Question about gordon growth and expected stock prices

Question

I have problems understanding the solution to the following question and would like to ask if you could help me with the interpretation?

A house is an asset which generates a benefit to its owners: a stream of housing services in the future. Similarly a stock generates a stream of dividends in the future. One measure of the value of this housing service is the rent one would have to pay on an equivalent house. For the purposes of this assignment, assume $D_{t+1}$ is the rent you have to pay over the next year (we know this number at time $t$), and $P_t$ is the price of the house today. The return on housing is $r_{t+1} = (P_{t+1}+D_{t+1}-P_t)/Pt$. If the discount rate and growth rate of rent were constant, this setup satisfies the Gordon equation, $P_t = D_{t+1} / (r – g)$, where $r$ is the expected return on housing and $g$ is the growth rate of rent. You can ignore issues about taxes and transaction costs on houses.

Suppose the Gordon equation were correct. In this world, looking at a specific city over time, suppose we start out at $D_{t+1} = \$10,000$ a year, $P_t =\$200,000$, $r$ is $10$% and $g$ is $5$%. Over time, would the price be constant, rising, or falling?

If I plug in the numbers into the Gordon formula I get $P=10.000/(10\text{%}-5\text{%})=200.000$. Thus the price stays constant.

However the answer is:

Since the expected return is $10$% but the “divided yield” is only $5$%, the price has to rise $5$% a year.

Could someone explain me if there is a way to understand the solution using the Gordon growth formula or some other intuitive approach?

Thank you very much in advance!

Answer 1

Plugging in the numbers into the Gordon's growth formula does not tell what will happen to the price of the house -it just verifies that the numbers given do satisfy the formula.

To see what will happen to the price in the future, you have to go to the definition of the return on an asset

$$r_{t+1} = \frac {P_{t+1}+D_{t+1}-P_t}{P_t} = \frac {\Delta P_{t+1}}{P_t} + \frac {D_{t+1}}{P_t} \tag{1}$$

The right-hand side now is clearly "Asset percentage appreciation(Capital Gain) + rate of return on asset value" which is indeed the Total Return on an asset expressed in percentages.

But we have assumed (and the numbers given verify the initial condition), that Gordon's formula holds so

$$P_t = \frac {D_{t+1}}{r-g} \implies P_t(r-g) = D_{t+1} \tag{2}$$

Substitute $(2)$ into $(1)$ to get

$$r_{t+1} = \frac {\Delta P_{t+1}}{P_t} + \frac {P_t(r-g)}{P_t} = \frac {\Delta P_{t+1}}{P_t} + r-g \tag{3}$$

Now, reasonably we should identify "expected (total) return per period" $r$ with "total return next period" $r_{t+1}$.

Then $(3)$ becomes

$$r_{t+1} = r = \frac {\Delta P_{t+1}}{P_t} + r-g \implies \frac {\Delta P_{t+1}}{P_t} = g \tag{4}$$

which in our case, means that the value of the house will appreciate $5$% per year. Intuitively, since expected Total Returns are $10$% and what we get out of the house as income ($D_{t+1}/P_t$) is $5$%, the other component of Total Returns, house price appreciation, must cover the distance between the two, which is another $5$%.

Note: Whether this increase in price refers to "asking" price or "equilibrium" price is another matter that involves a more elaborate modelling of the situation.