Dividend Discount Models ... for pricing stocks (or other equities)
a topic suggested by Keith B

Suppose I agree to pay you $25, two years from now. How much would you be willing to pay me for this risk-free investment?

>I have no idea, but I know I can get a 5% return so ...
A 5% risk-free return? Okay, then every dollar you invest at 5% would be worth 1.052 = 1.1025 so you would be willing to pay me 25/1.1025 or $22.68 because ...

>Because $22.68, invested at 5% for 2 years, would be worth $25. Right?
Right, and in general, if I agreed to pay you $D in N years then, at an annual discount of R (where, for 5%, we'd put R=0.05), that $D should be worth D/(1+R)N today - that's the Present or Discounted Value.

Further, if I agreed to pay you $D every year for N years then, at an annual rate of R, the Present Value or Discounted Value of this series of payments would be:

(1)       V = D + D/(1+R) + D/(1+R)2 + D/(1+R)3 + ... + D/(1+R)N-1 = D (1 - XN)/(1 - X)       where X = 1/(1+R)

>This has something to do with stock prices, eh?
Well, suppose you were interested in General Electric stock which currently pays a dividend of D = $0.72 so, at a 5% annual Discount Rate, we might estimate a "fair price" for GE stock by using Equation (1), namely:

      X = 1/1.05 = 0.9524 so       V = 0.72 { 1 - (0.9524)N }/(1-.9524)

which, of course, depends upon the number of years, N, and R, the assumed Discount Rate so Figure 1 shows several scenarios and ...

>Are you saying GE stock should be worth, say $10 - $15?
Not exactly. GE currently trades at about $30 so ...

Figure 1

>Aha! This Dividend Discount thing is lousy!
Aah, but what it's worth is what investors are willing to pay and they may expect the dividends to increase from year to year. Suppose, for example, that the current $D dividend is expected to increase by a Gain Factor G per year (an 8% annual increase means Gain Factor G = 1.08).

Note #1: for convenience, we'll call the annual growth rate g and the annual growth FACTOR G = 1 + g so for an 8% growth rate then g = 0.08 and G = 1.08 and ...

>Okay, I get it. Please continue.

Okay. With dividends of D, GD, G2D, G3D etc., then Equation (1) becomes:

(2)       V = D + GD/(1+R) + G2D/(1+R)2 + G3D/(1+R)3 + ... + GN-1D/(1+R)N-1 = D (1 - XN)/(1 - X)       where X = G/(1+R)
For the GE example, and an expected 5% annual increase in dividends, we can compare the more sophisticated
Equation (2) (which incorporates increases in dividends)
with Equation (1) (with constant dividends)

>$30 is looking better, for GE. Have dividends increased by 5%, historically?
I don't know, but the point is that we must estimate future dividends and future Discount Rates and dividends may increase with a company's increase in earnings and, even then, a company may not put all increased earnings into dividends and ...

>And may not even pay dividends!

Figure 2
The Dividend Discount Model we've been discussing is the Williams Model, after John Burr Williams who published "The Theory of Investment Value" in 1938, saying:
"Let us define investment value of a stock as the present worth of all dividends to be paid upon it .
To appraise the investment value then it is necessary to estimate the future payments.
The annuity of payments, adjusted for changes in the value of money itself, may be discounted at the pure interest rate demanded by the investor."

Anyway, we'll modify this model in order to reflect the fact that dividends should depend upon earnings and the Discount Rate should be related to expected returns and ...

>So do it!

Yes, but before we do it, note that if X = G/(1+R) is less than "1", in Equation (2), then XN gets smaller and smaller as N increases and this term may be ignored for long time periods (meaning large values of N) and Equation (2) simplifies to:

(2a)       V = D/(1 - X) where X = G/(1+R)

Indeed, it's convenient to note that, for an infinite sum:

    1 + X + X2 + X3 + ... = 1/(1 - X) provided -1 < X < 1.

