According to the economist Robert Shiller, a perfectly rational market would maintain a constant ratio between earnings growth and the earnings yield. That is, Shiller argues not that earnings growth ought to remain constant or that the earnings yield (the inverse of the P/E ratio) should remain at a constant level but that there is a natural equilibrium between the two. In the real world, however, where the market fails to adhere to that equilibrium, the gods or fate or sanity must ultimately intervene to correct the imbalance, if only temporarily. (See Shiller's *Irrational Exuberance*, especially Chapters One and Ten).

In more concrete terms, Shiller has it that, where the rate of earnings growth rises faster than the earnings yield (to be clear, the yield derived from the ratio between non-cyclical monthly values of earnings and stock prices (SPY), the market is out of balance and stock prices must suffer. Where the rate of earnings growth falls faster than the earnings yield, the market will eventually recover. Not immediately, but soon enough for those who are patient and disciplined.

As an initial thought experiment, we can think about how this compares with Peter Lynch's PEG ratio, which is the ratio between the P/E ratio and the rate of earnings growth (i.e., "G"). Where the P/E ratio is greater than earnings growth, the market is deemed to be expensive; where it is equal to or less than earnings growth, a bargain is expected to be found.

In simpler terms, if we were to compare Shiller's view and Lynch's view and imagine a market where the ratio between stock prices and earnings remained constant (so that the P/E ratio and earnings yield did not change), it would seem, if my description of their respective positions is correct, that under the former approach, a rise in earnings ought to be problematic in the long term, while Lynch's methodology suggests that a rise in earnings growth is a boon (again, all else being equal).

There are a number of important differences that I have papered over, however. The PEG ratio tends to be concerned with future nominal earnings growth, whereas Shiller's P/E10 is backward-looking and uses real rather than nominal earnings.

There is another key difference that becomes apparent once we get into the details of the respective formulations of the relationship between earnings growth and the earnings yield (or, in Lynch's case, the P/E ratio). We can begin by reflecting on how the P/E10 (also called the Cyclically Adjusted P/E ratio, or CAPE) is calculated.

For our purposes, we will calculate Shiller's ratio as follows:

a) P/E10 = (E∕E10) ∕ (E∕P).

In other words, the P/E10 is an expression of the relationship between earnings growth and the earnings yield.

*(Source: **Robert Shiller**)*

It could also be expressed as:

b) P/E10 = (P∕E) x (E∕E10),

which is rather the opposite of Lynch's PEG relationship.

Lynch divides the P/E ratio by earnings growth whereas Shiller multiplies the P/E ratio by earnings growth (keeping in mind the important differences between how "earnings growth" is defined in the respective systems). The other big difference is that in Shiller's system, earnings growth is expressed as a ratio while in Lynch's system, earnings growth is an annualized percentage.

Out of curiosity, I decided to compare how Shiller's and Lynch's conceptions played out historically as best I could, but before getting into that, I wanted to step back to the relationship Shiller posits between earnings growth and the earnings yield, as in equation A. As I said at the outset, Shiller's theory on equity prices can be read to be a basic statement on the relationship between these two factors.

What is compelling about looking at the P/E10 in this way is that it drags out from under the carpet and puts on full display precisely the thing that Shiller is trying to suppress: short-term fluctuations, especially with respect to earnings volatility. The E/E10 ratio is effectively a semi-smoothed measurement of the five-year rate of earnings growth, because if we were to transform the E/E10 ratio to a compound annual rate of growth (CAGR), which we will do shortly, the "E10" must be thought of as a centered five-year moving average rather than as a ten-year moving average. Shiller's P/E10, therefore, is nothing other than the ratio between a five-year earnings growth rate and the (highly volatile) earnings yield calculated from incremental earnings values (rather than a moving average of trailing earnings).

Shiller's demonstration that the P/E10 ratio has historically been negatively correlated with subsequent returns is therefore a demonstration that where the growth-to-yield ratio (let's call it the "GYR") is too high, it must eventually fall, and where it is too low, it must eventually rise. (We might assume that the equilibrium level, if it exists, is somewhere around the long-term historical average of the P/E10, but there is no way to be sure).

If you are like me, you might be curious as to whether the GYR (remember, this is simply a longer formula for describing the P/E10) has historically been pushed up by very high earnings growth or sharp declines in the earnings yield (or both) and whether historically low GYR episodes have been marked by low growth rates or high earnings yields. And, like me, you might be especially keen on knowing how our currently high P/E10 levels look under the light of history from this perspective.

