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The ability of the Value Line Investment Survey to forecast "Probable twelve months market performance.. 1966

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THE ABILITY OF THE VALUE LINE INVESTMENT SURVEY TO FORECAST "PROBABLE TWELVE MONTHS MARKET PERFORMANCE RANK" by DONALD ROSS STALEY B. Com., University of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Faculty of Commerce and Business Administration We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1966 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i ] a b l e f o r reference and study, I f u r t h e r agree t h a t p e r m i s s i o n - f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g ain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date ABSTRACT In the thesis the author attempts to discover whether or not The Value Line Investment Survey shows evidence of an a b i l i t y to f o r e - cast "Probable Twelve Months Market Performance Rank," a ranking of stocks according to t h e i r probable r e l a t i v e p r i c e performance w i t h i n the succeeding twelve months. To t e s t the a b i l i t y to for e c a s t , the author determines the s i g n i f i c a n c e of the c o r r e l a t i o n between the ranking of stocks according to the forecast and the ranking of stocks according to the observed r e l a t i v e p r i c e performance within the year. The conclusion drawn i s that The Value Line Investment Survey does not show evidence of a consistent a b i l i t y to forecast "Probable Twelve Months Market Perform- ance Rank." The author also presents a model of the process which may underly the generation of stock market p r i c e changes. The author te s t s the assumption of the independence of p r i c e changes, a part of the model, on the data of the thesis and f i n d s that the t e s t r e s u l t s do not refute the assumption. The model, the "Random Walk Hypothesis," i s related to the a b i l i t y of The Value Line Investment Survey to forecast "Probable Twelve Months Market Performance Rank." I t i s concluded that The Value Line Investment Survey has f a i l e d to show that i t s forecasts are superior to forecasts based sol e l y on past prices where the market i s assumed to follow a random walk. TABLE OF CONTENTS Chapter Page I. INTRODUCTION 1 I I . METHOD USED BY THE VALUE LINE INVESTMENT SURVEY TO FORE- CAST "PROBABLE TWELVE MONTHS MARKET PERFORMANCE RANK". . 3 I I I . NATURE, METHOD AND FINDINGS OF THE RESEARCH PROJECT . . . 8 Nature of the project . . . . . . . . . . . . . . . . . 8 Method of the project . . . . . . . . . . . 10 Findings of the project 12 IV. THE ABILITY OF FORECASTERS TO FORECAST 15 V. THE "RANDOM WALK HYPOTHESIS" AND ITS RELATION TO THE PRESENT STUDY 19 The hypothesis 19 Findings which may support the hypothesis . . . . . . . 27 Relation to the present study . . . . . . . . . . . . . 30 VI. CONCLUSION 32 SELECTED BIBLIOGRAPHY 34 CHAPTER I INTRODUCTION I t i s the purpose of t h i s paper to present the method and findi n g s of a research project * undertaken to t e s t the a b i l i t y of The Value Line Investment Survey ^ to forecast "Probable Twelve Months Market Perfor- mance Kank" (hereafter c a l l e d " p r e d i c t i o n s " ) . The findings are i n t e r - preted, and then related to the "Random Walk Hypothesis," a model -which may describe the generation of market p r i c e changes. Chapter II describes the method used by The Value Line Investment Survey to make "predi c t i o n s " which are pr i m a r i l y based on a calculated " i n t r i n s i c value." This method i s then compared with some generally received procedures f o r appraising the values of stocks. In Chapter III the method and findings of the author's research project are presented. The project adopts rank c o r r e l a t i o n methods to compare d i f f e r e n t rankings of a set of stocks. In Chapter IV the findi n g s of the study are interpreted to deter- mine the a b i l i t y of The Value Line Investment Survey to fo r e c a s t . *"' ',•„•..;m .* •> •'•The research project was conducted by the author i n 1965 on data f o r the period 1961 to 1964. ^The Value Line Investment Survey, New York, Arnold Bernhard & Co., Inc. (a weekly p u b l i c a t i o n ) . ^A ranking of stocks according to t h e i r probable r e l a t i v e market p r i c e performance within the succeeding twelve months. 1 2 In Chapter V the "Random Walk Hypothesis" i s discussed. Some earlier findings concerning the adequacy of the hypothesis to describe actual stock price changes are presented. A test of the independence assumption for stock price changes, which i s part of the "Random Yfalk Hypothesis," is carried out on the findings of the research project. The "Random Walk Hypothesis" i s then related to the study of The Value Line Investment Survey's a b i l i t y to ""predict." The conclusion follows; i t includes some inferences made in the earlier chapters and discusses further some subjects already considered. The present study should be of interest to those persons wishing to adopt the forecasts of short-run price changes as presented in The Value Line Investment Survey. It should also serve students interested in a description of the random walk hypothesis of stock market price changes. CHAPTER II METHOD USED BY THE VALUE LINE INVESTMENT SURVEY TO FORECAST "PROBABLE TWELVE MONTH MARKET PERFORMANCE RANK" The method used i n The Value Line Investment Surveysince November 1957 to f o r e c a s t "Probable Twelve Months Market Performance Rank" i s to assign ranks to stocks according to the r a t i o of the most recent 52-week average p r i c e of the stock to i t s "Value Line Rating of I n t r i n s i c Value" projected 12 months ahead, ( " i n t r i n s i c value" i s described as "the average p r i c e f o r a 12 month period that i s s t a t i s t i c a l l y determined to be the normal p r i c e i n r e l a t i o n to the subject stock's dividend-paying a b i l i t y i n that year.") 1 The " i n t r i n s i c value" i s obtained by performing a multiple c o r r e l a - 2 3 t i o n analysis on a group of s i m i l a r " q u a l i t y " stocks and applying the r e s u l t s to the p a r t i c u l a r stock. The c o r r e l a t i o n i s performed on time seri e s data f o r approximately 20 years. In the a n a l y s i s , p r i c e i s considered to depend upon earnings, dividends, past p r i c e , and a market sentiment f a c t o r . •'•Arnold Bernhard, The Evaluation of Common Stocks, (New York, Simon and Schuster, 1959), p. 120. ^Although described as a m u l t i p l e c o r r e l a t i o n analysis by Bernhard, the analysis should more f i t t i n g l y be described as a regression analysis as " i n t r i n s i c value" i s assumed to depend upon the values of the other variables involved. Quality" r e f e r s to a composite index of past performance i n growth and s t a b i l i t y of earnings and dividends. For a d e r i v a t i o n of the " q u a l i t y " index see Arnold Bernhard, The Evaluation of Common Stocks, pp. 41-57. 