Chapter 8, Problem 14QP

Stock Valuation [LO1] Bayou Okra Farms just paid a dividend of $2.65 on its stock. The growth rate in dividends is expected to be a constant 4.5 percent per year indefinitely. Investors require a return of 15 percent for the first three years, a return of 13 percent for the next three years, and a return of 11 percent thereafter. What is the current share price?

Summary Introduction

To determine: The current price of the 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 $37.21.

Explanation of Solution

Given information:

BO Company has currently paid dividends of $2.65. The constant growth rate of dividends is 4.5%. The investor’s required rate of return for the first three years is 15%, for the next three years is 13%, and a return of 11% for subsequent years.

Formulae:

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,

P₆ refers to the price of stock in 6 years

D_o refers 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 dividend value of Year 7 is computed to determine the price of stock in Year 6.

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, then the stock price in Year 3 is 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_o refers to the present price of 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:

$\begin{array}{c}{\text{P}}_6={\text{D}}_o\times \frac{{\left(1+\text{g}\right)}^n}{\left(\text{R}-\text{g}\right)}\\ =\$2.65\times \frac{{\left(1+\frac{4.5}{100}\right)}^7}{\left(\frac{11}{100}-\frac{4.5}{100}\right)}\end{array}$

$\begin{array}{c}=\$2.65\times \frac{{\left(1.045\right)}^7}{\left(0.11-0.045\right)}\\ =\$2.65\times \frac{1.36086}{0.065}\\ =\$2.65\times 20.93630\\ =\$55.48\end{array}$

Hence, the stock price of Year 6 is $55.48.

Compute the stock price of Year 3:

$\begin{array}{c}{\text{P}}_3=\frac{{\text{D}}_1{\left(1+\text{g}\right)}^4}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2{\left(1+\text{g}\right)}^4}{{\left(\text{1+R}\right)}^2}+\frac{{\text{D}}_3{\left(1+\text{g}\right)}^6}{{\left(\text{1+R}\right)}^3}+\frac{{\text{P}}_6}{{\left(\text{1+R}\right)}^3}\\ =\frac{\$2.65{\left(1+\frac{4.5}{100}\right)}^4}{{\left(1+\frac{13}{100}\right)}^1}+\frac{\$2.65{\left(1+\frac{4.5}{100}\right)}^5}{{\left(1+\frac{13}{100}\right)}^2}+\frac{\$2.65{\left(1+\frac{4.5}{100}\right)}^6}{{\left(1+\frac{13}{100}\right)}^3}+\frac{\$55.48}{{\left(1+\frac{13}{100}\right)}^3}\\ =\frac{\$2.65{\left(1.045\right)}^4}{1.13}+\frac{\$2.65{\left(1.045\right)}^5}{{\left(1.13\right)}^2}+\frac{\$2.65{\left(1.045\right)}^6}{{\left(1.13\right)}^3}+\frac{\$55.48}{{\left(1.13\right)}^3}\end{array}$

$\begin{array}{c}=\frac{\$2.65\times 1.19251}{1.13}+\frac{\left(\$2.65\times 1.24618\right)}{1.2769}+\frac{\left(\$2.65\times 1.30226\right)}{1.44289}+\frac{\$55.48}{1.44289}\\ =\$3.16015+\frac{\$3.30237}{1.2769}+\frac{\$3.45098}{1.44289}+\frac{\$55.48}{1.44289}\\ =\$3.16015+\$2.58624+\$2.39171+\$38.45060\\ =\$46.58\end{array}$

Hence, the stock price of Year 6 is $46.58.

Compute the current price of the share:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1{\left(1+\text{g}\right)}^1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2{\left(1+\text{g}\right)}^2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{D}}_3{\left(1+\text{g}\right)}^3}{{\left(\text{1+R}\right)}^3}+\frac{{\text{P}}_6}{{\left(\text{1+R}\right)}^3}\\ =\frac{\$2.65\times {\left(1+\frac{4.5}{100}\right)}^1}{{\left(1+\frac{15}{100}\right)}^1}+\frac{\$2.65\times {\left(1+\frac{4.5}{100}\right)}^2}{{\left(1+\frac{15}{100}\right)}^2}+\frac{\$2.65\times {\left(1+\frac{4.5}{100}\right)}^3}{{\left(1+\frac{15}{100}\right)}^3}+\frac{\$46.58}{{\left(1+\frac{15}{100}\right)}^3}\\ =\frac{\$2.65\times {\left(1.045\right)}^1}{{\left(1+0.15\right)}^1}+\frac{\$2.65\times {\left(1.045\right)}^2}{{\left(1+0.15\right)}^2}+\frac{\$2.65\times {\left(1.045\right)}^3}{{\left(1+0.15\right)}^3}+\frac{\$46.58}{{\left(1+0.15\right)}^3}\\ =\frac{\$2.76925}{1.15}+\frac{\left(\$2.65\times 1.09202\right)}{{\left(1.15\right)}^2}+\frac{\left(\$2.65\times 1.14116\right)}{{\left(1.15\right)}^3}+\frac{\$46.58}{{\left(1.15\right)}^3}\end{array}$

$\begin{array}{c}=\$2.40804+\frac{\$2.89385}{1.3225}+\frac{\$3.02407}{1.52087}+\frac{\$46.58}{1.52087}\\ =\$2.40804+\$2.18816+\$1.98838+\$30.62720\\ =\$37.21178\end{array}$

Hence, the current price of the share is $37.21.