As this impressive new cyclical bull market in the American stock markets continues to relentlessly march higher, the crucial issue of general-stock valuations remains at the forefront of the perpetual debate between the bulls and bears.

Today the bulls and bears are staking out very different opposing positions on stock-market valuations, which are simply the way of attempting to quantify what stocks ought to be worth today based on the underlying earnings and cashflow that they are able to spin off for their owners.

The bulls recognize that current stock-market valuations remain very high by historical standards, but they are not worried. Rather than fret about the annoying precedent of history, the bulls simply believe that we are in a New Era so New Era valuation metrics are needed to redefine when a stock is overpriced or not. Not surprisingly these New Era valuation philosophies unanimously suggest that the markets have plenty of room left to head higher from here.

The bears also recognize that current stock-market valuations remain very high by historical standards, and quite frankly it terrifies us. You can look at almost any market, in any country, during any era, and the stock-market results going forward off of valuations as high as todays are always ugly. Periods of rampant overvaluation, like today, are inevitably followed by dismal periods of undervaluation like clockwork, like summer is followed by winter.

The bulls generally believe in financial relativism, that the financial laws governing markets today are totally new and unique and are not constrained by anything in the past. The bears take the opposite position and believe in financial absolutism, that unchanging financial laws do exist, that valuable lessons can be learned from the past, and that patterns echo through history that we ought to prudently heed.

The leading example of the bullish push for financial relativism today is the idea that stock-market valuations are dependent on interest rates. While there are already many strains of this theory loose in the wild infecting Wall Street analysts and academics, the most common variety is simply known as The Fed Model of Valuation.

You have certainly heard of the Fed Model, as it is the most common rationalization offered by the bulls for the seductive hypothesis that today's markets overvalued by all historical standards are nothing to fear. The Fed Model family of theories states that the stock markets are not overvalued even at very high P/Es as long as interest rates remain low. Hence, if you believe in the Fed Model, current stellar P/E ratios and dismal dividend yields are "natural" and not a dire warning sign as the contrarians perceive.

In my essay this week, I would like to take a brief look at the philosophical underpinnings of the Fed Model family of valuation theories in light of modern market history.

Please realize that this particular essay is not about the formal Fed Model in particular, but all the theories falling under that general umbrella of inversely relating general-stock P/E ratios with bond yields. I do fully intend to write another essay in the future specifically focusing on the formal Fed Model exclusively, and this background information presented today is a necessary prerequisite for that endeavor.

It is easiest to understand the Fed Model family of valuation theories if you can see their logical origins leap out of a graph. Our first chart this week shows the S&P 500 P/E ratio graphed under interest rates as represented by 1-Year US Treasury Bills. Before we dive into this graph though, there are a couple of technical notes to consider.

First, all of the data used in this essay is weekly, so the monthly P/E numbers we usually use in my essays on valuation have been replaced with weekly ones from a separate data provider. These weekly P/E numbers are partially based on S&P 500 estimated earnings, which I don't care for, but they provided us with the highest resolution for this particular essay.

In addition, the formal Fed Model also uses forward earnings rather than trailing earnings, right or wrong. So, while the general strategic trend in P/Es between our usual monthly valuation data and this new weekly valuation data is identical, some of the tactical week-to-week details differ.

Second, the formal Fed Model is based on long-term interest rates, specifically US 10-Year Treasury Notes. In my future essay on the formal Fed Model specifically, we will use its own 10y Treasuries. But, for the general foundational background today, 1y rates are technically more accurate.

Why? P/E ratios are always defined in one-year blocks, the current stock price divided by the prior 12-months' earnings. Interest rates and P/Es are more comparable if they both cover the same time period, a single year. Comparing 10y yields with 1y P/Es is questionable at best, almost apples and oranges.

As you examine this chart, it is readily apparent why the Fed Model family of valuation theories is growing so popular with the bulls. For the last 23 years or so, encompassing the entire Great Bull market in equities, there has been a generally negative correlation between general-stock P/E ratios and interest rates. As interest rates grinded lower, the stock markets relentlessly marched to ever-higher valuations.

