An Empirical Test of the Theory of Random Walks In

AN EMPIRICAL TEST OF THE THEORY OF RANDOM WALKS IN

STOCK MARKET PRICES: THE MOVING AVERAGE STRATEGY

GARRY CRAIG YIP

B.Com., University of British 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 Administration

We accept this thesis as conforming to the required standard

THE UNIVERSITY OF BRITISH COLUMBIA

August, 1971 In presenting this thesis in partial fulfilment of the requirements for

an advanced degree at the University of British Columbia, I agree that

the Library shall make it freely available for reference and study.

I further agree that permission for extensive copying of this thesis

for scholarly purposes may be granted by the Head of my Department or

by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.

Department of UCWUMU- ftjntl BwiJUv\U& ftcliwinjqJj^YU

The University of British Columbia Vancouver 8, Canada

Date fliuxuit • ABSTRACT

This study investigates the independence assumption of the theory of random walks in stock market prices through the simulation of the moving average strategy. In the process of doing so, three related questions are examined: (1) Does the past relative volatility of a stock furnish a useful indication of its future behavior? (2) Is the performance of the decision rule improved by applying it to those securities which are likely to be highly volatile? (3) Does positive dependence in successive monthly price changes exist?

The purpose of Test No. 1 was to gauge the tendency for a stock's relative volatility to remain constant over two adjacent intervals of time. As measured by the coefficient of variation, the volatility of each of the 200 securities was computed over the 1936 to 19^5 and 1946 to 1955 decades. In order to evaluate the strength of the relationship between these paired observations, a rank correlation analysis was performed. The results indicated a substantial difference in relative volatility for each security over the two ten-year periods.

In Test No. 2 a different experimental design was employed to determine whether the relative volatility of a stock tended to remain within a definite range over time. According to their volatility in the

1936 to 1945 period, the 200 securities were divided into ten groups. i i i

Portfolio No. 1 contained the twenty most volatile securities while

Portfolio No. 2 consisted of the next twenty most volatile, etc. An average coefficient of variation was calculated for each group over the periods, 1936 to 19^5 and 19^6 to 1955. The rank correlation analysis on these ten paired observations revealed that the most volatile

securities, as a group, tended to remain the most volatile.

Test No. 3 consisted of the application of the moving average

strategy (for long positions only) to forty series of month-end prices covering the interval, 1956 to 1966. These securities had demonstrated

a high relative volatility over the previous decade and, on the basis of

the findings reported in Test No. 2, it was forecasted that they would

be the most volatile of the sample of 200 in the period under investi•

gation. Four different moving averages ranging from three to six months,

and thirteen different thresholds ranging from 2 to 50 per cent were

simulated. The results of the simulation showed the moving average

strategy to be much inferior to the two buy-and-hold models. Every

threshold regardless of the length of the moving average yielded a nega•

tive return. In addition, the losses per threshold were spread

throughout the majority of stocks. Altogether, therefore, considerable

evidence was found in favour of the random walk theory of stock price

behav i or. 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 Filter 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 Volatility of Portfolios of Stocks Selected on the Basis of Prior Volatil ity 42

TABLE II. Selected Sample of Forty Securities .... 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 Price Subject to Random Movement Within Fixed Limits

Hypothetical Chart of a Stock Price Subject to Random Movement Within Periodically Changing Limits ACKNOWLEDGMENT

I would like to express my gratitude to Dr. William Wood for his valuable guidance in the preparation of this thesis. I am also indebted to Mr. Vincent S. Manis for his programming assistance and

Miss Susan Lew for typing both the rough and completed drafts. r

THE THEORY OF RANDOM WALKS AND ITS IMPLICATIONS

FOR FUNDAMENTAL AND TECHNICAL ANALYSIS

For decades the dynamics of price formation in the stock market

has been a subject of study for a sizeable segment of the investing

public. The reason for such overwhelming interest in this capital market

is attributable to several factors, most of them interrelated. To

begin with, it is recognized that this market is among the best organized

of the multitude and almost always ranks as the largest in terms of

value of sales. Also, it is widely known that speculative prices are

extremely sensitive to all events, both real and imagined, providing an

insight into the future. As one might expect, these elements operate

in conjunction with one another to cause a considerable total of gains

and losses to be experienced each trading day. Needless to say,

fluctuations such as these may present the opportunity for large profits

to be realized within a very short time horizon. This has undoubtedly

provided a constant attraction to men with dreams of instant wealth.

With numerous participants in the equities market it would be

natural to find a vast array of "magic" formulas for predicting stock

prices and indeed, such is the case. Investors have correlated the

movement of share prices to every conceivable phenomenon for forecasting

purposes. While an infinite number of formulas exist, there are only two basic approaches to investing, fundamental analysis and technical analysis. Under the fundamental philosophy the analyst, in searching

for bargains, bases his valuation of a security upon those factors which he believes other rational investors use to assess value. In general

terms these factors consist of the expanse of information contained in economic and financial statistics. More specifically they include earnings, dividends and assets of the firm represented by the security

under observation. All such data, past or present, is taken into account to appraise the value of the share or as it is commonly referred

to, the intrinsic value of the share. The belief of the fundamentalist

that undervalued securities exist, implies an imperfect market. He attempts to capitalize on the asset price disequilibrium situation by purchasing a bargain and expects the market later to adjust to an equilibrium state. Implicitly the fundamentalist is assuming that

imperfect knowledge characterizes the market place and that trends will develop from the gradual spread of awareness of the relevant facts.

Unlike the fundamental approach, technical analysis is con• cerned with predicting share prices solely on the basis of past

information. A direct result is that rational investment behavior is not a critical assumption for the technician. The market continually and automatically weighs all factors including not only those relevant

to the fundamentalist but also opinions, moods, and guesses. In one

important respect a parallel can be drawn between fundamental and

technical analysis. They both infer that the market is not perfect by assuming that facts at one moment of time will govern prices at some future date. For the technician this is in the form of a belief that stock prices tend to move in trends which persist for an appreciable length of time and that these trends can be uncovered by studying price movements of the immediate past.

While numerous professional practitioners endorse the technical approach to investing, academics have not followed suit. Rather, they suggest that stock prices behave as a random walk and that future movements of prices could just as well be determined by tossing a coin as by any sophisticated analysis of past price changes. This serious threat to the technician's livelihood has engendered a commensurate degree of controversy between the respective parties.

