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Jim
Your first demonstration confuses me. It seems wrong.
You write:
Imagine that every stock in the world were trading at fair value tomorrow, and we all agreed on that fair value.
The sum total of the market cap is vastly more than the aggregate amount paid for the shares by their current holders.
The vast majority of the stock's current holders bought their stock sometime in the last few years. A quick google turns up about 36 days to turn over the entire market cap in daily trading volume, and an estimate of 5.5 months for the average (probably median) time a stock is held in a portfolio. So it seems virtually certain that when the Market is overvalued for a few years at a time, the aggregate paid by the current holders of the shares will be on average higher than the current fair market value of the shares, and that therefore, in the aggregate, larger than fair market value.
Am I missing something here?