Journal of Technical Analysis (JOTA). Issue 31 (1988, Winter)

MARKET TECHNICIANS ASSOCIATION

JOURNAL Issue 31 Winter 1988-89

STATISTICS, MATH & THE TECHNICIAN

From the Editor:

This issue, in a first for the MTA Journal, brings together sune of the best articles frun past Journals (up to 1985). The thene is the statistics and the math of Tech&al AMQ!3iS. While this may off-put aune, mast will acknowledge that especially with the advent of canputers, the better we ur&rstarxI the use of mathenatical and statistical techniques, the better we canpete. Put another way, without statistics - or with the inproper use thereof - it could be hazanlous to your wealth and/or job!

We have attenpted here to bring the reader a variety of topics, but intentionally have left the door open to further issues on statistics. Plans are afoot for periodic issues on other topics as well, subjects such as Cycles, Sentinent, and Indicators to mention a few. Newer readers of the Journal will be exposed to the “gold” of the past; older readers should appreciate the opportimity to review the articles as well as having them brought together in one place.

We begin this issue with a live interview, written especially for the Journal, on Chaos. While it is known Box-Jenkins doesn’t apply to the markets because of their serial correlation, the debate is ongoing whether Chaos actually - or accurately - describes the markets. If it is decided Chaos does describe the markets, it will be a black day for cycles Factitioners because the two cannot coexist. Sane feel a better description of how the markets work would allow for alternate periods of Chaos and of cyclic activity. In an effort to raise the readers’ level of understanding, Hank Pruden poses sune of the most asked Chaos questions to Dick Orr in the first MTA Journal Interview.

Sane additional references on Chaos: Probing Strange Attractions of Chaos, Scientific American, 7/87, p.108 Chaos, Scientific American, 12/86, p.46 Strange Attractors, Scientific American, 11.81, p.22 When Randun is not Randan, Journal of Futures Markets #8, p.271-289 -

In future issues, the Journal will publish a series of articles on Chaos as applied to the market, enanating frun Hank Pruden’s Chaos group. Those interested in their efforts can contact Hank at PO Box 1348, Ross, CA 94957. In addition, we tzannend to you a recent NOVA pogram on Public Television which covered Chaos in Gscinating detail.

Next is the first of two articles by Arthur Merrill whose ability to explain canplicated matters in the tiplest tears is unparalleled. Our other resident MTA statistics expert, Dick OIT, follows with possibly the final nail in the cofpin of randun walk. The nunber of studies since Cootner which have thoroughly discredited the basic tenets of the market randan walk hypothesis are legion. There are randun events - such as exactly when two sell oniers hit the floor - but they do not change the facts: a coin cannot mr its previous toas, but stock @ces are created by people wlth memories. One parenthetical note: people often confuse randun walk with efficient market theory, or even canbine the two, emors of great magnitude. The two are entirely separate subjects. The Journal encourages letters of discussion on these issues.

Fibonacci Ratios are so mlstmderstood and n&raFplied (there is one canputer program which seems never to have heard of the 1.618 Golden ratio) we thought Bob Prechter’s article should be included here, not so nuch for the market disc&on, but for the teclniques applied.

MTA JOURNAL, WINTER 1988/1989 1

Many model makers will find Dick OR’S Indicator Synergy instructive. There are pitfalls in combining indicators; for instance for every Indicator you add, you lose (have to add) one degree of freedun. To prevent the reader frun glazing over when Dick mentions the statistical test Chi2, we follow with Arthur Metrill’s description of Chi2 and a simplified table for its use. Two statistics textbooks on my shelf here contain an average of 20 pages on chi2; Arthur does it in two pages! The table can be kept handy.

Volatility and Variability are two terns often used interchangably. In same circles the term Volatility is 1L9ed in canparison to sune outside Index (your indicator vs. the S&P) while Variability is the variation in the indicator itself. Letters &an readers on this topic are invited. Dick 01~‘s article proposes first a volatility meesure and then a multiplicative filter easily used by the reader. Among other reasons, this disc-on is tnportant because the most weighty variable in the Black-8choles options equation is the volatility estimate. Same call it the “fudge factor”. Understanding volatility is crucial.

Finally Joln Carder’s definitive discourse on nxndng averages and their use is required reading for all beginners and fir others to maintain proper perspective on the oft- forgotten fact it is basically a data snoothing technique, not a @talHng device. John suggests scme moving average reflnfzments as well as a new one.

Additional articles on the tecmiques of Teclr&al Analysis are most eagerly sought. Letters to the Editor should beccrne a vital part of this wheation. Even reviews, criticizns, additions to articles are inportant.

Topic for Papers

If you get angry easily, do NOT read How Useful is the Sentiment Index? plblished in the September-October 1988 issue of the Financial Analysts doumal. A direct reply to the article would be sour grapes and unproductive; but published herein, scrne corrected aplzroaches by market technicians to the authors’ flawed testing procedure Hlould not only be informative, but instruction some acadenics seen to have sorely needed over the years - and could profit (sic.) fran. For example, one requirement might be that a rational person would use the indicator in this way, or alternatively that no thinking person would EVER use it in this other way. Another might be to utilize all perk& of t%ne covered by the data, not canplex timespans chosen to make the subject indicator look useless. After all, one does not buy stocks in “overlapping time periods”, why should one test an indicator this way? Etc. Please suhnit your conTnents/papers to the Journal.

The MTA Journal would also like to publish longer papers on the subject of proper indicator testing as well as an analgan of sane shorter pieces/contributions to the subject. The sooner the better!

On the other hand, Calendar Ananalies: Abnormal Returns at Calendar Mg Pojnts in the FAF Journal, Never/December 1988 is highly recurme nded, especially its vole bibliography.

Those with ccmnents, Ideas, additions to anything printed in this Journal, are most cordially invited to reply.

John R. is cGin.ley, Editor Winter 1988-89

MI’A J- b?-lXER 1988/1989

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MTA JOURNALWlXTlZ 1988/1989

STYLE SHEET FOR TEE SUBXISBION OF ARTICLEB

MTA Editorial Policy

The MARKET TECHNICIANS ASSOCIATION JOURNAL is published by the Market Technicians Association, 70 Pine Street, New York, New York 10005 to promote the investigation and analysis of price and volume activities of the world's financial markets. The MTA Journal is distributed to individuals (both academic and practi- tioner) and libraries in the United States, Canada, Europe and several other countries. The Journal is copyrighted by the Market Technicians Association and registered with the Library of Congress. All rights are reserved. Publication dates are Feb- ruary, May, and November.

Style for the YTA Journal

All papers submitted to the MTA Journal are requested to have the following items as prerequisites to consideration for publication:

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Short (one paragraph) biographical presentation for inclusion at the end of the accepted article upon publication. Name and affiliation will be shown under the title.

All charts should be provided in camera-ready form and be properly labeled for text reference.

Paper should be submitted typewritten, double-spaced in completed form on 8 l/2 by 11 inch paper. If both sides are used, care should be taken to use sufficiently heavy paper to avoid reverse side images. Footnotes and references should be put at the end of the article.

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Mail your manuscripts to: John R. McGinley, Jr. Technical Trends 55 Liberty Street Wilton, CT 06897

MI'A JOUIQ?AT-, WIXUD 1988/1989

BARKET TECHNICIANS ASSOCIATION

Board of Directors, 1988-89

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MI'A JOURNAL WIlIEER 1988/1989

From The Editor: Statistics, Math & The Technician

John R. McGinley. . . . . . . . . . . . . . . . . . . 1

MTA Member And Affiliate Information. . . . . . . . . . . . . . 3

Style Sheet For The Submission Of Articles. . . . . . . . . . . 4

MTA Officers And Committee Chairpersons. . . . . . . . . . . . 5

Searching For Chaos In The Market: An Interview with Richard C. Orr, Ph.D. . . . . . . . . . 7

Trends And The Log Scale Arthur A. Merrill. . . . . . . . . . . . . . . . . . . . 18

The Timely Demise Of The Random Walk Richard C. Orr. . . . . . . . . . . . . . . . . . . . . . 21

Fibonacci In The Dow - Some Examples Robert R. Prechter, Jr. . . . . . . . . . . . . . . . . . 30

Indicator Synergy Richard C. Orr, Ph.D. & Jack Y.aarayan, Ph.D. . . . . . .36

Significance: What Is It?? Arthur A. Merrill. . . . . . . . . . . . . . . . . . . . .41

A Simple Approach To Volatility Richard C. Orr, Ph.D. . . . . . . . . . . . . . . . . . . 44

Moving Averages And Their Variations John Carder. . . . . . . . . . . . . . . . . . . . . . . .52

6 M‘l'A JOURWL wItHlER 1988/1989

Searching for Chaos in the Market: An interview with Richard C. Orr, Ph.D.

At the MTA Seminar this year, Dick Orr and Hank Pruden found enough interest in

the emerging science of chaos theory to warrant founding a group to study this subject.

Since then, a number of members of the group have exchanged ideas and references,

which will be complied and distributed soon. In an attempt to give market technicians a

broad overview of chaos theory Hank Pruden, former editor of the MTA Journal, recently

conducted the following interview with Dick Orr.

JOURNAL: Dick, can you give us some brief insight into the notion of chaos as it is

currently being used?

RCO: Until recently, chaos was a term used to describe disorder. Chaos, as we will

use the term, implies a certain order within what appears to be disorder. This

phenomenon occurs in an incredible variety of situations across many disciplines. We

suspect that it may be of considerable importance in modeling stock market action. For

a broad perspective, one might read James Gleick’s marvelous account of the

development of chaos theory over the past 25 years (1).

.

JOURNAL: Gleick was a science editor for the New York Times wasn’t he?

RCO: Yes, and his book, “Chaos”, reads like a novel rather than a physics book. If an

MTA member, or any layperson, wants one book to read on the subject , this is it!

7

JOURNAL: In a sense I see this whole interview as an attempt to define chaos and to

see what interest it might have to market technicians. Have you come across any

intuitive descriptions in the literature which can be simply stated?

RCO: As a matter of fact two come to mind. Hao Bai-Lin, who has written extensively

on chaos and has compiled most of the early important papers into a single volume (2)

has written, l . . . we emphasize that chaos is not to be equated simply with disorder. It is

more appropriate to consider chaos as a kind of order without periodicity.” Douglas

Hofstadter, who had a column in Scientific American for a number of years, wrote an

article in 1981 on chaos theory. He has stated, ‘Chaos is disorder on the boundary of

order, yet an eerie order exists within the depths of chaos.”

JOUR?JR:: Ord-r posing as discrder sounds like trouble for the efficient market

theorists.

RCO: Definitely. This, of course, is of more than passing interest to me. As you recall, I

wrote a paper for this journal in 1980 demonstrating ovewhelmingly that the market is

not a random walk. Chaos theory could well challenge the fundamental notion of an

efficient market because small changes in knowledge could produce large changes in

valuations. This would mean that from a strategic viewpoint the market does not

disseminate information efficiently.

JOURNAL: Why should a practicing market technician care whether or not prices are

chaotic? It may be nice to have a response available to “efficient-market” criticism, but

what practical application would this knowledge have?

RCO: First of all, let’s be clear about one thing. We don’t know one way or the other

about the chaotic behavior of stock prices. Some work has been done in this area,

maybe we can touch on that later in this interview, but we really have yet to attack this

problem. We are beginning the process! Having said that, several things are clear if it

turns out that stock market behavior is chaotic. Any specific long-range forecast of the

a MICA J(3uFOUC hTINIER i988/1989

type we see on Wall Street Week: “Where will the Dow Jones Industrials be on

December 31?“, would be impossible. Long-term use of cycles in the strict periodic

sense (a cycle of exactly 21 days has existed over the past 20 years, for example),

rather than in the sense of Hurst, would be useless. One might be able to fit sine waves

to prices for a few cycles, but not for one hundred of them. These time-specific

processes: on day x the price will be p(x), or on day x we will have a bottom in the

market, would be ruled out.

JOURNAL: I presume you‘re implying that if this were the case, technicians could focus

on other more productive areas.

RCO: Yes. A good deal of research involves narrowing the possibilities under

consideration. By the way, all the news here wouldn’t be negative. Point and figure

charts aren’t time-specific. Neither is Elliot Wave theory. In fact, the use of a trendline

isn’t ruled out either. When we draw a trendline, we assume that it will remain in force

until broken, but we don’t predict when that break will occur.

JOURNAL: I know that Elliot Wave theory has another facet to it which is related to

chaos theory. Tell us a little about fractal geometry.

RCO: When I was in graduate school about 25 years ago, we would run across certain

mathematical curiosities from time to time. An example would be the snowflake curve

of Koch. This “curve” is formed by taking an equilateral triangle, replacing the middle

one-third of each side by an equilateral triangle one third as large as the original, and

repeating this process indefinitely. The result is a figure with finite area but infinite

perimeter. At that time, these strange curves were simply treated as special cases.