If we play with Equation (2) and solve for R, we get:

(2b)       R = (g + D/V)/(1-D/V) where we've put G = 1 + g (as per Note #1, above)

Note that D/V (Dividend/StockPrice) is the Dividend Yield and is normally pretty small.
(Example: For GE, at a price of $30 and a Dividend of $0.72, the Dividend Yield is 0.72/30 = 0.024 or 2.4%)

Anyway, if we ignore the D/V in the denominator, in Equation (2b), we get the Gordon Growth Model relating the stock price, the current dividend, the expected dividend growth rate and the discount rate (as popularized in a book "The Investment, Financing and Valuation of the Corporation", written in 1962 by Myron J. Gordon).

(2c)       R = g + D/V = Dividend Growth Rate + Dividend Yield

For GE, assuming that g = 0.08 and D = 0.72 and V = 30 we get R = 0.08 + 0.72/30 = 0.104 and ...

>Which means ... what?
It suggests a Discount Rate of R = 10.4% and ...
>Which means ... what?
It means that an investor who pays $30 for this stock is expecting a return of 10.4%.

>I have a stock which pays NO dividends!
The Dividend Growth Rate is 0%.
The Dividend Yield is 0%.
So I should expect a 0% return, eh?

Okay, we'll take another tack where, now, g stands for the Earnings Growth.
(These days, it seems more fashionable to consider earnings growth rather than dividends.)

Here's what we'll do:

  • Instead of D, in Equation (2), we put E, the current Earnings per Share (EPS).
  • We'll assume an Earnings Growth (at the Rate g) for N years, then constant EPS thereafter.
  • We consider the Present or Discounted Value of all Earnings: growing for N years, then constant thereafter.
  • After N years, Earnings are E(N) = E GN and the Discounted or Present Value of all Earnings beyond N years is:
    E(N)/(1+R)N + E(N)/(1+R)N+1 + E(N)/(1+R)N+2 + ... = E GN/(1+R)N/{1 - 1/(1+R)} = E XN {1+1/R}
  • Equation (2) then becomes:

(2d)       V = E (1 - XN)/(1 - X) + E XN(1+1/R)       where X = (1+g)/(1+R) = G/(1+R)

We can estimate the "fair value" of the (current) Stock Price, using Equation (2d), like so:
V = E (1 - XN)/(1 - X) + E XN(1+1/R)       where X = (1+g)/(1+R) = G/(1+R)

Current Earnings per Share = $       that's E
Discount Rate (as a percent) = %       so R = (Discount Percent)/100
Earnings Growth Rate (g, as a percent) = %       so G = 1 + g/100
Years N =       so you expect Earnings Growth just this many years into the future
Current Stock Price = V = $       some estimate of a "fair" current stock price

Consider this interesting determination of a Discount Rate R (or Expected Stock Return) ... to use in Equation (2), above:

>I thought "g" was the Dividends growth rate. Now it's the Earnings Growth rate!
Uh ... yes. Now it's the Earnings Growth Rate. Didn't I already say that?

Okay, we suppose the current P/E ratio is P/E = 30 and only 25% of earnings are paid as dividends, so f = 0.25, and earnings are expected to grow at 7%, so g = 0.07 so ...

>So R = g + f/(P/E) = 0.07 + 0.25/30 = 0.078, right?
Yes, meaning we'd use a 7.8% Discount Rate. Of course, the return we'd want (that's R) should be above some risk-free rate, say the 30-year Bond Rate. This is the Risk Premium; it justifies the risk we're taking in buying the stock, instead of the bond.
(Example: bond rate = 5.0% and R = 7.8% so Risk Premium = 7.8 - 5.0 = 2.8%)

>Slick! That means P/E can be calculated as P/E = ... uh ...
That means:

P/E = f/(R-g)

where f = earnings payout rate and R = (risk-free rate) + (risk premium) and g = Earnings Growth Rate

>Is that accurate?
Sometimes, but there's another P/E valuation model dubbed the Fed Model ... but we're drifting from our DDM discussion ...

Consider the Discount Rate given by:     R = g + fE/P = g + f/(P/E)
It requires estimating P/E ratios and these ratios depend upon the liquidity of the stock ... whether we're talking about large or small cap stocks. Further, the relationship between growth rates, payout rates, risk levels, and liquidity - this relationship changes in ways that are ... uh ... unpredictable. Further, an investor expects to make not only a series of dividends but also an increase in Stock Price.