3 4 The relationship formulated is that the independent variable, the logarithm of the annual average price for the group, is dependent upon the regressor variables, the logarithms of: A. A fraction of average annual earnings, which i s combined with average annual dividends, where the fraction is determined by the ratio of the standard deviation in the dividends to the 4 standard deviation in the earnings for the period considered. B. The annual average lagged price (the annual average of prices in the preceding year). C. The annual mean yields of 45 "representative" stocks over the period considered. Once the net regressor coefficients for the group have been deter- mined they are applied as weighting factors to the similar variables for the particular stock i n arriving at the stock's "intrinsic value." The logarithm of the "Value Line Rating of Intrinsic Value" for a particular stock i s calculated by weighting, with the appropriate net regressor coefficients (obtained from the group multiple correlation analysis), the variables of: 1. a combination of earnings and dividends projected 12 months ahead, where earnings are multiplied by a ratio (the ratio of the standard deviation in earnings of the particular stock for *The fraction, Bernhard states, (Arnold Bernhard, The Evaluation of Common Stocks, p. 99) tends to "equalize the variations in the earn- ings and dividend series." 5 The yields are included as a market sentiment factor. 5 a 10 year period) before being combined with dividends, and 2. the present average annual price, and combining them together with a constant which is for the particular stock equal to a 10-year average of i t s log price, minus the appropriate g net regressor coefficient times the 10-year average of log lagged price, minus the appropriate net regressor coefficient times the 10-year 7 average of log combined earnings and dividends. The market sentiment factor (the average yield of 45 representa- tive stocks) is not included in arriving at the "Value Line Rating of Intrinsic Value" for a particular stock "since market sentiment is held Q constant." Once the "Rating of Intrinsic Value" for a l l the stocks considered i s determined, the stocks are ranked according to the ratio of the 52- week average price of the stock to i t s "Rating of Intrinsic Value," to arrive at the forecast of "Probable Twelve Months Market Performance Rank." It should be noted that the ranks are not based solely on past information. To arrive at "intrinsic value," both earnings and d i v i - dends must be projected for a year into the future and the results of the projection used i n the formula. °The appropriate net regressor coefficient i s the net regressor coefficient of the similar variable determined for the group analysis. ^Earnings are multiplied by the ratio of the standard deviation of dividends to the standard deviation of earnings for the particular stock for a 10 year period before being added to dividends. g Arnold Bernhard, The Evaluation of Common Stocks, p. 106. 6 The method for arriving at "intr i n s i c value" differs i n form rather than substance from another generally accepted method for q appraising stock values. Graham, Dodd and Cottle assume that the basic components in common-stock valuation ares 1. The expected future earnings. 2. The expected future dividend. S. The capitalization rate - or multiplier - of dividends and earnings. 4. The asset value. They present a model for the valuation of common stocks which iss V = M(D + E) -V A 1 0 where? V is the value of the stock. M i s the earnings multiplier assumed appropriate for the type of stock. D i s the expected dividend. E is the Expected earnings. A i s an adjustment for asset value i f necessary. In both methods, once the values of the parameters (based on "quality* 1 factors) have been determined, i t s t i l l remains to project both earnings and dividends for the coming year. The choice of projecting earnings rather than or together with ^B. Graham, D. Dodd and S. Cottle, Security Analysis, 4th ed. (New York, McGraw-Hill Book Company, Inc., 1962). 1 0 I b i d . , p. 518. 7 dividends for a period as indicated by Sauvain ^ serves only as a short-cut to the valuation of stocks based on a consideration of a l l future dividends. A method which does treat of a l l future dividends and which i s generally considered more theoretically sound than the 12 methods already considered is the J.B. Williams model, Williams assumes that the intrinsic value of a stock i s equal to the present 13 value of a l l future dividends. The model may be adjusted to take into account the risk of divergence from expected values in the stream of dividends. The model, although theoretically sound, i s d i f f i c u l t , i f not impossible, to apply, as i t i s impossible to project the expected dividends for a l l future years. A procedure more generally applied than any of the preceding i s simply to multiply a price earnings ratio (often the average price earnings ratio of the stock for that year) by a projection of earnings for the following year. ^H. Sauvain, Investment Management 2nd ed. (New Jersey, Prentice- Hall, Inc. 1960), p. 310. 1 2J.B. Williams, The Theory of Investment Value (Amsterdam, North- Holland Publishing Company, 1956). l^The discount rates used to arrive at present values are deter- mined by the interest rates sought by the investor. CHAPTER III NATURE, METHOD AND FINDINGS OF THE RESEARCH PROJECT NATURE OF THE PROJECT The purpose of the research project is to determine the s i g n i f i - cance of the correlations between given series of stock price changes as viewed in The Value Line Investment Survey, during the period 1961- 1964. F i r s t , the correlation, or the degree of correspondence, of the ranking of stocks according to their "Probable Twelve Month Market Per- formance Rank" ("predictions") with the "observed" 1 ranking for the twelve months following the date of the "predictions" i s investigated. A significant correlation may indicate that a system successful in fore- casting price changes has been devised. Secondly, the correlation of the "predictions" with the "actual performance" (the ranking of stocks according to their percentage price appreciations) for the six months preceding the date of the "predictions" is determined. A significant correlation may indicate that "predictions" are primarily based upon past price performance, iThe "observed" ranking of stocks, according to their twelve months market performance, i s a ranking of the stocks according to the maximum percentage price increase i n the stocks achieved during the twelve months considered. The maximum prices are taken from the charts of stock prices as presented in The Value Line Investment Survey, 8 9 Thirdly, the correlation between the "actual performance" for the six month period preceding and