Take any two pieces of the figure, look at them under magnification and you see the

same structure. This characteristic is referred to as self-similarity. A mathematician

named Benoit Mandelbrot developed an entire geometry based on the notion of

FITA JOLIRTIX IXEYlE3 1988/1989 9

self-similarity to which he gave the name fractal geometry (3). He then began to show

that the world in which we live can be described better by the zigs and zags of fractals

than by the straight lines of Euclidean geometry.

JOURNAL: How is Elliot Wave theory fractal?

RCO: Mandelbrot distinguishes between what he calls fractal sets and natural fractals.

Fractal sets are mathematical constructs dealing with space filling cuwes and fractional

dimensions. Natural fractals are entities appearing in nature with fractal-like properties

which could be hydra-like, grainy, tangled or branching. The lunar surface, a coastline,

the human circulatory system and physical Brownian motion are all examples of natural

fractals. Elliot Wave theory has branches, has rules that apply at different scales (from

grand supercycle to sub-minuend) and, although it is a theoretical construct, it has the

properties of self-similarity and scaling which are common to so many natural fractals.

JOURNAL: Will Elliot Wave theory live or die depending on the chaotic or non-chaotic

behavior of stock prices?

RCO: No, but it certainly is a natural model to investigate if chaos exists.

JOURNAL: How did you first become interested in chaos theory?

RCO: Several years ago, the Investment Technology Association (a subgroup of the

New York Society of Security Analysts) initiated an annual brainstorming session which

it entitled “A Fuzzy Day”, after a particular type of geometry which involves fuzzy sets. I

had the good fortune to attend this first conference. The featured speaker was a

physicist from Cornell named Mitchell Feigenbaum. While I had never heard of this

person, I was very impressed with his talk. He spoke rather abstractly about problem

solving in general, but did suggest some interesting approaches to potential forecasts

of stock prices. He spoke about the fractal geometry of Mandelbrot and mentioned

10 MC.A JCXJFCKL wlNTEFt 1988/1989

some possibility that stock prices might have fractal dimension 3.7. Later. I was

introduced to chaos theory by my colleague John Gutman who spent time discussing at

some length with me a number of articles written for the New York Times by James

Gleick. They were fascinating articles, and when I heard that he was publishing a book

on chaos, I kept an eye out for it. Just about a year ago, on a trip to Princeton, I ran

across the book. It was so riveting that I couldn’t put it down. This was partly due to the

fact that both the prologue and a major chapter of the book dealt with a very deep

thinker, a person of a somewhat bizarre nature, who had become one of the major

figures in chaos theory. His name was Mitchell Feigenbaum!

JOURNAL: Are there any features which distinguish chaotic behavior from other

seemingly similar behavior?

RCO: Yes. If you have some sort of time series with values given every second or day

or month, chaotic behavior is characterized by extreme sensitivity to initial conditions,

the so-called butterfly effect. The name comes from the concept that weather behaves

in a chaotic fashion and that the air current generated by the flapping of a butterfly’s

wings in China today could make a dramatic change in the weather in Nebraska next

week!

JOURNAL: That’s hard to believe, isn’t it? Doesn’t that fly in the face of scientists’

earlier visions of the way physical systems worked?

RCO: It is exactly the opposite of what was perceived as true until about twenty-five

years ago. The prevailing scientific thought had been that, in forecasting, processes

converged nicely so that if you started with two points rather close together and tracked

their paths through this process, the points tended to stay close together or move apart

very gradually.

M!PA J- P3INtER 1988/1989 11

JOURNAL: Like laminar flow in liquid.

RCO: Yes, but to use your example, the whole picture changes when you move from

laminar flow to turbulence. Points that were previously very close together can

suddenly be separated by vast distances. Conversely, points that are now very dose

could have been anywhere previously. Predicting where a point will be sometime later

or deciding where it came from sometime earlier becomes impossible, since

measurements are always approximate, not perfect.

JOURNAL: I’ve heard the terms attractor and strange attractor used in conjunction with

chaos theory. Just what are they?

RCO: The short answer is: read the fifth chapter in Gleick’s book (1). However, let’s

take a brief look at attractors. We begin with the notion of a phase space. A phase

space has in it enough dimensions to completely characterize the motion of some body.

For example. picture a circle in the xy plane. If the y direction represents velocity and

the x direction represents position left or right of center, this circle represents the phase

space of the pendulum of a clock which is kept wound. Picture the pendulum frozen at

the far left position with zero velocity. This is the 9 o’clock position on the circle. If

rightward motion is considered negative then, as the pendulum swings back to the

center, its position is traced on the circumference of the circle from 9 o’clock down to 6

o’clock. We are now at maximum negative velocity. As the pendulum continues to

swing to the right, its position is traced on the circle from 6 o’clock to 3 o’clock. We are

now at the extreme right hand position with zero vel9city.

JOURNAL: And now, since the direction is reversed, the swing back is represented by

tracing out the top half of the circle from 3 o’clock back around to 9 o’clock.

RCO: Right! The circle represents the phase space for the pendulum and is called an

attractor because all future motion is locked to that same path.

12 PlTA JCXJ?UGC wIll.Nm 1988/1989

JOURNAL: You said that we were to keep winding the clock. Suppose we don’t?

RCO: Then then phase space changes. Instead of being a circle, it starts spiraling in

toward what was previously the center of the circle. You see, if you don’t wind the

clock, then the left and right movement of the pendulum, as well as its velocity, begin to

dampen slightly until eventually we have a stationary pendulum with no left or right

position and no velocity. The phase space has become a spiral and, since all future

motion is stopped once the pendulum stops, the attractor is the single point at the center

of the spiral.

JOURNAL: So with a phase space, once you know where you are in the space, you

also know everything about where you will be at any point in the future.

RCO: Now you see why technicians would love to be able to construct a stock market

phase space!

JOURNAL: How does chaos theory cloud this picture?

RCO: By making the phase space so complicated that apart from the recent past and

the near future, you have no idea as to where you have been or where you are going.

JOURNAL: How is that possible?

RCO: Suppose you are aboard a train in a huge switching yard. You are on track

which is lost in a maze of hundreds of tracks which run together in this yard. None of

these tracks has been marked. You have a map of the entire railroad system, but you

really can’t at this moment determine your exact location, even though you know

approximately where you are. You can look back the track a short distance to see

where you have been, and you can look forward a short distance to see where you are

going, but otherwise you are really lost. You know the location of the railroad yard in

““‘??a J- WINTER 1988/1989

which you presently rgside, but if the train now moves either forward or backward at

normal speed, you have no idea where you will be a day from now. This is my intuitive

picture of what chaos theorists call a strange attractor. Picture a phase space where

instead of converging to a circle or a point, as was the case of the pendulum, our motion

is converging to an infinitely long path in, say, three dimensions which has limited

volume and never intersects itself. A sketch of such a path might look like this:

These space filling curves are the mathematical oddities to which I referred early.

JOURNAL: Since this is an attractor, I presume that wherever we are in the phase

space, we will be attracted to some part of this curve.

RCO: Right, but because there are so may loops so close together in a strange

attractor, we can’t tell exactly to which loop we will be attracted.

JOURNAL: Why can’t these loops ever cross each other?

RCO: Remember that a phase space, by definition, contains all information necessary

to discern all future motion. Therefore each point can be on at most one path. If two

loops actually intersected each other, there would be more than one choice for future

movement.

P!“?A JOUF3U PZINFER 1988/1989

JOURNAL: Mandelbrot has published (3, p, 340) some early results showing apparent

scaling properties of commodity prices, so it is conceivable that stock price behavior

may have fractal properties. But how can we hope to discover whether price movement

has a phase space which is controlled by a strange attractor?

RCO: A number of approaches suggest themselves as one reads the literature. Let me

indicate one that I intend to pursue. If one assumes that once the right dimension is

reached, the data itself may reveal a strange attractor, we might proceed as follows.

Suppose we are looking for an attractor of dimension more than 3, but less than 4.

Remember, Feigenbaum alluded to the possibility of an attractor of dimension 3.7.

Create a space of points (dl, d3, d3, dq), where each coordinate represents the price

change from one day to the next, for the past four price changes. Since this data is four

dimensional we would want to take a slice of it in order to represent it in three

dimensions. As we move from day to day every five consecutive days gives 4 price

changes which we can plot. Our slice could be as follows: whenever the last

coordinate changes from positive to negative or negative to positive drop it and plot the

other three coordinates. Symbolically, we could denote the above process as:

when d4 = 0, then (dl, d2, d3, d4) -> (dl, d2, d3).

Three dimensions are easier to visualize than four! There are some great sketches of

this process for other kinds of data in Gleick’s book. If price changes are random, the

points we’ve plotted should be concentrated toward the center of our picture, but

scattered evenly in all directions away from the center. However, if there is a strange

attractor lurking somewhere, and the noise in the system isn’t overwhelming, it should

show itself. If it does, we will try to set the whole process up on a computer for the next

MTA seminar.

15

JOURNAL: Speaking of computers, John Bollinger, the current chairman of the

computer group for the MTA, has made some initial investigations in this area.

RCO: Yes. He’s a member of our chaos group, as you know. I hope we can coordinate

our efforts in making some work on chaos available in computer graphic form. Certain

fairly simple chaos experiments are hard to describe but are compelling when viewed

on a computer screen!

JOURNAL: This sounds like the initial phase of a long journey. Best wishes, and thank

you for sharing your insights with us.

RCO: Thank you. It was my pleasure.

MTA JOUFWG WrJTER 1988/1989

References

1. Gleick, James. Chaos. Viking Press.

2. Hao Bai-Lin. Chaos. World Scientific Publishing.

3. Mandelbrot, Benoit B. The Fractal Geometry of Nature. W. H. Freeman and Co.

MI’?L JOURNAL VZINIER 1988/1989 17

TRENDS AND THE LOG SCALE

ARTHUR A. MERRILL Merrill Analysis, Inc.

04/02/83

In these notes, trends are compared on arithmetic and logarithmic scales. Exactly the same data are presented in the left and right-hand charts of each pair.

When stock price changes are considered, percent change is much more important, of course, than change in points. A move from 10 to 12 is more important than a move from 100 to 110.

Figure 1.

I00

90

80

70

60

50

40

30

20

IO

ARITHMETIC: COGARITHMIC: I00

80

60

50

40

30

I- 20%

I- 20%

k2or

% 20%

% 20%

The arithmetic scale exaggerates moves at the higher price levels. The log scale keeps moves in proportion. A certain percen! change is the same vertical distance anywhere on a log chart.

18 &PITA JOURXAL wINTc,s 1988/1989

Constant Points Change - Figure 2.

ARITHMETIC LOGARITHMIC 100

so

80

70

60

50

40

30

20

IO

0

POINTS CHANGE

10t

8t

3(

2(

IC I I L I , I I I I

A straight line on an arithmetic chart is deceptive. It presents a steady rise; the log chart shows that the rate of price change has been slowing down. When you are looking at a chart with an arithmetic scale, think twice before you reach for a ruler.

Constant Percent Change - Figure 3.

ARlTHMETtC:

‘0°: lot

6C

6C

50

40

30

LOGARITHM I C :

This is called a geometric trend. The arithmetic scale gives a deceptive skyrocket effect. The log scale gives the true picture of steady sustainable growth.

MIX JC)UFX.. wlINT% 1988/1989 19

Parabolic Trend - Figure 4.

ARITHMETIC: 1001

70- “-“‘BOLIC . rL\n&a

CHA _- 60’

50 -

40’

/

30 -

IO

0’ L b _a , I I

so-

BO-

LOGARITHMIC:

too- 30’

20’

IO :

7-

5- 4-

3’

This curve (y = axxx) doesn’t make much sense on a stock chart, but we occasionally hear the term.

The chart, though, illustrates the danger of the trends drawn on an arithmetic scale. The left-hand chart is eubhoric; actually. the growth rate has been slowing down.

Beware trends on an arithmetic scale!

20

The Timely Demise of the Random Walk

Richard C. Orr’

1. Introduction The foundations of technical analysis lie in several key assumptions.

Among these are the existence of areas of support and resistance, which are horizontal boundaries to stock price movement, and the existence of trend lines together with their more general forms such as moving averages or cycles. The original random walk theory as applied to the movement of stock prices directly challanged these assumptions and hence the whole area of technical analysis. The random walk model asserts that at any given time the probability is p that a stock price will rise and the probability is l-p that it will fall, where p is some fixed constant. If p is either less than l/3 or greater than 2/3, so that

the probability of a move in one direction is at least twice that of the probability of a move in the other direction, then, if the random walk model is valid, we can easily determine stock trends. However, if p is l/2, or some value close to l/2, then the validity

of the random walk model for stock price movement would imply that most of technical analysis and fundamental analysis are useless as methods for forecasting stock prices. The statistical experiment described in this paper, therefore, has been designed to refute

the applicability of this model as a realistic description of stock price movements. Subsequent modifications of the random walk model, such as Cootner’s random walk within reflecting barriers, are much more compatable with technical analysis and will not

be discussed here. * l

‘Dr. Orr is Associate Professor and former chairman of the Department of Mathematics,

State University of New York, Oswego, New York. He is currently a consultant to Pershing & Company, New York.

