<p>Let us define a P/E ratio as <span class="math-container">$$\phi(P,E)=\frac{P}{E},E\ne{0},$$</span> where <span class="math-container">$P$</span> and <span class="math-container">$E$</span> are random variables.  Any function of data is a statistic, that is the definition of a statistic, so <span class="math-container">$\phi$</span> is a statistic.</p>
<p>Under mild assumptions, you can show that <span class="math-container">$P$</span> is normally distributed around the equilibrium.  However, let us simply assume that <span class="math-container">$P$</span> is normally distributed for brevity.  Likewise, it is reasonable to believe that the Board of Directors and management of firms targeted <span class="math-container">$E$</span> as <span class="math-container">$E^*$</span>.  Depending on the method used to form the estimates, <span class="math-container">$E$</span> may also be normally distributed around the target.  If not, you end up with a headache but should still have a solvable, if not analytic, problem.  Again, for brevity, let us assume that <span class="math-container">$E$</span> is normally distributed around the target.  I would note that such as statement does not have to be true.</p>
<p>By well-known theorems in the field of probability and statistics, <span class="math-container">$\phi$</span> cannot have a mean or a variance if it is the ratio of two normal distributions.  There is no mean to revert to.  Requiring truncation at zero, though some things have pierced the zero boundary at times, and assuming a stochastic budget constraint does not alter this outcome.  It does make the discussion very long.</p>
<p>With that said, under mild assumptions, the median or the mode would be the basin of attraction. Again, length makes that discussion prohibitive in so short a space.  Nonetheless, what is basically being said is that if there is a stationary <span class="math-container">$\mu$</span> somewhere in the system, even if it is not a mean, then it drives the system.</p>
<p>See, for example,</p>
<blockquote>
<p>Weisstein, Eric W. &quot;Normal Ratio Distribution.&quot; From MathWorld--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/NormalRatioDistribution.html" rel="nofollow noreferrer">https://mathworld.wolfram.com/NormalRatioDistribution.html</a> although this oversimplifies the question.</p>
</blockquote>
<p>Under mild assumptions, it never was mean-reverting.  Instead, it looks like Gull's Lighthouse Problem and there is a median but calling it <em>reverting</em> might be mathematically misleading.  Instead, it might be better to think in terms of a see-saw and the median is trivial instance that the seesaw is level.  The problem is also related to Poincare's double pendulum problem that helped set chaos theory in motion.</p>
<p>See</p>
<blockquote>
<p>Gull, S.F. (1988) Bayesian Inductive Inference and Maximum Entropy. Kluwer Academic Publishers, Berlin.</p>
</blockquote>
<p>You should also be able to pick up this method in any first semester textbook in mathematical statistics for undergraduate statistics majors as this is often assigned as homework because of its weird properties.</p>
<p>I think this version of Freund's statistics textbook assigns this question as homework.</p>
<p><a href="https://smile.amazon.com/John-Freunds-Mathematical-Statistics-6th/dp/013123613X/ref=sr_1_9?crid=3GUMF7LFT4KPE&amp;keywords=freund+statistics&amp;qid=1653020853&amp;sprefix=freund+statistics%2Caps%2C201&amp;sr=8-9" rel="nofollow noreferrer">https://smile.amazon.com/John-Freunds-Mathematical-Statistics-6th/dp/013123613X/ref=sr_1_9?crid=3GUMF7LFT4KPE&amp;keywords=freund+statistics&amp;qid=1653020853&amp;sprefix=freund+statistics%2Caps%2C201&amp;sr=8-9</a></p>
<p>I would properly cite it with page numbers, but I do not remember where my copy is at.  It isn't on a shelf.</p>
<p>Also,</p>
<blockquote>
<p>Curtiss, J.H. (1941) On the Distribution of the Quotient of Two Chance Variables. Annals of Mathematical Statistics, 12, 409-421.</p>
</blockquote>

CC BY-SA 4.0
2022-05-20T04:23:15.643Z
2022-05-20T04:34:57.107Z
2022-05-20T04:34:57.107Z
Dave Harris
<p>It looks like Shiller's P/E stopped being mean reverting after the year 2000. I have read online that this could be due to some changes in the way companies started to account their earnings.</p>
<p>Is P/E ratio in general not mean reverting and therefore it doesn't matter that Shiller's P/E is not mean reverting? Is there some sort of augmented version that fixes CAPE? I found this <a href="https://www.philosophicaleconomics.com/2013/12/shiller/" rel="nofollow noreferrer">article</a>, but it is from 2013 and the chart was not kept up to date. It uses Pro-Forma earnings data instead of GAAP (Generally Accepted Accounting Principles). However, I can't recreate the chart since I can't find the Pro-Forma earnings data. Has anybody looked into this recently?</p>
<p><a href="https://i.sstatic.net/Jbqxy.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Jbqxy.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-05-17T02:39:07.370Z
I'd wait a few months. With interest rates rising (i.e. as an alternative investment) it is likely that the PE ratio will fall -- maybe with stock prices falling, but more likely with stock prices flat while earnings rise over the next few years.
Daniel
CC BY-SA 4.0
2022-05-18T10:11:18.617Z
My feeling is that it's because central banks have been more active in supporting the stock market, either through super-low interest races (which boosts trading on margin) or through buying shares themselves. Certainly they have been heavily involved since 2000 but I am not putting this as a detailed answer because I'm uncertain about how much they intervened in the market before 2000.
Mick
CC BY-SA 4.0
2022-05-20T01:39:10.800Z
How exactly did you come up with the idea that is was mean reverting and no longer is? I assume you tested this and looked up online material like https://www.tker.co/p/cape-ratio-not-mean-reverting-goldman-sachs?s=r before posting here?
Alex
<stock-market>
Why Shiller P/E stopped being mean reverting?
spacemonkey
# Why Shiller P/E stopped being mean reverting?

