Compute the standard deviation given these four economic states, their likelihoods, and the potential returns:
$$\begin{array}{l}
\begin{array} { c c r }
\text { Eccrorric State }&\text { Probability } & \text { Return } \
\text { fast growth }&0.20 & 100 \% \
\text {slow growth }&0.50 & 10\% \
\text { recession }&0.20 & - 1\%\
\text { depression }&0.10 & - 10\%
\end{array}
\end{array}$$
A) 12.19 percent
B) 23.8 percent
C) 38.65 percent
D) 88.06 percent

Question

Compute the standard deviation given these four economic states, their likelihoods, and the potential returns:
$$\begin{array}{l}
\begin{array} { c c r } 
\text { Eccrorric State }&\text { Probability } & \text { Return } \
\text { fast growth }&0.20 & 100 \% \
\text {slow growth }&0.50 & 10\% \
\text { recession }&0.20 & - 1\%\
\text { depression }&0.10 & - 10\%
\end{array}
\end{array}$$
A) 12.19 percent
B) 23.8 percent
C) 38.65 percent
D) 88.06 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 ($E(R)$):$$E(R) = (0.20 \times 100) + (0.50 \times 10) + (0.20 \times -1) + (0.10 \times -10) = 20 + 5 - 0.2 - 1 = 23.8\%$$2. Calculate each state's squared deviation from the expected return:- Fast growth: $(100 - 23.8)^2 \times 0.20 = 5803.68$- Slow growth: $(10 - 23.8)^2 \times 0.50 = 95.16$- Recession: $(-1 - 23.8)^2 \times 0.20 = 123.84$- Depression: $(-10 - 23.8)^2 \times 0.10 = 144.64$3. Sum the squared deviations to find the variance:$$Variance = 5803.68 + 95.16 + 123.84 + 144.64 = 6167.32$$4. Calculate the standard deviation ($\sigma$) as the square root of the variance:$$\sigma = \sqrt{6167.32} \approx 78.53\%$$It appears there was a mistake in the calculation. Let's correct the process:1. **Expected Return**:$$E(R) = (0.20 \times 100) + (0.50 \times 10) + (0.20 \times -1) + (0.10 \times -10) = 20 + 5 - 0.2 - 1 = 23.8\%$$2. **Variance Calculation** involves finding the squared differences from the mean, multiplied by their probabilities:$$Variance = 0.20(100-23.8)^2 + 0.50(10-23.8)^2 + 0.20(-1-23.8)^2 + 0.10(-10-23.8)^2$$$$= 0.20(76.2)^2 + 0.50(-13.8)^2 + 0.20(-24.8)^2 + 0.10(-33.8)^2$$$$= 0.20(5803.64) + 0.50(190.44) + 0.20(615.04) + 0.10(1142.44)$$$$= 1160.728 + 95.22 + 123.008 + 114.244 = 1493.2$$3. **Standard Deviation** is the square root of the variance:$$\sigma = \sqrt{1493.2} \approx 38.65\%$$The correct calculation leads to a standard deviation of approximately 38.65%, which matches option C.

Compute the Standard Deviation Given These Four Economic States, Their