We might try to incorporate some of these things as follows:

  • Predict the P/E ratio in, say, N = 5 years. Call it PE(5) ... where PE(0) would be the current P/E ratio.
    Example: PE(5) = 30
  • Predict Earnings after 5 years, E(5), using an assumed annual growth factor, G.
    That'd make E(5) = E(0)G5, after 5 years, where E(0) = current earnings.
    Example: G = 1.08 would make E(5) = E(0)(1.05)5 = 1.28E(0)
  • The future Stock Price would then be P(5) = (P/E Ratio)(Earnings) = PE(5) * E(5) = PE(5)E(0)G5.
    Example: P(5) = 30 (1.28 E(0))
  • The Present Value of this Stock Price (with Discount Rate R) is then P(0) = P(5)/(1+R)5 = PE(5)E(0)G5/(1+R)5
The Present Value of the series of dividends PLUS the present value of the Stock Price is then:

(3)       V = D (1 - XN)/(1 - X) + PE(N)E(0)XN       where X = G/(1+R) = (1+g)/(1+R) and we're considering N = 5

where the first term is what you'd pay for the dividends and the second term is what you'd pay for the capital gains.

>Somehow, all this sounds familiar.
Yes, we're treating stocks much like bonds where there's a series of dividends (taking the place of bond coupons) and some final value (taking the place of the bond maturity value).

>I don't remember any of that, because ...
Then read this

>How about GE stock?
Okay, we'll assume PE(5) = 30 (the P/E ratio in 5 years) and G = 1.07 (a 7% annual earnings growth) and R = 1.08 (an 8% Discount Rate) and D = 0.72 ($0.72 is current dividend) and E(0) = $1.65 (current earnings per share) and we get
X = G/(1+R) = 1.07/1.08 = 0.9907 so V = 0.72 (1 - 0.99075)/(1 - 0.9907) + 30 (1.65) (0.99075) = $50.78 so ...

>And the current stock price is $30?
Well, play yourself ... and see if you can identify appropriate parameters:
V = D (1 - XN)/(1 - X) + PE(N)E(0)XN       where X = G/(1+R) = (1+g)/(1+R)

Current Dividend = $       that's D
Current EPS = $       that's E(0)
Discount Rate (as a percent) = %       so R = (Discount Percent)/100
Earnings Growth Rate (g, as a percent) = %       so G = 1 + g/100
Years N =       so you predict P/E this many years into the future
Estimated P/E Ratio (in N years) =       that's PE(N)
Current Stock Price = V = $       some kind of "fair value", eh?

>And does this stuff work with some Index like, say, the S&P 500 ?
Well, we could try ... maybe like so:

  • The S&P 500 is currently 850 and has a dividend yield of 1.5% and 1.5% x 850 = $12.75. That's D, our Dividend.
  • The operating Earnings (according to Standard and Poor's) is about $38. That's our E(0).
  • We'll assume PE(5) is 20 and and Earnings Growth Rate of 6% or 7% ( that's our g ) and ...

>So what's a fair value for the S&P 500?
It depends upon the assumed Discount Rate, R

Try your hand at this ritual:
Current Index =
Current Dividend Yield = %
Current Earnings = $
Discount Rate = %
Earnings Growth Rate = %
Years N =
Estimated P/E Ratio (in N years) =
Current "Fair Value" = $

>So the S&P is fairly priced ... almost, eh?
How would I know? I just draw pictures, write tutorials, sleep ...

>Yeah, yeah. So, where would I get all the numbers to stick into ...?
You might try

Okay. If we divide Equation (3) by current Earnings, E(0),
then it has the form:

(3a)     V/E = (D/E) (1 - XN)/(1 - X) + (PE)XN

V/E = (Stock Price)/(Current Earnings)
D/E = (Current Dividend)/(Current Earnings)
PE = P/E Ratio after N years
G = 1 + g = 1 + (Earnings Growth Rate)
R = Discount Rate
and X = (1+g)/(1+R)
and here's a picture (for constant P/E Ratio)

Figure 3
For Exxon (symbol XOM):
D = 0.92
E = 1.79
PE = 20 (assumed constant)
g = 0.09 (meaning an assumed 9% Earnings Growth Rate)
R = 0.08 (meaning an assumed 8% Discount Rate)
then X = 1.09/1.08 so (3a) gives V/E = 23.46 so the stock value should be 23.46 * Earnings = 23.46 * 1.79, about $42.

