The Dividend Valuation Model
Normal Growth Stocks
"Super" Growth Stocks

The Dividend Valuation Model

When you buy a share of common stock you expect it to be "profitable" -- you expect to earn a return made up of one or two parts: the dividends you receive from the company and/or the capital gain you earn by being able to sell the stock in the future at a price higher than the price you paid. But as we'll see, the only part that matters is the expected dividend stream starting from when you buy the stock all the way through infinity. The only thing that gives the stock certificate value is the stream of cash payments from the company to the owners of the certificate. But what about the capital gain or price appreciation? Very simply, the price that another investor will pay you for the stock at some point in the future depends on the portion of that infinite dividend stream that the second investor will receive plus his or her capital gain. But that capital gain will depend on what a third investor will pay and that's dependent on their share of the dividend stream plus their expected capital gain. Each time the stock is sold, theoretically the buyer is looking to the future and computing the present value of their expected dividend stream plus the present value of their capital gain. But the capital gain is always dependent on the dividend stream from that point through infinity.

Let's assume that you decide to buy a share of common stock and that you anticipate holding the stock for an arbitrary three years. We'll assume that yesterday the company paid a dividend per share of D₀. Does this mean that you were dumb for waiting one day too many to buy the stock? No! If you had bought the stock yesterday morning, you would have had to have paid an additional D₀ for the stock. The date of record for the dividend payment and the amount of the dividend are both known well before it is paid. There are not very many surprises.

OK, so you decide to buy a share and hold if for three years before selling it. Yesterday the stock paid a dividend per share of D₀. Let's assume that the market requires a rate of return on a common stock of this risk class of k_e%. Where the heck did that come from?? Lots of different ways to arrive at an estimate of k_e. k_e can be gotten from our "beta model". You remember that, right! k_e = i + B * ( k_m - i ), where i is the risk-free rate of interest, B is the stock's beta coefficient and k_m is the expected return of the stock market.

Conceptually, a good way to estimate k_e is to find out the rates of return on other similar firms. If you're buying a share of Exxon, find out what the return has been on Chevron, Sunoco, Shell, etc. In any case, we'll use a required return of k_e.

You plan to buy the stock, hold if for three years and then sell it at the prevailing market price at the end of year 3. What cash flows do you expect to receive? The cash dividends at the end of years 1, 2 and 3 (remember, D₀ was paid yesterday -- you need to wait a year for your first dividend) and the market price at the end of year 3. Each of these four cash flows needs to be discounted back to year 0 (today) at your required return of k_e%. Then add them up and that's the price you're willing to pay now, i. e., its intrinsic value to you.

Let's do it with symbols.

P₀ = D₁ / ( 1 + k_e )¹ + D₂ / ( 1 + k_e )² + D₃ / ( 1 + k_e )³ + P₃ / ( 1 + k_e )³      ( Equation 1 )

The day you receive your third and last dividend is the day you expect to unload the stock. Are you being a financial whiz by getting that last dividend before selling? No! If you sell it a day or two before the third dividend is to be paid, you would expect to receive a higher price for the stock since the buyer would then receive the dividend and he or she would be willing to pay you the extra amount.

The key question now becomes: How do we estimate P₃, the price you expect to receive at the end of the third year? You will receive what some other investor is willing to pay. And how do we figure that out? Well, if we assume that the second investor was as lucky as you and got to take Fin 125, we know that he or she would look to the future and take the present value of his or her expected cash flows. What is the first cash flow that the second investor expects to receive? It's D₄. But from their vantage point, D₄ is only one year in the future. The second investor will discount D₄ back only 1 period. The next expected cash flow is D₅ which needs to be discounted back only 2 periods to when the second investor is buying the stock.

Now we could assume that the second investor holds it for only two years and then sells it to a third investor who also looks to the future for his or her expected dividends and then holds it for a long time. Then they sell it and so on and so on. But putting all of this into web page HTML is somewhat tedious, so let's assume that the second investor, the one who buys the stock from you, plans to hold the stock for a very, very long time, passing it down to his or her heirs, etc. In other words, the second investor expects to hold the stock forever. This assumption does not change our results but it enables me to finish this section before my summer vacation is completely over.

