CAPE and math

In order for price-to-earnings (P/E) ratios to be useful for long-term asset allocators, they should be somewhat more stable and stationary over time.  This allows one to make and act on a decision, without the ratio confusingly oscillating about, too much, in-between.  A surprise to most, the high variance of the P/E ratio comes nearly as much from the numerator, as it does from the denominator (the first chart below will show the denominator).  In this article we discuss one particular P/E ratio, branded CAPE, and show some interesting statistical properties concerning it.

To start, let's explore both the numerator and the denominator separately, and then combine them.  One can best evaluate the current price -numerator- by seeing how far away the P/E is above or below its long run average of ~16.  The other part of the ratio, the earnings -denominator- we noted has similar volatility, and so this is where we can focus our efforts for P/E smoothing.  We can do this by substituting the prior 10-year earnings average, for the more conventional 1-year earnings.  Now the correlation between price changes and earnings changes is 0.3, while against the 10-years earnings changes the correlation disappears altogether.  And only the former is likely to have copula properties in the tail.  Just the coordinated movements between price changes and earnings changes would normally argue for not smoothing earnings, but for the standard deviation of earnings change is about 40%; for the 10-year earnings change it is about 4%.

Note that using 10 years of earnings also gives a somewhat equal comparison for a long-term investment manager who is also deciding between the risky asset, and the performance of a 10-year government bond.  With this backdrop, Professors Campbell and Shiller created the smoothed price-to-earnings (P/E) ratio, which they named CAPE.  Even though this "cyclically-adjusted" measure is advertised to incorporate information over a business cycle, it should be noted that recent research shows the business cycle length typically is far less than 10 years.  Having such an otherwise arbitrarily-large window can have both positive and negative effects, both of which must be weighed.

Professor Shiller (briefly discussed here a year ago) wrote in the New York Times (NYT) last weekend that their CAPE ratio is now above 25.  These high levels were only previously seen, since the late 19th century, in the years near 1929, 1999, and 2007.  Also for every month of about the past 20 years, we had an above-20 CAPE for all but nearly 20 months (or less then 2 years out of 20).  There are a number of ideas outside of price and earnings that he puts forward to help assess whether this high valuation value may or may not be justified.  One area that probably should be better covered by any analyst, is why probability and statistics on these variables can't help explain a greater portion of the CAPE valuation levels.

Professor Shiller is of course incredibly creative, intelligent, and approachable.  I discussed with him both his housing mortgage research and the smoothing mechanism of CAPE, a couple years ago, when he visited the U.S. Department of the Treasury.  More recently he was helpful in pointing out a couple aspects of his website where you can freely explore and analyze the time series data.  You should do that, since becoming familiar with the dataset is very helpful.  For example, the CAPE ratio has a few moving parts that differentiate it from more traditional pricing ratios.

To be complete, it is worth noting here that the earnings used in CAPE lags price, and does so even more in any most recent period.  Additionally, off-balance sheet accounting adjustments -from disclosure filings- takes considerable time to work their way into the earnings stream.  So during the flurry of major market volatility, for example at the start of 2009, one can be utilizing in CAPE a large mismatch in earnings relative to price (e.g., 2007 earnings and 2008 quarterly estimates, versus early 2009 price).

Even without this issue, the CAPE ratio can jump around because of the 10-year earnings window.  See the annual change in earnings (in orange) on the chart below.  Then in red see what the worst annual earnings change is, over the recent 10-year window.  Notice how in roughly the past 20 years (the vertical gridlines are every 10-years), there have been a number of weak or deteriorating earnings collapses that keep lingering in a 10-year window?  We'll show that the CAPE ratio may overemphasize this.  This is from the way the calculation mechanically works, and also in the collective investor's psyche of being ok to discount a high CAPE, if enough important market participants feel that the risks of earnings shocks won't come back with such ferocity anytime soon.

Now let's continue to explore more of the probability and statistics aspects of this ratio.  First we can consider each vertical gridline, in the chart above, to be 10-year partitions of the data.  The period prior to 1996 has a typical CAPE of ~15.  If we take the average of each of these 11.5 decade long periods (stretching to the earliest part of the time series), then the standard deviation of these 10-year partition averages is 3.6.  So -from a combinatorial perspective- seeing a subsequent 10-year period with an average CAPE of above-20, let alone two of them in subsequent adjoining periods, would be well less than 5%.  This aligns with Professor Shiller's position that the recent 20-year period presents valuations that are unusually lofty.  He uses the quarterly version of his data, but that doesn't change the conclusions.  For example, using quarterly data, he suggested an above-20 CAPE for all but about 20 quarters in the past 20 years.  This is probabilistically equivalent to roughly 2 years worth at best.  The annual CAPE -from 1996 onwards- also has just 2 years that are not above-20.  And the annual CAPE data shows 1929 as above-25, and 1930 as above-20.  This mix of annual statistics appropriately fits the distribution characteristics of the quarterly data.

The various moving parts in the CAPE ratio include changes in consumer prices for the earlier data, and the information inside the 10-year averaging window on earnings, and lastly how these two variables combine together.  From a statistical perspective, the inflation adjustment absolutely isn't the most significant factor for understanding the CAPE levels, and for the lay person it just unnecessarily obscures the main engine of the changes versus targeting the analysis on the the near-term nominal values as we show in our article here.

Of the parts that remains, we see in the chart below an interesting relationship with CAPE.  On the horizontal axis we show our independent variable as the number of years the "worst earnings of the recent 10-years" that  has been weak or deteriorating.  This is basic arithmetic and connects to the annual spreadsheet download (from late 19th century onward) on the link above.  What it reveals is that statistics on the earnings show that it is "hiding" some properties that asset allocators appreciate.  That is, there can at times be a high CAPE ratio that is "forgiven" or discounted, when the 10-year window contains unusually depressed values in it.  Similar to those we have seen from two recent recessions.

We also color-highlight some of the time periods that are significant.  Those are the past 20 years (this appears more statistically significant and is in red) and the cluster about 1929 (this appears to be of more tenuous significance and is in blue).  The more that depressed earnings linger in the 10-year window, the higher the CAPE generally rises with these low earnings thrown into the denominator.  If these dynamics noted in this web log article are not real, then it would be hard to argue that the red and blue data are mixed in well with the green data, and that there is no clear convex relationship(s) existing in the data pattern of the chart.  Momentum is evidenced through convexity (with a non-linear correlation square of three times that of just price to one-year earnings).

Of course eventually prices will severely underperform earnings, and the financial crisis earnings forever pass from the 10-year window.  Both would, on their own, cause the CAPE to regroup back into the upper-left of the main green cloud of chart data.  The CAPE isn't a tool for conjecturing on crash timings.  Professional prognosticators can't even get simple things right, let alone CAPE, such as when they again fumbled their at least 10% market correction call at the end of July.  Though it can take about a decade where prices generally underperform earnings, in order to force the CAPE to sustainably stay close to its normal long run, ~16 average (visualize the middle-left of the green cloud above).