4 Stock Characteristics

Earlier in the text we saw that firms can raise capital by using debt, that is to say, by borrowing. You learned that one way firms can borrow from investors is by issuing bonds. You examined the pricing and returns of bonds in detail. We now turn to a second way that firms can raise capital. Firms can also raise capital by issuing (selling) equity (stocks). Equity represents an ownership claim on a firm. This is in contrast to bonds which are a debt claim. You have encountered the term equity when you learned basic financial accounting. From the balance sheet we have

          Assets = Liabilities + Owners’ Equity

The balance sheet identity is just an accounting way to emphasize where the firm gets the money to buy the firm’s assets, and who owns those assets. Before we learn one way to determine the price of a stock and its returns, we first look at some characteristics of stocks.

4.1 Characteristics of Stock

Stock represents partial ownership in a company.

You will often hear the term “shares” of stock when referring to ownership. For example, “Warren Buffet owns 20,000 shares of XYZ company,” or “Bill Gates own 1.3% of the shares of Microsoft.”

The term “share” should help you remember that since stock represents partial ownership, your ownership of stock is your, “share of the company.”

Ownership implies control of how the company is operated through voting rights.

Ownership of a stock gives you certain rights with respect to a company, and one of those rights is who will run the company. As a shareholder you get to select the Board of Directors which is the governing body (group of people) that hire the management of the company and make strategic decisions about what the company will do and how.

You select the Board of Directors by voting, where your voting rights are proportional to the percentage of the company’s stock you own. For example, if you own 1,000 shares of XYZ company and I only own 500 shares, you have two times as many votes as I do, and therefore, have twice as much influence on how the company is run.

For many company’s the number of shares of stock can be many millions. For example, Exxon-Mobile Company consists of millions of shares of stock. With such a large number of shares, for most of its shareholders the number of shares represents such a small fraction of the shares that investors often don’t bother voting.

Sometimes management or institutional owners will request that individual shareholders give their votes to them so that they can combine the votes of smaller shareholders into a voting block. This process is known as proxy voting.

Stock represents a residual claim on the firm’s assets.

Most commonly, the most important right owning a stock gives you is the right to the assets of the firm AFTER all of the other liabilities of the company have been paid. Since the stockholders get paid after everyone else, we say that the stockholders are “residual claimants.”

The periodic cash-flows paid to the owner of a stock are called dividends.

For tax purposes, dividends paid to stockholders are NOT income tax deductible

While an interest payment to a bondholder is a tax deductible expense, when the firm makes a dividend payment to the stockholders, the dividend is NOT a tax deductible expense. This is just a feature of current (2018) US tax regulations.

Dividends are not legally required to be paid to the stockholders.

Recall that the coupons and face value from a bond are required to be paid to the bondholders on scheduled days. Failure to do so can trigger a bankruptcy event for a firm. This is not the case with dividends. Even though the firm may pay regular dividends, it is NOT required to pay a dividend. However, failure to pay a scheduled dividend or pay a dividend that is smaller than anticipated, will usually cause a drop in the firm’s stock price.

4.2 Stock Valuation

Stocks are just another type of asset that an investor can buy. In this section we examine how to determine the price of a stock using its cash-flows. We will consider three alternative cash-flow patterns for the stock. Note that there are other methods for estimating stock prices that are based on other company features instead of cash-flows, but we will leave these for others to cover.

You know that the price of any asset is the present value of the cash-flows the asset is expected to pay its owner using a discount rate appropriate for the asset’s riskiness. This same principle applies to stocks, so we begin our study of the price of a stock by writing down the cash-flows of a stock.

\[P_0=\frac{D_1}{(1+R)^1}+\frac{D_2}{(1+R)^2}+\frac{D_3}{(1+R)^3}+... \]

where each “D” is the dividend the stock is expected to pay in a particular period and R is the discount rate for the stock.

First note that, unlike most bonds, the cash-flows of a stock (its dividends) are expected to go on forever (assuming the company doesn’t go bankrupt). Also, unlike most bonds, where the coupon payments are the same throughout the life of the bond, with a stock, each dividend has the possibility of being unique. Unfortunately, when a stock’s dividends are so general (go on forever and can be anything) we cannot find the price of the stock. To address this problem, we must assume some structure or pattern for the dividends. We will look at three possible patterns for the dividends.

1. Dividends are constant forever (the perpetuity case)
2. Dividends grow at a constant rate forever (the Gordon Model)
3. Dividends grow for a finite amount of time at one growth rate and then at a constant rate forever (supernormal growth)

Dividends are constant forever (the perpetuity case)

The simplest pattern for a stock’s dividends is to assume that the dividends are constant and are expected to go on forever.

\[P_0=\frac{D}{(1+R)^1}+\frac{D}{(1+R)^2}+\frac{D}{(1+R)^3}+... \] The formula looks very similar to what we had before, but all of the dividends are the same. This particular cash-flow pattern is known as a perpetuity and fortunately, we have a formula for the price. The price of a perpetuity is:

\[P_0=\frac{D}{R} \]

We simply take the dividend the company pays, D, and divide it by the appropriate discount rate, R. For example, assume that JMG Corporation has paid a constant yearly dividend of $5.00 for the last 20 years. Consequently, you estimate that it will continue to pay the same dividend into the future. In addition, JMG Corporation is considered very safe so its discount rate, R, is equal to 3%. We can summarize these facts as:

\[P_0=\frac{$5.00}{(1+0.03)^1}+\frac{$5.00}{(1+0.03)^2}+\frac{$5.00}{(1+0.03)^3}+... \]

So the price of JMG Corporation is

\[P_0=\frac{$5.00}{0.03}=$166.67 \]

Watch the following for a summary of the above principles.



