Fair Value for the S&P500?
Jarrod Wilcox, November 15, 2002

The record of those who seek to get in and out of the stock market at the right times is as dismal today as it was in the wild days before the formation of modern stock exchanges.  Undismayed, I recently tried to look at the market with a fresh eye, to see whether the outlandish valuations of the recent stock market bubble had gotten back into "fair value" range.  The result surprised me.

If we mean by fair value that the US stock market reflects the underlying economic realities of today, the answer is yes.  We can stop worrying about getting soaked in the collapse of the soap bubble of unrealistic investor optimism.  Now we sporting types can bet on whether the bottom has been reached in US company profitability.  Or maybe on the impact of a weak US dollar on interest rates.  And we can spend less time worrying about the bubble's aftermath in terms of falling investor confidence.

My favorite tool for this kind of investigation is the PB-ROE model.  You'll see why when you see how convincing it is in explaining both the recent bull market and the subsequent climb down over the last couple of years.  Let's take a look at what it is, and then show the results in a couple of very illuminating charts.

The basic idea behind PB-ROE is that company equity ownership is capitalized as the compound result of the difference between return on equity and a comparable interest rate.  That is, take the viewpoint that the investor is buying a partner's share in the ownership equity already put into the company.  It is worth more if the company's return on that capital is higher.  It is worth less if the investor could get a good return on a bond from a similar company.  The form of this relationship has to recognize that any excess profits will be compounded, resulting in exponential growth.  It is the logarithm of PB that should relate most simply to ROE and bond interest rates. (To learn more about PB-ROE, click here.)

How much of the great bull and bear markets we have just been through can the PB-ROE model explain?  If it can explain much, can we learn anything more from deviations of actual price-to-book ratios from it?  To check this out, I looked at a history of profitability of the S&P 500 index, measured as return on equity as compared to AAA 10 year bond interest rates.  (Data sources: BARRA and St. Louis Federal Reserve Bank.) The data are over 300 monthly observations through October 2002.

What the chart below shows is a dramatic variation over the last two decades in the gap between profitability and its bond interest competition.  The gap almost closed in the early 1980's, only to grow wider and wider, reaching a peak in 1999.   During this period, interest rates came down.  (Much of this, though by no means all, was based on the reduction of high monetary inflation.)  At the same time, though it fluctuated with the business cycle, profitability not only held up but grew!  The increased profitability remains rather under-explained, although we now know some of it was based on optimistic accounting.  Nevertheless, the PB-ROE model should predict a strongly rising stock market during this period.

For the most recent two years, the pattern has been quite different.  Profitability has fallen sharply, although it still remains not far from the average experienced over the last two decades.  Interest rates have fallen slightly, but nowhere near enough to make up for the drop in ROE.  The PB-ROE model would therefore expect a decline in the market's price-to-book ratio, and since book value moves only slowly, a negative return for the S&P 500.

Now let's take a look at actual PB (the yellow circles in the chart that follows).  Sure enough, the peak in the market's capitalization relative to ownership equity actually invested in the 500 companies within the S&P 500 list came at the same time as the peak in profitability.  A straight line best fit between log PB and the difference ROE-AAA would tell much the same story, but a slightly better fit can be gotten by allowing the sensitivities to ROE and AAA to be different. This OLS multiple regression technique allows us to reflect the possibility that ROE differences are discounted for uncertainty or that we have measured a bond rate whose duration is different than the investment horizon.  The result gives us a simple "in-sample" explanation of PB, plotted as red triangles in the chart below.

A statistics expert would point to problems in the chart above.  The worst is something called autocorrelation of residuals.  It means either that I have left out additional explanatory variables or that the whole model's fit is more or less a coincidence.  I choose to believe the former.  Another problem is the lack of fit around and before 1980.  I suspect this has to do with a spike in inflation during that period, with current ROE being suspect based on accounting difficulties in matching today's revenues with older depreciation expenses denominated in dollars of lesser value.

What is the conclusion one might draw from the chart?  First, to me, it appears as though most of the variation in pricing levels has been an expression of changes in the underlying fundamentals rather than in investor psychology.  The "bubble" was more a bubble in profits relative to cheap interest rates than in speculative fever by day-traders.  Second, however, if we accept the model, the persistent more moderate differences from it are meaningful.  They may well be due to market psychology -- the periods of investor overconfidence that we tend to mean when we talk about a "bubble."  This helps us distinguish between a bubble as irrational exuberance and a bubble in the underlying fundamentals. Third, when we look at the chart at the end of October 2002, it looks like the market is near fair value based on current fundamentals.

In other words, I can stop paying so much attention to what is happening to investor confidence -- that part of the bubble bursting appears over.  One less thing to worry about.  There are more aspects to consider -- whether company profitability will decline further as the result of economic recession or whether international investors have slowed the lending to the US which has helped keep interest rates here low.  But at least I know where to focus my attention.