Can the equity premium help us identify stock market bubbles?

The S&P 500 (a commonly used index of large publicly traded companies) has declined in value by around 30% over the last month. How much of this is an event-driven response to the expected pandemic-related reduction in trade, and how much of this is a bubble popping, conveniently blamed on the pandemic? 

A popular measure of the stock market's bubbliness, is the CAPE ratio. The ordinary PE, or price-to-earnings ratio, represents the ratio between the price of the stock, and its share of a company's annualized earnings. For instance, if Acme Corp earns an annual $5 per share, and trades for $100 per share, then its PE ratio is 20.  You can also invert this to get an earnings yield of 1/20, or 5%. If earnings are accurately measured, a company's value ought to increase on average by about its earnings yield each year.

But market-wide earnings can fluctuate a lot from year to year, so the CAPE compares the current price to the inflation-adjusted average of the last ten years of earnings, for the S&P 500 . If you look at the CAPE alone, it appears that it was at an unusually high (and therefore presumably bubbly) level a few months ago:

But, unusually high compared to what? Haven't interest rates at historic lows for a while now? If yields on bonds have gotten quite low compared to a few decades ago, why shouldn't yields on equities decine too? (Remember, as yields decline, prices rise.)

Fortunately, the same website that provides CAPE numbers provides US treasury yield numbers. So I compared historical treasury yields with historical CAPE earnings yields. How to do this?

It's well known that stocks tend to yield more than bonds. Stocks are much more volatile than bonds, and there's no guarantee that you'll get your principal back ever, and perhaps that's why they tend to yield more. This is called the equity premium.

I subtracted the yield on the 10-year treasury note (same time period as the 10-year earnings smoothing) from the 10-year average earnings yield, and got a median difference of 2.32%. That means that on average, the 10-year earnings yield of a stock is 2.32 percentage points higher than the yield on a 10-year treasury note. So if the 10-year interest rate is 2%, then we should expect on average a stock earnings yield of 4.32%.*

If we accept this calculation, it should allow us to calculate a predicted CAPE, based only on the interest yield on the 10-year treasury note, by taking the inverse of the sum of that yield and the equity premium. If I do that for the available data, I get this:

Instead of spoon-feeding you conclusions, I invite you to look at it yourself and see what you find. You can also play with CAPE spreadsheet I used to put this chart together.

The main thing I like about this view is how much more obvious it makes it, that market movements are complicated, and not just a morality tale of bubbles and bursts. Prices can be depressed relative to fundamentals for extended periods of time, during war. One time, the equity premium dropped to zero briefly at the apex of a speculative bubble driven by self-validating expectations of growth. Another time, it hung around zero for a while as rising inflation made investments that weren't nominally fixed much more attractive, and then stayed low for a growth period that culminated in a bubble. There's just a lot going on, and the nominal multiple of earnings can only be interpreted in context.

It looks to me like the CAPE was racing somewhat ahead of interest rates in the last couple of years, but that interest rates have now caught up, so as long as that persists it should be a comparatively good time for long-term investors to buy stocks and a bad time to buy treasuries. There is, however, no clear guarantee as to when - if ever - the two will reequilibrate.

* I'm using nominal and not "real" (i.e. inflation-adjusted) interest rates here, because earnings are nominal, and the whole financial economy is nominal.

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