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Bloomberg, the financial services firm, presents a bar graph of monthly performances of equity indexes on most stock markets over a 10-year period. Analysis shows that on the main markets, the 120 monthly variations of the index rather faithfully obey the Gaussian law of probability. Bloomberg even provides the bell curve corresponding to the average and to standard deviation.
For example, over the final 10 years of the twentieth century (1989 – 99), the Paris Bourse rose by an average of 0.85 percent a month, with a standard deviation of 5.75 percent. If the prices indeed follow a normal distribution, it means that in 68 percent of the cases, the monthly variation was neither lower than – 4.9 percent nor higher than + 6.6 percent. Since 68 percent represents approximately two-thirds, this means that in two out of every three months, price variations neither go up by over 6.6 percent nor go down by more than 4.9 percent. Over two-thirds of the observations are to be found in a range from – 4.9 percent to + 6.6 percent. One month out of six, prices drop by more than 4.9 percent. One month out of six, prices soar at a rate of over 6.6 percent. Very similar results appear on other major stock markets. Bernstein analyzes the evolution of Standard and Poor’s index of 500 US stocks for January 1926 through December 1995, that is to say 840 observations of monthly price changes.23 The average monthly variation in New York over 70 years was + 0.6 percent compared with + 0.85 percent in Paris, but over only 10 years. Standard deviation in New York was 5.8 percent compared with 5.75 percent in Paris. You will agree that the difference is minuscule. In New York two-thirds of the 840 monthly price fluctuations observed ranged from – 5.2 percent to + 6.4 percent. In Paris, two-thirds of the 120 observations went from – 4.9 percent to + 6.6 percent. The very different stock markets studied over a protracted period turn out to be closely related.
On both markets, prices behaved as though obeying a normal law. This is likewise the case for the indexes of highly liquid stock markets, for the Dow Jones Industrial Average or Standard and Poor’s in the US, the Stoxx, Euronext and FTSE indexes in Europe. This is highly understandable. On an efficient market, prices reflect the available information. And yet the latest news concerning a firm is often unforeseen (strike, accident, merger, technological breakthrough and so on). These events hinge on chance. Influenced by such news, prices generally take a random walk. The prices of particular shares may follow different distributions, but Laplace’s central limit theorem tells us that the average price must comply with the normal law; and stock indexes are in fact structured as averages. There is also another reason for price indexes being made to take a random walk. If there were a way of forecasting them, if some technique allowed us to think that in days or weeks to come they would rise, then they would rise immediately. Price evolution has no more memory than Pascal’s throw of the dice. Each variation is independent of the past. Price variation regularly changes signs and there are few consecutive months in which the market goes either up or down. Variations in the same direction for five consecutive months occur in only one in ten cases. There is no way to use the price tendency in the past to predict their future direction. A price trend negates itself once it becomes known. This is another fundamental difference between the roulette wheel and the stock market; following the Martingale system in the latter – when you double your stake after a loss – leads to self-destruction.