the "observed" performance for the twelve month period following the date of the "predictions" i s determined. A significant correlation may indicate that stock market price changes are not independent of past price changes, although they are assumed to 2 be independent i n the "Random Walk Hypothesis." The correlation between the "predicted" and "observed" rankings is considered at four phases i n the market cycle: the bottom, the top, the middle of a f a l l i n g market, and the middle of a rising market. The dates of the phases have been discovered by viewing weekly figures of Standard and Poor's Index of Total Common Stock Prices. The number of dates for which correlation studies have been possible has been limited by the number of surveys available for study. The actual computations to obtain correlation coefficients have been performed on the I.B.M. 7040 computer at the University of British Columbia. To determine the significance of the correlation between two rank- ings, i t i s f i r s t necessary to compute a correlation coefficient between the rankings. The formula used i s presented i n the following section together with a note regarding a test of the significance of the coeffi- cients. For a description of the "Random Walk Hypothesis" see Chapter V. 10 METHOD OF THE PROJECT The following is a description of the method and formula used to determine the degree of correspondence between the different rankings in the study. The correlation between two rankings i s obtained from Spearman's formula: P where j J t < n 3 - f O - t 1 d i a - T ' \ / < : i / a n 3 - n ) - 2 T ' X ' ^ C n 3 - n ) ) P = Spearman's coefficient of rank correlation n = number of ranks used d^ = difference i n ranks T* = II , where t is a tied set of t members and 4 m typifies a set of ties i n the ranking. The formula takes tied ranks into account i n spite of the fact that ties should not occur i n an objective ordering of ranks. In the survey the ranks of 1 to 5 are allocated to stocks according to their prospect for relative price performance. The method used to rank tied sets i s to allocate to each member the average of the rank which the members would have possessed had they 3M.G. Kendall, Rank Correlation Methods, (London, Charles G r i f f i n & Co. Ltd., 1948) p. 28. 4Ibid., p. 29. 5Ibid., p. 28. 11 been distinguishable. For example, i f a total of six stocks had the "predicted" rank of 1, the rank allocated to stocks with the highest prospect for relative price performance, then eaoh would be allocated the rank 3.5, the average of the numbers 1 to 6. The resulting coefficients of rank correlation, P, are tested for significance. As each sample size n (the number of stocks considered in each test) i s greater than 50, the distribution of P is assumed to be normal. (For a sample size n greater than twenty, the assumption of normality i s lik e l y justified.) ^ If there were no correlation between two rankings the maximum likelihood value of P would be zero; however, for a normally distributed variable, the maximum likelihood value and the mean coincide, so that the mean of the distribution of P would be zero. \ The variance of the distribution of P under the assumption of 1 8 normality and no correlation i s given approximately by var P s — - . Hence the standard normal form of the variable (assuming zero population correlation) i s given by Z = P \ n-1 • 6Kendall, Rank Correlation Methods, p. 25. 7Ibid., p. 47. 8Ibid., p. 48. 12 FINDINGS OF THE PROJECT Presented below are the findings of the study, that i s , the signi- ficance of the correlations between the rankings studied. F i r s t , i t i s found that i n only three cases out of thirteen are there significant ̂  correlations between "predicted" and "observed" market performance. Also, i n no given stage of the market cycle do the number of significant relations seem large enough to distinguish that stage from any other. At the "Bottom of a Market Break," in only two out of six cases tested are the correlations significant. At the "Top of a Market Break" only one out of the three cases studied i s significant. At the "Middle of a Falling Market" and"Middle of a Rising Market,"' in which only two cases each are tested, none of the correla- tions are s t a t i s t i c a l l y significant. Second, the correlation between "actual performance" for the six months preceding publication of the survey and the "predicted" perfor- mance i s found to be significant in three out of the four tests made. Third, in only one test out of three i s the correlation between "actual performance" for the six months preceding publication of the sur- vey and the "observed" market performance found to be significant. The findings are summarized and presented i n the following pages. The Spearman rank correlation coefficients P, and the P's in standard form (Z » P ^n-1) are summarized together with the level of significance CC which would make the absolute value of ± Z, h e n c e t h e a b s o l u t e v a l u e of-t P significant. An asterisk beside an 0\- indicates that the P i s significant at a o r i t i c a l level of .050. ^Significance is considered to exist at the c r i t i c a l level of .050. { 13 Correlation between "predicted" and "observed" market performancej 1. Bottom of a Market Break: Date P Z OL July 2, 1962 0.1375 1.05 .294 Oct. 29, 1962 -0.0984 -0.78 .435 Mar. 4, 1963 0.0837 0.65 .516 July 29, 1963 0.2512 2.10 .036 ^ Oct. 4, 1963 -0.1405 -1.27 .204 Nov. 22, 1963 0.3640 3.30 .000 ^ 2. Top of a Market Break: Date P Z (X May 22, 1961 0.2416 1.96 .050 Aug. 20, 1962 0.1427 1.20 .230 Feb. 11, 1963 -0.0161 -0.14 .889 3. Middle of a Falling Market: Date P Z May 7, 1962 0.1746 1.45 .147 Oct. 8, 1962 0.0924 0.76 .447 4. Middle of a Rising Market: Date P Z Dec. 24, 1962 0.1623 1.42 .156 Apr. 8, 1963 0.0251 0.23 .818 14 Correlation between "actual performance" for the six months preceding publication of the survey and the "predicted" performance: Date May 22, 1961 Oct. 29, 1962 Mar. 4, 1963 Dec. 20, 1963 0.1587 0.2761 0.2649 0.2490 0.14 2.36 2.40 2.32 (X .889 .018 .016 % .020 ft Correlation between "actual performance" for the six months preceding publication of the survey and the "observed" market performance: Date May 22, 1961 Oct. 29, 1962 Mar. 4, 1963 -0.0146 -0.3516 0.1981 -0.12 -2.78 1.68 <X .005 -TfT .093 In Chapters IV and V the findings of the tests are related to the abi l i t y of the Value Line Investment Survey to predict "Probable Twelve Months Market Performance Rank." I CHAPTER IV THE ABILITY OF FORECASTERS TO FORECAST This chapter is concerned with the ab i l i t y of forecasters of the stock market to predict price changes. Presented f i r s t i s a summary of some research attempted to test the forecasting a b i l i t y of different concerns. Following this summary i s the interpretation of the author's research findings, which should indicate whether or not The Value Line Investment Survey shows any ab i l i t y to predict "Probable Twelve Months Market Performance Rank." Alfred Cowles III, in his study of the forecasting a b i l i t y of forty-five professional agencies for the years 1928-1932, found that these agencies f a i l e d to demonstrate any s k i l l i n forecasting stock price changes. ^ Of sixteen financial services, only six showed better returns than the market average, and even the best record did not indicate a performance better than that to be expected by chance. In Cowles* study of twenty-five f i r e insurance companies, each of which carried more than twenty percent of their total investments i n common stocks, only six showed evidence of forecasting a b i l i t y . Again i t was found that the records could not be attributed to s k i l l . He found that, of ninety forecasts of the stock market as a whole over the period 1904-1929 made by William P. Hamilton using the Dow *Alfred Cowles, "Can Stock Market Forecasters Forecast?" (Econome- tric a , vol. 1, July 1933, pp. 309-324). 