Chapter 8, Problem 14QP

Summary Introduction

To determine: The current price of a share.

Introduction:

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

Answer to Problem 14QP

The current share price is $56.85.

Explanation of Solution

Given information:

GP Company has currently paid dividends of $3.20. The constant growth rate of dividends is 5%. The investor’s required rate of return for the first three years is 15%, for the next three years are 13%, and a return of 11% for subsequent years.

Formula to calculate the stock price of Year 6:

${\text{P}}_6={\text{D}}_o\times \frac{{\left(1+\text{g}\right)}^n}{\left(\text{R}-\text{g}\right)}$

Where,

P₆refers to the price of stock in 6 years

D_orefers to the dividend paid presently

R refers to the required return on the stock

g refers to the constant growth rate

n refers to the number of years

Note: The price of stock (P₆) will be the dividend in Year 7.

Formula to calculate the stock price of Year 3:

${\text{P}}_3=\frac{{\text{D}}_1{\left(1+\text{g}\right)}^4}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2{\left(1+\text{g}\right)}^5}{{\left(\text{1+R}\right)}^2}+\frac{{\text{D}}_3{\left(1+\text{g}\right)}^6}{{\left(\text{1+R}\right)}^3}+\frac{{\text{P}}_6}{{\left(\text{1+R}\right)}^3}$

Where,

P₃ refers to the price of the stock in Year 3

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

R refers to the required rate of return on its stock

g refers to the constant growth rate

Note: At the time when the required return changes frequently, the stock price in Year 3 will be the present value of the dividends in Year 4, Year 5, and Year 6 plus the present value of stock of Year 3.

Formula to calculate the current price of the share:

${\text{P}}_o=\frac{{\text{D}}_1\left(1+\text{g}\right)}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2{\left(1+\text{g}\right)}^2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{D}}_3{\left(1+\text{g}\right)}^3}{{\left(\text{1+R}\right)}^3}+\frac{{\text{P}}_6}{{\left(\text{1+R}\right)}^3}$

Where,

P_orefers to the present price of a stock

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

R refers to the required return on the stock

g refers to the constant growth rate

Compute the stock price of Year 6:

$\begin{array}{c}{\text{P}}_6={\text{D}}_o\times \frac{{\left(1+\text{g}\right)}^n}{\left(\text{R}-\text{g}\right)}\\ =\$3.20\times \frac{{\left(1+\frac{5}{100}\right)}^7}{\left(\frac{11}{100}-\frac{5}{100}\right)}\end{array}$

$\begin{array}{c}=\$3.20\times \frac{{\left(1.05\right)}^7}{\left(0.11-0.05\right)}\\ =\$3.20\times \left(\frac{1.4071004}{0.06}\right)\\ =\$3.20\times 23.451673\\ =\$75.04\end{array}$

Hence, the stock price of Year 6 is $75.04.

Compute the stock price of Year 3:

$\begin{array}{c}{\text{P}}_3=\frac{{\text{D}}_1{\left(1+\text{g}\right)}^4}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2{\left(1+\text{g}\right)}^5}{{\left(\text{1+R}\right)}^2}+\frac{{\text{D}}_3{\left(1+\text{g}\right)}^6}{{\left(\text{1+R}\right)}^3}+\frac{{\text{P}}_6}{{\left(\text{1+R}\right)}^3}\\ =\frac{\$3.20{\left(1+\frac{5}{100}\right)}^4}{{\left(1+\frac{13}{100}\right)}^1}+\frac{\$3.20{\left(1+\frac{5}{100}\right)}^5}{{\left(1+\frac{13}{100}\right)}^2}+\frac{\$3.20{\left(1+\frac{5}{100}\right)}^6}{{\left(1+\frac{13}{100}\right)}^3}+\frac{\$75.04}{{\left(1+\frac{13}{100}\right)}^3}\\ =\frac{\$3.20{\left(1.05\right)}^4}{{\left(1.13\right)}^1}+\frac{\$3.20{\left(1.05\right)}^5}{{\left(1.13\right)}^2}+\frac{\$3.20{\left(1.05\right)}^6}{{\left(1.13\right)}^3}+\frac{\$75.04}{{\left(1.13\right)}^3}\end{array}$

$\begin{array}{c}=\frac{\left(\$3.20\times 1.21550\right)}{1.13}+\frac{\left(\$3.20\times 1.27628\right)}{1.2769}+\frac{\left(\$3.20\times 1.34009\right)}{1.44289}+\frac{\$75.04}{1.44289}\\ =\frac{\$3.8896}{1.13}+\frac{\$4.084096}{1.2769}+\frac{\$4.288288}{1.44289}+\frac{\$75.04}{1.44289}\\ =\$3.44212+\$3.19844+\$2.97201+\$52.00673\\ =\$61.62\end{array}$

Hence, the stock price of Year 3 is $61.62.

Compute the current price of the share:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1{\left(1+\text{g}\right)}^1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2{\left(1+\text{g}\right)}^2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{D}}_3{\left(1+\text{g}\right)}^3}{{\left(\text{1+R}\right)}^3}+\frac{{\text{P}}_6}{{\left(\text{1+R}\right)}^3}\\ =\frac{\$3.20\times {\left(1+\frac{5}{100}\right)}^1}{{\left(1+\frac{15}{100}\right)}^1}+\frac{\$3.20\times {\left(1+\frac{5}{100}\right)}^2}{{\left(1+\frac{15}{100}\right)}^2}+\frac{\$3.20\times {\left(1+\frac{5}{100}\right)}^3}{{\left(1+\frac{15}{100}\right)}^3}+\frac{\$75.04}{{\left(1+\frac{15}{100}\right)}^3}\\ =\frac{\$3.20\times {\left(1.05\right)}^1}{{\left(1+0.15\right)}^1}+\frac{\$3.20\times {\left(1.05\right)}^2}{{\left(1+0.15\right)}^2}+\frac{\$3.20\times {\left(1.05\right)}^3}{{\left(1+0.15\right)}^3}+\frac{\$75.04}{{\left(1+0.15\right)}^3}\\ =\frac{\$\text{3}\text{.36}}{{\left(1.15\right)}^1}+\frac{\left(\$3.20\times \text{1}\text{.1025}\right)}{{\left(1.15\right)}^2}+\frac{\left(\$3.20\times \text{1}\text{.157625}\right)}{{\left(1.15\right)}^3}+\frac{\$75.04}{{\left(1.15\right)}^3}\end{array}$

$\begin{array}{c}=\$2.92173+\frac{\$3.528}{1.3225}+\frac{\$3.7044}{1.52087}+\frac{\$75.04}{1.52087}\\ =\$2.40804+\$2.66767+\$2.43571+\$49.34018\\ =\$56.85\end{array}$

Hence, the current price of the share is $56.85.