Michael Mandel writes,

an investment in the stock market becomes a bet on innovation. Do you think that the U.S. or the global economy has another wave of disruptive innovation coming? Then buy stocks. But if you think that we are stuck in permanent stagnation mode, then stay away from the stock market.

Pointer from

Phil Izzo.

The most controversial algebra that I have ever written is this:

P/Y = (P/E) * (E/Y)

where P is the value of the stock market, E is the earnings of shareholder-owned corporations, and Y is GDP. Every time I write down this algebra, somebody says that I am a fool who is missing something. I am sure I will take grief again, but just keep in mind that the harsher the tone of your insults the more they apply to you rather than to me. And, no, dividends do not make a difference, so don't throw that one at me. Been there, done that.

The point of this algebra is that for the value of the stock market to rise faster than GDP, one or both of the following has to happen: the price/earnings ratio has to rise, meaning that people are willing to apply a lower discount rate to earnings; or the ratio of corporate earnings to GDP has to rise, meaning that profits of shareholder-owned corporations increase as a proportion of overall income.

The historical return on stocks has been high in large part because P/E has trended upward. Past performance is not necessarily predictive of future returns.

I read Mandel as saying that a bet on the stock market is a bet on GDP growth. If so, then I agree.

I am one of those who has given you grief in the past, but this time you did not make the same mistake. So no grief from me, congratulations.

You are absolutely right that the "value of the stock market" cannot rise faster than GDP in the long run. We have never disagreed with that. (There is some argument to be made about domestic vs world GDP growth rates, but you are basically correct.)

Our past disagreements came about because you claimed that stock *returns* could not be higher than GDP growth. Stock "returns" are simply not the same thing as the growth in the "value of the stock market." To find stock returns, you have to add in dividends and repurchases, and remove dilution due to share issues and new firms.

My reading of the S&P500 is that none of those three expressions has changing fundamentally over decades. When you add dividends, etc., to get total return, I think you are looking at, roughly, the average corporate return on invested capital.

Two points

1) Dividends do indeed matter. On a long enough timescale the survival rate of every corporation falls to zero. Eventually some unforeseen event WILL wipe every company out. In this case as an investor limited liability is your friend.

A company that's been generously paying dividends means most of the accumulated earnings are protected in the investors' pocket. In contrast a company that's been hoarding retained earnings means that in that case the creditors and not the shareholders will get the money.

Even absent this company's with powerful management and weak shareholders often hoard retained earnings to give the executives power. This clearly diminishes the shareholder value, especially if those retained earnings are parked in 1) low-yield cash (Google investors if they could re-invest in equities would get a much higher yield on that cash pile) or 2) negative NPV prestige projects (like acquisitions, which are almost always value destroying).

If you don't believe dividends don't matter, why is that almost every single non-dividend paying company with a large cash pile gets a strong bounce on the day they announce dividends.

2) Your choice of variables here is ambiguous. Let's talk about GDP, is it global GDP or US GDP. And in that case are we talking about global markets or the US market. Because US stocks can experience strong growth on weak US GDP by selling to a fast-growing global economy and/or stealing market share from other nations' multinationals.

Furthermore you call P the price of the stock market and assume that if P/E's stay constant and Y grows quick because of innovation therefore investors must do well. Well the thing is companies drop out of and enter the stock market at arbitrary times.

So there's a basis mis-match between any index and the performance of the economy as a whole. In cases of high innovation disruption small-caps are likely to perform much better than large caps (Studebaker Automobile vs Intel). In this a market-cap weighted index like the S&P 500 will underperform.

Going even a step further a large number of new companies are likely to be private and hence outside public investors' domain until already matured. No doubt Facebook is a huge innovation, yet its contribution to public investors has been strongly negative. Almost all the wealth gains were reaped by a handful of early private investors/founders.

Electric utilities, KO, MCD, PEP all pay 3% or 4% dividends and the dividends should go up with inflation. T-bills and cash pay less and will not go up with inflation.

