Chapter 11, Problem 7QP

Summary Introduction

To determine: The expected return of Stock A and Stock B.

Introduction:

Expected return refers to the return that the investors expect on a risky investment in the future

Answer to Problem 7QP

The expected return of Stock A is 0.103 or 10.3% and the expected return of Stock B is 0.147 or 14.7%.

Explanation of Solution

Given information:

Stock A’s rate of return is 2 percent when the economy is in a recession, 10 percent when the economy is normal, and 15 percent when the economy is in a boom.

Stock B’s rate of return is −30 percent when the economy is in a recession, 18 percent when the economy is normal, and 31 percent when the economy is in a boom.

The probability of having a recession is 0.15, the probability of having a normal economy is 0.55, and the probability of having a booming economy is 0.30.

The formula to calculate the expected return on the stock:

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

Where,

R₁ refers to the rate of returns during the recession economy,

R_n refers to the rate of returns for “n” number of items,

P₁ refers to the probability of having a recession economy,

P_n refers to the probability of having “n” number of economy.

Compute the expected return on Stock A:

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(0.02\times 0.15\right)+\left(0.10\times 0.55\right)+\left(0.15\times 0.30\right)\\ =0.003+0.055+0.045\\ =0.103\end{array}$

Hence, the expected return on Stock A is 0.103 or 10.3%.

Compute the expected return on Stock B:

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(\left(-0.30\right)\times 0.15\right)+\left(0.18\times 0.55\right)+\left(0.31\times 0.30\right)\\ =\left(-0.045\right)+0.099+0.093\\ =0.147\end{array}$

Hence, the expected return on Stock B is 0.147 or 14.7%.

Summary Introduction

To determine: The standard deviation of Stock A and Stock B.

Introduction:

Standard deviation refers to the variation in the actual returns from the expected returns.

Answer to Problem 7QP

The standard deviation of Stock A is 4.124% and the standard deviation of Stock B is 17.65%.

Explanation of Solution

Given information:

Stock A’s rate of return is 2 percent when the economy is in a recession, 10 percent when the economy is normal, and 15 percent when the economy is in a boom.

Stock B’s rate of return is −30 percent when the economy is in a recession, 18 percent when the economy is normal, and 31 percent when the economy is in a boom.

The probability of having a recession is 0.15, the probability of having a normal economy is 0.55, and the probability of having a booming economy is 0.30.

The formula to calculate the standard deviation of the stock:

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

Compute the standard deviation of Stock A:

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\begin{array}{l}\left[{\left(0.02-0.103\right)}^2\times 0.15\right]+\left[{\left(0.10-0.103\right)}^2\times 0.55\right]\\ +\left[{\left(0.15-0.103\right)}^2\times 0.30\right]\end{array}}\\ =\sqrt{\begin{array}{l}\left[{\left(-0.083\right)}^2\times 0.15\right]+\left[{\left(-0.003\right)}^2\times 0.55\right]\\ +\left[{\left(0.047\right)}^2\times 0.30\right]\end{array}}\\ =\sqrt{\begin{array}{l}\left[0.006889\times 0.15\right]+\left[0.000009\times 0.55\right]\\ +\left[0.002209\times 0.30\right]\end{array}}\end{array}$

$\begin{array}{c}=\sqrt{0.00103335+0.00000495+0.0006627}\\ =\sqrt{0.001701}\\ =0.04124\text{or}\text{4}\text{.124\%}\end{array}$

Hence, the standard deviation of Stock A is 4.124%.

Compute the standard deviation of Stock B:

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\begin{array}{l}\left[{\left(\left(-0.30\right)-0.147\right)}^2\times 0.15\right]+\left[{\left(0.18-0.147\right)}^2\times 0.55\right]\\ +\left[{\left(0.31-0.147\right)}^2\times 0.30\right]\end{array}}\\ =\sqrt{\begin{array}{l}\left[{\left(-0.447\right)}^2\times 0.10\right]+\left[{\left(-0.033\right)}^2\times 0.50\right]\\ +\left[{\left(0.163\right)}^2\times 0.40\right]\end{array}}\end{array}$

$\begin{array}{c}=\sqrt{0.0199809+0.0005445+0.0106276}\\ =0.1765\text{or}17.65\text{\%}\end{array}$

Hence, the standard deviation of Stock B is 17.65%.