Question

Hi, How to proof The Constant Growth Dividend Model with a Finite Horizon equals Gordon Model (if it is assumed that the growth rate of dividends and the required rate of return of the next owner, (after n years) remain the same)? [By deriving the equation]

Thanks!

Answer 1

Let's assume that:

D1 is the dividend in the first year

g = dividend growth rate that continues for the horizon period of n years and the same growth rate continues for the next owner

Ks = required rate of return for the horizon period and beyond the horizon period for the new owner

Hence, the growth rate and required rate of return continues at a constant level for an infinite period

Price or value of a common stock should be present value of all the future cash flows (dividends) it’s expected to generate.

Let’s assume the stock pays a dividend D_t in the time period t for an infinite period of time (going concern basis). Then,

OO Dt P Vs = (1Ks)t t-1

where V_s = Intrinsic value of a share of common stock; D_t = Expected dividend per share on the common stock in period t; K_s = Required rate of return on the common stock (cost of equity calculated by CAPM).

Constant Growth Dividend Model: In this case the dividend grows by a constant growth “g” which implies D_t+1 = D_t x (1 + g) and if D₁ = first dividend due at the end of first year = D₀ x (1 + g) then the above formula changes to

оо Di x (1+9) , D, х (1 +g)2 Dt D1 Vs ... oo (1Ks) (1 Ks)2 R EER ви ии (1+ К)2 (1+ К,)1 + t-1

If we multiply both side by (1 + g) / (1 + K_s) we get:

D1 x (1g) D1 x (1 g)2 (1+ К)? Рх (1 + g) оd (1+ К,)? ив (1Ks)

Subtract this resultant equation from the original equation above and simplify

Do x (1+ g) (Ks g) D1 D1 Рx (1+9) Р- P = V = (Ks g) (1+ K5)1 (1 Ks)

The last equation is the Gordan constant growth model.

Thus, the Constant Growth Dividend Model with a Finite Horizon equals Gordon Model (if it is assumed that the growth rate of dividends and the required rate of return of the next owner, (after n years) remain the same)