In a blog post **two days ago**, I assessed the WSJ’s claim that Shiller’s cyclically adjusted P/E ratio (CAPE) understates the case for equities by overstating the extent to which their valuation is stretched. My conclusion was that the WSJ argument was not necessarily wrong, but that it arrives a bit late in the rally and is unconvincing.

I mentioned then that there was another case for stocks, which was much more interesting, important if true, and hard to assess. I said I would take a shot at documenting aspects of it in a follow-up post.

I do so here, but I don’t nearly answer the issue. I stumble at resolving Modigliani Miller with the Gordon Growth model. Maybe somebody smarter could resolve that, but I’m going to show some charts.

I see via Brad Delong that **Antonio Fatas believes that US equities** are quite cheap once we take account of the fact that the risk-free real return available in Treasurys is very low, just over zero at the 10-year maturity.

In an effort to be intuitive, Fatas converts the inherently-real equity yield into a nominal yield to be compared with nominal Treasurys. But his argument is equivalent to claiming that the earnings yield exceeds the real Treasury yield by a healthy spread.

I have made this point myself, although being careful to call that spread (agnostically) the equity spread or equity yield spread, rather than the “equity premium.” To get an accurate measure of the equity premium, we would need a more reliable proxy of the real return to equities than merely the earnings yield.

Not that Fatas uses the wrong semantics, but he does strongly imply that the earnings yield is a good proxy for the real return to equities and is effectively invariant to the real Treasury yield and whatever information might be in that.

And it is here that I get nervous about his conclusion. Under some strong assumptions of rationality and market efficiency, the earnings yield is in principle a measure of the real yield to stocks, so long as earnings are measured correctly. Here is Brad Delong, who has actually been a pretty good real-time student of the market, on Modigliani Miller and **Haig-Simons earnings.**

Ok, that is fine, in part because arguing with it is beyond my competence. But central to that approach is the idea that earnings yields are a proxy of the (expected) real return, irrespective of what the bond market is telling us about growth. I am not a believer in the reliability of the consumption Euler relationship and its implication that real yields and real growth should be closely related. But it seems to me intuitive that if you are going to take account of low Treasury yields, then you also need to take into account the growth environment that they *might *reflect.

I am not sure how to do this rigorously, so I will drop to my chatty, simple-charts approach and give a sense of why I find Fatas’ argument unconvincing.

We all know that real bond yields have fallen and I will show a picture of that further below. But let’s take a look at how potential GDP growth, a rough proxy for the (domestic) corporate real top line, has evolved in recent years.

According to the CBO’s (current-vintage) estimates, potential growth in the US has fallen from about 3% during the three decades ending in the mid-2000s to only about 1.5% now. To assume that this would have no effect on the relationship between the earnings yields and the return to stocks seems extreme to me, even though the finance theory that relates the earning yield to returns explicitly takes no account of macroeconomics.

As a consistency check, I tried to proxy the CBO’s estimate of potential by comparing the 10-year growth rate of GDP with the 10-year change of the employment/labor force ratio. I promise this did not involve data mining. What I show in the chart above is the very first thing I looked at. Ok, it was the second thing, because at first I neglected to “center” the proxy, that is to assign the result of the 10-year calculation ending in year x to year x-5.

Anyhow, I think I know roughly what CBO is doing. Basically, they are assuming some sort of mean reversion in the capital/labor ratio and extrapolating the Solow residuals. Correct me if I am wrong! As for me, I will skip the pseudoscience and work with my simple proxy, which has no historical look-back bias and is smoother.

What I want to do then is to “adjust” the equity spread for the recent slowdown of estimates of potential growth, this time without the centering. If you don’t make this adjustment and if the low bond yield is related to slower prospective growth, then misinterpreting the equity spread as the equity risk premium could be a serious problem.

On the other hand, I concede that just adding potential growth (expressed as a deviation from its sample average) is pretty crude. I would say it is about as crude as not doing so, and assuming the bond market tells us NOTHING about the relationship between the current-period earnings yield and the expected return to equities.

*Chart corrected and updated on Dec. 13. Original version of lower-right panel showed variation of real GDP, rather than real PCE, and was mislabeled*.

So with that in mind, take a look at this chart. I will describe each of the panels:

The top left panel shows real Treasury yields are below my proxy of the potential growth rate, which means all else equal that P/Es should be higher now than in, say, the early 1980s. But the post-Crisis decline in potential growth roughly matches that in real yields.

The top right panel shows the Shillerized earnings yield. I discussed in my last post on this subject how I calculate that. It is not quite just the inverse of Shiller’s CAPE as I use operating rather than GAAP earnings and (far less importantly) convert nominal to real concepts with the PCE deflator rather than the CPI.

The bottom left panel is just the spread between my earnings yield and proxy of the 10-year real yield in 10-year Treasurys. Fatas would tell you that this is generous and that equities are therefore cheap.

The bottom right panel shows the conventional equity spread plus the potential growth rate expressed as a deviation from its post-1959 average. Right now that deviation is -190 basis points, because my proxy of potential growth is currently 1.3% vs 3.2% for the entire period since 1959.

The bottom right panel contains my conclusion, such as it is. A simple proxy* of the equity risk premium is lower than average, showing that stocks are not a screaming buy, even relative to bonds, whose yields are very low. I show as an addendum a simple measure of underlying macro volatility which might, in principle, bear some relation to the equity risk premium.

Again, there was no data mining here, although the relationship is not perfect, particularly for the 1960s. But on balance, this leaves me where I started. Equities are not demonstrably overvalued, but they look to be roughly fairly priced to deliver the low returns one might expect in a world of low-growth, low-risk-free real returns, and a seemingly serene**economic environment that might be depressing the equity premium.

One issue I have not discussed here is the durability of low Treasury yields. Maybe Treasury yields are due to spike. That is not my call, but if it is yours then I guess you could interpret this as suggesting that equities are not demonstrably overvalued relative to bonds but area about fair are slated to deliver weakish relative returns.

Goldman has a formal measure of the equity risk premium, which incorporates macro considerations indirectly. If I recall correctly, their measure gives the same *meh* signal as my crude proxy. Their strategist is now bearish, again if I recall correctly. But that too would be a separate discussion. Our indicators are in the same ball park is all.

** The Gordon Growth Model says that the dividend yield, rather than the earnings yield is related to the gap between the real interest rate and real growth. However, in that context growth refers to the growth of dividends, not the growth of the economy. I think this justifies using the earnings yield, but the result is a bid of a hodge lodge, awkwardly blending the two approaches mentioned at the top of this note. *

*** Some call this the Not So Great Moderation.*