The negative correlation between valuations and interest rates in the past couple decades is indeed remarkable. Generally higher interest rates appear to lead to lower valuations and vice versa, with even many of the troughs and peaks of the two data series inversely matching as if they were lock and key. Using this graph alone it would not be difficult at all to convince someone that the Fed Model valuation theories have great validity.

The formal Fed Model was introduced in the Fed's Monetary Policy report accompanying Alan Greenspan's July 22nd, 1997 Humphrey-Hawkins testimony before the US Congress. The actual background surrounding its introduction is fascinating too, but I will have to save that for my future essay on the formal Fed Model specifically. The reason I bring up its 1997 birth now, however, is to highlight the body of actual historical data used by the staff at the Federal Reserve to devise this model.

In the chart above there is a yellow-highlighted block of time running from 1982 to the end of 1996. This 15-year chunk of time, all of which happened to be in a mighty secular bull market, forms the entire historical statistical basis for the whole family of Fed Model valuation theories.

Not surprisingly, if you examine the S&P 500 P/E ratios and interest rates within that unique period of market history, their negative correlation is striking and amazingly precise. There is no doubt that the Fed Model worked reasonably well in the 15 years used in hindsight to create it.

Statistically we can easily measure the degree of correlation between these two data series using correlation coefficients, which mathematically quantify just how closely two variables are related. A correlation coefficient (C) of +1 means two data series tend to move together in perfect lockstep, while a C of -1 means they move away from each other in perfect opposition.

The yellow-shaded area above that was used by the Fed staff to create its valuation model, not surprisingly, exhibits a very strong negative correlation between P/Es and interest rates of -0.91. Depending on how you attribute causation, a nontrivial issue in itself, it appears that lower interest rates led to higher stock valuations or higher stock valuations led to lower interest rates.

Naturally this was a very convenient conclusion for the Fed research folks to reach for a couple key reasons. First, if high stock valuations were "normal" and acceptable during low interest-rate periods, then the Fed in 1997 did not have to worry that soaring equity prices were feeding a historical speculative mania and bubble. While this assertion was subsequently proven dead wrong historically, I have little doubt that the Fed Model provided much peace of mind for Alan "Irrational Exuberance" Greenspan and Wall Street alike during the bubble years.

Second, if the Fed Model was correct and correlation really did imply causation, another nontrivial assumption, then perhaps the Fed could indirectly affect stock-market valuations and hence stock-index levels by actively manipulating interest rates. If valuations ever reached bubble levels, which they did, the Fed Model suggested that slashing interest rates would give stocks breathing room and in effect "justify" their very high valuations without the need for the usual ugly post-bubble bust period. The raw anti-free-market hubris of such a notion would certainly appeal to bureaucrats!

Now in 2003 we can look back and laugh at these quaint anti-historical notions, both that soaring stock prices in the late 1990s into bubble-like valuations were nothing to fear and that the Fed could brazenly manipulate its way out of a bubble. But, to be fair, these notions that seem so silly now were untested in modern US history until only the past few years.

So the 1982-1996 block of time that defined the formal Fed Model and its many derivative valuation theories did indeed yield some impressive data supporting these theses. Its strong -0.91 inverse correlation fed an R-squared value (RR) of 83%, which is pretty darned impressive in statistics.

If you skipped your college statistics class to ski or hit the beaches, the R-squared value multiplies the correlation coefficient by itself, in this case -0.91 x -0.91. The 83% result tells us that 83% of the variation in valuations during this key period was predictably related to interest rates, and vice versa. This high degree of precision between valuations and interest rates is also quite evident visually in the yellow-shaded Fed-Model-foundational area shown above.

Now, as I vividly remember my own college statistics professors thundering down from their pulpits, it is crucial to remember that correlation does not necessarily imply causation. Just because two data series move the same way, it does not necessarily prove that they are related. They may be, they may not be, or there may be an entirely unknown third force driving both behind the scenes.

