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A portfolio formed from risky securities can have a lower standard deviation than either of the individual securities. The benefits of diversification, i.e., the reduction in risk, depends upon the correlation coefficient (or covariance) between the returns on the securities comprising the portfolio.
Consider stocks C and D. Stock C has an expected return of 8% and a standard deviation of 10%. Stock D has an expected return of 16% and a standard deviation of 20%. The concept of diversification will be illustrated by forming portfolios of stocks C and D under three different assumptions regarding the correlation coefficient between the returns on stocks C and D.
The table below provides the expected return and standard deviation for portfolios formed from stocks C and D under the assumption that the correlation coefficient between their returns equals 1.
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Weight of |
Portfolio |
Portfolio |
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100% |
8% |
10% |
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90% |
8.8% |
11% |
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80% |
9.6% |
12% |
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70% |
10.4% |
13% |
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60% |
11.2% |
14% |
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50% |
12% |
15% |
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40% |
12.8% |
16% |
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30% |
13.6% |
17% |
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20% |
14.4% |
18% |
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10% |
15.2% |
19% |
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0% |
16% |
20% |
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Opportunity Set and Efficient Set |
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Opportunity Set - The opportunity set depicts the set of risk return choices that can be achieved by forming a portfolio of stocks C and D. It is represented by the entire curves plotted on the graohs on this page. Efficient Set - The efficent set (or efficient frontier) is the positively sloped portion of the opportunity set. It is the set of risk return choices which offer the highest expected return for a given level of risk. |
When the correlation coefficient between the returns on two securities is equal to +1 the returns are said to be perfectly positively correlated. As can be seen from the table and the plot of the opportunity set, when the returns on two securities are perfectly positively correlated, none of the risk of the individual stocks can be eliminated by diversification. In this case, forming a portfolio of stocks C and D simply provides additional risk/return choices for investors.
The table below provides the expected return and standard deviation for portfolios formed from stocks C and D under the assumption that the correlation coefficient between their returns equals -1.
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Weight of |
Portfolio |
Portfolio |
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100% |
8% |
10% |
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90% |
8.8% |
7% |
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80% |
9.6% |
4% |
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70% |
10.4% |
1% |
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66.67% |
10.67% |
0% |
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60% |
11.2% |
2% |
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50% |
12% |
5% |
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40% |
12.8% |
8% |
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30% |
13.6% |
11% |
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20% |
14.4% |
14% |
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10% |
15.2% |
17% |
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0% |
16% |
20% |
When the correlation coefficient between the returns on two securities is equal to -1 the returns are said to be perfectly negatively correlated or perfectly inversely correlated. When this is the case, all risk can be eliminated by investing a positive amount in the two stocks. This is shown in the table above when the weight of Stock C is 66.67%.
The table below provides the expected return and standard deviation for portfolios formed from stocks C and D under the assumption that the correlation coefficient between their returns equals 0.
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Weight of |
Portfolio |
Portfolio |
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100% |
8% |
10% |
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90% |
8.8% |
9.22% |
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80% |
9.6% |
8.94% |
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70% |
10.4% |
9.22% |
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60% |
11.2% |
10% |
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50% |
12% |
11.18% |
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40% |
12.8% |
12.65% |
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30% |
13.6% |
14.32% |
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20% |
14.4% |
16.12% |
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10% |
15.2% |
18.03% |
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0% |
16% |
20% |
When the correlation coefficient between the returns on two securities is equal to 0 the returns are said to be uncorrelated. In this case, some risk can be eliminated via diversification. Notice that when the weight of Stock C is between 100% and 60% the portfolios have a higher expected return than Stock C and a lower standard deviation than either Stocks C or D. This is depicted in the graph by the inward curve in the opportunity set.
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