Chapter 11, Problem 37QP

a.

Summary Introduction

To determine: Choosing between the two stocks.

Introduction: Expected Return is a process of estimating the profits and losses an investor earns through the expected rate of returns. Standard deviation is apportioned of distribution of a collection of figures from its mean.

Answer to Problem 37QP

Solution: Stock B should be chosen.

Explanation of Solution

Determine the Stock Return for each state of economy

Generally a risk-averse investor expects greater return and less risks. A risk-averse investor who holds a portfolio that is well-diversified, beta is the fitting measure of the risk of an individual security. To choose between the two stocks we need to calculate the beta and expected return of the two stocks. As Stock A does not pay any dividend and the return on Stock A is,

$\begin{array}{c}\mathrm{Re}tur{n}_{\mathrm{Re}cession}=\left[\frac{\left(\mathrm{Pr}iceNextYea{r}_{StockA}-CurrentStock\mathrm{Pr}ic{e}_{StockA}\right)}{CurrentStock\mathrm{Pr}ic{e}_{StockA}}\right]\\ =\left[\frac{\$64-\$75}{\$75}\right]\\ =\left[\frac{-\$11}{\$75}\right]\\ =-0.146667or-14.67\%\end{array}$

$\begin{array}{c}\mathrm{Re}tur{n}_{Normal}=\left[\frac{\left(\mathrm{Pr}iceNextYea{r}_{StockA}-CurrentStock\mathrm{Pr}ic{e}_{StockA}\right)}{CurrentStock\mathrm{Pr}ic{e}_{StockA}}\right]\\ =\left[\frac{\$87-\$75}{\$75}\right]\\ =\left[\frac{\$12}{\$75}\right]\\ =0.16or16\%\end{array}$

$\begin{array}{c}\mathrm{Re}tur{n}_{Expansion}=\left[\frac{\left(\mathrm{Pr}iceNextYea{r}_{StockA}-CurrentStock\mathrm{Pr}ic{e}_{StockA}\right)}{CurrentStock\mathrm{Pr}ic{e}_{StockA}}\right]\\ =\left[\frac{\$97-\$75}{\$75}\right]\\ =\left[\frac{\$22}{\$75}\right]\\ =0.293333or29.33\%\end{array}$

Therefore the Stock Return for Recession is -14.70%, Normal is 16% and Expansion is 29.30%

Determine the Expected Return for Stock A

$\begin{array}{c}Expected\mathrm{Re}turn{\left(E_{Rp}\right)}_{StockA}=\left[\begin{array}{l}\left(\mathrm{Pr}obabilit{y}_{\mathrm{Re}cession}\times \mathrm{Re}tur{n}_{StockA}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Normal}\times \mathrm{Re}tur{n}_{StockA}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Expansion}\times \mathrm{Re}tur{n}_{StockA}\right)\end{array}\right]\\ =\left[\left(0.20\times \left(-0.147\right)\right)+\left(0.60\times 0.16\right)+\left(0.20\times 0.293\right)\right]\\ =\left[\left(-0.0294\right)+0.096+0.0586\right]\\ =0.12526or12.53\%\end{array}$

Therefore the Expected Return for Stock A is 12.53%

Determine the Variance for Stock A

$\begin{array}{c}Variance{\left({\sigma }^2\right)}_{StockA}=\left[\begin{array}{l}\left(StateofEconom{y}_{Bust}\times {\left(\mathrm{Re}tur{n}_{Bust}-{\left(E_R\right)}_{StockA}\right)}^2\right)+\\ \left(StateofEconom{y}_{Normal}\times {\left(\mathrm{Re}tur{n}_{Normal}-{\left(E_R\right)}_{StockA}\right)}^2\right)+\\ \left(StateofEconom{y}_{Boom}\times {\left(\mathrm{Re}tur{n}_{Boom}-{\left(E_R\right)}_{StockA}\right)}^2\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(20\%\times {\left(-0.1470-0.1253\right)}^2\right)+\left(60\%\times {\left(0.16-0.1253\right)}^2\right)+\\ \left(20\%\times {\left(0.293-0.1253\right)}^2\right)\end{array}\right]\\ =\left[\left(0.20\times 0.073944\right)+\left(0.60\times 0.001207\right)+\left(0.20\times 0.028249\right)\right]\\ =\left[0.014789+0.000724+0.00565\right]\\ =0.021163\end{array}$

Therefore the Variance for Stock A is 0.021163

Determine the Standard Deviation for Stock A

$\begin{array}{c}S\tan dardDeviation\left({\sigma }_{StockA}\right)=\sqrt{Variance{\left({\sigma }^2\right)}_{StockA}}\\ =\sqrt{0.021163}\\ =0.145474or14.55\%\end{array}$

