Chapter 8, Problem 35QP

Nonconstant Growth [LO1] This one’s a little harder. Suppose the current share price for the firm in the previous problem is $44.50 and all the dividend information remains the same. What required return must investors be demanding on Storico stock? (Hint: Set up the valuation formula with all the relevant cash flows, and use trial and error to find the unknown rate of return.)

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 13.10%.

Explanation of Solution

Given information:

SC Company has paid off dividends of $2.65 per share. The company increased dividends by 20% for the next year and the company even reduced its dividends’ growth rate by 5%.

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

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

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 constant rate of growth in second year

g₃ refers to the constant rate of growth in third year

g_n refers to the constant rate of growth in 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 + g}}_{\text{1}}\right)}{{\left(1+\text{R}\right)}^1}\right]+\left[\frac{{\text{D}}_{\text{0}}\times \left({\text{1 + g}}_{\text{1}}\right)\times \left({\text{1+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 + g}}_{\text{1}}\right)\\ \times \left({\text{1 + g}}_{\text{2}}\right)\times \left({\text{1 + 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+g}\right)\times \left({\text{1 +g}}_{\text{2}}\right)\times \left({\text{1+g}}_{\text{3}}\right)\times \left({\text{1+g}}_4\right)}{\left(\text{R – g}\right)}}{{\left(1+\text{R}\right)}^3}\right]\end{array}\right\}\\ \$44.50=\left[\begin{array}{l}\frac{\$2.65\times \left(1+\frac{20}{100}\right)}{\left(1+\text{R}\right)}+\frac{\$2.65\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{15}{100}\right)}{{\left(1+\text{R}\right)}^2}\\ +\frac{\$2.65\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{\$2.65\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]\\ \$44.50=\left[\begin{array}{l}\frac{\$2.65\times 1.20}{\left(1+\text{R}\right)}+\frac{\$2.65\times 1.20\times 1.15}{{\left(1+\text{R}\right)}^2}+\\ \frac{\$2.65\times 1.20\times 1.15\times 1.10}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\$2.65\times 1.20\times 1.15\times 1.10\times 0.05}{\text{R}-\text{0}\text{.05}}\right)}{{\left(1+\text{R}\right)}^3}\end{array}\right]\\ \$44.50=\frac{\text{\$3}\text{.18}}{\left(1+\text{R}\right)}+\frac{\text{\$3}\text{.657}}{{\left(1+\text{R}\right)}^2}+\frac{\text{\$4}\text{.0227}}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\text{\$4}\text{.223835}}{\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) has 13%. Then, substitute the value of the required rate of return in Equation (1) which is shown below:

$\begin{array}{l}\$44.50=\frac{\$3.18}{\left(1+\text{R}\right)}+\frac{\$3.657}{{\left(1+\text{R}\right)}^2}+\frac{\$4.0227}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\$4.223835}{\text{R}-0.05}\right)}{{\left(1+\text{R}\right)}^3}\\ \$44.50=\frac{\$3.18}{\left(1+\frac{13}{100}\right)}+\frac{\$3.657}{{\left(1+\frac{13}{100}\right)}^2}+\frac{\$4.0227}{{\left(1+\frac{13}{100}\right)}^3}+\frac{\left(\frac{\$4.223835}{\frac{13}{100}-0.05}\right)}{{\left(1+\frac{13}{100}\right)}^3}\\ \$44.50=\frac{\$3.18}{\left(1+0.13\right)}+\frac{\$3.657}{{\left(1+0.13\right)}^2}+\frac{\$4.0227}{{\left(1+0.13\right)}^3}+\frac{\left(\frac{\$4.223835}{0.13-0.05}\right)}{{\left(1+0.13\right)}^3}\\ \$44.50=\frac{\$3.18}{1.13}+\frac{\$3.657}{{\left(1.13\right)}^2}+\frac{\$4.0227}{{\left(1.13\right)}^3}+\frac{\left(\frac{\$4.223835}{0.08}\right)}{{\left(1.13\right)}^3}\end{array}$

$\begin{array}{l}\$44.50=\$2.81415+\frac{\$3.657}{1.2769}+\frac{\$4.0227}{1.44289}+\frac{\$52.79793}{1.44289}\\ \$44.50=\$2.81415+\$2.86396+\$2.78794+\$36.59179\\ \$44.50\ne \$45.05\end{array}$

Hence, the required rate of return is not 13% since the current stock price, $44.50, 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 has 13.10% to the same Equation (1), which is shown below:

$\begin{array}{l}\$44.50=\frac{\$3.18}{\left(1+\text{R}\right)}+\frac{\$3.657}{{\left(1+\text{R}\right)}^2}+\frac{\$4.0227}{{\left(1+\text{R}\right)}^3}+\frac{\left(\frac{\$4.223835}{\text{R}-0.05}\right)}{{\left(1+\text{R}\right)}^3}\\ \$44.50=\frac{\$3.18}{\left(1+\frac{13.10}{100}\right)}+\frac{\$3.657}{{\left(1+\frac{13.10}{100}\right)}^2}+\frac{\$4.0227}{{\left(1+\frac{13.10}{100}\right)}^3}+\frac{\left(\frac{\$4.223835}{\frac{13.10}{100}-0.05}\right)}{{\left(1+\frac{13.10}{100}\right)}^3}\\ \$44.50=\frac{\$3.18}{\left(1+0.131\right)}+\frac{\$3.657}{{\left(1+0.131\right)}^2}+\frac{\$4.0227}{{\left(1+0.131\right)}^3}+\frac{\left(\frac{\$4.223835}{0.131-0.05}\right)}{{\left(1+0.131\right)}^3}\\ \$44.50=\frac{\$3.18}{1.131}+\frac{\$3.657}{{\left(1.131\right)}^2}+\frac{\$4.0227}{{\left(1.131\right)}^3}+\frac{\left(\frac{\$4.223835}{0.081}\right)}{{\left(1.131\right)}^3}\end{array}$

$\begin{array}{l}\$44.50=\$2.81167+\frac{\$3.657}{1.279161}+\frac{\$4.0227}{1.44673}+\frac{\$52.1461}{1.44673}\\ \$44.50=\$2.81167+\$2.85890+\$2.78054+\$36.04411\\ \$44.50=\$\text{44}\text{.50}\end{array}$

Hence, the required rate of return is 13.10%. It is because the current stock price, $44.50, is equal to the derived (stock price) value to the assumed required rate of return (R) as per the trial and error method.