Chapter 7, Problem 22QP

Stock Valuation. According to the 2015 Value Line Investment Survey, the growth rate in dividends for IBM for the next five years is expected to be 7.5 percent. Suppose IBM meets this growth rate in dividends for the next five years and then the dividend growth rate falls to 5 percent indefinitely. Assume investors require a return of 10 percent on IBM stock. Is the stock priced correctly? What factors could affect your answer?

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 next five years, which is 7.5%. In case, the company meets the specified growth rate on dividends for next five years; then dividend growth rate will decline to 5%. The required rate of return is 10% 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:

$\begin{array}{c}{\text{P}}_5={\text{D}}_6\times {\left(1-{\text{g}}_1\right)}^t\times \frac{\left(1-{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}\\ =\$4.25\times {\left(1+\frac{7.5}{100}\right)}^5\times \frac{\left(1+\frac{5}{100}\right)}{\left(\frac{10}{100}-\frac{5}{100}\right)}\\ =\$4.25\times {\left(1.075\right)}^5\times \frac{1.05}{\left(0.10-0.05\right)}\\ =\$4.25\times 1.43562\times \left(\frac{1.05}{0.05}\right)\end{array}$

$\begin{array}{c}=\$4.25\times 1.43562\times 21\\ =\$128.13\end{array}$

Hence, the stock price in Year 5 is $128.13.

Compute the current stock price:

$\begin{array}{c}{\text{P}}_5=\left[\begin{array}{c}\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^1}{{\left(1+\text{R}\right)}^1}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^2}{{\left(1+\text{R}\right)}^2}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^3}{{\left(1+\text{R}\right)}^3}\right)\\ +\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^4}{{\left(1+\text{R}\right)}^4}\right)+\left({\text{D}}_6\times \frac{{\left(1-{\text{g}}_1\right)}^5}{{\left(1+\text{R}\right)}^5}\right)+\frac{{\text{P}}_5}{{\left(1+\text{R}\right)}^5}\end{array}\right]\\ =\left[\begin{array}{c}\$4.25\times \frac{{\left(1+\frac{7.5}{100}\right)}^1}{{\left(1+\frac{10}{100}\right)}^1}+\$4.25\times \frac{{\left(1+\frac{7.5}{100}\right)}^2}{{\left(1+\frac{10}{100}\right)}^2}+\$4.25\times \frac{{\left(1+\frac{7.5}{100}\right)}^3}{{\left(1+\frac{10}{100}\right)}^3}\\ +\$4.25\times \frac{{\left(1+\frac{7.5}{100}\right)}^4}{{\left(1+\frac{10}{100}\right)}^4}+\$4.25\times \frac{{\left(1+\frac{7.5}{100}\right)}^5}{{\left(1+\frac{10}{100}\right)}^5}+\frac{\$128.13}{{\left(1+\frac{10}{100}\right)}^5}\end{array}\right]\\ =\left[\begin{array}{c}\left(\$4.25\times \frac{{\left(1.075\right)}^1}{{\left(1.10\right)}^1}\right)+\left(\$4.25\times \frac{{\left(1.075\right)}^2}{{\left(1.10\right)}^2}\right)+\left(\$4.25\times \frac{{\left(1.075\right)}^3}{{\left(1.10\right)}^3}\right)\\ +\left(\$4.25\times \frac{{\left(1.075\right)}^4}{{\left(1.10\right)}^4}\right)+\left(\$4.25\times \frac{{\left(1.075\right)}^5}{{\left(1.10\right)}^5}\right)+\frac{\$128.13}{{\left(1.10\right)}^5}\end{array}\right]\\ =\left[\begin{array}{c}\left(\$4.25\times 0.97727\right)+\$4.25\times \left(\frac{1.155625}{1.21}\right)+\left(\$4.25\times \frac{1.24229}{1.331}\right)\\ +\left(\$4.25\times \frac{1.33546}{1.4641}\right)+\left(\$4.25\times \frac{1.43562}{1.61051}\right)+\left(\frac{\$128.13}{1.61051}\right)\end{array}\right]\end{array}$

$\begin{array}{c}=\left[\begin{array}{c}\left(\$4.25\times 0.97727\right)+\left(\$4.25\times 0.95506\right)\\ +\left(\$4.25\times 0.93335\right)+\left(\$4.25\times 0.91213\right)\\ +\left(\$4.25\times 0.89140\right)+\$79.55864\end{array}\right]\\ =\$4.15339+\$4.059005+\$3.96673+\$3.87655+\$3.78845+\$79.55864\\ =\$99.40\end{array}$

Hence, the current stock price is $99.40. 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