# Return Distributions and the Shiller P/E Ratio

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This article expands on ideas developed by Joseph A. Tomlinson in a series of recent articles for Advisor Perspectives on the topic of the Shiller P/E Ratio as a predictor of future returns in the stock market (See Advisor Perspectives: Shiller P/Es and Modeling Stock Market Returns, January 19, 2010).    Specifically, this article looks at the distribution of three-year returns in the stock market following different starting points for the Shiller P/E ratio on a monthly basis since 1884 to illustrate that the historical distribution of rolling three-year returns in the stock market is not random.

Imagine if I challenged you to a simple game: I fill a jar with 50 black marbles and 50 red marbles and propose to draw 10 marbles from the jar.  For each black marble in the draw, I agree to pay you whatever dollar amount you choose, provided that you will pay me the same amount for each red marble pulled from the jar.

Knowing that the distribution of black and red marbles is 50/50, most rational people would decline to play this game for real money.  But what if I agreed to remove 10 red marbles from the jar before we started?  With the distribution now 50/40 in favor of black marbles, it becomes sensible, even wise to play this game for money.  If I remove 20 red marbles to make the distribution 50/30, a rational person should be willing to raise the value of their wager.  And if I remove 40 red marbles before starting the game it becomes sensible to “bet big,” whatever “big” means for the player involved.

The logic of this sequence is straightforward – when the mix of marbles is 50/50, the likelihood of winning or losing the game is purely random, but once a few of the red marbles have been removed from the jar the distribution of possible outcomes becomes skewed – any given draw of 10 marbles is more likely to contain more blacks than more reds.  The most probable outcome is no longer random.

A recent article in this publication by Joseph A. Tomlinson reminds readers that the models used by most financial planners assume the distribution of future returns in the asset markets is always random, like drawing marbles from a jar with an equal mix of reds and blacks.  Tomlinson goes on to suggest that this assumption of randomness in the asset markets may be flawed.  He supports his point with a study of historical correlations between starting valuation and subsequent returns in the stock market over rolling periods of 1 and 10 years.

Tomlinson’s work on this subject inspired the studies I will describe in this article, which address the same question – are returns in the stock market random? – in a slightly different way.  Rather than measuring correlations between starting valuations and subsequent returns, I have measured the distributions of returns that follow various starting-point valuation levels in the stock market.  Like Tomlinson’s research, the studies in this article measure the valuation of the stock market with the Shiller P/E ratio.1

1 The “Shiller P/E Ratio” was designed by Yale economist Robert J. Shiller to reflect a normalized valuation multiple for the U.S. stock market.  It is calculated as the price of the market index divided by the average inflation adjusted earnings for the index over the previous 10-years.  The Shiller P/E Ratio is designed to smooth out the short-term swings in corporate earnings caused economic cycles.