The price the second investor is willing to pay you is:

P₃ = D₄ / ( 1 + k_e )¹ + D₅ / ( 1 + k_e )² + ... + D_n / ( 1 + k_e )^n-3 + P_n / ( 1 + k_e )^n-3.      ( Equation 2 )

But we're assuming that n goes to infinity. If we substitute the P₃ from Equation 2 into Equation 1, we end up with:

P₀ = D₁ / ( 1 + k_e )¹ + D₂ / ( 1 + k_e )² + D₃ / ( 1 + k_e )³ + [ 1 / ( 1 + k_e )³ ] * [ D₄ / ( 1 + k_e )¹ + D₅ / ( 1 + k_e )² + ... + D_n / ( 1 + k_e )^n-3 + P_n / ( 1 + k_e )^n-3 ) ]

Simplifying, we get P₀ = D₁ / ( 1 + k_e )¹ + D₂ / ( 1 + k_e )² + ... + D_n / ( 1 + k_e )ⁿ

P₀ = S [ D_t / ( 1 + k_e )^t ]    for t = 1 to infinity.   ( Equation 3 )

It is the present value of the infinite dividend stream that determines the value of a share, regardless of the expected holding period.

Remember Equation 3; it is the basis for the rest of these notes. While Equation 3 is theoretically perfect, it's also totally useless. It's hard enough to forecast D₅, let alone D₅₀. And what about D_5,000,000. That could be a lot of money, assuming that there are still humans on the planet. But of course, the present value of that D_5,000,000 today is about $.0000001. So estimating it may not be that important. But I ramble.

In order to be able to use our dividend valuation model, (aka Equation 3) we need to make some assumption(s) about the behavior of that infinite dividend stream. We do just that in the two sections that follow. The normal growth model is by far the more important of the two, but the "super"growth model is usually worth about 30 points on the third hourly. Guess that makes it pretty important too, doesn't it!

Normal Growth Stocks

Most companies are not Microsoft or Intel (or IBM many years ago or Polaroid many, many years ago). Over time they tend to grow at a fairly dull, steady rate roughly equal to the growth rate of the economy. Let's assume that this so-called "normal" growth rate is g_n percent each year. It doesn't have to be a constant rate each and every year. But it will average out to an annual growth of about g_n percent per year. That means that the dividend at the end of year 1, D₁, can be thought of as being equal to the previous dividend, D₀, compounded one year at g_n. In other words,

D₁ = D₀ * ( 1 + g_n )¹.

Likewise, D₂ = D₁ * ( 1 + g_n )¹.

Substituting, we get D₂ = D₀ * ( 1 + g_n )².

In general, we have D_t = D₀ * ( 1 + g_n )^t.

Let's repeat Equation 3: P₀ = S [ D_t / ( 1 + k_e )^t ]    for t = 1 to infinity.

Now write Equation 3 out for a couple of terms:

P₀ = D₁ / ( 1 + k_e )¹ + D₂ / ( 1 + k_e )² + ... +

Now put each year's dividend in terms of D₀ and the normal growth rate, g_n:

P₀ = D₀ * ( 1 + g_n )¹/ ( 1 + k_e )¹ + D₀ * ( 1 + g_n )² / ( 1 + k_e )² + ... +

Rewriting in sigma notation:

P₀ = S[ D₀ * ( 1 + g_n )^t / ( 1 + k_e )^t ]     for t = 1 to infinity.

This is a geometric progression and solving it is beyond the scope of the course. As it turns out, the sum of the progression is

P₀ = D₁ / ( k_e - g_n )      ( Equation 4 )

That's not an HTML typo. The numerator is D₁ and not D₀.

Equation 4 is the most important equation for the remainder of the entire course!

P₀ = 2.00 * ( 1 + .06 )¹ / ( .16 - .06 ) = 21.20.

Most common stocks can be approximated by the normal growth model. And the best part is that it's simple to use. But what about companies like Microsoft and Intel (and IBM many years ago and Polaroid many, many years ago)? How do we value them -- they don't grow at the same rate as the economy? What about them? And what about the 30 points on the third hourly?

"Super" Growth Stocks

"No company can grow faster than the economy forever or it would become the economy." (Prof. Eli Schwartz, 1970) Know who Eli Schwartz is? Your Lehigh education is not complete until you talk finance and economics with the CBE's one true scholar. Seek him out -- he's lurking someplace on the fourth floor of Rauch. Tell him I sent you.

Even Microsoft's phenomenal growth rate will eventually peter out (probably after the release of Windows 2098). It may be far into the future but it will happen. For simplicity we're going to assume that the company we are trying to value will grow at a "super" rate of g_s for N years and then the bottom will drop out and the growth rate will plummet over night until it reaches the same growth rate as the economy, namely our "normal" rate of g_n. It will then grow for the indefinite future at g_n.

Let's figure out what a share of the stock should sell for today under these assumed conditions. All we need to do is apply Equation 4. We'll assume that the investor plans to hold the stock until the end of the "super" period of N years and then sell it for P_N. Again, this N year holding period is arbitrary.

