Chapter 13, Problem 7QP

Calculating Returns and Standard Deviations [LO1] Based on the following information, calculate the expected return and standard deviation for the two stocks:

Summary Introduction

Introduction:

Expected return refers to the return that the investors expect on a risky investment in the future. Standard deviation refers to the variation in the actual returns from the expected returns.

To determine: The standard deviation of Stock A.

Answer to Problem 7QP

The standard deviation of Stock A is 4.49 percent.

Explanation of Solution

Given information:

Stock A’s return is 4 percent when the economy is in a recession, 9 percent when the economy is normal, and 17 percent when the economy is in a boom. The probability of having a recession is 15 percent, the probability of having a normal economy is 55 percent, and the probability of having a booming economy is 30 percent.

The formula to calculate the expected return on the stock:

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

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 expected return on Stock A:

“R₁” refers to the returns during the recession. The probability of having a recession is “P₁”. “R₂” is the returns in a normal economy. The probability of having a normal economy is “P₂”. “R₃” is the returns in a booming economy. The probability of having a booming economy is “P₃”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)+\\ \left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(0.04\times 0.15\right)+\left(0.09\times 0.55\right)+\left(0.17\times 0.30\right)\\ =0.006+0.0495+0.051\\ =0.1065\text{ or }10.65\%\end{array}$

Hence, the expected return on Stock A is 10.65 percent.

Compute the standard deviation of Stock A:

“R₁” refers to the returns during the recession. The probability of having a recession is “P₁”. “R₂” is the returns in a normal economy. The probability of having a normal economy is “P₂”. “R₃” is the returns in a booming economy. The probability of having a booming economy is “P₃”. The expected return on Stock A “E(R)” is 10.65 percent.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\begin{array}{l}\left[{\left(0.04-0.1065\right)}^2\times 0.15\right]+\left[{\left(0.09-0.1065\right)}^2\times 0.55\right]+\\ \left[{\left(0.17-0.1065\right)}^2\times 0.30\right]\end{array}}\\ =0.0449\text{ or }4.49\%\end{array}$

Hence, the standard deviation of Stock A is 4.49 percent.

Summary Introduction

To determine: The standard deviation of Stock B.

Answer to Problem 7QP

The standard deviation of Stock B is 13.92 percent.

Explanation of Solution

Given information:

Stock B’s return is (17 percent) when the economy is in a recession, 12 percent when the economy is normal, and 27 percent when the economy is in a boom. The probability of having a recession is 15 percent, the probability of having a normal economy is 55 percent, and the probability of having a booming economy is 30 percent.

The formula to calculate the expected return on the stock:

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

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 expected return on Stock B:

“R₁” refers to the returns during the recession. The probability of having a recession is “P₁”. “R₂” is the returns in a normal economy. The probability of having a normal economy is “P₂”. “R₃” is the returns in a booming economy. The probability of having a booming economy is “P₃”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)+\\ \left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\end{array}\right]\\ =\left(\left(0.17\right)\times 0.15\right)+\left(0.12\times 0.55\right)+\left(0.27\times 0.30\right)\\ =\left(0.0255\right)+0.066+0.081\\ =0.1215\text{ or }12.15\%\end{array}$

Hence, the expected return on Stock B is 12.15 percent.

Compute the standard deviation of Stock B:

“R₁” refers to the returns during the recession. The probability of having a recession is “P₁”. “R₂” is the returns in a normal economy. The probability of having a normal economy is “P₂”. “R₃” is the returns in a booming economy. The probability of having a booming economy is “P₃”. The expected return on Stock B “E(R)” is 12.15 percent.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\end{array}\right)}\\ =\sqrt{\begin{array}{l}\left[{\left(\left(0.17\right)-0.1215\right)}^2\times 0.15\right]+\left[{\left(0.12-0.1215\right)}^2\times 0.55\right]+\\ \left[{\left(0.27-0.1215\right)}^2\times 0.30\right]\end{array}}\\ =0.1392\text{ or }13.92\%\end{array}$

Hence, the standard deviation of Stock B is 13.92 percent.