Chapter 12, Problem 6QP

Market Model The following three stocks are available in the market:

	E(R)	β
Stock A	10.5%	1.20
Stock B	13.0	.98
Stock C	15.7	1.37
Market	14 .2	1.00

Assume the market model is valid.

1. a. Write the market model equation for each stock
2. b. What is the return on a portfolio with weights of 30 percent Stock A, 45 percent Stock B, and 25 percent Stock C?
3. c. Suppose the return on the market is 15 percent and there are no unsystematic surprises in the returns. What is the return on each stock? What is the return on the portfolio?

Summary Introduction

To determine: The Market Model equation for each stock.

Introduction:

Systematic Risk is acknowledged as non diversifiable risks or market risk. Such category of risk is not intended to be separated by distinguishing assets. Systematic risk leads on how a particular investment in a distinguished portfolio that support financially to the total or aggregate risk of a business's financial funding. Unsystematic Risk is acknowledged as diversifiable or residual or particular risk. The proportion of a corporation’s total or aggregate risk which can be barred by holding such risks in a distinguished or as diversified asset portfolio.

Explanation of Solution

Determine the Market Model equation for each stock

Stock A:

$\begin{array}{c}{R}_{StockA}=\left[{\overline{R}}_{StockA}+{\beta }_{StockA}\times \left(R_m-{\overline{R}}_m\right)+{\epsilon }_{StockA}\right]\\ R_{StockA}=\left[10.5\%+1.2\times \left(R_m-14.2\%\right)+{\epsilon }_{StockA}\right]\end{array}$

Stock B:

$\begin{array}{c}{R}_{StockB}=\left[{\overline{R}}_{StockB}+{\beta }_{StockB}\times \left(R_m-{\overline{R}}_m\right)+{\epsilon }_{StockB}\right]\\ R_{StockB}=\left[13\%+0.98\times \left(R_m-14.2\%\right)+{\epsilon }_{StockB}\right]\end{array}$

Stock C:

$\begin{array}{c}{R}_{StockC}=\left[{\overline{R}}_{StockC}+{\beta }_{StockC}\times \left(R_m-{\overline{R}}_m\right)+{\epsilon }_{StockC}\right]\\ R_{StockC}=\left[15.7\%+1.37\times \left(R_m-14.2\%\right)+{\epsilon }_{StockA}\right]\end{array}$

Summary Introduction

To determine: The Return on Portfolio Equation.

Explanation of Solution

Determine the Return on Portfolio Equation

By substituting the portfolio weights of each stock with the market model equation for each stock we find the return on portfolio equation.

$\begin{array}{c}\mathrm{Re}turn\left(R_p\right)=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \left(10.5\%+1.2\times \left(R_m-14.2\%\right)+{\epsilon }_{StockA}\right)\right)+\\ \left(Weigh{t}_{StockB}\times \left(13\%+0.98\times \left(R_m-14.2\%\right)+{\epsilon }_{StockB}\right)\right)+\\ \left(Weigh{t}_{StockC}\times \left(15.7\%+1.37\times \left(R_m-14.2\%\right)+{\epsilon }_{StockC}\right)\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(30\%\times \left(10.5\%+1.2\times \left(R_m-14.2\%\right)+{\epsilon }_{StockA}\right)\right)+\\ \left(45\%\times \left(13\%+0.98\times \left(R_m-14.2\%\right)+{\epsilon }_{StockB}\right)\right)+\\ \left(25\%\times \left(15.7\%+1.37\times \left(R_m-14.2\%\right)+{\epsilon }_{StockC}\right)\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(\left(0.30\times 0.105\right)+\left(0.45\times 0.13\right)+\left(0.25\times 0.157\right)\right)+\\ \left(\left(0.30\times 1.2\right)+\left(0.45\times 0.98\right)+\left(0.25\times 1.37\right)\right)\times \left(R_m-0.142\right)+\\ \left(0.30\times {\epsilon }_{StockA}\right)+\left(0.45\times {\epsilon }_{StockB}\right)+\left(0.25\times {\epsilon }_{StockC}\right)\end{array}\right]\\ =\left[\begin{array}{l}0.12925+\left(1.1435\times \left(R_m-0.142\right)\right)+\\ \left(0.30\times {\epsilon }_{StockA}\right)+\left(0.45\times {\epsilon }_{StockB}\right)+\left(0.25\times {\epsilon }_{StockC}\right)\end{array}\right]\end{array}$

Summary Introduction

To determine: The Return on each stock and Return on Portfolio.

Explanation of Solution

Determine the Return on each stock

$\begin{array}{c}\mathrm{Re}tur{n}_{StockA}=\left[\begin{array}{l}Expected\mathrm{Re}turn{\left(E_r\right)}_{StockA}+Beta{\left(\beta \right)}_{StockA}\times \\ \left(Risk\mathrm{Pr}emium\left(R_m\right)-ExpectedMarket\mathrm{Re}turn\left(\overline{R}\right)\right)\end{array}\right]\\ =\left[10.50\%+1.20\times \left(15\%-14.2\%\right)\right]\\ =\left[0.1050+1.20\times 0.008\right]\\ =\left[10.50\%+\text{0}\text{.0096}\right]\\ =\text{0}\text{.1146}or11.46\%\end{array}$

$\begin{array}{c}\mathrm{Re}tur{n}_{StockB}=\left[\begin{array}{l}Expected\mathrm{Re}turn{\left(E_r\right)}_{StockB}+Beta{\left(\beta \right)}_{StockB}\times \\ \left(Risk\mathrm{Pr}emium\left(R_m\right)-ExpectedMarket\mathrm{Re}turn\left(\overline{R}\right)\right)\end{array}\right]\\ =\left[13\%+0.98\times \left(15\%-14.2\%\right)\right]\\ =\left[0.13+0.98\times 0.008\right]\\ =\left[0.13+\text{0}\text{.00784}\right]\\ =\text{0}\text{.13784}or13.78\%\end{array}$

$\begin{array}{c}\mathrm{Re}tur{n}_{StockC}=\left[\begin{array}{l}Expected\mathrm{Re}turn{\left(E_r\right)}_{StockC}+Beta{\left(\beta \right)}_{StockC}\times \\ \left(Risk\mathrm{Pr}emium\left(R_m\right)-ExpectedMarket\mathrm{Re}turn\left(\overline{R}\right)\right)\end{array}\right]\\ =\left[15.70\%+1.37\times \left(15\%-14.2\%\right)\right]\\ =\left[0.157+1.37\times 0.008\right]\\ =\left[0.157+\text{0}\text{.01096}\right]\\ =\text{0}\text{.16796}or16.80\%\end{array}$

Therefore the Return on Stock A is 11.46%, Stock B is 13.78% and Stock C is 16.80%.

Determine the Return on Portfolio

$\begin{array}{c}\mathrm{Re}turn\left(R_p\right)=\left[\begin{array}{l}\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)+\\ \left(Weigh{t}_{StockC}\times \mathrm{Re}tur{n}_{StockC}\right)\end{array}\right]\\ =\left[\left(30\%\times 11.46\%\right)+\left(45\%\times 13.78\%\right)+\left(25\%\times 16.80\%\right)\right]\\ =\left[0.03438+0.06201+0.042\right]\\ =0.13839or13.84\%\end{array}$

Therefore the Return on Portfolio is 13.84%.