《投资学原理 Investments》课程教学资源（阅读资料）filter rules and stock-market trading（fama&blume,1966）.pdf_大学文库

FILTER RULES AND STOCK-MARKET TRADING* EUGENE F FAMAT AND MARSHALL E. BLUMEt " N THE recent literature there has been tcy? profits than a buy-and-hold pol- I. INTRODUCTION pected a considerable interest in the theory On the other hand, the statistician has of random walks in stock-market different though prices. The basic hypothesis of the theory tions of what constitutes an important is that successive price changes in indi- violation of the independence assump- vidual securities are independent random tion of the random-walk model. He will variables plies, of course, that the past history of a series of changes implies that the history of a price series can hanges cannot be used to predict future not be used to increase expected gains, the revers changes in any"meaningful"way. proposition does not hold. It is possible to construct What constitutes a"meaningful""pre- models where successive price changes are depend iction depends, of course, on the pur- used to increase expected profits.In fact, Mandelbrot di pose for which the data are being ex- 19 and Samuelson (12 show that, under fairly gen. to know whether the history of prices can tingle"which may or may not have the independ be used to increase expected gains. In a the martingale property mplies only that the random-walk market, with either zero of expecled values of future prices will be independent positive drift, no mechanical trading rule of the values of past prices; the distributions of applied to an individual security would future prices, however may very well depend on consistently outperform a policy of the values of past prices. In a martingale, though simply buying and holding the security. cannot be used by the trader to increase his expected Thus, the investor who must choose be- profits. A random walk is a martingale, but a mar- tween the random-walk model and a more complicated model which assumes the behavior of stock-market prices came about be- the existence of an excessive degree of fore the theoretical importance of the martingale either persistence (positive dependence) ly concerned with the theory of random walks. In or reaction (negative dependence)in suc- practice, this is not serious, since in most cases it is cessi ive price changes, should accept the probably impossible to distinguish a that fol- heory of random walks as the better series that follows a random walk. In most cases the model if the actual degree of dependence degree of dependence shown by a martingale will be cannot be used to produce greater e ors have bene. random-walk model fited from discussions with Professors Lawrence The terminology used in this paper will be that Fisher, Benoit Mandelbrot, Merton Miller, Peter of the more familiar theory of random walks rather Pashigian, and Harry Roberts of the University of than the more general(but perhaps simpler)theor Chicage of martingale processes. The reader will note, how Assistant professor of finance graduate School ever, that the bulk of our discussions remain valid if of Business, University of Chicago. e word“ martingale" is substituted for“rand Lecturer in applied mathematics, Graduate walk"and the words"the martingale property"are School of Business, University of Chicago substituted for"independe

FILTER RULES AND STOCK-MARKET TRADING 227 typically be interested in whether the de- and Morgenstern [7], and Godfrey, gree of dependence in successive changes Granger, and Morgenstern [6] also lend is sufficient to account for some particu- support to the independence assumption lar property of the distribution of price of the random-walk model changes or whether the dependence is Nevertheless, it is difficult to deter- sufficient to invalidate the results pro- mine whether these results indicate that duced by statistical tools applied to the the random-walk model is adequate for data. For example, price changes may be the investor. For example, there is ne one variable in a regression analysis and obvious relationship between the mag- the statistician will want to determine nitude of a serial correlation coefficient hether dependence in the series might and the expected profits of a mechanical produce serial dependence in the resid trading rule. Moreover, the market pro- als. If the amount of dependence is low, fessional would probably object that he will probably conclude that it will not common statistical tools cannot measure seriously damage his results. From the the types of dependence that he sees in it of however, the de- the data. For pendence may make the expected profits relationships that underlie the serial cor from some mechanical trading rule relation model are much too unsophist greater than those of a simple buy-and- cated to identify the complicated"pat hold policy. terns that the chartist"sees in stock 6b It is important to note, however, that prices. Similarly, runs tests are too rigid dependence"is always specific to the case and downward movements in prices. A at hand, the ultimate criterion is always run is considered terminated wheneve practical. In an encounter with a more there is a change in sign in the sequence complicated alternative, the theory of of successive price changes, regardless of random walks is overthrown only if the the magnitude of the price change that alternative leads to a better action than causes the reversal in sign. The market the random-walk theory would have sug- professional would require a more sophis gested ticated method to identify movements Previously the independence assump- a method that does not always predict tion of the random-walk model has been the termination of the movement simply tested primarily with standard statistical because the price level has temporarily tools, and in most cases the results have changed direction tended to uphold the model. This is true, Not all the published empirical tests of of Cootner [3], Fama [4], Kendall [8], and statistical models, however: Most no Moore [11]. In these studies the sample table, for example, is the work of Sidney serial correlation coefficients computed S. Alexander [1, 2]. Professor Alexander's for successive daily, weekly, and monthly filter technique is a mechanical tradin price changes were extremely close to rule which attempts to apply more so zero-evidence against"important" de- phisticated criteria to identify move- pendence in price changes. Similarly, ments in stock prices. An a per cent filter Fama's [4] analysis of runs of successive is defined as follows: If the daily closing price changes of the same sign and the price of a particular security moves up at spectral analysis techniques of Granger. least a per cent, buy and- hold the securi