**For an excellent exposition of much of the early work supporting and rejecting the

random walk model see Cootner, Paul, The Random Character of Stock Market Prices, ----

M.I.T. Press, 1964.

H’I’A JoURN?% P?lXITER 1988/1989 21

2. A Brief Review of Hypothesis Testing. Before describing our experiment in detail, we turn first to a simpler

example as a means of reviewing the relevant statistics. l Suppose in observing a particular

roulette wheel over a reasonably long period of time we feel that black numbers are winners much more frequently than are red numbers. How do we test to see if the wheel

is biased in favor of black numbers? First, we formulate the null hypothesis, namely: that

the wheel is fair; black should occur as often as does red “in the long run”. Next, we

formulate the alternative hypothesis, namely: the wheel is unfair. The occurrences of red

and black numbers in random spins of a roulette wheel which is fair are described by a

particular probability distribution and there is a small chance, even with a fair wheel, that we might still have, say, twenty black numbers occur in a row. Therefore, we must also

set the tolerance for error which we will accept, known as the level of significance of the experiment. Suppose that we decide to test the hypothesis that black and red numbers

have an equal probability of occurring at the .Ol level of significance by recording a total of 100 red or black numbers, while ignoring any green numbers.** If we are able to reject

the null hypothesis, that the wheel is fair, at the .Ol kvel of significance then we will have a result which 99 out of 100 times is due to the fact that the wheel is unfair, but 1 out of 100 times is due to an unusual occurrence in the experiment. This kind of result is similar

to the conviction of a defendant in a court of law because the jury was convinced of his guilt “beyond a reasonable doubt”. Our “reasonable doubt” in this example is the .Ol probability of error. If, on the other hand, we fail to reject the null hypothesis at the .Ol

level of significance, this does not mean that the null hypothesis is correct with probability .99. To continue our analogy to a courtroom situation, the defendant is

presumed innocent until proven guilty beyond a reasonable doubt. Failure to convict could mean that the defendant was innocent, but it could also mean that while he was

guilty, there was simply not enough evidence to be found to allow a conviction, given that the jury must be 99 percent sure before finding the defendant guilty. Note that juries

don’t find defendants “innocent”, they find them “not guilty”. Returning to our

experiment, suppose that our sample yields 70 black winners and 30 red winners. The wheel could be unfair if it tends to give us either too many black numbers or not enough

black numbers as winners. Thus our .Ol chance for error must be divided into a -005

chance for too many black numbers and a .005 chance for too few of them. A test of this

type with two alternatives is called two-sided. Checking statistical tables we find that for

a fair wheel the probability of 70 black winners in 100 spins is less than .005 and hence we reject the null hypothesis in favor of the alternative hypothesis that the wheel is unfair.***

*This material can be found in any introductory text for mathematical statistics, for example, Johnson, R., Elementary Statistics, Second Edition, Duxbury Press, 1976.

“Recall that the house’s edge in roulette is created by placing a 37th number, 0, on the wheel in Europe and, additionally, a 39th number, 00, on the wheel in the United States. In our experiment, any spin resulting in 0 or 00 as a winner will simply be disregarded.

“‘This binomial distribution can be approximated by a normal distribution with mean 50 and standard deviation 5, so 70 is about 4 standard deviations above the mean, after

a continuity correction.

22 "YTA JOtJR.WL wItNTEX 1988/1989

On the other hand, an outcome of 60 black winning numbers would not give us sufficient evidence to reject the null hypothesis at the .Ol level of significance. Again, such a result would not imply that the wheel was fair, but merely that we were unable to be 99

percent sure that it was unfair.

The important idea to keep in mind, then, is that in testing stock price movements against a random walk model, failure to find a significant difference in the

two does not imply that they are the same.

3. Description of the Main Experiment.

It seems clear that the variation of stock prices, defined mathematically to be the sum of the absolute value of the changes in these prices, is so large compared to

the net change, even over a fairly short period of time, that any statistical test based entirely on the correlation between successive changes would not yield meaningful

results. If the hypothesis that stock prices change in a random manner is to be decisively tested, we need to find a way to prevent the stock’s deviation in price movement from a random walk model while the stock is in an uptrend from nullifying the deviation which

occurs when the stock is in a downtrend. The following experiment solves this problem by focusing on time rather than on price change.

We will base the null hypothesis on the assumption that at any time a

given index or stock price has an equal probability of rising or falling. A more general case will be mentioned later. We will attempt to reject the null hypothesis at the 600001 level

of significance. In other words, we will only allow a chance for error in this experiment of one in one million.* We will ignor any changes which are equal to zero, so in-our

sequence of prices each term is either higher or lower than its predecessor. A peak in

prices is defined to be any price which is higher than both its predecessor and its successor, while a trough is defined to be a price which is lower than both its predecessor

and its successor. We measure the average number of time intervals (days, weeks, months, etc.) either between successive peaks or between successive troughs. A graph of the

random walk model appears on the next page. Assuming the null hypothesis to be correct, the distribution of the number of days between peaks or troughs should have

mean equal to 4 and standard deviation equal to 2. A derivation of these results appears in an appendix to this paper. The length of time between successive peaks or troughs is certainly less strongly correlated to the trend of the prices than would be changes in the

prices themselves. Hence, at the outset, we would appear to have a better chance of resolving this issue than have others in the past. The null hypothesis is that in the population from which we sample, the mean number of days between successive peaks

(or troughs) is 4. The standard deviation for this population is 2.

*This is ample for our purposes. The actual prob-value, the best possible level of significance at which we could still reject the null hypothesis, is closer to 10e20-

E’A JotlFK% WIPJTER 1988/1989 23

Graph of the Random Walk Model with p =.5

M is a peak which is actually formed the succeeding day (week, month, etc.) when the stock price reaches S. The next peak will be formed on the first day that the price falls after having risen. For example, the peak at N is formed one day later when the stock price reaches F. In this example, the “distance” between M and N is three days. Because the number of days between peaks is equal to the number of days between the points at which the peaks were actually formed, we could measure the distance from S to F rather than from M to N, if it was more convenient. By symmetry, if the above graph is reflected so that all moves up become moves down, and conversely, then we have a model describing the distances between successive troughs.

24 ML’A JCXJRII k7INTER 1988/1989

Our sample consists of 1369 consecutive values, from July 5, 1973 to

December 29, 1978, of the Indicator Digest unweighted index of 1500 NYSE stocks, better known as IDA. This is an appropriate set of data, since under the null hypothesis,

if stock prices move at random then so must the IDA. Another experiment using only

data on individual stock prices appears in the next section. The IDA data contains 246 troughs and 245 peaks (these numbers can never differ from each other by more than

one). Because a trough appears before a peak in the data, we measure the distance

between consecutive troughs, but equally impressive results would be obtained by measuring the distance between consecutive peaks, instead. The first trough in our data

occurs on July 9, 1973 while the last trough occurs on December 28, 1978. This yields 245 distances with a sample mean equal to 5.57. The Central Limit Theorem tells us that a sampling distribution of sample means is normally distributed (within our tolerances) with mean equal to 4 and standard error equal to 2/m This implies that our sample mean of 5.57 is more than 12 standard deviations above the population mean!* As this is a two-sided test we need to ascertain whether or not a z score this large can occur with

probability greater than .0000005. A quick check of a normal distribution table tells us that indeed it cannot occur with that large a probability. We therefore reject the null hypothesis that these prices behave like a random walk with p = l/2.

Using a derivation similar to the one given in our appendix, we can show that the generalized random walk model, with the probability of a stock rising equal to p,

has its mean number of days between successive peaks (or troughs) equal to l/p(l-p). Note that if the probability of a stock falling is p, then its probability of rising is l-p and

we again have the mean number of days between peaks (or troughs) equal to 1/(1-p) (1-(1-p)) = l/(1-p)p. While the IDA data previously discussed would not rejsct the

null hypothesis that the mean is 1 /p(l-p) for, say, p=3/4, some of the data in the next

section will place even this hypothesis in jeopardy. Not withstanding that, I suspect that most of us would be delighted to have a 3 to 1 edge in forecasting current price changes.

l Recall that the equation for z score is

where 2 =5.57, # =4, G =2, and 3t =245.

Hence, 2 =12.287.

PITA JoUEX.FL iCllTE3 1988/1989 25

4. Individual Stocks: Preliminary Results. In the following experiments, approximately 120 consecutive prices (daily

or weekly) for individual stocks are investigated. This sample size is barely large enough to give any results which might be considered meaningful. Time constraints forced the use of data which, for the most part, was readily accessable by our computer. Any reader with an interest in this kind of investigation who has rapid access to perhaps one or two thousand consecutive prices for a fairly large number of stocks is urged to repeat these experiments with his or her own data.

The first experiment uses weekly closing prices for each of the current DJIA stocks for the 119 weeks ending December 28, 1979. Each stock’s prices are used to test the general hypothesis that stock prices behave like a random walk with p=1/2. As before, this is done by assuming that the mean number of days,&, between peaks is equal to 4. For 27 of the 30 stocks involved, no significant results exist. The results for the remaining 3 stocks are interesting but only mildly significant. They are summarized as follows:

Two-sided Test of the Hypothesis: = 4 (Weekly Data) P

Stock

AA AC TX

Mean Number of Da?ls Between Peaks z-score

Reject Null Hypothesis at Level of Significance

.l .05 .02 .Ol

3.375 -1.768 Yes No No No 5.091 +2.558 Yes Yes Yes No 3.294 -2.058 Yes Yes No No

The second experiment involves daily closing prices for the first six months of 1968 for each of three randomly selected stocks from the S&P 500”. We test the null hypothesis:+ =4 at the .Ol level of significance. This experiment leads to a stronger conclusion than did the last one. We have the following results:

Two-sided Test of the Hypothesis: P

=4 (Daily Data)

Stock Mean Number of

Days Between Peaks z-score Disposition of Null Hypothesis at the .Ol Level of Significance

SCE 4.45 CUM 3.088 ITT 4.2

+1.01 Fail to Reject - 2.66 Reject

+0.5 Fail to Reject

*These prices were obtained from the ISL Daily Stock Price Index, Standard & Poors Corp., 1968.

26 ?Tl’P. ~JOURXL Vi- 1988/1989

One interesting point to note is that since, for any p between 0 and 1, l/p(l-p) is never less than 4, (in hindsight) we could have rejected a null hypothesis that stocks behave like a random walk for any fixed p, using the sample of prices for Cummins Engine Co. stock.

5. Summary The daily price study of CUM allows us to reject the hypothesis that stock

prices behave like a random walk. The level of significance I.011 is good,* but experiments involving a much larger set of data would probably yield more powerful

results. For completeness, the DJIA weekly price experiment is included, although the results are not really satisfactory. On the other hand, the much larger set of IDA data

gives us a sample which overwhelmingly rejects the random walk model for stock price movement.

‘By way of contrast, many early investigations of stock price movement were carried

out at levels of significance between .05 and .Ol.

eFTL”P- JWFX?iL WINTER 1988/1989 27

Appendix: Derivation of the Mean and Standard Deviation for the Number of Days Between Successive Peaks.

Let d represent the number of days between any two successive peaks. The mean of d, denoted E (d), is calculated by finding the (infinite) sum of the product of d, the number of paths of length d, and the probability that a path of length d occurs.

Note that the probability of a given path occurring is just the product of the probabilities for each segment of the path. (See the earlier graph.)

Days Paths

2

3 4

1 (l/2)(1/2) = l/4 l/2 2 (l/2)(1/2)(1/2) = l/8 314 3 (l/2)(1/2)(1/2)(1/2) = l/l6 314

.

Probability Product

. .

. . .

We then have:

E(d) = To find the actual value of this sum, let

Then,

and hence,

.

28 MI’A JOLJFN?L ‘h’IXER 1988/1989

Dividing by 4, we have:

From elementary statistics we know that:

where 6 represents the standard deviation of d. In a manner similar to that which we used to find E (d), we obtain:

.

To calculate this sum, let

Then,

and hence,

Dividing by 2, we have:

*.

so that 6 = 2. Our conclusion is, therefore, that the distribution of the number of days between successive peaks has a mean of 4 and a standard deviation of 2. As previously

mentioned, the symmetry of this model leads us to the same mean and standard deviation

for the number of days between successive troughs.

Mm JCIURNL I- 1988/1989 29

FIEJONACCI IN THE DU4 - S@E Dt#PES

ROBERT R. PRECHTER, JR. Editor, Elliott Wave Theorist Financial Letter

Whv should a concept in theoretical mathematics discovered by an Italian mathematician in the 12th Century be applicable to today’s stock market? Starting with the late R.N. Elliott, many analysts have become convinced that it is. Bob Prechter, an MTA member, along with A.J. Fast, with whom he wrote Elliott Wave Principle, is the leading contemporary interpreter of Eliiott’s work. In this article, he explores the Fibonacci or “Golden” ratio and applies it to recent trading. Bob is the editor of the Elliott Wave Theorist Financial Letter.

