Chapter 8, Problem 35QP

Summary Introduction

To determine: The required rate of return.

Introduction:

Rate of return is a loss or gain incurred on the investment made by the investors. It is expressed in terms of percentage.

Stock is a type of security in a company which denotes ownership. On issuing stocks, the company can raise capital.

Answer to Problem 35QP

The required rate of return is 12.75%.

Explanation of Solution

Given information:

SC Company has paid off dividends of $2.25 per share. The company increased dividends by 20% for the next year and the company even reduced its dividends’ growth rate by 5%. The required rate of return is 12%. The current share price of the firm is $39.52.

The formula to calculate the required rate of return using stock price formula:

${\text{P}}_{\text{0}}=\left\{\begin{array}{l}\left[\frac{{\text{D}}_{\text{0}}\times \left(\text{1}+{\text{g}}_{\text{1}}\right)}{{\left(1+\text{R}\right)}^1}\right]+\left[\frac{{\text{D}}_{\text{0}}\times \left(\text{1}+{\text{g}}_{\text{1}}\right)\times \left(\text{1}+{\text{g}}_{\text{2}}\right)}{{\left(1+\text{R}\right)}^2}\right]+\left[\frac{\left(\begin{array}{l}{\text{D}}_{\text{0}}\times \left(\text{1}+{\text{g}}_{\text{1}}\right)\\ \times \left(\text{1}+{\text{g}}_{\text{2}}\right)\times \left(\text{1}+{\text{g}}_{\text{3}}\right)\end{array}\right)}{{\left(1+\text{R}\right)}^3}\right]\\ +\left[\frac{\frac{{\text{D}}_{\text{0}}\times \left(\text{1}+\text{g}\right)\times \left(\text{1}+{\text{g}}_{\text{2}}\right)\times \left(\text{1}+{\text{g}}_{\text{3}}\right)\times \mathrm{...}\times \left(\text{1}+{\text{g}}_n\right)}{\left(\text{R}–\text{g}\right)}}{{\left(1+\text{R}\right)}^3}\right]\end{array}\right\}$

Where,

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

D_orefers 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 second year

g₃ refers to the constant rate of growthin third year

g_n refers to the constant rate of growthin n number of year

Compute the required rate of return using stock price formula:

$\begin{array}{c}{\text{P}}_{\text{0}}=\left\{\begin{array}{l}\left[\frac{{\text{D}}_{\text{0}}\times \left(\text{1}+{\text{g}}_{\text{1}}\right)}{{\left(1+\text{R}\right)}^1}\right]+\left[\frac{{\text{D}}_{\text{0}}\times \left(\text{1}+{\text{g}}_{\text{1}}\right)\times \left(\text{1}+{\text{g}}_{\text{2}}\right)}{{\left(1+\text{R}\right)}^2}\right]\\ +\left[\frac{\left(\begin{array}{l}{\text{D}}_{\text{0}}\text{×}\left(\text{1}+{\text{g}}_{\text{1}}\right)\\ \times \left(\text{1}+{\text{g}}_{\text{2}}\right)\times \left(\text{1}+{\text{g}}_{\text{3}}\right)\end{array}\right)}{{\left(1+\text{R}\right)}^3}\right]\\ +\left[\frac{\frac{{\text{D}}_{\text{0}}\times \left(\text{1}+\text{g}\right)\times \left(\text{1}+{\text{g}}_{\text{2}}\right)\times \left(\text{1}+{\text{g}}_{\text{3}}\right)\times \left(\text{1}+{\text{g}}_4\right)}{\left(\text{R}–\text{g}\right)}}{{\left(1+\text{R}\right)}^3}\right]\end{array}\right\}\\ \$39.52=\left[\begin{array}{l}\frac{\$2.25\times \left(1+\frac{20}{100}\right)}{\left(1+\text{R}\right)}+\frac{\$2.25\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{15}{100}\right)}{{\left(1+\text{R}\right)}^2}\\ +\frac{\$2.25\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{15}{100}\right)\times \left(1+\frac{10}{100}\right)}{{\left(1+\text{R}\right)}^3}\\ +\frac{\frac{\$2.25\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{15}{100}\right)\times \left(1+\frac{10}{100}\right)\times \left(1+\frac{5}{100}\right)}{\left(\text{R}-\frac{5}{100}\right)}}{{\left(1+\text{R}\right)}^3}\end{array}\right]\\ \$39.52=\left[\begin{array}{l}\frac{\$2.25\times 1.20}{\left(1+\text{R}\right)}+\frac{\$2.25\times 1.20\times 1.15}{{\left(1+\text{R}\right)}^2}+\\ \frac{\$2.25\times 1.20\times 1.15\times 1.10}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\$2.25\times 1.20\times 1.15\times 1.10\times 1.05}{\text{R}-\text{0}\text{.05}}\right)}{{\left(1+\text{R}\right)}^3}\end{array}\right]\\ \$39.52=\frac{\text{\$2}\text{.7}}{\left(1+\text{R}\right)}+\frac{\text{\$3}\text{.105}}{{\left(1+\text{R}\right)}^2}+\frac{\text{\$3}\text{.4155}}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\text{\$3}\text{.586275}}{\text{R}-\text{0}\text{.05}}\right)}{{\left(1+\text{R}\right)}^3}\end{array}$