15 16 Theory, only half were successful. He also investigated twenty-four forecasting publications for the period 1928-1932. His analysis failed to indicate better than random successes for these publications. 2 In his "Stock Market Forecasting," Cowles extends the records of eleven of the twenty-four financial services he had previously studied. This study covers the period from 1928-1943. He found that the records of these services, during varying periods ranging from ten to fifteen years after 1927, f a i l e d to disclose any a b i l i t y to predict the stock market. In another study, "A Study of Mutual Funds," by Brown, Herman and Vickers, i t was found that the average performance of the funds was about the same as to be expected from an unmanaged fund consisting of similar securities. "Both for balanced funds and common stock funds separately, the distribution of funds classified by the number of years in which they demonstrated above-average performance seem completely random or conforming to chance." 5 Wu, in following the transactions on the New York Stock Exchange 2A. Cowles, "Stock Market Forecasting," (Econometrice, Nos. 3-4, July-October, 1944, pp. 206-214). 3 I . Friend, F.E. Brown, E. Herman and D. Vickers, "A Study of Mutual Fundss Investment Policy and Investment Company Performance," (in H. Wu and A.J. Zakon, ed. Elements of Investments, New York, Holt, Rinehart and Winston, Inc., 1965). 4Ibid., p. 383. 5H. Wu, "Corporate Insider Trading Profits and the Ab i l i t y to Fore- cast Stock Prices," (in H. Wu and A.J. Zakon, ed. Elements of Invest- ments, New York, Holt, Rinehart and Winston, Inc., 1965). 17 of f i f t y common stocks from October to December 1959, finds that "there i s very l i t t l e evidence that a definite relationship exists between insider transactions and subsequent price movements in relation to the general market trend." His tests indicate that the insiders i n the f i f t y companies did not outperform the market. As indicated, there i s evidence that stock market traders exhibit l i t t l e , i f any, s k i l l i n forecasting the future prices on the stock market. However, i t i s s t i l l possible that a system has been devised which may accurately prediot future stock price changes. It i s the pur- pose of this section to interpret the author's research findings regard- ing the forecasting a b i l i t y of the Investment Line Survey in presenting the "Probable Twelve Months Market Performance Rank" of stocks. To test the forecasting a b i l i t y , the "predicted" ranking is com- pared with the actual "observed" ranking (a ranking of the maximum per- centage price appreciations achieved by the stocks during the year) of the stocks. A test of the correlation between the two rankings shows that in only three out of thirteen tests performed on data taken from d i f f e r - ent periods are there s t a t i s t i c a l l y significant correlations. The results f a i l to indicate any consistent a b i l i t y of the survey to forecast the relative performance of stocks over the period of 12 months. In the book describing his survey Bernhard states, "It i s of ^Every person who i s the beneficial owner of ten percent of the stock or i s a director or officer or issuer of the security i s a cor- porate insider. 7Wu, "Corporate Insider Trading Profits and the Ability to Forecast Stock Prices," p. 448. 8Arnold Bernhard, The Evaluation of Common Stocks. 18 l i t t l e practical value, under a method based upon yearly evaluation of earnings and dividends to determine that a stock is overvalued or under- valued i n a particular year, i f prices do not perform i n accordance with the recommendation within that year." ^ It would seem then, by Bern- hard's own admission, that the survey's predictions of the "Probable Twelve Months Market Performance Rank" are of l i t t l e practical value, as they f a i l to indicate any consistent a b i l i t y of the survey to forecast. In commencing the research project the author thought that the "predictions" might be more accurate at one stage i n the market cycle than another. However, the results of the study f a i l e d to support this assumption; at no stage of the cycle was a consistent a b i l i t y to fore- cast discovered. The forecasting a b i l i t y of stock market forecasters i s further considered i n the following chapter. 9Bernhard, The Evaluation of Common Stocks, p. 86. CHAPTER V THE "RANDOM WALK HYPOTHESIS" AND ITS RELATION TO THE PRESENT STUDY THE HYPOTHESIS In this chapter a model i s presented of a process which may gener- ate stock prices. The model, the "Random Walk Hypothesis," once dis- cussed, i s related to The Value Line Investment Survey's "predictions." The random walk hypothesis for price changes i s actually composed of two separate hypotheses? 1. That successive price changes are independent of preceding price changes, that i s , that P (X t = Xt, / X t - i , X t - 2 > " . ) = P (X t = X t) where P (Xt = Xt) i s the unconditional probability that the variable Xt, the price change during the period t, w i l l assume the value X-fc and where P (X_t = Xt / Xt-1, Xt -2>»««) i s t n e conditional probability that Xt w i l l assume the value Xt« 2 . That the individual price changes conform to probability dis- tributions each with mean zero. * The random walk process i t s e l f i s given by: Xt - Zt - Zt_i where Zt is a variable generated at time t by the process in such a way that Xt forms, for successive periods t, a sequence of random, *The price changes for successive periods need not necessarily conform to identical probability distributions. 