Chapter 13, Problem 13.1CTF

Summary Introduction

To determine: The standard deviation of the stock.

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.

Answer to Problem 13.1CTF

The standard deviation of the stock is 3.82 percent.

Explanation of Solution

Given information:

A stock’s return is 15 percent when the economy is in a boom and 7 percent when the economy is normal. The probability of having a booming economy is 35 percent, and the probability of having a normal economy is 65 percent.

The formula to calculate the expected return on the stock:

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

The formula to calculate the standard deviation of the stock:

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

Compute the expected return:

“R₁” is the returns in a booming economy. The probability of having a booming economy is “P₁”. Similarly, “R₂” is the returns in a normal economy. The probability of having a normal economy is “P₂”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(0.15\times 0.35\right)+\left(0.07\times 0.65\right)\\ =0.0525+0.0455\\ =0.098\text{ or }9.8\%\end{array}$

Hence, the expected return on the stock is 9.8 percent.

Compute the standard deviation:

“R₁” is the returns in a booming economy. The probability of having a booming economy is “P₁”. Similarly, “R₂” is the returns in a normal economy. The probability of having a normal economy is “P₂”.

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\end{array}\right)}\\ =\sqrt{\left[{\left(0.15-0.098\right)}^2\times 0.35\right]+\left[{\left(0.07-0.098\right)}^2\times 0.65\right]}\\ =\sqrt{0.0009464+0.0005096}\end{array}$

$\begin{array}{c}=\sqrt{0.001456}\\ =0.0382\text{ or }3.82\%\end{array}$

Hence, the standard deviation of the stock is 3.82 percent.