Concept explainers
a
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.
a
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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
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