Chapter 11, Problem 30QP

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.

Explanation of Solution

Given information:

The probability of having a recession, normal economy, and irrational exuberance is 0.20, 0.55, and 0.25 respectively. Stock I will yield 2%, 32%, and 18% when there is a recession, normal economy, and irrational exuberance respectively.

Stock II will yield −20%, 12%, and 40% 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:

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

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:

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

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.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(\left(0.20\right)\times 0.02\right)+\left(0.55\times 0.32\right)+\left(0.25\times 0.18\right)\\ =\left(0.004+0.176+0.045\right)\\ =0.225\end{array}$

Hence, the expected return on Stock I is 0.225 or 22.5%.

Compute the beta of Stock I:

$\begin{array}{c}E(R_I)=R_f+\left[E(R_M)-R_f\right]\times {\beta }_I\\ 0.225=0.04+\left[0.07\right]\times {\beta }_I\\ \left(0.225-0.04\right)=\text{0}\text{.07}{\beta }_I\\ \frac{0.185}{0.07}={\beta }_I\\ 2.64={\beta }_I\end{array}$

Hence, the beta of Stock I is 2.64.

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.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[\begin{array}{l}{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\\ \times \text{}\text{ Probability}\left(P_1\right)\end{array}\right]\\ +\left[\begin{array}{l}{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns}E\left(R\right)\right)}^2\\ \times \text{Probability}\left(P_2\right)\end{array}\right]\\ +\left[\begin{array}{l}{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns}E\left(R\right)\right)}^2\\ \times \text{Probability}\left(P_3\right)\end{array}\right]\end{array}\right)}\\ =\sqrt{\left[\begin{array}{l}\left[{\left(0.02-0.225\right)}^2\times 0.20\right]+\left[{\left(0.32-0.225\right)}^2\times 0.55\right]\\ +\left[{\left(0.18-0.225\right)}^2\times 0.25\right]\end{array}\right]}\\ =\sqrt{\left({\left(-0.205\right)}^2\times 0.20\right)+\left({\left(0.095\right)}^2\times 0.55\right)+\left({\left(-0.045\right)}^2\times 0.25\right)}\\ =\sqrt{\left(0.04203\times 0.20\right)+\left(0.009025\times 0.55\right)+\left(0.002025\times 0.25\right)}\end{array}$

$\begin{array}{c}=\sqrt{0.008406+0.00496375+0.00050625}\\ =\sqrt{0.013876}\\ =0.1178\end{array}$

Hence, the standard deviation of Stock I is 0.1178 or 11.78%.

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.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(\left(-0.20\right)\times \left(0.20\right)\right)+\left(0.12\times 0.55\right)+\left(0.40\times 0.25\right)\\ =\left(-0.04+0.066+0.1\right)\\ =0.126\end{array}$

Hence, the expected return on Stock II is 0.126 or 12.6%.

Compute the beta of Stock II:

$\begin{array}{c}E(R_{II})=R_f+\left[E(R_M)-R_f\right]\times {\beta }_{II}\\ 0.126=0.04+\left[0.07\right]\times {\beta }_{II}\\ \left(0.126-0.04\right)=0.07{\beta }_{II}\\ \frac{\text{0}\text{.086}}{0.07}={\beta }_{II}\\ 1.23={\beta }_{II}\end{array}$

Hence, the beta of Stock II is 1.23.

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.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[\begin{array}{l}{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\\ \times \text{}\text{ Probability}\left(P_1\right)\end{array}\right]\\ +\left[\begin{array}{l}{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns}E\left(R\right)\right)}^2\\ \times \text{Probability}\left(P_2\right)\end{array}\right]\\ +\left[\begin{array}{l}{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns}E\left(R\right)\right)}^2\\ \times \text{Probability}\left(P_3\right)\end{array}\right]\end{array}\right)}\\ =\sqrt{\left[\begin{array}{l}\left[{\left(\left(-0.20\right)-0.126\right)}^2\times 0.20\right]+\left[{\left(0.12-0.126\right)}^2\times 0.55\right]\\ +\left[{\left(0.40-0.126\right)}^2\times 0.25\right]\end{array}\right]}\\ =\sqrt{\left({\left(-0.326\right)}^2\times 0.20\right)+\left({\left(0.006\right)}^2\times 0.55\right)+\left({\left(0.274\right)}^2\times 0.25\right)}\\ =\sqrt{\left(0.106276\times 0.20\right)+\left(0.000036\times 0.55\right)+\left(0.075076\times 0.25\right)}\end{array}$

$\begin{array}{c}=\sqrt{\left(0.0212552+0.0000198+0.018769\right)}\\ =\sqrt{0.040044}\\ =0.20011\end{array}$

Hence, the standard deviation of Stock II is 0.20011 or 20.01%.

Interpretation:

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. Therefore, Stock II has higher unsystematic risk than Stock I.

Summary Introduction

To discuss: The riskier Stock among Stock I and Stock II.

Introduction:

Risk refers to the movement or fluctuation in the value of an investment. The movement can be positive or negative. A positive fluctuation in the price benefits the investor. The investor will lose money if the price movement is negative.

Explanation of Solution

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