A stock has the following returns over 3 years: -10%, +15%, +25%. The annual geometric rate of return over the 3 years is ________.
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A. B. C. D. E. F. G. H.Explanation
The annual geometric rate of return equals [(1-10%)(1+15%)(1+25%)]^(1/3) - 1 = (0.9 * 1.15 * 1.25)^0.33 - 1 = 0.0896 = 8.96%
To calculate the annual geometric rate of return over multiple periods, you need to use the following formula:
Geometric rate of return = [(1 + r1) * (1 + r2) * (1 + r3) * ... * (1 + rn)]^(1/n) - 1
Where:
In this case, you have the following individual returns over 3 years: -10%, +15%, +25%.
First, convert the percentage returns to decimal form by dividing each return by 100:
-10% = -0.10 +15% = 0.15 +25% = 0.25
Next, apply the formula:
Geometric rate of return = [(1 - 0.10) * (1 + 0.15) * (1 + 0.25)]^(1/3) - 1
Calculating the numerator:
(1 - 0.10) * (1 + 0.15) * (1 + 0.25) = 0.9 * 1.15 * 1.25 = 1.21875
Taking the cube root of the numerator:
(1.21875)^(1/3) ≈ 1.08079
Finally, subtract 1 from the result and convert it to a percentage:
1.08079 - 1 = 0.08079 = 8.08%
Therefore, the annual geometric rate of return over the 3 years is approximately 8.08%.
Among the provided answer choices, the closest option is H. 8.96%, which is the correct answer.