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# Mea Culpa

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So, in this debate with commenters over stock returns vs. GDP growth, I may have to concede.

The ratio of stock prices to GDP is P/Y. Assuming no dividends, that means that P/Y has to increase for stock returns to exceed GDP growth.

I write P/Y = (P/E)(E/Y) where E is earnings. In a steady state, E/Y is constant.

My intuition was that P/E cannot grow indefinitely. If P/E is constant and E/Y is constant, then P/Y is constant. Which would make my claim, that stock returns cannot exceed GDP growth, a theorem.

But my intuition that P/E cannot grow indefinitely is not necessarily true. My intuition treats P/E as the inverse of an interest rate, and when P/E goes to infinity, the interest rate goes to zero. I worry that my intuition is based on a more complex world, in which implicitly there are other securities with positive real interest rates.

I think I can imagine P/E rising indefinitely. The machine costs \$100 today. It earns \$1. So now it is worth \$101. If I bought it for \$100, I can sell it for \$101 and spend the \$1. Then next year, the person who bought it for \$101 can sell it for \$102. And so on.

There is still something quasi-Ponzi about this. Suppose that GDP is \$200. That means that in one hundred years the price of the machine is going to be more than GDP.

In any case, the commenters who have been telling me that it is possible for stock returns to exceed GDP growth forever have convinced me that my basic proof does not work.

### Comments and Sharing

COMMENTS (14 to date)
Grant Gould writes:

The denominator of GDP is years; there is no reason that something costing one year of GDP is impossible but something costing one month of GDP is not.

(However in your one-sale-per-year model the sale of the machine must presumably count toward GDP anyway and so even the apparent contradiction would not appear.)

azmyth writes:

I think you conceed too much. You are correct in saying that fixing an interest rate does pin down a P/E ratio.

Imagine your machine made \$5 a year and cost \$100 and because people had a time preference of 5%, they were indifferent to building more of the machines. The return on owning a machine is 5%. The capital income as a fraction of GDP does not change, nor does the price of the machine.

The growth of the value of the stock market cannot exceed GDP growth, but it doesn't have to for the return on capital to exceed GDP growth.

writes:

If the rate of growth of GDP is increasing over time then the NPV of GDP can (will?) grow faster than GDP. All the value of capital is, a fraction of the claims on future output, so the value of the capital stock can grow faster than GDP forever. All you need is ever faster rates of growth.

Azmyth, when you say, "The growth of the value of the stock market cannot exceed GDP growth", you mean in the long run? Because certainly in the short run the stock market has low correlation with GDP growth.

Tahtweasel writes:

The key is the no dividends assumption. With that assumption, Arnold is correct. But with dividends (as they exist in the real world) he is very, very wrong.

The total value invested in the stock market can't grow at a rate different from GDP in the very long run. Just imagine if stock prices, with no dividends, grew at 8% and GDP grew at 3%. Let's say that the total market cap of all companies in the stock market is twice GDP. In a century, it will be 200 times GDP. In three centuries, it will be 2 million times GDP. Something doesn't seem right there. For any two different rates of exponential growth, that will happen eventually.

However, if stocks grow at 3%, and return dividends of 5%, then you have a perfectly acceptable (and accurate to real life) model, with investors getting a ROI better than the growth rate of GDP.

writes:

Tahtweasel, baked within your conclusion is that there is a non accelerating rate of growth for the economy. If growth in the future is faster than today, the stock market can grow faster than the economy.

It seems unreasonable to assume ever faster growth, but it doesn't seem impossible.

azmyth writes:

@OneEyedMan: Yes, in the very long run - 50+ years. There are tons of things which can cause the relationship to be very loose in the short-medium run: Interest rate changes, capital/labor income changes, monetary policy, etc.

Also, growth is not accelerating. It's been remarkably stable in the long run. If you have accelerating growth, this whole discussion flys out the window.

ed writes:

Arnold, now that you're in a humble mood, please please PLEASE redo you model including dividends. As Tahtweasel and many other excellent commenters have pointed out, you don't need any change in P/E or even any growth in GDP to have positive stock returns.

Assuming a world in which firms reinvest all earnings, never paying out in dividends or stock-repurchases, is a very different world than the one we live in. You can't just ignore the fact that some of firms earnings are paid out to shareholders and are effectively consumed rather than reinvested.

And let's not confuse the issue by worrying about scenarios where interest rates go to zero or growth explodes. Positive stock returns are perfectly consistent with constant interest rates and constant (or even zero) GDP growth.

Eric Ulm writes:

I think your intuition that the price of the machine can't increase forever is correct, but all you've proved is that a company (even Microsoft) can't avoid paying dividends forever. Your machine above has to start producing hamburgers at some point. Person A buys it for \$100. At the end of the year, it produces a hamburger which he sells for \$1 to some hungry guy and then sells the machine for \$100 to person B. Paid \$100, made \$101, 1% return. Person B buys it for \$100. At the end of the year, it produces a hamburger. He sells it to a hungry person for \$1 and sells the machine to person C for \$100. Paid \$100, made \$101, 1% return. The machine never sells for more than the \$200 GDP.

Various writes:

Wait a second. You are missing an essential link. I believe you are correct in your original assertion that the value of "businesses" (substitute "productive assets" if you like) should increase in relative proportion to GDP. However, businesses are financed with both debt and equity. The debt piece, being inherently safer, should, ex ante, yield a rate of return lower than GDP growth. By definition, the equity piece (i.e., stock prices, adjusted for dividends) should generate a yield in excess of GDP growth. This is an ex ante assumption, and should hold true in the long-run. In the short-run, there can be significant differences as actual GDP growth either exceeds or is below expectations (and as short-run changes in things like P/E multiples move stocks around).

ed writes:

No, Various, it's not the case that the real Interest rate on debt must be lower than GDP growth.

This would all be much simpler if everyone would consider the case of an economy with zero GDP growth.

Various writes:

Ed, I didn't say "must" be lower. I said, ex ante, the real interest rate for corporate debt would typically be lower than real GDP growth. That indeed has been the case over the last 60 years, although I'm not sure it's been true over the last 10 years, a period in which real GDP growth has been relatively crummy.

kebko writes:

Arnold, I appreciate your conciliatory tone, but you're not addressing your error at all. Why must you assume no dividends? You might as well say, "if we assume no returns, there can be no returns." If returns on capital can never be consumed, there would be no point in investing. Please consider the no growth scenario we have suggested.

rpl writes:
The machine costs \$100 today. It earns \$1. So now it is worth \$101.
Is the assumption here that the dollar that the machine earned conveys along with the machine? In that case, it is not correct to say that the machine is worth \$101 because \$101 buys you the machine plus a dollar. You could take that dollar out and put it in your bank account right away, in which case you really only spent \$100 for the machine. The end result is identical to what it would have been if the original owner kept the dollar that the machine earned and sold the machine to the buyer for \$100. There is nothing "quasi-Ponzi" about either scenario.
writes:

PE Ratios do not have to grow for stock returns to exceed GDP growth rates indefinitely. The difference is consumption, and in one way or another you have to postulate some form of distributions from corporations, whether in the form of dividends, stock buybacks, mergers & acquisitions, or whatever. A zero consumption society makes no sense at all, and is incompatible with the continued life of its members.

As long as there is consumption, long run stock returns not only can, but MUST, exceed GDP growth rates. MUST.

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