There was a problem in my lecture and I'm having difficulty understanding the solution. It is this:
A common stock was paid a dividend of $2. The dividend is expected to grow at 8% for three years, then it will grow at 4% in perpetuity. What is the stock worth? The discount rate is 12%.
So far I've been given these equations (the first is for zero growth in perpetuity and the second is constant growth in perpetuity):
$$P_0=\frac{Div_1}{(1+R)^1}+\frac{Div_2}{(1+R)^2}+\frac{Div_3}{(1+R)^3}+...=\frac{Div}{R}$$
$$P_0=\frac{Div_1}{R-g}$$
I would have thought that the solution would be the following:
$$P_0=\frac{\$2.16}{1.12}+\frac{\$2.33}{1.12^2}+\frac{\$2.52}{1.12^3}+\frac{\$2.72}{0.08}$$
I found the dividends using the equations $Div_1=Div_0(1+g)$, $Div_2=Div_1(1+g)$, and so on. The final term is using the dividend from the 5th time period rather than the 4th because the equation for constant growth perpetuity says to use the dividend from the next time period which would be $P_4=\frac{Div_5}{R-g}=\frac{2.72}{0.12-0.04}$
However, that is not the solution, the solution is:
$$P_0=\frac{\$2.16}{1.12}+\frac{\$2.33}{1.12^2}+\frac{\$2.52+\$32.75}{1.12^3}$$
I think that this equation comes out to look like:
$$P_0=\frac{Div_1}{1+R}+\frac{Div_2}{(1+R)^2}+\frac{Div_3+\frac{Div_4}{R-g_2}}{(1+R)^3}$$
I can see that it appears to be a combination of the zero growth model and constant growth, but I really don't understand why the numerator of the last equation has the stock value at $P_4$ in the numerator. The lectures aren't clear about this so I'm looking for some guidance. Why is that the solution?
Edit to add: R is the discount rate and g is the growth rate.