19 20 independent numbers, each conforming to p r o b a b i l i t y d i s t r i b u t i o n s with mean zero. For the random walk process as hypothesized to hold f o r stock p r i c e changes, successive p r i c e changes should e x h i b i t the same charac- t e r i s t i c s as a sequence of random independent numbers, each conforming to p r o b a b i l i t y d i s t r i b u t i o n s with mean zero* 2 I t was Louis Bachelier who f i r s t hypothesized that the process underlying speculative p r i c e changes was the random walk. However, hi s pioneering work remained i n obscurity u n t i l l a t e r (about 1934) when H. Yforking hypothesized that the revealed speculative p r i c e changes were generated by a random walk process. Bachelier tested the hypothesis on the French Bond Market, assum- ing a " f a i r game," that i s , a zero expectation of gain. A close corres- pondence was found to exi s t between the d i s t r i b u t i o n of p r i c e changes to be expected from h i s theory and that of the observed p r i c e changes. For h i s study, Bachelier chose to consider only one type of secur- i t y at a time. Had Bachelier been working with various types of secur- i t i e s as Osborne did l a t e r f o r common stocks, he might have been l e d to the percentage form of p r i c e changes. In t h i s form, i f a single p r o b a b i l i t y d i s t r i b u t i o n f o r a given period were hypothesized to generate ^L. Bachelier, "Theorie de l a Speculation" (Annales de 1'Boole Wormale Superieure, Ser. 3, XVII (1900), pp. 21-86. 3H. Working, "A Random-Difference Series f o r Use i n the Analysis of Time S e r i e s , " (journal of the American S t a t i s t i c a l Association, XXIX (1934), pp. 11-24). %.F.M. Osborne, "Brownian Motion i n the Stock Market," (Opera- tions Research, V o l . 7 (March-April, 1959), pp. 145-173. 21 equal p r o b a b i l i t i e s f o r given percentage pr i c e changes f o r a l l s e c u r i - t i e s , then a " f a i r game" would r e s u l t from the model. That i s , i t would be equally l i k e l y that a $100 security would r i s e or f a l l $10, or that a |10 security would r i s e or f a l l $1. Empirical f i n d i n g s of approximate normality i n the d i s t r i b u t i o n of percentage p r i c e changes lend support to the hypothesis that there i s equal p r o b a b i l i t y f o r given percentage p r i c e changes. However, the apparent long run trend i n stock market prices as a whole does not lend support to the assumption of a " f a i r game" which i s part of Bachelier's model. Osborne, i n h i s study of the random character of the market was led to consider the logarithm of p r i c e as the v a r i a b l e of concern i n the "Random Walk Hypothesis." However, i f the d i s t r i b u t i o n of changes i n the logarithms of prices f o r a given period i s hypothesized to gener- ate equal p r o b a b i l i t i e s f o r given changes i n the logarithm of p r i c e , then the model i s no longer a " f a i r game." Empirical f i n d i n g s concerning the changes i n the logs of p r i c e s f o r a number of d i f f e r i n g stocks from one period to the next, tend to support the hypothesis that there i s equal p r o b a b i l i t y f o r given changes i n the 7 logs of p r i c e s . Such evidence would indi c a t e that the model i s other ^S. Alexander, "Price Movements i n Speculative Markets: Trends or Random Walks," (I n d u s t r i a l Management Review, 2 No. 2 (May 1961), pp. 7-26) p. 467. 6 I f there i s an equal p r o b a b i l i t y of a change of one i n the log-^, on one hundred d o l l a r s the expected amount of gain i s : | ($10 + $1000) - $100 = $405. A.B. Moore, "Some C h a r a c t e r i s t i c s of Changes i n Common Stock P r i c e s " ( i n P.H. Cootner, ed. The Random Character of Stock Market P r i c e s , Cambridge, Mass., M.I.T. Press, 1964). 22 than a " f a i r game." In f a c t , using l og pr i c e s as the v a r i a b l e of concern i n the "Random Walk Hypothesis" allows f o r expectation of gain i n the model. It i s apparent that a long-run trend i n stock market prices as a Q whole does e x i s t . The market i s not a " f a i r game," but rather i s weighted i n favour of the investor (at the moment the long-run trend i s f o r the market to r i s e ) . The choice of logarithm of p r i c e as the va r i a b l e of concern i n the "Random Walk Hypothesis" does allow f o r a r i s i n g trend i n p r i c e s . However, i t seems u n l i k e l y that the "Random Walk Hypothesis" as formulated, using logs of p r i c e s , i s the "best" choice to describe the random nature of the market where a r i s i n g trend i n p r i c e s i s apparent. A revised form of the "Random Walk Hypothesis" which takes trend into account i s the "Random Walk Hypothesis" incorporating d r i f t , that i s : Ẑ . - Z-t-1 ~ £t Wh e r e the symbols r e f e r to the v a r i a b l e s previously indicated, save that Xj- i s now assumed to conform to d i s t r i b u t i o n s with means other than zero, chosen so that the expected value of X-̂  accounts f o r the trend i n the market. So regarded, a t e s t of the random nature of stock market prices may be made by comparing actual performance i n stock prices under a systema- t i c trading r u l e with p r i c e performance possible through a"buy-and-hold" plan. I f r e s u l t s indioate that s i g n i f i c a n t l y greater " p r o f i t s " are av a i l a b l e to a systematic trading plan, which takes into account only 8M.G. Kendall might d i f f e r , (M.G. Kendall, "The Analysis of Economic Time-Series, Part I: P r i c e s , " (Journal of the Royal S t a t i s t i c a l Society, V o l . 96, Part I (1953), pp. 11-25) p. 11), f o r he found that "In a series of p r i c e s which are observed at f a i r l y close i n t e r v a l s the random changes from one term to the next are so large as to swamp any systematic e f f e c t which may be present." 23 past prices, than to a "buy-and-hold" system on the same stocks, then a dependency between past and future prices may be established, and grounds may be made on which to refute the "Random Walk Hypothesis." 9 10 Alexander has suggested such a scheme, a f i l t e r scheme, and 11 has tested i t on some commonly used price indices. His original results have indicated that, generally, a l l sizes of f i l t e r s yield greater profits than a simple "buy-and-hold" policy. 12 However, as Mandelbrot points out, Alexander's trading system 13 1 4 has incorporated certain computational biases. In his second paper, Alexander has tested the "Random Yfelk Hypothesis" after attempting to remove a l l biases from his f i l t e r scheme. These later results indicate that the apparent pr o f i t a b i l i t y of the f i l t e r technique i s drastically reduced but s t i l l indicate that the technique should be more profitable than a "buy-and-hold" policy for f i l t e r sizes up to about 10% and over about 40$. ^S. Alexander, "Price Movements in Speculative Markets" (in Wu) and "Price Movements in Speculative Markets: Trends or Random Walks, No. 2," (in P.H. Cootner, ed. Random Character of Stock Market Prices, Cambridge, Mass., M.I.T. Press, 1964). •^Alexander's scheme i s to select a stock and watch i t . If i t s price goes up X percent, buy i t . Hold onto i t until i t s price f a l l s X percent from a subsequent high price then s e l l the stock and go short. Buy the stock when i t s price rises X percent from a subsequent low price. **A non-statistical test i s used. •^B. Mandelbrot, "The Variation of Certain Speculative Prices," (Journal of Business, Vol. 36, No. 4 (October, 1963), pp. 394-419). 