Your firm's analyst believes that economic conditions during the next year will be either strong,normal,or weak,and she thinks that Crary Inc.'s returns will have the probability distribution shown below.What's the standard deviation of Crary's returns as estimated by your analyst? (Hint: Use the formula for the standard deviation of a population,not a sample.) $30%−15.75%\begin{array}{l}\text { Economic }\\\begin{array}{lrr}{\text { Conditions }} & \text { Prob. } &{\text { Return }} \\\hline \text { Strong } & 30 \% & 32.50 \% \\\text { Normal } & 40 \% & 10.25 \% \\\text { Weak } & 30 \% & -15.75 \%\end{array}\end{array}$

A) 17.77%
B) 18.71%
C) 19.65%
D) 20.63%

Question

Your firm's analyst believes that economic conditions during the next year will be either strong,normal,or weak,and she thinks that Crary Inc.'s returns will have the probability distribution shown below.What's the standard deviation of Crary's returns as estimated by your analyst? (Hint: Use the formula for the standard deviation of a population,not a sample.)

30%−15.75%\begin{array}{l}\text { Economic }\\\begin{array}{lrr}{\text { Conditions }} & \text { Prob. } &{\text { Return }} \\\hline \text { Strong } & 30 \% & 32.50 \% \\\text { Normal } & 40 \% & 10.25 \% \\\text { Weak } & 30 \% & -15.75 \%\end{array}\end{array}

A) 17.77%
B) 18.71%
C) 19.65%
D) 20.63%

Ivory Davies · Accepted Answer

Final Answer : B, Explanation :   To calculate the standard deviation of Crary's returns, we first need to find the expected return (mean) and then use it to calculate the variance before taking the square root to get the standard deviation.1. Calculate the expected return (mean):  E ( R ) = ( 0.30 × 32.50 % ) + ( 0.40 × 10.25 % ) + ( 0.30 × − 15.75 % ) = 9.75 % + 4.10 % − 4.725 % = 9.125 % E(R) = (0.30 	imes 32.50\%) + (0.40 	imes 10.25\%) + (0.30 	imes -15.75\%) = 9.75\% + 4.10\% - 4.725\% = 9.125\% E ( R ) = ( 0.30 × 32.50% ) + ( 0.40 × 10.25% ) + ( 0.30 × − 15.75% ) = 9.75% + 4.10% − 4.725% = 9.125%  2. Calculate the variance:  V a r ( R ) = ( 0.30 × ( 32.50 % − 9.125 % ) 2 ) + ( 0.40 × ( 10.25 % − 9.125 % ) 2 ) + ( 0.30 × ( − 15.75 % − 9.125 % ) 2 ) Var(R) = (0.30 	imes (32.50\% - 9.125\%)^2) + (0.40 	imes (10.25\% - 9.125\%)^2) + (0.30 	imes (-15.75\% - 9.125\%)^2) Va r ( R ) = ( 0.30 × ( 32.50% − 9.125% ) 2 ) + ( 0.40 × ( 10.25% − 9.125% ) 2 ) + ( 0.30 × ( − 15.75% − 9.125% ) 2 ) V a r ( R ) = ( 0.30 × ( 23.375 % ) 2 ) + ( 0.40 × ( 1.125 % ) 2 ) + ( 0.30 × ( − 24.875 % ) 2 ) Var(R) = (0.30 	imes (23.375\%)^2) + (0.40 	imes (1.125\%)^2) + (0.30 	imes (-24.875\%)^2) Va r ( R ) = ( 0.30 × ( 23.375% ) 2 ) + ( 0.40 × ( 1.125% ) 2 ) + ( 0.30 × ( − 24.875% ) 2 ) V a r ( R ) = ( 0.30 × 0.0547 ) + ( 0.40 × 0.0013 ) + ( 0.30 × 0.0619 ) Var(R) = (0.30 	imes 0.0547) + (0.40 	imes 0.0013) + (0.30 	imes 0.0619) Va r ( R ) = ( 0.30 × 0.0547 ) + ( 0.40 × 0.0013 ) + ( 0.30 × 0.0619 ) V a r ( R ) = 0.01641 + 0.00052 + 0.01857 = 0.0355 Var(R) = 0.01641 + 0.00052 + 0.01857 = 0.0355 Va r ( R ) = 0.01641 + 0.00052 + 0.01857 = 0.0355  3. Calculate the standard deviation:  S D ( R ) = V a r ( R ) = 0.0355 = 0.1884  or  18.84 % SD(R) = \sqrt{Var(R)} = \sqrt{0.0355} = 0.1884 	ext{ or } 18.84\% S D ( R ) = Va r ( R ) ​ = 0.0355 ​ = 0.1884  or  18.84%  The closest answer provided is 18.71%, which is option B.

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