228 THE JOURNAL OF BUSINESS ty until its price moves down at least t Mandelbrot [10, pp. 417-18] pointed per cent from a subsequent high, at out, however, that Alexander's computa- which time simultaneously sell and go tions incorporated biases whichled to seri hort The short ion is maintained ous overstatement of the profitability of until the daily closing price rises at least the filters. In each transaction Alexander w per cent above a subsequent low at assumed that his hypothetical trader could which time one covers and buys. Moves always buy at a price exactly equal to the less than a per cent in either direction are low plus a per cent and sell at the high r cent. In fact because of the Alexander formulated the filter tech- frequency of large price jumps, the pur nique to test the belief, widely held chase price will often be somewhat higher among market professionals, that prices than the low plus a per cent, while the adjust gradually to new information. sale price will often be below the high The professional analysts operate in the be- minus a per cent lief that there exist certain trend generating In his later paper [2, Table 1] Alexan knowable today, that will guide a specu- der reworked his earlier results to take to profit if only he can read them correctly. account of this source of bias. In the rather than instantaneous jumps because most corrected tests the profitability of the of those trading in speculative markets have filter technique was drastically reduced imperfect knowledge of these facts, and the fu. However, though his later work takes ant of discontinuities in the spread of awareness of these facts throughout series, Alexander's results are still very rpre For the filter technique, this means that arise because it is impossible to adjust for some values of a we would find that the commonly used price indexes for the if the stock market has moved up x per effects of dividends. This will later be cent it is likely to move up more than x shown to introduce serious biases int per cent further before it moves down by filter results x per cent”[1,p.26]」 In his earlier article 1, Table 7 Alex II. THE FILTER RULE AND TRADING ander reported tests of the filter tech nique for filters ranging in size from 5 to Alexander's filter technique has been 50 per cent. The tests covered different applied to series of daily closing prices time periods from 1897 to 1959 and in- for each of the individual securities of the volved closing"prices"for two indexes, Dow-Jones Industrial Average. The ini the Dow-Jones Industrials from 1897 to tial dates of the samples vary from Janu 1929 and Standard and Poor's Indus- ary, 1956, to April, 1958, but are usually trials from 1929 to 1959. In general, about the end of 1957. The final date is filters of all different sizes and for all the always September 26, 1962. Thus there different time periods yielded substan- are thirty samples with 1, 200 to 1,700 tial profits--indeed profits significantly observations per sample greater than those of the simple buy-and wenty-four different filters ranging from 0.5 per cent to 50 per cent ha hold policy. This led Alexander to co been simulated. Table 1 shows, for each clude that the independence assumpti 2 The is of central the of the random-walk model was not up- the stable Paretian hypot beld by his data. cussion and empirical evidence, see Fama [4]