Therefore the Standard Deviation for Stock A is 14.55%

Determine the Beta for Stock A

$\begin{array}{c}Beta{\left(\beta \right)}_{StockA}=\left[\frac{Correlation{\left(\rho \right)}_{\left(StockA\right)}\times S\tan dardDeviation{\left(\sigma \right)}_{StockA}}{S\tan dardDeviation\left({\sigma }_P\right)}\right]\\ =\left[\frac{0.70\times 0.1455}{0.18}\right]\\ =\left[\frac{\text{0}\text{.10185}}{0.18}\right]\\ =\text{0}\text{.56583}\text{or}\text{0}\text{.57}\end{array}$

Therefore the Beta for Stock A is 0.57

Determine the Beta for Stock B

$\begin{array}{c}Beta{\left(\beta \right)}_{StockB}=\left[\frac{Correlation{\left(\rho \right)}_{\left(StockB\right)}\times S\tan dardDeviation{\left(\sigma \right)}_{StockB}}{S\tan dardDeviation\left({\sigma }_P\right)}\right]\\ =\left[\frac{0.24\times 0.34}{0.18}\right]\\ =\left[\frac{0.0816}{0.18}\right]\\ =0.453333\text{or}\text{0}\text{.45}\end{array}$

Therefore the Beta for Stock A is 0.45

Result:

Stock A has an expected return lower than Stock B. A stock’s beta is determined using the beta of the stock. Stock A’s beta is higher than Stock B. A risk-averse investor who holds a well-diversified portfolio should chose Stock B because in this condition it explains that at least one of the stocks is not priced correctly due to greater beta or greater risk. The stock has a low return than the low risk of the stock.

b.

Summary Introduction

To determine: The Expected Return and Standard Deviation of Portfolio.

Answer to Problem 37QP

Solution: The Expected Return is 12.97% and Standard Deviation of Portfolio is 16.81%.

Explanation of Solution

Determine the Expected Return on Portfolio

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_{rp}\right)=\left[\left(Weigh{t}_{StockA}\times \mathrm{Re}tur{n}_{StockA}\right)+\left(Weigh{t}_{StockB}\times \mathrm{Re}tur{n}_{StockB}\right)\right]\\ =\left[\left(70\%\times 12.53\%\right)+\left(30\%\times 14\%\right)\right]\\ =\left[0.08771+0.042\right]\\ =0.12971or12.97\%\end{array}$

Therefore the Expected Return on Portfolio is 12.97%

Determine the Variance on Portfolio

$\begin{array}{c}Variance\left({\sigma }_P^2\right)=\left[\begin{array}{l}{\left(Weigh{t}_{StockA}\right)}^2\times {\left({\sigma }_{StockA}\right)}^2+{\left(Weigh{t}_{StockB}\right)}^2\times {\left({\sigma }_{StockB}\right)}^2+\\ \left(2\times Weigh{t}_{StockA}\times {\sigma }_{StockA}\times Weigh{t}_{StockB}\times {\sigma }_{StockB}\times {\rho }_{\left(A,B\right)}\right)\end{array}\right]\\ =\left[\begin{array}{l}{\left(70\%\right)}^2\times {\left(14.55\%\right)}^2+{\left(30\%\right)}^2\times {\left(34\%\right)}^2+\\ \left(2\times 0.70\times 0.1455\times 0.30\times 0.34\times 0.36\right)\end{array}\right]\\ =\left[\left(0.49\right)\times \left(0.02117\right)+\left(0.09\right)\times \left(0.1156\right)+0.010373\right]\\ =\left[0.010373+0.010404+0.00748\right]\\ =0.028257\end{array}$

Therefore the Variance on Portfolio is 0.028257

Determine the Standard Deviation on Portfolio

$\begin{array}{c}S\tan dardDeviation\left({\sigma }_P\right)=\sqrt{Variance\left({\sigma }_P^2\right)}\\ =\sqrt{0.028257}\\ =0.168099or16.81\%\end{array}$

Therefore the Standard Deviation is 16.81%.

c.

Summary Introduction

To determine: The Beta of Portfolio.

Answer to Problem 37QP

Solution: The Beta of Portfolio is 0.532.

Explanation of Solution

Determine the Beta of Portfolio

$\begin{array}{c}PortfolioBeta\left({\beta }_P\right)=\left[\left(Weigh{t}_{StockA}\times Bet{a}_{StockA}\right)+\left(Weigh{t}_{StockB}\times Bet{a}_{StockB}\right)\right]\\ =\left[\left(70\%\times \text{0}\text{.56583}\right)+\left(30\%\times 0.453333\right)\right]\\ =\left[0.396081+0.136\right]\\ =0.532081or0.532\end{array}$

Therefore the Beta of Portfolio is 0.532