Chapter 11, Problem 7QP

Calculating Returns and Standard Deviations. Based on the following information, calculate the expected return and standard deviation for the two stocks.

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 11.20 percent.

The expected return of Stock B is 18.40 percent.

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 10 percent, the probability of having a normal economy is 50 percent, and the probability of having a booming economy is 40 percent.

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:

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(0.02\times 0.10\right)+\left(0.10\times 0.50\right)+\left(0.15\times 0.40\right)\\ =0.002+0.05+0.06\\ =0.1120\end{array}$

Hence, the expected return on Stock A is 0.1120 or 11.20 percent.

Compute the expected return on Stock B:

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(\left(-0.30\right)\times 0.10\right)+\left(0.18\times 0.50\right)+\left(0.31\times 0.40\right)\\ =\left(-0.03\right)+0.09+0.124\\ =0.1840\end{array}$

Hence, the expected return on Stock B is 0.1840 or 18.40 percent.

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 3.87 percent.

The standard deviation of Stock B is 17.26 percent.

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 10 percent, the probability of having a normal economy is 50 percent, and the probability of having a booming economy is 40 percent.

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:

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\begin{array}{l}\left[{\left(0.02-0.1120\right)}^2\times 0.10\right]+\left[{\left(0.10-0.1120\right)}^2\times 0.50\right]\\ +\left[{\left(0.15-0.1120\right)}^2\times 0.40\right]\end{array}}\\ =\sqrt{\begin{array}{l}\left[{\left(-0.092\right)}^2\times 0.10\right]+\left[{\left(-0.012\right)}^2\times 0.50\right]\\ +\left[{\left(0.038\right)}^2\times 0.40\right]\end{array}}\\ =\sqrt{\begin{array}{l}\left[0.008464\times 0.10\right]+\left[0.000144\times 0.50\right]\\ +\left[0.001444\times 0.40\right]\end{array}}\end{array}$

$\begin{array}{c}=\sqrt{0.0008464+0.000072+0.0005776}\\ =\sqrt{0.001496}\\ =0.0387\end{array}$

Hence, the standard deviation of Stock A is 0.0387 or 3.87 percent.

Compute the standard deviation of Stock B:

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\begin{array}{l}\left[{\left(\left(-0.30\right)-0.1840\right)}^2\times 0.10\right]+\left[{\left(0.18-0.1840\right)}^2\times 0.50\right]\\ +\left[{\left(0.31-0.1840\right)}^2\times 0.40\right]\end{array}}\\ =\sqrt{\begin{array}{l}\left[{\left(-0.484\right)}^2\times 0.10\right]+\left[{\left(-0.004\right)}^2\times 0.50\right]\\ +\left[{\left(0.126\right)}^2\times 0.40\right]\end{array}}\\ =\sqrt{\begin{array}{l}\left[0.234256\times 0.10\right]+\left[0.000016\times 0.50\right]\\ +\left[0.015876\times 0.40\right]\end{array}}\end{array}$

$\begin{array}{c}=\sqrt{\begin{array}{l}0.0234256+0.000008\\ +0.0063504\end{array}}\\ =\sqrt{\begin{array}{l}0.0234256+0.000008\\ +0.0063504\end{array}}\\ =\sqrt{0.029784}\\ =0.1726\end{array}$

Hence, the standard deviation of Stock B is 0.1726 or 17.26 percent.