For example, if computer laptop sales are up this year and ice cream sales in Paris, France are also up, does this positive correlation mean that laptop buyers fly to Paris to eat ice cream to celebrate their new toys? Does it mean that Parisian ice cream connoisseurs feel compelled to buy a new laptop after they delight in a sundae? Obviously not!

So the strong correlation between valuations and interest rates, even within the Fed Model formative period, is certainly impressive but it doesn't prove a concrete relationship between the two. Correlation does not guarantee causation by any means. Statistical analysis is a wonderful tool, which every financial-market analyst on Earth including I use all the time, but its inherent limitations must also be understood.

Now if you are still with me after all of this statistical mumbo jumbo, you probably are wondering with glazed eyes why I veered down this tedious path. The answer is we needed an indisputable mathematical way to compare the 1982-1996 Fed Model formative period with the rest of modern market history. The correlation coefficient and R-squared numbers grant us a way to empirically define the precise relationship, or lack thereof, between general-stock valuations and interest rates.

What if the Fed Model valuation theories' ideas only hold true for that one unique period of history? If this proves to be the case statistically, perhaps the bulls ought to start wondering if a theory born entirely within a Great Bull cycle still applies to our current vastly different Great Bear market cycle.

From 1997 to the present for example, about half of which was Great Bull market conditions and half of which was Great Bear conditions, the Fed Model hypothesis flounders badly. Valuations and interest rates only had a negative correlation of -0.53, leading to an RR of 28%. This is a pretty poor relationship and, at least in my opinion, not statistically significant enough to brazenly bet against all of financial market history by staying long the equity markets on the premise that historical valuation methodologies no longer matter.

If we consider a much longer period, the entire chart above from 1980 to today, valuations and interest rates manage a C of -0.82 and an RR of 67%. This is still on the high side as far as correlations go, but it is a lot lower than the 1982-1996 Fed Model formative period by itself. It is troubling that the Fed Model theories, when not only artificially limited to this one little window of history, tend to fall apart during the rest of market history.

This creates an enormous problem for the bulls. The best valuation methodologies, the old P/E ratios and dividend yields, are so prized by the contrarians and bears because they have withstood the whole test of history. They have worked for many centuries, in both Great Bull and Great Bear markets, and they have easily weathered countless full market cycles.

Market cycles, as I explained in depth last year in "Valuation Wave Reversion", are a lot like the annual seasonal temperature cycles with which we all are familiar, except over a far greater duration. In the past century or so long valuation cycles generally run about 33 years in duration, so any valuation model attempting to replace classical valuation methodologies really needs to prove itself over at least 33 years, and more preferably two whole cycles or almost seven decades just to be sure.

The Fed Model, on the other hand, is only supported by 15 years of data, not even half of a traditional valuation cycle. This curious practice of using such a historically unproven theory to manage money is like trying to forecast weather if you only had six months of temperature data to work with. It is fundamentally flawed and pretty silly to try and forecast valuation cycles using data not even encompassing one single cycle!

To compare the Fed Model to the seasons, imagine if a long-range weather forecaster only had data from April to September to consider. He would be totally justified in believing that temperatures always ran between 60 and 90 in most of the US since that is what his data showed. But the flaw is obvious. If a weather forecaster only uses summer data to build his models, obviously winter is going to slaughter him when temperatures plummet below zero and his inadequate sub-cycle models weren't even close to predicting such a calamity.

It is really odd and probably dangerous to build valuation models for the financial markets that don't consider at least one full 33-year valuation cycle of data, preferably two or more cycles. To forecast the probable temperature you really need at least a whole year of seasonal data, otherwise your weather model is doomed to fail when the seasons shift. It works the same way in the financial markets, but over a longer period of time.

So, what if we expand our time sample out a bit? What if we run back to 1950, a half century or so, and look at the Fed Model theories' assumptions over about one-and-half valuation cycles? Provocatively, this theory that looked so promising with 15-years of sub-cycle data all out of a Great Bull season (like summer) does not look so tough when it is run over entire valuation cycles, including valuation winter (7xish P/Es).

