UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An empirical test of the theory of random walks in stock market prices : the moving average strategy Yip, Garry Craig 1971

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1971_A4_5 Y56.pdf [ 2.19MB ]
Metadata
JSON: 831-1.0101827.json
JSON-LD: 831-1.0101827-ld.json
RDF/XML (Pretty): 831-1.0101827-rdf.xml
RDF/JSON: 831-1.0101827-rdf.json
Turtle: 831-1.0101827-turtle.txt
N-Triples: 831-1.0101827-rdf-ntriples.txt
Original Record: 831-1.0101827-source.json
Full Text
831-1.0101827-fulltext.txt
Citation
831-1.0101827.ris

Full Text

AN EMPIRICAL TEST OF THE THEORY OF RANDOM WALKS IN STOCK MARKET PRICES: THE MOVING AVERAGE STRATEGY by GARRY CRAIG YIP B.Com., Un iver s i t y of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION in the Faculty of Commerce and Business Admin is t rat ion We accept th i s thes i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1971 In present ing th i s thes i s in pa r t i a l f u l f i lmen t o f the requirements for an advanced degree at the Un iver s i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying of th i s thes i s fo r s cho la r l y purposes may be granted by the Head of my Department or by h i s representat ives . It is understood that copying or pub l i c a t i on o f th i s thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of UCWUMU- ftjntl BwiJUv\U& ftcliwinjqJj^YU The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date fliuxuit • ABSTRACT This study invest igates the independence assumption of the theory of random walks in stock market pr ices through the s imulat ion of the moving average s trategy. In the process of doing so, three re lated questions are examined: (1) Does the past r e l a t i v e v o l a t i l i t y of a stock furn i sh a useful ind ica t ion of i t s future behavior? (2) Is the performance of the dec i s ion ru le improved by applying it to those s e c u r i t i e s which are l i k e l y to be h igh ly v o l a t i l e ? (3) Does po s i t i ve dependence in successive monthly p r i ce changes ex i s t ? The purpose of Test No. 1 was to gauge the tendency for a s tock ' s r e l a t i v e v o l a t i l i t y to remain constant over two adjacent in terva l s of time. As measured by the c o e f f i c i e n t of v a r i a t i o n , the v o l a t i l i t y of each of the 200 s e c u r i t i e s was computed over the 1936 to 19^5 and 1946 to 1955 decades. In order to evaluate the strength of the re l a t i onsh ip between these paired observat ions, a rank co r re l a t i on ana lys i s was performed. The resu l t s indicated a substant ia l d i f f e rence in r e l a t i v e v o l a t i l i t y for each secur i ty over the two ten-year per iods . In Test No. 2 a d i f f e r e n t experimental design was employed to determine whether the r e l a t i v e v o l a t i l i t y of a stock tended to remain with in a d e f i n i t e range over time. According to the i r v o l a t i l i t y in the 1936 to 1945 per iod, the 200 s e c u r i t i e s were d iv ided into ten groups. i i i P o r t f o l i o No. 1 contained the twenty most v o l a t i l e s e c u r i t i e s while P o r t f o l i o No. 2 cons isted of the next twenty most v o l a t i l e , e t c . An average c o e f f i c i e n t of va r i a t i on was ca lcu la ted for each group over the per iods, 1936 to 19^5 and 19^6 to 1955. The rank co r re l a t i on ana lys i s on these ten paired observations revealed that the most v o l a t i l e s e c u r i t i e s , as a group, tended to remain the most v o l a t i l e . Test No. 3 consisted of the app l i ca t i on of the moving average strategy (for long pos i t ions only) to fo r ty ser ies of month-end pr ices covering the i n t e r v a l , 1956 to 1966. These s e c u r i t i e s had demonstrated a high r e l a t i v e v o l a t i l i t y over the previous decade and, on the basis of the f indings reported in Test No. 2, i t was forecasted that they would be the most v o l a t i l e of the sample of 200 in the period under i n v e s t i -gat ion. Four d i f f e r e n t moving averages ranging from three to s ix months, and th i r teen d i f f e r e n t thresholds ranging from 2 to 50 per cent were s imulated. The resu l t s of the s imulat ion showed the moving average strategy to be much i n f e r i o r to the two buy-and-hold models. Every threshold regardless of the length of the moving average y ie lded a nega-t i ve re turn . In add i t i on , the losses per threshold were spread throughout the major i ty of stocks. A l together, therefore , considerable evidence was found in favour of the random walk theory of stock p r i ce behav i o r . TABLE OF CONTENTS PAGE CHAPTER I. THE THEORY OF RANDOM WALKS AND ITS IMPLICATIONS FOR FUNDAMENTAL AND TECHNICAL ANALYSIS . . . 1 CHAPTER I I. FILTER RULES 12 CHAPTER III. THE MOVING AVERAGE STRATEGY APPLIED TO HIGHLY VOLATILE SECURITIES lh Selected Techniques for Improving the Performance of the F i l t e r Rule and Moving Average Strategy 2k The Data 29 Test No. 1: The Experimental Procedure and the Resul ts 30 Test No. 2: The Experimental Procedure and the Results 31 Test No. 3: The Experimental Procedure and the Results 32 CHAPTER IV. CONCLUSIONS AND FUTURE RESEARCH 37 Summary 37 Suggested Research Topics kO BIBLIOGRAPHY 50 Page TABLE I. Subsequent V o l a t i l i t y of Po r t f o l i o s of Stocks Selected on the Basis of Pr ior V o l a t i l i ty 42 TABLE II. Selected Sample of Forty Secu r i t i e s . . . . 43 > TABLE III. Three Month Moving Average Results 44 TABLE IV. Four Month Moving Average Results 45 TABLE V. Five Month Moving Average Results 46 TABLE VI. Six Month Moving Average Results 47 Hypothetical Chart of a Stock Pr ice Subject to Random Movement Within Fixed L imits Hypothetical Chart of a Stock Pr i ce Subject to Random Movement Within P e r i o d i c a l l y Changing L imits ACKNOWLEDGMENT I would l i ke to express my grat i tude to Dr. Wi l l iam Wood for h is valuable guidance in the preparat ion of th i s thes i s . I am a l so indebted to Mr. Vincent S. Manis for h is programming ass i s tance and Miss Susan Lew for typing both the rough and completed d r a f t s . r THE THEORY OF RANDOM WALKS AND ITS IMPLICATIONS FOR FUNDAMENTAL AND TECHNICAL ANALYSIS For decades the dynamics of p r i ce formation in the stock market has been a subject of study for a s izeab le segment of the invest ing p u b l i c . The reason for such overwhelming interest in th i s cap i ta l market is a t t r i b u t a b l e to several f a c to r s , most of them i n t e r r e l a t e d . To begin with, i t is recognized that th i s market is among the best organized of the mult itude and almost always ranks as the largest in terms of value of sa le s . A l so , i t is widely known that speculat ive pr ices are extremely sens i t i ve to a l l events, both real and imagined, providing an ins ight into the fu ture . As one might expect, these elements operate in conjunction with one another to cause a cons iderable tota l of gains and losses to be experienced each trading day. Needless to say, f l uc tua t i ons such as these may present the opportunity for large p r o f i t s to be rea l i zed with in a very short time hor izon. This has undoubtedly provided a constant a t t r a c t i o n to men with dreams of instant wealth. With numerous pa r t i c ipan t s in the equ i t i e s market i t would be natural to f ind a vast array of "magic" formulas for p red i c t ing stock pr i ces and indeed, such is the case. Investors have cor re la ted the movement of share pr ices to every conceivable phenomenon for forecas t ing purposes. While an i n f i n i t e number of formulas e x i s t , there are only two basic approaches to invest ing, fundamental ana lys i s and technica l ana l y s i s . Under the fundamental philosophy the ana lys t , in searching for bargains, bases his va luat ion of a secur i ty upon those factors which he bel ieves other ra t iona l investors use to assess va lue. In general terms these fac tors cons i s t of the expanse of information contained in economic and f i nanc i a l s t a t i s t i c s . More s p e c i f i c a l l y they include earnings, dividends and assets of the f i rm represented by the secur i t y under observat ion. A l l such data, past or present, is taken into account to appraise the value of the share or as i t is commonly referred to, the i n t r i n s i c value of the share. The b e l i e f of the fundamentalist that undervalued s e c u r i t i e s e x i s t , implies an imperfect market. He attempts to c a p i t a l i z e on the asset pr i ce d i sequ i l i b r ium s i tua t i on by purchasing a bargain and expects the market l a ter to adjust to an equ i l i b r ium s t a te . Imp l i c i t l y the fundamentalist is assuming that imperfect knowledge character izes the market place and that trends w i l l develop from the gradual spread of awareness of the relevant f a c t s . Unl ike the fundamental approach, technica l ana lys i s is con-cerned with p red i c t i ng share pr ices so le l y on the basis of past information. A d i r e c t resu l t is that ra t iona l investment behavior is not a c r i t i c a l assumption for the techn i c i an . The market cont inua l l y and automat ica l ly weighs a l l f ac tors inc luding not only those relevant to the fundamentalist but a l so opin ions, moods, and guesses. In one important respect a p a r a l l e l can be drawn between fundamental and technica l ana l y s i s . They both infer that the market is not per fect by assuming that fac t s at one moment of time w i l l govern pr ices at some future date. For the technic ian th i s is in the form of a b e l i e f that stock pr i ces tend to move in trends which per s i s t for an appreciable length of time and that these trends can be uncovered by studying p r i ce movements of the immediate past. While numerous profess ional p r a c t i t i o n e r s endorse the technica l approach to invest ing, academics have not followed s u i t . Rather, they suggest that stock pr ices behave as a random walk and that future movements of pr i ces could jus t as well be determined by toss ing a coin as by any soph i s t i cated ana lys i s of past p r i ce changes. This ser ious threat to the techn i c i an ' s l i ve l i hood has engendered a commensurate degree of controversy between the respect ive p a r t i e s . "The theory of random walks in stock pr ices is based on two