1 3Ibid», p. 417. 14 S. Alexander, Price Movements in Speculative Markets: Trends or Random Walks, No. 2." 24 However, as Fama and Blume point out, Alexander has not adjusted the p r i c e indices f o r dividends which are l o s t through a c t i v e trading 16 of the stocks. The r e s u l t s of tests c a r r i e d out by Fama and Blume indicate that only f o r very small f i l t e r s , 1.5% and l e s s , i s the f i l t e r technique more p r o f i t a b l e than a "buy-and-hold" p o l i c y before commissions. They suggest that there i s a p o s i t i v e dependence i n very small p r i c e changesj and evidence of a negative dependence i n intermediate sized changes (1.5$ to 12% f i l t e r s ) . However, when commissions and the time funds w i l l be i d l e are taken i n t o account, the apparent dependencies are not l i k e l y to increase the p r o f i t a b i l i t y of the f i l t e r scheme to more than that of the simple "buy-and-hold" p o l i c y . 17 In another paper, "A new Look at Clustering of Stock P r i c e s , " dependencies are found to e x i s t i n terms of the exact prices at which stocks are sold. I t i s found that stocks trade at even eighths more often than at odd eighths, and most often at whole number prices l i k e 10, 25, 50, 75 and 100, a l l even eighths. The same phenomenon has been dem- 18 onstrated e a r l i e r by Osborne i n h i s work, A P h y s i c i s t Looks at Stock P r i c e s . 1 5 E . Fama and M. Blume, "'Filter Rules and Stock-Market Trading," (The Journal of Business, V o l . XXXIX (January, 1966), pp. 226-241). •l^Fama and Blume point out (E. Fama and M. Blume, " F i l t e r Rules and Stock-Market Trading," p. 235) that "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 f o r any dividends that are paid while the short p o s i t i o n i s outstanding. Thus adjusting f o r dividends w i l l reduce the p r o f i t a b i l i t y of short sales and thereby reduce the p r o f i t a b i l i t y of the f i l t e r technique r e l a t i v e to buy-and- hold." 1 7 V . Niederhoffer, "A New Look at Cl u s t e r i n g of Stock P r i c e s , " (Journal of Business, V o l . 39, No. 2 ( A p r i l , 1966), pp. 309-313). ^Osborne, A P h y s i c i s t Looks at Stock Prices ( c i t e d i n A.B. Moore, "Some C h a r a c t e r i s t i c s of Changes i n Common Stock P r i c e s , " p. 153). 2 5 A recent paper by Mandelbrot introduces the thought that stock prioes may follow a "Martingale" rather than a random walk. The "Mar- tingale" property "implies only that the expected values of future prices w i l l be independent of the values of past prices: the distribu- tions of future prices, however, may very well depend on the values of past prices." 2 0 The random walk is a "Martingale,1" but the "Martingale" is not necessarily a random walk. However, as Fama and Blume point out, the degree of dependence shown by the "Martingale" i s li k e l y so small that for practical purposes the random walk hypothesis i s not greatly violated for stock price changes. Some st a t i s t i c a l tests do not reject the assumption of independence in the "Random Walk Hypothesis" for stock price changes. Kendall found for British stock price averages that "stock-exchange movements revealed l i t t l e serial correlation within series &nd l i t t l e lag correlation between series." 2 1 Moore 2 2 found that, although there was slight posi- tive serial correlation in the differences of the logs of prices for successive periods, non-parametric tests of the runs in the signs of Standard and Poor's 5 0 0 Stock index tended to support the hypothesis of independence in runs. * 9B. Mandelbrot, "Forecasts of Future Prices, Unbiased Markets, and 'Martingale' Models," Journal of Business, v o l . X X X I X (January, 1 9 6 6 ) , pp. 2 4 2 - 2 5 5 . 2 0 E . Fama and M. Blume, " F i l t e r Rules and Stock-Market Trading," p. 2 2 6 . 21M.G. Kendall, "The Analysis of Economic Time-Series, Part I ; Prices," p. 1 1 . 2 2 A . B . Moore, "Some Characteristics of Changes in Common Stock Prices," p. 1 4 5 . 26 As only slight, i f any, dependence i n stock price changes has been observed, for a l l practical purposes the assumption that successive price changes are independent of preceding price changes may be an adequate description of reality where i t seems that the slight dependencies can- not be used to produce greater profits than a "buy-and-hold" policy. The nature of the probability distribution l i k e l y to underly stock price changes for any given period has been explored by some researchers. 23 Kendall, in analyzing changes in both stock price averages and com- modity prices on the London market, found approximate normality in the 24 changes. Osborne, in his study of the New York Stock Exchange, found approximate normality in changes in the logarithms of stock prices. 25 Mandelbrot questions the findings of normality in stock price changes. He finds that more observations l i e near the mean and more l i e in the extreme t a i l s i n the frequency distribution of price changes than would be expected i f the underlying distribution were normal. He suggests that, not the mormal distribution which i s a member of the Pareto family of distributions, but rather, some other member of the family would better " f i t " the empirical data. However, those candidates of the Pareto family which might better " f i t " the empirical findings, although they have the 26 additive properties associated with the normal distribution, do not have f i n i t e variances. Tests of significance have not been develogeddfor the 23M.G. Kendall, "The Analysis of Economic Time-Series, Part I: Prices," p. 17. Osborne, "Brownian Motion i n the Stock Market." 2 5B. Mandelbrot, "The Variation of Certain Speculative Prices." 2^A linear combination of normal independent yariables is i t s e l f normal. 27 Pareto family as a whole. As such is the case, i f a Pareto distribution with i n f i n i t e variance i s assumed to underly price changes, then para- metric significance tests of the independence of stock price changes 27 cannot be applied. However, Fama indicates that the mean deviation, rather than the variance, may be the basis for future significance tests for the Pareto distribution. FINDINGS WHICH MAY SUPPORT THE HYPOTHESIS Now, in the present study, many differing stocks are considered at once so that percentage price changes or the difference in the logarithms of successive prices would be the acceptable variables for testing the independence assumption of the "Random Walk Hypothesis" where i t is assumed that single probability distributions underly the 28 changes i n common stock prices for given periods. The method chosen to test the independence assumption i s non-parametric and hence does not involve the parameters of the underlying distributions. The ques- tion as to whether or not the underlying distributions are normal, Paretian, or even symmetric, or whether or not the means of the under- lying distributions are zero or some other value to account for d r i f t in the prices i s circumvented by performing the rank correlation study described i n the paper. The choice of percentage price changes or differences i n logarithms of prices would yield the same results i n the test. In fact, the non-parametric rank correlation method chosen to 2 7Eugene F. Fama, "The Behaviour of Stock Market Prices." 