1. Who needs stock prices to rise faster than inflation?

2. That is not to mention KO, MCD, PEP can grow by expanding to other countries without any new tech.

3. Small incremental tech changes are sufficient.

4. I do not see a basket of those stocks as much more risky that T-bills.

So why should I not invest.

But I do see possible breakthroughs in biotech and in energy.

Addendum after reading the linked article:

I am fine with any return above 3% + inflation. I agree that 6.6% seems unlikely and should fall over time.

Note: I see bonds as no safer than diversified equity in dividend paying companies with very low or no debt.

You are analyzing fundamentals. But fundamentals are only one part of the equation; risk appetite is the other. And the vast majority of movements in the markets are due to risk appetite changes, not fundamental news.

It's entirely conceivable that valuations will rise significantly without any new "wave"...all the money that has piled into treasuries has to go somewhere eventually, after all...

To someone with a hammer, everything is a nail, which has created the following scene in my head:

Friend 1: sees Arnold slapping a rock with a fish.

Friend 1, to friend 2: What's Arnold doing?

Friend 2: Oh. He thinks the rock is a nail.

Friend 1: Mm. OK. Why is he hitting it with a fish?

Friend 2: He thinks the fish is a hammer.

Your math is wrong for a number of reasons. Its makes a number of poor assumptions

1. All companies are listed on the market, they are not. Companies listed on the market are different from private companies, and maybe represent a more profitable part of the economy. Maybe private companies are lagging. Most bankruptcies are for private companies not public ones.

2. Earnings between country and within country problems, as well as GDP between companies etc...

That said, there is some basic truth to your identity. However, there is another possibility, our measurements of GDP and inflation are way off, and the market measures them better, but they are very hard to disentangle.

Generally agree, with one exception: you may also be betting that *established companies* -- that is, those listed on the exchange -- will grow faster than non-established companies not on the exchange.

Soft drink GDP could go up 5%, but you might bet that Coke goes up 6% and Acme Cola stays, um, flat.

The most interesting assertion to me is:

"... the harsher the tone of your insults the more they apply to you rather than to me."

Wow! I hope you can explain that more fully at some point.

The math isn't wrong, per se, but I can see why the formulation would be controversial (the problem is really the notation):

P/Y = (P/E) * (E/Y)

Using the typical definition of the P/E Ratio for stocks (where N = Number of Shares):

P = Price per Share (units of $/N)

E = Earnings per Share (units of $/N)

Meanwhile:

Y = GDP (units of $)

With that notation, P/Y doesn't work, because its units are [($/N)/$) or just 1/N], which has no meaning.

If you want to get around that notation problem and make the math actually work, you really want:

P = Market Capitalization = Price per Share * Number of Shares (units of $)

E = Aggregate Market Earnings (units of $)

Once you make that shift, you'll have much less trouble trying to shoehorn GDP into a relationship between stock prices and earnings because all the units of all the components are now the same ($).

Alternatively, we could set Y to be equal to "GDP per share" and use the rest of the original formulation.

Three thoughts:

1. Many commenters appear to be confusing price of specific stocks (which is not what Professor Kling defined as P) with the value of the entire stock market (which is what he defined as P).

2. A specific investment in particular stocks could be a bet on something other than GDP rising. For example, one could be betting on one's ability to predict the future better than other investors. We can confidently predict that at least a few of these bets will pay out, though the pay-out might very well occur even though the prediction turned out to be false.

3. Professor Kling is talking in aggregates! That made my head spin momentarily because I associate him with warnings about the dangers of thoughtless aggregation (as opposed to, say, patterns of sustainable specialization and trade). There's not really a conflict there (especially as indexes allow closer tracking of an aggregate), just a moment of dissonance.

Max

"just keep in mind that the harsher the tone of your insults the more they apply to you rather than to me"

I'm rubber and you're glue...

The worst abuses are really material: wasting someone's time with lousy arguments, refusing to shoulder BOPs, offhand dismissals of evidence, etc.

