A stock had annual returns of 5 percent, 25 percent, and −25 percent over a three-year period. Based on this information, what is the 68 percent probability range for any given year?

Question

BlackTom AI · Accepted Answer

Compute the sample mean and the sample standard deviation, then apply the empirical rule (approximately ±1 standard deviation ≈ 68% probability).

1) Relevant concepts
- Mean (average) of returns gives the center.
- Sample standard deviation (using n−1) estimates the volatility from a small sample.
- Empirical rule: for a roughly normal distribution, about 68% of observations lie within one standard deviation of the mean.

2) Step-by-step reasoning and calculations
- Returns: 5%, 25%, −25%. Number of observations n = 3.
- Mean = (5 + 25 + (−25)) / 3 = 5 / 3 = 1.6666667% ≈ 1.67%.

Compute deviations from the mean and their squares:
- 5 − 1.6667 = 3.3333 → (3.3333)^2 = 11.1111
- 25 − 1.6667 = 23.3333 → (23.3333)^2 = 544.4444
- (−25) − 1.6667 = −26.6667 → (−26.6667)^2 = 711.1111

Sum of squared deviations = 11.1111 + 544.4444 + 711.1111 = 1,266.6666.

Sample variance (divide by n − 1 = 2):
- s^2 = 1,266.6666 / 2 = 633.3333.

Sample standard deviation:
- s = sqrt(633.3333) = 25.1661% ≈ 25.17%.

Apply the 68% range (mean ± 1 s):
- Lower = 1.6667% − 25.1661% = −23.4994% ≈ −23.50%.
- Upper = 1.6667% + 25.1661% = 26.8328% ≈ 26.83%.

3) Why this is correct
Using the sample standard deviation (n−1) is appropriate when estimating the underlying volatility from a small sample; the empirical rule then gives the one-standard-deviation range wherein about 68% of single-year returns would fall if the return distribution is approximately normal.

4) Why other choices are incorrect
- −48.67 to 52% corresponds to roughly ±2 sample standard deviations (≈95% range), not the 68% range.
- −22.28 to 20.28 does not match the mean ± either the population or sample standard deviation computed above.
- −5 to 5% is just ±5% around zero and ignores the computed mean and volatility.

Conclusion: The 68% probability range is −23.50% to 26.83%, which matches the provided correct answer.

A stock had annual returns of 5 percent, 25 percent, and −25 percent over a three-year period. Based on this information, what is the 68 percent probability range for any given year?单项选择题

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