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S&P 500 Shiller-forward Price-Earnings ratio (P/E) versus 10 year Treasury bond rates

How does one estimate an appropriate Price-Earnings ratio (P/E) should be at a given time? This may help in judging whether stocks, or in the case here, the Standard and Poor's 500 index (S&P 500), are perhaps [over, under]-valued, and whether you need to take a closer look to make sure that other factors justify going ahead with their [sale, purchase]. Sometimes my thinking is based on out-moded "rules of thumb" that fail when applied to rapidly changing market conditions, so this is of interest to me.

In my view, fundamental analysis is a very important basis for any such decision, so criteria such as PE ratios are only supplementary to that.

A very [simple, basic] model is shown below as a starting point. While something far more [sophisticated, proven] could be pulled out of a standard text-book, by doing it "blind" by myself, I can better appreciate what others have done before me (when and if I get around to going further with this), and it's a powerful way to learn when complimented with further reading. It's also a great way to make a fool of myself (been there many times, done that).

This web-page was [inspired by, based on] :
  1. How Much Do Interest Rates Affect the Market's P/E Ratio?, Taking a look at the relationship between the two, Ben Reynolds, July 22, 2016
  2. P/E Ratios & Interest Rates: A Formula for Fair P/E Ratios Incorporating Interest Rates, Ben Reynolds, Updated August 22nd, 2018
  3. multpl.com

17Apr2021 update : see also!! :

Image of one result from the model :

Don't pay too much attention to the results. As with any [parametric, statistical] model, you can often fudge things to get what you want. More important is just a "feel" for how parameters MIGHT affect results in a very rough sense, and getting a feel for what might done to improve the [concepts, model]. While "A picture is worth a thousand words", I say that "A good question is worth a thousand answers".

Howell's "[simplest, basic] model" derivation :

(This needs to be cleaned up, as it went through 3 iterations.)

Nomenclature : Do present value calculations for simplicity : Treasury bonds :
S&P 500 index : Notice that this model looks ONLY to the future using projected earnings, rather than taking into account the Shiller P/E ratios and current market price! Consideration of those two factors could greatly improve the reliability of the model, but that is for some time in the future... (just a maybe - not so likely for me).

Now look at "balance" of [Treas_PV, SP500_PV] for equal 1 k$ investment : This is the model used.

LibreOffice Calc spreadsheet implementation :

see Howell - SP500 PE Shiller ratios versus 10 year Treasury bond yields, with earnings growth & discount factors.ods

Note that Microsoft Exel will often complain that it cannot load the file, but go ahead with the loading anyways. There is a good degree of compatibility between Exel and LibreOffice Calc, with the important exception of macros, which have to be re-written. I have not used macros in thispreadsheet.

The spreadsheet is very simple, allowing changes to the discount factors for [earnings growth, T-bills], resulting in immediate updating of the graph. Other changes are also easy for those familiar with using spreadsheets, such as : Spreadsheets are NOT [flexible, powerful] for much more complex models than the [simple, basic] model shown above, unless one uses macros. Or perhaps better stated, I prefer to work with programming languages that I know can easily get the job done, and that make it [easy, fast] to [think, work] at more abstract levels. If I ever find the time, I will continue instead with the QNial programming language to do complex models. Explanations and the resulting graphs would then be posted to this web-page. Don't hold your breath - this isn't a focus of mine, and I need to focus on project priorities in completeeldifferent subject areas (notably [artificial neural networks, fundamental theoretical physics]).


First of all, I have zero [expertise, experience] in this area, even though I've always been interested in [business, economics, finance, markets] but never found time to look into it. On occasion over past decades, I have done fundamental analysis of individual companies (financial statements, business and market outlook, competitive scene, etc) but only with limited time. My favourite was always tiny [technology, biotech] startup companies with an interesting scientific concept and not much of an organisation. But most of theose are killed off quickly, both in a scientific and business sense. I had terrible investment results, but of fun, and I got to know very interesting people with dreams!

This model : Obviously, do NOT this for any real work that you have! It's just a toy!

Future possibilities :

A key interest that I have is with concepts that identify and apply basic DRIVERs of phenomena, beyond conventional approaches. For example, you will have noticed that weather, climate]modelling ignores the most basic drivers of the phenomena that they are dealing with (weather - largely a Navier-Stokes engine with [heat, mass transfer], blind to most fundamentals, climate - what a disgrace!).
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