When approaching the discovery of mathematical relationships in the markets, the Wave Principle offers a mental foothold for the practical thinker. If studied carefully, it can satisfy even the most cynical researcher. A side element of the Wave Principle is the recognition that the Fibonacci ratio- one of the primary governors of the price movements in the averages.

Some of the advantages of Elliott’s use of Fibonacci ratios as compared to currently popular numerological studies are as follows:

1. Fibonacci ratios are independent of the unit of price measurement , the unit of time measurement, and chart scale.

2. Fibonacci ratios are few. The only ratio which occurs often enough in anrkets to be of practical importance is 1.618. Of secondary importance are .50, 1.00 (equality), and 2.618, which are all ratios faux inthe Fibonacci sequence. Thenverses of these ratios are alternate expressions of the same relationships.

3. The Fibonacci ratio’s occurrence in markets may have mystical

30

connotations, * but it’s expectation is not based upon mystical assertions. The Wave Principle is based on empirical evidence, which led to a working model, which subsequently led to a tentatively developed theory. In a nutshell, the portion of the theory which applies to anticipating the occurrence of Fibonacci ratios in the market can be stated this way:

The Wave Principle describes the movement of markets. The numbers of waves in each degree of trend corre- spond to the Fibonacci sequence. The Fibonacci ratio is the governor of the Fibonacci sequence. The Fibonacci ratio has reason to be evident in the market.

Obviously, the intellectual acceptance that Fibonacci ratios should be expected to appear in markets is based upon satisfaction that point “a” above is a valid statement. Two books have been published on the subject, and an interested analyst can learn the model quickly. As for satisfying oneself that the Wave Principle is valid, some effort must be spent attack- ing the charts. The bulk of that task must be left up to you. The purpose of this article is merely to present evidence that the Fibonacci ratio expresses itself often enough in the averages to make it clear that it is indeed a governing force on aggregate stock prices. These examples are presented in hopes that interest in the Wave Principle will be stimulated and that new research may be undertaken in this field.

“Orthodox” Turninq Points

One concept that must be mentioned at the outset is that of the “orthodox” top or bottom of a move in the averages. According to the Wave Principle, a wave which should be composed of five waves officially ends when the fifth wave has been completed, preceding or ensuing minor new highs/lows notwithstanding. Similarly, corrective patterns are measured from the beginning of the pattern to the end to calculate the orthodox “distance” covered by the pattern, intervening minor new highs /lows notwithstanding. While most of the following examples were chosen to avoid the concept, the August-December 1980 pattern does involve it. If a measuring point appears to ignore what looks like a higher or lower price point, in fact it is taken at an orthodox turning point. If anything, the fact that calculations are accurate when measured from the ends of the textbook wave patterns serves further to validate the theory of the Wave Principle.

‘This ratio describes the only price relationship whereby the length of the shorter wave under consideration is to the length of the longer wave as the length of the longer wave is to the length of the entire distance traveled, thus creating an interlocking wholeness to the price structure. It was this propery that led early mathematicians to dub 1.618 the “Golden Ratio.”

AyTA JOURKAL ‘ViTNTE? 1988/1989 31

When the Ratio Occurs

A Fibonacci relationship between adjacent waves occurs more often within corrective patterns. A Fibonacci relationship between unconnected waves which are nevertheless part of a single pattern occurs more often within five wave patterns when the third wave is the longest. Relationships are calculated only with reference to vertical points traveled.

Past Examples

Other publications have pointed out, sometimes in advance for forecasting purposes, the 1.618 relationships found in the 1921-November 1928 period, where the final wave (1926-1928) of the sequence is 1.618 tunes as long as the first three (1921-1925), the 1932-1937 period, where the finaI wave (1934-1937) is 1.618 times as long as the first three (1932-19221, the 193b 1939 period, which contains four swings, each of which is related to the ensuing swing by 1.618, the 1949-1966 period, where 1957-1966 is 1.618 times 194%1956, and the 1966-1974 period where the distance from 1966 to 1974 is 1.618 times the length of the 1966 decline

The period from 1974 to the present has been less documented, but de- serves mention for the frequent occurrence of the 1.618 ratio during this period. All such relationships in all the interlocking waves during this time span could not possibly be mentioned in a short article. In order to give an osen5ew Gf re cer;t Fibonacci history and bring us to the present, I have outlined both the bigger picture, 1974-1981, and on a closer basis, the period from August 1980 to December 1981.

The Fibonacci Ratio in the Dow From 1974

1021.86 1024.05

lOOO-

900 -

800 -

739.59 700-

600- / Hourly Turning Points TY’CLLT)l r WAK TYonlsl

c.0. &, IhI (1 indicates deviation 572.20 Glrrwwr. CA )fnO) lrom ideal kngth .

b ‘ 1975 ’ 1976 ’ 1977 ’ 1978 ’ 1978 ’ IQ80 ’ 1981 ’

Chart 1

32 MTA J- WIEvTEIi 1988/1989

1974-1981 (Charts 1 a 2) Each wave in this pattern is related to an adjacent wave by 1.618 (or in the center of the pattern by equality), within the percentage emrs listed. The only large deviation is the “overshoot” in the 1978 October massacre. During the October-December 1978 period, the first ideal retracement level at 806 was penetrated eight times in whipsaw action. All of these penetra- tions were extremely volatile, so much so that five of them left intraday

aps in the Dow. Among the three that did not leave gaps were a 9-point p” opening hour and a 17-point “down” opezg hour. That kind of action

my indicate that the market sensed the importance of the exact Fibonacci retracement level despite the overshoot. The next largest deviation occur- red because of the mild overshoot at the November 1979 low, which was again due to a “massacre” type market.

Although not a forecast, if the down wave which began in 1981 were to fall 1.618 times the length of ae preceding advance, it would bottom at 563.79, and create a symmetrical pattern from 1974.

The Fibonacci Ratio Using Average Turning Points

6QQ- 572

' 1975 ' 1976 ' 1977 ' 1976 ' 1979 ' 1960 ' 1961 '

Chart 2

A second way of looking at this period is to average the tops in 1976, producing 1013.08, and the tops in 198&81,producing 1011.51. It so happens that a rounded average of the two, which gives a central peak point of 1012, is the exact point which provides 0% error for the Fibonacci relation&i! involved. Similarly, the average of the tops in 1978-1980 is 908 and the average of the bottoms in 1978-1979 is 789. The three average price points makes the Fibonacci ratio relationships perfect except for the average of the 1978-1979 bottoms, of 804.

which is 15 points below the ideal low

LMTA JCXJFXPL VlIX- 1988/1989 33

August 12, 1980 - December 11, 1980 (Chart 3) This sideways pattern began from the point at which the orthodox top of five waves from April 21 to August 12, 1980 ended. The height of each A-B-C pattern in this sequence is related to the preceding A-B-C pattern by 1.618 (with 1% error and 6% error respectively). The height of the final A-B-C is 2.618 times the height of the first. IF the December low ( 899.57) had been 905.84, the second calculation would have had 0% error. If we label the first pattern 33”, the second “Y” and the third “Z”, note that X:Y as Y”Z; X:Y as Y:(X+Y); Y :Z as Z”(Y+Z).

Tripk Three-AugrDec. 1980

2nd ABC = 1.618 x 1st ABC (1% error).. 3rd ABC = 1.618 x 2nd ABC (6% error). 3rd ABC = 2.618 x 1st MC (5% errot).

chart3

with division at 894.37, all calculations have 0% error. December 1960 low at 899.57 is a compromise between an ideal 905.84 for the XYZ pattern and an ideal 894.37 for the PQR pattern.

Chart 4

December 11, 1980 - December 4, 1981 (Chart 4) Th A fl-September d&in . 1 618 times the December-April advance (4: erk) , and the DecembeeEApkl advance is 1.618 times the September- December advance (3% error). The April-September decline is 2.618 times

34 INTA JOUFWE WlXtER 1988/1989

the September-December advance (1% error). Each calculation would have had 0% error if both the December 1980 low (899.57) and the December 1981 high (893.55) had been 894.37. In other words, the entire height of the pattern is divided into a Golden Section at 894. Lf we label these lengths P, Q and R, note that R:P as P:Q; R:P as P:(R+P); P:Q as Q:(P+Q)s

The 792 level may provide support and/or resistance for the current decline since it is the halfway level for the entire pattern and since it has acted as end-of-decline support for three declines since 1974. It also happens to be the level at which the decline from December 1981 would be 1.618 times the length of the April-May 1981 decline. These two declines can be labeled as fifth and first waves respectively. To see another example of this type of relationship, see the 1980 five-wave advance on Chart 4, where wave 5 = 1.618 x wave 1.

35

INDICATOR SYNERGY

RICHARD C. ORR, PH.D.

and

JACK Y. NARAYAN, PH.D.

The familiar “model, ” or combination of market indicators, is a predictive technique that has been utilized by many technicians. It has long been theorized that this combination produces a syner- gistic effect, i.e. , the combination produces better results than the best of the individual indicators. The authors in this article demonstrate mathematically why this should be the case, drawing on the actual results obtained from two widely-used indicators. The appendix to the article provides a non-technical discussion of the Chi-squared technique of assessing significance. This technique is often of great value in testing stock market indicators.

Dick Orr, an MTA member, is an associate professor of mathematics at the State University of New York, Oswego, and a consultant to Pershing & Co. His co-author, Jack Narayan, is also a S.V.N.Y. professor.

INTRODUCTION

Market indicators measure a variety of disparate rhythms of the market such as breadth, momentum, cycles, trends, and sentiment. Market technicians generally use a combination of indicators from a number of these categories, rather than relying only on a single “favorite” indicator. Two approaches which immediately come to mind are the informal combination of indicators based upon the considerable experience of the technician (e.g. Stan Weinstein of The Professional Tape-Reader) and the formal combination of indicators to obtain the “weight of the evidence” (e.g. Art Merrill of Technical Trends). This effort to maintain a balance between various cateEories of indicators is curcial. For examDIe. momen- turn indicators all ten>, roughly, to measure a common character&tic of the market, albeit in different ways. While there is an ,advantage to the

36 KW. JOUXAL WIVIE? 1988/1989

use of more than one momentum indicator, one must keep momentum in proper perspective. It is only one facet of the market. As the market approaches a top, one’s stable of momentum indicators may be very bullish, in which case they may be giving an incorrect “signal.” At the same time, another category of indicators may be warning that the market is severely overbought, while yet another may show that sentiment is too bullish. The real power of these indicators, then, lies in their interaction.

The purpose of this paper is to demonstrate by means of a simple model the synergy generated by the simultaneous use of two indicators. The model by no means represents the state of the art in market timing, but is merely a device by which we will explore the added power obtained by the joint use of indicators. We wish to express our appreciation to Tony Tabell of Delafield, Harvey, Tabell for his generous assistance in obtain- ing up /down volume and DJIA historical data.

DESCRIPTION OF THE VARIABLES TO BE USED

Market behavior will be measured by using the daily close of the DJIA in the following manner. On a given day, the market will be considered bullish if, from that time forward, it rises at least 10 percent before cor- recting 5 percent. The market will be considered bearish if it falls at least 10 percent before rallying 5 percent. Should the market move less than 10 percent in a given direction before moving 5 percent in the other direction, it will be considered neutral. No time limit is prescribed. How- ever, historically these moves take weeks or months, not years.

One of the indicators we will use is the well-known percentage of ali NYSE stocks above their lo-week moving average, published by Abe Cohen in Investors Intelligence. When the percentage figure rises above 70 percent the market is considered overbought, but it may go higher. Hence, no action is recommended until the percentage figure again drops below the 70 percent level. At this point a bearish signal is given. Similarly, when the percentage figure drops below and then rises above the 30 percent level, a buIlish signal is given. If the figure is either below 30 percent or above 70 percent, we will consider it to be neutral. As this data is compiled weekly, the state of this indicator is assumed to hold for an entire week.

The other indicator we will use employs an exponentially smoothed average of NYSE up volume minus down volume such that the data has a 16-day “half-life. ” A 25-day moving average of up volume minus down volume would probably serve the same purpose. The current value of this exponentially smoothed average is compared with the value 10, 20, and 30 trading days earlier. If the current value is larger than all three of the previous values, then the indicator is considered bullish, while it is considered bearish if the current value is smaller than all three previous values. Any other combination gives a neutral reading for this indicator.