hypotheses: (1) successive pr i ce changes in an indiv idual secur i ty are independent, and (2) the p r i ce changes conform to some p r o b a b i l i t y d i s t r i b u t i o n . " ^ Of the two, the f i r s t is more important. If dependence in successive increments of pr i ces ex i s t s then the theory is not v a l i d . The second hypothesis is not nearly as d e f i n i t i v e because the form or shape of the d i s t r i b u t i o n need not be s p e c i f i e d . Consequently, any d i s t r i b u t i o n which s a t i s f i e s the condi t ion of c o r r e c t l y charac te r i z ing the process generating the p r i ce movements is cons is tent with the theory. Eugene F. Fama, 'The Behaviour of Stock Market P r i c e s , " Journal of Business, 38 (January, 1965), P- ^ 0». In s t a t i s t i c a l terms independence means that the p r o b a b i l i t y d i s t r i b u t i o n for the p r i ce change during time period t is independent of the sequence of p r i ce changes during previous time per iods . That i s , know-ledge of the sequence of p r i ce changes leading up to time period t is of no help in assessing the p r o b a b i l i t y d i s t r i b u t i o n for the p r i ce change during time period t.2 The fact that there is probably no hope of ever f ind ing a time ser ies which exh ib i t s per fect independence precludes the theory of random walks from being an exact desc r ip t i on of stock pr i ce behavior. Nevertheless, the independence assumption of the model may for p rac t i ca l purposes be accepted as long as the dependence in the ser ies of successive pr i ce changes is below some spec i f i ed l e v e l . For the s t a t i s t i c i a n the d e f i n i -t ion of what cons t i tu tes a s i g n i f i c a n t v i o l a t i o n of the independence assumption depends upon the p a r t i c u l a r s t a t i s t i c a l tool app l i ed . Since the market p r a c t i t i o n e r is p r imar i l y interested in p r o f i t a b i l i t y , the relevant c r i t e r i o n for him in judging the v a l i d i t y of the independence assumption may be stated accord ing ly : the random walk model is va l i d as long as the knowledge of the actual degree of dependence in the ser ies of p r i ce changes cannot be used to increase expected gains above that which is a t ta inab le under a buy-and-hold p o l i c y . There are a number of market s i tua t ions compliant with independence. Louis Bachel ier (1900), who is c red i ted with introducing the notion of speculat ive pr ices fo l lowing random walks, submitted in a nebulous manner the simplest explanation for the independence assumption. This was la ter re f ined and stated e x p l i c i t l y by M. F. M. Osborne. Both theoret i c i ans bel ieved that: " . . . If successive b i t s of new information a r i se independently across time, and i f noise or uncerta inty concerning i n t r i n s i c values does not tend to fo l low any cons i s tent pa t tern , then successive p r i ce changes in a common stock w i l l be 3 independent." In th i s instance a per fect market is env i s ioned. When viewed with in the context of the market place i t s e l f the assumptions embodied in the Bachelier-Osborne model appear rather extreme. As advocated by Eugene F. Fama in h is study, "The Behaviour of Stock Market 4 P r i c e s " , there are no strong reasons for expecting e i ther each i n d i -vidual 's estimates of i n t r i n s i c values to be independent of the estimates made by others , or successive b i t s of new information to be generated independently across time. There may be opinion leaders in the market and there may be a tendency for news with a b u l l i s h inf luence to be followed by the same type of news more often than by pes s im i s t i c news. A log i ca l extension of th i s l i ne of reasoning would be to acknowledge the p o s s i b i l i t y of such dependencies causing pr i ce changes to be dependent. In defense of the independence assumption in stock pr ices i t is often contended that the e f f e c t s of systematic dependencies may be neut ra l i zed by o f f s e t t i n g mechanisms. This ra t iona le has been elaborated upon by Fama in the aforementioned a r t i c l e and takes the fo l lowing form. 31 b i d . , p. 37. ^1 b id , pp. 34 -105 . If there are opinion leaders in the market (dependence in the noise generation process) , "bubbles" (dependence in successive p r i ce movements) would tend to a r i s e in a p r i ce s e r i e s . In other words the accumulation of the same type of noise would cause the pr i ce level to run well above or below the i n t r i n s i c va lue. Recognition of th i s occurrence by soph i s t i ca ted traders would provide them with an incent ive to s e l l the secur i ty or to s e l l i t short in the f i r s t case and buy i t in the second, s ince they expect actual pr i ces to move eventua l ly back toward i n t r i n s i c va lues. Provided that there are enough of these t raders , the e f f e c t of the i r act ions would be to e l iminate the "bubbles " . Correspondingly, p r i ce movements would approximate a random walk. If there are dependencies in the process generating new informa-t ion th i s w i l l a l so tend to create dependence in successive pr i ce changes of a s e c u r i t y . The act ions of many sophis t icated ana lys t s , as a resu l t of in terpret ing both the p r i ce e f f e c t s of current new information and of the future information implied by the dependence in the information generation process, would again tend to make p r i ce changes independent. Even in s i tua t ions where there is cons is tent vagueness or uncerta inty surrounding new information, independence in successive p r i ce changes could s t i l l occur. For example, uncerta inty concerning the importance of new information may cause the market to cons i s ten t l y underestimate (or overestimate) i t s e f f e c t s on i n t r i n s i c va lues. Since pr ices genera l ly w i l l not adjust instantaneously to the i r i n t r i n s i c values the astute trader with knowledge of the underestimation phenomenon could p r o f i t by purchasing s e c u r i t i e s when the new information is o p t i m i s t i c . If several traders attempt to c a p i t a l i z e on th i s opportunity any con-s i s tent lags in the adjustment of actual pr i ces to changes in i n t r i n s i c values w i l l tend to be removed. With regard to the above d i scus s ion , i t is i m p l i c i t l y assumed that actual p r i ces w i l l adjust almost instantaneously to the f u l l e f f e c t s of new information, provided there are a s u f f i c i e n t number of astute traders in the market. However, as recognized by Fama, " instantaneous adjustment" r e a l l y bears two s i g n i f i c a n t impl icat ions because there i s , in f a c t , ambiguity regarding new information. F i r s t , actual pr i ces w i l l i n i t i a l l y overadjust to the new i n t r i n s i c values as often as they w i l l underadjust. Second, the lag in the complete adjustment of actual pr ices to successive new i n t r i n s i c values w i l l i t s e l f be an independent random va r i ab le , sometimes preceding the new information which is the basis of change ( i . e . , when the information is ant i c ipa ted by the market before it a c tua l l y appears) and sometimes fo l lowing.5 In e f f e c t then, the actual p r i ce of a secur i ty w i l l f l uc tuate randomly about i t s i n t r i n s i c va lue. Although i t has been demonstrated that the stock market may conform to the independence assumption of the random walk model, even i f systematic dependencies are present, the question of whether or not th i s ra t iona le is an accurate portrayal of r e a l i t y remains. Its v a l i d i t y is contingent upon the existence of several soph i s t icated traders who are well informed and well endowed. If one uses the c r i t e r i o n of cons is tent gains in the market to detect these soph i s t i cated traders he w i l l probably f i nd that very few, i f any e x i s t . For example, in a study on mutual fund performance, Fama discovered that funds in general seem to do no better than the market. Another s i g n i f i c a n t resu l t of the inves t i ga t ion was that indiv idual funds do not seem to outperform con-s i s t e n t l y the i r competitors. With such evidence in mind, one must conclude that th i s complex e f f o r t to j u s t i f y the independence assumption suf fers from the same weakness as the simple ra t iona le proposed by Bachel ier and Osborne, that i s , i t is not substantiated by f a c t . Since most a r t i c l e s on the theory of random walks have been wr i t ten by and for academics, the investment community has encountered d i f f i c u l t y in assessing i t s r ami f i ca t i ons , two of which require a t t e n t i o n . The random walk theory does not imply that superior invest -ment performance is impossible. In a dynamic economy there w i l l always be new information to e f f e c t changes in i n t r i n s i c values over time. Consequently, the opportunity for above-average p r o f i t s w i l l forever be ava i l ab l e to those who can cons i s ten t l y pred ic t the appearance of new information and evaluate i t s e f f e c t s on i n t r i n s i c values better than others . The fact that the a c t i v i t i e s of these superior i n t r i n s i c - v a l u e analysts (fundamentalists) serve to produce independence in successive p r i ce changes does not imply that the i r expected p r o f i t s cannot exceed those of the investor who pursues some buy-and-hold p o l i c y . What the theory does suggest is that superior performance is extremely d i f f i c u l t to a t t a i n for the average investor. Under the random walk theory pr i ces are, on the basis of a l l a va i l ab le information, the best estimates of i n t r i n s i c va lues. Thus fundamental ana lys i s is a useless tool to the a v e r a g e a n a l y s t whose o n l y a i d c o n s i s t s o f p u b l i c l y d i s t r i b u t e d i nformat i o n . As p r e v i o u s l y m e n t i o n e d , the b a s i c c h a l l e n g e o f the t h e o r y i s t o the t e c h n i c i a n s . A l t h o u g h h i s a c t i v i t i e s may h e l p t o make s u c c e s s i v e p r i c e i n c r e m e n t s independent) once independence i s e s t a b l i s h e d h i s chance f o r e x c e s s p r o f i t i s l o s t . The