Loose-Leaf Essentials of Investments
10th Edition
ISBN: 9781259604966
Author: Kane, Alex, Marcus Professor, Alan J., Bodie Professor, Zvi
Publisher: McGraw-Hill Education
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Chapter 6, Problem 4WM
a
Summary Introduction
To determine: The efficient frontier is the combination of portfolios that best meets the expectation of optimum risk and return tradeoff thus making the combination efficient. Now to make a efficient frontier we have to choose the time period from 2008 to 2018. For this visit the website www.portfoliovisualizer.com and feed the data by choosing the asset class and weights of them on historical portfolio.
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Expert Solution
Explanation of Solution
The efficient frontier:
We have chosen five asset class such as US stock market, European stock market, short term treasuries, REIT and Gold.
The Result is hereunder:
| Efficient Frontier | |||||||||
| Start Year | 2008 | ||||||||
| End Year | 2018 | ||||||||
| Geometric Returns | No | ||||||||
| Efficient Frontier Assets | |||||||||
| # | Asset | CAGR | Expected Return (*) | Standard Deviation | Sharpe Ratio (*) | Min. Weight | Max. Weight | ||
| 1 | US Stock Market | 8.28% | 9.56% | 15.31% | 0.598 | 0.00% | 100.00% | ||
| 2 | European Stocks | 0.42% | 2.50% | 20.14% | 0.104 | 0.00% | 100.00% | ||
| 3 | Intermediate Term Treasury | 3.19% | 3.29% | 4.41% | 0.653 | 0.00% | 100.00% | ||
| 4 | REIT | 7.05% | 10.33% | 24.23% | 0.409 | 0.00% | 100.00% | ||
| 5 | Gold | 3.14% | 4.86% | 18.31% | 0.243 | 0.00% | 100.00% | ||
| Tangency Portfolio | |||||||||
| Asset Class | Allocation | ||||||||
| US Stock Market | 21.06% | ||||||||
| Intermediate Term Treasury | 78.94% | ||||||||
| Asset Correlations | |||||||||
| Asset | US Stock Market | European Stocks | Intermediate Term Treasury | REIT | Gold | ||||
| US Stock Market | 1 | 0.88 | -0.22 | 0.74 | 0.03 | ||||
| European Stocks | 0.88 | 1 | -0.13 | 0.67 | 0.12 | ||||
| Intermediate Term Treasury | -0.22 | -0.13 | 1 | 0.05 | 0.41 | ||||
| REIT | 0.74 | 0.67 | 0.05 | 1 | 0.08 | ||||
| Gold | 0.03 | 0.12 | 0.41 | 0.08 | 1 | ||||
| Based on monthly returns from Jan 2008 to Nov 2018 | |||||||||
| Efficient Frontier Points | |||||||||
| # | US Stock Market | European Stocks | Intermediate Term Treasury | REIT | Gold | Expected Return (*) | Standard Deviation (*) | Sharpe Ratio (*) | |
| 1 | 0.00% | 100.00% | 0.00% | 0.00% | 0.00% | 2.50% | 20.14% | 0.104 | |
| 2 | 0.00% | 90.30% | 9.70% | 0.00% | 0.00% | 2.58% | 18.14% | 0.119 | |
| 3 | 0.00% | 80.60% | 19.40% | 0.00% | 0.00% | 2.66% | 16.14% | 0.139 | |
| 4 | 0.00% | 70.89% | 29.11% | 0.00% | 0.00% | 2.73% | 14.16% | 0.164 | |
| 5 | 0.00% | 61.19% | 38.81% | 0.00% | 0.00% | 2.81% | 12.21% | 0.196 | |
| 6 | 0.00% | 51.49% | 48.51% | 0.00% | 0.00% | 2.89% | 10.30% | 0.24 | |
| 7 | 0.00% | 41.79% | 58.21% | 0.00% | 0.00% | 2.96% | 8.46% | 0.301 | |