P₀ = PV of the "super" dividends + PV of P_N.   ( Equation 5 )

The present value of the "super" dividends is easy. But they need to be done by hand; there is no short-cut. If D₀ was the dividend that was paid yesterday and k_e is the market's required rate of return, then the present value of the "super" dividends is:

D₀ * ( 1 + g_s )¹ / ( 1 + k_e )¹ + ... + D₀ * ( 1 + g_s )^N / ( 1 + k_e )^N

Add up those N terms and the total is the present value of the N years worth of "super" dividends.

What about the second part of Equation 5, the present value of P_N? We need to once again look at the situation through the eyes of the investor who is buying the stock at time N, the end of the "super" period and the beginning of the "normal" period. This investor only looks to the future -- any dividends that have been already paid, super or otherwise, are history. And the second investor could not care less about this history. The second investor needs to be concerned only with what he or she is forecasting about the company's future: [1 ] "yesterday (from the second investor's vantage point) the company paid a dividend of D_N, [2] its earnings and dividends are expected to grow at a rate of g_n indefinitely, and [3] the market requires a return of k_e.

Those conditions sure sound the same as the mini-problem at the end of the previous section and solving that was cake. We just whipped out Equation 4 which tells us that the price of a share of stock at time t, P_t, is equal to the next dividend, D_t+1, divided by the difference between the required return, k_e, and the growth rate, g_n.

P_t = D_t+1 / ( k_e - g_n )

From the vantage point of the second investor, that means he or she would be willing to pay:

P_N = D_N+1 / ( k_e - g_n )      ( Equation 6 )

Don't forget that this P_N is not received by the first investor until the end of the N^th year and so it still needs to be discounted back N years at k_e.

We now have both parts of Equation 5: the PV of the "super" dividends and the PV of P_N. Let's put them together.

P₀ = D₀ * ( 1 + g_s )¹ / ( 1 + k_e )¹ + D₀ * ( 1 + g_s )² / ( 1 + k_e )² + ... + D₀ * ( 1 + g_s )^N / ( 1 + k_e )^N + P_N / ( 1 + k_e )^N, where P_N = D_N+1 / ( k_e - g_n ).

Ready for a problem?

Yesterday a firm paid a dividend per share of 4.00, its earnings and dividends are expected to grow at a rate of 20% for another 5 years before dropping over night to a growth rate of 5%. Investors require a return of 15%. Find P₀, P₂ and P₂₀.

Let's first find P₀.

P₀ = D₁ / ( 1 + k_e )¹ + ... + D₅ / ( 1 + k_e )⁵ + P₅ / ( 1 + k_e )⁵ where P₅ = D₆ / ( k_e - g_n ).

P₀ = 4.00(1+.20)¹/(1+.15)¹ + ... + 4.00(1+.20)⁵/(1+.15)⁵ + { [4.00(1+.20)⁵(1+.05)¹] / [ k_e - g_n ] } / ( 1 + k_e )⁵

Want to check your answers? The PV of the dividends are 4.17 + 4.36 + 4.54 + 4.74 + 4.95 + the PV of the terminal price of 104.51 is 51.96 for a total of 74.72. The stock's price today is 74.72.

The two most common errors are: [1] not calculating the D_N+1 as D₀ compounded 5 years at the super rate and 1 year at the normal rate; and [2] forgetting to discount the P_N back N years to time 0. Don't make either of them!

Now, what about P₂? Look to the future and take the present value of what an investor at year 2 would receive. The first dividend will be D₃ but it's only discounted back 1 period. Your already calculated numerators stay the same -- only the exponents of the denominators decrease by 2 and the first two terms are history.

P₂ = 4.00(1.20)³/(1+.15)¹ + ... + 4.00(1.20)⁵/(1+.15)³ + ... + { [4.00(1+.20)⁵(1+.05)¹] / [ k_e - g_n ] } / ( 1 + k_e )³

I got 87.54.

What about P₂₀? Easiest one yet. You can still use anytime that our necessary conditon holds: the growth rate must be constant and be expected to continue forever. Well that is true in year 20. It's true in any year after the fifth year. So...

P₂₀ = D₂₁ / ( k_e - g_n )

P₂₀ = 4.00(1+.20)⁵(1+.05)¹⁶ / ( .15 - .05 )

I got 217.27.

What annual rate of return did the investor who bought the stock today for 74.72 and who sold it for 217.27 make over the 20 years if all the growth forecasts came to pass? Think about it.

Hope this all helps!