Peaks in the P/E10 (defined arbitrarily here as breaches above the 20 level) occurred roughly in 1898-1903, 1928-1930, 1936-1937, 1961-1969, 1993-2008, and 2010-present. Generally, however, the major peaks are regarded as having occurred in 1901, 1929, 1966, and 2000. (I believe that another peak will be formed in 2016-2017, but more on that later).

The surprising thing about these episodes is that the common element in these peaks was *not a decline in the earnings yield* (which is what we might expect from a Shillerian emphasis on stock prices) but *a surge in earnings growth*. If we transform the E/E10 ratio into a five-year annualized growth rate, we will find that in each peak, the growth rate (as with Shiller, adjusted for inflation) rose above the earnings yield.

That is, the spread between Shiller's growth rate and the earnings yield went positive. These have been (or had been, at any rate) relatively rare occurrences: 1899-1903, 1928-1930, 1937, 1956, 1966, 1995-2001, 2004-2007, 2011-present (with a brief break two years ago). These generally match the P/E10 peaks above. Again, the common element in each case was a surge in earnings. In many cases, this surge coincided with a decline in the yield, but not in all, and in 2011, the yield was the highest it had been since 1989.

If we assume that the spread between the earnings growth rate and the earnings yield is subject to the same dynamics as the growth-to-yield ratio (the GYR, and therefore, the P/E10 ratio) - and the correlation between the two suggests that they are - there is a symmetry here again between the way the P/E10 functions and the way the PEG ratio functions.

First, we have to point out that there is no good reason for the PEG to be treated as a ratio rather than a spread. If the ratio is above 1.0, this is "bad," and if it is below 1.0, it is "good," but that is mathematically no different from saying that the *spread* between P/E and growth is bad when it is above zero and good when it is below zero. This brings us a bit closer to tying the PEG ratio into this but let's hold off on that a bit longer.

I think the fact that peaks in the P/E10 are marked *primarily* by surges in earnings is significant and is detrimental to Shiller's "psychological" explanation of stock prices. Remember, once we strip it of the garb of behavioral finance, the implication of Shiller's thesis is that the ratio between growth and the yield should remain constant. That position can be mentally split into two components: a) that the distance between the two should be constant, and b) that the correlation between the two should be 1.0. Shiller has emphasized the first half of this relationship, how deviations from the ideal ratio predict returns, but what really is the justification for assuming that there is an equilibrium ratio between the two? What if the real story of the relationship between growth and yields is not one of ratios but of correlations?

The reason this may make a difference is that if the fundamental relationship between growth and yield is one of correlations rather than ratios (or spreads), then new equilibriums could be achieved at different ratios at different times. That could accommodate scenarios wherein returns are high despite periods of persistently high ratio (as has been the case over much of the last twenty years).

In the charts below, I show how an approach that emphasizes the correlative aspect of the growth-to-yield relationship rather than the Shillerian emphasis on the magnitude of the growth-to-yield ratio seems to do a better job of adjusting to changes in the relationship over time. This grants that there is, as Shiller insists, a necessary relationship between growth and the yield without dictating where the Shiller equilibrium, if one exists, must lie.

Under Shiller's system, what force ought to keep the relationship between earnings growth and the yield at a constant equilibrium? It is simple, of course: that the market today should react no differently to the most recent quarter's earnings than to earnings in the second quarter of 2005, the fourth quarter of 2008, the first quarter of 2011, or the third quarter of 2013 and react no differently from one age to the next.

Certainly, I cannot disprove Shiller's insistence that it is ultimately the ratio between growth and the yield that is decisive and that the ratio is driven by euphoria (or lack thereof) rather than by other factors. What I hope to show and believe I have already begun to show is that there are dynamics on the earnings side of things that help us to understand the evolution of markets at both the cyclical and supercyclical levels.

