About the denominator in the future value formula

Question

To get the present value of a stock that gives divined in the future we use the formula: P = Div+P1/1+r with P1 is the selling price of the stock after a period. So it's considering what you can get from the stock (dividend + market value) and discount it with 1+r. This is pretty clear. However, If the stock won't be sold in the future, there would be no P1 included in the future, and so it's only the dividend you get. If this dividend is growing, it should be included in the denominator. This also clear. so, the formula to get the present value of a stock with growing dividend and no market value (selling price) in the future is P= Div/r-g. My question is, why the 1 is excluded in the denominator in this case?

Answer 1

Let $G=1+g$ and $q=1+r$. Then the future value of the dividends is

$C_n=D\cdot q^{n-1}+ D\cdot q^{n-2}\cdot G+\ldots + D\cdot q\cdot G^{n-2}+D\cdot G^{n-1}$

$D\cdot q^{n-1}:$ The first dividend has to be compounded n-1 times and no growth.

$D\cdot G^{n-1}:$ The n-th dividend has grown n-1 times and no compounding.

$C_n=D\cdot \left[ q^{n-1}+ q^{n-2}\cdot G+\ldots + q\cdot G^{n-2}+ G^{n-1} \right]$

$q\cdot C_n=D\cdot \left[ q^{n}+ q^{n-1}\cdot G+\ldots + q^2\cdot G^{n-2}+ q \cdot G^{n-1} \right] \quad \quad \quad (1)$

$G\cdot C_n=D\cdot \left[ \ \ \quad q^{n-1}\cdot G+ q^{n-2}\cdot G^2+\ldots + q\cdot G^{n-1}+ G^{n} \right] \quad (2)$

Substracting (2) from (1)

$(q-G)\cdot C_n=D\cdot (q^n-G^n)$

$C_n=D\cdot\frac{q^n-G^n}{q-G}$

The present value is

$C_0=D\cdot\frac{q^n-G^n}{q-G}\cdot \frac{1}{q^n}$

$C_0=D\cdot \frac{1}{q-G}-\left(\frac{G}{q}\right) ^n\cdot \frac{1}{q-G}$

Suppose that $q >G\Rightarrow 1+r>1+g \Rightarrow r>g$

$$\lim_{n \to \infty} C_0=\lim_{n \to \infty} D\cdot \frac{1}{q-G}-\left(\frac{G}{q}\right) ^n\cdot \frac{1}{q-G}=D\cdot \frac{1}{q-G}-0$$

$$\lim_{n \to \infty} C_0=D\cdot \frac{1}{1+r-(1+g)}=\boxed{D\cdot \frac{1}{r-g}}$$