| 8 | 0.00% | 32.09% | 67.91% | 0.00% | 0.00% | 3.04% | 6.75% | 0.389 | |
| 9 | 0.00% | 22.39% | 77.61% | 0.00% | 0.00% | 3.12% | 5.28% | 0.512 | |
| 10 | 0.00% | 12.68% | 87.32% | 0.00% | 0.00% | 3.19% | 4.33% | 0.643 | |
| 11 | 0.52% | 6.96% | 92.52% | 0.00% | 0.00% | 3.27% | 4.14% | 0.69 | |
| 12 | 1.67% | 6.13% | 92.20% | 0.00% | 0.00% | 3.35% | 4.11% | 0.714 | |
| 13 | 2.82% | 5.29% | 91.88% | 0.00% | 0.00% | 3.42% | 4.08% | 0.738 | |
| 14 | 3.98% | 4.46% | 91.56% | 0.00% | 0.00% | 3.50% | 4.05% | 0.763 | |
| 15 | 5.13% | 3.63% | 91.25% | 0.00% | 0.00% | 3.58% | 4.02% | 0.787 | |
| 16 | 6.28% | 2.79% | 90.93% | 0.00% | 0.00% | 3.65% | 3.99% | 0.812 | |
| 17 | 7.43% | 1.96% | 90.61% | 0.00% | 0.00% | 3.73% | 3.97% | 0.836 | |
| 18 | 8.59% | 1.12% | 90.29% | 0.00% | 0.00% | 3.81% | 3.95% | 0.86 | |
| 19 | 9.74% | 0.29% | 89.97% | 0.00% | 0.00% | 3.89% | 3.93% | 0.884 | |
| 20 | 10.96% | 0.00% | 89.04% | 0.00% | 0.00% | 3.96% | 3.91% | 0.908 | |
| 21 | 12.23% | 0.00% | 87.77% | 0.00% | 0.00% | 4.04% | 3.91% | 0.929 | |
| 22 | 13.49% | 0.00% | 86.51% | 0.00% | 0.00% | 4.12% | 3.91% | 0.947 | |
| 23 | 14.75% | 0.00% | 85.25% | 0.00% | 0.00% | 4.20% | 3.93% | 0.963 | |
| 24 | 16.01% | 0.00% | 83.99% | 0.00% | 0.00% | 4.27% | 3.96% | 0.975 | |
| 25 | 17.27% | 0.00% | 82.73% | 0.00% | 0.00% | 4.35% | 4.00% | 0.985 | |
| 26 | 18.53% | 0.00% | 81.47% | 0.00% | 0.00% | 4.43% | 4.05% | 0.992 | |
| 27 | 19.79% | 0.00% | 80.21% | 0.00% | 0.00% | 4.51% | 4.11% | 0.996 | |
| 28 | 21.06% | 0.00% | 78.94% | 0.00% | 0.00% | 4.58% | 4.18% | 0.998 | |
| 29 | 22.32% | 0.00% | 77.68% | 0.00% | 0.00% | 4.66% | 4.26% | 0.997 | |
| 30 | 23.58% | 0.00% | 76.42% | 0.00% | 0.00% | 4.74% | 4.35% | 0.994 | |
| 31 | 24.84% | 0.00% | 75.16% | 0.00% | 0.00% | 4.82% | 4.45% | 0.99 | |
| 32 | 26.10% | 0.00% | 73.90% | 0.00% | 0.00% | 4.90% | 4.56% | 0.984 | |
| 33 | 27.36% | 0.00% | 72.64% | 0.00% | 0.00% | 4.97% | 4.67% | 0.977 | |
| 34 | 28.62% | 0.00% | 71.38% | 0.00% | 0.00% | 5.05% | 4.79% | 0.969 | |
| 35 | 29.88% | 0.00% | 70.12% | 0.00% | 0.00% | 5.13% | 4.91% | 0.96 | |
| 36 | 31.15% | 0.00% | 68.85% | 0.00% | 0.00% | 5.21% | 5.05% | 0.95 | |
| 37 | 32.41% | 0.00% | 67.59% | 0.00% | 0.00% | 5.29% | 5.18% | 0.94 | |
| 38 | 33.67% | 0.00% | 66.33% | 0.00% | 0.00% | 5.37% | 5.33% | 0.93 | |
| 39 | 34.93% | 0.00% | 65.07% | 0.00% | 0.00% | 5.44% | 5.47% | 0.919 | |
| 40 | 36.16% | 0.00% | 63.72% | 0.00% | 0.12% | 5.52% | 5.62% | 0.909 | |
| 41 | 37.35% | 0.00% | 62.26% | 0.00% | 0.39% | 5.60% | 5.78% | 0.898 | |
| 42 | 38.55% | 0.00% | 60.80% | 0.00% | 0.65% | 5.68% | 5.94% | 0.887 | |
| 43 | 39.74% | 0.00% | 59.34% | 0.00% | 0.92% | 5.76% | 6.10% | 0.877 | |
| 44 | 40.93% | 0.00% | 57.88% | 0.00% | 1.19% | 5.84% | 6.26% | 0.866 | |
| 45 | 42.13% | 0.00% | 56.42% | 0.00% | 1.45% | 5.92% | 6.43% | 0.856 | |
| 46 | 43.32% | 0.00% | 54.96% | 0.00% | 1.72% | 5.99% | 6.59% | 0.846 | |
