# Reversion to the Mean Coin Flippers

*By A.T. On July 27, 2000*

**Steve Berman: **“You quote towards the bottom of your column that 9% is the average annual return for US stocks including dividends but before inflation. I thought the IBBOTSON data for large U.S. stocks was much closer to 11% a year — going back at least to 1925 or so — and even higher for small stocks. What gives?”

☞ I was looking at the traditional number *before* this amazing 18-year run-up. Only when you add that in does the 9% or so jump to 11% (and perhaps even a bit more when you include small stocks). So the question is — is 11% or so the new “mean” we can expect going forward, or is 9% or so the traditional mean to which the market, with time, will revert?

I obviously don’t know for sure, but my hunch is that 11%, pre-tax, pre-inflation, will not be the new standard. (And note that even if it is, that would be a lot lower than we have come to expect, lo these last boffo 18 years.)

I hope I’m wrong!

**Dean Cardno:** “You slightly misrepresented ‘Reversion to the Mean’ in today’s column. As applied to tossing coins, it does *not* mean that an observed string of heads will be followed by slightly more tails. Even after an improbable streak of heads, the forward-looking expectation remains 50/50 heads/tails. The results will ‘revert to the mean’ only when an infinite number of trials are averaged in with all the past results (necessarily a finite number).

☞ You’re right. I used a bad example. But I came close!

**Sergei Slobodov:** “If you flip a coin 100 times, you are likely to find the number of heads (and tails) deviate from the mean by about 10 or so, in one direction or another. If you flip the coin another 300 times (total of 400), the deviation would be about 20, or square root of 400. This is known as ‘random walk.’ After a million flips, you are likely to ‘walk’ about 1000 away from the mean. And since the coin does not remember the past, there is no reason to believe that you get more tails (or heads) in the next million flips to ‘compensate’ for the past history. Thus, no reversion to the mean.”

☞ You’re right. I used a bad example. But I came close!

**Jonathan Levy:** “Assuming a fair (balanced) coin, the expected distribution of future coin flips is 50/50 regardless of past history. Coins do not have memories. Neither do dice.”

☞ You’re right. I used a bad example. But I came close!

If a bright child flipped 7 heads out of 10 the first time he tried it, he might conclude that there’s something about the coin that tends to make it heads-heavy, and to expect 70% heads in the future. As he kept flipping, of course, the overall result would inexorably revert back to 50% — but not, as you say, because a run of heads can be expected to be followed by a run of tails.

I feel as if the last 18 years in the market, because of falling interest rates and rising price/earnings ratios — which have magnified the earnings gains — have been atypical. Rates will not fall forever, p/e’s will not expand forever, and thus earnings gains will not be magnified and translate into outsize stock-price gains forever.

So a lot of bright young investors who’ve come to expect 70% heads just because it’s “always” been that way in their 18-year experience might be expecting too much going forward.

A “string of tails” would result if price/earnings ratios — or earnings growth — fell for a while. That may not happen for a long time, or might conceivably never happen. But though I’m an optimist, I’m not *that* much of an optimist.

I expect great things for our planet, our country, and the stock market. But I doubt the stock market will do nearly as well these coming 18 years as it has over the past 18.

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