A very commonly cited valuation metric is the stock market's P/E ratio. It describes the price of stocks relative to their earnings. A recent article by economist John Huston brings attention to a different way of measuring the price of stocks. It compares the P/E with the yield one can receive on a 10-year Treasury note. The ratio proposed is i*(p/e), where i is the yield on the 10-year bond. This ratio adds yield to measure the relative attractiveness of bonds vs. equities. As a matter of behavioral indifference, this ratio is proposed to signal equilibrium when it is at its mean which should be constant throughout time. Other than our own irrational behavior, are there fundamental forces that could cause this metric to drift?

Before examining the possibilities, a fundamental illustration of the relationship and the theory behind it:

It is supposed that equilibrium in this metric is achieved when this series is at its mean, and that mean is stationary over time. That is to say, increases or decreases in the yield of Treasury bonds should offset changes in the price to earnings ratio of stocks. When there are deviations from the ratio, demand will have a tendency to return to equilibrium in the asset pair and the ratio will correct to the mean.

As a demand problem - demand for Treasuries and demand for equities should be the same directionally at any given time. Divergences in demand would cause the rate to move from equilibrium, or the long-term mean. For example, if rates are low (high bond demand), the price to earnings ratio of stocks should be high (high equity demand) to offset it. These assumptions are required for the series to be stationary, as proposed.

To make a reasonable guess about the validity of the theory, we should try to identify forces that affect demand from all angles of the market.

First, the Federal Reserve has been actively involved in the credit economy, and will impact the demand equilibrium on Treasury bonds. The Fed not only controls the baseline interest rate, but it has buoyed credit markets with QE purchases. By purchasing both long and short-term Treasuries, the Fed directly invests into government credit. So there are two ways the Fed can impact at least one variable in the relationship:

- With the Fed Funds target rate, it directly controls the risk-free interest rate on the USD
- Equilibrium in the Treasury market. More QE buying means less private demand needed to balance the market at a lower rate.

So Fed open market participation and control of the risk-free rate have a direct impact on this model. Would the model be more a meaningful indicator of private demand indifference if we could normalize for the Fed's non-market controls? Would using the interest rate expressed as the spread above the risk-free rate be an improvement to the equation?

When studying the ^TNX-minus-Fed Funds rate series, we see that this metric is unusable in the analysis. This is because of time periods with an inverted yield curve - when longer maturity government debt yields less than short dated. Negative series values do not make sense, and directionally a more negative yield would impact our test ratio incorrectly.

Still, it seems like normalizing for the Fed Funds rate's direct impact on yield in our equation could be a helpful improvement to the model - after all, the equation should reflect *demand* indifference. So for the sake of this analysis, I am creating a new interest rate metric: An average of ^TNX and ^TNX minus the Fed Funds rate. I propose substituting this interest rate metric in the Fed model for i. A 1954-2012 time series of the metric is shown below, in addition to the Shiller CAPE10 model and the Fed i*(p/e).

As indicators, these series are intended to be mean-reverting. That is, when assets are "overpriced" or "underpriced", market participants will buy what is cheap, and cause the ratio drift back to the mean. To compare the diffusion rates, and thus the mean-reverting tendencies of each model, I used a coefficient of variation test - standard deviation divided by mean. Without going into the mathematical proof, this is a good test to compare relative stationarity for multiple series. This is because by definition a mean-reverting series' up/down conditional probabilities would be impacted by its location relative to the mean. Disequilibrium points will possess drift opposite to their location relative to the mean, and increase the conditional probabilities of low residual measurements continuously. Therefore, a more stationary series should display a lower coefficient of variation. As shown below, averaging ^TNX with its spread above the Fed Funds rate made the model slightly more mean-reverting. This model also implies a nice discount in equities currently.

Unfortunately the statistics suggest that the model is only *maybe* a tiny improvement. Below is a comparison of the means and confidence intervals for squared residuals divided by squared means in each model (the same as coefficient of variation squared, used to account for negative residuals and give the statistic a sample size). This shows that the mean-reverting properties of all three models are more or less the same, with the fed model likely a better incremental improvement than my tweaked interest rate variable. Still, normalizing for the risk-free rate gave us the most mean-reverting equation from July 1954- June 2012 when tested with monthly observations.

So are any of these series actually stationary? I don't know if it can be proven one way or the other, but I tend to think that they are not. Perhaps on a long enough time horizon. Dr. Shiller displays a dotplot of P/E against 20 year returns on his page, but these test points include overlapping yearly return and valuation inputs. This makes the relationship look a bit stronger than it actually is, the price paths are 95% shared and the earnings predictors are 90% shared. Maybe if they *are* mean-reverting, it could be to a much smaller degree than their stochastic components.

So logically, why do I think these series are not necessarily stationary?

For the Shiller CAPE model, I think that earnings variance is an important factor that could cause trends in the series. Consistent with the capital asset pricing model, earnings variance serves as a risk adjustment to the P/E ratio. Curiously enough, there *was* a decently strong relationship (R-squared ~.31) that suggested variance up, P/E up - the opposite of my intuition. While this contradicts theory, it appears the regression was trumped by high variance, high multiple-growth observations from the dot-com bubble. High variance can also be manifested through high sequential earnings-growth observations, and could easily be perceived as a trend. I would expect P/E ratios to be less instructive in a high earnings variance environment. The other caveat is already expressed through the intention of the Fed model revision - there is no comparison to other domestic, yielding, paper assets. Shown below is the 10-year moving coefficient of variation for S&P500 earnings.

For either of the interest-rate-adjusted models, there is not complete independence of factors. Total credit market debt owed in the U.S. economy stands at nearly 55 trillion dollars today. So any significant increase in borrowing costs would have a pronounced negative effect on underlying corporate earnings. It follows that in a low rate environment, earnings from our equation would be directly improved by lower credit costs, therefore related. It could, however, be argued that the interest rate is an assumption of economic growth prospects, therefore offsetting credit costs through business opportunity. While I am sure it does to an extent, this is an implicit assumption whereas the cost of credit can be explicitly measured and is accounted for in net earnings. This would suggest less mean-reversion in the Fed model than the rate spread model because the non-market interest rate forces are not being discounted from its equation at all. In my view, both exacerbate the interest rate factor, double counting it to a degree.

For investors who think these series are inherently stationary, the Fed model is giving the strongest S&P500 buy signal, followed by the Guttenberger i=(^TNX+(^TNX-Fed Funds Rate))/2 model, while the Shiller model suggests equities are slightly overpriced. Finally as a word of caution - both models suggesting equities are cheap are probably overly sensitive to interest rate moves. That means without future QE, these ratios could easily correct because of rising interest rates, as opposed to higher equity valuations.

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