.?n’-A JOURNX KXI!EX 1988/1989 37

DESCRIPTION OF THE EXPERIMENT

As we have now characterized, on a daily basis,the state of each of the two indicators, as well as the state of the market, we are ready to formulate hypotheses and test them. Our testing procedure will involve the selec- tion of a random sample of 144 of the 3,000 trading days from April 22, 1968 through April 15, 1980. We are sampling without replacement which means that no day. will be represented more than once in the sample. The sampling will be accomplished by the use of a table of random numbers .l The date corresponding to each of these numbers will then be determined and the state of both indicators as well as the market will be recorded in tabular form. From this ‘first table, we will create three contingency tables to measure the interaction of the indicators, both individually and jointly, against the market. We will then calculate the LA (Chi-squared) value for each contingency table. The significance of XL and its computation are described in an appendix to this paper. Beyond this, we hope to gain an insight into the reliability of these indicators individually and jointly by simply observing the distributions in the several tables. The states of the indicators and the market will be denoted as follows: U for a bullish state, N for a neutral state and D for a bearish state. In the case of the joint use of indicators, we will hypothesize that if both indicators are in state U, then the market should be in state U abnormally often. If both indicators are in state D , or if the net volume indicator is in state N while the NYSE percent indicator is in state U, then the market should be in state D a disproportionate percentage of the time. This dissimilarity takes into account the old adage that “stocks will fall of their own weight .” If the NYSE percentage h is maved above 30 percent and net volume either remains neutral or falls to neutral, we would expect this to be bearish more often than not. We will further hypothesize that any combination of indicator states other than the three described above will tend to produce a market state of N.

RESULTS OF THE EXPERIMENT AND CONCLUSIONS

The experiment produced the following results. Our original hypotheses are listed next to the frequencies.

Net Vol. NYSE Market Hypothesis

D

D

‘D

N

N

N

U

U

u

D

N

U

D

N

U

D

N

U

D N U

5 7 1

2 15 5

1 1 1

2 9 2

5 19 2

9 6 4

0 0 0

3 18 3

3 9 12

D

N

N

N

N

D

N

N

U

38 MTA JcxJps;AL-- 1988/1989

The contingency tables produced by this table are:

Table 1

Market

D N U -

Table 2

Market

DN U

Table 3

Market

In DN U

D

N

U

L ?c = 7.3

1 7L = 18.7

= 27.7

For a detailed explanation of the calculation of e values the reader is reminded to see the appendix. The larger the value of ti the more dependent the market state is on the state of the indicator(s) listed on the left of the table. In Table 1, a s value of 7.3 is not very significant, but the ‘$% values for the other two tables are very significant. Beyond this analysis, however, the reader will immediately notice a dramatic improvement in Table 3 of the ratio of good predictions (down the main diagonal) to the bad predictions (which are circled) when compared to Table 1 and Table 2. This technique shows real promise in its application to the formal combination of various market indicators. While we have restricted our attention to only two indicators, this technique may be generalized to handle combinations of a larger number of indicators. 2

MI’A JCXJIXAL lfIX’ER 1988/1989 39

FOOTNOTES

1 These tables are readily available. See, for example, the Handbook of Tables for Probability and Statistics, published by the Chemical Rubber Company.

2 The interested reader with a background in statistics should see Feinberg, The Analysis of Cross-Classified Categorical Date, M.I.T. Press.

PITA JOUFNL NIESTER 1988/1989

SIGNIFICANCE: WHAT IS IT??

ARTHUR A. MERRILL Merrill Analysis Inc.

Arthur Merrill is a charter member of the Market Technicians Association and was the 1977 winner of its Award for Distin- guished Contribution to Techncial Analysis. One of his fortes for many years has been the advocacy of standards of statistical significance , and his ability to explain the derivation of those standards in easily understandable terms. This article is one more piece of evidence of that ability.

The “witchcraft” appellation can be dispelled by the use of just two words: Significant Evidence.

The second word is easy. We all support our conclusions with evidence. But do we always check significance ? Are the examples we cite sufficient to dispel the suspicion of luck ? As John Heywood put it in 1577, “One swallowe prouveth not that summer is neare.”

To check for significance, some knowledge of statistics is needed. But don’t be alarmed ; the problem may have an easy solution. Many of our conclusions are simple two-way: “The price of this stock will probably be higher two weeks after this breakout. ” “The Dow will probably close higher next Friday. ” The result is two way : Up-Down, or Right-Wrong. The statistician’s buzz wqrd for this type is “binomial,” and tables giving the answers are quite easy to use.

The tables in this paper deal with the problem when the basic probabilities are about even money. The problem is more complicated if the odds are biased one way or the other; if, for example, we are forecasting a rise in an upward trend. There are methods, but we don’t discuss them here.

But suppose the odds are about even. Matching coins is a good parallel. Rou- lette red and black is another. Serious workers who use evidence Iike to see odds of 20-to-1 against simple luck before calling the results “probably significant . ” If the evidence shows lOO-to-l, it’s called “significant ;” lOOO-to-l is

,YI’A JOURNAL WINTER 1988/1989 41

c:llled “highly signilicrlnt .” The first is called the “95% confidence level. I1 The second, the “99% confidence level, ” the third, the “99.9% confidence level. ”

The ZO-to-1 doesn’t mean a test with 20 successes and 1 failure. It is reached when a test has six successes and no failure, or eight successes and one failure . You will see this when you use the following table. The loo-to-1 can be reached by a test with eight successes and no failures. The table gives the required ratios for various test sizes. It’s easy to use:

The Problem

In a situation with two solutions, with an expected 50/50 outcome (heads and tails, red and black in roulette, stock market rises and declines, etc .) are the results of a test significantly different from 50/50?

Solution

Call the two outcomes (such as number of reds, number of blacks) (a) and (b) . Let (a) be the smaller of the two. Find this number in the left hand column of the tabulation. If (b) exceeds the corresponding number in the 5% column, the difference from 50/50 is probably significant; the odds of it happening by chance are one in twenty. If (b) exceeds the number in the 1% column, the difference can be considered significant ; the odds are one in a hundred. If (b) exceeds the numbers in the 0.2% (one in five hundred) or 0.1% (one in a thousand), the difference is highly significant. Note that the actual numbers must be used for (a) and (b) , not the percentages.

Example

In the last 83 years, on the trading day before the 4th of July holiday, the stock market went up 65 times and declined 18 times. Is this significant? On the day following the holiday, the market rose 50 times and declined 33 times. Significant?

For the day before the holiday, (a) = 18 and (b) = 65. Find 18 in the left hand column; 65 far exceeds the 33, 39, 43, and 45 in the table, soThe record demonstrates aTighly significant bullish bias on that day.

For the day following the holiday, (a) = 33 and (b) = 50. Find 33 in the left hand column. The minimum requirement for (b) is 53: 50 falls short; no signifi- - cant bias is demonstrated.

Source

Some of the figures were developed from a 50% probability table by Russell Langley (in Practical Statistics Simply Explained, Dover 1971), for which he used binomial tables. Some of the figures I calculated using a formula for Chi- squared with the Yates correction. This table was distributed at our 1978 seminar.

43 ~YI'AJOUFEAT,WIWER 1988/1989

MERRILL ANALYSIS INC.

bx 228, c*ra, NY II14

Sigdficurce of Devi8tion from 8 50/50 Proportion: (8) + (b) = (a)

1 2 3 4

6” 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

-

(b) requirement: i% l%O. t%o.lq 6 8 10 11 8 11 13 14

lo 13 16 17 12 15 18 19 13 17 20 22 15 19 22 24 10 20 24 26 18 22 26 28 20 24 27 29 21 25 29 31 23 27 31 33 24 28 32 34 25 30 34 36 27 31 36 38 28 33 37 39 29 34 39 41 31 36 40 42 32 37 42 44 33 39 43 45 35 40 45 47 36 41 46 49 37 43 48 50 39 44 49 51 40 46 51 53 41 47 52 ,54 42 48 54 56 44 50 55 57 45 51 56 59 46 52 58 60 48 54 59 62 49 55 61 63 50 56 62 64 51 58 63 66 53 59 65 67 54 60 66 69 55 62 67 70 56 63 69 71 57 64 70 73 59 66 72 74 60 67 73 75

Jb) requirement: 18) 5% 1% 0.2% 0.1% 40 61 68 74 77 41 62 69 76 78 42 63 71 77 79 43 65’ 72 78 81 U 66 73 80 82 45 67 74 81 84 46 68 76 82 85 47 69 77 84 86 48 71 78 85 88 49 72 79 86 89 50 73 91 87 90 51 74 82 89 91 52 75 83 90 93 53 77 85 91 94 54 78 86 93 95 55 79 87 94 97 56 80 88 95 98 57 81 89 96 99 58 82 91 98 101 59 84 92 99 102 60 85 93 100 103~ 61 86 94 102 105 62 87 96 103 106 63 88 97 104 107 64 90 98 106 108 65 91 99 107 110 66 92 101 108 111 67 93 102 109 112 68 94 103 111 114 69 95 104 112 115 70 97 105 113 116 71 98 107 114 117 72 99 108 116 119 73 100 109 117 120 74 101 110 118 121 75 102 111 119 123 76 104 113 121 124 77 105 114 122 125 78 106 115 123 126 79 107 116 124 128

F?T.X JoIJ-RWG wlXT?ZR 1988/ .989 43

(b) reauirement: 18, 5% 1% 0.2% 0.1% 80 108 117 125 129 81 109 119 127 130 82 110 120 128 131 83 112 121 129 133 84 113 122 130 134 85 114 123 132 135 86 115 125 133 136 87 116 126 134 138 88 117 127 135 139 .e9 118 128 137 140 90 120 129 138 141 91 121 131 139 143 92 122 132 140 144 93 123 133 141 145 94 124 134 143 146 95 125 135 1U 148 96 127 137 145 149 97 128 138 146 150 98 129 139 148 151 99 130 140 149 152

100 131 141 150 154

04 05 78

A SIMPLE APPROACB TO VOLATILITY

RICHARD C. ORR, Ph.D.

INTRODUCTION .

One of the more elusive concept6 in technical analyeio ie that of volatility. The dictionary define6 volatility 86 the quality of being changeable or trinsient, that is: passing quickly or 6oon. The problem one face6 in trying to utilize volatility in forecasting the market is in trying to construct a volatility measure, which in itself i6 not too volatile. A mea6ure of volatility that give6 a low reading one day and a high reading the next would be useless. At the same time, simply constructing 8 volatility function that evolve6 slowly from high value6 to low value6 8CConIpli6he6 nothing if it doe6 not accurately reflect the likelihood of the future movement of the market.

In what follow6 we will propose an easily calculated measure of volatility, give the reader 8 feel for its behavior over more than 8 ten year period, and suggest a property that volatile market6 seem to have, which could be exploited in market timing work. We will 8160 di6CU66 multiplicative filter6 generslly, and look 8t 8 5 percent nultiplicati.ve filter on ihe S&P 500 index to 6ee how volatile the market has been at intermediate-term (about 2 months) top6 and bottom6 in the market over the past ten year6.

A CANDIDATE FOR A VOLATILITY HEASURE.

Definitions.

A classic .mathematical definition of volatility might be that of variance. calculate6

For 8 given bet of values (x,, x2, . . . . x,1 one it6 mean or average, ic, and then proceed6 to 6\rm the

squares of the difference6 between each value and th& mean value, finally dividing this 6um by n-l (one less than the number of term6 added)l. Algebraically stated, variance is given by:

VAR = --& 5 hi - I) 2

i=l

1The question 86 to when we divide by n-l or by n depend6 on whether we are dealing with 8 whole population or 8 sample of that population, but will not be important for our di6CU66iOn.

44 E’A JCiJFX?& KIWER 1988/1989

Notice that if all the values {xl, x , 1

. . . . 51 are equal to some value, a, then the mean is also l qua to a, and the variance would be 0. There is in that case no variability of the values around a. In all other cases variance vould be positive, with values far below the mean contributing just as much to the variance as values far above the mean. This latter feature is important since vith volatility the changeability of the market, but not the direction of such a change, is being measured.

One of the drawbacks of using variance as the measure of volatility is that it may be less sensitive than it should be to the extreme values, vhich are of the utmost importance vithin a given time interval. A much simpler measure of volatility, vhich ve now consider is the following: For some fixed period, say 25 days2, calculate the largest and smallest values of the data within that period. The measure of volatility ve will use will have the characteristics of the ratio of the largest piece of data to the smallest piece of data, but to make the function easier to vork vith we define it as follovs: Let MAX be the largest of 25 consecutive daily closing values of the S&P 500 index, while HIN represents the smallest value of the same set. Our volatility function is then defined as follows:

VOLATILITY - 100 l (!E-1) All this definition does is to simplify the interpretation of’its values, e.g., if VOLATILITY = 20 then MAX is 20 percent larger than MIN.

Notation.

In the tables that follow we will use a calendar notation YPMMDD, which is common to many databases. This is simply a six digit representation of a date where the first two digits represent the last two digits of the year, the second two digits represent the month and the last two digits represent the date. For example, 791008 represents October 8, 1979, while 800327 represents March 27, 1980.

MAIN RESULTS.

Baving defined volatility to be the percentage that the largest value in the 25 day range is above the smallest value in the same

2Any number could be used here. The larger the number of days considered the longer term this measure of volatility vould be. Units are daily, but could be hourly, veekly, monthly, or anything else.

1’R’A JOE%AL WINTER 1988/1989 45

range, let us look at the distribution of its values from 750206 through 850916. Table I on the next page displays a frequency distribution in half-percent increments for the 2679 days in question. For example, there vere 224 days in which the volatility value was at least 5.5 percent but less than 6.0 percent, while the total number of days with volatility less than 6.0 percent was 1597.

