A stock had returns of 16.15 percent, −6.15 percent, and 23.45 percent for the past three years. What is the standard deviation of the returns?

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BlackTom AI · Accepted Answer

We are given three yearly returns: 16.15%, -6.15%, and 23.45%. The task is to compute the standard deviation of these returns.
First, convert percentages to decimal form to keep arithmetic consistent: 0.1615, -0.0615, 0.2345.
Next, find the mean (average) of the three returns: (0.1615 − 0.0615 + 0.2345) ÷ 3 = 0.3345 ÷ 3 = 0.1115, which is 11.15%.
Then compute each deviation from the mean: 0.1615 − 0.1115 = 0.0500, −0.0615 − 0.1115 = −0.1730, 0.2345 − 0.1115 = 0.1230.
Square these deviations: (0.0500)^2 = 0.0025, (−0.1730)^2 ≈ 0.029929, (0.1230)^2 ≈ 0.015129.
Sum of squared deviations ≈ 0.0025 + 0.029929 + 0.015129 ≈ 0.047558.
Now decide whether to use population or sample standard deviation. In financial data with a sample of observations, the standard practice for a small sample is to use the sample standard deviation, which divides by n − 1.
- Sample variance = 0.047558 ÷ (3 − 1) = 0.047558 ÷ 2 ≈ 0.023779.
- Sample standard deviation = sqrt(0.023779) ≈ 0.1542, i.e., 15.42%.
If one were to use the population formula (dividing by n), the standard deviation would be sqrt(0.047558 ÷ 3) ≈ sqrt(0.0158527) ≈ 0.1260 or 12.60%, which is not the value given in the options.
Now evaluate each answer option:
- 2.57 percent: this is far too small relative to the computed deviations; it would correspond to a much smaller spread and is not in line with the data.
- 23.44 percent: this is close to the raw magnitude of one large deviation but far from the combined variability; it overstates the overall dispersion and does not align with either the population or sample calculation here.
- 8.90 percent: this value does not match the calculated spread using either standard approach; it undervalues the variability shown by the three numbers.
- 15.42 percent: this matches the standard deviation using the sample formula (n − 1 in the denominator), which is commonly used for a small set of observations like three years of returns.
Thus, the option that aligns with the standard statistical approach for a small sample and the provided data is 15.42 percent.

FIN 220 01 Final Exam _ Fall 2025

A stock had returns of 16.15 percent, −6.15 percent, and 23.45 percent for the past three years. What is the standard deviation of the returns?

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