# 51e: multistage growth

Consider a stock with dividends that are expected to grow at 20% per year for four years, after which they are expected to grow at 5% per year, indefinitely. The last dividend paid was $1 .00, and k = 10%. Calculate the value of this stock using the__multistage growth model. Okay, so I calculated that in years 1 2 3 4, the dividends are as follows D1 = 1.20. That’s the dividend after 12 months D2 = 1.44 D3 = 1.73 D4 = 2.08 Using the Gordon Growth model with the newly reduced dividend growth rate going forward, we find that 2.08/(0.10- 0.05 ) gives us a value of$41.60. The book states that P 3 = $41.60. Why isn’t it P 4? It isn’t until the end of the 4th year/beginning of the 5th year that we see a change in dividend growth rates? Obviously we have to discount the first 3 dividends based on a 20% growth rate. Discounted D1 = 1.2/(1.1) Discounted D2 = 1.44/(1.1^2) Discounted D3 = 1.73/(1.1^3) Now comes the tricky part. The book adamently states that we are to divide the$41.60 value by 1.1^ 3. Why is that, when it had 4 years to grow? The answer given is 1.09 + 1.19 + 1.30 + 31.25 = 34.83

Your question: The book states that P 3 = \$41.60. Why isn’t it P 4?

Because the D4 is the dividend which grows by the infinite rate.

D3 you discount with (1+k)^3 and the value calculated with the help of D4 you also discount with (1+k)^3. Think of this latter as a final value. As if it was an FV of a bond you receive as final payment.

This is a “final payment” in this case because you assume the permanent growth (g) from then on.

I have not read the Gordon growth model from cfa curriculum yet but the reason why its P3 and not P4 because in the original formula Gordon specified P0 = D1/(k-g) hence P3 = D4/(k-g). As for the second part, Moosey has explained well.

Remember the Gordon Growth model:

V0 = D1 / (r – g)

The value at time zero is the dividend at time one divided by (r – g); more generally, the value at time n is the dividend at time n + 1 , divided by (r – g).

Thus, if the dividend at time 4 grows at a constant rate, then n + 1 = 4, so n = 3: dividing D4 by (r – g) gives you the value at time 3.

this clarification was really helpful, thanks

Cool!

You’re welcome.

I never liked the way the Gordon Growth model is presented. I think it’s much clearer to put it as P(t) = D(t+1)/(r-g) – i.e. the price at any point in time is based on the next dividend.

Amen!

I’m putting the finishing touches on a video for my classes that covers this - maybe I’ll post it here when I’m done.

The way I approach this topic is that stock valuation involves calculating the PV of an infinite stream of dividends (or cash flows). The price at any point in time should be the PV of all dividends after that point. For Constant Dividend Growth stocks, the Gordon formula is just the PV of a growing perpetuity.

For the Holding Period Model, we’re still theoretically calculating the PV of an infinite dividend stream. Think of the terminal stock value (at time "t) as the PV as of time “t”) of all dividends occurring after time “t”. I use the analogy of a carpet - you first roll up the dividend stream to the terminal value point, and then in the second stage, roll the carpet the rest of the way home (i.e. calculate the PV of all cash flows up to the terminal value point, including the terminal price).

The 2-stage model is is using the Constant Growth Formula to calculate the terminal value at time “t” (i.e. rolling the carpet to that point in the first step). and then rolling it the rest of the way home in the second step.

Does the Gordon growth model only work if the dividend growth rate is absolutely constant?

If that’s the case, does this even apply in real life? When is a dividend growth rate actually fixed and predictable in the real world?

Yes.

No.

Never.

(Then . . . answer it.)

The Gordon Model is “wrong”. So is a road map (at least, in the pre-GPS/Smartphone world) where one inch doesn’t equal one inch. By definition, models throw detail out the window (like a road map). But even though they’re “wrong”, some models are useful, since they help us simplify the world.

The Gordon model is “approximately” right. It works best when dividend growth is fairly smooth. Obviously, the more variability in the growth rates, the further off the model’s predictions will be. Still, it provides an estimate that can be used along with other estimates. As to how good the estimate is, that’s a matter of judgement.

S2000magician,

In above example, the growth rate is 20% for four years and then the growth rate is 5%. Shouldn’t we calculate P4 and then calculate its present value P0? I have made a picture for this. I would appreciate your thoughts.

http://tinypic.com/r/347wnjn/8

any help experts?

Once you get to a dividend that will grow at a constant rate thereafter, you can use that dividend or any future dividend in the Gordon Growth model and the answer will be the same. The key is the “thereafter” part: here, dividends will start to grow at a constant rate after D4, so we can use D4 in Gordon Growth (to get P3). If you, instead, use D5 in Gordon Growth to get P4, you will get the same value for the stock today. (Why?) I encourage you to do this in Excel an see. Try discounting D1, D2, and (D3 + P3), then D1, D2, D3, and (D4 + P4), and so on.

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S2000magician,

You are a genius. I did get the same result as I did when I solved this problem. Yay!!! After reading your first post, I thought that I was lucky in that I got the same answer as the OA. Thank you so much!

However, I am a bit curious. Why is it that I got the same answer? I believe what I can think from this exercise is that when the growth rate of the dividend is constant, it doesn’t matter at which point we apply the gordon’s method. It is very similar to a slope of a line at any point. It doesn’t matter what point we choose. The slope remains the same as long as the point is on the same line.

To piggyback on S2000’s last post, the reason you get the same answer using either the first “good” dividend (the one where all subsequent dividends grow at a constant rate is that (are you ready/)

the price of the stock is the present value of ALL dividends.

In each case, you are calculating the PV of an infinite stream of dividends. I use the analogy of a carpet - in the first step, you “roll up” the carpet using the constant growth formula. Therefore, the price of the stock (the huge roll of carpet sitting at time (if you use D4) or at time 4 (if you use D5) represents the PV of all dividends occurring after that time.

Then in the second step, you roll the carpet the rest of the way home. In other words, you then calculate the PV of all cash flows up to and including the terminal stock price you calculated in time 1.

Whether you use P3 or P4 in the first step, it’s important to remember that the process involves calculating the PV of ALL the infinite stream of dividends. The rest is just technique.

Wowzer! You’re immensely welcome.

The reason that you get the same answer is that you’re pushing the value farther into the future, then discounting it (the same amount) farther to get to the present value. I like the analogy of the slope: the farther we go in x, the greater the change in y, but when we divide by the new Δx, we cancel out the greater Δy.