Dividends grow at a constant rate forever (the Gordon Model)

A second possible pattern for a stock’s dividends is the Gordon Model. With the Gordon Model the dividends are expected to grow at a constant rate forever. To see how this works we need to add some additional notation.

\(D_i\) is the dividend for period i

\(g\) is the growth rate in the dividends from one period to the next

\(R\) is the discount rate appropriate for the firm’s riskiness

With this notation we have that the dividend in any period is

\(D_i=D_0(1+g)^i\)

and the cash-flows of the stock are as shown.

\[P_0=\frac{D_1}{(1+R)^1}+\frac{D_2}{(1+R)^2}+\frac{D_3}{(1+R)^3}+... \] With some algebra, this formula for the stock price simplifies to

\[P_0=\frac{D_1}{R-g}=\frac{D_0(1+g)^1}{R-g}\]

the “Gordon Model”.

To use the Gordon model, it must be the case that R>g otherwise the model is not applicable. Let’s apply the formula to some examples.

Example 1 Gordon Model

JMG Corporation has introduce a new line of its hair care products and sales are through the roof. As a consequence, you expect that its dividends will grow at a yearly rate of 1% into the distant future. In addition to the increased dividends, the new product line has increased the riskiness of JMG and so the discount rate has increased to 3.5%.

If the current dividend for JMG Corp. is $5.00, what is the price of JMG’s stock according to the Gordon Model?

Using the basic cash-flow expression, we have

\[P_0=\frac{\$5.00(1+0.01)^1}{(1+0.035)^1}+\frac{\$5.00(1+0.01)^2}{(1+0.035)^2}+\frac{\$5.00(1+0.01)^3}{(1+0.035)^3}+... \]

\[P_0=\frac{\$5.00(1+0.01)^1}{0.035-0.01}=\frac{\$5.05}{0.035-0.01}=\$202\] JMG Corporation stock would sell for $202.

Example 2 Gordon Model

MJG Corp is a pharmaceutical company that developed a highly effective cholesterol drug. Starting next year, the company’s patent on the drug will expire and you expect that many generic versions of the drug will be available. As a consequence, you forecast that MJG’s dividends will decrease at a rate of 2% into the distant future.

If the current dividend for MJG Corp. is $6.00, what is the price of JMG’s stock according to the Gordon Model if MJG’s discount rate is 5%?

This is an example where the “growth rate” in dividends is negative, but we can proceed with the Gordon Model as usual.

Using the basic cash-flow expression, we have

\[P_0=\frac{\$6.00(1-0.02)^1}{(1+0.05)^1}+\frac{\$6.00(1-0.02)^2}{(1+0.05)^2}+\frac{\$6.00(1-0.02)^2}{(1+0.05)^2}+... \]

\[P_0=\frac{\$6.00(1-0.02)^1}{0.05-(-0.02)}=\frac{\$5.88}{0.05-(-0.02)}=\$84\]

MJG Corporation stock would sell for $84.

Watch the following for a summary of the above principles.



Step-by-step guide

Problem: Finding the price of a stock with constant dividend growth. Formula: Gordon Growth Model

\[P_0=\frac{D_1}{R-g}\] \(P_0\)=the stock price at time 0

\(D_1\)=the expected dividend payment at time 1

Note the dividend is always from the period following the stock price period

\(g\)=the expected growth rate in dividends for rest of time

Process:

Find \(D_1\) \(D_1\)=\(D_0(1+g)\)

Find \(D_2\) \(D_2=D_1(1+g)\)

Find \(P_0\) \(P_0=D_1/(R-g)\)

Find \(P_1\) \(P_1=D_2/(R-g)\)

Dividend yield = \(D_1/P_0\times100\%\)

Capital gain/loss =\(((P_1-P_0)/P_0 )\times100\%\)

Total Return = Dividend Yield + Capital Gain/Loss = \(R_s\)

Given the following information calculate the following:

(1) The price of the stock today.
(2) The dividend yield of the stock.
(3) The capital gains/loss on the stock 
(4) The total return for the stock

Example 1

\(D_0\)=$2.25 \(R\)=5% \(g\)=2%

Find \(D_1\): \(D_1=D_0(1+g) \rightarrow D_1=\$2.25(1+0.02)=\$2.2950\)

Find \(D_2\): \(D_2=D_1 (1+g) \rightarrow D_2=\$2.2950(1+0.02)=\$2.3409\)