The first question we deal vith is that of the transience of our volatility measure. By the vay ve have defined volatility, we have guaranteed that successive values vi11 be highly correlated. The data from which the volatility values are calculated overlaps until 25 days have elapsed. We consider the autocorrelation of the time series of volatility values with lag 25. This simply means that we look at the correlation between each pair of values separated by exactly 24 trading days. We would like to see a positive correlation between these two values of volatility even though 25 days have passed. The process begins on 750206, the 25th trading day of the year, and 750314, the 50th trading day of the year. The volatility values for all pairs of days that are 25 days apart are compared from this point until 850819 and 850924, the last pair considered. For these 2660 pairs of days ve have a correlation coefficient r=+.178, which is very significantly positive. Thus ve have shown that over the ten-plus year period in question volatility levels tend to maintain some stability into the future.

Next, it is our intention to show that periods of high volatility are charicterized by a certain pattern of market action. We claim that the following schematic describes the typical market behavior during periods of high volatility and beyond.

S&P 500 PRICES

SUBSEQUENT TOP

BEG IN PERIOD OF HIGH OF LOW VOLATILITY VOLATILITY ------- -----------------------------

SUBSEQDENT BOTTOM

46 ,Xl’A JOUFW& WIFT’lXR 1988/1989

At Least But Less Than count Cumulative 0.0 0.5 0 0 0.5 1.0 0 0 1.0 1.5 0 0 1.5 2.0 2 2 2.0 2.5 39 41 2.5 3.0 105 146 3.0 3.5 211 357 3.5 4.0 307 664 4.0 4.5 293 957 4.5 5.0 215 1172 5.0 5.5 201 1373 5.5 6.0 224 1597 6.0 6.5 154 1751 . 6.5 7.0 156 1907 7.0 7.5 141 2048 7.5 8.0 100 2148 8.0 8.5 107 2255 a.5 9.0 72 2327 9.0 9.5 42 2369 9.5 10.0 40 2409

10.0 10.5 56 2465 10.5 11.0 23 2488 11.0 11.5 42 2530 11.5 12.0 19 2549 12;o 12.5 15 2564 12.5 13.0 1.1 2575 13.0 13.5 27 2602 13.5 '14.0 16 2618 14.0 14.5 2 2620 14.5 15.0 6 2626 15.0 15.5 6 2632 15.5 16.0 19 2651 16.0 16.5 3 2654 16.5 17.0 6 2660 17.0 17.5 5 2665 17.5 18.0 2 2667 18.0 18.5 0 2667 18.5 19.0 1 2668 19.0 19.5 1 2669 19.5 20.0 7 2676 20.0 20.5 1 2677 20.5 21.0 0 2677 21.0 21.5 2 2679

Table I. Frequency Distribution of Volatility Values. 750206 through 850916

MTA JC)UIXX WINTER 1988/1989 47

For our purposes, high volatility will be any value of volatility that is at the 75th percentile or higher, while low volatility will be any value that is at the 25th percentile or lover. From the frequency distribution (Table I> we find these values to be VOLATILITY 2 7.5 and VOLATILITY < 4 respectivelg. We then study all intervals that begin with the first day that VOLATILITY is at least 7.5 and end with the first day that VOLATILITY is less than 4.0. Within each interval there is a largest price that we refer to as the TOP and a smallest price that we will refer to as the BOTTOM. What the above schematic asserts is that most of the time the BOTTON will precede the TOP and that the starting day will have a lover price than that of the ending day. Table II on the next page demonstrates this characteristic for the period 750206 through 850916.

A MULTIPLICATIVE FILTER

While for very short tern-market moves a standard additive filter may be fine, for most moves we feel that a multiplicative filter is superior. With an additive filter, moves up equal moves dovn: a move to 100 from 50 would have as its counterpart a move to 0 from 50 (50+50 and 50-50). A similar multiplicative filter would pair a move from 50 to 100 with a move from 50 to 25 (50=(2) and 50.('1/2)). Given that prices are bounded below by 0 but unbounded above, we feel that.a multiplicative filter handles this situation more appropriately.

For the period 750206 through 850924 consider a 5 percent multiplicative filter on the S&P 500 index. Under this filter a TOP is any largest price in an interval that is 1.05 times (larger than) some previous price and subsequent price in that interval. Similarly, a BOTTOM is any smallest price in an interval that is l/l.05 times (smaller than) some previous price and subsequent price in that interval. Table III, displayed on the two pages following Table II, lists all the extrema for a 5 percent multiplicative filter on the S&P 500 index. Also listed are the volatility values as well as their percentile values on the day of each TOP or BOTTOM. It is worth noting that on the average BOTTOMS tend to be more volatile than TOPS. Perhaps even more interesting is that both TOPS and BOTTOMS tend to occur in more volatile times. Of the 53 extrema listed in TABLE III, 20 of 27 TOPS and 21 of 26 BOTTOMS were above the 50th percentile in volatility. ~~

3For those who are interested, the distribution of volatility has a mean of 6.12 and a standard deviation of 2.92, but is not normal (in fact, it is skewed to the right), which is why we are using the frequency distribution to find the percentiles.

48

First Date

750206 78.56 750210 78.36 BOT 750715 760106 93.53 760106 93.53 BOT 760220 761022 99.96 761110 98.81 BOT 761215 780424 95.77 780424 95.77 BOT 760517 780802 102.92 780802 102.92 BOT 780906 781020 97.95 781114 92.49 BOT 790126 791015 103.36 791107 99.87 BOT 791217 800123 113.44 800327 98.22 BOT 800903 801015 133.70 801030 126.29 BOT 801128 810325 137.11 810325 137.11 TOP 800414 810827 123.51 810925 112.77 BOT 811130 820111 116.78 820129 120.40 TOP 820308 820604 110.09 820611 111.24 TOP 820707 820806 103.71 820812 102.42 BOT 830304 830428 162.95 830519 161.99 BOT 830525 840208 155.85 830223 154.29 BOT 830227 840529 150.29 840608 155.17 TOP 840615 840803 162.35 840808 161.75 BOT 840821 850121 175.23 850121 175.23 BOT 850213

S&P 500 First Extremum Date S&P 500

Second Extremum Date s&P 500

95.61 TOP 751111 89.87 102.10 TOP 760227 99.71 105.14 TOP 761223 104.84

99.60 TOP 780526 96.58 105.38 TOP 780906 105.38 101.86 TOP 790307 98.44 109.33 TOP 791231 107.94 126.12 TOP 800910 124.81 140.52 TOP 810225 128.52 132.68 BOT 810415 134.17 126.35 TOP 811228 122.27 107.34 BOT 820511 119.42 107.22 BOT 820709 108.83 153.67 TOP 830330 153.39 166.21 TOP 830531 162.39 159.30 TOP 840312 156.34 149.03 BOT 840716 151.60 167.83 TOP 840907 164.37 183.35 TOP 850227 180.71

Last Date S&P 500

Table II. Order of Precedence of Extreme Values During Moves from High to Lov Volatility. 750206 through 850916

??T. JCUEJX '+KXEZ? 1988/1989 49

INTERMEDIATE TOP DATE PRICE VOLAT

750317 86.01 9.76

750715 95.61 6.14

751117 91.k 3.83

760921 107.83 6.48 71

770103 107.00 4.80 48

770719 101.79 3.09

771125 96.69 6.59 72

780606 100.32 4.61 45

780912 106.99 3.58 27

790126 101.86 7.55 81

790410 103.34 4.98 51

791005 111.27 4.58 45

800213 118.44 7.78 82

PEBC 91

67

32

15

INTERMEDIATE BOTTOM DATE PRICE VOLAT

750407 80.35 7.04

750916 82.09 6.13

751205 86.82 5.34

761110 98.81 4.79

770531 96.12 4.35

771102 90.71 6.65

780306 86.90 4.52

780705 94.27 6.42

781114 92.49 13.95

790227 96.13 5.96

790514 98.06 5.38

791107 99.87 11.41

800327 98.22 17.12

Table III, Part I. Volatility Values for Extreme Values (5 Percent Filter). 750206 through 800630

PERC

77

67

57

48

41

73

44

70

98

65

58

95

100

50 MI'A ;rOU?NAL WTMI'EZ 1988/1989

INTERMEDIATE TOP DATE PRICE VOLAT

800922 130.40 6.82

801015 133.70 8.22

801128 140.52 11.27

810106 138.12 8.45

810325 137.11 8.32

810811 133.85 5.29

811130 126.35 6.93

820507 119.47 4.13

821109 143.02 13.54

821207 142.72 7.59

830622 170.99 5.97

831010 172.65 5.03

841106 170.41 5.41

850717 195.65 5.57

Average for TOP 6.53

PERC 75

85

95

87

86

56

76

38

98

81

65

52

58

60

64 Average for BOTTOM 7.70 73

INTERMEDIATE BOTTOM DATE PRICE VOLAT PEBC

800929 123.54 6.82 75

801030 126.29 8.22 85

801211 127.36 10.33 93

810220 126.58 6.47 71

810722 127.13 4.89 50

810925 112.77 14.60 98

820308 107.34 9.94 92

820812 102.42 8.90 88

821123 132.93 7.59 81

821215 135.24 7.36 79

830808 159.18 7.02 77

840724 147.82 4.75 69

841213 161.81 4.25 40

Table III, Part II. Volatility Values for Extreme Values (5 Percent Filter). 800701 through 850924

.m’R J’3JKS.L WINEI? 1988/1989 51

MOVING AVERAGES AND THEIR~VARIATIONS

John Carder

When markets are discussed, one of the most frequently used words ir “trend“. Dow Theory uses the concepts of Primary Trend, Secondary Trend and Minor Trend. Elliot Wave Theory (as practiced by Robert Prechter) mentions waves of degrees ranging from Grand Supercycle to Subminuette. Usually, the firstT,s,';pil; analyzing a market is to identify the primary trend. the trend that is of sufficiently large degree to be in effec; for the expected duration of your trade. Unfortunately, prices don't move in smooth curves. They are up one momentanddownthe next. Moving averages were developed in an attempt to tame the zigs and zags on a chart into a smooth curve. This paper will discuss some of the refinements to moving averages and suggest a new one.

Conventional Moving Averages (MA's):

Conventional moving averages are a simple idea. An "n period MA" is simply the average (mean) of the last n values. That is, take the sum of the last n values and divide by n. One of the most common examples is the 200-day moving average of the Dow-Jones Industrial Average (see Figure 1). This turns the daily gyrations of the stock market into a gentle curve. The most common usage of the indicator is to be long stocks when the DJIA is above its 200-day MA and out or short when the DJIA is below its 200-day MA. This example uses days for the periods. Any time period is possible for this and the other moving averages mentioned in this paper. I have seen periods ranging from 5 minute bar charts of futures prices to yearly charts. Whatever the period, the curve can be smoothed by using a moving average.

Exponential Moving Averages (EMA's):

The first variation is the exponential moving average. There is no standard way of defining an EMA. Martin Pring suggests the following formula in his book Technical Analysis Explained: To calculate an "n period EMA" for a range of values, you first calculate the "n period MA" for the first n values. Then, take the difference of the n+lst value and the "n period MA". Multiply the difference by t e "exponent", 2/n, and add the product to the previous 1 value. An example will probably make more sense than this explanation. We will take a four period EMA of the following values: Since n=4, the exponent is 2/4 or 0.5.

52 YTA JGiJX?AL WINTJZR 1988/1989

Table 1:

1 2 3 4 5 6 Previous Difference Difference EMA (co1.3+

Period Value EMA (co1.2-~01.3) times Exponent co1.5) P=====E--------------------- ---------------------=========~= --------P=='ftDI'====E====rI=2='============ --------

1 10 2 11 3 12 4 11 5 12 11 1 0.5 11.5 6 13 11.5 1.5 0.75 12.25 7 14.25 12.25 2 1.0 13.25 8 12 13.25 -1 -0.625 12.625 ----- ,,,,,,-----------------------------------=========----- ===============t====--------------- --------------------------

The previous EMA (column 3) of the fifth period is the average of the first four periods. Figure 1 also shows the 200-Day EMA of the Dow-Jones Industrial Average.

Two advantages are usually claimed for the EMA over the MA. The first is that it is easier to compute than the MA. Since you only use two values (the previous day's EMA and today's value) as opposed to all n values in an MA, it is supposed to be faster. However, you don't need to add the last n values to compute an MA. Most of the numbers in that sum are the same as in the prior period's calculation. You add one new value and subtract one old value. Again, a table is useful. We'll compute a 4 period M.A with the same values.

Table 2: 1 2 3 4 5 6 7

Previous Value four Difference MA (~01.3 Period Value MA periods ago col.2-~01.4 col.S/n +co1.6)

-w--- --------_---___-__------------------------------------- ================t==r-------------------------------------------------- 1 10 2 11 3 12 4 11 5 12 11 10 2 0.5 '11.5 6 13 11.5 11 2 0.5 12 7 14.25 12 12 2.25 0.5625 12.5625 8 12 12.5625 11 1 0.25 12.8125

----_--------__--_------- ------------------- ============-------------------- ,,,,,DPI=tt=='='PIS'tP=P-----------------------

The point of my discussion is that using the above formula, a conventional MA should take about as long to compute as an EMA.

