Anyone who has visited IFA’s office in Irvine will likely have a fond memory of Francis, IFA’s probability machine, named after Francis Galton, an English mathematician who was an expert in many scientific fields. For those needing a refresher or a first look, we have this YouTube video. Of IFA's 405 videos on YouTube, four of the ten most viewed videos are on probability machines, with about 156,000 views between them. The purpose of this article is to explain why the probability machine provides us with a reasonably good simulation of free market prices and risk appropriate market returns.

Nobel Laureate Eugene Fama summarizes his Efficient Market Hypothesis by stating that prices reflect all available information and that means that investors are always paying fair prices (in his own words). In our opinion, the objective of fair prices are fair returns, given the risk. Given this assumption of fair prices, what should the distribution of returns look like?

As shown in the chart below, when we observe 600 monthly returns for an Index Portfolio 100, they look very similar to a bell curve, otherwise known as the normal distribution. This means that every first of the month, about 10 million global investors agree upon fair prices for the globally diversified 13,000 stocks in Index Portfolio 100. From that fair price, each subsequent daily closing price should have an equal chance of going above or below a price that would result in a risk-appropriate daily return (for Index Portfolio 100 that has been about 0.03%/day, based on the last 50 years). This is the reason price changes follow a random walk down Wall Street and why IFA advises against market timing. The cumulative price change (plus dividends) over about a 30 day period results in a monthly return for each subsequent month, and applying the Law of Large Numbers, a large sample of those monthly returns results in a distribution that incorporates the characteristics of repeated fair prices throughout the month and the constant uncertainty of the future return over a one month period (this is opposed to a belief in the Law of Small Numbers or the Gambler's Fallacy that many investors rely on). The higher the risk, the higher the average return and width of the distribution of returns (measured as the standard deviation).

This means that from every fair price:

**1.** the most likely outcome is a fair return,

**2.** there is an equal chance that the future return will be above or below a fair return,

**3.** the further the monthly return is from the fair return, the less likely it is to occur and

**4.** an extreme negative return, like -23%/month (a black swan) is equally likely to an extreme positive return, like 23%/month (a green swan). Note the symetry of extreme outcomes for Index Portfolio 100 in this 50 year period.

Here is what the 600 monthly returns of Index Portfolio 100 looks like over various 600 month periods with an overlay of a normal distribution. For more information about IFA Index Portfolio 100, click here to visit the portfolio page.

At IFA, we often refer to the probability machine as the "fair price simulator." Our assumption is that the constant entry point at the top center of the array of pegs represents the constant fair price of a constant risk investment, positioning the outcome for a constant expected fair return. "Expected" includes the concept that there is a range of future returns, where the average (expected value) of a large sample of 50/50 chance events will be driven by the Central Limit Theorem to a bell-shaped curve. Wikipedia describes the Central Limit Theorem in a way that could be used to describe stock market distributions [our added comments appear in brackets], given the condition of the Efficient Market Hypothesis and more importantly, what we observe in the data:

"In probability theory, the central limit theorem (CLT) states that, given certain conditions [fair prices and constant risk], the arithmetic mean of a sufficiently large number of iterates of independent random variables [news and the resulting prices], each with a well-defined expected value [expected return given the risk] and well-defined variance [the risk or uncertainty of the return of the investment], will be approximately normally distributed, regardless of the underlying distribution.^{[1]}^{[2]} That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a "bell curve")."

As the beads bounce through the pegs in what appears to be chaos, they simulate subsequent news that impacts the companies and the resulting price changes as time moves forward over the next 30 days. The pegs are arranged like repeated sets of 5, as they appear on dice. The result is that for a given peg, there is an equal chance the the bead will bounce left or right, just like the fair price that is equally positioned to go up or down relative to the price that would result in a fair return, given the relevant information of the moment. As the beads bounce around among the pegs, it appears to be chaos in motion. But, according to the Law of Large Numbers, in large samples of chaos there is order representing the characteristics of the beads and pegs.

In 1886 Francis Galton wrote, "I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the law of error. ... The huger the mob and the greater the anarchy, the more perfect is its sway. Let a large sample of chaotic elements be taken and marshalled in order of their magnitudes, and then, however wildly irregular they appeared, an unexpectedï»¿ and most beautiful form of regularity proves to have been present all along."

We have found this machine so instructive for investors that we have recently engaged a design firm to create and manufacture about a 6" x 5" desk top version, so investors can be frequently reminded of the fair prices, market randomness, and fair returns. Stay tuned.

Please keep in mind that this is just a model, and like any model, this one is not perfect. As shown on the front panel explanation of the machine, the columns in the probability machine limit the extreme bead distributions to about plus or minus 15% around the average return of Index Portfolio 100, but the historical data has 6 out of 600 months that are outside of that, with three months that are below -15% and three months that is above 15%. This indicates that our model does not adequately explain about 1% of the outcomes, often referred to as extreme outliers which occur in kurtotic distributions (fat tail distributions) like we see in monthly stock market returns. That being said, it is also important to note that the extreme high months were approximately offset by the extreme low months in this sample and as buy and hold investors, these extreme outcomes should be ignored. Inspired by Nassim Nicholas Taleb, some observers call these outliers (particularly the negative ones) “black swans.” One objection we have voiced against this term is that there is no conclusive evidence that extreme negative events are more probable than their positive counterparts which we sometimes refer to as “green swans.”

As one of the great statistical minds of the 20^{th} century, George E.P. Box wrote^{1}, “Essentially, all models are wrong, but some are useful.” We at Index Fund Advisors unhesitatingly say that the probability machine provides a useful model of how to think about fair prices and the resulting market returns. From these critical lessons, investors can develop better investing strategies and simulations of future returns.

^{1}Box, George E.P.; Norman R. Draper (1987). Empirical Model-Building and Response Surfaces, p. 424, Wiley.

## About the Authors

Mark Hebner and additional IFA employees contributed to this article

Founder and President of Index Fund Advisors, Inc., and author of Index Funds: The 12-Step Recovery Program for Active Investors. He is a Wealth Advisor, with an MBA from the University of California at Irvine and a BS in Pharmacy from the University of New Mexico with a specialization in Nuclear Pharmacy.