2®S. Alexander, "Price Movements in Speculative Markets: Trends or Random Walks," p. 15. 28 test the forecasting a b i l i t y of the survey would yield the same results whether relative price changes, relative prices, the log of relative prices, or the difference i n the logs of prices were chosen to conduct the tests, as there i s a one to one ordered correspondence between each 29 of the variables l i s t e d . If the "Random Walk Hypothesis" is assumed and i f a single d i s t r i - bution does underly price changes for a given period, or i f single dis- tributions underly price changes for every fraction of a period, then, for any period chosen, a random ordering of stocks should result accord- 30 ing to percentage price changes. If single distributions underly a l l price changes for different periods throughout a year, then a random ordering of stocks according to the greatest percentage price apprecia- tions achieved during the year should result from cumulations of the random percentage changes. 2 9Log of relative prices equals the difference i n the logs of prices, i . e . log = log P t w - log P t ?t Log of relative prices has a one to one ordered correspondence to relative prices (relative prices i n a l l cases are or can be made positive), Relative prices have a one to one ordered correspondence to relative price changes, that i s , has a one to one correspondence p t + i - ^ h P *•»" P p "fc*1 t is the same as - 1 to Pt Proof» Pt Pt Pt+1 " p t p t + l Hence and have a one to one ordered corres- p t P t pondence, as they d i f f e r only by a constant -1. 3 0 I f the single distributions are themselves each Paretian (including normal) then the percentage price changes over a number of periods w i l l appear to have been generated by a single Paretian d i s t r i - bution. 29 Assuming that a single distribution does underly stock price changes for different stocks in a market characterized by a random walk, and hence that a random ordering of stocks according to percentage price changes i s the result, the comparison of the orderings for successive periods should not reveal any significant correlation; that i s , they should indicate independence in price changes. Such comparisons were undertaken in the research project, and, as indicated i n the findings of Chapter III, the correlation between the "observed" (future) and actual past market performance was found to be significant at the b% level of significance in only one test out of three. These results are not adequate to refute the hypothesis that price changes are independent. However, i t might be f r u i t f u l to make more tests of the independence hypothesis between prices i n a period during which most prices are f a l - ling and a succeeding period in which most prices are rising, as was the case for the one test which was significant. Such a test may indicate that there i s a correlation between price changes in the two periods and thus may be grounds for refuting the assumption of independence. Then again, i f the prices are found to be dependent using the present test, but independent using other tests, the assumption that single distribu- tions underly a l l stock price changes, rather than the assumption of in- dependence, may be false. Benjamin King finds, through factor analysis, "that the movemeat of a group of security price changes can be broken 31 down into market and industry components; his results seem to indicate that the assumption of a single underlying distribution may not be correct. 3 1B.F. King, "Market and Industry Factors i n Stock Price Behavior," (Journal of Business, Vol. XXXIX (January, 1966), pp. 139-190), p. 163. 30 RELATION TO THE PRESENT STUDY Assuming that the "Random Walk Hypothesis" i s applicable to stock market price changes, i t should be clear that any forecast of future price changes based wholly on past price changes i s not likely to yield accurate results. In the author's study, the correlation coefficient between "actual past performance" and "predictions" of future price performance i s determined. It is found that, in three out of four tests, the correlation between the two performances is significant. Such con- sistent significant correlations might lead one to suppose that the Value Line "predictions" are heavily based on past performance in price changes. If such i s the case, one might expect that there should be l i t t l e significance to the correlation between "forecasted" and "observed" (future) price appreciations. The author finds that i n only three out of thirteen tests performed were correlation coefficients between the "predicted" and the "observed" price appreciations found significant at the five percent level of significance. The results do not differ greatly from the results one might expect had predictions been based on past prices alone, where the market i s assumed to follow a random walk. It would seem from the present tests, that the Value Line Investment Survey has failed to prove that i t s "predictions" of future "Twelve Months Mar- ket Performance Rank" are superior to "predictions" based solely on past prices. However, The Value Line Survey's "predictions" do not take into account solely the price appreciations over a six months preceding period (the Value Line method is described i n Chapter I I ) . This fact is borne out by the tests made on The Value Line Survey "predictions" for Oct. 29, 31 1962. Although the "predictions" are s i g n i f i c a n t l y c o r r e l a t e d with "actual performance** f o r the s i x months preceding p u b l i c a t i o n of the survey, and, although the "actual performance" f o r the s i x months pre- ceding p u b l i c a t i o n of the survey i s s i g n i f i c a n t l y correlated with the "observed" (future) market performance, the "predictions" are not s i g n i f i c a n t l y correlated with the "observed" (future) market performance. Also, i n the one case out of four i n which "actual past s i x month's performance" i s not found to be s i g n i f i c a n t l y correlated with "predic- ted" performance, ""predicted" performance i s found to be s i g n i f i c a n t l y correlated with "observed" (future) market performance. Although the method f