"The theory of random walks in stock prices is based on two hypotheses: (1) successive price changes in an individual security are independent, and (2) the price changes conform to some probability distribution."^ Of the two, the first is more important. If dependence in successive increments of prices exists then the theory is not valid.

The second hypothesis is not nearly as definitive because the form or shape of the distribution need not be specified. Consequently, any distribution which satisfies the condition of correctly characterizing the process generating the price movements is consistent with the theory.

Eugene F. Fama, 'The Behaviour of Stock Market Prices," Journal of Business, 38 (January, 1965), P- ^0». In statistical terms independence means that the probability distribution for the price change during time period t is independent of the sequence of price changes during previous time periods. That is, know• ledge of the sequence of price changes leading up to time period t is of no help in assessing the probability distribution for the price change during time period t.2

The fact that there is probably no hope of ever finding a time series which exhibits perfect independence precludes the theory of random walks from being an exact description of stock price behavior. Nevertheless, the independence assumption of the model may for practical purposes be accepted as long as the dependence in the series of successive price changes is below some specified level. For the statistician the defini• tion of what constitutes a significant violation of the independence assumption depends upon the particular statistical tool applied. Since the market practitioner is primarily interested in profitability, the relevant criterion for him in judging the validity of the independence assumption may be stated accordingly: the random walk model is valid as long as the knowledge of the actual degree of dependence in the series of price changes cannot be used to increase expected gains above that which is attainable under a buy-and-hold policy.

There are a number of market situations compliant with

independence. Louis Bachelier (1900), who is credited with introducing the notion of speculative prices following random walks, submitted in a nebulous manner the simplest explanation for the independence assumption. This was later refined and stated explicitly by M. F. M. Osborne. Both theoreticians believed that: "... If successive bits of new information arise independently across time, and if noise or uncertainty concerning intrinsic values does not tend to follow any consistent pattern, then successive price changes in a common stock will be 3 independent." In this instance a perfect market is envisioned. When viewed within the context of the market place itself the assumptions embodied in the Bachelier-Osborne model appear rather extreme. As advocated by Eugene F. Fama in his study, "The Behaviour of Stock Market 4

Prices", there are no strong reasons for expecting either each indi• vidual 's estimates of intrinsic values to be independent of the estimates made by others, or successive bits 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 bullish influence to be followed by the same type of news more often than by pessimistic news.

A logical extension of this line of reasoning would be to acknowledge the possibility of such dependencies causing price changes to be dependent.

In defense of the independence assumption in stock prices it is often contended that the effects of systematic dependencies may be neutralized by offsetting mechanisms. This rationale has been elaborated upon by Fama in the aforementioned article and takes the following form.

31 bid., p. 37.

^1 bid, pp. 34-105. If there are opinion leaders in the market (dependence in the noise generation process), "bubbles" (dependence in successive price movements) would tend to arise in a price series. In other words the accumulation of the same type of noise would cause the price level to run well above or below the intrinsic value. Recognition of this occurrence by sophisticated traders would provide them with an incentive to sell the security or to sell it short in the first case and buy it in the second, since they expect actual prices to move eventually back toward intrinsic values. Provided that there are enough of these traders, the effect of their actions would be to eliminate the "bubbles". Correspondingly, price movements would approximate a random walk.

If there are dependencies in the process generating new informa• tion this will also tend to create dependence in successive price changes of a security. The actions of many sophisticated analysts, as a result of interpreting both the price effects of current new information and of the future information implied by the dependence in the information generation process, would again tend to make price changes independent.

Even in situations where there is consistent vagueness or uncertainty surrounding new information, independence in successive price changes could still occur. For example, uncertainty concerning the importance of new information may cause the market to consistently underestimate

(or overestimate) its effects on intrinsic values. Since prices generally will not adjust instantaneously to their intrinsic values the astute trader with knowledge of the underestimation phenomenon could profit by purchasing securities when the new information is optimistic. If several traders attempt to capitalize on this opportunity any con• sistent lags in the adjustment of actual prices to changes in intrinsic values will tend to be removed.

With regard to the above discussion, it is implicitly assumed that actual prices will adjust almost instantaneously to the full effects of new information, provided there are a sufficient number of astute traders in the market. However, as recognized by Fama, "instantaneous adjustment" really bears two significant implications because there is, in fact, ambiguity regarding new information.

First, actual prices will initially overadjust to the new intrinsic values as often as they will underadjust. Second, the lag in the complete adjustment of actual prices to successive new intrinsic values will itself be an independent random variable, sometimes preceding the new information which is the basis of change (i.e., when the information is anticipated by the market before it actually appears) and sometimes following.5

In effect then, the actual price of a security will fluctuate randomly about its intrinsic value.

Although it has been demonstrated that the stock market may conform to the independence assumption of the random walk model, even if systematic dependencies are present, the question of whether or not this rationale is an accurate portrayal of reality remains. Its validity is contingent upon the existence of several sophisticated traders who are well informed and well endowed. If one uses the criterion of consistent gains in the market to detect these sophisticated traders he will probably find that very few, if any exist. For example, in a study on mutual fund performance, Fama discovered that funds in general seem to do no better than the market. Another significant result of the investigation was that individual funds do not seem to outperform con• sistently their competitors. With such evidence in mind, one must conclude that this complex effort to justify the independence assumption suffers from the same weakness as the simple rationale proposed by

Bachelier and Osborne, that is, it is not substantiated by fact.

Since most articles on the theory of random walks have been written by and for academics, the investment community has encountered difficulty in assessing its ramifications, two of which require attention. The random walk theory does not imply that superior invest• ment performance is impossible. In a dynamic economy there will always be new information to effect changes in intrinsic values over time.

Consequently, the opportunity for above-average profits will forever be available to those who can consistently predict the appearance of new information and evaluate its effects on intrinsic values better than others. The fact that the activities of these superior intrinsic-value analysts (fundamentalists) serve to produce independence in successive price changes does not imply that their expected profits cannot exceed those of the investor who pursues some buy-and-hold policy. What the theory does suggest is that superior performance is extremely difficult to attain for the average investor. Under the random walk theory prices are, on the basis of all available information, the best estimates of intrinsic values. Thus fundamental analysis is a useless tool to the average analyst whose only aid consists of publicly distributed

i nformat ion.

As previously mentioned, the basic challenge of the theory is to the technicians. Although his activities may help to make successive price increments independent) once independence is established his chance for excess profit is lost. The past history of the price series will no longer reveal any dependencies and as a result cannot be used to

increase expected profits.

