Consider the following information for three stocks,A,B,and C,and portfolios of these stocks.The stocks' returns are positively but not perfectly positively correlated with one another,i.e.,the correlation coefficients are all between 0 and 1.Portfolio AB has half of its funds invested in Stock A and half in Stock B.Portfolio ABC has one-third of its funds invested in each of the three stocks.The risk-free rate is 5%,and the market is in equilibrium,so required returns equal expected returns.Which of the following statements is correct? $\begin{array}{l}
\begin{array}{lll}
&&\text { Expected }\
\underline {\text { Stock }} & & \underline {\text { Return }} \
\text { Stock A } & & 10 \% \
\text { Stock B } & &10 \
\text { Stock C } & & 12
\end{array}
\begin{array}{lll}
\text { Standard }\
\underline { \text { Deviation } }& \underline { \text { Beta } }\
20 \%&1.0 \
10 &1.0\
12&1.4
\end{array}
\end{array}$
A)Portfolio AB has a standard deviation of 20%.
B)Portfolio AB's coefficient of variation is greater than 2.0.
C)Portfolio AB's required return is greater than the required return on Stock A.
D)Portfolio ABC's expected return is 10.67%.

Question

Consider the following information for three stocks,A,B,and C,and portfolios of these stocks.The stocks' returns are positively but not perfectly positively correlated with one another,i.e.,the correlation coefficients are all between 0 and 1.Portfolio AB has half of its funds invested in Stock A and half in Stock B.Portfolio ABC has one-third of its funds invested in each of the three stocks.The risk-free rate is 5%,and the market is in equilibrium,so required returns equal expected returns.Which of the following statements is correct? $\begin{array}{l}
\begin{array}{lll}
&&\text { Expected }\
\underline {\text { Stock }} & & \underline {\text { Return }} \
\text { Stock A } & & 10 \% \
\text { Stock B } & &10 \
\text { Stock C } & & 12
\end{array}
\begin{array}{lll}
\text { Standard }\
 \underline {   \text { Deviation }  }& \underline { \text { Beta }    }\
20 \%&1.0 \
10 &1.0\
12&1.4
\end{array}
\end{array}$
A)Portfolio AB has a standard deviation of 20%.
B)Portfolio AB's coefficient of variation is greater than 2.0.
C)Portfolio AB's required return is greater than the required return on Stock A.
D)Portfolio ABC's expected return is 10.67%.

Quizplus · Accepted Answer

To calculate portfolio AB's expected return and standard deviation, we use the following formulas:

$E(R_{AB}) = w_A	imes E(R_A) + w_B 	imes E(R_B)$

$\sigma_{AB} = \sqrt{w_A^2 	imes \sigma_A^2 + w_B^2 	imes \sigma_B^2 + 2w_Aw_Bho_{A,B}\sigma_A\sigma_B}$

where $w_A = w_B = 0.5$ (since the portfolio is equally invested in A and B), $ho_{A,B}$ is the correlation coefficient between stock A and B.

Substituting the values, we get:

$E(R_{AB}) = 0.5	imes 10\% + 0.5	imes 10\% = 10\%$

$\sigma_{AB} = \sqrt{0.5^2 	imes (20\%)^2 + 0.5^2 	imes (10\%)^2 + 2	imes 0.5	imes 0.5	imes ho_{A,B}	imes 20\%	imes 10\%}$

From the given information, we know that the correlation coefficient between A and B is between 0 and 1, which means $ho_{A,B}	imes 20\%	imes 10\%$ is also between 0 and 1. So, the standard deviation of portfolio AB must be less than 20%. Thus, statement A is incorrect.

To calculate the coefficient of variation, we use the formula:

$CV_{AB} = \frac{\sigma_{AB}}{E(R_{AB})} 	imes 100\%$

Substituting the values, we get:

$CV_{AB} = \frac{\sigma_{AB}}{E(R_{AB})} 	imes 100\% = \frac{\sigma_{AB}}{10\%} 	imes 100\%$

Since $\sigma_{AB}$ is less than 20%, the coefficient of variation must be less than 2.0. Thus, statement B is incorrect.

To calculate portfolio AB's required return, we use the CAPM formula:

$E(R_{AB}) = R_f + \beta_{AB}	imes (E(R_m) - R_f)$

where $R_f$ is the risk-free rate, $\beta_{AB}$ is the beta of the portfolio, and $E(R_m)$ is the expected return on the market.

From the given information, we know the risk-free rate is 5%. The beta of portfolio AB is calculated as:

$\beta_{AB} = w_A	imes \beta_A + w_B 	imes \beta_B = 0.5	imes 1.0 + 0.5	imes 1.0 = 1.0$

Since we are given that the market is in equilibrium, required returns equal expected returns. The expected return on the market is the same as the expected return on each of the stocks, which is 10% for A and B. Thus, the required return on portfolio AB is:

$E(R_{AB}) = R_f + \beta_{AB}	imes (E(R_m) - R_f) = 5\% + 1.0	imes (10\% - 5\%) = 15\%$

From the given information, we know that the expected return on stock A is 10%. Since the required return on portfolio AB is greater than 10%, statement C is correct.

To calculate portfolio ABC's expected return, we use the formula:

$E(R_{ABC}) = w_A	imes E(R_A) + w_B 	imes E(R_B) + w_C 	imes E(R_C)$

where $w_A = w_B = w_C = 1/3$ (since the portfolio is equally invested in A, B, and C).

Substituting the values, we get:

$E(R_{ABC}) = \frac{1}{3}	imes 10\% + \frac{1}{3}	imes 10\% + \frac{1}{3}	imes 12\% = 10.67\%$

Thus, statement D is correct.

Consider the Following Information for Three Stocks,A,B,and C,and Portfolios of These