Flame is just flame. And yet interjecting curse words earns instant excommunication from "intellectual" circles, while putzing around with dishonest arguments is tolerated till the cows come home.

At any rate, I thought about it and P/Y=(P/E)(E/Y) checks out to me. You say you get a lot of controversy out of this, but I hope you also get appreciation for tethering stock market earnings/value to the whole pie. There is definitely the common assumption that the stock market will always yield returns above general economic growth. This is a dangerous assumption, even for non-economists.

"

Y is GDP."Yes, but remember that the S&P 500 corporations receive a significant and rapidly growing portion of their earnings from abroad, so Y can include the GDP of Asia.

I am very confused. This appears to be a trivial accounting identity. Where is the controversy?

Assuming a constant population, the implication of the formula is that all earnings need to be reinvested in the economy to produce the same GDP. This means we will earn enough to maintain, but not improve, our lifestyle as long as we continue to work at a constant total rate. This is controversial?

What happens if productivity increases? We can work less and/or earn more. What happens if we can trade with other countries with which we have relative comparative advantages? It means we can work less and/or earn more. What happens if other countries grow and we have relative comparative advantages? What if population shrinks and GDP stays constant? Same, work less and/or earn more.

Aren't these all macroeconomic accounting identities? All you are saying is our standard of living will remain constant if it does not improve!

We do not know the future, so we are constantly guessing these future conditions. Hence we have market volatility as well as private return on investment volatility.

We do have a general idea that growth (really productivity---it need not be growth per se---but it tends to be) is good and no growth is less good. It is just difficult to guess, and therefore price it, accurately.

I am unclear why you think your readers will call you a fool for pointing this out or why you even think your point is insightful (sorry!).

What am I missing?

It seems that the reality is that investors are willing to accept a lower return for their risk. Companies can easily create paper profits and shareholder value must be reflected in those paper profits.

therefore the real situation is that people have no place to park their money so they accept lower return for their risk. this boosts the value of the corporation creating paper profits that are a higher proporation of overall income.

Fake numbers that one day will either have to create real profits or that money will drain into something that will create real wealth.

P/Y = (P/E) * (E/Y)

That looks like a tautology to me.

(P/E) * (E/Y) is another way of saying (P*E) / (E*Y) just as (4/9) * (9/2) is another way of saying (4*9) / (9*2) because you are allowed to multiply the numerator vs the numerator and the denominator vs the denominator.

P/Y equals P/Y just as 2 equals 2

I just realized that I agree with Mike Rulle's framing of your tautology.

sieben, and everyone else, you might want to notice that Arnold's identity does not by itself say anything about the *return* to owning stocks. That will be a function not just of earnings growth and payouts, but of the level of P (which generates P/E and P/Y), which is not restricted by the identity in any way.

If you make P twice as large, then steady state returns will be half as large. If you make P half as large, then steady state returns will be twice as large. (By steady state I mean that P changes over time to keep P/E and P/Y constant, and E/Y is also constant, as Arnold supposes.)

Thus, *any* level of steady state returns is theoretically possible, consistent with the identity. Regardless of the path of Y, and even with no growth in Y at all.

Arnold, Bill Gross, and many others keep confusing the *return* of stocks with the *value* of stocks. I don't know why this is so hard.

Mike Rulle, what you are missing is that, in past posts, Arnold claimed that the long-run return on stocks could not exceed GDP growth. That caused many of us to give him a hard time. The current post does not repeat that mistaken claim.

ed:

The odd thing is that the item he links to is about how different types of capital and different subsets of the pool of equity capital might react to different temporal contexts.

How this relates to algebra that does not account for the income portion of returns is a head scratcher.

Ok, here is another take on how Arnold Kling's take on his tautology is wrong, or more generously, incomplete.

First I will reframe what he said.

