You have been given this probability distribution for the holding-period return for a stock: $0.25−9%\begin{array}{lcc} \text { Stock of the Economy } & \text { Probability} & \text {HPR } \\ \text { Boom } &0.40&22\%\\ \text { Normal growth } &0.35&11\%\\\text { Recession }&0.25&-9\%\end{array}$

What is the expected standard deviation for the stock?

A) 2.07%
B) 9.96%
C) 7.04%
D) 1.44%
E) None of the options are correct.

Question

You have been given this probability distribution for the holding-period return for a stock:    Stock of the Economy   Probability HPR   Boom  0.40 22 %  Normal growth  0.35 11 %  Recession  0.25 − 9 % \begin{array}{lcc}  	ext { Stock of the Economy }  &  	ext {  Probability}  &  	ext {HPR  } \ 	ext { Boom }  & 0.40 & 22\%\ 	ext { Normal growth }  & 0.35 & 11\%\	ext { Recession } & 0.25 & -9\%\end{array}  Stock of the Economy   Boom   Normal growth   Recession  ​  Probability 0.40 0.35 0.25 ​ HPR  22% 11% − 9% ​ What is the expected standard deviation for the stock?A)  2.07%B)  9.96%C)  7.04%D)  1.44%E)  None of the options are correct.

Zybrea Knight · Accepted Answer

Final Answer : E, Explanation :   First, calculate the expected return (mean) using the given probabilities and returns:  E ( R ) = ( 0.40 × 22 % ) + ( 0.35 × 11 % ) + ( 0.25 × ( − 9 % ) ) = 8.8 % + 3.85 % − 2.25 % = 10.4 % E(R) = (0.40 	imes 22\%) + (0.35 	imes 11\%) + (0.25 	imes (-9\%)) = 8.8\% + 3.85\% - 2.25\% = 10.4\% E ( R ) = ( 0.40 × 22% ) + ( 0.35 × 11% ) + ( 0.25 × ( − 9% )) = 8.8% + 3.85% − 2.25% = 10.4%  Then, calculate the variance using the formula for variance of expected returns:  V a r ( R ) = ∑ [ P ( i ) × ( R ( i ) − E ( R ) ) 2 ] Var(R) = \sum [P(i) 	imes (R(i) - E(R))^2] Va r ( R ) = ∑ [ P ( i ) × ( R ( i ) − E ( R ) ) 2 ] V a r ( R ) = ( 0.40 × ( 22 % − 10.4 % ) 2 ) + ( 0.35 × ( 11 % − 10.4 % ) 2 ) + ( 0.25 × ( − 9 % − 10.4 % ) 2 ) Var(R) = (0.40 	imes (22\% - 10.4\%)^2) + (0.35 	imes (11\% - 10.4\%)^2) + (0.25 	imes (-9\% - 10.4\%)^2) Va r ( R ) = ( 0.40 × ( 22% − 10.4% ) 2 ) + ( 0.35 × ( 11% − 10.4% ) 2 ) + ( 0.25 × ( − 9% − 10.4% ) 2 ) V a r ( R ) = ( 0.40 × 0.0133 ) + ( 0.35 × 0.00036 ) + ( 0.25 × 0.0416 ) Var(R) = (0.40 	imes 0.0133) + (0.35 	imes 0.00036) + (0.25 	imes 0.0416) Va r ( R ) = ( 0.40 × 0.0133 ) + ( 0.35 × 0.00036 ) + ( 0.25 × 0.0416 ) V a r ( R ) = 0.00532 + 0.000126 + 0.0104 = 0.015846 Var(R) = 0.00532 + 0.000126 + 0.0104 = 0.015846 Va r ( R ) = 0.00532 + 0.000126 + 0.0104 = 0.015846  Finally, calculate the standard deviation, which is the square root of the variance:  S D ( R ) = V a r ( R ) = 0.015846 ≈ 12.59 % SD(R) = \sqrt{Var(R)} = \sqrt{0.015846} \approx 12.59\% S D ( R ) = Va r ( R ) ​ = 0.015846 ​ ≈ 12.59%  Since none of the provided options match the calculated standard deviation, the correct choice is:E

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