p a s t h i s t o r y o f the p r i c e s e r i e s w i l l no l o n g e r r e v e a l any de p e n d e n c i e s and as a r e s u l t cannot be used t o i n c r e a s e e x p e c t e d p r o f i t s . The second p o i n t w h i c h s h o u l d be made c l e a r i s t h a t the random w a l k t h e o r y i s not i n c o n s i s t e n t w i t h a r i s i n g t r e n d o f s t o c k p r i c e s . The independence h y p o t h e s i s i s c o n c e r n e d o n l y w i t h the sequence o f s u c c e s s i v e p r i c e changes and not t h e i r a v e r a g e magnitude. "Whereas the random w a l k h y p o t h e s i s does not imply t h a t any p r i c e change i s as l i k e l y t o be a f a l l as a r i s e , i t i s a p p r o x i m a t e l y t r u e t h a t any change i s as l i k e l y t o be below the s t o c k ' s a v e r a g e change as above." Most e m p i r i c a l i n v e s t i g a t i o n s o f t h e random w a l k t h e o r y have been o f a s t a t i s t i c a l n a t u r e i n v o l v i n g t e s t s o f t h e s e r i e s o f p r i c e s o v e r t i m e . For example, i n t e s t i n g t he v a l i d i t y o f the independence a s s u m p t i o n f r e q u e n t use has been made o f the s e r i a l c o r r e l a t i o n model which measures s t a t i s t i c a l l y t he c l o s e n e s s o f the r e l a t i o n s h i p between s u c c e s s i v e p r i c e changes. An a l t e r n a t i v e t e c h n i q u e t h a t has proved R i c h a r d A. B r e a l e y , An I n t r o d u c t i o n t o R i s k and R e t u r n from  Common S t o c k s (Cambridge, M a s s a c h u s e t t s : The M. I.T. P r e s s , 1969), p. 18. equal ly popular is a " runs " ana lys i s which assigns an equal weight to each p r i ce change. A " r u n " is def ined as a sequence of p r i ce changes of the same s ign. Since there are three d i f f e r e n t poss ib le types of p r i ce changes (+, 0, - ) , there are three d i f f e r e n t types of runs. If there is a tendency for runs to p e r s i s t , that i s , for a move in one d i r e c t i o n to be succeeded by a further such move, the average length of run w i l l be longer and the tota l number of runs w i l l be less than i f the moves were d i s t r i b u t e d randomly.^ S t a t i s t i c a l tests employing these techniques and others concerned with invest i gat ing probab i1 i t y d i s t r i b u t ions have la rge ly supported the hypothesis that stock pr i ces are independently d i s -t r ibuted random va r i ab le s . Since s t a t i s t i c a l analyses do not d i r e c t l y test the p r i n c i p l e s of technica l ana ly s i s , i t is d i f f i c u l t to infer that the random walk model is adequate for the investor. Moreover, the c h a r t i s t would probably argue that common s t a t i s t i c a l tools are i n s u f f i c i e n t l y powerful to detect the dependencies seen by him in the s e r i e s . The grounds for such a contention are indeed present. For example, the se r i a l c o r r e l a t i o n model is able to ident i f y simple l inear re l a t i onsh ip s only and not non-l inear ones. A runs te s t , on the other hand, is too r i g i d in determining the durat ion of upward and downward movements in p r i ce s , for i t ignores the magnitude of the p r i ce change that causes the reversal in s i gn. The technic ian would require a more soph i s t i ca ted method to ident i f y trends - - a method that does not always pred ic t the termination of the movement 8 simply because the p r i ce level has temporari ly changed d i r e c t i o n . The above problems have been solved by the second method of empir ica l te s t ing involv ing the use of mechanical trading ru l e s . If the market is character ized by independence of successive p r i ce changes, mechanical t rading ru les should not be able to produce p r o f i t s in excess of those obtained under a buy-and-hold p o l i c y . The evidence from th i s type of research has been far from conc lus i ve . Studies in which trading ru les are seemingly " p r o f i t a b l e " often contain ce r t a in biases and ignore t ransact ion cos t s . In the fo l lowing chapter on f i l t e r rules such er ror s w i l l be out l ined in d e t a i l . Eugene F. Fama and Marshall E. Blume, " F i l t e r Rules and Stock Market T r a d i n g , " Journal of Business: Special Supplement, 39 (January, 1966), p. 227. CHAPTER I I FILTER RULES One of the f i r s t empir ica l invest igat ions involv ing the use of a mechanical t rading ru le is the work of Professor Sidney S. Alexander. In h is study, "P r i ce Movements in Speculat ive Markets: Trends or Random Walks'1,-^ a trading rule c a l l ed the f i l t e r technique is employed to i dent i f y movements in stock p r i c e s . Alexander formulated th i s dec i s ion ru le to v e r i f y the market p ro fe s s i ona l s ' b e l i e f that noise may be generated in a dependent fashion causing pr ices to adjust gradual ly to new information. He, therefore, t en ta t i ve l y assumed the existence of trends but bel ieved them to be disguised by the j i g g l i n g of the market. 'The path of a speculat ive p r i ce might, accord ing ly , be repre-sented by a sum of two components, a smooth underlying trend or cyc le changing d i r e c t i o n only in frequent ly , and a much shorter cyc le of act ion and r e a c t i o n . " ^ (Reaction in th i s instance is associated with p r o f i t taking.) To test th i s hypothesis he suggested f i l t e r i n g out a l l movements smaller than a spec i f i ed s ize and examining the remaining movements. ^Sidney S. Alexander, " P r i ce Movements in Speculat ive Markets: Trends or Random Walks," The Random Character of Stock Market P r i ces , ed. Paul H. Cootner (Cambridge, Massachusetts: The M. l .T . Press, 1964), pp. 199-218. An x per cent f i l t e r can be def ined as fo l lows: Open a long pos i t i on i f the d a i l y c lo s ing p r i ce of a pa r t i cu l a r secur i t y r i ses at least x per cent, and maintain th i s pos i t i on un t i l i t s p r i ce f a l l s at least x per cent from a subsequent high, at which time s e l l and go short an equivalent amount. The short pos i t i on is held un t i l the da i l y c lo s ing p r i ce r i ses a minimum of x per cent above a subsequent low at which time one covers and buys. Moves less than x per cent in e i ther d i r e c t i o n are ignored. A trend, therefore, once es tab l i shed is considered to be in e f f e c t un t i l a p r i ce change causes a reversal in d i r e c t i o n by some predetermined amount. As noted by Alexander, the choice of a f i l t e r s i ze involves a t radeof f between r i s k and expected returns: Thus, as the f i l t e r s i ze is increased, the number of t ransact ions is reduced, and losses on small moves are e l im inated, gains on large moves are reduced, and some moves which would y i e l d gains with a small f i l t e r w i l l y i e l d losses with a large. This example i l l u s t r a t e s the f am i l i a r t radeoff between r e l i a b i l i t y of the i n f o r -mation and the cost of the information. The more s tr ingent the f i l t e r , the higher the r e l i a b i l i t y , but the more of the move that is s a c r i f i c e d in i dent i f y ing i t both in gett ing in and in gett ing out. In the study, f i l t e r s ranging in s i ze from 5 to 50 per cent were tested on the data. The sample consisted of c lo s ing " p r i c e s " for two indexes, the Dow-Jones Industr ia l s from 1897 to 1929 and Standard and Poor 's Industr ia l s from 1929 to 1959. Genera l ly , f i l t e r s of a l l d i f f e r e n t s izes and for a l l d i f f e r e n t time periods y ie lded superior p r o f i t s t o t h o s e o f a buy-and-hold p o l i c y . (The l a r g e s t p r o f i t s were a s s o c i a t e d w i t h the s m a l l e s t f i l t e r s . ) From t h e s e r e s u l t s A l e x a n d e r c o n c l u d e d t h a t s t o c k p r i c e changes c o u l d not have been g e n e r a t e d by a random w a l k . I t was l a t e r r e c o g n i z e d by B e n o i t M a n d e l b r o t t h a t the e s t i m a t e d p r o f i t s from the use o f the f i l t e r s were s u b j e c t t o a b i a s e x a g g e r a t i n g the p r o f i t a b i l i t y . In each t r a n s a c t i o n A l e x a n d e r assumed t h a t h i s h y p o t h e t i c a l t r a d e r c o u l d a l w a y s buy or s e l l a t a t r o u g h or peak t x per c e n t . I t i s a p p a r e n t t h a t a t r a d e r o p e r a t i n g i n the market p l a c e c o u l d not consummate h i s t r a n s a c t i o n s a t t h e s e p r i c e s due t o the f r e q u e n c y o f l a r g e p r i c e jumps ( c h a n g e s ) . As a r e s u l t the p u r c h a s e p r i c e w i l l o f t e n exceed the low p l u s x per c e n t , w h i l e the s a l e p r i c e w i l l o f t e n be 1 3 below the h i g h minus x per c e n t . A f t e r c o r r e c t i o n f o r the b i a s , i n a f o l l o w i n g s t u d y e n t i t l e d , " P r i c e Movements i n S p e c u l a t i v e Markets:'' 14 T r e n d s or Random Walks, No. 2", A l e x a n d e r found t h e p r o f i t a b i l i t y o f t h e f i l t e r t e c h n i q u e s u b s t a n t i a l l y r e d u c e d . More s p e c i f i c a l l y , the f i l t e r s o n l y r a r e l y compared f a v o r a b l y w i t h the buy-and-hold model. No a l l o w a n c e was made f o r t r a n s a c t i o n c o s t s i n the c o m p u t a t i o n o f p r o f i t s . 