Dividend Valuation Model If dividends are constant forever, the value of a share of stock is the

Dividend Valuation Model If dividends are constant forever, the value of a share of stock is the

present value of the dividends per share per period, in perpetuity. Let D represent the constant dividend per share of common stock expected

next period and each period thereafter, forever, P 0 represent the price of

a share of stock today, and r e the required rate of return on common stock. The required rate of return (RRR) is the return shareholders demand to compensate them for the time value of money tied up in their investment and the uncertainty of the future cash flows from these investments.

The current price of a share of common stock, P 0 , is:

D D D P 0 = --------------------- + --------------------- + … + ---------------------

( 1 + r x e ) which we can write using summation notation,

THE FUNDAMENTALS OF VALUATION

-------------------- ∑ D

The summation of a constant amount discounted from perpetuity sim- plifies to:

P 0 = D/r e

As an example, if the current dividend is $2 per share and the required rate of return is 10%, the value of a share of stock is:

P 0 = $2/0.10 = $20

Therefore, if you pay $20 per share and dividends remain constant at $2 per share, you will earn a 10% return per year on your investment every year. But dividends on common stock often change through time.

If dividends grow at a constant rate, the value of a share of stock is the present value of a growing cash flow. Let D 0 indicate this period’s (i.e., end of period 0) dividend. If dividends grow at a constant rate, g, forever, the present value of the common stock is the present value of all future dividends:

D 0 ( 1 + g 1 D 2 ) ∞ 0 ( 1 + g ) D 0 ( 1 + g ) P 0 = ---------------------------- + ---------------------------- + … + -----------------------------

Pulling today’s dividend D 0 , from each term,

P 0 = D 0 --------------------- ( ) + --------------------- ( ) +

( ) … + ----------------------

1 + r 1 1 + r 2 1 + r ( ∞ e ) ( e ) ( e ) Using summation notation:

0 ∑ --------------------

which simplifies to:

Valuation of Securities and Options

( 1 + g ) P 0 = D 0 ------------------ ( r e – g )

If we represent the next period’s dividend, D 1 , in terms of this period’s dividend, D 0 , compounded one period at the rate g (that is, D 1 =

D 0 (1+ g)) and substitute for D 0 :

0 = -------------

This equation is referred to as the Dividend Valuation Model (DVM). 1 As an example, consider a firm expected to pay a constant dividend of $2 per share, forever. If this dividend is capitalized at 10%, the value of a share is $20. If, on the other hand, the current dividend is $2 but these dividends are expected to grow at a rate of 6% per year, forever, the value of a share of stock is $53:

( + 0.06 --------------- P $2.12 = = $53

Does this make sense compared to the constant amount case where divi- dends are unchanged at $2 per year? Yes: If dividends are expected to grow in the future, the stock is worth more than if the dividends are expected to remain the same.

If today’s value of a share is $53, what are we saying about the value of the stock next year? If we move everything up one year, D 1 is no longer $2.12, but the current dividend of $2 grows at 6% to $2(1 +

0.06) 2 = $2.2472. Therefore, we expect the price of the stock at the end of one year, P 1 , to be $5:

1 = ------------------------------------ = --------------------- = $56.18

1 The Dividend Valuation Model is attributed to Myron Gordon, who popularized the constant growth model. A more formal presentation of this model can be found

in published works by Gordon entitled “Dividends, Earnings and Stock Prices,” ( Review of Economics and Statistics, May 1959, pp. 99–105) and The Investment Financing and Valuation of the Corporation (Homewood, IL: R. D. Irwin, 1962). However, the foundation of common stock valuation is laid out—for both constant and growing dividends—by John Burr Williams in The Theory of Investment Value (Amsterdam: North-Holland Publishing Company, 1938), Chapters V, VI, and VII.

