Derivation of dividend discount model

1 I. Summation of Infinite Geometric Series

Summation of geometric series can be defined as:

(A1)

Multiplying both sides of equation (A1) by R, we obtain

(A2)

Subtracting equation (A1) by equation (A2), we obtain

It can be shown

(A3)

If R is smaller than 1, and n approaches to ∞, then R ⁿ approaches to 0 i.e.,

(A4)

then,

(A5)

2 II. Dividend Discount Model

Dividend Discount Model can be defined as:

(A6)

Where P ₀ = present value of stock price per share

D _t = dividend per share in period t (t = 1, 2, …,n)

If dividends grow at a constant rate, say g, then, D ₂ = D ₁(1 + g), D ₃ = D ₂(1+ g) = D ₁(1+ g)², and so on.

Then, equation (A6) can be rewritten as:

(A7)

Comparing equation (A7) with equation (A4), i.e., P ₀ = S _∞, D ₁/1+k = A, and 1+g/1+k=R as in the equation (A4).

Therefore, if 1+g/1+k<1 or if k>g, we can use equation (A5) to find out P ₀ i.e.,