CAPM formula:** \[ R_p = R_f + \beta (R_m - R_f) \] **Step 2: Plug in known values:** \[ 12.5\% = 5\% + \beta (10\% - 5\%) \] Simplify: \[ 12.5\% = 5\% + \beta \times 5\% \] **Step 3: Isolate \( \beta \):** Subtract 5% from both sides: \[ 12.5\% - 5\% = \beta \times 5\% \] \[ 7.5\% = 5\% \times \beta \] Divide both sides by 5%: \[ \beta = \frac{7.5\%}{5\%} = \frac{7.5}{5} = 1.5 \] **Conclusion:** The beta corresponding to an expected return of 12.5% is **1.5**. **Rechecking confirms the initial calculation.**

Let's re-express and verify each step carefully, including the calculations: --- ### Given Data: - Risk-free rate, \( R_f = 5\% \) - Market return, \( R_m = 10\% \) - Portfolio expected return, \( R_p = 12.5\% \) --- ### Step 1: Write the CAPM equation \[ R_p = R_f + \beta (R_m - R_f) \] ### Step 2: Plug in the known values \[ 12.5\% = 5\% + \beta (10\% - 5\%) \] Calculate the difference \( R_m - R_f \): \[ 10\% - 5\% = 5\% \] So, \[ 12.5\% = 5\% + \beta \times 5\% \] --- ### Step 3: Isolate \( \beta \) Subtract 5% from both sides: \[ 12.5\% - 5\% = \beta \times 5\% \] \[ 7.5\% = 5\% \times \beta \] --- ### Step 4: Solve for \( \beta \) Divide both sides by 5%: \[ \beta = \frac{7.5\%}{5\%} \] Express percentages as decimals to clarify calculations: \[ \beta = \frac{0.075}{0.05} = 1.5 \] --- ### **Final Answer:** **The beta of the portfolio is \(\boxed{1.5}\).** This indicates the portfolio has 1.5 times the systematic risk compared to the market. --- ### **Summary:** - The calculation confirms that, given the expected return and the market data, the beta is **1.5**. - The steps involve plugging the values into the CAPM equation and solving algebraically. Let me know if you'd like further clarification!