THE FUNDAMENTALS OF VALUATION

EXHIBIT 9.2 The Price of a Share of Stock with a Current Dividend of $2, a 6%

Growth in Dividends, and a 10% Required Rate of Return

At the end of two years, the price will be $59.55. Since we expect dividends to grow each year, we also are expecting the price of the stock to grow through time as well. In fact, the price is expected to grow at the same rate as the dividends: 6% per period.

g, and the price of the stock expected in the future is illustrated in Exhibit 9.2. For a given required rate of return and dividend—in this case r e = 10% and

The relation between the growth rate of dividends,

D 0 = $2—we see that the price of a share of stock is expected to grow each period at the rate g. What if the dividends are expected to decline each year? That is, what if g is negative? We can still use the Dividend Valuation Model, but each dividend in the future is expected to be less than the one before it. For example, suppose a stock has a current dividend of $2 per share and the required rate of return is 10%. If dividends are expected to decline 6% each year, what is the value of a share of stock today? We

know that D 0 = $2, r e = 10%, and g= −6%. Therefore,

$2 1 0.06 -------------------------------- ( – ) $1.88

P 0 = = --------------- = $11.75

Two periods from now, the expected price is even lower:

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P 1 = ----------------------------------- = --------------------- = $11.045

Let’s look at another situation, one in which growth is expected to change but at different growth rates as time goes on. Consider a share of common stock whose dividend is currently $3.00 per share and is expected to grow at a rate of 8% per year for five years and afterward at

a rate of 4% per year after five years. To tackle this problem, let’s break it into two manageable parts: the first five years and after five years, or:

P 0 = Present value of dividends in the first five years + Present value of dividends received after the first five years to infinity

Assuming a required rate of return of 10%,

----------------------------- 1 ----------------------------- 2 D D P D 0 = + + ----------------------------- 3 + ----------------------------- 4 + ----------------------------- 5

Dividends growing at a rate of 8% per year

D 6 D 7 D ----------------------------- -----------------------------

Dividends growing at a rate of 4% per year The present value of the dividends in the first five years is:

Present value of dividends received during the first five years $3.24 $3.4992 $3.7791 $4.0815 $4.4080

= ------------------ + --------------------- + --------------------- + --------------------- + --------------------- 1.1000 1.2100

= $2.9455 + $2.8919 + $2.8393 + $2.7877 + $2.7370 = $14.2014 The present value of dividends received after the fifth year—evaluated five

years from today—is the expected price of the stock in five years, P 5 :

D 5 ( 1 + 0.04 )

P 5 = ---------------------------------- = --------------------- = $76.4053

THE FUNDAMENTALS OF VALUATION

The price expected at the end of five years is $76.4053, which we trans- late into a value today by discounting it five years at 10%:

Present value of dividends to be received after the first five years = ----------------------------- $76.4053 = $76.4053 ------------------------- = $47.4420

Putting together the two pieces, P 0 = $14.2014 + $47.4420 = $61.6434

The value of a share of this stock is $61.6434. We can represent the Dividend Valuation Model in terms of a share’s price to earnings ratio (P/E ratio). Let’s start with the Dividend Valuation Model with constant growth in dividends:

D P 0 = ------------- 1

If we divide both sides of this equation by earnings per share, we can rep- resent the dividend valuation model in terms of the price-earnings (P/E) ratio:

D ------------ 1

------------ 0 = EPS ------------- 1

EPS 1 r e – g

Dividend payout ratio P/E = --------------------------------------------------------- r e – g

This tells us the P/E ratio is influenced by the dividend payout ratio, the required rate of return on equity, and the expected growth rate of divi- dends.

The Dividend Valuation Model makes some sense regarding the relation between the value of a share of stock, the growth in dividends, and the discount rate:

■ The greater the current dividend, the greater the value of a share of stock.

Valuation of Securities and Options

■ The greater the expected growth in dividends, the greater the value of a share of stock. ■ The more uncertainty regarding future dividends, the greater the dis- count rate and the lower the value of a share of stock.

However, the DVM has some drawbacks. How do you deal with dividends that do not grow at a constant rate? As you can see in the last example, this model does not accommodate nonconstant growth easily.

What if the firm does not pay dividends now? In that case, D 0 would be zero and the expected price would be zero. But the price of a share of stock cannot be zero. Therefore, the DVM may be appropriate to use to value the stock of companies with stable dividend policies, but it is not applicable for all firms.

Despite its drawbacks, the DVM captures the valuation for many companies’ securities. We can use the DVM to take a closer look at investors’ required rate of return and the expected rate of growth in future dividends. Moreover, the DVM has been modified to allow for different types of dividend patterns. 2