We have seen that every peak in the P/E10 has occurred with a surge in earnings growth rates, particularly in excess of the yield level. The reason for this is that, even with Shiller's effort to strip earnings of their volatility by using ten-year moving averages, this surge in earnings occurs after severe growth recessions. In 1894, Shiller's annualized growth rate for earnings was -9%. In 1921, it was -25%. In 1932, it was -13%. In 1958, it was only -1%, but this was, by far, the lowest rate in the post-War boom. From 1970-1994, the annualized growth rate in real earnings was barely above 1%. For most of the 1980s and early 1990s, it was around -4%. Any earnings euphoria must have come very late in the game. We see this again today: although profit margins have been notoriously high in the last twenty years, Shiller's annualized growth rate was -34% in 2009. I think everyone who has been paying attention to the markets over the last few years would have a hard time ascribing the stock market boom to unbounded enthusiasm. It is only in recent months where "hard data" from the "real economy" have begun to provide a rational for exuberance.

In sum, earnings booms arrive relatively late in the supercycle. In fact, it seems as if there is a symmetry between the size and extent of the earnings droughts and the subsequent booms. In the 1894-1901, 1921-1929, and 1932-1937 cycles, the booms were as brief and as sharp as the initiatory earnings recessions. In the 1960s, the P/E10 top was as flat and drawn out as the previous earnings recession was shallow. And, the elevated P/E10 levels since the mid-1990s have been as extensive as the earnings drought of the 1970s-1990s. The unprecedented collapse in 2009 would seem to indicate that the present cycle will be both robust and brief.

Earnings booms that drive the P/E10 to nosebleed levels occur only very late in the supercycle, and only after the market has long been climbing a wall of worry.

What I am hinting at and will attempt to describe in more positive and detailed terms in subsequent articles is that P/E supercycles progress along fairly predictable paths. For now, let me suggest the broad outlines of the relationship:

As we saw in equation B above, the P/E10 can be calculated as the product of the non-adjusted P/E ratio and the rate of earnings growth (i.e., (P/E)*(E/E10)). Peaks in the P/E10 always coincide with peaks in earnings growth and never with the P/E ratio, although earnings growth alone can never produce a peak in the P/E10. Almost without exception, those cyclical earnings peaks are *preceded* by peaks in the earnings yield seven to eight years in advance (1894, 1921, 1961, 1992, and, I believe, 2009). Once we combine these observations with a method that emphasizes correlations rather than levels and ratios, the nature of these supercycles becomes rather more apparent.

For now, however, let's conclude with a comparison of the PEG spread and the growth-to-yield spread (GYS), which, as discussed above, is an approximation of the P/E10. I will use annualized seven-year growth rates in nominal earnings for the PEG spread. In the following chart, I compare the GYS to the PEG spread calculated from backwards-looking earnings.

From this vantage point, PEG has only been useful when earnings growth has been relatively stable and positive, as in the post-War years up until the 1990s. When earnings were much more volatile and the cyclicality was therefore enhanced, it was worse than useless.

If we look at the PEG calculated from forward earnings, there is a relatively high correlation between it and the P/E10 and its derivative spread. To apply the PEG to historical markets, you simply have to know which earnings are going to turn over the medium term. The P/E10, as it turns out, has done a fairly decent job of predicting PEG.; that is, (P/E * G) tends to predict ((P/E)/G). The reason for that is not due to any inherent virtues in the P/E10 but entirely due to the cyclicality of earnings - which is still evident when we compare the five-year growth rate in smoothed, inflation-adjusted earnings to the seven-year growth rate in raw, nominal earnings - and the way that cyclicality fits in with the supercyclicality of the P/E ratio. In 2000, for example the P/E ratio times past growth predicts P/E divided by future growth, because the jump in earnings that culminated in 2000 will almost inevitably be followed by an earnings bust, the value of which will be divided from the P/E ratio in the calculation of the PEG ratio.

The PEG ratio is, therefore, superfluous, certainly as a predictive tool, except insofar as it helps us illustrate the critical importance of earnings volatility and cyclicality. The past, one might say, is more helpful in predicting the future than the future is. I will try to make good use of that dictum in the next article, where I will attempt to construct a positive framework and apply it to markets moving forward into 2020.

In the meantime, the theoretical question that is left hanging in the air is why earnings growth and the earnings yield must bear a necessary relationship with one another. I can accept such a relationship on utilitarian, analytical grounds, but this assumption is the entire basis for Shiller's theory on stock market prices. It is impossible to talk about the P/E10 without implicitly talking about the relationship between growth and yields.

**Disclosure:** The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it. The author has no business relationship with any company whose stock is mentioned in this article.