The second point which should be made clear is that the random walk theory is not inconsistent with a rising trend of stock prices. The

independence hypothesis is concerned only with the sequence of successive price changes and not their average magnitude. "Whereas the random walk hypothesis does not imply that any price change is as likely to be a fall as a rise, it is approximately true that any change is as likely to be below the stock's average change as above."

Most empirical investigations of the random walk theory have been of a statistical nature involving tests of the series of prices over time. For example, in testing the validity of the independence assumption frequent use has been made of the serial correlation model which measures statistically the closeness of the relationship between successive price changes. An alternative technique that has proved

Richard A. Brealey, An Introduction to Risk and Return from Common Stocks (Cambridge, Massachusetts: The M. I.T. Press, 1969), p. 18. equally popular is a "runs" analysis which assigns an equal weight to each price change. A "run" is defined as a sequence of price changes of the same sign. Since there are three different possible types of price changes (+, 0, -), there are three different types of runs. If there is a tendency for runs to persist, that is, for a move in one direction to be succeeded by a further such move, the average length of run will be

longer and the total number of runs will be less than if the moves were distributed randomly.^ Statistical tests employing these techniques and others concerned with investigating probabi1ity distribut ions have

largely supported the hypothesis that stock prices are independently dis• tributed random variables.

Since statistical analyses do not directly test the principles of technical analysis, it is difficult to infer that the random walk model is adequate for the investor. Moreover, the chartist would probably argue that common statistical tools are insufficiently powerful to detect the dependencies seen by him in the series. The grounds for such a contention are indeed present. For example, the serial correlation model is able to identify simple linear relationships only and not non•

linear ones. A runs test, on the other hand, is too rigid in determining the duration of upward and downward movements in prices, for it ignores the magnitude of the price change that causes the reversal in sign. The technician would require a more sophisticated method to identify trends --a method that does not always predict the termination of the movement 8 simply because the price level has temporarily changed direction.

The above problems have been solved by the second method of empirical testing involving the use of mechanical trading rules. If the market is characterized by independence of successive price changes, mechanical trading rules should not be able to produce profits in excess of those obtained under a buy-and-hold policy. The evidence from this type of research has been far from conclusive. Studies in which trading rules are seemingly "profitable" often contain certain biases and ignore transaction costs. In the following chapter on filter rules such errors will be outlined in detail.

Eugene F. Fama and Marshall E. Blume, "Filter Rules and Stock Market Trading," Journal of Business: Special Supplement, 39 (January, 1966), p. 227. CHAPTER I I

FILTER RULES

One of the first empirical investigations involving the use of a mechanical trading rule is the work of Professor Sidney S. Alexander.

In his study, "Price Movements in Speculative Markets: Trends or

Random Walks'1,-^ a trading rule called the filter technique is employed to identify movements in stock prices. Alexander formulated this decision rule to verify the market professionals' belief that noise may be generated in a dependent fashion causing prices to adjust gradually to new information. He, therefore, tentatively assumed the existence of trends but believed them to be disguised by the jiggling of the market. 'The path of a speculative price might, accordingly, be repre• sented by a sum of two components, a smooth underlying trend or cycle changing direction only infrequently, and a much shorter cycle of action and reaction."^ (Reaction in this instance is associated with profit taking.) To test this hypothesis he suggested filtering out all movements smaller than a specified size and examining the remaining movements.

^Sidney S. Alexander, "Price Movements in Speculative Markets: Trends or Random Walks," The Random Character of Stock Market Prices, ed. Paul H. Cootner (Cambridge, Massachusetts: The M.l.T. Press, 1964), pp. 199-218. An x per cent filter can be defined as follows: Open a long position if the daily closing price of a particular security rises at

least x per cent, and maintain this position until its price falls at

least x per cent from a subsequent high, at which time sell and go short an equivalent amount. The short position is held until the daily closing price rises a minimum of x per cent above a subsequent low at which time one covers and buys. Moves less than x per cent in either direction are ignored. A trend, therefore, once established is considered to be in effect until a price change causes a reversal in direction by some predetermined amount. As noted by Alexander, the choice of a filter size involves a tradeoff between risk and expected

returns:

Thus, as the filter size is increased, the number of transactions is reduced, and losses on small moves are eliminated, gains on large moves are reduced, and some moves which would yield gains with a small filter will yield losses with a large. This example illustrates the familiar tradeoff between reliability of the infor• mation and the cost of the information. The more stringent the filter, the higher the reliability, but the more of the move that is sacrificed in identifying it both in getting in and in getting out.

In the study, filters ranging in size from 5 to 50 per cent were tested on the data. The sample consisted of closing "prices" for

two indexes, the Dow-Jones Industrials from 1897 to 1929 and Standard

and Poor's Industrials from 1929 to 1959. Generally, filters of all

different sizes and for all different time periods yielded superior profits to those of a buy-and-hold policy. (The largest profits were associated with the smallest filters.) From these results Alexander concluded that stock price changes could not have been generated by a random walk.

It was later recognized by Benoit Mandelbrot that the estimated profits from the use of the filters were subject to a bias exaggerating the profitability. In each transaction Alexander assumed that his hypothetical trader could always buy or sell at a trough or peak t x per cent. It is apparent that a trader operating in the market place could not consummate his transactions at these prices due to the frequency of large price jumps (changes). As a result the purchase price will often exceed the low plus x per cent, while the sale price will often be

1 3 below the high minus x per cent. After correction for the bias, in a following study entitled, "Price Movements in Speculative Markets:'' 14

Trends or Random Walks, No. 2", Alexander found the profitability of the filter technique substantially reduced. More specifically, the filters only rarely compared favorably with the buy-and-hold model.

No allowance was made for transaction costs in the computation of profi ts.

1 3 To eliminate the bias, Alexander assumed that transactions were made at the closing price of the confirmation day rather than at x per cent away from trough or peak. 14 Sidney S. Alexander, "Price Movements in Speculative Markets: Trends or Random Walks, No. 2," The Random Character of Stock Market Pr ices, ed. Paul H. Cootner (Cambridge, Massachusetts: The M.l.T. Press, 1964), pp. 338-372. In addition to reworking his earlier results, Alexander modified the filter rule slightly and tested it on a different sample. On this occasion he applied logarithmic filters to the daily closing prices of

Standard and Poor's Industrials for the 9,592 trading days from

January 3, 1928 to December 29, 1961, inclusive. Ten separate filters were used with the smallest being 1 per cent and the largest, 45.6 per cent. Excluding commissions nearly all of the filters produced profits above those of buy-and-hold. The 1 per cent filter proved to be the most profitable, with terminal capital forty-one times as large as the

initial capital. Buy-and-hold yielded a terminal capital about five

times the initial capital. When transaction costs were included, only

the largest filter beat buy-and-hold.

