Let's go step-by-step to solve the problem. --- # Step 1: Understand the problem - Portfolio weights are originally 50% for Zenon and 50% for Dynamics. - Ten years of annual returns are provided for each stock and the portfolio. - The goal: - Calculate the original portfolio return and risk (standard deviation). - Then, analyze how the return and risk change if weights are adjusted to: - 40% Zenon / 60% Dynamics - 30% Zenon / 70% Dynamics --- # Step 2: Calculate average returns Using the data: | Year | Zenon Return | Dynamics Return | |-------|--------------|-----------------| | 2018 | 9.89 | -47.67 | | 2017 | -12.34 | 30.79 | | 2016 | 13.56 | 24.78 | | 2015 | 34.56 | 7.89 | | 2014 | -15.23 | 24.42 | | 2013 | 20.09 | 34.56 | | 2012 | 7.56 | 67.56 | | 2011 | 16.47 | 44.67 | | 201 | 18.34 | 78.56 | | 2009 | 15.56 | 51.00 | ### Calculate average returns: \[ \text{Average Zenon Return} = \frac{9.89 - 12.34 + 13.56 + 34.56 - 15.23 + 20.09 + 7.56 + 16.47 + 18.34 + 15.56}{10} \] \[ = \frac{108.61}{10} = 10.86\% \] Similarly for Dynamics: \[ \text{Average Dynamics Return} = \frac{-47.67 + 30.79 + 24.78 + 7.89 + 24.42 + 34.56 + 67.56 + 44.67 + 78.56 + 51.00}{10} \] \[ = \frac{315.56}{10} = 31.56\% \] --- # Step 3: Calculate the portfolio returns for the original weighting (50/50) \[ R_{portfolio} = .5 \times R_{Zenon} + .5 \times R_{Dynamics} \] Calculate for each year: | Year | Portfolio Return | |-------|------------------| | 2018 | .5 \times 9.89 + .5 \times (-47.67) = -18.89 | | 2017 | .5 \times (-12.34) + .5 \times 30.79 = 9.225 | | 2016 | .5 \times 13.56 + .5 \times 24.78 = 19.17 | | 2015 | .5 \times 34.56 + .5 \times 7.89 = 21.225 | | 2014 | .5 \times (-15.23) + .5 \times 24.42 = 4.595 | | 2013 | .5 \times 20.09 + .5 \times 34.56 = 27.325 | | 2012 | .5 \times 7.56 + .5 \times 67.56 = 37.56 | | 2011 | .5 \times 16.47 + .5 \times 44.67 = 30.57 | | 201 | .5 \times 18.34 + .5 \times 78.56 = 48.45 | | 2009 | .5 \times 15.56 + .5 \times 51.00 = 33.28 | --- # Step 4: Calculate the average portfolio return \[ \text{Average Portfolio Return} = \frac{-18.89 + 9.225 + 19.17 + 21.225 + 4.595 + 27.325 + 37.56 + 30.57 + 48.45 + 33.28}{10} \] \[ = \frac{212.505}{10} = 21.25\% \] --- # Step 5: Calculate the portfolio risk (standard deviation) Calculate the variance: \[ \sigma^2 = \frac{1}{n-1} \sum_{i=1}^n (R_i - \bar{R})^2 \] Where \( R_i \) is the annual return, and \( \bar{R} = 21.25\% \). Compute squared deviations: | Year | \( R_i \) | \( R_i - \bar{R} \) | \( (R_i - \bar{R})^2 \) | |-------|-----------|---------------------|------------------------| | 2018 | -18.89 | -40.14 | 1609.2 | | 2017 | 9.225 | -12.025 | 144.6 | | 2016 | 19.17 | -2.08 | 4.33 | | 2015 | 21.225 | .975 | .95 | | 2014 | 4.595 | -16.655 | 277.36 | | 2013 | 27.325 | 6.075 | 36.9 | | 2012 | 37.56 | 16.31 | 266.4 | | 2011 | 30.57 | 9.32 | 86.88 | | 201 | 48.45 | 27.2 | 739.84 | | 2009 | 33.28 | 12.03 | 144.7 | Sum of squared deviations: \[ 1609.2 + 144.6 + 4.33 + .95 + 277.36 + 36.9 + 266.4 + 86.88 + 739.84 + 144.7 = 3309.16 \] Variance: \[ \sigma^2 = \frac{3309.16}{10 - 1} = \frac{3309.16}{9} \approx 368.79 \] Standard deviation (risk): \[ \sigma = \sqrt{368.79} \approx 19.21\% \] --- # **Summary for 50/50 weights:** | Metric | Value | |---------|--------| | Return | **21.25%** | | Risk (Std Dev) | **19.21%** | --- # Step 6: Recalculate for other weights --- ## **Case 1: 40% Zenon / 60% Dynamics** 1. **Portfolio Return per Year:** \[ R_{portfolio} = .4 \times R_{Zenon} + .6 \times R_{Dynamics} \] Calculate for each year: | Year | Return Calculation | Result | |-------|----------------------|---------| | 2018 | .4 \times 9.89 + .6 \times (-47.67) | -23.781 | | 2017 | .4 \times -12.34 + .6 \times 30.79 | 9.872 | | 2016 | .4 \times 13.56 + .6 \times 24.78 | 20.712 | | 2015 | .4 \times 34.56 + .6 \times 7.89 | 14.712 | | 2014 | .4 \times -15.23 + .6 \times 24.42 | 6.578 | | 2013 | .4 \times 20.09 + .6 \times 34.56 | 28.43 | | 2012 | .4 \times 7.56 + .6 \times 67.56 | 46.2 | | 2011 | .4 \times 16.47 + .6 \times 44.67 | 32.862 | | 201 | .4 \times 18.34 + .6 \times 78.56 | 55.628 | | 2009 | .4 \times 15.56 + .6 \times 51.00 | 36.824 | 2. **Calculate average:** \[ \text{Average} = \frac{-23.781 + 9.872 + 20.712 + 14.712 + 6.578 + 28.43 + 46.2 + 32.862 + 55.628 + 36.824}{10} = \frac{227.039}{10} = 22.70\% \] 3. **Calculate standard deviation:** Calculate squared deviations from the mean (22.70%). Sum of squared deviations: | Year | \( R_i \) | \( R_i - 22.70 \) | \( (R_i - 22.70)^2 \) | |--------|--------------|---------------------|------------------------| | 2018 | -23.781 | -46.481 | 216.7 | | 2017 | 9.872 | -12.828 | 164.5 | | 2016 | 20.712 | -1.988 | 3.95 | | 2015 | 14.712 | -7.988 | 63.8 | | 2014 | 6.578 | -16.122 | 260.4 | | 2013 | 28.43 | 5.73 | 32.8 | | 2012 | 46.2 | 23.5 | 552.25 | | 2011 | 32.862 | 10.162 | 103.3 | | 201 | 55.628 | 32.928 | 1084.9 | | 2009 | 36.824 | 14.124 | 199.5 | Sum of squared deviations: approximately 4627.1 Variance: \[ \sigma^2 = \frac{4627.1}{9} \approx 514.12 \] Standard deviation: \[ \sigma \approx \sqrt{514.12} \approx 22.68\% \] --- ## **Summary for 40/60 weights:** | Metric | Value | |---------|--------| | Return | **22.70%** | | Risk (Std Dev) | **22.68%** | --- ## **Case 2: 30% Zenon / 70% Dynamics** 1. **Portfolio Return:** | Year | Calculation | Result | |-------|--------------|---------| | 2018 | .3 \times 9.89 + .7 \times (-47.67) | -31.41 | | 2017 | .3 \times -12.34 + .7 \times 30.79 | 8.65 | | 2016 | .3 \times 13.56 + .7 \times 24.78 | 20.37 | | 2015 | .3 \times 34.56 + .7 \times 7.89 | 12.12 | | 2014 | .3 \times -15.23 + .7 \times 24.42 | 10.37 | | 2013 | .3 \times 20.09 + .7 \times 34.56 | 30.52 | | 2012 | .3 \times 7.56 + .7 \times 67.56 | 50.4 | | 2011 | .3 \times 16.47 + .7 \times 44.67 | 34.68 | | 201 | .3 \times 18.34 + .7 \times 78.56 | 55.28 | | 2009 | .3 \times 15.56 + .7 \times 51.00 | 39.4 | 2. **Average return:** \[ \frac{-31.41 + 8.65 + 20.37 + 12.12 + 10.37 + 30.52 + 50.4 + 34.68 + 55.28 + 39.4}{10} = 31.07\% \] 3. **Standard deviation:** Squared deviations from 31.07%: | Year | \( R_i \) | \( R_i - 31.07 \) | \( (R_i - 31.07)^2 \) | |--------|--------------|---------------------|------------------------|