Chapter 11, Problem 33QP

Systematic versus Unsystematic Risk Consider the following information about Stocks I and II:

The market risk premium is 7.5 percent, and the risk-free rate is 4 percent. Which stock has the most systematic risk? Which one has the most unsystematic risk? Which stock is “riskier”? Explain.

Summary Introduction

To determine: The Stock which has more systematic risk, unsystematic risk and stock which is riskier.

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.

Answer to Problem 33QP

Stock I have more systematic risk, Stock II has more unsystematic risk and Stock I is riskier.

Explanation of Solution

Determine the Expected Return for Stock I

$\begin{array}{c}Expected\mathrm{Re}turn{\left(E_r\right)}_{StockI}=\left[\begin{array}{l}\left(\mathrm{Pr}obabilit{y}_{\mathrm{Re}cession}\times \mathrm{Re}tur{n}_{StockI}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Normal}\times \mathrm{Re}tur{n}_{StockI}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{IrrationalExuberance}\times \mathrm{Re}tur{n}_{StockI}\right)\end{array}\right]\\ =\left[\left(15\%\times 0.11\right)+\left(55\%\times 0.18\right)+\left(30\%\times 0.08\right)\right]\\ =\left[0.0165+0.099+0.024\right]\\ =0.1395or13.95\%\end{array}$

Therefore the Expected Return for Stock I is 13.95%

Determine the Beta for Stock I

Using CAPM formula we calculate the beta for Stock I as,

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_r\right)=\left[RiskfreeRate\left(R_f\right)+Beta\left(\beta \right)\times MarketRisk\mathrm{Pr}emium\left(R_m\right)\right]\\ 13.95\%=\left[4\%+\beta \times 7.5\%\right]\\ 13.95\%-4\%=0.075\beta \\ \beta =\left[\frac{0.095}{0.075}\right]\\ =1.32667or1.33\end{array}$

Therefore the Beta for Stock I is 1.33

Determine the Variance for Stock I

$\begin{array}{c}Variance{\left({\sigma }^2\right)}_{StockI}=\left[\begin{array}{l}\left(StateofEconom{y}_{\mathrm{Re}cession}\times {\left(\mathrm{Re}tur{n}_{StockI}-{\left(E_r\right)}_{StockI}\right)}^2\right)+\\ \left(StateofEconom{y}_{Normal}\times {\left(\mathrm{Re}tur{n}_{StockI}-{\left(E_r\right)}_{StockI}\right)}^2\right)+\\ \left(StateofEconom{y}_{IrrationalExuberance}\times {\left(\mathrm{Re}tur{n}_{StockI}-{\left(E_r\right)}_{StockI}\right)}^2\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(15\%\times {\left(0.11-0.1395\right)}^2\right)+\left(55\%\times {\left(0.18-0.1395\right)}^2\right)+\\ \left(30\%\times {\left(0.08-0.1395\right)}^2\right)\end{array}\right]\\ =\left[\left(0.15\times 0.00087\right)+\left(0.55\times 0.00164\right)+\left(0.30\times 0.00354\right)\right]\\ =\left[0.000131+0.000902+0.001062\right]\\ =0.002095\end{array}$

Therefore the Variance for Stock I is 0.002095

Determine the Standard Deviation for Stock I

$\begin{array}{c}S\tan dardDeviation\left(\sigma \right)=\sqrt{Variance{\left({\sigma }^2\right)}_{StockI}}\\ =\sqrt{0.002095}\\ =0.04576or4.58\%\end{array}$

Therefore the Standard Deviation for Stock I is 4.58%

Determine the Expected Return for Stock II

$\begin{array}{c}Expected\mathrm{Re}turn{\left(E_r\right)}_{StockII}=\left[\begin{array}{l}\left(\mathrm{Pr}obabilit{y}_{\mathrm{Re}cession}\times \mathrm{Re}tur{n}_{StockII}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{Normal}\times \mathrm{Re}tur{n}_{StockII}\right)+\\ \left(\mathrm{Pr}obabilit{y}_{IrrationalExuberance}\times \mathrm{Re}tur{n}_{StockII}\right)\end{array}\right]\\ =\left[\left(0.15\times \left(-0.25\right)\right)+\left(0.55\times 0.11\right)+\left(0.30\times 0.31\right)\right]\\ =\left[\left(-0.0375\right)+0.0605+0.093\right]\\ =0.116or11.60\%\end{array}$

Therefore the Expected Return for Stock II is 11.60%

Determine the Beta for Stock II

Using CAPM formula we calculate the beta for Stock II as,

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_r\right)=\left[RiskfreeRate\left(R_f\right)+Beta\left(\beta \right)\times MarketRisk\mathrm{Pr}emium\left(R_m\right)\right]\\ 11.60\%=\left[4\%+\beta \times 7.5\%\right]\\ 11.60\%-4\%=0.075\beta \\ \beta =\left[\frac{0.076}{0.075}\right]\\ =1.0133or1.01\end{array}$

Therefore the Beta for Stock II is 1.01

Determine the Variance for Stock II

$\begin{array}{c}Variance{\left({\sigma }^2\right)}_{StockII}=\left[\begin{array}{l}\left(StateofEconom{y}_{\mathrm{Re}cession}\times {\left(\mathrm{Re}tur{n}_{StockI}-{\left(E_r\right)}_{StockII}\right)}^2\right)+\\ \left(StateofEconom{y}_{Normal}\times {\left(\mathrm{Re}tur{n}_{StockI}-{\left(E_r\right)}_{StockII}\right)}^2\right)+\\ \left(StateofEconom{y}_{IrrationalExuberance}\times {\left(\mathrm{Re}tur{n}_{StockI}-{\left(E_r\right)}_{StockII}\right)}^2\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(15\%\times {\left(-0.25-0.1160\right)}^2\right)+\left(55\%\times {\left(0.11-0.1160\right)}^2\right)+\\ \left(30\%\times {\left(0.31-0.1160\right)}^2\right)\end{array}\right]\\ =\left[\left(0.15\times 0.133956\right)+\left(0.55\times 0.000036\right)+\left(0.30\times 0.037636\right)\right]\\ =\left[0.020093+0.0000198+0.011291\right]\\ =0.031404\end{array}$

Therefore the Variance for Stock II is 0.031404

Determine the Standard Deviation for Stock II

$\begin{array}{c}S\tan dardDeviation\left(\sigma \right)=\sqrt{Variance{\left({\sigma }^2\right)}_{StockII}}\\ =\sqrt{0.031404}\\ =0.17721or17.72\%\end{array}$

Therefore the Standard Deviation for Stock II is 17.72%

Conclusion

• Stock I have less systematic risk than Stock II since the beta of Stock II is lesser than Stock I and Stock II is said to have more total risk.
• Hence Stock I have more systematic risk and Stock II have more unsystematic risk and greater total risk.
• Stock I is riskier because unsystematic risk can be diversified due to lack of volatility in the stock return.
• Additionally Stock I have higher expected return and higher risk premium.