A second bias present in both of Alexander's works was pointed out by Eugene F. Fama and Marshall E. Bl ume in their paper, "Filter

Rules and Stock-Market Trading". Since Alexander tested the filters on common price indices it was impossible to include the effect of dividends in the computation of returns. This shortcoming reduced the

profitability of the buy-and-hold policy as the total return on an

investment consists of the capital appreciation for the time period plus any dividends that have been paid. Furthermore, it overstated the profitability of the filters by failing to take account of the fact that

Eugene F. Fama and Marshall E. Blume, "Filter Rules and Stock Market Trading," Journal of Business: Special Supplement, 39 (January, 1966), pp. 226-241. in a short sale the borrower of the securities typically reimburses the lender for any dividends paid while the short position is outstanding.

Fama and Blume applied twenty-four different percentage filters ranging from 0.5 per cent to 50 per cent to series of daily closing prices for each of the individual securities of the Dow-Jones Industrial

Average. The initial dates of the samples varied, but most covered the period from 1957 to 1962. Returns under the filter technique were computed in several different ways: gross and net of brokerage fees, with and without dividends, etc. They analyzed the results first by security (the nominal annual rates of return by company: averaged over all filters) and then by filters (the nominal annual rates of return by filter: averaged over all companies).

When commissions were included none of the thirty securities exhibited returns in excess of those obtained under the buy-and-hold policy. In fact, only four securities had positive returns. A compari• son of the profits before commissions under the two strategies indicated that the filter technique was inferior to buy-and-hold for all but two securities. The discrepancy between this last result and the findings of Alexander when he employed logarithmic filters is explained by the bias resulting from his use of price indices. Fama and Blume found that the adjustment for dividends increased the average advantage of buy-and- hold over the filter technique by at least two percentage points. Had such an adjustment been applied to Alexander's data, the profitability of his filters would probably have decreased markedly. Further evidence negating the value of the filter technique (in the Fama and Blume study) was provided by a breakdown of gross returns for long and short transactions. Only one security had positive returns per filter for short positions initiated by the rule. For all securities, the average return on short transactions was -12.79 per cent while the average return from buy-and-hold was 9.86 per cent, a spread of 22.65 per cent. On long positions thirteen securities demonstrated higher average returns per filter than the corresponding returns from buy-and-hold. However, averaging over-all securities, the return on these transactions also failed to surpass the average return from buy- and-hold, the difference in this case being 1,6k per cent.

The analysis of results by filter disclosed that virtually every filter produced net returns below those of buy-and-hold. This, together with the preceding observations, supported the conclusion that the filter technique could not be used to increase the expected profits of the investor who is charged regular brokerage commissions.

The breakdown of gross returns (by filter) for long and short transactions unveiled very slight amounts of dependence in the price changes. Fama and Blume applied the following criterion to detect dependence:

Note that if successive price changes conformed strictly to the random-walk model, the average returns per security on long positions should be approximately equal to the average returns from buy-and-hold while the average returns on short positions should be approximately equal ^ to the negative of the average returns from buy-and-hold. For three filter sizes, 0.5;, 1.0 and 1.5 per cent, the average returns

per security on long positions exceeded the average return from buy-and-

hold. In addition the same filter sizes showed losses on short positions of a lesser magnitude than the gains from buy-and-hold. This behavior of returns on the smallest filters constituted evidence of positive dependence in very small movements of stock prices. When the average

returns per security on long and short positions for filters 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

size price movements was also furnished.

The dependencies uncovered by the two authors were used to

suggest trading procedures for a floor trader. Although the rules pro•

duced greater gross returns than those of buy-and-hold, they could not

be used to increase expected profits. The clearing-house fees and costs

of search were more than sufficient to erase any advantage of the filter

rules over buy-and-hold. Thus Fama and Blume's results added further to

the evidence that for practical purposes the random walk model is an

adequate description of price behavior.

In a related study entitled, "Random vs. Systematic Changes",^

Paul H. Cootner used a variation of the filter technique to test his

hypothesis that stock prices behave like a restricted random walk. The

'7paul H. Cootner, "Stock Prices: Random vs. Systematic Changes", The Random Character of Stock Market Prices, ed. Paul H. Cootner (Cambridge, Massachusetts: The M. I.T. Press, 1964), pp. 231-252. rationale10 for this supposition pertains to the activities of two groups of investors—amateurs and professionals. Since amateurs are engaged in other types of employment in which they possess a comparative advantage, it is very costly 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 result, they tend to regard present prices as roughly representing true differences in value, and choose between securities on the basis of their liquidity require• ments and risk preferences. Those who do utilize information about future prospects as a criterion for selection among stocks are just as likely to be incorrect as not in their forecasts of prices. Thus, the activities of amateur investors tend to induce randomness in a price series.

Because professionals are more familiar with analyzing securities and possess access to privileged information, their oppor• tunity cost of research is minimal. They attempt to predict events in the future, but profits cannot be gained through the use of such insight unless the current price diverges from the expected price by an amount in excess of their opportunity costs. Their profits will come from observing the random walk of stock market prices produced by amateurs and recognizing the situation where the price has wandered sufficiently far from the expected price to warrant the prospect of an adequate return. Competition among these professionals will tend to limit the potential profit to opportunity costs. If it is assumed that all professionals have identical expecta•

tions and the same opportunity costs, then prices should behave as a

random walk with reflecting barriers (Figure 1). As long as prices are within the upper and lower boundaries, they are determined by the actions of amateurs and therefore tend to move like a random walk. However, if prices reach either limit, the activities of professionals will prevent

them from continuing in the same direction. In such an undertaking a measure of risk is incurred by these participants, for they cannot be

certain that their estimates of values are correct or that other professionals share their estimates. Moreover, even if their expecta•

tions are correct, their rate of return on investment is still a

stochastic variable since the rate at which the price converges upon the expected price is subject to the random process operating between the

barriers.

Another sort of random walk environment is liable to occur

because of random changes in the price expectations of professionals.

This could be caused by independence in the process generating new

information. Given such a phenomenon there probably would also be

random changes in the trends around which the random walk takes place.

That is, the path of stock prices over any substantial period of time would be composed of a random number of trends, each of which is a

random walk with reflecting barriers (Figure 2).

