ABSTRACT
Probably the most widely used interpretation of the constant-growth model developed in §4-14 is the so-calledGordonGrowthModel.We can develop this interpretation of the constant-growth model by defining E to be the firm’s earnings during the current (first) year, that is, the firm’s earnings over the time period from t = 0 until t = 1. Moreover, suppose the firm re-invests a constant proportion, 0≤ b≤ 1, of its earnings in each year, in which case the dividend payment at the end of the first yearwill be d1 = (1−b)E. Now assuming that retained earnings are re-invested at the rateR, it then follows that the firm’s earnings during the second year will be E+RbE =E(1+Rb). At the end of the second year, the firm will pay a dividend amounting to d2 = (1− b)E(1+Rb) and re-invest bE(1+Rb). This means that its earnings during the third year will be E(1+Rb)+RbE(1+Rb) = E(1+Rb)(1+Rb) = E(1+Rb)2. The dividend at the end of the third year will thus be d3 = (1 − b)E(1 + Rb)2, and so on. Generalizing this procedure shows that the dividend payment at time t will be dt = (1 − b)E(1 + bR)t−1. This in turn implies that the present value of the future stream of dividends will be
P0 = N∑ t=1
(1− b)E(1+ bR)t−1 (1+ ke)t =
E(1− b) 1+ bR
( 1+ bR 1+ ke
)t where, as previously, ke is the cost of equity capital. Now, if we set d1 = E(1− b), g = bR and let N → ∞ in the constant-growth model developed in §4-14 then the above equation reduces to
lim N→∞
P0 = E(1− b) 1+ bR
( 1+ bR 1+ ke
)t = lim
1+ g N∑ t=1
( 1+ g 1+ ke
)t = d1
ke − g = (1− b)E ke − bR
In other words, the dividend paymentmade at time onewill be d1 =E(1−b), whilst dividends will grow each year by the product of the retention rate for earnings b and the rate R at which retained earningswill be re-invested. That is, g = bR in the constant-growthmodel formulated in §4-14.