The second advantage claimed for an EMA is that it is front weighted. All of the last n values are equally weighted in a conventional MA. The most recent value is the most heavily

,~AJCURNALWINTEP 1988/1989 53

weighted in an EMA. In fact, an EMA is always moving toward the most recent unsmoothed value.

Figure 2 contrasts the weig ting for a10 periodEMAwiththatof a conventional 10 period ?! MA. As you can see, the four most recent periods are more heavily weighted in the EMA than in the MA, giving it its front weighting. Unfortunately, an EMA is also "tail weighted". Over ten percent of the weight is assigned to values that are more than ten periods old. The situation gets worse for longer term EMA's, as you can see in Table 3.

Table 3: DAY EMA: TAIL W-EIGHT ------------------I== ------------------ : 10 10.7% : : 15 :: 11.7% : : 20 : 12.2% : . . 30 : 12.6% : : 50 13.0% : : 75 : 13.2% : : 100 13.3% : : 200 :: 13.4% : PI====================

This means that while an EMA is always trending toward the most recent unsmoothed value, it is being held back by a substantial amount of stale values. When you have a large move in prices, the EMA is held back by its "tail weighting".

Front Weighted Moving Averages (FWMA's)

In order to avoid the problem of "tail weighting", you must use a real front weight moving average (FWMA). The most common way to construct an "n period FWMA" follows. This kind of smoothing has been called the Coppock Guide. Multiply the most recent value by n, the next most recent by n-l, the third most recent by n-2, an so on. Sum the products. Divide by the sum n+(n-l)+(n- 2)+...+2+1. (That sum equals n*(n+l)/2.) The result is the most common FWMA. Tables 4 and 5 illustrate the calculation for a four period FWPIA.

54 MTA JOUR'Z& WIKI'EX 1988/1989

Table 4: Weights

Day 1 2 3 4 ===========t==t====tt===tl=='ttt==IDrt=D============ Weight 4 3 2 1 Sum of weights=4*(4+1)/2=10

Table 5: 1 2 3 4 5 6 7 8

4 times 3*last 2*2nd 1*1st Sum of FWMA Period Value most recent value value value products col.7/1cT --------------------============================== -------------------- E=p=tr=tPPI=L=rt=tI=tD=L

1 10 2 11 3 12 4 11 44 36 22 10 112 11.2 5 12 48 33 24 11 116 11.6 6 13 52 36 22 12 122 12.2 7 14.25 57 39 24 11 131 13.1 8 12 48 42.75 26 12 128.75 12.875

-----------------------------'--""""----------= _---------_----_---_------------------------------ ===t====='=============~

Figure 3 contrasts the weighting of a lo-day EMA and a lo-day FWMA. As you ca.n see, the FWMA loses the tail weighting of the EMA. The FWMA is truly front weighted and is not "tail weighted". Its main disadvantage is that it involves a lot of computation. That has been overcome by the advances in computers. We can eliminate the step of dividing each sum of products by dividing weight by that sum before we begin. That is, we would multiply by 0.4 instead of 4 in column 3 of table 5. This means that an n period FWMA requires n multiplications and n-l additions. The FMA required one subtraction, one multiplication and one addition. The MA required one subtraction, one division and one addition. For long period FWMA's, a computer becomes essential.

Figure 1 also illustrates the 200-day FWMA of the DJIA. As in the cases of the MA and EMA, shorter period FWMA's are less smooth than longer period FWMA's.

A problem arises with all three of these moving averages when markets make large sudden shifts, as the gold market did in early 1983. All those old numbers will tend to draw the average away from current values for some time after the shift.

A shorter FWMA will lose the old values more quickly, but it won't be as smooth as a longer term FWMA. I have devised a simple system to vary the front weighting of an FWMA continuously from one extreme of a simple MA to the other extreme of the most recent value.

Variable FWMA's:

Consider the function Y=X raised to the power P, with X ranging

MTA JCURN?G WIRIER 1988/1989 55

from 0 to 1.3 Figure 4 shows this function for several values of P. To construct an n period FWMA, you begin by constructing the weights. You will need n of them. Begin with the fractions l/n, 2/n, . . . (n-1)/n, n/n. Raise each of them to the power P. These are'your unadjusted weights. Unfortunately, they do not add up to one (unless n=l), so we do one last thing to obtain our weights. Sum them and divide each of them by the sum. Now you have your weights. 'Let's construct the weights for a 3 period FWMA with P=2 as an example:

Table 6: Weight: 1 2 3 ='t=I=====r=='t==='I===='rrPS==IPLPtPD=I===== l/3 l/3 213 1 >>>>>>>> the fractions squared 1/g 4/g 1 >>>>>>>> the unadjusted summed = 14/9 = (l/9+4/9+1) weights ====t=='PE=tllPft"t=="==t==Pllt=ltPtDI===== The weights l/14 4/14 9/14 >>>>>>>> the unadjusted

weights divided by the sum

To determine the n-period FWMA for a given period, you multiply the last n values by the corresponding weights and then sum their products. Table 7 provides an example:

Table 7: 1 2 3 4 5 6

9/14 times most 4/14 times l/14 times FWMA (~01's Period Value recent value 2nd value 3rd value 3+4+5) -------------------------------- -------------------------------- --------------------------------------====: --------------------------------------

1 10 2 11 3 12 108/14 44/14 lo-/l4 162/14 (11.57) 4 11 99/14 48/14 11/14 158/14 (11.29)

i 12 108/14 44/14 12/14 164/14 (11.71) 13 . 117/14 48/14 11/14 176/14 (12.57)

-----------------------------------------------------------------=========: -----------------------------------------------------------------

Figure 5 shows some adjusted weighting curves for various values. of P. If you compare Figure 5 with Figures 2 and 3 you will notice that P=O yields the weighting curve of a conventional moving average and that P=l yields the weighting curve of the Coppock Guide described above. Values of P between 0 and 1 yield FWMA's that are less front weighted than the Coppock Guide, while values greater than 1 yield FWMA's that are more front weighted than the Coppock Guide. Also, there is none of the tail weighting of the EMA.

Why bother with variable FWMA's? Conventional MA'S are slow, and a sudden shift in underlying values makes too sudden shifts in the MA; one when it enters the sum and one when it departs the sum. EMA's are tail weighted, and I don't think that that is what we want in a smoothing. Probably the most compelling reason is that the ability to vary P gives us another degree of freedom

56 ,XTA JOUFNAL WINTEZ 1988/1989

in attempting to find the "right" moving average for a given application. The second part of this paper shows how that degree of freedom is used to improve a valued indicator.

An Example:

In his excellent Winter 1980/81 Chartbook, a tool to illustrate the four year cycle.'

Ian McAvity introduced He started with the

monthly average of the daily closing level of the S&P 500. He took the 12 month percentage change of these values. He then used a lo-month Coppock Guide (10 month FWMA with P=l) to smooth the % change. A buy signal was generated whenever this smoothed value was falling and negative and then turned up. Figures 6, 7, and 8 illustrate this tool. To quote from his chartbook: "Yes, it's always LATE on the upturn (but sure)". I want to thank Mr. McAvity for his generous assistance in the work that follows. He was kind enough to send me the underlying data, along with a worksheet explaining the Coppock Guide and how it is constructed. The refinements which follow are not intended as a criticism of his insightful work, rather they are an attempt to "fine tune" an already excellent tool.

First, I varied the percent change from 10 to 14 month changes. Then for each of these I computed 42 FWMA's on the data. P varied over 0.8, 0.9,. 1.0, 1.1, 1.2, 1.3, 1.4. Also, I computed 9, 10, 11, 12, 13 and 14 FWMA's for each of the P's. I will use the terms fast and slow in the following discussion. One FWMA is faster tha=otherit is shorter in period or has a greater P. I firstnoticedthatthe faster FWMA's were whipsawed by bear market rallies, and the slower ones were even later in their signals. Figure 9 illustrates this effect. How do we optimize this tool? Which combination is best? Is there a best combination? The problem here is that by using a faster combination to get an earlier (more profitable) buy signal, we risk being whipsawed. In order to compare the various values, I calculated the average gain of the signals generated by each FWMA. I assumed a 28 month holding period, and if a subsequent signal was generated during those 28 months, the holding period was extended to 28 months from the subsequent buy signal, and it was considered one long signal. Table -8 shows the result of averaging these calculations. As you can see, the lo-month FWMA _ with P=l that Mr. McAvity used produced an excellent average gain (34.46%). The best average gain (36.92%) came from a 13 month FWMA with P11.3 of the ll-month percent change. Figures 10, 11 and 12 illustrate it and its signals.

There is another way of looking at the signals. Cumulative gains might yield different results. Table 9 shows the results of compounding your gains at each signal, that, is, as if you had invested $1.00 at the first signal/sold it as above, and invested the proceeds at the next signal, etc. The major difference in the results is in the very slow signals. They generated fewer signals and thus had lowercumulative returns. The best result was once again a 13-month FWMA with Ptl.3 of the

MTA JOEN& WINTER 1988/1989 57

ll-month percent change.

There were three other combinations that should be mentioned, since they had good results n both average and cumulative gains. They are 12-month FWMA with P=O.9 and 1.0 of ll-month percentage change and 12-month FWMA with Pr1.3 of 12-month percentage change. Tables 10,11,12 and 13 show the signals generated by these three combinations and the best one. None of the four combinations generated a losing signal. The fourth one was odd, though. It was generally late and generated two extra signals. It does illustrate the problem of relying solely on gains. I would describe it as a lucky anomaly and would place little faith in it as a technical tool. The remaining three smoothings all were of ll-month percentage change. The 13 month FWMA with Pf0.9 was generally slower, but August 1938 was a lower place to buy than July 1938. This kept it in the running. Obviously, none of them is best for all purposes. The 13-month FWMA with P-l.3 has yielded the best results.

The difference in gains does not seem particularly significant to me. Most of the improvement is due to its having given a signal a month earlier here and there. A shorter conventional FWMA would have signalled earlier too, but would also have been easier to whipsaw. The Coppock Guide was trapped by bear rallies in March 1948 and February 1958, while the 13-month FWMA with Pz1.3 did not generate signals in either bear market rally. The additional degree of freedom has allowed us to find an FWMA t= -s---p generates more relzble signaT and it generates them sooner. --

It could-be argued that this has been merely an exercise in curve-fitting. If you examine Tables 8 and 9 in the area of this FWMA, you willseethatany of the FWMA's near it would have also produced good results. . I think that this continuity of the results indicates that we haven't found an anomaly. Why should we expect this to continue to be effective? Mr. McAvity introduced it in 1980. We have had two signals since then, in 1982 and 1984. At this point, they appear to me to be good signals, confirming the indicator's predictive val<e.

Figures 13 and 14 show one more refinement. I havetakenthe13- month FWMA with Po1.3 of ll-month percent change and converted it to daily figures. As you can see, the original Coppock Guide and the three smoothings above all gave buy signals in January 1985. There is no guarantee, but it indicates that the bull market is going to be alive and well for some time. No indicator is perfect. It is still late, but it is not quite as late. It still can be whipsawed, but if markets are similar to those of the last 60 years, this is not likely to happen. That is the benefit that the additional degree of freedom provides. It gives one more way to refine good tools.

Moving averages are the technician's tools for smoothing. Each has its advantages and disadvantages. A conventional MA is simple, easy to explain and easy to use. It is the slowest of

58 -PITA XtJFNiG WINI'ER 1988/1989

the n-period MA's that I've covered. The EMA is faster and easy to calculate. It does have that "tail weighting". The main disadvantage of the FWMA is its computation time. Luckily, computers are overcoming that disadvantage. By adding the FWMA with parameter P to your collection of smoothing tools, I hope that you will be able to refine your indicators.

MTA JOUFKAL WlXIER 1988/1989 59

FOOTNOTES :

1) Often, EMA's are described not as an n period EMA, but as an m% EMA, where m is the exponent viewed as a percent. The conversion formulas follow:

An n period EMA is a (ZOO/n)% EMA. An m% EMA is a (200/m) period EMA.

One advantage of the percent nomenclature is that it allows a wider range of values for the exponent. On the other hand, I have seen no standard way to start an EMA with an exponent equivalent to a non-integral number of days.

