A simulation-based valuation of the S&P 500: July 2022

The figure below shows two simulation-based valuations of the S&P 500. They assume a fair price-to-earnings (PE) ratio for the S&P 500 that is the inverse of half of the 10-year U.S. Treasury yield. The price (at the top) is the most recent top value of the S&P 500.

The numbers on the left consider a rather benign scenario: S&P 500 earnings in 2022 are up by 10% from the previous year, and the 10-year U.S. Treasury yield is at 2.50%. The numbers on the right refer to a more likely scenario: S&P 500 earnings are down by 10%, and the 10-year U.S. Treasury yield is at 3.50%.

The second scenario takes us to a fair price for the S&P 500 of 2,638.16, which is 45.25% down from the most recent high. A sobering thought, given the rally that we are in right now, which many believe to be nothing more than another bear market rally.

The recent negative GDP growth figure was revised down to -1.6%

As it turned out, the negative GDP growth figure mentioned in our last post was revised down to -1.6%, from -1.5%. This is related to our recent post on the Atlanta Fed. From Tradingeconomics.com:

  

"The American economy contracted an annualized 1.6% on quarter in Q1 2022, slightly worse than a 1.5% drop in the second estimate. It is the first contraction since the pandemic-induced recession in 2020 as record trade deficits, supply constraints, worker shortages and high inflation weigh. Imports surged more than anticipated (18.9% vs 18.3% in the second estimate), led by nonfood and nonautomotive consumer goods and exports dropped less (-4.8% vs -5.4%). Also, consumer spending growth was revised lower (1.8% vs 3.1%), as an increase in spending on services, led by housing and utilities was partly offset by a decrease in spending on goods, namely groceries and gasoline. Meanwhile, private inventories subtracted 0.35 percentage points from growth, much less than a 1.09 percentage points drag in the second estimate. Fixed investment growth remained robust (7.4%) but housing investment was subdued (0.4%, the same as in the second estimate)."

Is the Atlanta Fed predicting an imminent recession?

Fed funds rates, 10-year Treasury rates, and recessions

Can 10-year compound annual S&P 500 returns help predict market crashes?

As someone interested in applying data analytics techniques to finance and economics, I tend to look for leading indicators that contribute new insights when compared to existing ones.

It also helps if they are not obvious, and go somewhat against consensus; e.g., in interviews on media covering financial markets issues, frequently experts express the opinion that recent market gains do not influence future performance.

If we put this opinion to the test against data, we find that the opposite is what usually happens, as you can see from the graph below, which shows annualized 10-year compound changes in the S&P 500.

Dividends are not considered in this depiction. The graph is based on monthly data points from January 1910 to March 2022, where the annualized changes are calculated for each month as follows.

SP10(T) = ( SP(T) – SP(T-10y) ) ^ (1/10) – 1

In this equation we have:

SP10(T): the annualized 10-year compound change in the S&P 500 at a given point in time (T).

SP(T): the value of the S&P 500 at a given point in time (T). For this illustration, we used the average value of the S&P 500 for each month.

SP(T-10y): the value of the S&P 500 exactly 10 years prior to the time T.

This is a fairly easy calculation, which relies on a single variable. The graph seems to show that periods in which the SP10 moves quickly above 10% typically precede market crashes.

As a leading indicator (not a cause), the SP10 may not be precise enough to be used alone. Nevertheless, it can be used together with other indicators (e.g., Shiller’s PE10 ratio) in a more complete predictive model.

Unemployment rates and recessions in the US

Often one hears in interviews on media covering business issues and financial markets that a recession cannot happen if the unemployment rate is low.

As it turns out, the opposite is what usually happens, as you can see from the graph below, which shows unemployment rates over time.

The graph shows that periods of low unemployment usually precede recessions, even though recessions cause high unemployment!

Is there a relationship between interest rates and PE ratios?

Summary

- We look at the relationship between 10y Treasury yields () and Shiller PE10 ratios () from 1971 to 2021.

- When these two measures are compared and correlated, without time lags, there seems to be no relationship.

- When we consider time lags, a relationship becomes apparent: periods of tightening, when yields go up, seem to be followed by contractions in Shiller PE10 ratios.

10y Treasury yields vs. Shiller PE10 ratios from 1971 to 2021

The graph below shows the relationship between 10y Treasury yields and Shiller PE10 ratios during the period going from 1971 to 2021. The low Shiller PE10 ratios shown at the bottom generally occur during market crashes. The R-squared for the relationship is shown next to the best-fitting inverted J curve.

As we can see, the 10y Treasury yield explains only 5.1 percent of the variance in the Shiller PE10 ratio, even after nonlinear function transformation (aka “warping”). The relationship is weaker if it is modeled as a linear association. This goes against the idea that there is a relationship between 10y Treasury yields and Shiller PE10 ratios.

Arguably the time period considered is too large to be representative of what might happen today. Notably, the S&P 500 has been much more strongly influenced by high growth companies since 2003, after the crash of the “tech bubble” and the emergence of a few large and very successful technology companies.

10y Treasury yields vs. Shiller PE10 ratios from 2003 to 2021

The graph below shows the relationship between 10y Treasury yields and Shiller PE10 ratios during the more recent period going from 2003 to 2021. Again, the low Shiller PE10 ratios at the bottom occur during the market crash of 2008 (those ratios did not drop as much during the COVID crash), and the R-squared for the relationship is shown next to the best-fitting J curve.

As we can see, the 10y Treasury yield explains 12.3 percent of the variance in the Shiller PE10 ratio, with nonlinear function transformation (aka “warping”). Still, this is a small number, which goes somewhat against the idea that there is a relationship between 10y Treasury yields and Shiller PE10 ratios.

However, closer inspection of the data suggests that market corrections and crashes, where typically Shiller PE10 ratios contract quickly, follow periods of tightening (top-right part of the graph). Also, PE10 expansion appears to occur after easing (top-left part of the graph).

Time series of 10y Treasury yields vs. Shiller PE10 ratios from 2003 to 2021

The graph below shows a time series with 10y Treasury yields in blue and Shiller PE10 ratios in orange, during the period going from 2003 to 2021. Since this is a time series graph, with time varying along the x axis, we can more easily spot lagged relationships.

Here we can see that periods of tightening (blue arrows up) appear to be followed by periods where the Shiller PE10 ratios drop (orange arrows down). This suggests that there is a relationship between 10y Treasury yields and Shiller PE10 ratios.

Conclusion

There seems to be a relationship between 10y Treasury yields and Shiller PE10 ratios. Periods of tightening, when yields go up, seem to be followed by drops in the Shiller PE10 ratios drop. And, given that yields rising precede PE10 ratios dropping, there is a predictive “flavor” to the relationship.

The problem is that as yields go up, so do PE10 ratios, until a point is reached where PE10 ratios drop precipitously. While it is not clear when the tipping point is reached, it seems to occur after a 1 to 2 percent point rise in yields.