Well, the charts stop at 30 or so years, so obviously the gap won't vanish entirely. The point is that it gets smaller and smaller as the period increases, which is the definition of convergence (assuming it can get as close to 0 as you wish).

Your argument is like saying, 2 ^ -x doesn't converge to 0 as x -> infinity, with the proof that "draw a chart from 0 to 10, and you'll see the curve NEVER reaches 0".

Long term market returns probably converge to a single value by the central limit theorem, i.e. after throwing a large number of random outcomes, the standard deviation of the empirical mean approaches 0. It's always a question of how long you wait. Of course, in an investing lifetime of ~50 years, the standard deviation of the mean is not necessarily small or negligible, which is trivial IMO.

Gerard, I'm not sure if we agree or disagree. And I agree the visual gap between the worst and best cases narrow if you look further into future.

However, if your understanding is that either the two cases "coverge" to a single value or as a practical matter tighten into a very tight range that means individuals get nearly the same outcome, your understanding differs from my understanding.

Two or three hopefully constructive comments:

1. Even tiny differences compounded over long periods of time result in large differences in the final "future values."

2. The history surely shows that wide variability has occurred in past.

I sort of hesitate to veer into the weeds about this--I'm far more concerned about new investors understanding this point than about experienced investors polishing their expertise, but I think a monte carlo simulation someone can play with on their own perhaps moves the discussion forward in a helpful way.

Accordingly, I constructed a simple century long monte carlo simulation with Excel that looks at a 1000 simulations of the case where someone invests $1,000 when they're born into the stock market and then lets that investment compound for a century. Just to keep things simple but reasonably accurate, I used a 10% nominal rate of return and then also a 10% standard deviation.

With those inputs, I get the following results after 100 years:

Statistic | Dollars | Returns |

Mean | $14,017,736.79 | 10.019% |

Median | $9,271,956.09 | 9.565% |

St. Dev. | $14,868,015.08 | NA |

Minimum | $433,632.93 | 6.260% |

Maximum | $134,498,934.17 | 12.535% |

Three features I'd like to point out:

1. Though the mean and median nominal returns are very close to each other, after a century of compounding, even that small half a percent difference produces a really large difference in the future value.

2. Wide variability in nominal rates of returns and future values occurs even with a century of "evening out" or "averaging out".

3. Anyone looking at this data needs to really take note of the fact that these values are nominal values (so greatly, greatly inflated after a century of inflation.)

I've attached a skinny-ed down version of the Excel spreadsheet. Note four things... First, the 1000 scenarios version of the spreadsheet is too big to attach so the one people can download only does a 100 scenarios. Second, someone can expand the 100 scenarios version to a 1000 scenarios version by copying the range CK16..ALM16 into the range CK17..ALM114. Third, you can press the F9 key to recalculate the spreadsheet's values and run another simulation. Fourth, the MAX function, if it gets too large, returns the #NUM error. Sorry...