P is the value of the stock market, E is the earnings of shareholder-owned corporations, and Y is GDP.should read:

P is the value of the globe's stock market and privately held equity, E is the earnings of the globe's shareholder-owned corporations and privately held equity, and Y is GWP, with W standing for World.The reason for adding privately held equity is because if you use Kling's definition of P, then a change in the percentage of companies going public or private changes the value of his P without changing the value of his Y.

The reason for including World to each of the definitions is because it would be wrong if he meant US instead of world, so I wanted to clarify. It would be wrong because international earnings are rising as a percentage of US corporate earnings, while foreign corporations earnings as a percentage of earnings generated in the US is also rising.

With that reframing out of the way, Kling still left out corporate debt.

Imagine a corporation, XYZ1, that is worth $1 billion, is debt-free, and adds debt, uses it to build something worthwhile, and over time earns a positive return from that debt-financed new project after taking into account debt costs. $500 million of that return goes to bondholders, and $500 million to shareholders

Imagine a clone of that corporation, XYZ2 worth $1 billion, that is debt-free and doesn't add debt, but expands by issuing new shares (equity), and has the same positive pre-debt return on capital from the new project that XYZ1 has from its project, namely $1 billion dollars.

Neither company adds more to GWP than the other one. However XYZ2 adds more to its market cap than XYZ1, while XYZ1 has a higher

percentageincrease in value for pre-existing shareholders (before each company's capital injection) than XYZ2.There is no free lunch for a company increasing debt, it is merely leveraging both its gains and losses. Nevertheless adding debt to the world's corporations in aggregate could either add or subtract value to P without changing Y relative to a world without a change in corporate debt.

I laughed when I read this Arnold.

In the late 80's I was doing Money and Banking for the first time. And I wrote something very similar to this on the overhead projector - and made the comment that the present value of an stock is determined (in part) by its expected cash flows.

A student in the back stood up (!!!) and announced this was nonsense - he then argued that capital gains were the only reason people bought stocks. So I started explaining that capital gains come from ... expected cash flows.

The student muttered an expletive under his breath, and stormed out. It's funny now, but it wasn't then.

Wait I'm totally confused. People might be able to argue with your verbal analysis but surely no one actually disbelieves your equation.

It's a pure algebraic identity. It doesn't matter what P, E and Y even measure or how they change over time. So long as they are all real numbers (or even members of any multiplicative group) the equation is valid.

The analysis you give implies, though doesn't explicitly state, that one's return in the interval [t0,t1] from investment in the stock market has (expected) value proportional to your expectation of the difference P(t1)-P(t0). This is the only claim you make that doesn't directly follow from a pure algebraic identity. While it is false if one understands P to be the total valuation of the stocks in the market (this can increase as a result of a greater percentage of societal resources being held in corporations as opposed to directly by individuals without growing returns to stock owners). If P is an appropriately weighted stock market index (I don't think price-weighted indexes like the dow will cut it) and one assumes that that the expected future dividends component of P is constant. This last point is reasonable if one thinks that dividends are either a trivial fraction of returns or that dividends paid per dollar per year remains relatively constant and this is reflected accordingly in P (investors rationally assess the value of dividends). To see that crazy fluctuations in payment of dividends can break the proportionality of P with return on stock investment (assumed to be an appropriate market index) consider the limiting case where, all corporate holdings are perfectly liquid and at time t0 they are all solid with all profits returned to investors as dividends. In such a case P(t1)=0 but investing at t0 results in 0 profits rather than a lose.

Still, the math and all your explicit statements are correct. However, the implied proportionality of P and return on stocks seems to fail in edge cases. You claim to have been here before and I'm no expert so I would appreciate if you would point out where my analysis went wrong or I misinterpreted you.

The algebra seems to me to be a not very controversial generalization. But the last paragraph of Mandel's piece could be restated to point out the differences between the economic worldviews of our choices this November:

Do you trust that the U.S. or the global economy has another wave of disruptive innovation coming...even if you cannot now imagine what it will be? Then vote Republican. But if you think that we are stuck in permanent stagnation mode, in which the future will rely on government directed health care and education employment, then vote Democrat.

