Chapter 13, Problem 26QP

Systematic versus Unsystematic Risk [LO3] Consider the following information about Stocks I and II:

The market risk premium is 7 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 that has the most systematic risk and 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 15.90 percent, the beta is 1.70, and the standard deviation is 7.87%. The expected return on Stock II is 9.15 percent, the beta is 0.74, and the standard deviation is 18.90%.

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 depends 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.

Explanation of Solution

Given information:

The probability of having a recession, normal economy, and irrational exuberance is 0.15, 0.70, and 0.15 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:

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

The formula to calculate the beta of the stock:

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

Where,

“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_i” refers to the beta coefficient of the risky asset relative to the market portfolio

The formula to calculate the standard deviation:

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

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”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(\left(0.15\right)\times 0.02\right)+\left(0.70\times 0.21\right)+\left(0.15\times 0.06\right)\\ =0.1590\text{ or }15.90\%\end{array}$

Hence, the expected return on Stock I is 15.90 percent.

Compute the beta of Stock I:

$\begin{array}{c}E(R_I)=R_f+\left[E(R_M)-R_f\right]\times {\beta }_I\\ 0.1590=0.04+\left[0.07\right]\times {\beta }_I\\ {\beta }_I=1.70\end{array}$

Hence, the beta of Stock I is 1.70.

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”.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \mathrm{\dots }+\\ \left[{\left(\text{Possible returns}\left(R_n\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_n\right)\right]\end{array}\right)}\\ =\sqrt{\left[\begin{array}{l}\left[{\left(0.02-0.1590\right)}^2\times 0.15\right]+\left[{\left(0.21-0.1590\right)}^2\times 0.70\right]+\\ \left[{\left(\left(0.06\right)-0.1590\right)}^2\times 0.15\right]\end{array}\right]}\\ =\sqrt{0.00619}\\ =0.0787\text{ or }7.87\%\end{array}$

Hence, the standard deviation of Stock I is 7.87%.

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”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(\left(0.15\right)\times \left(0.25\right)\right)+\left(0.70\times 0.09\right)+\left(0.15\times 0.44\right)\\ =0.0915\text{ or }9.15\%\end{array}$

Hence, the expected return on Stock II is 9.15 percent.

Compute the beta of Stock II:

$\begin{array}{c}E(R_{II})=R_f+\left[E(R_M)-R_f\right]\times {\beta }_{II}\\ 0.0915=0.04+\left[0.07\right]\times {\beta }_{II}\\ {\beta }_{II}=0.74\end{array}$

Hence, the beta of Stock II is 0.74.

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”.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \mathrm{\dots }+\\ \left[{\left(\text{Possible returns}\left(R_n\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_n\right)\right]\end{array}\right)}\\ =\sqrt{\left[\begin{array}{l}\left[{\left(\left(0.25\right)-0.0915\right)}^2\times 0.15\right]+\left[{\left(0.09-0.0915\right)}^2\times 0.70\right]+\\ \left[{\left(\left(0.44\right)-0.0915\right)}^2\times 0.15\right]\end{array}\right]}\\ =\sqrt{0.03571}\\ =0.1890\text{ or }18.90\%\end{array}$

Hence, the standard deviation of Stock II is 18.90%.