Consider the last equation above as Equation (1). This last equation can be even simplified by using trial-and-error method, so assume the required rate of return (R) as 13%. Then, substitute the value of the required rate of return in Equation (1) as shown below:

$\begin{array}{l}\$39.52=\frac{\$2.7}{\left(1+\text{R}\right)}+\frac{\$3.105}{{\left(1+\text{R}\right)}^2}+\frac{\$3.4155}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\$3.586275}{\text{R}-0.05}\right)}{{\left(1+\text{R}\right)}^3}\\ \$39.52=\frac{\$2.7}{\left(1+\frac{13}{100}\right)}+\frac{\$3.105}{{\left(1+\frac{13}{100}\right)}^2}+\frac{\$3.4155}{{\left(1+\frac{13}{100}\right)}^3}+\frac{\left(\frac{\$3.586275}{\frac{13}{100}-0.05}\right)}{{\left(1+\frac{13}{100}\right)}^3}\\ \$39.52=\frac{\$2.7}{\left(1+0.13\right)}+\frac{\$3.105}{{\left(1+0.13\right)}^2}+\frac{\$3.4155}{{\left(1+0.13\right)}^3}+\frac{\left(\frac{\$3.586275}{0.13-0.05}\right)}{{\left(1+0.13\right)}^3}\\ \$39.52=\frac{\$2.7}{1.13}+\frac{\$3.105}{{\left(1.13\right)}^2}+\frac{\$3.4155}{{\left(1.13\right)}^3}+\frac{\left(\frac{\$3.586275}{0.08}\right)}{{\left(1.13\right)}^3}\\ \$39.52=\$2.3893+\frac{\$3.105}{1.2769}+\frac{\$3.4155}{1.44289}+\frac{\$44.82843}{1.44289}\\ \$39.52=\$2.3893+\$2.4316+\$2.3671+\$31.0685\\ \$39.52\ne \$38.2565\end{array}$

Hence, the required rate of return is not 13% since the current stock price is $39.52, and is not equal to the derived value (stock price) to the assumed required rate of return (R) as per the trial-and-error method. Next, assume the required rate of return as 12.75% to the same Equation (1 as shown below:

$\begin{array}{l}\$39.52=\frac{\$2.7}{\left(1+\text{R}\right)}+\frac{\$3.105}{{\left(1+\text{R}\right)}^2}+\frac{\$3.4155}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\$3.586272}{\text{R}-0.05}\right)}{{\left(1+\text{R}\right)}^3}\\ \$39.52=\frac{\$2.7}{\left(1+\frac{12.75}{100}\right)}+\frac{\$3.105}{{\left(1+\frac{12.75}{100}\right)}^2}+\frac{\$3.4155}{{\left(1+\frac{12.75}{100}\right)}^3}+\frac{\left(\frac{\$3.586272}{\frac{12.75}{100}-0.05}\right)}{{\left(1+\frac{12.75}{100}\right)}^3}\\ \$39.52=\frac{\$2.7}{\left(1+0.1275\right)}+\frac{\$3.105}{{\left(1+0.1275\right)}^2}+\frac{\$3.4155}{{\left(1+0.1275\right)}^3}+\frac{\left(\frac{\$3.586272}{0.0775}\right)}{{\left(1+0.1275\right)}^3}\\ \$39.52=\frac{\$2.7}{1.1275}+\frac{\$3.105}{{\left(1.1275\right)}^2}+\frac{\$3.4155}{{\left(1.1275\right)}^3}+\frac{46.2744}{{\left(1.1275\right)}^3}\\ \$39.52=\$2.39467+\frac{\$3.105}{1.271}+\frac{\$3.4155}{1.433}+\frac{\$46.2744}{1.433}\\ \$39.52=\$2.395+\$2.443+\$2.38+\$32.30\\ \$39.52=\$39.52\end{array}$

Hence, the required rate of return is 12.75%. It is because the current stock price is $39.52, which is equal to the derived (stock price) value to the assumed required rate of return (R) as per the trial and error method.