Two implicit assumptions were made by Cootner: --that

dependence in the noise generation process exists, and, that a lag in the

information distribution process causes prices to adjust gradually to new information. In an attempt to profit from the alleged shifts in the whole trading range (major price movements initiated by the changes in expectations of professionals), he employed a modification of the filter technique whereby a forty week moving average is substituted for reference peaks and troughs. The rule is defined accordingly: If the current price is higher than the moving average, buy and then hold the stock until the price falls below the moving average, at which time, sell. If the current price is less than the moving average, sell short, and cover when the price rises above the moving average. Two advantages of this strategy over the filter procedure are:

First, it would enable a follower to sell (buy) a stock when it stopped rising (falling) along the previously defined trend, rather than waiting for a substantial reversal. Second, it would permit an investor the alter• native of holding cash rather than adopting a position in either direction--as the filter rule (though not Houthakker's stop-loss) requires.^

The rule was applied to weekly observations of forty-five stocks listed on the New York Stock Exchange which, except for six series, spanned the interval from 1956 to i960. When commissions were excluded, the moving average strategy proved to be far superior to buying and holding. While this was indicative of non-randomness, the extent of the non-randomness could not be used to increase the expected profits of the investor; after allowance for commissions, the moving average strategy yielded inferior returns.

Ibid., 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 fell 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

still 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 filter technique and moving average strategy cannot be used to

uHowever, as with Alexander's filter technique, a tradeoff was involved since a larger part of the move was sacrificed for this greater reliability. increase the expected profits of the typical investor. All of the

investigations revealed that the rules generated inferior net returns to those of a buy-and-hold policy. Nevertheless, because the tests have not exhausted every possibility, it is still conceivable that these mechanical trading rules could produce above-average profits. An approach which looks promising will 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 Filter Rule and Moving Average Strategy

The profitability of both the filter technique and moving average strategy is determined by the size of the filter (threshold) and the magnitude of the swings in prices which establish the peaks and troughs. A better perspective of this is gained when the path traced by security prices is viewed within the following context. As suggested by

Seymour Smidt, a wave-like motion around the long-term upward trend could account for the symmetric pattern in the stock price sequence implied by the filter results of the Fama and Blume paper. For almost every filter size the actual returns earned by long and short positions are displaced from their expected returns by approximately the same absolute magnitude. The fact that the number of transactions decreases as the filter size increases suggests a mixture of waves of different amplitudes. Where the amplitude of the wave is less than the filter size, transactions do not take place for such waves are filtered out, and excluded from consideration:

^'Seymour Smidt, "A New Look at the Random Walk Hypothesis," Journal of Financial and Quantitative Analysis, 3 (September, 1968), pp. 235-261. Suppose that the trough of a wave occurs when the stock is priced at 100 dollars per share and the peak when the stock rises to 100 + A dollars. The amplitude of the wave is A per cent. With an I per cent filter (assuming A>l), a buy signal will be given when the price reaches 100 + I dollars. A sell signal will occur when the price passes the peak and declines to 100 + A - I dollars. Thus the profits on this long open position will be A - 21 dollars or A - 2 I per cent before commissions. In general, the trading rule will produce gains (losses) if the average amplitude of waves that exceed the filter size is greater than (less than) twice the filter size.22

A similar but not identical mathematical relationship holds for the moving average strategy. The slight difference arises because the buy and sell signals for this rule are interpreted relative to the trend defined by the moving average.

In the preceding discussion the term 'wave' has been used in a metaphorical sense:

It is not suggested that if the price series were plotted that one would necessarily be able to observe a persistent wave-like pattern. The waves must be defined statistically in terms of the conditional pro• babilities of price changes given that the price is already in a certain relation to the previous peak or trough.23

It is apparent that the necessary conditions for the maximization of returns are: the existence of a high degree of positive dependence

in movements of stock prices and the fluctuation of prices over a wide range. Since the volatility aspect of security prices has been over•

looked, an opportunity for improved performance of both trading rules

^jbid., p. 248.

23!bid., p. 249. remains unexploited. |n the aforementioned studies and all others of a

related nature known by the author, no overt attempt is made to restrict the sample of those securities which are likely to be highly volatile.

The correction for such a factor provides the foundation for this thesis.

Whether or not an empirical test of the subject matter under consideration is warranted depends of course on the extent to which the price variation of securities can be predicted. In this regard it has been suggested that the relative volatility exhibited by any stock may be consistent over successive intervals of time. As submitted by

Richard A. Brealey such a phenomena could result from a direct

relationship between uncertainty and price variability, wherein the causes of uncertainty tend to persist over time. The margin for error in forecasting company prospects is liable to be greater if either the range of possible outcomes is wide or there is little information on which to base a forecast. The former condition will arise when the concern is, in the broadest sense, highly leveraged, the latter condition when either the business is very individualistic in character or management is very secretive about operations. It seems improbable that these characteristics are typically transitory. For example, most metal-refining companies are likely to continue to possess high operating leverage, advanced-technology businesses should continue to be very individualistic, and companies working on classified contracts should continue to be secretive. If this reasoning is correct and the causes of uncertainty do persist over time, it is probable that stocks that are most volatile in one period will tend to be the most volatile in the next.25

^Richard A. Brealey, An Introduction to Risk and Return from Common Stocks (Cambridge, Massachusetts: The M.l.T. Press, 1969). Brealey found empirical evidence supporting his hypothesis in

26 an unpublished doctoral dissertation by Shannon P. Pratt who investi• gated the subsequent behavior of 3^8 sets of five portfolios of stocks selected on the basis of prior volatility. In each set, portfolio A consisted of the 20 per cent of the New York Stock Exchange stocks that displayed the least variation in returns over the previous three years, while portfolio B consisted of the next 20 per cent, etc. The average subsequent experience (in index form) of the 348 sets revealed that the ranking of the five portfolios did not change over time. That is, portfolio A continued to demonstrate the least variation in returns, while portfolios B, C, D and E continued to exhibit higher levels of variation respectively. Statistics regarding the probability of subse• quent loss on the portfolios also favoured Brealey's hypothesis. If, on the average, portfolio A tended to show less violent changes in value than portfolio E, it is probable that over any one period the former would have been less likely to suffer an actual loss. As expected, those portfolios composed of stocks that had been less variable in former years resulted less frequently in loss. In its entirety 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 portfolios formed after 1931 were considered, the observa• tions were similar. Changing the lengths of the periods over which the

^"Shannon P. Pratt, Relationship Between Risk and Rate of Return for Common Stocks (unpublished D.B.A. dissertation, Indiana University, 1966). volatility was measured did not affect the conclusions either.