1 3 To e l i m i n a t e t h e b i a s , A l e x a n d e r assumed t h a t t r a n s a c t i o n s were made a t the c l o s i n g p r i c e o f the c o n f i r m a t i o n day r a t h e r than a t x per c e n t away from t r o u g h or peak. 14 S i d n e y S. A l e x a n d e r , " P r i c e Movements i n S p e c u l a t i v e M a r k e t s : Trends or Random Walks , No. 2," The Random C h a r a c t e r o f S t o c k Market  Pr i c e s , e d . Paul H. C o otner (Cambridge, M a s s a c h u s e t t s : The M.l.T. P r e s s , 1964), pp. 338-372. In add i t ion to reworking his e a r l i e r r e su l t s , Alexander modified the f i l t e r ru le s l i g h t l y and tested i t on a d i f f e r e n t sample. On th i s occasion he appl ied logar ithmic f i l t e r s to the d a i l y c l o s ing pr ices of Standard and Poor 's Industr ia l s for the 9,592 trading days from January 3, 1928 to December 29, 1961, i nc lu s i ve . Ten separate f i l t e r s were used with the smallest being 1 per cent and the la rges t , 45 .6 per cent. Excluding commissions nearly a l l of the f i l t e r s produced p r o f i t s above those of buy-and-hold. The 1 per cent f i l t e r proved to be the most p r o f i t a b l e , with terminal cap i ta l forty-one times as large as the i n i t i a l c a p i t a l . Buy-and-hold y ie lded a terminal cap i ta l about f i ve times the i n i t i a l c a p i t a l . When transact ion costs were included, only the largest f i l t e r beat buy-and-hold. A second bias present in both of A lexander ' s works was pointed out by Eugene F. Fama and Marshall E. Bl ume in the i r paper, " F i l t e r 15 Rules and Stock-Market T rad ing " . Since Alexander tested the f i l t e r s on common pr i ce indices i t was impossible to include the e f f e c t of dividends in the computation of returns . This shortcoming reduced the p r o f i t a b i l i t y of the buy-and-hold po l i cy as the tota l return on an investment cons i s t s of the cap i t a l apprec iat ion for the time period plus any dividends that have been pa id . Furthermore, it overstated the p r o f i t a b i l i t y of the f i l t e r s by f a i l i n g to take account of the fact that Eugene F. Fama and Marshall E. Blume, " F i l t e r Rules and Stock Market T r a d i n g , " Journal of Business: Special Supplement, 39 (January, 1966), pp. 226-241. in a short sale the borrower of the s e c u r i t i e s t y p i c a l l y reimburses the lender for any dividends paid while the short pos i t i on is outstanding. Fama and Blume appl ied twenty-four d i f f e r e n t percentage f i l t e r s ranging from 0.5 per cent to 50 per cent to ser ies of d a i l y c lo s ing p r i ces for each of the indiv idual s e c u r i t i e s of the Dow-Jones Industr ia l Average. The i n i t i a l dates of the samples var ied , but most covered the period from 1957 to 1962. Returns under the f i l t e r technique were computed in several d i f f e r e n t ways: gross and net of brokerage fees, with and without d iv idends, e t c . They analyzed the resu l t s f i r s t by secur i t y (the nominal annual rates of return by company: averaged over a l l f i l t e r s ) and then by f i l t e r s (the nominal annual rates of return by f i l t e r : averaged over a l l companies). When commissions were included none of the t h i r t y s e c u r i t i e s exh ib i ted returns in excess of those obtained under the buy-and-hold p o l i c y . In f a c t , only four s e c u r i t i e s had po s i t i ve returns . A compari-son of the p r o f i t s before commissions under the two s t ra teg ies indicated that the f i l t e r technique was i n f e r i o r to buy-and-hold for a l l but two s e c u r i t i e s . The discrepancy between th i s last resu l t and the f ind ings of Alexander when he employed logar i thmic f i l t e r s is explained by the bias re su l t i ng from h is use of p r i ce ind ices . Fama and Blume found that the adjustment for dividends increased the average advantage of buy-and-hold over the f i l t e r technique by at least two percentage po in t s . Had such an adjustment been appl ied to Alexander ' s data, the p r o f i t a b i l i t y of h is f i l t e r s would probably have decreased markedly. Further evidence negating the value of the f i l t e r technique (in the Fama and Blume study) was provided by a breakdown of gross returns for long and short t ransact ions . Only one secur i ty had po s i t i ve returns per f i l t e r for short pos i t ions i n i t i a t e d by the r u l e . For a l l s e c u r i t i e s , the average return on short t ransact ions was - 1 2 . 7 9 per cent whi le the average return from buy-and-hold was 9.86 per cent, a spread of 22.65 per cent. On long pos i t ions th i r teen s e c u r i t i e s demonstrated higher average returns per f i l t e r than the corresponding returns from buy-and-hold. However, averaging o v e r - a l l s e c u r i t i e s , the return on these transact ions a l so f a i l e d to surpass the average return from buy-and-hold, the d i f f e rence in th i s case being 1,6k per cent. The ana lys i s of resu l t s by f i l t e r d i sc losed that v i r t u a l l y every f i l t e r produced net returns below those of buy-and-hold. Th i s , together with the preceding observat ions, supported the conclus ion that the f i l t e r technique could not be used to increase the expected p r o f i t s of the investor who is charged regular brokerage commissions. The breakdown of gross returns (by f i l t e r ) for long and short t ransact ions unvei led very s l i gh t amounts of dependence in the pr i ce changes. Fama and Blume appl ied the fo l lowing c r i t e r i o n to detect dependence: Note that i f successive p r i ce changes conformed s t r i c t l y to the random-walk model, the average returns per secur i ty on long pos i t ions should be approximately equal to the average returns from buy-and-hold whi le the average returns on short pos i t ions should be approximately equal ^ to the negative of the average returns from buy-and-hold. For three f i l t e r s i ze s , 0.5;, 1 .0 and 1.5 per cent, the average returns per secur i ty on long pos i t ions exceeded the average return from buy-and-ho ld . In add i t ion the same f i l t e r s izes showed losses on short pos i t ions of a lesser magnitude than the gains from buy-and-hold. This behavior of returns on the smallest f i l t e r s const i tuted evidence of po s i t i ve dependence in very small movements of stock p r i c e s . When the average returns per secur i ty on long and short pos i t ions for f i l t e r s greater than 1.5 per cent and less than 12 per cent were compared to the average return from buy-and-hold, proof of negative dependence in intermediate s i ze p r i ce movements was a l so furn i shed. The dependencies uncovered by the two authors were used to suggest trading procedures for a f l oo r t rader . Although the ru les pro-duced greater gross returns than those of buy-and-hold, they could not be used to increase expected p r o f i t s . The c lear ing-house fees and costs of search were more than s u f f i c i e n t to erase any advantage of the f i l t e r ru les over buy-and-hold. Thus Fama and Blume's re su l t s added further to the evidence that for p r ac t i c a l purposes the random walk model is an adequate desc r ip t i on of p r i ce behavior. In a re la ted study e n t i t l e d , "Random vs. Systematic Changes " ,^ Paul H . Cootner used a va r i a t i on of the f i l t e r technique to test h i s hypothesis that stock pr ices behave l i k e a r e s t r i c t e d random walk. The '7paul H . Cootner, "Stock P r i ce s : Random vs. Systematic Changes", The Random Character of Stock Market P r i ce s , ed. Paul H . Cootner (Cambridge, Massachusetts: The M. I.T. Press, 1964), pp. 231-252. r a t i o n a l e 1 0 for th i s supposit ion perta ins to the a c t i v i t i e s of two groups of investors—amateurs and p ro fe s s i ona l s . Since amateurs are engaged in other types of employment in which they possess a comparative advantage, i t is very co s t l y for them (at least in terms of opportunity cost per unit of valuable information uncovered) to devote time to the relevant kind of stock market research. As a r e s u l t , they tend to regard present pr ices as roughly representing true d i f fe rences in value, and choose between s e c u r i t i e s on the basis of the i r l i q u i d i t y requ i re -ments and r i s k preferences. Those who do u t i l i z e information about future prospects as a c r i t e r i o n for se lec t ion among stocks are jus t as l i k e l y to be incorrect as not in the i r forecasts of p r i c e s . Thus, the a c t i v i t i e s of amateur investors tend to induce randomness in a pr ice s e r i e s . Because profess iona l s are more f ami l i a r with analyzing s e c u r i t i e s and possess access to p r i v i l e ged information, the i r oppor-tun i ty cost of research is minimal. They attempt to pred ic t events in the future, but p r o f i t s cannot be gained through the use of such ins ight unless the current pr i ce diverges from the expected p r i ce by an amount in excess of the i r opportunity cos t s . Their p r o f i t s w i l l come from observing the random walk of stock market pr ices produced by amateurs and recognizing the s i t ua t i on where the pr i ce has wandered s u f f i c i e n t l y far from the expected p r i ce to warrant the prospect of an adequate re turn . Competition among these profess iona l s w i l l tend to l im i t the potent ia l p r o f i t to opportunity cos t s . If i t is assumed that a l l profess iona l s have ident ica l expecta-t ions and the same opportunity costs , then pr ices should behave as a random walk with r e f l e c t i n g bar r ie r s (Figure 1). As long as pr i ces are with in the upper and lower boundaries, they are determined by the act ions of amateurs and therefore tend to move l i ke a random walk. However, i f p r i ces reach e i ther l i m i t , the a c t i v i t i e s of profess iona l s w i l l prevent them from cont inuing in the same d i r e c t i o n . In such an undertaking a measure of r i s k is incurred by these pa r t i c i pan t s , for they cannot be ce r t a in that the i r estimates of values are correct or that other pro fess iona l s share the i r est imates. Moreover, even i f the i r expecta-t ions are co r rec t , t he i r rate of return on investment is s t i l l a s tochas t i c var i ab le s ince the rate at which the pr i ce converges upon the expected p r i ce is subject to the random process operat ing between the b a r r i e r s . Another sort of random walk environment is l i a b l e to occur because of random changes in the pr i ce expectat ions of p ro fe s s i ona l s . This could be caused by independence in the process generating new information. Given such a phenomenon there probably would a l so be random changes in the trends around which the random walk takes p lace. That i s , the path of stock pr ices over any substant ia l period of time would be composed of a random number of trends, each of which is a random walk with r e f l e c t i n g ba r r i e r s (Figure 2). Two imp l i c i t assumptions were made by Cootner: - - tha t dependence in the noise generation process e x i s t s , and, that a lag in the information d i s t r i b u t i o n process causes pr ices to adjust gradual ly to new information. In