Thorp may have been the first to discover that Madoff could not have been hedging his customers accounts because the actual reported volume of trading in stock options was a small fraction of the trades that he alone was claiming.

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Perhaps a lot better, since our scheme gets your portfolio management costs down to a small fraction of the 1.5% or so that stock funds charge on average.

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The dividend yield on the index (dividends divided by the stock price) is just 2.1% a fraction of historic averages, and barely enough to keep pace with current, subdued inflation.

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Without stock options, high-technology America would have developed at only a fraction of the pace it's done for the past 20 years.

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On Friday, its shares ended the day in Tokyo at 1, 149 yen - a fraction of their peak of 16, 950 yen at the height of the Japanese stock market bubble in 2000.

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Still, they are a small fraction of overall trading, accounting for 2.4% of all U.S. stock trading in June, according to Rosenblatt Securities.

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Schwab doesn't do stock underwritings, and it rates 30% of the 3, 500 stocks it follows a sell, the same fraction it rates a buy.

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