Find \(P_0\): \(P_0=D_1/(R-g) \rightarrow P_0=\$2.2950/(0.05-0.02)=\$76.50\)

Find \(P_1\): \(P_1=D_2/(R-g) \rightarrow P_1=\$2.3409/(0.05-0.02)=\$78.03\)

\(Dividend\;yield=D_1/P_0 \times 100\% \rightarrow \$2.2950/\$76.50 \times 100\%=3.00\%\)

\(Capital\;gain/loss=((P_1-P_0)/P_0 )\times100\% \rightarrow ((\$78.03-\$76.50)/\$76.50)x100\%=2.00\%\)

\(Total\;Return = Dividend\;Yield + Capital\;gain/loss = R\)

\(5\% =3.00\%+2.00\%=R\)

Example 2

\(D_0\)=$3.50 \(R\)=7% \(g\)=-3%

Find \(D_1\): \(D_1=D_0(1+g) \rightarrow D_1=\$3.50(1-0.03)=\$3.3950\)

Find \(D_2\): \(D_2=D_1 (1+g) \rightarrow D_2=\$3.3950(1-0.03)=\$3.2932\)

Find \(P_0\): \(P_0=D_1/(R-g) \rightarrow P_0=\$3.3950/(0.07-(-0.03))=\$33.950\)

Find \(P_1\): \(P_1=D_2/(R-g) \rightarrow P_1=\$3.2932/(0.07-(-0.03))=\$32.932\)

\(Dividend\;yield =D_1/P_0 \times 100\% \rightarrow \$3.3950/\$33.950 \times 100\%=10.00\%\)

\(Capital\;gain/loss =((P_1-P_0)/P_0 )\times100\% \rightarrow ((\$32.932-\$33.950)/\$33.950)\times100\%=-3.00\%\)

\(Total\;Return =Dividend\;Yield+Capital\;gain/loss =R\) \(7\% =10.00\%+(-3.00\%)=R\)

Click here for: Worked Problems 14: Gordon Model

Click here for: In-Class Exercise 17: Stock Valuation – Constant Dividend Growth

Click here if you want to navigate to the end of the chapter, where you can get some practice on all calculations for stocks with constant dividend growth, using the practice questions widget.

4.3 Dividends grow for a finite amount of time at one growth rate and then at a constant rate forever (supernormal growth)

Under some circumstances, a firm’s dividends will grow at a rate faster than the discount rate. In terms of the Gordon Model, this would mean, R<g and the Gordon Model would not be applicable since it would predict a negative price for the stock.

\(P_0=D_1/(R-g)\) (the denominator would be negative).

We can accommodate dividends growing faster than the discount rate (we call this “supernormal growth”) as long as the supernormal growth does not continue forever. The way we address supernormal growth is to break the firm’s dividend pattern into sections, some with supernormal growth and some with constant growth. Let’s look at an example.

Example 1 Supernormal Growth

JMG Corporation runs an advertising campaign that will increase sales and cause dividends to grow at a rate of 5% from today until year three. After year 3, its dividends will grow at a constant rate of 1% for the rest of time. JMG’s current dividend is $5.00 and its discount rate is 3%. With the above information, what is the current price of JMG?

First, draw the timeline for the dividends.

Make an indication of the growth in dividends between each period. Make sure to understand that the 1% growth continues forever from period 3 onward.

Calculate the lump sum value of all the dividends from period 4 onward using the Gordon Model. You can use the Gordon Model because the growth rate is constant (at 1%) and less than the discount rate of 3%. The Gordon model converts the infinite amount of dividends to a single lump sum value of $292.30 using the formula shown. The lump sum value is as of period 3. This value is shown in period 3 on the time line.

Next, add up the dividend and the lump sum in period 3 to get a cash-flow of $298.09 in period 3.

Finally, discount each lump sum (from periods 1, 2 and 3) back to today at the firm’s discount rate. This gives the price of JMG Corp stock as $283.09.

With this summary in hand, let’s do another example.

Example 2

MJG Corporation has developed a new drug and will have exclusive rights to sell it for the next three years. During those three years, its dividends will grow at a rate of 7% per year. After year 3, its dividends will grow at a constant rate of 2% for the rest of time. MJG’s current dividend is $3.00 and its discount rate is 4%. With the above information, what is the current price of MJG?

First draw the time line and put in the dividends.

Next use the Gordon Model to calculate the lump sum of the infinite stream of dividends.

Next, add up the dividend and the lump sum in period 3 to get a cash-flow of $191.1051 in period 3. Finally, discount each lump sum (from periods 1, 2 and 3) back to today at the firm’s discount rate.

This gives the price of MJG Corp stock as $176.15

Watch the following to review the principles just described.



Click here for: Worked Problems 15: Supernomral Growth

Click here for: In-Class Exercise 18: Stock Valuation – Supernormal Growth

Click here if you want to navigate to the end of the chapter, where you can get some practice on calculating the price for stocks with supernormal dividend growth, using the practice questions widget.

4.4 Interactive Practice Questions: Characteristics of Stocks