an attempt to p r o f i t from the a l leged s h i f t s in the whole trading range (major p r i ce movements i n i t i a t e d by the changes in expectations of p ro fe s s i ona l s ) , he employed a modi f icat ion of the f i l t e r technique whereby a fo r ty week moving average is subst i tuted for reference peaks and troughs. The rule is defined accord ing ly : If the current p r i ce is higher than the moving average, buy and then hold the stock un t i l the p r i ce f a l l s below the moving average, at which time, s e l l . If the current pr i ce is less than the moving average, s e l l short, and cover when the pr i ce r i ses above the moving average. Two advantages of th i s strategy over the f i l t e r procedure are: F i r s t , i t would enable a fo l lower to s e l l (buy) a stock when it stopped r i s i n g ( f a l l i n g ) along the prev ious ly defined trend, rather than wait ing for a substant ia l r e v e r s a l . Second, i t would permit an investor the a l t e r -nat ive of holding cash rather than adopting a pos i t i on in e i ther d i r e c t i o n - - a s the f i l t e r ru le (though not Houthakker's s top- loss ) r e q u i r e s . ^ The ru le was appl ied to weekly observations of f o r t y - f i v e stocks l i s t e d on the New York Stock Exchange which, except for s ix s e r i e s , spanned the interva l from 1956 to i 9 6 0 . When commissions were excluded, the moving average strategy proved to be far superior to buying and ho ld ing. While th i s was ind i ca t i ve of non-randomness, the extent of the non-randomness could not be used to increase the expected p r o f i t s of the investor; a f te r allowance for commissions, the moving average strategy y ie lded i n f e r i o r returns. Ib id. , p. 245. To reduce the excessive number of transactions and hence mitigate the effect of brokerage fees on returns, Cootner modified his decision rule to allow for transactions only when the moving average and the current price differed by more than a specified percentage. This adjustment increased the probability of participation in a shift in 20 trend instead of merely a movement between barriers. Under the new strategy the stock was to be purchased only when the price rose above the moving average by more than 5 per cent and would be sold whenever the price f e l l below the moving average by any amount: short sales would only be undertaken when the moving average rose above the price by more than 5 per cent but would be covered whenever the price moved above the moving average by any amount. The implementation of this rule provided a larger gross gain than the buy-and-hold model but the net gain was s t i l l smaller. In terms of average net weekly gain, however, variations of the moving average strategy (long positions only) outperformed simple investment. This is attributed to the fact that the investor under the 5 per cent threshold rule is free to devote his financial resources to other uses whenever the stock price shows no particular trend. Since the findings of Alexander, Fama and Blume, and Cootner are in basic accord with one another after certain biases have been eliminated, a relatively strong case has been built for the conclusion that the f i l t e r technique and moving average strategy cannot be used to uHowever, as with Alexander's f i l t e r technique, a tradeoff was involved since a larger part of the move was sacrificed for this greater r e l i a b i l i t y . increase the expected p r o f i t s of the typ i ca l investor. A l l of the invest igat ions revealed that the ru les generated i n f e r i o r net returns to those of a buy-and-hold p o l i c y . Nevertheless, because the tests have not exhausted every p o s s i b i l i t y , i t is s t i l l conceivable that these mechanical trading rules could produce above-average p r o f i t s . An approach which looks promising w i l l be presented in the next chapter. CHAPTER I I I THE MOVING AVERAGE STRATEGY APPLIED TO HIGHLY VOLATILE SECURITIES I. Selected Techniques for Improving the Performance  of the F i l t e r Rule and Moving Average Strategy The p r o f i t a b i l i t y of both the f i l t e r technique and moving average strategy is determined by the s ize of the f i l t e r (threshold) and the magnitude of the swings in pr i ces which e s tab l i sh the peaks and troughs. A better perspect ive of th i s is gained when the path traced by secur i ty pr i ces is viewed with in the fo l lowing context. As suggested by 21 Seymour Smidt, a wave- l ike motion around the long-term upward trend could account for the symmetric pattern in the stock pr i ce sequence implied by the f i l t e r resu l t s of the Fama and Blume paper. For almost every f i l t e r s i ze the actual returns earned by long and short pos i t ions are d i sp laced from the i r expected returns by approximately the same absolute magnitude. The fact that the number of t ransact ions decreases as the f i l t e r s i ze increases suggests a mixture of waves of d i f f e r e n t amplitudes. Where the amplitude of the wave is less than the f i l t e r s i z e , t ransact ions do not take place for such waves are f i l t e r e d out, and excluded from cons idera t ion : ^'Seymour Smidt, "A New Look at the Random Walk Hypothes i s , " Journal of F inanc ia l and Quant i tat ive Ana ly s i s , 3 (September, 1968), pp. 235-261. Suppose that the trough of a wave occurs when the stock is pr iced at 100 d o l l a r s per share and the peak when the stock r i ses to 100 + A d o l l a r s . The amplitude of the wave is A per cent. With an I per cent f i l t e r (assuming A > l ) , a buy s ignal w i l l be given when the pr i ce reaches 100 + I d o l l a r s . A s e l l s ignal w i l l occur when the p r i ce passes the peak and dec l ines to 100 + A - I d o l l a r s . Thus the p r o f i t s on th i s long open pos i t i on w i l l be A - 21 d o l l a r s or A - 2 I per cent before commissions. In general , the trading ru le w i l l produce gains ( losses) i f the average amplitude of waves that exceed the f i l t e r s i ze is greater than ( less than) twice the f i l t e r s i z e . 2 2 A s im i l a r but not ident ica l mathematical re l a t i onsh ip holds for the moving average s trategy. The s l i gh t d i f f e rence a r i se s because the buy and s e l l s ignals for th i s ru le are interpreted r e l a t i v e to the trend defined by the moving average. In the preceding d i scuss ion the term 'wave' has been used in a metaphorical sense: It is not suggested that i f the pr i ce ser ies were p lot ted that one would necessar i l y be able to observe a per s i s tent wave- l ike pa t tern . The waves must be defined s t a t i s t i c a l l y in terms of the condi t iona l pro-b a b i l i t i e s of p r i ce changes given that the p r i ce is already in a ce r t a in re l a t i on to the previous peak or trough.23 It is apparent that the necessary condi t ions for the maximization of returns are: the existence of a high degree of po s i t i ve dependence in movements of stock pr ices and the f l uc tua t i on of pr ices over a wide range. Since the v o l a t i l i t y aspect of secur i ty pr ices has been over-looked, an opportunity for improved performance of both trading ru les ^ j b i d . , p. 248. 2 3 ! b i d . , p. 249. remains unexplo i ted. |n the aforementioned studies and a l l others of a re la ted nature known by the author, no overt attempt is made to r e s t r i c t the sample of those s e c u r i t i e s which are l i k e l y to be h ighly v o l a t i l e . The cor rec t ion for such a factor provides the foundation for th i s t he s i s . Whether or not an empir ica l test of the subject matter under cons iderat ion is warranted depends of course on the extent to which the p r i ce va r i a t i on of s e c u r i t i e s can be p red i c ted . In th i s regard i t has been suggested that the r e l a t i v e v o l a t i l i t y exh ib i ted by any stock may be cons i s tent over successive in terva l s of time. As submitted by 24 Richard A. Brealey such a phenomena could resu l t from a d i r e c t re l a t i onsh ip between uncerta inty and pr i ce v a r i a b i l i t y , wherein the causes of uncerta inty tend to per s i s t over t ime. The margin for er ror in forecas t ing company prospects is l i a b l e to be greater i f e i ther the range of poss ib le outcomes is wide or there is l i t t l e information on which to base a fo recas t . The former condi t ion w i l l a r i s e when the concern i s , in the broadest sense, h igh ly leveraged, the l a t t e r condi t ion when e i ther the business is very i n d i v i d u a l i s t i c in character or management is very secret ive about operat ions. It seems improbable that these c h a r a c t e r i s t i c s are t y p i c a l l y t r a n s i t o r y . For example, most meta l - re f in ing companies are l i k e l y to continue to possess high operating leverage, advanced-technology businesses should continue to be very i n d i v i d u a l i s t i c , and companies working on c l a s s i f i e d contracts should continue to be s e c r e t i v e . If th i s reasoning is correct and the causes of uncerta inty do per s i s t over time, i t is probable that stocks that are most v o l a t i l e in one period w i l l tend to be the most v o l a t i l e in the next.25 ^ R i c h a r d A. Brealey, An Introduction to Risk and Return from  Common Stocks (Cambridge, Massachusetts: The M. l .T . Press, 1969) . Brealey found empir ica l evidence supporting his hypothesis in 26 an unpublished doctoral d i s s e r t a t i on by Shannon P. Pratt who i n v e s t i -gated the subsequent behavior of 3^8 sets of f i v e p o r t f o l i o s of stocks se lected on the basis of p r io r v o l a t i l i t y . In each set, p o r t f o l i o A cons isted of the 20 per cent of the New York Stock Exchange stocks that d isp layed the least v a r i a t i on in returns over the previous three years, whi le p o r t f o l i o B consisted of the next 20 per cent, e t c . The average subsequent experience (in index form) of the 348 sets revealed that the ranking of the f i v e p o r t f o l i o s did not change over time. That i s , p o r t f o l i o A continued to demonstrate the least v a r i a t i on in returns, whi le p o r t f o l i o s B, C, D and E continued to exh ib i t higher leve l s of v a r i a t i on re spec t i ve l y . S t a t i s t i c s regarding the p r o b a b i l