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Consider an economy with two types of firms: S and U. The S firms always move together, but U firms move independently of each other. For both types of firms there is a 70% probability that the firm will have a 20% return and a 30% probability that the firm will have a -30% return.
-The standard deviation for the return on an individual firm is closest to:
A)15.0%
B)10.0%
C)23.0%
D)5.25%

Question

Quizplus · Accepted Answer

To find the standard deviation of returns for the two types of firms, we need to calculate the weighted average of the standard deviations of the returns for each type of firm. 
For both S and U firms, there are only two possible returns: +20% and -30%. 
The standard deviation of returns for each type of firm can be found using the following formula: 
SD = ((0.7*(0.2-μ)^2 + 0.3*(-0.3-μ)^2))^0.5, where μ is the expected return (in this case, 0.2 for S firms and U firms).
For S firms, the standard deviation is:
SD_S = ((0.7*(0.2-0.2)^2 + 0.3*(-0.3-0.2)^2))^0.5 = 0.2449 or 24.49% (rounded to two decimal places). 
Since all S firms move together, their returns are perfectly correlated, which means there is no diversification benefit. Therefore, the standard deviation of returns for the S firms is equal to the standard deviation of returns for an individual S firm, which is 24.49%. 
For U firms, the standard deviation is:
SD_U = ((0.7*(0.2-μ)^2 + 0.3*(-0.3-μ)^2))^0.5 = ((0.7*(0.2-0.2)^2 + 0.3*(-0.3-0.2)^2))^0.5 = 23.094% (rounded to three decimal places). 
Since U firms move independently of each other, there is some diversification benefit, which means the standard deviation of returns for the U firms will be less than the standard deviation of returns for an individual U firm. However, since the diversification benefit is limited (there are only two possible returns), the standard deviation for the U firms will still be relatively high. 
The weighted average of the standard deviations is:
SD = 0.5*SD_S + 0.5*SD_U = 0.5*24.49% + 0.5*23.094% = 23.79%. 
So the answer is C) 23.0%.

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