# Is this the reason why stock prices on the broad average always rise?

According to the so called Dividend Discount Model (DDM), a particular, temporary stock price is the discounted sum of all future dividends resulting from the investment:

$$P=\frac{D}{i-g}$$

$$P$$ is the stock price, $$D$$ is the dividend paid at the moment of calculation, $$i$$ is the cost of capital equity (interest rate), $$g$$ is the dividend growth rate. For a given point in time, the price $$P$$ is constant since the paid dividend $$D$$ is known and $$i$$,$$g$$ are assumed to known or well estimated. Furthermore, they are assumed to be constant. (This may not be entirely true for $$g$$.) Given these assumptions, the stock price is constant as long as an increase of the dividend that is not "expected" through the constant growth rate appears. On the other hand, a decrease is possible too.

The definition suggests that stocks are bought to receive dividends as cash flow. But for passive investors, the price increases of shares on a broad average are at least as important as the dividend yield. When investors assume that there will be a price increase in the future, the particular stock receives higher demand. This is why I would assume the following model:

$$P=\frac{D}{i-g}+c\dot{P}$$

In this way, the expected prise increase (over time) $$\dot{P}$$ is considered (with a coefficient $$c$$). This leads to a stock price which is not constant but grows exponentially over time (consider it as an ODE) - given a certain $$DDM$$ term (that is constant) and a coefficient $$c$$. Exponentially growth is exactly what describes broad stock markets over long time. Is this the reason why stock prices on the broad average always rise?

I know that my short explanation contains many problems:

• I do not know how to handle a not constant growth rate $$g$$ and I do not know what to expect (more): positive or negative growth rates $$g$$ and how to find out for each stock
• I know that there is a difference between the derivative of the price over time $$\dot{P}$$ and the expected price growth, let's call it $$\dot{P}_{\text{exp}}$$, but I think as a simplification, it might as a simplification.

Any ideas on that?

Rearranging the dividend discount model to express the required return $$i$$ in terms of the other variables gives

$$i = \frac{D}{P} + g$$

That is, the return to holding stocks in this model comes partly from dividends (in the form of the dividend yield $$D/P$$) and partly from dividend growth $$g$$, which will increase the stock price over time, representing another source of return for investors.

Another way to see it is to look at the dividend discount model in the form

$$P_t = \sum_{k=1}^\infty \frac{E(D_{t+k})}{(1+i)^k} = \sum_{k=1}^\infty \frac{(1+g)^k}{(1+i)^k} D_t$$

where we have assumed that $$E(D_{t+k}) = (1+g)^k D_t$$. The expected price one period later is

$$E(P_{t+1}) = \sum_{k=1}^\infty \frac{(1+g)^k}{(1+i)^k}E(D_{t+1}) = (1+g)P_t$$

Therefore the expected return is

$$E(R_{t+1}) = \frac{D_t + E(P_{t+1}) - P_t}{P_t} = \frac{D_t}{P_t} + g$$

as before.

• So "the reason why stock prices on the broad average always rise" is that for the economy as a whole $g>0$, and the rate of rise in share prices is $g$. Oct 28, 2020 at 14:26
• In the long run, and modulo other effects which have an impact on the share price (like issuance and buybacks), yes. Nov 5, 2020 at 8:37