Chapter 8, Problem 37QP

Two-Stage Dividend Growth [LO1] Regarding the two-stage dividend growth model in the chapter, show that the price of a share of stock today can be written as follows;

  $P_0=\frac{D_0\times (1+g_1)}{R-g_1}\times \left[1-{\left(\frac{1+g_1}{1+R}\right)}^t\right]+{\left(\frac{1+g_1}{1+R}\right)}^t\times \frac{D_0\times (1+g_2)}{R-g_2}$

Can you provide an intuitive interpretation of this expression?

Summary Introduction

To prove: The current price of a share under the two-stage dividend growth model expressed as:

${\text{P}}_{\text{o}}=\left[\frac{{\text{D}}_{\text{o}}\text{×}\left({\text{1 + g}}_{\text{1}}\right)}{\left(\text{R}-{\text{g}}_1\right)}\right]\times \left[1-{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\right]+{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\times \frac{{\text{D}}_{\text{o}}\times \left(1+{\text{g}}_2\right)}{\left(\text{R}-{\text{g}}_2\right)}$

Where,

P_o refers to the price of the stock of current year

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 growth rate of one period

t refers to the number of times

Introduction:

Stock price is a price of a particular stock of the company. It is a financial asset of the company.

Two-stage dividend growth model is used to determine the value of a stock. This model is used to express the two stages of growth. The first stage of growth has a high growth rate and second stage has a constant growth rate.

Answer to Problem 37QP

As per the two-stage dividend growth model, the current stock price can be denoted as:

${\text{P}}_{\text{o}}=\left[\frac{{\text{D}}_{\text{o}}\text{×}\left({\text{1 + g}}_{\text{1}}\right)}{\left(\text{R}-{\text{g}}_1\right)}\right]\times \left[1-{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\right]+{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\times \frac{{\text{D}}_{\text{o}}\times \left(1+{\text{g}}_2\right)}{\left(\text{R}-{\text{g}}_2\right)}$

Where,

P_o refers to the price of the stock of current year

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 growth rate of one period

t refers to the number of times

Explanation of Solution

In the two-stage dividend growth model, the current stock price can be determined by considering the present value (PV) of first time’s (t) dividends which is the PV of a rising annuity. Then, add PV of stock price at the number of times (t) which can be expressed as shown below:

${\text{P}}_{\text{o}}=\text{PV of t dividends}+\text{PV}\left({\text{P}}_{\text{t}}\right)$

Where,

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

PV refers to thepresent value

P_t refers to the stock price at time t

t refers to the number of times

Substitute the equation for PV of a rising annuity by using the expected growth rate of dividend (g₁) which can be expressed as:

${\text{P}}_{\text{o}}={\text{D}}_{\text{1}}\times \left[\frac{1-{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}}{\left(\text{R}-{\text{g}}_1\right)}\right]+\text{PV}\left({\text{P}}_{\text{t}}\right)$

Where,

P_o refers to the price of the stock of current year

PV refers to thepresent value

P_t refers to the stock price at time t

t refers to the number of times

D₁ refers to thenext period dividend per share

R refers to the required rate of return on its stock

g₁ refers to the expected growth rate of dividend

Later, the dividend in one year can be increased at an expected growth rate of dividend (g₁) so the equation of current stock price can be expressed as:

${\text{P}}_{\text{o}}={\text{D}}_{\text{o}}\times \left(1+{\text{g}}_1\right)\times \left[\frac{1-{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}}{\left(\text{R}-{\text{g}}_1\right)}\right]+\text{PV}\left({\text{P}}_{\text{t}}\right)$

Then, rearrange the equation as follows: [Consider the equation as Equation (1)]

${\text{P}}_{\text{o}}=\frac{{\text{D}}_{\text{o}}\times \left(1+{\text{g}}_1\right)}{\left(\text{R}-{\text{g}}_1\right)}\times \left[1-{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\right]+\text{PV}\left({\text{P}}_{\text{t}}\right)$

The constant dividend growth model equation can be used to determine the price of the stock at time (t). The equation of constant dividend growth model is given below: [Consider the equation as Equation (2)]

${\text{P}}_{\text{t}}=\frac{{\text{D}}_{\text{t+1}}}{\left(\text{R}-{\text{g}}_2\right)}$

Where,

P_t refers to the price of the stock at time t

t refers to the number of times

D_t refers to thedividend per share at time t

R refers to the required rate of return on its stock

g₂ refers to the growth rate of one period

Now, consider the dividend at $\left(\text{t}+1\right)$ which can grow at an expected growth rate of dividend (g₁) for number of time/period (t) and even at growth rate of one period (g₂). In view of these factors, the equation can be expressed as follows:

${\text{D}}_{\text{t +1}}={\text{D}}_{\text{o}}\times {\left(1+{\text{g}}_1\right)}^{\text{t}}\times \left(1+{\text{g}}_2\right)$

Substitute, the equation of $\left({\text{D}}_{\text{t +1}}\right)$ in the Equation (2) which can be written as:

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

Rearrange, the equation to determine the present value of future stock price which can be expressed as follows:

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

Where,

PV (P_t) refers to the present value of the future stock price

D_o refers to the current year dividend paid

R refers to the required rate of return on its stock

t refers to the number of times

g₁ refers to the expected growth rate of dividend

g₂ refers to the growth rate of one period

Simplify the current equation into the simplest form: [Consider the equation as Equation (3)]

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

After that, substitute Equation (3) into Equation (1). The present equation will be expressed as:

${\text{P}}_{\text{o}}=\left[\frac{{\text{D}}_{\text{o}}\text{×}\left({\text{1 + g}}_{\text{1}}\right)}{\left(\text{R}-{\text{g}}_1\right)}\right]\times \left[1-{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\right]+{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\times \frac{{\text{D}}_{\text{o}}\times \left(1+{\text{g}}_2\right)}{\left(\text{R}-{\text{g}}_2\right)}$

The interpretation of equation emphasizes that the right-hand side of equation denotes the PV of the first “t” dividends. On the other hand, the left-hand side denotes the PV of stock price at the time after the dividend growth continues to be constant.

Hence, the current stock price can be denoted as:

${\text{P}}_{\text{o}}=\left[\frac{{\text{D}}_{\text{o}}\text{×}\left({\text{1 + g}}_{\text{1}}\right)}{\left(\text{R}-{\text{g}}_1\right)}\right]\times \left[1-{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\right]+{\left(\frac{\left(1+{\text{g}}_1\right)}{\left(1+\text{R}\right)}\right)}^{\text{t}}\times \frac{{\text{D}}_{\text{o}}\times \left(1+{\text{g}}_2\right)}{\left(\text{R}-{\text{g}}_2\right)}$