Chapter 13, Problem 26QP

Summary Introduction

To determine: The stock that has the most systematic risk and the most unsystematic risk.

Introduction:

Systematic risk refers to the market-specific risk that affects all the stocks in the market. Unsystematic risk refers to the company-specific risk that affects only the individual company.

Answer to Problem 26QP

The expected return on Stock I is 12.5 percent, the beta is 1.21, and the standard deviation is 0.0861 or 8.61%. The expected return on Stock II is 9.25 percent. The beta is 0.75 and the standard deviation is 24.39%.

Explanation of Solution

Given information:

The probability of having a recession, normal economy, and irrational exuberance is 0.25, 0.50, and 0.25 respectively. Stock I will yield 2%, 21%, and 6% when there is a recession, normal economy, and irrational exuberance respectively.

Stock II will yield (−25%), 9%, and 44% when there is a recession, normal economy, and irrational exuberance respectively. The market risk premium is 7% and the risk-free rate is 4%.

The formula to calculate the expected return on the stock:

$\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\mathrm{...}+\left(\text{Possible returns}\left(R_n\right)\times \text{Probability}\left(P_n\right)\right)\end{array}\right]$

The formula to calculate the beta of the stock:

$E(R_i)=R_f+\left[E(R_M)-R_f\right]\times {\beta }_i$

E (R_i) refers to the expected return on a risky asset

R_f refers to the risk-free rate

E (R_M) refers to the expected return on the market portfolio

β refers to the beta coefficient of the risky asset relative to the market portfolio

The formula to calculate the standard deviation:

$\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\mathrm{...}+\left[{\left(\text{Possible returns}\left(R_n\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_n\right)\right]\end{array}\right)}$

Compute the expected return on Stock I:

R₁ is the returns during the recession. The probability of having a recession is P₁. Similarly, R₂ is the returns in a normal economy. The probability of having a normal is P₂. R₃ is the returns in irrational exuberance. The probability of having an irrational exuberance is P₃.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(0.02\times 0.25\right)+\left(0.21\times 0.50\right)+\left(0.06\times 0.25\right)\\ =0.005+0.105+0.015\\ =0.125\end{array}$

Hence, the expected return on Stock I is 0.125 or 12.5 percent.

Compute the beta of Stock I:

$\begin{array}{c}E(R_I)=R_f+\left[E(R_M)-R_f\right]\times {\beta }_I\\ 0.125=0.04+\left[0.07\right]\times {\beta }_I\\ 0.125-0.04=0.07{\beta }_I\\ \text{0}\text{.085}=0.07{\beta }_I\end{array}$

$\begin{array}{c}\frac{\text{0}\text{.085}}{0.07}={\beta }_I\\ \text{1}\text{.21}={\beta }_I\end{array}$

Hence, the beta of Stock I is 1.21.

Compute the standard deviation of Stock I:

R₁ is the returns during the recession. The probability of having a recession is P₁. Similarly, R₂ is the returns in a normal economy. The probability of having a normal is P₂. R₃ is the returns in irrational exuberance. The probability of having an irrational exuberance is P₃.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\left[\begin{array}{l}\left[{\left(0.02-0.125\right)}^2\times 0.25\right]+\left[{\left(0.21-0.125\right)}^2\times 0.50\right]\\ +\left[{\left(0.06-0.125\right)}^2\times 0.25\right]\end{array}\right]}\\ =\sqrt{\left[{\left(-\text{0}\text{.105}\right)}^2\times 0.25\right]+\left[{\left(\text{0}\text{.085}\right)}^2\times 0.50\right]+\left[{\left(-\text{0}\text{.065}\right)}^2\times 0.25\right]}\\ =\sqrt{\left[\text{0}\text{.011025}\times 0.25\right]+\left[\text{0}\text{.007225}\times 0.50\right]+\left[\text{0}\text{.004225}\times 0.25\right]}\end{array}$

$\begin{array}{c}=\sqrt{\text{0}\text{.00275625}+\text{0}\text{.0036125}+\text{0}\text{.00105625}}\\ =\sqrt{\text{0}\text{.007425}}\\ =\text{0}\text{.0861}\end{array}$

Hence, the standard deviation of Stock I is 0.0861 or 8.61%.

Compute the expected return on Stock II:

R₁ is the returns during the recession. The probability of having a recession is P₁. Similarly, R₂ is the returns in a normal economy. The probability of having a normal is P₂. R₃ is the returns in irrational exuberance. The probability of having an irrational exuberance is P₃.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(\left(-0.25\right)\times 0.25\right)+\left(0.09\times 0.50\right)+\left(0.44\times 0.25\right)\\ =-0.0625+0.045+0.11\\ =0.0925\end{array}$

Hence, the expected return on Stock II is 9.25 percent.

Compute the beta of Stock II:

$\begin{array}{c}E(R_{II})=R_f+\left[E(R_M)-R_f\right]\times {\beta }_{II}\\ 0.0925=0.04+\left[0.07\right]\times {\beta }_{II}\\ 0.0925-0.04=0.07{\beta }_{II}\\ \text{0}\text{.0525}=0.07{\beta }_{II}\end{array}$

$\begin{array}{c}\frac{\text{0}\text{.0525}}{0.07}={\beta }_{II}\\ \text{0}\text{.75}={\beta }_{II}\end{array}$

Hence, the beta of Stock II is 0.75.

Compute the standard deviation of Stock II:

R₁ is the returns during the recession. The probability of having a recession is P₁. Similarly, R₂ is the returns in a normal economy. The probability of having a normal is P₂. R₃ is the returns in irrational exuberance. The probability of having an irrational exuberance is P₃.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]+\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\left[\begin{array}{l}\left[{\left(-\left(0.25\right)-0.0925\right)}^2\times 0.25\right]+\left[{\left(0.09-0.0925\right)}^2\times 0.50\right]+\\ \left[{\left(\left(0.44\right)-0.0925\right)}^2\times 0.25\right]\end{array}\right]}\\ =\sqrt{\left[{\left(-\text{0}\text{.3425}\right)}^2\times 0.25\right]+\left[{\left(-\text{0}\text{.0025}\right)}^2\times 0.50\right]+\left[{\left(\text{0}\text{.3475}\right)}^2\times 0.25\right]}\\ =\sqrt{\left(\text{0}\text{.11730625}\times 0.25\right)+\left(\text{0}\text{.00000625}\times 0.50\right)+\left(\text{0}\text{.12075625}\times 0.25\right)}\end{array}$

$\begin{array}{c}=\sqrt{\text{0}\text{.0293265625}+\text{0}\text{.000003125}+\text{0}\text{.0301890625}}\\ =\sqrt{\text{0}\text{.05951875}}\\ =0.2439\end{array}$

Hence, the standard deviation of Stock II is 24.39%.

Interpretation of the results:

The beta refers to the systematic risk of the stock. Stock I has higher beta than Stock II. Hence, the systematic risk of Stock I is higher. The standard deviation indicates the total risk of the stock. The standard deviation is high for Stock II despite having a low beta. Hence, a major portion of the standard deviation of Stock II is the unsystematic risk.

Stock II has higher unsystematic risk than Stock I. The formation of a portfolio helps in diversifying the unsystematic risk. Although Stock II has a higher unsystematic risk, it can be diversified completely. However, the beta cannot be eliminated. Hence, Stock I is riskier than Stock II.

The expected return and the market risk premium depend on the beta of the stock. As Stock I has a higher beta, the expected return and market risk premium of the stock will be higher than Stock II.