o r "forec a s t i n g " "Twelve Months Market Per- formance Rank" does not take i n t o account only changes i n past prices (which, f o r a l l p r a c t i c a l purposes seem to be independent of future p r i c e s ) , the r e s u l t s of i t s a p p l i c a t i o n have not shown any consistent accuracy. The author i s led to conclude that the accuracy of predic- tions by t h i s method i s no greater than to be expected by basing "pre- d i c t i o n s " of a "Twelve Months Market Performance Rank" on a random ranking of stocks. CHAPTER VI CONCLUSION The r e s u l t s of the author's study do not refute the "Random Walk Hypothesis" f o r stock p r i c e s . However, though adopted i n the present study, the assumption that a single d i s t r i b u t i o n underlies a l l stock price changes f o r a given period may not be v a l i d generally, King f i n d s "that the movement of a group of security p r i c e changes can be broken down i n t o market and industry components." ^ In the present study, the "actual past performance" i s not s i g n i f i c a n t l y correlated with "observed" performance, even though both rankings involve stocks from d i f f e r e n t i n d u s t r i e s . I t i s possible that, at l e a s t f o r the present project, the assumption of a single d i s t r i b u t i o n underlying percentage pr i c e changes f o r a l l stocks i s an adequate assumption. "Predictions" i n a market following a random walk may be accurate where the "predictions" are based upon an appraisal of new information which a f f e c t s p r i c e s . Dependencies i n the information should not lead to dependencies i n p r i c e changes, as sophisticated traders w i l l take such dependencies into account, and t h e i r actions should tend to eliminate the dependencies i n p r i c e changes. Those analysts who can ^B.F. King, "Market and Industry Factors i n Stock P r i c e Behavior," p. 163. 32 33 c o n s i s t e n t l y p r e d ict the appearance of new information and determine i t s e f f e c t s on stock prices should show evidence of an a b i l i t y i n making "predictions.''" The Value Line Investment Survey f a i l s to evince a consistent a b i l i t y to forecast "Probable Twelve Months Market Performance Rank," the ranking of stocks according to t h e i r probable r e l a t i v e market pr i c e performance f o r the year. I t seems that r e s u l t s comparable to those based on the "predictions" could be gained by randomly ranking the same stocks. 34 SELECTED BIBLIOGRAPHY Alexander, S. "Price Movements i n Speculative Markets: Trends or Random Walks." Industrial Management Review, 2, no. 2 (May 1961), pp. 7-26. Alexander, S. "Price Movements in Speculative Markets: Trends or Random Walks, No. 2," in P.H. Cootner, ed. Random Character of Stock Market Prices. Cambridge, Mass., M.I.T. Press, 1964. Bachelier, L. "Theorie de l a Speculation" (doctoral dissertation i n mathematics, University of Paris, March 29, 1900). Annales de l'Ecole Normale Superieure, Ser. 3, XVII (1900), pp. 21-86. An English translation appears as pp. 17-75 i n P.H. Cootner, ed. The Random Character of Stock Market Prices. Cambridge, Mass., M.I.T. Press, 1964. Bernhard, Arnold. The Evaluation of Common Stocks. New York, Simon and Schuster, 1959. Board of Governors of the Federal Reserve System. Federal Reserve Bulletin. Washington, 1960-1964. Cootner, P.H. "Stock Prices: Random vs. Systematic Changes." Indus- t r i a l Management Review, vol. 3, no. 2 (May 1961), pp. 7-26. Cootner, P.H., ed. The Random Character of Stock Market Prices. Cambridge, Mass., M.I.T. Press, 1964. Cowles, A. "Can Stock Market Forecasters Forecast?" Econometrica, vol. 1 (July, 1933), pp. 309-324. Cowles, A. "Stock Market Forecasting." Econometrica, v o l . 12, nos. 3-4 (July-October, 1944), pp. 206-214. Fama, E.F. "Mandelbrot and the Stable Paretian Hypothesis." Journal of Business, vol. 36, no. 4 (October, 1963), pp. 420-4297 Fama, Eugene F. "The Behaviour of Stock Market Prices." Journal of Business, vol. XXXVIII (January, 1965), pp. 34-105. Fama, E. and Blume, M. " F i l t e r Rules and Stock-Market Trading." Journal of Business, vol. XXXIX (January, 1966), pp. 226-241. Fredrikson, E.B., ed. Frontiers of Investment Analysis. Pennsylvania, International Textbook Company, 1965. 35 Friend, I., Brown, F.E., Herman, E. and Vickers, D. "A Study of Mutual Fundsi Investment Policy and Investment Company Performance," in H, Wu and A.J. Zakon, ed. Elements of Investments. New York, Holt, Rinehart and Winston, Inc., 1965. Freund, J»E. Mathematical Statistics. New Jersey, Prentice-Hall, Inc., 1962. Graham, B., Dodd, D.L., and Cottle, S. Security Analysis. 4th ed. New York, McGraw-Hill Book Company, Inc., 1962. Granger, C.W.J, and Morgenstern, 0., "Spectral Analysis of New York Stock Exchange Prices." Kyklos, 16, (1963), pp. 1-27. Kendall, M.G. Rank Correlation Methods. London, Charles G r i f f i n and Co. Ltd., 1948. Kendall, M.G. "The Analysis of Economic Time-Series, Part Is Prices." Journal of the Royal Sta t i s t i c a l Society, vol. 96, Part I (1953), pp. 11-25. King, B.F. "Market and Industry Factors i n Stock Price Behavior." Journal of Business, vol. XXXIX (January, 1966), pp. 139-190. Lorie, J . "Some Comments on Recent Quantitative and Formal Research on the Stock Market." Journal of Business, v o l . XXXIX (January, 1966), pp. 107-110. Mandelbrot, B. "The Variation of Certain Speculative Prices." Journal of Business, vol. 36, no. 4 (October, 1963), pp. 394-419. Mandelbrot, B. "Forecasts of Future Prices, Unbiased Markets, and 'Martingale' Models." Journal of Business, vol. XXXIX (January, 1966), pp. 242-255. Moore, A.B. "Some Characteristics of Changes in Common Stock Prices" in P.H. Cootner, ed. The Random Character of Stock Market Prices. Cambridge, Mass., M.I.T. Press, 1964. Niederhoffer, V. "A New Look at Clustering of Stock Prices." Journal of Business, vol. 39, no. 2 (April, 1966), pp. 309-313. Osborne, M.V.M. "Brownian Motion i n the Stock Market." Operations Research, vol. 7 (March-April, 1959), pp. 145-173. Roberts, H. "Stock-Market 'Patterns* and Financial Analysis: Method- ological Suggestions" Journal of Finance, vol. 14, no. 1 (March, 1959), pp. 1-10. Sauvain, H. Investment Management, 2nd ed. New Jersey, Prentice-Hall, Inc., 1960. 36 The Value Line Investment Survey. New York, Arnold Bernhard and Co., Inc., 1960-1964. Williams, J.B. The Theory of Investment Value, Amsterdam, North- Holland Publishing Company, 1956. Working, H. "A Random-Difference Series." Journal of the American S t a t i s t i c a l Association, XXIX (1934), pp. 11-24. Wu, H. and Zakon, A. Elements of Investments. New York, Holt, Rine- hart and Winston, Inc., 1965. Wu, H. "Corporate Insider Trading P r o f i t s and the A b i l i t y to Forecast Stock P r i c e s , " i n H. Wu and A.J. Zakon, ed. Elements of Invest- ments. New York, Holt, Rinehart and Winston, Inc., 1965.

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