Chapter 8, Problem 16P

Summary Introduction

To determine: Expected return on portfolio and the risk of the portfolio.

Explanation of Solution

Calculation of expected return on security A:

$\begin{array}{c}\text{E}\left({\text{R}}_{\text{A}}\right)=\text{Probability}\times \text{expected}\text{return}\\ =\left(0.25\right)\times 0.10+\left(0.50\right)\times 0.12+\left(0.25\right)\times 0.14\\ =12\%\end{array}$

Therefore, expected return on security A is 12%

Calculation of expected return on security B:

$\begin{array}{c}\text{E}\left({\text{R}}_{\text{B}}\right)=\text{Probability}\times \text{expected}\text{return}\\ =\left(0.30\right)\times 0.13+\left(0.35\right)\times 0.16+\left(0.35\right)\times 0.19\\ =16.15\%\end{array}$

Therefore, expected return on security B is 16.15%

Calculation of expected return on security C:

$\begin{array}{c}\text{E}\left({\text{R}}_{\text{C}}\right)=\text{Probability}\times \text{expected}\text{return}\\ =\left(0.4\right)\times 0.14+\left(0.3\right)\times 0.18+\left(0.3\right)\times 0.22\\ =17.6\%\end{array}$

Therefore, expected return on security C is 12%

Calculation of expected return on portfolio:

$\begin{array}{c}\text{E}\left({\text{R}}_{\text{P}}\right)=\text{Probability}\times \text{expected}\text{return}\\ =\left(0.4\right)\times 0.12+\left(0.3\right)\times 0.1615+\left(0.3\right)\times \mathrm{0.17.6}\\ =14.93\%\end{array}$

Therefore, expected return on portfolio is 14.93%

Calculation of standard deviation of stock A, B and C

$\begin{array}{c}\text{Standard deviation of A}=\sqrt{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{w}}_{\text{1}}{\left(\text{x}-\widehat{\text{x}}\right)}^2}}\\ ={\left[{\left(10-12\right)}^2\times 0.25+{\left(12-12\right)}^2\times 0.5+{\left(14-12\right)}^2\times 0.25\right]}^{0.5}\\ =1.41\%\end{array}$

$\begin{array}{c}\text{Standard deviation of B}=\sqrt{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{w}}_{\text{1}}{\left(\text{x}-\widehat{\text{x}}\right)}^2}}\\ ={\left[{\left(13-16\right)}^2\times 0.3+{\left(16-16\right)}^2\times 0.35+{\left(19-16\right)}^2\times 0.35\right]}^{0.5}\\ =2.41\%\end{array}$

$\begin{array}{c}\text{Standard deviation of C}=\sqrt{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{w}}_{\text{1}}{\left(\text{x}-\widehat{\text{x}}\right)}^2}}\\ ={\left[{\left(14-18\right)}^2\times 0.40+{\left(18-18\right)}^2\times 0.30+{\left(22-18\right)}^2\times 0.30\right]}^{0.5}\\ =3.32\%\end{array}$

Calculation of standard deviation of portfolio:

$\begin{array}{c}{\text{σ}}_{\text{p}}=\left[\begin{array}{l}{\left(w_1\right)}^2\times {\left({\sigma }_1\right)}^2+{\left(w_2\right)}^2\times {\left({\sigma }_2\right)}^2+{\left(w_3\right)}^2\times {\left({\sigma }_3\right)}^2\\ +2\times \left(w_1\right)\times \left(w_2\right)\times \left(w_3\right)\times \left(p_1\right)\times \left({\sigma }_1\right)\times \left({\sigma }_2\right)\times \left({\sigma }_3\right)\end{array}\right]\\ ={\left[\begin{array}{l}{\left(0.40\right)}^2\times {\left(1.41\%\right)}^2+{\left(0.30\right)}^2\times {\left(2.41\%\right)}^2+{\left(0.30\right)}^2\times {\left(3.32\%\right)}^2\\ +2\times \left(0.4\right)\times \left(0.3\right)\times \left(0.7\right)\times \left(1.41\%\right)\times \left(2.41\%\right)\\ +2\times \left(0.4\right)\times \left(0.3\right)\times \left(0.6\right)\times \left(1.41\%\right)\times \left(3.32\%\right)\\ 2\times \left(0.3\right)\times \left(0.3\right)\times \left(0.85\right)\times \left(2.41\%\right)\times \left(3.32\%\right)\end{array}\right]}^{0.50}\\ =2.07\%\end{array}$

Therefore, standard deviation of the portfolio is 2.07%