i t y of subse-quent loss on the p o r t f o l i o s a l so favoured Brea ley ' s hypothesis . If, on the average, p o r t f o l i o A tended to show less v i o len t changes in value than p o r t f o l i o E, i t is probable that over any one period the former would have been less l i k e l y to suf fer an actual los s . As expected, those p o r t f o l i o s composed of stocks that had been less var iab le in former years resu l ted less f requent ly in lo s s . In i t s e n t i r e t y the study was concerned with analyzing the behavior of approximately 1,000 stocks over the twenty-nine year period from January 1929 to December 1957. When only the p o r t f o l i o s formed a f te r 1931 were considered, the observa-t ions were s i m i l a r . Changing the lengths of the periods over which the ^"Shannon P. P r a t t , Re lat ionship Between Risk and Rate of Return  for Common Stocks (unpublished D.B.A. d i s s e r t a t i o n , Indiana Un iver s i t y , 1966) . v o l a t i l i t y was measured did not a f f ec t the conclusions e i t h e r . Although P r a t t ' s resu l t s provide substant ia l evidence that the r e l a t i v e v o l a t i l i t y exh ib i ted by any stock has tended to per s i s t over time, further conf irmation seems appropriate s ince such a c h a r a c t e r i s t i c may have only held true for the p a r t i c u l a r sample and methodology chosen. In deference to th i s cons iderat ion tests of a somewhat d i f f e r e n t nature have been undertaken here to determine the relevance of h i s t o r i c a l v o l a t i l i t y as a basis for forecas t ing future pr i ce v a r i a t i o n . It w i l l l a ter be shown that the past v o l a t i l i t y of a stock does in fact furnish a useful ind ica t ion of i t s future behavior. While co r rec t i on for the v o l a t i l i t y aspect may improve the per-formance of both trading rules so a l so might the use of monthly data. For th i s to occur i t is e s sent ia l that the bulk of relevant information, guiding a speculator to p r o f i t , d i f f u se throughout the market place over a lengthy interva l of time. Such a prospect appears l i k e l y i f one considers the time element involved in the d i s t r i b u t i o n of research reports by profess ional ana lys t s . Pr ior to dispensat ion the analyst must seek the approval of h is investment committee, which general ly meets once a week. If i t is assumed that the large i n s t i t u t i o n a l investors (e.g. mutual funds, l i f e insurance companies, e tc . ) are the f i r s t to be n o t i f i e d , then another week w i l l have passed before any act ion is taken on the recommendation as the i r po l i cy committees a l so convene on a weekly bas i s . By the time the information is released to the amateurs (odd-lot investors) through the branch o f f i c e s of brokerage f i rms, and is acted upon, four to s ix weeks could have elapsed s ince the report was i n i t i a l l y submitted by the ana lys t . If th i s type of dependency e x i s t s , then a gradual adjustment of pr ices over a monthly time horizon is a d i s t i n c t p o s s i b i l i t y . The tes t ing of such a hypothesis adds yet another dimension to th i s t h e s i s . I I. The Data The basic data f i l e u t i l i z e d in th i s ana lys i s was compiled by the Center for Research in Secur i ty Pr ices at the Un iver s i t y of Chicago. It was made ava i l ab le for use in the form of a magnetic tape at the Un ivers i ty of B r i t i s h Columbia Computer Center. Included in the f i l e are 27 month-end pr ices (adjusted for stock s p l i t s and stock dividends ) of a l l s e c u r i t i e s l i s t e d on the New York Stock Exchange over the period January, 1926 to June, 1966. 28 From the en t i r e population of 1,952 secu r i t i e s a sample of 200 was drawn. A l l of the se lected ser ies covered the three successive i n te rva l s : January, 1936 to December, 19^ +5 (period |); January, 1946 to December, 1955 (period ||); January, 1956 to January, 1966 (period M l ) . The data over the periods 1936-1945 and 1946-1955 were examined in Test No. 1 and Test No. 2 to detect any tendency for the r e l a t i v e v o l a t i l i t y of a common share to remain constant. Observations over the 1956-1966 horizon were employed in Test No. 3 to simulate a trading r u l e . ^ D a t a a f te r March, I966 are unadjusted for cap i ta l changes. 2^A11 of the empir ica l r e su l t s , inc luding a l i s t i n g of the sample, are ava i l ab le in the computer.statements for th i s study. A copy can be obtained from Dr. Wi l l iam Wood, Department of Finance, Faculty of Commerce and Business Admin i s t ra t ion , Un iver s i t y of B r i t i s h Columbia. The c o e f f i c i e n t of va r i a t i on was used as a measure of p r i ce v a r i a b i l i t y . It is def ined as the standard dev iat ion d iv ided by the mean (OT/x) and as such, expresses the magnitude of the d i spers ion r e l a t i v e to the quant ity being gauged. The fact that th i s s t a t i s t i c furnishes a basis for comparing d i f f e r e n t frequency d i s t r i b u t i o n s permitted the ranking of stocks according to v o l a t i l i t y . III. Test No. 1; The Experimental  Procedure and the Results The v o l a t i l i t y of each of the 200 s e c u r i t i e s was computed for both period I and period ||. In order to test the strength of the r e l a -t ionsh ip between these paired observat ions, a rank c o r r e l a t i o n ana lys i s was performed. The resu l t s are presented below: Spearman Rank Test of Number of Cor re l a t i on C o e f f i c i e n t S i gn i f icance Degrees of Freedom .309 4.57 198 As revealed by the degree of c o r r e l a t i o n (.309) s i g n i f i c a n t at the .0010 l e v e l , the r e l a t i v e v o l a t i l i t y of each of the 200 s e c u r i t i e s d i f f e r e d subs tan t i a l l y over the two per iods . However, i t should be noted that th i s f ind ing does not necessar i l y contrad ic t the evidence reported by P ra t t . The discrepancy between resu l t s may be a t t r i bu ted to the more s tr ingent nature of the test used here. While Pratt invest igated the subsequent behavior of f i v e p o r t f o l i o s of stocks se lected on the basis p r io r v o l a t i l i t y , th i s experiment deal t with the p r i ce va r i a t i on of 200 indiv idual s e c u r i t i e s . It is conceivable that a tendency ex i s t s for the r e l a t i v e v o l a t i l i t y of any secur i ty to remain within some d e f i n i t e range as sub-s tant ia ted by P r a t t ' s ana lys i s even though a tenuous re l a t i on sh ip ;iis indicated by the above s t a t i s t i c . For example, the twenty most v o l a t i l e s e c u r i t i e s in period I may have been the twenty most v o l a t i l e in period II. Furthermore, with in th i s group a material s h i f t in rankings may have occurred. If such a phenomenon preva i led throughout other groups of stocks i t could account for the low rank co r re l a t i on c o e f f i c i e n t of .309. In th i s p a r t i c u l a r case the test appl ied is i n s u f f i c i e n t l y powerful to detect any propensity for the r e l a t i v e v o l a t i l i t y of a secur i ty to stay with in s p e c i f i a b l e l i m i t s . To .cor rec t for such a de f i c i ency , a method-ology s imi la r to that employed by Pratt is u t i l i z e d in Test No. 2. IV. Test No. 2: The Experimental  Procedure and the Results On the 'bas i s of t he i r v o l a t i l i t y ( c o e f f i c i e n t of va r i a t ion ) in period I, the 200 s e c u r i t i e s were c l a s s i f i e d into ten groups. P o r t f o l i o No. 1 included the twenty most v o l a t i l e stocks while each of the remain-ing p o r t f o l i o s in turn consisted of another twenty stocks exh ib i t i n g lower leve l s of p r i ce v a r i a t i o n . As be f i t s i t s p o s i t i o n , P o r t f o l i o No. 10 contained the twenty least v o l a t i l e s e c u r i t i e s . For each group an average c o e f f i c i e n t of v a r i a t i on was ca lcu la ted over both period I and period I I while a rank c o r r e l a t i o n ana lys i s was conducted for the purpose of tes t ing the closeness of the re l a t i onsh ip between these ten paired observat ions. The resu l t s are as fo l lows: Spearman Rank Test of Number of Cor re la t ion C o e f f i c i e n t S ign i f icance Degrees of Freedom .903 5.95 8 As indicated by the high degree of co r re l a t i on ( . 9 03 ) , s i g n i f i -cant at the .0010 l e v e l , there was a strong tendency for the demonstrated r e l a t i v e v o l a t i l i t y of each p o r t f o l i o to pe r s i s t over time (see Table I). That i s , as a group the most v o l a t i l e s e c u r i t i e s in period I tended to be the most v o l a t i l e in period I i . Consequently, th i s f ind ing adds further to the evidence (submitted by Pratt) that the r e l a t i v e v o l a t i l i t y of a secur i ty tends to remain with in a f a i r l y narrow range. Since there is no reason for expecting th i s c h a r a c t e r i s t i c to diminish in the future, the past va r i a t i on in pr ices should continue to provide a useful ind icat ion of the future v a r i a t i o n . V. Test No. 3: The Experimental  Procedure and the Results While the performance of both the f i l t e r technique and moving average strategy might be improved by r e s t r i c t i n g the sample to h ighly v o l a t i l e s e c u r i t i e s , only the l a t te r hypothesis has been tested here. The trading ru le employed is defined as fo l lows: the stock is to be purchased when the pr i ce r i ses above the moving average by more than x per cent and is to be sold whenever the pr i ce f a l l s below the moving average by any amount. No allowance was made for short sales because the p o s i -t i ve d r i f t in stock pr ices over the period examined l imi ted the prospects of gain in th i s case. The rule was appl ied to fo r ty ser ies of month-end pr ices extending over the i n t e r v a l , January, 1956 to January, 1966 (period III). These s e c u r i t i e s were se lected because of the i r high r e l a t i v e v o l a t i l i t y in period II. On the basis of the resu l t s in the preceding te s t , i t was predicted that the most v o l a t i l e stocks in period II would be the most v o l a t i l e in period III. Four d i f f e r e n t moving averages ranging from