Chapter 7, Problem 32QP

a)

Summary Introduction

To determine: The current share price.

Introduction:

Stock price refers to the price of a particular share of a company. It is the financial asset of a company.

Answer to Problem 32QP

The current stock price is $56.53.

Explanation of Solution

Given information:

A Company has currently paid dividends of $3.20 per share. The constant rate of dividend is 6% and the required rate of return is 12%.

Formula to calculate the current stock price:

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

Where,

P_o refers to the present price of stock,

D_o refers to the current year dividend paid,

R refers to the required rate of return on its stock,

g refers to the constant growth rate.

Compute the current stock price:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$3.20\times \frac{\left(1+\frac{6}{100}\right)}{\left(\frac{12}{100}-\frac{6}{100}\right)}\\ =\$3.20\times \frac{1+0.06}{\left(0.12-0.06\right)}\\ =\$3.20\times \frac{1.06}{\text{0}\text{.06}}\end{array}$

$\begin{array}{c}=\$3.20\times 17.667\\ =\$56.53\end{array}$

Hence, the current price of the stock is $56.53.

b)

Summary Introduction

To determine: The value of the current share price.

Introduction:

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

Answer to Problem 32QP

The value of the current share price is $59.02.

Explanation of Solution

Given information:

The current annual dividend paid by the company is $3.20. The constant rate of dividend is 6% and dividends are paid in equal quarterly instalments to the company’s shareholders.

Note:

Determine the value of current share price:

• Firstly, determine the quarterly dividend, when the company pays quarterly dividend instead of annual dividends.
• Then, find the effective quarterly rate and effective annual dividend. This effective annual dividend will be the future value of annuity of quarterly dividend payments at an effective quarterly rate.
• Finally, determine the current stock price using the constant growth model.

Formulae:

Formula to calculate the quarterly dividends:

$\text{Quarterly dividend}=\frac{\text{Annual dividend }\times \left(1+\text{g}\right)}{4}$

Where,

g refers to the constant growth rate.

Formula to calculate effective quarterly rate:

$\text{Effective quarterly rate}={\left(1+\text{R}\right)}^{\frac{1}{4}}-1$

Where,

R refers to the required rate of return on the stock of the company.

Formula to calculate effective annual dividend:

$\text{Effective annual dividend}=\text{Quarterly dividend }\times \left(\begin{array}{l}\text{Future value interest factor of }\\ \text{an ordinary annuity of \$1 per period }\\ \text{at i \% for n periods}\end{array}\right)$

Formula to calculate the current stock price:

${\text{P}}_o=\frac{\text{Effective annaul dividend}}{\left(\text{R}-\text{g}\right)}$

Where,

g refers to the constant growth rate,

R refers to the required rate of return on the stock of the company,

P_o refers to the present price of stock.

Compute the quarterly dividends:

$\begin{array}{c}\text{Quarterly dividend}=\frac{\text{Annual dividend }\times \left(1+\text{g}\right)}{4}\\ =\frac{\$3.20\times \left(1+\frac{6}{100}\right)}{4}\\ =\frac{\$3.20\times \left(1.06\right)}{4}\end{array}$

$\begin{array}{c}=\frac{3.392}{4}\\ =\$0.848\end{array}$

Hence, the quarterly dividends are $0.848.

Compute the effective quarterly rate:

$\begin{array}{c}\text{Effective quarterly rate}={\left(1+\text{R}\right)}^{\frac{1}{4}}-1\\ ={\left(1+\frac{12}{100}\right)}^{0.25}-1\\ ={\left(1.12\right)}^{0.25}-1\end{array}$

$\begin{array}{c}=1.02874-1\\ =0.02874\end{array}$

Hence, the effective quarterly rate is 0.02874 or 2.874%.

Compute the effective annual dividend:

Note: To determine the future value of interest factor of an ordinary annuity of $1 period at 2.874% for 4 periods, refer the future value of an interest factor annuity table.

Here, the value for the rate 2.874% and 4 years period is 4.1758.

$\begin{array}{c}\text{Effective annual dividend}=\left[\begin{array}{c}\text{Quarterly dividend }\\ \times \left(\begin{array}{l}\text{Future value interest factor of }\\ \text{an ordinary annuity of \$1 per period }\\ \text{at i \% for n periods}\end{array}\right)\end{array}\right]\\ =\left[\$0.848\times \left(\begin{array}{l}\text{Future value interest factor of }\\ \text{an ordinary annuity of \$1 per period }\\ \text{at 2}\text{.873\% for 4 periods}\end{array}\right)\right]\\ =\$0.848\times \text{4}.\text{1758}\\ =\$\text{3}\text{.541}\end{array}$

Hence, the effective annual dividend is $3.541.

Compute the current stock price:

$\begin{array}{c}{\text{P}}_o=\frac{\text{Effective annaul dividend}}{\left(\text{R}-\text{g}\right)}\\ =\frac{\$3.541}{\left(\frac{12}{100}-\frac{6}{100}\right)}\\ =\frac{\$\text{3}\text{.541}}{\left(0.12-0.06\right)}\end{array}$

$\begin{array}{c}=\frac{\$\text{3}\text{.541}}{0.06}\\ =\$59.02\end{array}$

Hence, the current stock price is $59.02.

Interpretation on the model of stock valuation:

The stock valuation cannot be determined by computing the effective required return and growth rate, because it will assume that the dividends will increase in each quarter and not in each year.

However, the model would be appropriate if the investor reinvests the dividends at the stock’s required rate of return.