Compute the standard deviation given these four economic states, their likelihoods, and the potential returns:
$$\begin{array} { l c r }
\text { Economic State } & \text { Probability } & \text { Retum } \
\text { Fast Growth } & 0.35 & 40 \% \
\text { Slow Growth } & 0.45 & 10 \% \
\text { Recession } & 0.10 & - 10 \% \
\text { Depression } & 0.10 & - 100 \%
\end{array}$$
A) 7.5 percent
B) 12.65 percent
C) 39.48 percent
D) 113.69 percent

Question

Compute the standard deviation given these four economic states, their likelihoods, and the potential returns:
$$\begin{array} { l c r } 
\text { Economic State } & \text { Probability } & \text { Retum } \
\text { Fast Growth } & 0.35 & 40 \% \
\text { Slow Growth } & 0.45 & 10 \% \
\text { Recession } & 0.10 & - 10 \% \
\text { Depression } & 0.10 & - 100 \%
\end{array}$$
A) 7.5 percent
B) 12.65 percent
C) 39.48 percent
D) 113.69 percent

Quizplus · Accepted Answer

To compute the standard deviation of the returns, we first calculate the expected return, then use it to find the variance, and finally take the square root of the variance to get the standard deviation.1. Calculate the expected return (ER):$$ER = (0.35 \times 40\%) + (0.45 \times 10\%) + (0.10 \times -10\%) + (0.10 \times -100\%) = 14\% + 4.5\% - 1\% - 10\% = 7.5\%$$2. Calculate each state's squared deviation from the expected return:- Fast Growth: $(40\% - 7.5\%)^2 = 1056.25$- Slow Growth: $(10\% - 7.5\%)^2 = 6.25$- Recession: $(-10\% - 7.5\%)^2 = 306.25$- Depression: $(-100\% - 7.5\%)^2 = 11656.25$3. Calculate the expected variance:$$Variance = (0.35 \times 1056.25) + (0.45 \times 6.25) + (0.10 \times 306.25) + (0.10 \times 11656.25) = 370.1875 + 2.8125 + 30.625 + 1165.625 = 1569.25$$4. Calculate the standard deviation:$$Standard\ Deviation = \sqrt{Variance} = \sqrt{1569.25} \approx 39.61\%$$The correct answer is not listed exactly as calculated (39.61%), but the closest provided option is C) 39.48 percent, which seems to be a minor discrepancy likely due to rounding in the calculation process or a slight error in the provided options.

Compute the Standard Deviation Given These Four Economic States, Their