## Monday, January 6, 2014

### Dividend Discount Model

This is part of a series on valuation techniques.

The fundamental reason why stocks are a vehicle for investment is that they represent a fractional ownership of a company and thus allow you to partake in that fraction of the profits. These profits, called dividends, are typically distributed once per quarter (i.e. four times a year) and are directly proportional to the number of shares that you own. If we have perfect information of future dividends, then we can compute the present value of a share of the company via discounting.

Suppose I have a constant cost of capital (also called the discount rate) of $$r$$, i.e. the opportunity cost of 1 dollar over one year is $$1+r$$ dollars. And for simplicity, let's say dividends are distributed yearly, starting tomorrow, at $$D_0, D_1, D_2, \dots$$ dollars per share. Then the value (to me) of a share is $V = D_0 + \frac{D_1}{1+r} + \frac{D_2}{(1+r)^2} + \dots$ If the dividends are constant at $$D$$, then this simplifies to a simple geometric series \begin{align*} V &= D \left(1 + \frac{1}{1+r} + \frac{1}{(1+r)^2} + \dots\right) \\ &= \left(\frac{1}{1 - \frac{1}{1+r}}\right) D \\ &= \boxed{\left(\frac{1+r}{r}\right) D} \end{align*} If instead the dividends grow linearly at a rate of $$m$$, then we have that $V = D + \frac{D+m}{1+r} + \frac{D+2m}{(1+r)^2} + \dots$ Then we use the standard technique for simplifying such expressions \begin{align*} \left(\frac{1}{1+r}\right) V &= \frac{D}{1+r} + \frac{D+m}{(1+r)^2} + \dots \\ \left(1 - \frac{1}{1+r}\right) V &= D + \frac{m}{1+r} + \frac{m}{(1+r)^2} + \dots \\ \left(\frac{r}{1+r}\right) V &= D + \frac{m}{r} \\ V &= \boxed{\left(\frac{1+r}{r}\right) \left(D + \frac{m}{r}\right)} \end{align*} Finally, let's consider the case where the dividends grow exponentially at a rate of $$g$$ \begin{align*} V &= D + \frac{(1+g) D}{1+r} + \frac{(1+g)^2 D}{(1+r)^2} + \dots \\ &= D \left(1 + \frac{1+g}{1+r} + \frac{(1+g)^2}{(1+r)^2} + \dots \right) \\ &= \boxed{\left(\frac{1+r}{r-g}\right) D} \end{align*} It is worth noting that these computations only reflect the value of a stock for a given person's or organization's discount rate. The actual price of a stock is a function of supply and demand, i.e. the distribution of values as computed by everyone in the market.

Furthermore, having perfect knowledge of future dividend distributions is, of course, impossible. However, it can be reasonably approximated for certain classes of stocks, such as blue chips. For example, energy companies like Pepco (POM) and PG&E (PCG) have had very consistent dividends over the course of their lifetimes and can be expected to continue such trends in the future.

Perhaps also of interest, we assumed that the first dividend would be distributed the very next day. This reflects the maximum value of the stock to me. The minimum value is achieved the day after a dividend distribution. And the difference between these two values is given by $$D_i$$ (i.e. the value will fall by $$D_i$$ after the dividend is distributed). This can give rise to some arbitrage opportunities if the market is inefficient at such pricing.