As long as you consider GDP to be the weight adjusted GDP for the markets the company invests in, the formula makes sense with the following modifications.

The market attempts to look forward. So,

P should be current price

Y should be expected future GDP.

E should be expected future earnings.

Sorry to be late to the party but I have a different take on this than anything I have seen above. I would like to present a different approach and solicit comment from those who care to follow my logic.

Arnold’s equation is an identity but the importance is what questions we might be able to answer from its use. My problem is the equation helps us determine P/E but we are really interested in what we can expect from future market performance. If we deploy a little very simple calculus and algebra we can come up with some interesting results.

I will refrain from getting into what Arnold likes to call “math porn” and just state the results. If anyone wants more detailed derivations just ask.

If we take the total derivative of the left and right-hand-side of his equation and use the convention of G(x) to indicate growth in variable x we get:

G(P/Y) = G(P) – G(Y)

A similar approach to the equation’s right-hand-side yields:

G(P/Y) = G(P/E) + G(E/Y).

Combining the two gives us:

G(P) – G(Y) = G(P/E) + G(E/Y)

Presumably, we are interested in the growth in the value of stocks so we can restate this in a more useful form as:

G(P) = G(Y) + G(P/E) + G(E/Y)

Now, we know from what Larry Klein called the “great ratios of economics” that as a first approximation labor and capital income shares of GDP are relatively constant and sum to one. This is the basis for the popular use of the Cobb-Douglas production function in macroeconomics. If we accept that argument then G(E/Y) must be zero.

Furthermore, if we assume the total stock market in the aggregate can be represented as a constant growth stock it can be shown that the Fed Model can be used to describe fair market value for the overall stock market. This result can be shown with a little algebra and will be shown upon request. The model was developed by the Fed staff to help Allen Greenspan make his case for “irrational exuberance” back in the late 1990s (if memory serves). The model result becomes:

P/E = 1 / r

“r” is the nominal risk-free long-term rate of interest which is proxied by the 10 year Treasury bond yield. Now from growth theory we know that “r” converges to the long-term growth in the labor force plus productivity growth plus inflation. As an approximation we can assume the pre-2008 market meltdown values were about 1%, 2% and 3% respectively so the base case value of “r” was about 6% which would correspond to a P/E of about 16.7 which is consistent with most analyst views.

Using this information, what would we expect the stock market to do? Specifically, let’s see what would happen to stock prices if the “new normal” for productivity growth were to either increase to 3% or fall to 1%. I will assume a three year transition to the new steady-state which could be the period 2009-2012 or under a new fiscal policy the period 2012-2015. The following table summarizes the model results.

r, P/E, 3 yr avg G(P/E), G(Y), 3 yr avg G(P), New Normal

6% 16.67 n/c 6% 6% 6%

7% 14.29 -5.0% 7% 2% 7%

5% 20.00 6.2% 5% 12.2% 5%

These results surprised me and I wanted to ask for comments as to the result. In words it says prices will ultimately grow at the rate of the economy which is not a surprising result. However, during the transition to the “new normal” we should be prepared for underperformance when moving to a better new normal and outperformance when moving to a less desirable growth track.

Any comments and suggestions for revising the methodology would be much appreciated.

A little quiz:

In 2010, Arnold earned $100,000.

In 2011, Arnold earned $110,000.

How much did Arnold earn in 2011?

Answer: $110,000.

As long as you keep insisting that the answer to that question is $10,000, you will not understand our criticism.

There's nothing wrong with your equation, but GDP is not the same as the growth in GDP, and the growth in stock values is not the same as the total return on stocks (and total return far exceeds capital gains plus dividends because of other cash distributions, such as mergers and buybacks, that are not included).

The P/E expansion argument is untenable: a TRIPLING of the P/E ratio over the course of the 20th century would only have added 1% to annual returns. But we don't need it: we just need to understand that ALL of GDP is product, not just the GROWTH in GDP from the previous year.