three to s ix months, and th i r teen d i f f e r e n t thresholds ( f i l t e r s ) ranging from 2 to 50 per cent were s imulated. Although the i n i t i a l i z a t i o n date for the ru le var ied according to the length of the moving average, the termination date was always January, 1966. At th i s time a l l outstanding pos i t ions were c losed out. For each t ransact ion which consisted of a purchase and a sale an annualized rate of return, adjusted for round-lot commissions, was ca l cu la ted from the fo l lowing formula: P t + 1 - P t - C 12 Y t - Ft X -Y t = the annualized rate of return Pt+l - the sale p r i ce P t = the purchase p r i ce C - the combined round-lot commission on a purchase and a sale n s the number of months the secur i ty is held These returns were averaged to y i e l d a mean annualized rate of return for each secur i t y and threshold s i z e . A l i s t i n g of the sample is shown in Table II. Corresponding returns for two buy-and-hold s t ra teg ies were com-puted to provide a standard of comparison. The Fisher Investment 30 , Performance Index (for a l l s e c u r i t i e s l i s t e d on the New York Stock Exchange over the span, January, 1926 to June, 1966) was used to simulate one of the s t ra teg ies while a re la ted p r i ce index was employed to simulate the other. Although cash dividends were taken into account in the formulat ion of the f i r s t index, the commissions on the reinvestment of these dividends were omitted. Consequently, the ca l cu la ted average annual rate of return for th i s buy-and-hold model i s , to some degree, overs tated. The resu l t s of the s imulat ion of the trading ru le appear in Tables III, IV, V, and VI for moving averages ranging from three to s ix months re spec t i ve l y . For each threshold an average annual rate of return is shown together with the tota l number of t ransact ions , the average number of t ransact ions per stock, and the number of p r o f i t a b l e s e c u r i t i e s . The corresponding return for the investment performance index and pr i ce index is a l so included. The r e su l t s , without except ion, indicate that the moving average strategy is vas t ly i n f e r i o r to both buy-and-hold models. Rather than producing above-average returns, the trading ru le led to a negative average annual rate of return for every thresho ld. The losses per thresho ld, as revealed by an ana lys i s of the returns for each secur i t y , ^Lawrence F i sher , "Some New Stock-Market Indexes," Journal of  Business: Special Supplement, 39 (January, 1966) , pp. 1 91 -225 . are spread throughout most of the fo r ty stocks. In the majority of instances fewer than 25 per cent of the s e c u r i t i e s (for which there is at least one recorded transact ion) demonstrate any p r o f i t whatsoever under the r u l e . It is usua l ly the same stocks that f a l l into th i s ca te -gory, regardless of the threshold s i ze or moving average length. Of the l imi ted number which exh ib i t a po s i t i ve return, only two s e c u r i t i e s , Texas Instruments and Rexall Drug and Chemical Co., f requent ly show gains in excess of those generated by the investment performance index. As d i sc lo sed by a comparison of the resu l t s of the d i f f e r e n t moving averages, the returns for small thresholds (2 to 10 per cent) are only marginal ly improved by using a longer period to def ine the t rend. Such is not the case, however, for the large thresholds s ince the losses are s i g n i f i c a n t l y reduced by employing the s ix month moving average. With respect to the p r o f i t a b i l i t y of the thresholds a major discrepancy ex i s t s between the d i f f e r e n t sets of r e s u l t s . The large thresholds under the three month moving average show losses far above those of the other thresholds . In contras t , the large thresholds under the s ix month moving average show the smallest losses of a l l thresholds . These con-f l i c t i n g resu l t s suggest that no one pa r t i cu l a r threshold is superior to the others . , A number of explanations can be submitted to account for the poor performance of the moving average strategy in th i s study. To begin with, i t is conceivable that each of the se lected s e c u r i t i e s did not exh ib i t a r e l a t i v e v o l a t i l i t y in period III cons istent with that in period ||. Such a p o s s i b i l i t y , however, appears remote in l i gh t of the evidence reported in the preceding test (and by P r a t t ) . Stocks which were the most v o l a t i l e in period I tended to be the most v o l a t i l e in period II. A more p l aus ib le ra t iona le - -and one which can be inferred d i r e c t l y from the ava i l ab le r e s u l t s — i s that po s i t i ve dependence does not predominate in successive monthly pr i ce changes. In th i s respect the f ind ings support the random walk theory of stock pr i ce behavior. Both Fama/Blume and Cootner found evidence suggestive of po s i t i ve dependence in d a i l y and weekly pr i ce changes re spec t i ve l y , although of an i n s u f f i c i e n t extent to lead to a remunerative f i l t e r and moving average s trategy. That i s , the i r trading rules for ce r t a in f i l t e r s and thresholds were superior to simple buying and holding on a gross p r o f i t basis on ly. Even i f brokerage commissions had been ignored in the inves t i ga t ion here, the dec i s ion ru le , regardless of the s ize of thres -hold or length of moving average, would probably have f a i l e d to outperform the buy-and-hold models. The impl icat ion which a r i se s out of th i s comparison is that the flow of relevant information from profess ional analyst to amateur does e x i s t , but occurs with in a much shorter time span than one month. CHAPTER IV CONCLUSIONS AND FUTURE RESEARCH I. Summary The moving average strategy is one of the mechanical trading ru les used by academicians to invest igate the theory of random walks. It can be defined accord ing ly : buy the stock when the p r i ce exceeds the moving average by more than the threshold amount and s e l l the stock when-ever the p r i ce f a l l s below the moving average by any amount. A short sale is indicated whenever the pr i ce f a l l s below the moving average by more than the threshold amount and is covered whenever the p r i ce r i ses above the moving average by any amount. If the dec i s ion ru le is to produce above average returns then po s i t i ve dependence in successive p r i ce changes must be preva lent . In essence there are two types of systematic dependencies which could account for the existence of po s i t i ve dependence in p r i ce movements: dependence in the noise generation process together with a lag in the d i s t r i b u t i o n of relevant information (that i s , the presence of opinion leaders ) , and dependence in the process generating new information where op t im i s t i c news tends to be followed by op t im i s t i c news more often than by pess imi s t i c news (or pess imi s t i c news tends to be fol lowed by pess imi s t i c news more often than by op t im i s t i c news). Of the two, the f i r s t is probably more c h a r a c t e r i s t i c of the market p lace . Both Alexander and Cootner incorporated th i s s p e c i f i c type of systematic dependency in the i r models of stock pr i ce behavior. Alexander appl ied the f i l t e r technique (which bears a c lose resemblance to the moving average strategy) to the d a i l y c lo s ing pr ices of two p r i ce indexes and i n i t i a l l y found that the gross gains from his ru le subs tan t i a l l y exceeded those of a buy-and-hold p o l i c y . In h is second paper, corrected for ce r ta in biases, the gains were s i g n i f i c a n t l y reduced. To correct for yet another bias present in both s tud ies , Fama and Blume employed the f i l t e r rule on the d a i l y c lo s ing pr ices of t h i r t y indus t r i a l s tocks. Their resu l t s revealed a s l i gh t amount of po s i t i ve dependence in small movements of stock p r i ce s , but the extent of the dependence could not be used as a basis for a p r o f i t a b l e dec i s ion r u l e . In general , f i l t e r s of a l l d i f f e r e n t s izes f a i l e d to produce net returns in excess of those of the buy-and-hold model. A s im i l a r set of f ind ings was unvei led by Cootner when he used the moving average strategy on weekly p r i ce data. The po s i t i ve dependence suggested by the above average gross returns was not not iceable enough to lead to a remunera-t i ve s trategy. That i s , the returns a f te r the deduction of t ransact ion costs were far below those of the simple buying and holding procedure. In tes t ing the independence assumption of the theory of random walks through the use of such mechanical trading ru le s , the aforementioned authors overlooked one important feature in se lec t ing the i r sample. It is apparent that the largest returns would be rea l i zed in the case where po s i t i ve dependence in successive pr i ce changes ex i s ted and pr ices f luctuated over a wide range. In v i r t u a l l y a l l of the studies summarized, no conscious attempt was made to l im i t the sample to those s e c u r i t i e s which were l i k e l y to exh ib i t a high degree of v o l a t i l i t y . The cor rec t i on for such a de f i c i ency furnished the basis for th i s t h e s i s . An i nd i ca -t ion that the approach could be adopted was found in a study by Pra t t , who detected a tendency for the r e l a t i v e v o l a t i l i t y demonstrated by any secur i ty to pe r s i s t over time. With regard to the presence of po s i t i ve dependence in pr i ce movements, i t was bel ieved that pr i ces adjusted gradual ly to new in fo r -mation over a month-long per iod . The ra t iona le underlying such a hypothesis pertained to the time factor involved in: the ana lys i s of a secur i t y , the pub l i ca t ion of the relevant information, and the consequent release of the report to the general invest ing p u b l i c . To test th i s assumption, monthly stock pr i ce data was used in the s imulat ion of the trading r u l e . Test No. 1 was conducted to determine whether the r e l a t i v e v o l a t i l i t y of a secur i t y tended to remain constant over successive in terva l s of time (1936 to 19^ +5 and 1946 to 1 955 ) . The resu l t s did not support the evidence