# Modigliani Ratio Calculation The **Modigliani Ratio (M²)** measures risk-adjusted of a portfolio investment by converting Sharpe ratio percentage return units, using the market standard deviation. ## **Given Data** - **Stock annualized return (\(R_p\))**: 8.45% - **Risk-free rate (\(R_f\))**: 2.25% - **Market return (\(R_m\))**: 9.00% - **Std. dev. of stock (\(\sigma_p\))**: 13.30% - **Std. dev. of market (\(\sigma_m\))**: 11.40% ## **Steps** ### 1. Calculate Sharpe Ratio \[ \text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p} \] \[ \text{Sharpe Ratio} = \frac{8.45\% - 2.25\%}{13.30\%} = \frac{6.20\%}{13.30\%} \approx .4662 \] ### 2. Calculate Modigliani Ratio (M²) \[ M^2 = \text{Sharpe Ratio} \times \sigma_m + R_f \] \[ M^2 = .4662 \times 11.40\% + 2.25\% \] \[ M^2 = 5.3127\% + 2.25\% = 7.56\% \] ### 3. Calculate the Difference to Market Return \[ \text{M}^2_\text{relative} = M^2 - R_m \] \[ \text{M}^2_\text{relative} = 7.56\% - 9.00\% = -1.44\% \] ## **Answer** **b. -1.44 percent** --- **Note:** M² is sometimes reported as the difference to the market return, depending on convention. Here, since answer choices are negative or small, the calculation above matches the typical exam intent. --- **Summary Table** | Step | Value | |---------------------|------------| | Sharpe Ratio | .4662 | | M² | 7.56% | | Market Return | 9.00% | | M² relative | -1.44% | --- **Correct answer: b. -1.44 percent**

# Modigliani Ratio Calculation The **Modigliani Ratio (M²)** is a performance measure that assesses the risk-adjusted return of an investment. It takes into account the Sharpe ratio and adjusts it according to the market's standard deviation. This ratio is particularly useful in determining whether an investment is offering adequate returns for its level of risk. ## Introduction Investors often seek to understand the relationship between risk and return when evaluating potential investments. The Modigliani Ratio provides a means of comparing the returns of a specific investment against the risk-free rate and the market's performance. In this case, the stock’s performance is assessed against the risk-free rate of return and compared to the overall market returns, factoring in the volatility of both the stock and the market. ### Explanation The Modigliani Ratio is derived from the Sharpe ratio, which measures the excess return per unit of risk. By incorporating the market's volatility, M² gives a clearer picture of how well an investment performs relative to its risk when compared to the market. This insight is crucial for investors aiming to optimize their portfolios. ## Identifying the Correct Option Among The Choices The correct choice among the provided options is **b. -1.44 percent**. ### Explanation The calculation of the Modigliani Ratio revealed a value of -1.44 percent when comparing the adjusted return of the stock to the market return. This indicates that the stock underperformed relative to the market when considering its risk. ## A Step-by-Step Calculation ### Step 1: Calculate the Sharpe Ratio The formula for the Sharpe Ratio is: \[ \text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p} \] Where: - \(R_p\) = Stock annualized return = 8.45% - \(R_f\) = Risk-free rate = 2.25% - \(\sigma_p\) = Standard deviation of stock returns = 13.30% #### Calculation \[ \text{Sharpe Ratio} = \frac{8.45\% - 2.25\%}{13.30\%} = \frac{6.20\%}{13.30\%} \approx 0.4662 \] ### Explanation The Sharpe ratio quantifies the excess return earned for each unit of risk taken. In this case, the stock's return exceeds the risk-free rate by 6.20%, relative to its standard deviation of 13.30%. ### Step 2: Calculate Modigliani Ratio (M²) The formula for M² is: \[ M^2 = \text{Sharpe Ratio} \times \sigma_m + R_f \] Where: - \(\sigma_m\) = Standard deviation of market returns = 11.40% #### Calculation \[ M^2 = 0.4662 \times 11.40\% + 2.25\% \] \[ M^2 = 5.3127\% + 2.25\% = 7.56\% \] ### Explanation This step converts the excess return of the stock, adjusted for risk, into a form comparable with market returns, yielding an M² of 7.56%. ### Step 3: Calculate the Difference to Market Return The formula for the relative Modigliani Ratio is: \[ \text{M}^2_\text{relative} = M^2 - R_m \] Where: - \(R_m\) = Market return = 9.00% #### Calculation \[ \text{M}^2_\text{relative} = 7.56\% - 9.00\% = -1.44\% \] ### Explanation This final step shows how the stock's adjusted return compares to the market return. A negative result indicates underperformance relative to market expectations. ## Analysis of Why Other Options are Incorrect - **a. -2.75 percent**: This value does not align with the calculated M² and indicates a misunderstanding of the risk-return assessment. - **c. 1.72 percent**: This option misrepresents the relationship between the stock's returns and the market, suggesting an overperformance that is unsupported by the calculations. - **d. 4.41 percent**: This value incorrectly suggests a significantly positive return relative to the market, which contradicts the computed negative result. ## Conclusion The thorough calculation and analysis indicate that the correct answer is **b. -1.44 percent**. This result signifies that the investment does not provide sufficient returns for the level of risk taken when compared to the market benchmark. The methodology employed ensures accurate insights into the stock's performance relative to market expectations.