Although Pratt's results provide substantial evidence that the relative volatility exhibited by any stock has tended to persist over time, further confirmation seems appropriate since such a characteristic may have only held true for the particular sample and methodology chosen.

In deference to this consideration tests of a somewhat different nature have been undertaken here to determine the relevance of historical volatility as a basis for forecasting future price variation. It will later be shown that the past volatility of a stock does in fact furnish a useful indication of its future behavior.

While correction for the volatility aspect may improve the per• formance of both trading rules so also might the use of monthly data.

For this to occur it is essential that the bulk of relevant information, guiding a speculator to profit, diffuse throughout the market place over a lengthy interval of time. Such a prospect appears likely if one considers the time element involved in the distribution of research reports by professional analysts. Prior to dispensation the analyst must seek the approval of his investment committee, which generally meets once a week. If it is assumed that the large institutional investors

(e.g. mutual funds, life insurance companies, etc.) are the first to be notified, then another week will have passed before any action is taken on the recommendation as their policy committees also convene on a weekly basis. By the time the information is released to the amateurs

(odd-lot investors) through the branch offices of brokerage firms, and

is acted upon, four to six weeks could have elapsed since the report was initially submitted by the analyst. If this type of dependency exists, then a gradual adjustment of prices over a monthly time horizon is a distinct possibility. The testing of such a hypothesis adds yet another dimension to this thesis.

I I. The Data

The basic data file utilized in this analysis was compiled by the Center for Research in Security Prices at the University of Chicago.

It was made available for use in the form of a magnetic tape at the

University of British Columbia Computer Center. Included in the file are 27 month-end prices (adjusted for stock splits and stock dividends ) of all securities listed on the New York Stock Exchange over the period

January, 1926 to June, 1966. 28

From the entire population of 1,952 securities a sample of

200 was drawn. All of the selected series covered the three successive intervals: January, 1936 to December, 19^+5 (period |); January, 1946 to

December, 1955 (period ||); January, 1956 to January, 1966 (period Ml).

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 relative volatility of a common share to remain constant. Observations over the 1956-1966 horizon were employed in Test No. 3 to simulate a trading rule.

^Data after March, I966 are unadjusted for capital changes.

2^A11 of the empirical results, including a listing of the sample, are available in the computer.statements for this study. A copy can be obtained from Dr. William Wood, Department of Finance, Faculty of Commerce and Business Administration, University of British Columbia. The coefficient of variation was used as a measure of price variability. It is defined as the standard deviation divided by the mean

(OT/x) and as such, expresses the magnitude of the dispersion relative to the quantity being gauged. The fact that this statistic furnishes a basis for comparing different frequency distributions permitted the ranking of stocks according to volatility.

III. Test No. 1; The Experimental Procedure and the Results

The volatility of each of the 200 securities was computed for both period I and period ||. In order to test the strength of the rela• tionship between these paired observations, a rank correlation analysis was performed. The results are presented below:

Spearman Rank Test of Number of Correlation Coefficient S i gn i f icance Degrees of Freedom

.309 4.57 198

As revealed by the degree of correlation (.309) significant at the .0010 level, the relative volatility of each of the 200 securities differed substantially over the two periods. However, it should be noted that this finding does not necessarily contradict the evidence reported by Pratt. The discrepancy between results may be attributed to the more stringent nature of the test used here. While Pratt investigated the subsequent behavior of five portfolios of stocks selected on the basis prior volatility, this experiment dealt with the price variation of 200

individual securities. It is conceivable that a tendency exists for the relative

volatility of any security to remain within some definite range as sub•

stantiated by Pratt's analysis even though a tenuous relationship ;iis

indicated by the above statistic. For example, the twenty most volatile

securities in period I may have been the twenty most volatile in period

II. Furthermore, within this group a material shift in rankings may have occurred. If such a phenomenon prevailed throughout other groups of

stocks it could account for the low rank correlation coefficient of .309.

In this particular case the test applied is insufficiently powerful to

detect any propensity for the relative volatility of a security to stay within specifiable limits. To.correct for such a deficiency, a method•

ology similar to that employed by Pratt is utilized in Test No. 2.

IV. Test No. 2: The Experimental Procedure and the Results

On the'basis of their volatility (coefficient of variation) in

period I, the 200 securities were classified into ten groups. Portfolio

No. 1 included the twenty most volatile stocks while each of the remain•

ing portfolios in turn consisted of another twenty stocks exhibiting

lower levels of price variation. As befits its position, Portfolio No. 10

contained the twenty least volatile securities. For each group an

average coefficient of variation was calculated over both period I and

period I I while a rank correlation analysis was conducted for the purpose

of testing the closeness of the relationship between these ten paired

observations. The results are as follows: Spearman Rank Test of Number of Correlation Coefficient S ign i f icance Degrees of Freedom

.903 5.95 8

As indicated by the high degree of correlation (.903), signifi• cant at the .0010 level, there was a strong tendency for the demonstrated relative volatility of each portfolio to persist over time (see Table I).

That is, as a group the most volatile securities in period I tended to be the most volatile in period Ii. Consequently, this finding adds further to the evidence (submitted by Pratt) that the relative volatility of a security tends to remain within a fairly narrow range. Since there

is no reason for expecting this characteristic to diminish in the future, the past variation in prices should continue to provide a useful

indication of the future variation.

V. Test No. 3: The Experimental Procedure and the Results

While the performance of both the filter technique and moving average strategy might be improved by restricting the sample to highly volatile securities, only the latter hypothesis has been tested here.

The trading rule employed is defined as follows: the stock is to be purchased when the price rises above the moving average by more than x per cent and is to be sold whenever the price falls below the moving average

by any amount. No allowance was made for short sales because the posi•

tive drift in stock prices over the period examined limited the prospects of gain in this case. The rule was applied to forty series of month-end

prices extending over the interval, January, 1956 to January, 1966 (period III). These securities were selected because of their high relative volatility in period II. On the basis of the results in the preceding test, it was predicted that the most volatile stocks in period

II would be the most volatile in period III.

Four different moving averages ranging from three to six months, and thirteen different thresholds (filters) ranging from 2 to 50 per cent were simulated. Although the initialization date for the rule varied according to the length of the moving average, the termination date was always January, 1966. At this time all outstanding positions were closed out. For each transaction which consisted of a purchase and a sale an annualized rate of return, adjusted for round-lot commissions, was calculated from the following formula:

Pt+1 - Pt - C 12

Y X t - Ft -

Yt = the annualized rate of return

Pt+l - the sale price

Pt = the purchase price

C - the combined round-lot commission on a purchase and a sale

n s the number of months the security is held

These returns were averaged to yield a mean annualized rate of return for each security and threshold size.

