The quick and dirty rule of thumb is that the relationship between bond yields and the S&P 500 PE is one-to-one.
That is, 100 basis point change in bond yields should cause about a 100 basis point change in the PE.
Since WWII, the long term trend for earnings growth has been some 7%, or about the same as nominal GDP growth. I’ve long thought that the PE is an expression of the present value of a perpetual stream of 7% earnings growth
In the 1990s bubble , investors came to believe that long term earnings growth had moved to a permanently higher level and that justified the higher valuation.
Now, I wonder if the market, in its wisdom is already discounting a lower level of earnings growth, maybe about 4%. That could explain why the market looks cheap. Interestingly, in this recovery nominal GDP growth has been very stable around 4%.