There are several relationships here:
1) leverage seems to be related to nominal interest rates
2) to compare equity earnings yields to the bonds yields, I think the appropriate comparison is between the earnings yield and the real interest rate.
3) the effect of leverage on the earnings yield should be reasonable without any inflation adjustments.
Here is the original graph I posted in the earlier post, which has nominal interest rates, unadjusted earnings yield, and leverage.
Below is the same graph with real interest rates instead of nominal interest rates. This is the best version of the graph for looking at the earnings yield relationships. I have shortened the time frame because inflation adjustments before the modern era are very volatile.
I think this makes a much stronger case for the earnings yield being associated with leverage. There may be an inverse relationship between earnings yield and real bond rates. And, this does suggest that in low interest rate contexts, risk insensitive investors are capturing some premium from the risk averse debt investors. Looking at it in the way I framed it in the previous post, if the entire universe of savers shifted to a more risk averse position, the total return to assets (risk free rate + equity premium, which is the forward looking version of the earnings yield) might need to rise, in which case the equity premium would rise by even more than the real risk free interest rate was declining. This looks like it may have been the case in the late 1970s & early 1980s. So far, though, in this cycle, the total required return to assets doesn't seem to be unusually high.
Maybe that is the difference between low inflation and high inflation, and thus low leverage and high leverage. The high risk aversion we see today might have a slight effect on equity earnings yields, but the low leverage of corporations is a stronger effect, and so equity earnings yields remain in the normal range.
In a regression, both leverage and real rates are strong variables for estimating the earnings yield, with leverage being the strongest. But, this may not mean much because leverage and earnings yield both have equity market values in the denominator of the variables. Let me know in the comments if you have an idea for avoiding that problem.
I would like to make a subtle distinction from the typical use of that relationship. I think it is generally seen as a sort of arbitrage situation, where investors should rebalance to equities until the return on equities is bid down to something close to the return to bonds.
But, I don't think this is an arbitrageable relationship. The difference between these rates of return is a product of a public sentiment toward risk. There will probably be a change in sentiment at some point, but the difference between these rates of return will come mostly from a rise in long term rates as that sentiment changes. So, the equity investor does receive the higher return on investment, and avoids the low returns on bonds. But the convergence of this rates of return isn't so much an arbitrage as it is a speculative position on changing sentiment.
In some ways, it's a distinction without a difference, since a risk insensitive investor would still tend to want to add weight to equities in this context, all else equal.
As I noted in earlier posts, historically, the 1940 to 1970 period was the golden era of equity risk premiums (see chart to the right). That comes through very clearly in the chart above, as the earnings yield was consistently above the real interest rate throughout that period.
With regard to monetary policy, a looser policy that brought inflation to the 3-4% range might not create much of a boost for investors through changing yields. The earnings yield would probably rise slightly along with some corporate releveraging, and that increase in required returns would keep equity prices from increasing. But, that releveraging would be partly the product of short term debt markets that would be freed of the frictions caused by the zero lower bound and of long term debt markets freed of the frictions still remaining in the mortgage market. Improvements from looser monetary policy wouldn't, therefore, lead to higher stock prices because of the first order effect of changing rates. They would move higher because the more optimal monetary policy would be leading to more real economic activity due to the removal of those frictions. (edit: TravisV points out in the comments that I neglected to note that the higher NGDP growth that would come from a 3-4% inflation range would also be likely to reduce equity risk premiums because it would reduce the risks associated with very low interest rates, which would have all sorts of positive effects, including rising equity prices.)