Returns for the Alcoff Company over the last 3 years are shown below.What's the standard deviation of the firm's returns? (Hint: This is a sample, not a complete population, so the sample standard deviation formula should be used.) $\begin{array}{rr}
\text { Year } & \text { Return } \
2010 & 21.00 \% \
2009 & -12.50 \% \
2008 & 25.00 \%
\end{array}$
A) 20.08%
B) 20.59%
C) 21.11%
D) 21.64%
E) 22.18%

Question

Quizplus · Accepted Answer

First, calculate the mean (average) return: $(21.00 + (-12.50) + 25.00) / 3 = 33.50 / 3 = 11.17\%$. Then, calculate the variance using the sample standard deviation formula: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1}$, where $x_i$ are the values, $\mu$ is the mean, and $n$ is the number of observations. The variance calculation is $\sigma^2 = \frac{(21.00 - 11.17)^2 + (-12.50 - 11.17)^2 + (25.00 - 11.17)^2}{3-1}$. This simplifies to $\sigma^2 = \frac{(9.83)^2 + (-23.67)^2 + (13.83)^2}{2} = \frac{96.4889 + 560.3889 + 191.2689}{2} = \frac{848.1467}{2} = 424.07335$. The standard deviation ($\sigma$) is the square root of the variance: $\sigma = \sqrt{424.07335} \approx 20.59\%$.

Returns for the Alcoff Company Over the Last 3 Years Year Return 201021.00%2009−12.50%200825.00%\begin{array}{rr}\text { Year } & \text { Return } \\2010 & 21.00 \% \\2009 & -12.50 \% \\2008 & 25.00 \%\end{array} Year 201020092008​ Return 21.00%−12.50%25.00%​

Returns for the Alcoff Company Over the Last 3 Years $\begin{array}{rr}\text { Year } & \text { Return } \\2010 & 21.00 \% \\2009 & -12.50 \% \\2008 & 25.00 \%\end{array}$