Chapter 7, Problem 22QP

Summary Introduction

To determine: Whether the stock price is correct and the factors that could affect it.

Introduction:

Stock is a type of security in a company which denotes ownership. On issuing stocks, the company can raise the capital.

Stock price is the cost incurred to purchase a security on an exchange. Every investor will be careful on purchasing a stock of the company because the stock price will fluctuate based on the economic market conditions.

Explanation of Solution

Given information:

IB Company has an expected dividend growth rate for the next five years, which is 12.5%. In case, the company meets the specified growth rate on dividends for the next five years; then dividend growth rate will decline to 5%. The required rate of return is 11% on the company’s stock.

The formula to calculate price of stock in Year 5:

${\text{P}}_5={\text{D}}_6\times {\left(1-{\text{g}}_1\right)}^t\times \frac{\left(1-{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}$

Where,

P₅refers to the price of stock of Year 5,

D₆ refers to the next period dividend per share that is Year 6,

R refers to the required return on the stock,

g₁ refers to the expected growth rate of dividend,

g₂ refers to the constant rate of growth,

t refers to the number of years.

The formula to calculate the current stock price:

${\text{P}}_5=\left[\begin{array}{l}\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^1}{{\left(1+\text{R}\right)}^1}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^2}{{\left(1+\text{R}\right)}^2}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^3}{{\left(1+\text{R}\right)}^3}\right)\\ +\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^4}{{\left(1+\text{R}\right)}^4}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^5}{{\left(1+\text{R}\right)}^5}\right)+\frac{{\text{P}}_5}{{\left(1+\text{R}\right)}^5}\end{array}\right]$

Where,

P₅refers to the price of stock of Year 5,

D₆ refers to the next period dividend per share that is Year 6,

R refers to the required return on the stock,

g₁ refers to the expected growth rate of dividend,

g₂ refers to the constant rate of growth.

Compute the price of stock in Year 5:

$\begin{array}{c}{\text{P}}_5={\text{D}}_6\times {\left(1-{\text{g}}_1\right)}^t\times \frac{\left(1-{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}\\ =\$3\times {\left(1+\frac{12.5}{100}\right)}^5\times \frac{\left(1+\frac{5}{100}\right)}{\left(\frac{11}{100}-\frac{5}{100}\right)}\\ =\$3\times {\left(1.125\right)}^5\times \frac{1.05}{\left(0.11-0.05\right)}\\ =\$3\times 1.8020\times \left(\frac{1.05}{0.06}\right)\end{array}$

$\begin{array}{c}=\$3\times 1.8020\times 17.5\\ =\$94.605\end{array}$

Hence, the stock price in Year 5 is $94.605.

Compute the current stock price:

$\begin{array}{c}{\text{P}}_5=\left[\begin{array}{c}\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^1}{{\left(1+\text{R}\right)}^1}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^2}{{\left(1+\text{R}\right)}^2}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^3}{{\left(1+\text{R}\right)}^3}\right)\\ +\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^4}{{\left(1+\text{R}\right)}^4}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^5}{{\left(1+\text{R}\right)}^5}\right)+\frac{{\text{P}}_5}{{\left(1+\text{R}\right)}^5}\end{array}\right]\\ =\left[\begin{array}{c}\$3\times \frac{{\left(1+\frac{12.5}{100}\right)}^1}{{\left(1+\frac{11}{100}\right)}^1}+\$3\times \frac{{\left(1+\frac{12.5}{100}\right)}^2}{{\left(1+\frac{11}{100}\right)}^2}+\$3\times \frac{{\left(1+\frac{12.5}{100}\right)}^3}{{\left(1+\frac{11}{100}\right)}^3}\\ +\$3\times \frac{{\left(1+\frac{12.5}{100}\right)}^4}{{\left(1+\frac{11}{100}\right)}^4}+\$3\times \frac{{\left(1+\frac{12.5}{100}\right)}^5}{{\left(1+\frac{11}{100}\right)}^5}+\frac{\$94.605}{{\left(1+\frac{11}{100}\right)}^5}\end{array}\right]\\ =\left[\begin{array}{c}\left(\$3\times \frac{{\left(1.125\right)}^1}{{\left(1.11\right)}^1}\right)+\left(\$3\times \frac{{\left(1.125\right)}^2}{{\left(1.11\right)}^2}\right)+\left(\$3\times \frac{{\left(1.125\right)}^3}{{\left(1.11\right)}^3}\right)\\ +\left(\$3\times \frac{{\left(1.125\right)}^4}{{\left(1.11\right)}^4}\right)+\left(\$3\times \frac{{\left(1.125\right)}^5}{{\left(1.11\right)}^5}\right)+\frac{\$94.605}{{\left(1.11\right)}^5}\end{array}\right]\\ =\left[\begin{array}{c}\left(\$3\times 1.01351\right)+\$3\times \left(\frac{1.26562}{1.2321}\right)+\left(\$3\times \frac{1.42382}{1.36763}\right)\\ +\left(\$3\times \frac{1.60181}{1.51807}\right)+\left(\$3\times \frac{1.80203}{1.68506}\right)+\left(\frac{\$94.605}{1.68506}\right)\end{array}\right]\end{array}$

$\begin{array}{c}=\left[\begin{array}{c}\left(\$3\times 1.01351\right)+\left(\$3\times 1.02721\right)\\ +\left(\$3\times 1.04109\right)+\left(\$3\times 1.05516\right)\\ +\left(\$3\times 1.06942\right)+\$56.14\end{array}\right]\\ =\$3.04053+\$3.08163+\$3.12327+\$3.16548+\$3.20826+\$56.14\\ =\$71.76\end{array}$

Hence, the current stock price is $71.76. The stock is overvalued as per the constant growth model. Therefore, the stock price does not seem to be correct.

The factors which affect the stock price are as follows:

• Non-constant growth rate
• Long-term growth rate
• Length of non-constant growth
• Required rate of return