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Using the following data: Scenario Probability return K1 return K2 0.2 -10% 5% W2 0.4 0% 30% W3 0.4 20% -5% compute the weights in the portfolio with minimum risk. What are the expected return and risk of this minimum risk portfolio?

Using the following data: Scenario Probability return K1 return K2 0.2 -10% 5% W2 0.4 0% 30% W3 0.4 20% -5% compute the weights in the portfolio with minimum risk. What are the expected return and risk of this minimum risk portfolio?

Essentials Of Investments
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
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### Portfolio Optimization Example #### Problem Statement Using the following data: | Scenario | Probability | Return \( K_1 \) | Return \( K_2 \) | |----------|--------------|-----------------|-----------------| | \( \omega_1 \) | 0.2 | -10% | 5% | | \( \omega_2 \) | 0.4 | 0% | 30% | | \( \omega_3 \) | 0.4 | 20% | -5% | Compute the weights in the portfolio with minimum risk. What are the expected return and risk of this minimum risk portfolio? #### Explanation: To tackle this problem, apply the principles of portfolio theory and optimization. This involves calculating expected returns, variances, and covariances for the assets, and then using these to find the portfolio weights that minimize risk (typically measured as the standard deviation of portfolio returns). 1. **Calculate the Expected Returns**: - The expected return for each asset can be calculated as the weighted sum of returns in each scenario. 2. **Determine Variance and Covariance**: - Variance and covariance measures give an understanding of the risk and its distribution between two assets. 3. **Optimization**: - Use optimization techniques to minimize the portfolio variance while maintaining the desired rate of return. 4. **Portfolio Weights**: - The portfolio weights are the proportion of the total investment allocated to each asset to achieve the minimum risk portfolio. With this information, students can methodically approach the calculation of portfolio weights to minimize risk and determine the expected return and risk (variance) of the portfolio. For a more detailed breakdown, proceed with the mathematical formulation and solution using portfolio theory methodologies.
Transcribed Image Text:### Portfolio Optimization Example #### Problem Statement Using the following data: | Scenario | Probability | Return \( K_1 \) | Return \( K_2 \) | |----------|--------------|-----------------|-----------------| | \( \omega_1 \) | 0.2 | -10% | 5% | | \( \omega_2 \) | 0.4 | 0% | 30% | | \( \omega_3 \) | 0.4 | 20% | -5% | Compute the weights in the portfolio with minimum risk. What are the expected return and risk of this minimum risk portfolio? #### Explanation: To tackle this problem, apply the principles of portfolio theory and optimization. This involves calculating expected returns, variances, and covariances for the assets, and then using these to find the portfolio weights that minimize risk (typically measured as the standard deviation of portfolio returns). 1. **Calculate the Expected Returns**: - The expected return for each asset can be calculated as the weighted sum of returns in each scenario. 2. **Determine Variance and Covariance**: - Variance and covariance measures give an understanding of the risk and its distribution between two assets. 3. **Optimization**: - Use optimization techniques to minimize the portfolio variance while maintaining the desired rate of return. 4. **Portfolio Weights**: - The portfolio weights are the proportion of the total investment allocated to each asset to achieve the minimum risk portfolio. With this information, students can methodically approach the calculation of portfolio weights to minimize risk and determine the expected return and risk (variance) of the portfolio. For a more detailed breakdown, proceed with the mathematical formulation and solution using portfolio theory methodologies.
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