A stock experienced returns of 4 percent, −17 percent, and 12 percent during the last three years. What is the standard deviation of the stock's returns for the three-year period?

To calculate the standard deviation of the stock's returns for the three-year period, we follow these steps:

### Step 1: Calculate the Mean (Average) Return
The three returns are: \(4\%\), \(-17\%\), and \(12\%\). 

First, calculate the mean return:
\[
\text{Mean} = \frac{4 + (-17) + 12}{3} = \frac{-1}{3} = -0.33\%
\]

### Step 2: Calculate the Squared Deviations from the Mean
Next, we calculate the deviation of each return from the mean, square these deviations, and then sum them up.

For \(4\%\):
\[
(4 - (-0.33))^2 = (4 + 0.33)^2 = 4.33^2 = 18.7489
\]

For \(-17\%\):
\[
(-17 - (-0.33))^2 = (-17 + 0.33)^2 = (-16.67)^2 = 278.0889
\]

For \(12\%\):
\[
(12 - (-0.33))^2 = (12 + 0.33)^2 = 12.33^2 = 152.0289
\]

### Step 3: Calculate the Variance
The variance is the average of these squared deviations:
\[
\text{Variance} = \frac{18.7489 + 278.0889 + 152.0289}{3} = \frac{448.8667}{3} = 149.6222
\]

### Step 4: Calculate the Standard Deviation
Finally, take the square root of the variance to get the standard deviation:
\[
\text{Standard Deviation} = \sqrt{149.6222} \approx 12.23\%
\]

So, the standard deviation of the stock's returns over the three-year period is approximately **12.23%**.

Would you like more details on any of these steps or calculations?

### Related Questions:
1. How do you interpret standard deviation in the context of stock returns?
2. What other statistical measures are important when analyzing stock performance?
3. How does volatility differ from standard deviation in financial terms?
4. What is the difference between variance and standard deviation?
5. How would the calculation change if we had more years of data?

### Tip:
In finance, a higher standard deviation indicates more volatility, meaning the stock's returns are spread out over a wider range.

Learn how to calculate the standard deviation of stock returns using a step-by-step approach. This example covers key statistics concepts like mean, variance, and standard deviation applied to stock performance with returns of 4%, -17%, and 12%. Ideal for finance and statistics students.

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