| 47 | 44.51% | 0.00% | 53.50% | 0.00% | 1.99% | 6.07% | 6.77% | 0.837 | |
| 48 | 45.71% | 0.00% | 52.04% | 0.00% | 2.25% | 6.15% | 6.94% | 0.827 | |
| 49 | 46.90% | 0.00% | 50.58% | 0.00% | 2.52% | 6.23% | 7.11% | 0.818 | |
| 50 | 48.09% | 0.00% | 49.12% | 0.00% | 2.79% | 6.31% | 7.29% | 0.809 | |
| 51 | 49.29% | 0.00% | 47.66% | 0.00% | 3.05% | 6.39% | 7.47% | 0.8 | |
| 52 | 50.48% | 0.00% | 46.20% | 0.00% | 3.32% | 6.47% | 7.65% | 0.792 | |
| 53 | 51.67% | 0.00% | 44.74% | 0.00% | 3.59% | 6.55% | 7.83% | 0.784 | |
| 54 | 52.87% | 0.00% | 43.28% | 0.00% | 3.85% | 6.63% | 8.01% | 0.776 | |
| 55 | 54.06% | 0.00% | 41.82% | 0.00% | 4.12% | 6.71% | 8.19% | 0.768 | |
| 56 | 55.25% | 0.00% | 40.36% | 0.00% | 4.39% | 6.78% | 8.38% | 0.761 | |
| 57 | 56.44% | 0.00% | 38.90% | 0.00% | 4.66% | 6.86% | 8.56% | 0.754 | |
| 58 | 57.64% | 0.00% | 37.44% | 0.00% | 4.92% | 6.94% | 8.75% | 0.747 | |
| 59 | 58.83% | 0.00% | 35.98% | 0.00% | 5.19% | 7.02% | 8.94% | 0.74 | |
| 60 | 60.02% | 0.00% | 34.52% | 0.00% | 5.46% | 7.10% | 9.12% | 0.733 | |
| 61 | 61.22% | 0.00% | 33.06% | 0.00% | 5.72% | 7.18% | 9.31% | 0.727 | |
| 62 | 62.41% | 0.00% | 31.60% | 0.00% | 5.99% | 7.26% | 9.50% | 0.721 | |
| 63 | 63.60% | 0.00% | 30.14% | 0.00% | 6.26% | 7.34% | 9.69% | 0.715 | |
| 64 | 64.80% | 0.00% | 28.68% | 0.00% | 6.52% | 7.42% | 9.88% | 0.709 | |
| 65 | 65.99% | 0.00% | 27.22% | 0.00% | 6.79% | 7.50% | 10.08% | 0.704 | |
| 66 | 67.18% | 0.00% | 25.76% | 0.00% | 7.06% | 7.58% | 10.27% | 0.698 | |
| 67 | 68.38% | 0.00% | 24.30% | 0.00% | 7.32% | 7.66% | 10.46% | 0.693 | |
| 68 | 69.57% | 0.00% | 22.84% | 0.00% | 7.59% | 7.74% | 10.65% | 0.688 | |
| 69 | 70.76% | 0.00% | 21.38% | 0.00% | 7.86% | 7.82% | 10.85% | 0.683 | |
| 70 | 71.96% | 0.00% | 19.92% | 0.00% | 8.12% | 7.90% | 11.04% | 0.678 | |
| 71 | 73.15% | 0.00% | 18.46% | 0.00% | 8.39% | 7.98% | 11.24% | 0.673 | |
| 72 | 74.34% | 0.00% | 17.00% | 0.00% | 8.66% | 8.06% | 11.43% | 0.669 | |
| 73 | 75.53% | 0.00% | 15.54% | 0.00% | 8.92% | 8.14% | 11.63% | 0.665 | |
| 74 | 76.73% | 0.00% | 14.08% | 0.00% | 9.19% | 8.22% | 11.83% | 0.66 | |
| 75 | 77.92% | 0.00% | 12.62% | 0.00% | 9.46% | 8.30% | 12.02% | 0.656 | |
| 76 | 79.11% | 0.00% | 11.16% | 0.00% | 9.72% | 8.38% | 12.22% | 0.652 | |
| 77 | 80.31% | 0.00% | 9.70% | 0.00% | 9.99% | 8.46% | 12.42% | 0.648 | |
| 78 | 81.50% | 0.00% | 8.24% | 0.00% | 10.26% | 8.54% | 12.62% | 0.644 | |
| 79 | 82.69% | 0.00% | 6.78% | 0.00% | 10.53% | 8.62% | 12.81% | 0.641 | |
| 80 | 83.89% | 0.00% | 5.32% | 0.00% | 10.79% | 8.70% | 13.01% | 0.637 | |
| 81 | 85.08% | 0.00% | 3.86% | 0.00% | 11.06% | 8.78% | 13.21% | 0.634 | |
| 82 | 86.27% | 0.00% | 2.40% | 0.00% | 11.33% | 8.87% | 13.41% | 0.63 | |
| 83 | 87.47% | 0.00% | 0.94% | 0.00% | 11.59% | 8.95% | 13.61% | 0.627 | |
| 84 | 88.84% | 0.00% | 0.00% | 0.00% | 11.16% | 9.03% | 13.81% | 0.624 | |
| 85 | 90.53% | 0.00% | 0.00% | 0.00% | 9.47% | 9.11% | 14.01% | 0.62 | |