Long-term investors who remain heavily long general stocks today because of the Fed Model theories really ought to pay very close attention to this data that utterly destroys the Fed Model's philosophical and historical foundation. Trillions of dollars are riding on the assumption that the Fed Model theories are correct, but even recent history shreds the flawed Fed Model theses with ease.

When the dataset used in the first graph is expanded back to a half century or so over one-and-half valuation cycles, the Fed Model disgracefully implodes. The same yellow Fed Model foundation area discussed above is also shaded yellow here again for easy visual reference. While the 1982-1996 period did provide a strong negative correlation between valuations and interest rates, it appears to be an exception and not the rule when viewed within a broader strategic perspective.

It is readily apparent that there have been times of low valuations and low interest rates, low valuations and high interest rates, and high valuations and low interest rates. For example in the early 1950s, one of the only other times in modern history when interest rates plunged this low, the S&P 500 was very undervalued at 7x earnings. Earth to bulls … low valuations and low interest rates are not mutually exclusive! If someone tells you that low interest rates always mean high valuations are expected and acceptable, they are a liar.

As you carefully examine the interplay of the valuation and interest-rate lines over time, it becomes blindingly obvious that they are not always related. Sometimes they correlate well, and other times they don't. Our statistical tools of correlations and R-squared numbers empirically quantify and verify this observation. Outside of that small yellow box above, there is zero statistical support for the philosophical basis of the Fed Model New Era valuation theories.

From 1950 to today, a valid comparison over a half-century of time including more than one whole valuation wave that is not limited to a single financial-market season, classic S&P 500 P/E valuations and interest rates were barely negatively correlated at -0.36. This is not statistically significant and is more or less random, as the RR of 13% indicates. Only 13% of the move in either valuation or interest rates over this entire modern history of the US markets could predict the move in the other variable.

The Fed Model theories look even worse in the pre-1982 period, the second half of which was a long nasty secular bear market with much more in common to today than the post-1982 Great Bull. From 1950 to the end of 1981 the valuations and interest rates only had a -0.28 correlation, hence a trivial 8% RR. Once again it is crucial to realize that correlation coefficients and R-squared numbers this low are essentially random. They certainly are not correlated well enough to base major valuation theories off of!

So while the Fed Model theories performed well from 1982 to 1996, that peculiar Great Bull period is history now. I suspect that the correlation even during this period is attributable to two more factors that cannot be repeated at the same time again. First, interest rates soared after Nixon severed the US Dollar from gold in the early 1970s, and much of the 1980s was spent with interest rates mean-reverting after this massive one-time currency-debasement disruption.

This massive interest-rate dislocation just happened to coincide with major stock-market-valuation lows in the early 1980s that spawned a Great Bull. If the Great Bull hadn't erupted just as the post-Gold-Standard interest-rate chaos was unfolding, odds are that the Fed Model foundation correlation would never have existed. It seems to be a rare historical anomaly, not a rule off of which to define New Era valuation models.

So, when someone suggests that high valuations are normal in a low interest-rate environment or that classic valuation metrics aren't meaningful anymore, be really skeptical. The foundation of all of these pernicious and easily refuted theories, the Fed Model, can only withstand historical scrutiny within one brief period of time. If you really want to bet on a New Era where the old laws of markets and finance are junked, you'd better find a better rationalization than the goofy Fed Model theories.

And rationalization is the right word for these falsehoods! They are merely excuses, rationalizations, the bulls make up to try and explain away chronically overvalued markets. Rather than face the probability that after summer comes winter, the bulls desperately need to believe that summer is endless, that high valuations and ever higher stock prices will exist forever despite history's crystal-clear precedent. Winter be damned, the bulls desperately cry!

The Fed Model, and all of its progeny, are simply financial fairy tales for adults. If you care at all about your precious long-term investment capital, you really ought to choose a classical valuation model that has worked for centuries, not New Era fiction.