A listing of the sample is shown in Table II. Corresponding returns for two buy-and-hold strategies were com• puted to provide a standard of comparison. The Fisher Investment

30 ,

Performance Index (for all securities listed on the New York Stock

Exchange over the span, January, 1926 to June, 1966) was used to simulate one of the strategies while a related price index was employed to simulate the other. Although cash dividends were taken into account in the formulation of the first index, the commissions on the reinvestment of these dividends were omitted. Consequently, the calculated average annual rate of return for this buy-and-hold model is, to some degree, overstated.

The results of the simulation of the trading rule appear in

Tables III, IV, V, and VI for moving averages ranging from three to six months respectively. For each threshold an average annual rate of return is shown together with the total number of transactions, the average number of transactions per stock, and the number of profitable securities.

The corresponding return for the investment performance index and price index is also included.

The results, without exception, indicate that the moving average strategy is vastly inferior to both buy-and-hold models. Rather than producing above-average returns, the trading rule led to a negative average annual rate of return for every threshold. The losses per threshold, as revealed by an analysis of the returns for each security,

^Lawrence Fisher, "Some New Stock-Market Indexes," Journal of Business: Special Supplement, 39 (January, 1966), pp. 191-225. are spread throughout most of the forty stocks. In the majority of instances fewer than 25 per cent of the securities (for which there is at least one recorded transaction) demonstrate any profit whatsoever under the rule. It is usually the same stocks that fall into this cate• gory, regardless of the threshold size or moving average length. Of the limited number which exhibit a positive return, only two securities,

Texas Instruments and Rexall Drug and Chemical Co., frequently show gains in excess of those generated by the investment performance index.

As disclosed by a comparison of the results of the different moving averages, the returns for small thresholds (2 to 10 per cent) are only marginally improved by using a longer period to define the trend.

Such is not the case, however, for the large thresholds since the losses are significantly reduced by employing the six month moving average.

With respect to the profitability of the thresholds a major discrepancy exists between the different sets of results. The large thresholds under the three month moving average show losses far above those of the other thresholds. In contrast, the large thresholds under the six month moving average show the smallest losses of all thresholds. These con• flicting results suggest that no one particular 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 this study. To begin with, it is conceivable that each of the selected securities did not exhibit a relative volatility in period III consistent with that in period ||. Such a possibility, however, appears remote in light of the evidence reported in the preceding test (and by Pratt). Stocks which were the most volatile in period I tended to be the most volatile in period II. A more plausible rationale--and one which can be inferred directly from the available results—is that positive dependence does not predominate in successive monthly price changes. In this respect the findings support the random walk theory of stock price behavior.

Both Fama/Blume and Cootner found evidence suggestive of positive dependence in daily and weekly price changes respectively, although of an insufficient extent to lead to a remunerative filter and moving average strategy. That is, their trading rules for certain filters and thresholds were superior to simple buying and holding on a gross profit basis only. Even if brokerage commissions had been ignored in the investigation here, the decision rule, regardless of the size of thres• hold or length of moving average, would probably have failed to outperform the buy-and-hold models. The implication which arises out of this comparison is that the flow of relevant information from professional analyst to amateur does exist, but occurs within 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 rules used by academicians to investigate the theory of random walks. It can be defined accordingly: buy the stock when the price exceeds the moving average by more than the threshold amount and sell the stock when• ever the price falls below the moving average by any amount. A short sale is indicated whenever the price falls below the moving average by more than the threshold amount and is covered whenever the price rises above the moving average by any amount. If the decision rule is to produce above average returns then positive dependence in successive price changes must be prevalent. In essence there are two types of systematic dependencies which could account for the existence of positive dependence in price movements: dependence in the noise generation process together with a lag in the distribution of relevant information (that is, the presence of opinion leaders), and dependence in the process generating new information where optimistic news tends to be followed by optimistic news more often than by pessimistic news (or pessimistic news tends to be followed by pessimistic news more often than by optimistic news). Of the two, the first is probably more characteristic of the market place.

Both Alexander and Cootner incorporated this specific type of systematic dependency in their models of stock price behavior. Alexander applied the filter technique (which bears a close resemblance to the moving average strategy) to the daily closing prices of two price indexes and initially found that the gross gains from his rule substantially exceeded those of a buy-and-hold policy. In his second paper, corrected for certain biases, the gains were significantly reduced. To correct for yet another bias present in both studies, Fama and Blume employed the filter rule on the daily closing prices of thirty industrial stocks. Their results revealed a slight amount of positive dependence in small movements of stock prices, but the extent of the dependence could not be used as a basis for a profitable decision rule.

In general, filters of all different sizes failed to produce net returns in excess of those of the buy-and-hold model. A similar set of findings was unveiled by Cootner when he used the moving average strategy on weekly price data. The positive dependence suggested by the above average gross returns was not noticeable enough to lead to a remunera• tive strategy. That is, the returns after the deduction of transaction costs were far below those of the simple buying and holding procedure.

In testing the independence assumption of the theory of random walks through the use of such mechanical trading rules, the aforementioned authors overlooked one important feature in selecting their sample. It

is apparent that the largest returns would be realized in the case where positive dependence in successive price changes existed and prices fluctuated over a wide range. In virtually all of the studies summarized, no conscious attempt was made to limit the sample to those securities which were likely to exhibit a high degree of volatility. The correction for such a deficiency furnished the basis for this thesis. An indica• tion that the approach could be adopted was found in a study by Pratt, who detected a tendency for the relative volatility demonstrated by any security to persist over time.

With regard to the presence of positive dependence in price movements, it was believed that prices adjusted gradually to new infor• mation over a month-long period. The rationale underlying such a hypothesis pertained to the time factor involved in: the analysis of a security, the publication of the relevant information, and the consequent

release of the report to the general investing public. To test this assumption, monthly stock price data was used in the simulation of the trading rule.

Test No. 1 was conducted to determine whether the relative volatility of a security tended to remain constant over successive

intervals of time (1936 to 19^+5 and 1946 to 1955). The results did not

support the evidence reported by Pratt, as there appeared to be very

little tendency for any security to exhibit the same relative volatility over the two ten-year periods.

