Chapter 25, Problem 7SP

a.

Summary Introduction

Determine: Expected return and standard deviation of the portfolio invested.

Explanation of Solution

Given information:

It is given that expected return of A is 0.07, expected return of B is 0.10, expected return of C is 0.20, standard deviation of A is 0.3311, standard deviation of B is 0.5385, standard deviation of C is 0.8944, correlation coefficient between A and B is 0.35, portfolio invested in A is 30% and in B is 70%.

Formula to calculate expected return is as follows:

$\text{Expected return}=\text{Return of A}\times \text{Portfolio invested in A}+\text{Return of B}\times \text{Portfolio invested in B}$

Substituting Equation with 0.10 for the return of A and 30% for the portfolio invested in A, 0.16 for the return of B and 70% for the portfolio invested in B to calculate expected return.

$\begin{array}{c}\text{Expected return}=\text{0}\text{.10}\times 30\%+\text{0}\text{.16}\times 70\%\\ =0.03+0.112\\ =0.142\text{ or }14.2\%\end{array}$

Hence, the expected return from portfolio investment of 30% in stock A and 70% in stock B is 14.2%.

Formula to calculate standard deviation of the portfolio invested is as follows:

$\sigma =\sqrt{\sum \left({\text{w}}_{\text{a}}^2{\sigma }_{\text{a}}^2+{\text{w}}_{\text{b}}^2{\sigma }_{\text{b}}^2+2{\text{w}}_{\text{a}}\times {\text{w}}_{\text{b}}\times {\text{r}}_{\text{ab}}\times {\sigma }_a\times {\sigma }_{\text{b}}\right)}$

Substituting Equation with 0.30 for W_a, 0.20 for σ_a, 0.70 for W_b, 0.40 for σ_b to calculate standard deviation of the portfolio invested.

$\begin{array}{c}\sigma =\sqrt{\sum \left({0.30}^2\times {0.20}^2+{0.70}^2\times {0.40}^2+2\times 0.30\times 0.70\times 0.35\times 0.20\times 0.40\right)}\\ =\sqrt{\sum \left(0.13+0.65+0.01176\right)}\\ =\sqrt{0.79176}\\ =0.8898\text{ or }88.98\%\end{array}$

Hence, standard deviation of the portfolio invested is 88.98%.

b.

Summary Introduction

Determine: Expected return and standard deviation of the portfolio invested.

Explanation of Solution

Given information:

It is given that expected return of A is 0.10, expected return of B is 0.16, standard deviation of A is 0.20, standard deviation of B is 0.40, correlation coefficient between A and B is 0.35, portfolio invested in A is 30% and in B is 70%.

Formula to calculate expected return is as follows:

$\text{Expected return}=\text{Return of A}\times \text{Portfolio invested in A}+\text{Return of B}\times \text{Portfolio invested in B}$

Substituting Equation with 0.10 for the return of A and 30% for the portfolio invested in A, 0.16 for the return of B and 70% for the portfolio invested in B to calculate expected return.

$\begin{array}{c}\text{Expected return}=\text{0}\text{.10}\times 30\%+\text{0}\text{.16}\times 70\%\\ =0.03+0.112\\ =0.142\text{ or }14.2\%\end{array}$

Hence, the expected return from portfolio investment of 30% in stock A and 70% in stock B is 14.2%.

Formula to calculate standard deviation of the portfolio invested is as follows:

$\sigma =\sqrt{\sum \left({\text{w}}_{\text{a}}^2{\sigma }_{\text{a}}^2+{\text{w}}_{\text{b}}^2{\sigma }_{\text{b}}^2+2{\text{w}}_{\text{a}}\times {\text{w}}_{\text{b}}\times {\text{r}}_{\text{ab}}\times {\sigma }_a\times {\sigma }_{\text{b}}\right)}$

Substituting Equation with 0.30 for W_a, 0.20 for σ_a, 0.70 for W_b, 0.40 for σ_b to calculate standard deviation of the portfolio invested.

$\begin{array}{c}\sigma =\sqrt{\sum \left({0.30}^2\times {0.20}^2+{0.70}^2\times {0.40}^2+2\times 0.30\times 0.70\times 0.35\times 0.20\times 0.40\right)}\\ =\sqrt{\sum \left(0.13+0.65+0.01176\right)}\\ =\sqrt{0.79176}\\ =0.8898\text{ or }88.98\%\end{array}$

Hence, standard deviation of the portfolio invested is 88.98%.

c.

Summary Introduction

Determine: Expected return and standard deviation of the portfolio invested.

Explanation of Solution

Given information:

It is given that expected return of A is 0.10, expected return of B is 0.16, standard deviation of A is 0.20, standard deviation of B is 0.40, correlation coefficient between A and B is 0.35, portfolio invested in A is 30% and in B is 70%.

Formula to calculate expected return is as follows:

$\text{Expected return}=\text{Return of A}\times \text{Portfolio invested in A}+\text{Return of B}\times \text{Portfolio invested in B}$

Substituting Equation with 0.10 for the return of A and 30% for the portfolio invested in A, 0.16 for the return of B and 70% for the portfolio invested in B to calculate expected return.

$\begin{array}{c}\text{Expected return}=\text{0}\text{.10}\times 30\%+\text{0}\text{.16}\times 70\%\\ =0.03+0.112\\ =0.142\text{ or }14.2\%\end{array}$

Hence, the expected return from portfolio investment of 30% in stock A and 70% in stock B is 14.2%.

Formula to calculate standard deviation of the portfolio invested is as follows:

$\sigma =\sqrt{\sum \left({\text{w}}_{\text{a}}^2{\sigma }_{\text{a}}^2+{\text{w}}_{\text{b}}^2{\sigma }_{\text{b}}^2+2{\text{w}}_{\text{a}}\times {\text{w}}_{\text{b}}\times {\text{r}}_{\text{ab}}\times {\sigma }_a\times {\sigma }_{\text{b}}\right)}$

Substituting Equation with 0.30 for W_a, 0.20 for σ_a, 0.70 for W_b, 0.40 for σ_b to calculate standard deviation of the portfolio invested.

$\begin{array}{c}\sigma =\sqrt{\sum \left({0.30}^2\times {0.20}^2+{0.70}^2\times {0.40}^2+2\times 0.30\times 0.70\times 0.35\times 0.20\times 0.40\right)}\\ =\sqrt{\sum \left(0.13+0.65+0.01176\right)}\\ =\sqrt{0.79176}\\ =0.8898\text{ or }88.98\%\end{array}$

Hence, standard deviation of the portfolio invested is 88.98%.