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]

232 THE JOURNAL OF BUSINESS NOTES TO TABLE Q In applying the filter technique, the data P(? the closing price of security j for the ermine whether the first position taken day on which transaction t for filter will be long or short. With an a per cent filter, an initial position is taken as soon as Ii?= the total dollar profit on transaction there is an up-move or a down-move(which t of filter i when applied to security er comes first) where the total price j. The profits ar tal change is equal to or greater than a per cent dividends, which are positive for he position is assumed to be taken on the long transactions and negative fc first day for which the price change equals or n(p= the duration in terms of total trad eeds the a per cent limit. Any positions ng days of transaction t for filter i open at the end of the sampling period are when applied to security j. disregarded. Thus only completed transac- N(= the total number of trading days tions are included in the calculations during which positio The closing price on the day a position is under filter i when al opened defines a reference price: a peak in ty j. Thus the case of a long transaction and a trough in the case of a short transaction. On each N() subsequent day it ssary to check whether the position should be closed, i. e, whether the current price is a per cent below where r(s is the total number of the reference (peak)price in a long position transactions initiated by filter i for security j price if the open position is short. If the cur- ri= the rate of return with daily com- rent position is not to be closed, it is then necessary to check whether the reference when applied to security j. It is price must be changed. In a long position this will be necessary when the current price P{2[1+2]y=P{+r exceeds the reference price so that a new peak has been attained, whereas in a short rp"=the over-all rate of return with daily. position a new trough will be defined when the current price is below the reference when applied to security j. It price. Of course, when the reference price changes all subsequent testing uses the new On ex-dividend days the reference price r()=I[1+r2]p/NY value as ba s adjusted by adding back the amount of the dividend. Such an adjustment is neces- Rp"= the nominal annual rate of return sary in order to insure that the filter will not for filter i when applied to com- iggered simply because the stock's pri pany ]. It is computed as typically falls on an ex-dividend date. In R4=260r4 addition, if a split occurs when a position is open, the price of the security subsequent to R( are the returns shown for the the split is adjusted upward by the appropri- filter technique() in Table 1 ate factor until the position is closed. R (=the nominal annual rate of return With this background discussion we shall from buy-and-hold during the time ow consider the rate-of-return calcula period for which filter i had open summarized in Table 1. The following are positions in security] e basic variables in the calculations

FILTER RULES AND STOCK-MARKET TRADING NOTES TO TABLE 1-C where bri 3) is defined as This roundabout procedure for comput- [1+r11292y) g buy-and-hold returns is necessary to in- sure that the buy-and-hold returns cover exactly the same time periods and are com- where u=ri? if the correspond outed on exactly the same basis as the re ing filter transaction is long, and turns under the filter technique if the correspondi Finally, it should also be noted that the filter transaction is short. bR j) are results for the largest filters are probably not the returns reported for the buy- reliable since for these filters the number of and-hold policy(B)in Table 1 transactions is very small. Cf. Table 3 security and filter size, the annual re- policy. The reported returns are vari turns, adjusted for dividends but not for ously adjusted for dividends and for brokerage fees, under both the filter commissions technique and a simple buy-and-hold When commissions are taken into ac- policy. For each security and filter size, count the largest profits under the filter for the perio returns are computed only technique are those of the broker. Only tions are open under the filter rule, which General Foods, Procter Gamble, and requires that multiple buy-and-hold fig- Sears) have positive average returns per ures be reported for each security. The filter when commissions are included a act procedure used to compute the re-(col. [2J). When commissions are omitted, s is discussed in the note to Table 1. the returns from the filter technique(col. Table 1 presents only a small fraction [1]) are, of course, greatly improved but of the results of this study. For example, are still not as large as the returns from returns under the filter technique have simply buying and holding Comparison been computed in many different ways: of the profits before commissions under gross and net of brokerage fees, with and the filter technique(col. [1])and under a without dividends, etc. Since presenting buy-and-hold policy(col. [6]) indicates all the empirical work would require a that, even ignoring transactions costs, small book of tables, we shall be con- the filter technique is inferior to buy-and- strained to concentrate on summary ver- hold for all but two securities: Alcoa and sions of the results-summarized by Union Carbide security and by filter size. Table 1 pre- This last result is inconsistent with sents the most important of the basic re- some of Alexander's latest empirical work sults in full detail, however, and permits 2, Tables 1 and 2]. When commissions the reader to verify conclusions that will are omitted, alexander finds that the fil be drawn from the summary statistics. ter technique is typically superior to a A. ANALYSIS OF RESULTS BY SECURITY buy-and-hold policy, at least for the pe riod 1928-61 a bias in Alexander's com- Table 2 summarizes the filter results putations, however, tends to overstate by security. For each security the table the actual profitability of the filter tech- shows average returns per filter under nique relative to buy-and-hold. This bias both the filter rule and the buy-and-hold arises because using common price in