To calculate the standard deviation of the returns, follow these steps:

1. Compute the mean return ($\bar{R}$): $\bar{R} = \frac{\text{Sum of all returns}}{\text{Number of returns}}$ Given returns: $17.47\%, -7.03\%, 23.78\%$.

   $\bar{R} = \frac{17.47 + (-7.03) + 23.78}{3} = \frac{34.22}{3} = 11.41\%$

2. Compute the squared deviations from the mean: For each return $R_i$, calculate $(R_i - \bar{R})^2$:

   • $(17.47 - 11.41)^2 = (6.06)^2 = 36.7236$
   • $(-7.03 - 11.41)^2 = (-18.44)^2 = 340.6336$
   • $(23.78 - 11.41)^2 = (12.37)^2 = 152.8769$

3. Find the variance ($\sigma^2$): $\sigma^2 = \frac{\text{Sum of squared deviations}}{\text{Number of returns - 1}}$ $\sigma^2 = \frac{36.7236 + 340.6336 + 152.8769}{3 - 1} = \frac{530.2341}{2} = 265.11705$

4. Take the square root to find the standard deviation ($\sigma$): $\sigma = \sqrt{265.11705} \approx 16.28\%$

Correct Answer: 16.28%

Would you like further explanation or more examples? Here are five related questions to practice:

1. How is the standard deviation of returns useful in financial analysis?
2. Calculate the variance of a stock with annual returns of 5%, 10%, and -3%.
3. What is the relationship between standard deviation and portfolio risk?
4. How do you calculate the mean return when the returns are given in decimals rather than percentages?
5. Why do we divide by $n-1$ instead of $n$ when calculating sample variance?

Tip: Always check whether you're working with a sample or population when calculating variance or standard deviation. This determines whether to divide by $n-1$ or $n$.