reported by Pra t t , as there appeared to be very l i t t l e tendency for any secur i ty to exh ib i t the same r e l a t i v e v o l a t i l i t y over the two ten-year per iods . In Test No. 2 an experimental design s imi la r to the one employed by Pratt was adopted to ascer ta in the tendency for a s tock ' s r e l a t i v e v o l a t i l i t y to stay with in some d e f i n i t e range. It was found that, as a group, the most v o l a t i l e s e c u r i t i e s in the 1936 to 19^ +5 period tended to remain the most v o l a t i l e in the succeeding per iod . This f ind ing was used as a basis for p red ic t ing which of the 200 s e c u r i t i e s would be most l i k e l y to d i sp lay a r e l a t i v e l y high level of p r i ce va r i a t i on over the fo l lowing 1956 to 1966 i n t e r v a l . The moving average strategy for long pos i t ions only was simu-lated on fo r ty ser ies of monthly pr ices covering the 1956 to 1966 period (Test No. 3 ) . These stocks were the fo r ty most v o l a t i l e of the e n t i r e sample of 200 in the preceding ten year span. The resu l t s of the s imulat ion revealed that the dec i s ion ru le was markedly i n f e r i o r to the two buy-and-hold models. Had brokerage fees been e l iminated from cons iderat ion the f ind ings would not have been a l te red to any s i g n i f i c a n t extent. Thus, the evidence presented d e f i n i t e l y supported the random walk theory of stock p r i ce behavior. I 1. Suggested Research Topics One of the major f ind ings of th i s study is that the r e l a t i v e v o l a t i l i t y of a stock tends to remain with in a f a i r l y narrow range over successive in terva l s of time. Whether or not th i s c h a r a c t e r i s t i c is an anomaly has yet to be determined s ince the test conducted here used monthly stock pr i ce observations on ly . If the phenomena is found not to ex i s t for other types of data, then perhaps a weighted moving average of the past va r i a t i on in pr ices can be adopted to pred ict the r e l a t i v e v o l a t i l i t y of a secur i ty in succeeding per iods . The p a r t i c u l a r end in mind is the r e s t r i c t i o n of the sample to those s e c u r i t i e s which are l i k e l y to be h igh ly v o l a t i l e and the app l i ca t i on of the moving average strategy to d a i l y or weekly p r i ce data. It is bel ieved that excess pro-f i t s could resu l t in such a case. The poor performance of the moving average strategy in Test No. 3 can be a t t r i bu ted to the absence of pos i t i ve dependence in successive monthly pr i ce movements which des-troyed any advantage to be gained from conf in ing the sample in the manner descr ibed. I n view of the fact that both Fama/Blume and Cootner detected evidence of s l i gh t po s i t i ve dependence in d a i l y and weekly p r i ce changes re spec t i ve l y , the tes t ing of the above hypothesis appears warranted. Another top ic worthy of cons iderat ion in future research e f f o r t s is the p o s s i b i l i t y of negative dependence in monthly movements of stock p r i c e s . This is implied by the magnitude of the losses for the moving average strategy reported in Tables III, IV, V, and VI. The development of an appropriate ra t iona le for th i s type of p r i ce behavior is l e f t to those interested readers. TABLE I SUBSEQUENT VOLATILITY OF PORTFOLIOS OF STOCKS SELECTED ON THE BASIS OF PRIOR VOLATILITY Average Coe f f i c i en t of Var i a t i on Rank*  Period | Period II Period I Period I I Por t fo l io No. 1 .889 .410 1 1 Por t fo l io No. 2 .636 .387 2 3 Port fo l io No. 3 .549 .380 3 4 Port fo l io No. 4 .460 .348 4 5 Port fo l io No. 5 .394 .393 5 2 Por t fo l io No. 6 .345 .297 6 7 Port fo l io No. 7 .303 .318 7 6 Port fo l io No. 8 .260 .268 8 9 Port fo l io No. 9 .208 .288 9 8 Port fo l io No. 10 .153 .265 10 10 Rank of 1 is assigned to the most v o l a t i l e p o r t f o l i o . SELECTED SAMPLE OF FORTY SECURITIES 1. C i t y Invest ing Co. 2. Conde Nast Pub l icat ions Inc. 3. Norfolk & Western Railway 4. Telautograph Corp. 5 . Reynolds Metals 6. Madison Square Garden Corp. 7. Fibreboard Paper Products Corp. 8. Boston & Maine Corp. 9 . Texas Instruments 10. Mack Trucks Inc. 11. Consol idated E l e c t ron i c s Industries Corp. 12. American & Foreign Power 13. Bayuk Cigars 14. Rexall Drug and Chemical Co. 15. Bigelow-Sanford Inc. 16. Chadbourn Gotham Inc. 17. Greyhound Corp. 18. Duplan Corp. 19. Lane Bryant 20 . Eastern S ta in les s Steel Corp. 21. . Northern P a c i f i c Railway 22. National D i s t i l l e r s & Chemical Corp. 23. Crown Cork & Seal Co. Inc. 24. Thatcher Glass Manufacturing 2 5 . Helme Products, Inc. 26 . American Broadcasting Paramount 27. Morrel l John and Co. Inc. 28. Penn D ix ie Cement Corp. 29 . Fawick Corp. 30. Howe Sound Co. 3 1 . Sperry-Rand Corp. 32. Hotel Corp. of America 33. Socony Mobil Oi l Co. Inc. 34. Dana Corp. 35. C o l l i n s and Aikman Co. 36 . United A i r c r a f t Corp. 37. American Zinc Lead & Smelting Co. 38. Johns Manvi l le Corp. 39. Transamerica Corp. 40. J . I . Case Co. TABLE I I I THREE MONTH MOVING AVERAGE RESULTS Average Annual Rate of Return Average Annual Average No. of No. of Investment Threshold Rate of Return Total No. of Transactions P ro f i t ab l e Performance Pr i ce Index Per Transaction Transactions Per Secur i ty Secu r i t i e s Index .2158 .1155 .02 - .2555 597 14.92 6/40 .03 - . 2 530 556 13.90 5/40 .04 - . 2 386 513 12.82 5/40 .05 - . 2315 479 11.97 6/40 .06 -.2171 450 11.25 7/40 .07 - . 1810 419 10.47 8/40 .08 - . 1647 388 9 .70 9/40 .09 - .1913 364 9 .10 9/40 .10 -.1841 347 8.67 9/40 .20 - . 2848 134 3.94 13/34 .30 - . 2083 55 2.29 13/24 .40 - . 5203 23 1.64 5/14 • 50 - . 6019 12 1.71 1/7 TABLE IV FOUR MONTH MOVING AVERAGE RESULTS Average Annual Rate of Return Average Annual Average No. of No. of Investment Threshold Rate of Return Total No. of Transactions P r o f i t a b l e Performance Pr i ce Index Per Transaction Transactions Per Secur i ty Secur i t i e s Index .2158 .1155 .02 -.2507 504 12.60 3/40 .03 -.2459 467 11.67 4/40 .04 -.2610 442 11.05 3/40 .05 -.2538 420 10.50 4/40 .06 -.2208 388 9.70 6/40 .07 -.1843 356 8.90 6/40 .08 -.1897 346 8.65 8/40 .09 -.1841 316 7.90 11/40 .10 -.1727 297 7.42 11/40 TABLE V FIVE MONTH MOVING AVERAGE RESULTS Average Annual Rate of Return Average Annual Average No. of No. of Investment Threshold Rate of Return Total No. of Transactions P r o f i t a b l e Performance Pr ice Index Per Transaction Transactions Per Secur i ty Secu r i t i e s Index .2158 .1155 .02 - . 2537 451 11.27 4/40 .03 - . 2325 416 10.40 8/40 .04 - . 2253 386 9.65 8/40 .05 -.2104 366 9.15 8/40 .06 - . 2 236 348 8.70 7/40 .07 - . 2026 330 8'. 25 9/40 .08 -.1791 309 7.72 10/40 .09 - . 1482 287 7.17 10/40 .10 -.1661 278 6.95 11/40 SIX MONTH MOVING AVERAGE RESULTS Average Annual Rate of Return No. of Investment P r o f i t a b l e Performance Pr i ce Index Secu r i t i e s Index .2158 .1155 .02 -.2422 412 10.30 6/40 .03 -.2231 381 9.52 7/40 .04 -.2025 359 8.97 7/40 .05 -.1952 338 8.45 7/40 .06 -.1772 321 8.02 9/40 .07 -.1849 306 7.65 8/40 .08 -.2016 289 7.22 9/40 .09 -.1791 271 6.77 9/40 .10 - . 1 577 257 6.42 12/40 .20 -.1341 141 3.71 18/38 .30 -.1136 73 2.43 14/30 .40 -.0331 42 1.83 13/23 .50 -.0353 20 1.43 8/14 Average Annual Average No. of Threshold .Rate of Return Total No. of Transactions Per Transact ion Transactions Per Secur i ty HYPOTHETICAL CHART OF A STOCK PRICE SUBJECT TO RANDOM MOVEMENT WITHIN FIXED LIMITS Figure 1. The p ro fe s s i ona l ' s opinion of a s tock ' s worth is represented by the dotted l i n e . The margin of p r o f i t that they require in order to e i ther buy or sel1 is represented by the d i f f e rence between th i s dotted l i ne and each of the two so l i d l i n e s . ^ R i c h a r d ^ . Brealey, An Introduction to Risk and Return from  Common Stocks (Cambridge, Massachusetts: The M. I .T. Press, 1969, p. 23. HYPOTHETICAL CHART OF A STOCK PRICE SUBJECT TO RANDOM MOVEMENT WITHIN PERIODICALLY CHANGING LIMITS i ! \ ^ • A . Figure 2. 32 BIBLIOGRAPHY A. Books Brealey, Richard A. An Introduction to Risk and Return from Common  Stocks. Cambridge, Massachusetts: The M. I.T. Press, 1969. Freund, John E., and Wi l l iams, Frank Je f fe r son . Freund and Wi l l i ams '  Modern Business S t a t i s t i c s . Revised by Benjamin Per les and Charles S u l l i v a n . Englewood C l i f f s , New Jersey: P r e n t i c e - H a l l , Inc., 1969. Levy, Robert A. The Re lat ive Strength Concept of Common Stock Pr i ce  Forecast ing ; An Evaluat ion of Selected App l i ca t ions of Stock  Market Timing Techniques, Trading T a c t i c s , and Trend Ana l y s i s . Larchmont, New York: Investors Intel 1igence, 1968 . Smith, Adam (pseud) . The Money Game. New York: Del l Publ i sh ing Co., Inc., 1969. Yamane, Taro. S t a t i s t i c s ; An Introductory Ana l y s i s . 2nd ed. New York: Harper & Row, 1967. B. A r t i c l e s in Books Alexander, Sidney S. " P r i ce Movements in Speculat ive Markets: Trends or Random Walks." The Random Character of Stock Market .Pr ices . Edi ted by Paul H. Cootner. Cambridge, Massachusetts: The M.I.T. Press, 1964. _ _ _ _ _ _ _ " P r i c e Movements in Speculat ive Markets: Trends or Random Walks, No. 2 . " The Random Character of Stock Market P r i c e s . Edi ted by Paul H. Cootner. Cambridge, Massachusetts: The M.I.T. Press, 1964. Cootner, Paul H. Stock P r i ce s : Random vs. Systematic Changes. Edited by Paul H. Cootner. Cambridge, Massachusetts: The M.T.T. Press, 1964. Roberts, Harry V. Stock Market "Pa t te rns " and F inanc ia l Ana ly s i s :  Methodological Suggestions. Edited by Paul H. Cootner. Cambridge, Massachusetts: The M. l .T . Press, 1964. C. A r t i c l e s in Journals Fama, Eugene F. "The Behaviour of Stock Market P r i c e s . " Journal of  Business, 38 (January, I 9 6 5 ) , pp. 3 4 - 1 0 5 . Fama, Eugene F., and Blume, Marshall E. " F i l t e r Rules and Stock Market T r a d i n g . " Journal of Business: Special Supplement, 39 (January, I 9 6 6 ) , pp. 226-241. F i sher , Lawrence. "Some New Stock-Market Indexes." Journal of Business:  Specia l Supplement, 39 (January, I 9 6 6 ) , pp. 191-225. Smidt, Seymour. "A New Look at the Random Walk Hypothes i s . " Journal of  F inanc ia l and Quant i tat ive Ana ly s i s , 3 (September, 1968), pp. 235-261. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0101827/manifest

Comment

Related Items