Consider the following probability distribution for stocks A and B: $10.158%8%20.2013%7%30.1512%6%40.3014%9%50.2016%11%\begin{array}{cccc}\text { State } & \text { Probability } & \text { Return on Stock A } & \text { Return on Stock B } \\1 & 0.15 & 8 \% & 8 \% \\2 & 0.20 & 13\% & 7\% \\3 & 0.15 & 12\%& 6\% \\4 & 0.30 & 14\%& 9\% \\5 & 0.20 & 16\% & 11 \%\end{array}$
The standard deviations of stocks A and B are _ and _, respectively.

A) 1.56%; 1.99%
B) 2.45%; 1.66%
C) 3.22%; 2.01%
D) 1.54%; 1.11%

Question

Consider the following probability distribution for stocks A and B:

10.158%8%20.2013%7%30.1512%6%40.3014%9%50.2016%11%\begin{array}{cccc}\text { State } & \text { Probability } & \text { Return on Stock A } & \text { Return on Stock B } \\1 & 0.15 & 8 \% & 8 \% \\2 & 0.20 & 13\% & 7\% \\3 & 0.15 & 12\%& 6\% \\4 & 0.30 & 14\%& 9\% \\5 & 0.20 & 16\% & 11 \%\end{array}

The standard deviations of stocks A and B are _____ and _____, respectively.

A) 1.56%; 1.99%
B) 2.45%; 1.66%
C) 3.22%; 2.01%
D) 1.54%; 1.11%

Destiny Scott · Accepted Answer

Final Answer : B, Explanation :   To calculate the standard deviation of the returns for stocks A and B, we first need to find the expected return (mean) for each stock, then calculate the variance, and finally take the square root of the variance to get the standard deviation.**For Stock A:**- Expected Return =   0.15 × 8 % + 0.20 × 13 % + 0.15 × 12 % + 0.30 × 14 % + 0.20 × 16 % = 12.7 % 0.15 	imes 8\% + 0.20 	imes 13\% + 0.15 	imes 12\% + 0.30 	imes 14\% + 0.20 	imes 16\% = 12.7\% 0.15 × 8% + 0.20 × 13% + 0.15 × 12% + 0.30 × 14% + 0.20 × 16% = 12.7%  - Variance =   0.15 × ( 8 % − 12.7 % ) 2 + 0.20 × ( 13 % − 12.7 % ) 2 + 0.15 × ( 12 % − 12.7 % ) 2 + 0.30 × ( 14 % − 12.7 % ) 2 + 0.20 × ( 16 % − 12.7 % ) 2 = 6.01 % 0.15 	imes (8\% - 12.7\%)^2 + 0.20 	imes (13\% - 12.7\%)^2 + 0.15 	imes (12\% - 12.7\%)^2 + 0.30 	imes (14\% - 12.7\%)^2 + 0.20 	imes (16\% - 12.7\%)^2 = 6.01\% 0.15 × ( 8% − 12.7% ) 2 + 0.20 × ( 13% − 12.7% ) 2 + 0.15 × ( 12% − 12.7% ) 2 + 0.30 × ( 14% − 12.7% ) 2 + 0.20 × ( 16% − 12.7% ) 2 = 6.01%  - Standard Deviation =   6.01 % ≈ 2.45 % \sqrt{6.01\%} \approx 2.45\% 6.01% ​ ≈ 2.45%  **For Stock B:**- Expected Return =   0.15 × 8 % + 0.20 × 7 % + 0.15 × 6 % + 0.30 × 9 % + 0.20 × 11 % = 8.35 % 0.15 	imes 8\% + 0.20 	imes 7\% + 0.15 	imes 6\% + 0.30 	imes 9\% + 0.20 	imes 11\% = 8.35\% 0.15 × 8% + 0.20 × 7% + 0.15 × 6% + 0.30 × 9% + 0.20 × 11% = 8.35%  - Variance =   0.15 × ( 8 % − 8.35 % ) 2 + 0.20 × ( 7 % − 8.35 % ) 2 + 0.15 × ( 6 % − 8.35 % ) 2 + 0.30 × ( 9 % − 8.35 % ) 2 + 0.20 × ( 11 % − 8.35 % ) 2 = 2.76 % 0.15 	imes (8\% - 8.35\%)^2 + 0.20 	imes (7\% - 8.35\%)^2 + 0.15 	imes (6\% - 8.35\%)^2 + 0.30 	imes (9\% - 8.35\%)^2 + 0.20 	imes (11\% - 8.35\%)^2 = 2.76\% 0.15 × ( 8% − 8.35% ) 2 + 0.20 × ( 7% − 8.35% ) 2 + 0.15 × ( 6% − 8.35% ) 2 + 0.30 × ( 9% − 8.35% ) 2 + 0.20 × ( 11% − 8.35% ) 2 = 2.76%  - Standard Deviation =   2.76 % ≈ 1.66 % \sqrt{2.76\%} \approx 1.66\% 2.76% ​ ≈ 1.66%  Therefore, the correct answer is B) 2.45%; 1.66%.

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