(Asked by: username_0 on 2022-05-16)

It looks like Shiller's P/E stopped being mean reverting after the year 2000. I have read online that this could be due to some changes in the way companies started to account their earnings.

Is P/E ratio in general not mean reverting and therefore it doesn't matter that Shiller's P/E is not mean reverting? Is there some sort of augmented version that fixes CAPE? I found this [article](https://www.philosophicaleconomics.com/2013/12/shiller/), but it is from 2013 and the chart was not kept up to date. It uses Pro-Forma earnings data instead of GAAP (Generally Accepted Accounting Principles). However, I can't recreate the chart since I can't find the Pro-Forma earnings data. Has anybody looked into this recently?

[![enter image description here](https://i.sstatic.net/Jbqxy.png)](https://i.sstatic.net/Jbqxy.png)

**Tags:** stock-market

> **Comments:**
> **username_1**: I'd wait a few months. With interest rates rising (i.e. as an alternative investment) it is likely that the PE ratio will fall -- maybe with stock prices falling, but more likely with stock prices flat while earnings rise over the next few years.  
> **username_2**: My feeling is that it's because central banks have been more active in supporting the stock market, either through super-low interest races (which boosts trading on margin) or through buying shares themselves. Certainly they have been heavily involved since 2000 but I am not putting this as a detailed answer because I'm uncertain about how much they intervened in the market before 2000.  
> **username_3**: How exactly did you come up with the idea that is was mean reverting and no longer is? I assume you tested this and looked up online material like https://www.tker.co/p/cape-ratio-not-mean-reverting-goldman-sachs?s=r before posting here?  

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## 1 answers

### Answer by username_4 on 2022-05-20. Score: 1

Let us define a P/E ratio as $$\phi(P,E)=\frac{P}{E},E\ne{0},$$ where $P$ and $E$ are random variables. Any function of data is a statistic, that is the definition of a statistic, so $\phi$ is a statistic.

Under mild assumptions, you can show that $P$ is normally distributed around the equilibrium. However, let us simply assume that $P$ is normally distributed for brevity. Likewise, it is reasonable to believe that the Board of Directors and management of firms targeted $E$ as $E^\*$. Depending on the method used to form the estimates, $E$ may also be normally distributed around the target. If not, you end up with a headache but should still have a solvable, if not analytic, problem. Again, for brevity, let us assume that $E$ is normally distributed around the target. I would note that such as statement does not have to be true.

By well-known theorems in the field of probability and statistics, $\phi$ cannot have a mean or a variance if it is the ratio of two normal distributions. There is no mean to revert to. Requiring truncation at zero, though some things have pierced the zero boundary at times, and assuming a stochastic budget constraint does not alter this outcome. It does make the discussion very long.

With that said, under mild assumptions, the median or the mode would be the basin of attraction. Again, length makes that discussion prohibitive in so short a space. Nonetheless, what is basically being said is that if there is a stationary $\mu$ somewhere in the system, even if it is not a mean, then it drives the system.

See, for example,

> Weisstein, Eric W. "Normal Ratio Distribution." From MathWorld--A Wolfram Web Resource. <https://mathworld.wolfram.com/NormalRatioDistribution.html> although this oversimplifies the question.

Under mild assumptions, it never was mean-reverting. Instead, it looks like Gull's Lighthouse Problem and there is a median but calling it *reverting* might be mathematically misleading. Instead, it might be better to think in terms of a see-saw and the median is trivial instance that the seesaw is level. The problem is also related to Poincare's double pendulum problem that helped set chaos theory in motion.

See

> Gull, S.F. (1988) Bayesian Inductive Inference and Maximum Entropy. Kluwer Academic Publishers, Berlin.

You should also be able to pick up this method in any first semester textbook in mathematical statistics for undergraduate statistics majors as this is often assigned as homework because of its weird properties.

I think this version of Freund's statistics textbook assigns this question as homework.

<https://smile.amazon.com/John-Freunds-Mathematical-Statistics-6th/dp/013123613X/ref=sr_1_9?crid=3GUMF7LFT4KPE&keywords=freund+statistics&qid=1653020853&sprefix=freund+statistics%2Caps%2C201&sr=8-9>

I would properly cite it with page numbers, but I do not remember where my copy is at. It isn't on a shelf.

Also,

> Curtiss, J.H. (1941) On the Distribution of the Quotient of Two Chance Variables. Annals of Mathematical Statistics, 12, 409-421.


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