>And what's XOM worth now?
It closed at $39.42 on June 7/02.

>You invented numbers so it'd look good, right?
Uh ... yes, but here are more examples (as of June 7/02)
where PE is the current P/E Ratio (assumed constant)
and we're using N = 5 years in Equation (3)
and g is the historic Earnings Growth Rate
and R is simply an invented figure (!)
and g and R are expressed as percentages (for sanitary reasons):
Microsoft    (MSFT)
D = $0.00
E = $1.82
PE = 28
g = 25.40%
R = 20%

V = $63.51
Current Price = $51.90

General Motors    (GM)
D = $2.00
E = $4.93
PE = 12
g = -8.80%
R = 5%

V = $36.94
Current Price = $58.20

Coca Cola    (KO)
D = $0.80
E = $1.79
PE = 30
g = -0.70%
R = 5%

V = $44.21
Current Price = $54.15

AT & T    (T)
D = $0.15
E = $0.13
PE = 87
g = -28.00%
R = 0%

V = $2.62
Current Price = $11.75

>Some are pretty lousy! Look at AT&T! It's ...
Yes, and it'd be worse if I hadn't picked R = 0%. However, if we pick a different N-value we'd get a different Stock Value. Remember, the above table is
for N = 5 years (so we're discounting for 5 years and the P/E Ratio is 5 years into the future).

If we leave the above parameters fixed, but vary N, we'd get this:

>So should we pick N = 0?
Sure, if you want the best number for the current stock price. It's bang on.

Figure 4

On the other hand, we've taken the Earnings Growth Rate as -28%, the historical rate. If we change this, we can get other stock values, as in Figure 5 where we've assumed R = 10% and a 15% Earnings Growth Rate
(so g = 0.15).

>But how would I know what R and N and g to pick?
Actually, it's quite easy. I'll lend you the appropriate tool.
>Very funny.

You can also play with the calculator above, stick in various values for R and g ... and see how close you can come to the current stock price. In particular, use for your R-value what you'd like to get as your stock return. It's a personal thing.
For example, if I want 10% (so that's the Discount Rate I'd use), I put:
D = $0.15    E = $0.13    N = 5    PE = 87    g = 15%    R = 10%
and I'd get V = $14.95 (the red dot in Figure 5).

Figure 5

>A spreadsheet?
Sure, just RIGHT-click on the picture below and select Save Target or Save Link:

>In Figure 2 you showed the effect of increasing the Dividend Gain Rate. How about a picture where we increase ...?
The Earnings Gain Rate? Good idea. Here's a picture

>That's confusing. The biggest Stock Value has the smallest R-value.
It means that, if you're satisfied with a smaller return (that means a smaller R), then you'd be willing to pay more for the stock (that means a larger V = Stock Value).

>It looks like, if you expect spectacular Earnings Growth, you'd pay big time for the stock, right?
I would? No, YOU would.

Figure 6
I actually own a stock that has had a Earnings Growth Rate of 25 - 30%, its current P/E Ratio is 13, its current EPS of $1.13 and I figure it should be worth at least $30.

>What does DDM say?
Here's a picture: Figure 7    

>Aha! So it should be worth over $25, right?

>What's it trading at, now?
It closed at $14.50, on June 7/02.

>You're assuming a PE of 13, in N = 5 years. That's low, isn't it??
Yes, so if we stick in a PE of, say, 18 (less than half the current P/E for the S&P 500), then we'd get Figure 8    

>Now we're at $35 - $40. What if we only want R = 8% or maybe ...
Just play with the calculator
... or, if there are no dividends, try that other calculator

Figure 7

Figure 8