# Step 1: Calculate Treynor’s Ratio and Sharpe’s Ratio ### **Given Data** - **Risk-Free Rate (\( R_f \))**: 6% - **Market Portfolio:** - Return (\( R_m \)): 12% - Std Dev: 16% - Beta: 1.00 - **Portfolio X:** - Return: 13% - Std Dev: 12% - Beta: .85 - **Portfolio Y:** - Return: 15% - Std Dev: 25% - Beta: 1.20 - **Portfolio Z:** - Return: 10% - Std Dev: 8% - Beta: .75 --- ### **Formulas** - **Treynor’s Ratio:** \[ \text{Treynor's Ratio} = \frac{R_p - R_f}{\beta_p} \] - **Sharpe’s Ratio:** \[ \text{Sharpe's Ratio} = \frac{R_p - R_f}{\sigma_p} \] --- ### **Calculations** #### **Market** - Treynor: \( \frac{12 - 6}{1.00} = 6.00 \) - Sharpe: \( \frac{12 - 6}{16} = .375 \) #### **X** - Treynor: \( \frac{13 - 6}{.85} \approx 8.24 \) - Sharpe: \( \frac{13 - 6}{12} = .583 \) #### **Y** - Treynor: \( \frac{15 - 6}{1.20} = 7.50 \) - Sharpe: \( \frac{15 - 6}{25} = .36 \) #### **Z** - Treynor: \( \frac{10 - 6}{.75} = 5.33 \) - Sharpe: \( \frac{10 - 6}{8} = .50 \) --- #### **Step 1 Explanation** We have calculated Treynor’s and Sharpe’s ratios for each portfolio using the provided returns, standard deviations, and betas. These ratios measure risk-adjusted performance, with Treynor using systematic risk (beta) and Sharpe using total risk (standard deviation). --- # Step 2: Calculate Jensen’s Alpha and Rank Performance ### **Formula for Jensen’s Alpha** \[ \alpha_p = R_p - [R_f + \beta_p (R_m - R_f)] \] --- ### **Calculations** #### **Market** \[ \alpha = 12 - [6 + 1 \times (12 - 6)] = 12 - 12 = \] #### **X** \[ \alpha = 13 - [6 + .85 \times (12-6)] = 13 - [6 + 5.1] = 13 - 11.1 = 1.9 \] #### **Y** \[ \alpha = 15 - [6 + 1.2 \times (12-6)] = 15 - [6 + 7.2] = 15 - 13.2 = 1.8 \] #### **Z** \[ \alpha = 10 - [6 + .75 \times (12-6)] = 10 - [6 + 4.5] = 10 - 10.5 = -.5 \] --- #### **Ranking the Managers (Based on Ratios & Alpha):** | Portfolio | Treynor | Sharpe | Alpha | Rank | |-----------|---------|--------|-------|------| | X | 8.24 | .583 | 1.9 | 1 | | Y | 7.50 | .36 | 1.8 | 2 | | Market | 6.00 | .375 | | 3 | | Z | 5.33 | .50 | -.5 | 4 | **Step 2 Explanation:** Jensen’s Alpha measures the excess return above the CAPM-predicted return. Higher alpha indicates better performance. Based on all three metrics, Portfolio X consistently outperforms, followed by Y, Market, and Z. --- ## **Final Answer** ### **a) Calculated Ratios:** | Portfolio | Treynor | Sharpe | Alpha | |-----------|---------|--------|-------| | Market | 6.00 | .375 | | | X | 8.24 | .583 | 1.9 | | Y | 7.50 | .36 | 1.8 | | Z | 5.33 | .50 | -.5 | ### **b) Performance Ranking:** **X > Y > Market > Z** **Portfolio X is the best performer based on risk-adjusted return and alpha, followed by Y, Market, and Z.**