"I resolved this inconsistency..."
The discrepancy is that the long-term historical (real) return has been 7% while the Gordon Equation indicates that the long-term (real) return is now 3%. If the Gordon Equation is correct, then the long-term return has changed.
In this case the long-term is 50 years or so.
If we start with an initial balance that grows for N years at an annualized rate of return r (without making any deposits or withdrawals), the final balance satisfies the following equation: [final balance/initial balance] = (1+r)^N.
If the annualized rate of return r is 7%, then the final balance should be 1.07^N times the initial balance. Letting N=50 years represent the long-term, the final balance should be 29.46 times the initial balance. This has been the long-term return of the stock market.
If the annualized rate of return is 3%, then the final balance will be 1.03^N times the initial balance. Again letting N=50 years to represent the long-term, the final balance should be only 4.38 times the initial balance. This is what the Gordon Model predicts.
One way to reconcile these different numbers is to look more closely at what we mean when we speak of a long-term return. Suppose that we are able to purchase stocks at half-price. Then the [final balance/initial balance] = twice as much as the calculations indicate. On the other hand, if you pay twice as much as you should, the [final balance/initial balance] = only one-half of what the calculations indicate.
Now let us see what that does to our calculations.
If the long-term annualized rate of return were 7%, then the final balance would be 1.07^N or 29.65 times the initial balance. But if you purchased stock at half-price, the ratio would be twice as big or 59.30. [Here is where
An Illusion of Numbers enters in.] We now use the formula to calculate r, (1+r)^N = 59.30 where N=50 years. Taking the 50th root on both sides of the equals sign, (1+r) = 1.085 or r = 8.5%.
Similarly, if you paid twice as much for the stock, the final balance would only be one-half of 29.65 times the initial balance or 14.825. Solving for r when N=50 years, the rate of return satisfies (1+r) = 1.055 or r = 5.5%.
Continuing, if the long-term annualized rate of return were 3% as predicted by the Gordon Equation, then the final balance would be 1.03^N or 4.38 times the initial balance. If you purchased stock at half-price, the ratio would be twice as big or 8.76. We now use the formula to calculate r, (1+r)^N = 8.76 where N=50 years. Taking the 50th root on both sides of the equals sign, (1+r) = 1.044 or r = 4.4%. If you paid twice as much for the stock, the final balance would only be one-half of 4.38 times the initial balance or 2.19. Solving for r when N=50 years, the rate of return satisfies (1+r) = 1.016 or r = 1.6%.
If the long-term rate of return continues to be 7%, which it has been historically, a reasonable amount of variation in the measured (annualized total) return of the stock market would be (of the order of) 5.5% to 8.5%. If the Gordon Equation is correct, a reasonable amount of variation is 1.6% to 4.4%.
Something has to give. It is the Gordon Equation.
The Gordon Equation is a mathematical theorem with assumptions about an initial dividend yield, dividends that grow (forever) at a constant rate and a constant ratio of what people will pay for prices relative to dividends. Obviously, this is an idealization. The formula is realistic enough to be helpful. The theorem is always true. It is not always applicable.
The thing about the Gordon Equation is that its mathematical calculations become better and better as the number of years increase. That is, if you compare the calculated rate of return (which corresponds to an infinite number of years) to the results after a finite number of years N, the difference between the two becomes smaller and smaller as N becomes larger and larger.
The assumptions behind the Gordon Equation become less and less reliable as the number of years N increases.
The Gordon Equation starts us in the right direction, but it fails later on.
It was
Ben Solar who pointed out that the Gordon Equation calculates a long-term return, not an intermediate return. When we see it applied, however, it is for the intermediate term.
Are you disagreeing with Bernstein that there is a 7 percent long-term return assumption embedded into the conventional methodology studies?
No. I agree that the long-term (annualized real) return of the stock market is 7%. That is the historical record. Any study that uses historical sequences has this (7%) return embedded in it.
Are you saying that the long-term return will be 7 percent from any possible starting point? If that is what you are saying, that's probably the part that I am having a hard time understanding. It's hard for me to understand why that would be so. Shouldn't the long-term return vary according to the valuation level that applies at the starting point?
The mathematical theorem is that the Nth root of any (fixed) positive number becomes closer and closer to 1 as N increases. This may require an exceedingly large value of N. The net effect is that the rate of return r at prices starting at twice as large and at prices starting at one-half as large as is typical come closer and closer together. THIS EFFECT IS FOR THE CALCULATED RATE OF RETURN r. THE RATIO OF THE FINAL BALANCE TO INITIAL BALANCE DOES NOT CHANGE. I refer to this as
An Illusion of Numbers.
The false notion that one's purchase price does not matter in the long-run is based upon looking at the rate of return r instead of the ratio of the final balance to the initial balance.
Have fun.
John R.