Compute the standard deviation given these four economic states, their likelihoods, and the potential returns:
$$\begin{array}{l}
\begin{array} { c c r }
\text { Eccrorric State }&\text { Probability } & \text { Return } \
\text { fast growth }&0.40 & 50 \% \
\text {slow growth }&0.40 & 10\% \
\text { recession }&0.10 & - 10\%\
\text { depression }&0.10 & - 5\%
\end{array}
\end{array}$$
A) 6.71 percent
B) 22.5 percent
C) 23.37 percent
D) 52.20 percent

Question

Compute the standard deviation given these four economic states, their likelihoods, and the potential returns:
$$\begin{array}{l}
\begin{array} { c c r } 
\text { Eccrorric State }&\text { Probability } & \text { Return } \
\text { fast growth }&0.40 & 50 \% \
\text {slow growth }&0.40 & 10\% \
\text { recession }&0.10 & - 10\%\
\text { depression }&0.10 & - 5\%
\end{array}
\end{array}$$
A) 6.71 percent
B) 22.5 percent
C) 23.37 percent
D) 52.20 percent

Quizplus · Accepted Answer

To calculate the standard deviation of the returns, we first need to find the expected return, then use it to calculate the variance, and finally take the square root of the variance to get the standard deviation.1. Calculate the expected return (mean):$$E(R) = (0.40 \times 50\%) + (0.40 \times 10\%) + (0.10 \times -10\%) + (0.10 \times -5\%) = 20\% + 4\% - 1\% - 0.5\% = 22.5\%$$2. Calculate each state's squared deviation from the mean, multiply by the probability, then sum these values to get the variance:$$Variance = (0.40 \times (50\% - 22.5\%)^2) + (0.40 \times (10\% - 22.5\%)^2) + (0.10 \times (-10\% - 22.5\%)^2) + (0.10 \times (-5\% - 22.5\%)^2)$$$$Variance = (0.40 \times 27.5\%^2) + (0.40 \times 12.5\%^2) + (0.10 \times 32.5\%^2) + (0.10 \times 27.5\%^2)$$$$Variance = (0.40 \times 0.075625) + (0.40 \times 0.015625) + (0.10 \times 0.105625) + (0.10 \times 0.075625)$$$$Variance = 0.03025 + 0.00625 + 0.0105625 + 0.0075625 = 0.054625$$3. Calculate the standard deviation by taking the square root of the variance:$$Standard\ Deviation = \sqrt{0.054625} \approx 0.2337 \text{ or } 23.37\%$$Therefore, the correct answer is 23.37 percent, which corresponds to choice C.

Compute the Standard Deviation Given These Four Economic States, Their