Dividend Discount Model

Dividend at the end of $i$-th period: $D_i={\left(1+\frac{g}{k}\right)}^i{D}_0$

• Assumption in this model is that dividend will grow at rate $g$ per year

Present value of Equity

• $r$ is the required rate
• $S_n$ is the stock price when you sell it (end of period $n$)
• $R:=\frac{1+\frac{g}{k}}{1+\frac{r}{k}}$

$$\begin{array}{rlr}{S}_0& =\frac{D_1}{\left(1+\frac{r}{k}\right)}+\frac{D_2}{{\left(1+\frac{r}{k}\right)}^2}+\cdots +\frac{S_n}{y_n}& \text{selling after period }n\\ S_{\infty }& =D_1\left(1+\frac{r}{k}\right)+\cdots =\sum _{i=1}^{\infty }\frac{{\left(1+\frac{g}{k}\right)}^i{D}_0}{{\left(1+\frac{r}{k}\right)}^i}=D_0\sum _{i=1}^{\infty }{R}^i& \text{holding stock forever}\\ & =D_0\cdot \frac{R}{1-R}& \text{when }r>g⟹R<1\text{ (geometric sum) }\\ & =D_0\frac{1+\frac{g}{k}}{\frac{r}{k}-\frac{g}{k}}=\frac{D_1}{\frac{r}{k}-\frac{g}{k}}& \text{DDM}\end{array}$$