Chapter 7, Problem 18QP

Summary Introduction

To determine: The current stock price.

Introduction:

Stock price is a price of a share on the number of saleable stock of a company.

Non-constant growth rate model is used when there are two growth rates of dividends. It is also known as the supernormal growth rate. This model can be used to determine the price of a stock.

Answer to Problem 18QP

The current stock price is $60.4524.

Explanation of Solution

Given information:

Y Company has paid dividends of $2.50 and the company expects the dividends to grow at the rate of 25% for the next three years, the required rate of return is 13%, and the diminishing constant growth rate is 6%.

Formula:

The formula to calculate the stock price in Year 3:

${\text{P}}_3={\text{D}}_o\times \frac{{\left(1+{\text{g}}_1\right)}^t\left(1+{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}$

Where,

P₃ refers to the price of the stock in Year 3

D_o refers to the current year dividend paid

R refers to the required rate of return on its stock

g₁ refers to the expected growth rate of dividend

g₂ refers to the constant rate of growth

The formula to calculate the current stock price:

${\text{P}}_o=\frac{{\text{D}}_1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2}{{\left(\text{1+R}\right)}^2}+\mathrm{...}+\frac{{\text{D}}_n}{{\left(\text{1+R}\right)}^n}+\frac{{\text{P}}_3}{{\left(\text{1+R}\right)}^3}$

Where,

P_o refers to the current price of the stock

D₁…D_n refers to the next period expected dividend per share

P₃ refers to the price of the stock in Year 3

R refers to the required rate of return on its stock

n refers to the number of years

Compute the stock price in Year 3:

$\begin{array}{c}{\text{P}}_3={\text{D}}_o\times \frac{{\left(1+{\text{g}}_1\right)}^t\left(1+{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.50\times \frac{{\left(1+\frac{25}{100}\right)}^3\left(1+\frac{6}{100}\right)}{\left(\frac{13}{100}-\frac{6}{100}\right)}\\ =\$2.50\times \frac{{\left(1.25\right)}^3\times 1.06}{\left(0.13-0.06\right)}\\ =\$2.50\times \frac{\left(1.95312\times 1.06\right)}{0.07}\end{array}$

$\begin{array}{c}=\$2.50\times 29.5758\\ =\$73.9395\end{array}$

Hence, the stock price in Year 3 is $73.9395.

Compute the current stock price:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1{\left(1+{\text{g}}_1\right)}^1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2{\left(1+{\text{g}}_1\right)}^2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{D}}_3{\left(1+{\text{g}}_1\right)}^3}{{\left(\text{1+R}\right)}^3}+\frac{{\text{P}}_3}{{\left(\text{1+R}\right)}^3}\\ =\frac{\$2.50{\left(1+\frac{25}{100}\right)}^1}{{\left(1+\frac{13}{100}\right)}^1}+\frac{\$2.50{\left(1+\frac{25}{100}\right)}^2}{{\left(1+\frac{13}{100}\right)}^2}+\frac{\$2.50{\left(1+\frac{25}{100}\right)}^3}{{\left(1+\frac{13}{100}\right)}^3}+\frac{\$73.9395}{{\left(1+\frac{13}{100}\right)}^3}\\ =\frac{\$2.50{\left(1.25\right)}^1}{\left(\text{1}\text{.13}\right)}+\frac{\$2.50{\left(1.25\right)}^2}{{\left(\text{1}\text{.13}\right)}^2}+\frac{\$2.50{\left(1.25\right)}^3}{{\left(\text{1}\text{.13}\right)}^3}+\frac{\$73.9395}{{\left(\text{1}\text{.13}\right)}^3}\\ =\frac{\$\text{3}\text{.125}}{1.13}+\frac{\left(\$2.50\times \text{1}\text{.5625}\right)}{\text{1}\text{.2769}}+\frac{\left(\$2.50\times \text{1}\text{.95312}\right)}{\text{1}\text{.4429}}+\frac{\$73.9395}{\text{1}\text{.4429}}\end{array}$

$\begin{array}{l}=\$2.7655+\$3.0592+\$3.3840+\$51.2437\\ =\$60.4524\end{array}$

Hence, the current stock price is $60.4524.