Chapter 8, Problem 19QP

Summary Introduction

To determine: The projected dividend for the upcoming year.

Introduction:

Dividend is a sum of money paid to the shareholders of the company. It is distributed among the investors from the portion of a company’s earnings.

Answer to Problem 19QP

The projected dividend for the future year is $1.82.

Explanation of Solution

Given information:

F Company has sold stock for $76 per share. The required rate of return is 10%. The dividends growth rate for the next three years is 30%, the subsequent years’ dividends growth rate is 20%, and the dividends growth rate for the next corresponding years is 6% per year.

The formula to calculate dividends of “n” number of Years:

${\text{D}}_{\text{n}}={\text{D}}_{\text{0}}\times {\left(1+\text{g}\right)}^{\text{n}}$

Where,

D_nrefers to the dividend paid in “n” number of years

D_o refers to the current year dividend paid

n refers to the number of years

g refers to the constant dividend’s growth rate

The formula to calculate current dividends using current stock price formula:

${\text{P}}_{\text{o}}=\frac{{\text{D}}_o\times {\left(1+g_1\right)}^1}{{\left(\text{1}+R\right)}^1}+\frac{{\text{D}}_o\times {\left(1+g_1\right)}^2}{{\left(\text{1}+R\right)}^2}+\frac{{\text{D}}_o\times {\left(1+g_1\right)}^3}{{\left(\text{1}+R\right)}^3}+\mathrm{...}+\frac{{\text{D}}_o\times {\left(1+g_1\right)}^n}{{\left(\text{1}+R\right)}^n}+\frac{{\text{P}}_{\text{4}}}{{\left(\text{1}+R\right)}^4}$

Where,

P_o refers to the price of the stock of the current year

D_o refers to the current year dividend paid

R refers to the required rate of return on its stock

g₁refers to the expected growth rate of dividend

g₂ refers to the constant rate of growth in the second year

g₃ refers to the constant rate of growth in the third year

g_n refers to the constant rate of growth in n number of year

n refers to the number of years

The formula to calculate projected dividend for the next year:

${\text{D}}_1={\text{D}}_o\times \left(1+{\text{g}}_1\right)$

Where,

D_o refers to the current year dividend paid

g₁refers to the expected growth rate of dividend

D₁ refers to the dividend payable in the future year

Compute the dividends in Year 3:

$\begin{array}{c}{\text{D}}_{\text{3}}={\text{D}}_{\text{0}}\times {\left(\text{1}+\text{g}\right)}^3\\ ={\text{D}}_{\text{0}}\times {\left(\text{1}+\frac{30}{100}\right)}^{\text{3}}\\ ={\text{D}}_{\text{0}}\times {\left(1.30\right)}^3\end{array}$

Hence, the dividends in Year 3 is (D_o × 1.30)³. Consider this as Equation (1).

Compute the dividend in Year 4:

$\begin{array}{c}{\text{D}}_{\text{4}}={\text{ D}}_{\text{0}}\times {\left({\text{1+ g}}_1\right)}^3\times \left(1+{\text{g}}_2\right)\\ ={\text{ D}}_{\text{0}}\times {\left(\text{1+}\frac{30}{100}\right)}^{\text{3}}\times \left(\text{1+}\frac{20}{100}\right)\\ ={\text{ D}}_{\text{0}}\times {\left(1.30\right)}^3\left(1.20\right)\end{array}$

Hence, the dividends in Year 4 is (D_o × 1.30)³ (1.20). Consider this as Equation (2).

As the stocks starts growing at a constant growth rate in Year 4, determine the stock price for the Year 4. The formula to calculate stock price for the Year 4 is given below: [Consider this formula as Equation (3)]

${\text{P}}_4={\text{D}}_4\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}$

Where,

P_orefers to the present price of a stock

D₄refers to the dividend per share in Year 4

Rrefers to the required return on the stock

g refers to the constant dividend’s growth rate

Substitute Equation (2) in Equation (3) and simplify it into the simplest form as shown below:

$\begin{array}{c}{\text{P}}_4={\text{D}}_4\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ ={\text{D}}_{\text{0}}\times {\left(1.30\right)}^3\left(1.20\right)\times \frac{\left(1+\frac{6}{100}\right)}{\left(\frac{10}{100}-\frac{6}{100}\right)}\\ ={\text{D}}_{\text{0}}\times {\left(1.30\right)}^3\left(1.20\right)\times \frac{\left(1.06\right)}{\left(0.10-0.06\right)}\\ ={\text{D}}_{\text{0}}\times {\left(1.30\right)}^3\left(1.20\right)\times \left(\frac{1.06}{0.04}\right)\end{array}$

$\begin{array}{c}={\text{D}}_{\text{0}}\times 2.197\times 1.20\times 26.5\\ ={\text{D}}_{\text{0}}\times \text{69}\text{.86}\end{array}$

Hence, the stock price in Year 4 is D_o× 69.86.

Compute the current dividends using current stock price formula:

$\begin{array}{c}{\text{P}}_{\text{o}}=\frac{{\text{D}}_o\times {\left(1+g_1\right)}^1}{{\left(\text{1}+R\right)}^1}+\frac{{\text{D}}_o\times {\left(1+g_1\right)}^2}{{\left(\text{1}+R\right)}^2}+\frac{{\text{D}}_o\times {\left(1+g_1\right)}^3}{{\left(\text{1}+R\right)}^3}+\frac{{\text{D}}_o\times {\left(1+g_1\right)}^4}{{\left(\text{1}+R\right)}^4}+\frac{{\text{P}}_{\text{4}}}{{\left(\text{1}+R\right)}^4}\\ \$76=\frac{{\text{D}}_o\times {\left(1+0.30\right)}^1}{{\left(1+0.10\right)}^1}+\frac{{\text{D}}_o\times {\left(1+0.30\right)}^2}{{\left(1+0.10\right)}^2}+\frac{{\text{D}}_o\times {\left(1+0.30\right)}^3}{{\left(1+0.10\right)}^3}+\frac{{\text{D}}_o\times {\left(1+0.30\right)}^4}{{\left(1+0.10\right)}^4}+\frac{{\text{D}}_o\times 69.85}{{\left(1+0.10\right)}^4}\\ \$76=\frac{\left({\text{D}}_o\times 1.30\right)}{1.10}+\frac{\left({\text{D}}_o\times 1.69\right)}{1.21}+\frac{\left({\text{D}}_o\times 2.197\right)}{1.331}+\frac{\left({\text{D}}_o\times 2.86\right)}{1.46}+\frac{\left({\text{D}}_o\times 69.85\right)}{1.46}\\ \$76=\left({\text{D}}_o\times 1.18\right)+\left({\text{D}}_o\times 1.40\right)+\left({\text{D}}_o\times 1.65\right)+\left({\text{D}}_o\times 1.96\right)+\left({\text{D}}_o\times 47.84\right)\\ \$76={\text{D}}_o\times \left(1.18+1.40+1.65+1.96+47.84\right)\\ \$76={\text{D}}_o\times \text{54}\text{.03}\\ \frac{\$76}{\text{54}\text{.03}}={\text{D}}_o\\ \$1.40={\text{D}}_o\end{array}$

Hence, the current dividend is $1.40.

Compute the projected dividend for the next year:

$\begin{array}{c}{\text{D}}_1={\text{D}}_o\times \left(1+{\text{g}}_1\right)\\ =\$1.40\times \left(1+\frac{30}{100}\right)\\ =\$1.40\times \left(1+0.30\right)\\ =\$1.82\end{array}$

Hence, the projected dividend for the future year is $1.82.