| 86 | 92.23% | 0.00% | 0.00% | 0.00% | 7.77% | 9.19% | 14.23% | 0.617 | |
| 87 | 93.92% | 0.00% | 0.00% | 0.00% | 6.08% | 9.27% | 14.45% | 0.613 | |
| 88 | 95.62% | 0.00% | 0.00% | 0.00% | 4.38% | 9.35% | 14.68% | 0.609 | |
| 89 | 97.31% | 0.00% | 0.00% | 0.00% | 2.69% | 9.43% | 14.92% | 0.605 | |
| 90 | 99.00% | 0.00% | 0.00% | 0.00% | 1.00% | 9.51% | 15.16% | 0.6 | |
| 91 | 95.62% | 0.00% | 0.00% | 4.38% | 0.00% | 9.59% | 15.44% | 0.595 | |
| 92 | 84.99% | 0.00% | 0.00% | 15.01% | 0.00% | 9.68% | 15.88% | 0.583 | |
| 93 | 74.37% | 0.00% | 0.00% | 25.63% | 0.00% | 9.76% | 16.50% | 0.566 | |
| 94 | 63.75% | 0.00% | 0.00% | 36.25% | 0.00% | 9.84% | 17.28% | 0.545 | |
| 95 | 53.12% | 0.00% | 0.00% | 46.88% | 0.00% | 9.92% | 18.20% | 0.522 | |
| 96 | 42.50% | 0.00% | 0.00% | 57.50% | 0.00% | 10.00% | 19.23% | 0.499 | |
| 97 | 31.87% | 0.00% | 0.00% | 68.13% | 0.00% | 10.08% | 20.37% | 0.475 | |
| 98 | 21.25% | 0.00% | 0.00% | 78.75% | 0.00% | 10.16% | 21.59% | 0.452 | |
| 99 | 10.63% | 0.00% | 0.00% | 89.37% | 0.00% | 10.25% | 22.88% | 0.43 | |
| 100 | 0.00% | 0.00% | 0.00% | 100.00% | 0.00% | 10.33% | 24.23% | 0.409 |
Here the tangency portfolio has a Sharpe ratio of 0.998 and the allocation is given hereunder:
| Asset Class | Allocation |
| US Stock Market | 21.06% |
| Intermediate Term Treasury | 78.94% |
Increase in expected return increased the weight of US equity in Tangency portfolio. This shows that weight of any asset in a portfolio having maximum Sharpe ratio have direct proportional relationship with its expected return.
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Chapter 6 Solutions
Loose-Leaf Essentials of Investments
Ch. 6 - Prob. 1PSCh. 6 - When adding a risky asset to a portfolio of many...Ch. 6 - A portfolio’s expected return is 12%, its standard...Ch. 6 - An investor ponders various allocations to the...Ch. 6 - The standard deviation of the market-index...Ch. 6 - Suppose that the returns on the stock fund...Ch. 6 - Use the rate-of-return data for the stock and bond...Ch. 6 - Prob. 8PSCh. 6 - Prob. 9PSCh. 6 - Prob. 10PS
Ch. 6 - Prob. 11PSCh. 6 - Prob. 12PSCh. 6 - Prob. 13PSCh. 6 - Suppose that many stocks are traded in the market...Ch. 6 - You can find a spreadsheet containing annual...Ch. 6 - Assume expected returns and standard deviations...Ch. 6 - Prob. 17PSCh. 6 - Prob. 18PSCh. 6 - A project has a 0.7 chance of doubling your...Ch. 6 - Investors expect the market rate of return this...Ch. 6 - The following figure shows plots of monthly rates...Ch. 6 - Prob. 22PSCh. 6 - Prob. 23PSCh. 6 - Prob. 25CCh. 6 - Prob. 1CPCh. 6 - Prob. 2CPCh. 6 - Abigail Grace has a $900,000 fully diversified...Ch. 6 - Prob. 4CPCh. 6 - Prob. 5CPCh. 6 - Prob. 6CPCh. 6 - Prob. 7CPCh. 6 - Prob. 1WMCh. 6 - Following the procedures in the previous question,...Ch. 6 - Prob. 3WMCh. 6 - Prob. 4WM
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