In Test No. 2 an experimental design similar to the one employed

by Pratt was adopted to ascertain the tendency for a stock's relative

volatility to stay within some definite range. It was found that, as a

group, the most volatile securities in the 1936 to 19^+5 period tended

to remain the most volatile in the succeeding period. This finding was

used as a basis for predicting which of the 200 securities would be

most likely to display a relatively high level of price variation over the following 1956 to 1966 interval.

The moving average strategy for long positions only was simu• lated on forty series of monthly prices covering the 1956 to 1966 period

(Test No. 3). These stocks were the forty most volatile of the entire sample of 200 in the preceding ten year span. The results of the simulation revealed that the decision rule was markedly inferior to the two buy-and-hold models. Had brokerage fees been eliminated from consideration the findings would not have been altered to any significant extent. Thus, the evidence presented definitely supported the random walk theory of stock price behavior.

I 1. Suggested Research Topics

One of the major findings of this study is that the relative volatility of a stock tends to remain within a fairly narrow range over successive intervals of time. Whether or not this characteristic is an anomaly has yet to be determined since the test conducted here used monthly stock price observations only. If the phenomena is found not to exist for other types of data, then perhaps a weighted moving average of the past variation in prices can be adopted to predict the relative volatility of a security in succeeding periods. The particular end in mind is the restriction of the sample to those securities which are likely to be highly volatile and the application of the moving average strategy to daily or weekly price data. It is believed that excess pro• fits could result in such a case. The poor performance of the moving average strategy in Test No. 3 can be attributed to the absence of positive dependence in successive monthly price movements which des•

troyed any advantage to be gained from confining the sample in the manner described. I n view of the fact that both Fama/Blume and Cootner detected evidence of slight positive dependence in daily and weekly price changes

respectively, the testing of the above hypothesis appears warranted.

Another topic worthy of consideration in future research efforts

is the possibility of negative dependence in monthly movements of stock prices. 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 rationale for this type of price behavior is left to

those interested readers. TABLE I

SUBSEQUENT VOLATILITY OF PORTFOLIOS OF STOCKS SELECTED ON THE BASIS OF PRIOR VOLATILITY

Average Coefficient of Variation Rank* Period | Period II Period I Period I I

Portfol io No. 1 .889 .410 1 1

Portfol io No. 2 .636 .387 2 3

Portfol io No. 3 .549 .380 3 4

Portfol io No. 4 .460 .348 4 5

Portfol io No. 5 .394 .393 5 2

Portfol io No. 6 .345 .297 6 7

Portfol io No. 7 .303 .318 7 6

Portfol io No. 8 .260 .268 8 9

Portfol io No. 9 .208 .288 9 8

Portfol io No. 10 .153 .265 10 10

Rank of 1 is assigned to the most volatile portfolio. SELECTED SAMPLE OF FORTY SECURITIES

1. City Invest ing Co. 2. Conde Nast Publications 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. Consolidated Electronics 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 Stainless Steel Corp. 21. . Northern Pacific Railway 22. National Distillers & Chemical Corp. 23. Crown Cork & Seal Co. Inc. 24. Thatcher Glass Manufacturing 25. Helme Products, Inc. 26. American Broadcasting Paramount 27. Morrell John and Co. Inc. 28. Penn Dixie Cement Corp. 29. Fawick Corp. 30. Howe Sound Co. 31. Sperry-Rand Corp. 32. Hotel Corp. of America 33. Socony Mobil Oil Co. Inc. 34. Dana Corp. 35. Collins and Aikman Co. 36. United Aircraft Corp. 37. American Zinc Lead & Smelting Co. 38. Johns Manville 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 Profitable Performance Price Index Per Transaction Transactions Per Security Securities Index

.2158 .1155 .02 -.2555 597 14.92 6/40 .03 -.2530 556 13.90 5/40 .04 -.2386 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

.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

.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 -.2236 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 Average Annual Average No. of No. of Investment Threshold .Rate of Return Total No. of Transactions Profitable Performance Price Index Per Transaction Transactions Per Security Securities 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 -.1577 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 HYPOTHETICAL CHART OF A STOCK PRICE SUBJECT TO

RANDOM MOVEMENT WITHIN FIXED LIMITS

Figure 1. The professional's opinion of a stock's worth is represented by the dotted line. The margin of profit that they require in order to either buy or sel1 is represented by the difference between this dotted line and each of the two solid lines.

^Richard ^. 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 .

32 Figure 2. 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 Williams, Frank Jefferson. Freund and Williams' Modern Business Statistics. Revised by Benjamin Perles and Charles Sullivan. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1969.

Levy, Robert A. The Relative Strength Concept of Common Stock Price Forecasting; An Evaluation of Selected Applications of Stock Market Timing Techniques, Trading Tactics, and Trend Analysis. Larchmont, New York: Investors Intel 1igence, 1968 .

Smith, Adam (pseud) . The Money Game. New York: Dell Publishing Co., Inc., 1969.

Yamane, Taro. Statistics; An Introductory Analysis. 2nd ed. New York: Harper & Row, 1967.

B. Articles in Books

Alexander, Sidney S. "Price Movements in Speculative Markets: Trends or Random Walks." The Random Character of Stock Market.Prices. Edited by Paul H. Cootner. Cambridge, Massachusetts: The M.I.T. Press, 1964.

______"Price Movements in Speculative Markets: Trends or Random Walks, No. 2." The Random Character of Stock Market Prices. Edited by Paul H. Cootner. Cambridge, Massachusetts: The M.I.T. Press, 1964.

Cootner, Paul H. Stock Prices: Random vs. Systematic Changes. Edited by Paul H. Cootner. Cambridge, Massachusetts: The M.T.T. Press, 1964. Roberts, Harry V. Stock Market "Patterns" and Financial Analysis: Methodological Suggestions. Edited by Paul H. Cootner. Cambridge, Massachusetts: The M.l.T. Press, 1964.

C. Articles in Journals

Fama, Eugene F. "The Behaviour of Stock Market Prices." Journal of Business, 38 (January, I965), pp. 34-105.

Fama, Eugene F., and Blume, Marshall E. "Filter Rules and Stock Market Trading." Journal of Business: Special Supplement, 39 (January, I966), pp. 226-241.

Fisher, Lawrence. "Some New Stock-Market Indexes." Journal of Business: Special Supplement, 39 (January, I966), pp. 191-225.

Smidt, Seymour. "A New Look at the Random Walk Hypothesis." Journal of Financial and Quantitative Analysis, 3 (September, 1968), pp. 235-261.