When applying the dividend discount model, we must decide about the expected growth rate of the dividends. Depending on whether we assume a constant growth rate or a variable growth rate, an analyst either applies the Gordon growth model or the multi-stage dividend discount model.

## Gordon growth model

The Gordon growth model (also called the constant growth model) is a special case of the dividend discount model which assumes a constant dividend growth rate. It is appropriate for the valuation of stock of companies who have achieved a mature growth rate and are insensitive to the business cycle.

Under the Gordon growth model, the value of a stock equals expected cash dividend next year D1 divided by the excess of the required rate of return (r) over the growth rate (g):

V_0=\frac{D_1}{r-g}

D1 equals current dividend D0 multiplied by (1 + g)

V_0=\frac{D_0×(1+g)}{r-g}

## Estimating the dividend growth rate

The above expression is a formula for growing perpetuity. The growth rate g can be estimated using a number of ways: (a) using the median growth rate of the industry, or (b) using the sustainable growth rate, which equals the product of the retention rate and the return on equity.

### Assumptions and weaknesses of the Gordon growth model

The Gordon growth model is based on the assumption that (a) the dividends are the relevant measure of cash flows, (b) the required rate of return and growth rate are constant forever and (c) the required rate of return is always greater than the growth rate.

These assumptions are problematic because there are companies that may not be currently paying dividends either because they have profitable reinvestment opportunities, or they are currently in a bad financial position. One method would be to defer the start of dividend payments, but such as approach results in very uncertain forecasts.

Due to these weaknesses, the analysts might (a) use a multi-stage dividend discount model, in which he assumes different growth rates for different periods, (b) use a cash flow measure such as free cash to equity (instead of dividends) and/or (c) use other models such as the multiplier models.

## Multi-stage dividend discount model

In the multi-stage dividend discount models such as two-stage or three-stage dividend discount models, discount growth can be broken down into multiple stages each with a separate dividend growth rate assumption. It is appropriate for rapidly growing companies because it allows us to model a rapid growth phase followed by a stabilized mature growth phase.

The intrinsic value of a stock under the two-stage model (V0) equals the sum of the present value of the dividends in the rapid growth phase (VO,R) and the present value of the stable growth phase dividends (VO,S).

The present value in the rapid growth phase can be worked out by individually forecasting dividends for each period by growing it at the rapid growth rate gR and discounting them at the required rate of return r. This is expressed mathematically as follows:

V_{0,R}=\sum_{t=1}^{n}\frac{D_0\times{(1+g_R)}^t}{{(1+r)}^t}

The present value of the stable growth dividends at time 0 is calculated by first using the Gordon growth model to calculate the intrinsic value at the end of the rapid growth phase (Vn,S) and then discounting it back to time 0.

V_{n,S}=\frac{D_n\times(1+g_S)}{r-g_S}
V_{0,S}=V_{n,S}\times\frac{1}{{(1+r)}^n}=\frac{D_n\times(1+g_S)}{r-g_S}\times\frac{1}{{(1+r)}^n}

Dn equals the dividend in the last year of the rapid growth phase, which equals current dividend D0 compounded at the rapid growth rate gR for n periods (the number of periods in the rapid growth phase)

V_{0,S}=\frac{D_0\times{(1+g_R)}^n\times(1+g_S)}{r-g_S}\times\frac{1}{{(1+r)}^n}

Combining these two components gives us the intrinsic value of the stock at time 0.

V_0=V_{0,R}+V_{0,S}
V_0=\sum_{t=1}^{n}\frac{D_0\times{(1+g_R)}^t}{{(1+r)}^t}+\frac{D_0\times{(1+g_R)}^n\times(1+g_S)}{r-g_S}\times\frac{1}{{(1+r)}^n}

### Three-stage dividend discount model

In the three-stage dividend growth model, which is appropriate for a fairly young company, the three phases are typical: growth (initial rapid growth), transition (lower finite growth) and maturity (lower sustainable indefinite growth).

### Application of two-stage dividend discount model

The two-stage model is appropriate for a company that has already past the high growth phase but its dividends have not yet stabilized. However, it is not just dependent on age. For example, if an older company engages in innovation, expansion, acquisition, etc. it might be able to enjoy an above-average growth rate. Similarly, it might be applied to a company that is expected to have a subnormal growth in a certain period.