# Multi-stage Dividend Discount Models

In the previous articles, we learned about how a dividend discount model can be use to value a stock using the future cash flows.  However, in the constant growth model, we made an assumption that the dividends will grow at a constant rate. However, in reality this may not be the case. A firm may experience a period of very high growth and then after a few years, the growth rate may fall to a lower sustainable growth rate. In such a case we cannot apply the simple formula that we saw in the Gordon Growth Model that assumed a constant growth rate.

A stock with such characteristics can be valued by adding the present value of cash flows in the initial period of high growth and the present value of the stock after this high growth period assuming a constant growth rate.

$V_{0}=\frac{D1}{1+k}+\frac{D2}{\left ( 1+k \right )^{2}}+ \cdot \cdot \cdot +\frac{Dn}{\left ( 1+k \right )^{n}}+\frac{Pn}{\left ( 1+k \right )^{n}}$

Here, Pn is calculated using the Gordon Growth Model formula:

$P_{n}=\frac{D_{n+1}}{k-g}$

## Example

Let’s take a simple example to understand this:

Assume that a stock that pays dividends is expected to grow at a high rate of 15% per year for the first 3 years, after which it will grow at 6% per year.

The last dividends pay were $1, and the required rate of return in 8%. Let’s see how multi-stage growth model can be used to value this stock. First we will calculate the dividends in the high growth period. D1 = 1*1.15 =$1.15

D2 = 1.15*1.15 = 1.3225

D3 = 1.3225*1.15 = 1.521

Note that after D3, the dividends are expected to grow at a constant growth rate of 6%. So, D3 will grow at a constant rate of 6%. We can use D3 and the constant growth rate to calculate P2, that is, the value of the stock at t=2.

P2 = D3/(k-g) = 1.521/(8% – 6%) = 76.05

The present value of the stock will now be the sum of D1, D2, and P2.

V0 = 1.15/(1.08) + 1.3225/(1.08)^2 + 76.05/(1.08)^2

= 67.34

Note that you could also calculate D1, D2, D3, and P3 and then take the present value of all these to arrive at the value of the stock. The answer will be the same.

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• Dan

is stated here as the formula for the Gordon growth model however if you derive the Gordon growth model using a convergent geometric series you actually get Pn = [Dn(1 + g)] / [k – g] which will only equal if the constant growth begins at t = n and NOT be equivalent if it begins at t = n + 1. Therefore I would submit that the calculation presented here is less correct than using the method described in the note at the end and not actually the same.

In other words Gordon Growth Model can actually only be used if D1 = D0(1+g) and so it cannot be used to find P2 in this example, only P3 even tough the method described here matches that in books. If someone could mathematically prove this otherwise I am very interested in seeing how it can be proved.

• Dan

I see pasting the equation failed, here is te post again with the equation i tried to paste from the post:

Pn = [Dn+1] / [k – g] is stated here as the formula for the Gordon growth model however if you derive the Gordon growth model using a convergent geometric series you actually get Pn = [Dn(1 + g)] / [k – g] which will only equal if
the constant growth begins at t = n and NOT be equivalent if it begins
at t = n + 1. Therefore I would submit that the calculation presented
here is less correct than using the method described in the note at the
end and not actually the same.

In other words Gordon Growth Model
can actually only be used if D1 = D0(1+g) and so it cannot be used to
find P2 in this example, only P3 even tough the method described here
matches that in books. If someone could mathematically prove this
otherwise I am very interested in seeing how it can be proved