2) Where does the tail weighting come from? It is inherent in the construction of the EMA. First note that in the construction of the EMA described in Table 1, we multiplied the difference (column 4) by the exponent and added that to the previous EMA. That is equivalent to multiplying the Value (column 2) by the exponent, then multiplying the Previous EMA (column 3) by (l- exponent) and finally adding the two products. We will use this equivalent construction in the example below. An example shows how an EMA gets its tail weighting: Assume a lo-day EMA. Its expoent n equals 2/10=0.2. So 20% of Tuesday's unsmoothed value is added to 80% of Monday's EMA. So Tuesday's unsmoothed value has a weighting of 20% in Tuesday's EMA. On Wednesday, we repeat the process, adding 20% of Wednesday's unsmoothed value to 80% of Tuesday's EMA. In so doing we have reduced the weighting of Tuesday's unsmoothed value from 20% to 16% (80% of 20%). On Thursday, we repeat the process, adding 20% of Thursday's unsmoothed value to 80% of Wednesday's EMA. The weighting of Tuesday's unsmoothed value is thereby reduced from 16% to 12.8% (80%of 16%). As each day goes by, the weighting of Tuesday's unsmoothed value is reduced to 80% of what it was the day before. While repeatedly reducing the weight to 80% of its previous value does shrink its value, it NEVER goes to zero. So any day's unsmoothed value will be reflected in the EMA, albeit to a small degree.

In general for an EMA with exponent n, the weight of the dth period's value in the EMA is given by:

(d-1) n(l-n)

where the most recent period is considered the first period, the previous period is the second period, etc.

3) If the reader is unfamiliar with exponents, the following explanation maybe helpful. In allthatfollows assumethatxis a positive number. And if P is a natural number (i.e. 1,2,3,4 ,...), then X to the power P is the product of P X's. If P=O then X to the power P is 1 for any positive X. This definition has been expanded to cover any number P (even infinite

60 Rl'A JOUPYAL, Wl3l?ER 1988/1989

cardinals). If you are writing a program in BASIC you only need type X-P and the language will perform the calculations for you. Some calculators offer LOG and/or LN functions, they usually also offer their inverses. If you insist on doing it by hand, LOG and LN tables can be found in the Handbook of Chemistry and Physics published annually by the Chemical Ruaero. 18901 Cranwood Parkway, Cleveland, Ohio, 44128. To calculate X to the power P, first take the LOG (or LN) of X, then multiply the result by P and apply the inverse function to the product.

4) Ian McAvity edits DELIBERATIONS, published twice monthly by Deliberations Research, Inc., P>O> Box 182, Adelaide Street Station, Toronto, Ontario, Canda M5C 251

REFERENCES:

The Dow Theory Today, Richard Russell, Edit= 1981 a division of:

Fraser Publishing Company

Fraser Management Associates Box 494 Burlington, Vermont 05402 Library of Congress Catalog Card Number: 81-68858 ISBN: 0-87034-061-l

The Elliott Wave Principle, Robert Rougelot Prechter, Jr. and Alfred John Frost

Second Edition 1981 New Classics Library, Inc. P.O. Box 1618 Gainesville, Georgia 30503 Library of Congress Catalog Card Number: 81-80170 ISBN: o-932750-02-8

Technical Analysis Explained, Martin J. Pring, McGraw-Hill Book Company 1980 Edition

ISBN: O-07-050871-2

BIOGRAPHY

John Carder received a B.A. in Mathematics from the University of Colorado at Boulder. He currently acts as an advisor to several of his family's trusts. He is the author of the programs used to draw the charts appearing in this paper.

61

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62 MIX JCIUXCRL h. 1988/1989

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64 YTA Zou?;NAL, h’l2.m 1988/1989

FIGURE 4 1.00

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em JOUXtAL wIIXIE2 1988/1989 65

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KrA JOLJFSAL WIXEX 1988/1989 67

FIGURE 3

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'-* JOUSTAL, wETIE 1988/1989

FIGURE 5

Adj usted weighting curves for vario

0.50

0.00 0.00 0.25 0.50 0.75 1.00

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us P’S

70 >J'.JA JCUFWJL Xl3lTE2 1988/1989

12 month % change (monthly)

30.

20.

10-

100%

40%

20x

-0%

-2 0%

-4 0%

T 22 24 26 21 ,I8 Ii! 34 36 31 ,48 42 44

71

.

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FIGURE 7 150-

- S&P 500 - Monthly Rverage of the Monthly Rverage of the

Dally Clorlng Level Dally Clorlng Level

n 40%

12 month % change 12 month % change

30%

10%

I

V V w 0% 9 -10%

10 month MM ofth P-l 10 month MM ofth P-l

72 MTA JOfJFSAL PXlXER 1988/1989

r S&P 500 Monthly Rverage of the Dally Closing Lsvel

FIGURE 8

40%

12 month % change

10 month FWMA ulth P-1

30%

20%

10%

\ I 0%

-10%

XI!.% JC3JRX.U iGlXI%X 19881 1989 73

P = l 2 . 4 . 6 . 8 1 1.2 1.4

4

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14

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if t h e March 1935 bottom was signalled *late? than May 1935

if was trapped by the 1957 bear market rally

if was trapped by a 1974 bear market rally

(FWMR's of 12 month percent change only)

FIGURE 9

/

74 XI’A JOLJFSAL ?TINTE3 1988/1989

4 ! ! ! ! ! ! ! 1 ! 1 ! ! ! ! ! ! ! ! ! ! ! ! 1

11 month X change 100% (monthly)

40%.

13 month FWMA wfth P-l.3

l~~ mmm WINTER 1988/1989 75

- S&P 500 80-

- flonthly Average of the flonthly Average of the Dal ly Closing Law I Dal ly Closing Law I

60-

40-

FIGURE 11

11 month % change

42 44 46 48 ,5E .

76 cTA ,XUFN.. wIlX'l3 1988/1989

c . . . . . . . I - - . t S&P 500 t flonthly Rverrge of the

t

Dally Closing Level

FIGURE 12 125 1

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13 month fHtR utith P-1.3

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77

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FIGURE 13

1978 1979 1980 1981 .

drawn daily from 2

MTA JOLITWU WETIER 1988/1989

1982 1983 1984 1985

I

180

160

140

120

100

ilAY MOVING AVERAGE Jan 1978 to 3 Ott 1985

YTA JOUEL'iL TIJCITER 1988/1989 79

c I I I

1

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FIGURE 14

1978 1979 1980 1981

I I 1 I I I I 1 I I 1 I 1 I 1

231 DflY % CHANGE OF THE 21 DRY Mfl OF THE

drawn daily from 2 3

80 m'?i JOUXGG WII- 1988/1989

1983 1984 1985

20

0

1 I I I I 1 I I I 1 I I 1 I I

SW 500 WITH ITS 273 DAY FWMfl USING P-1.3

40

an 1978 to 3 Ott 1985

XTA JCXIXW wllT?ER 1988/1989 81

10 MONTH CHANGE WITH MONTH P FWMA .E .9 1.0 1.1 1.2 1.3 1.4 .------.............................*......................................*...*

9 30.39% 30.39% 35.48% 35.51% 34.933 34.01% 33.79% 10 31.17% 31.172 31.97% 30.64% 35.57% 35.25% 35.22% 11 30.11% 30.43% 31.172 31.58% 31.58% 30.64% 30.64% 12 30.29% 30.62% 30.70% 30.43% 31.58% 31.58% 31.58% 13 27.53% 27.53% 27.81% 28.15% 30.43% 31.58% 31.58% 14 29.40% 27.53% 27.53% 27.53% 28.05% 28.15% 27.96%

.-*--..........*.....*.....*........*....*.........*....*..*..*.................

11 RONTH CHANGE WITH MONTH P FWMA .E .9 1.0 1.1 1.2 1.3 1.4 . . . . . ..t.*.*........*..*...............*........*.......*....*..................

9 32.89% 32.51% 31.41% 30.12% 35.33% 35.33% 35.33%

10 31.62% 31.82% 33.93% 32.09% 31.46% 31.46% 30.93%

11 33.46% 53.15% 33.15% 33.93% 33.55% 3:.02x 31.78%

12 34.48% 36.a3z 36.87% 32.54% 34.05% 33.13% 33.13%

13 35.29% 32.29% 34.56% 33.92% 34.39% 36.92% 32.54%

14 31.60% 33.66% 33.08% 32.95% ii. 15% 33.92% 34.79% . . . . . . . . . . . . . . . . . . . . . . . ..*........................................*.............

12 MONTH CHANGE WITH MONTH P FWMA .E .9 1.0 1.1 1.2 1.3 1.4 . . . ..t....*......................*...............*..*.............*...........*.

9 34.01% 34.86% 30.66% 30.25% 30.82% 28.54% 28.58%

10 35.31% 34.86% 34:46% 34.86% 34.86% 30.77% 30.?7%

11 34.93% 35.75% 35.75% 34.86% 34.86% 34.86% 30.05% 12 35.61% 35.08% 34.60% 35.75% 35.75% 36.31% 35 56% .

13 33.42% 33.21% 33.75% 34.60% 34.84% 35.41% 35.75%

14 30 25% . 31.08% 31.08% 32.89% 33.40% 33.40% 34.94:: . . ..P.......PI........................................*.........................

13 MONTY CHANGE WITH MONTH P

FWMA .8 .9 1.0 1.1 1.2 1.3 1.4 ..-- --*......t..................................*.................................

9 27.20% 29.22% 28.93% 29.50% 29.33% 29.41% 29.67%

10 29.33% 23.9ax 28.74% 27.79% 28.34% 28.93% 29.50%

11 32.13% 31.73% 29.15% 28.76% 28.62% 29.29% 28.71%

12 31.51% 31.70% 31.70% 27.43% 27.40% 27.40% 27.07%

13 29.87% 30.83% 31.06% 31.64% 27.00% 27.61% 27.40%

14 28.67% 30.28% 30.28% 30.77% 31.13% 30.88% 27.18% ~.......~..*.~~...............................................~~~.~.~...~~~~....

14 MONTH CHANGE WITH MONTH P

FWMA .E .9 1.0 1.1 1.2 1.3 1.4 .*...*.........*...........*.*..*..*................*...........*........-=-====

9 31.21% 31.34% 32.08% 32.08% 32.08% 32.93% 32.93%

10 30.28% 31.21% 30.98% 31.28% 31.34% 31.29% 31.88%

11 26.90% 26.90% 27.92% 27.42% 27.60% 31.73% 31.73%

12 26.08% 25.88% 25.41% 26.71% 27.06% 28.71% 28.99%

13 24.79% 25.96% 21.42% 21.23% 25.41% 25.41% 27.14%

14 24.81% 25.96% 25.96% 21.42% 20.95% 20.42% 21.22% . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*............................*....-.....=-=--=-===

82 M!I!A JCURNAL k?INlXR 1988/1989

TFIBI F !=I CUMULFITIUF GFtIN

10 MONTH CHPINGE .WITH RONTH P FWMA .8 .9 1.0 1.1 1.2 1.3 1.4 1*.1111*11111111-111~===**~~~~~~.~~.~~*~*~=~*=.**~~~~~~~*~~=~=~~~=~~~.=~~*~=~=~~

9 14.7 14.7 16.1 16.2 15.5 14.7 14.4 10 16.1 16.1 17.1 15.0 16.1 15.8 15.8 11 14.6 14.9 16.1 16.7 16.7 15.0 15.0 12 14.6 15.2 15.3 14.9 16.7 16.7 16.7 13 14.2 14.2 12.1 12.6 14.9 16.7 16.7 14 21.2 14.2 14.2 14.2 15.1 12.6 12.3

111*1111111111-*111*=~*-~~**==~***=~~==~**~~==.~~~~-.*~--*--~~-*=~~-.~~**.~~*---

1 I MONTH CE3NGE UJITH MONTH P FWilrl .s .9 1.0 1.1 1.2 1.3 1.4 lPtllltlrltlllPIIItl=~~=~=~*=~*~~~*~~~=~*=~**=~~~==~~~=~*-~*~~=~=~~==~**==*~*=~-

9 13.3 18.7 17.1 15.2 17.3 17.3 17.3 10 17.6 17.6 21.5 18.1 17.2 17.2 16.4 11 20.5 20.1 20.1 21 .s 20.9 18.1 17.8 12 30.5 36.6 36 . 3 18.6 21.7 20.2 20.2 13 34 9 33.0 30.6 28.9 30.0 36.8 18.8 14 29:; 35.4 33.5 33.2 33.9 28.9 31.3

llPIIIItl*lllltl*lll*~=~~~*==~~~~~=~-~*==~~*~**=~~.**~~--~*-==~~*==*~~=~==~=~~==

it MONTH CHANfE 'JITH MONTH P FWMA .8 .9 1.0 1.1 1.2 1.3 1.4 III*II*IIIPIIIIPIIII*====~~~==~=~~-=~~*=~~*==~~~~*=~==-=-*-=~~*~~~~~~=~=~=~=~==-

9 29.1 31.6 16.9 15.3 17.3 13.4 13.5 10 32.4 31.2. 30.3 31.6 31.6 17.2 17.2 11 31.5 33.6 53.6 31.2 31.2 31.6 16.3 12 33.7 32.2 30.7 33.6 33.6 35.8 33.6 13 33.4 32.7 35.8 30.7 31.2 32.8 33.6 14 24.5 26.5 26.5 31.7 34.6 34.6 31.2

ImIIIII.II*PIIPIIII- -~IIIEIIPtlltll~ll*l=~~~~~~~~~~=~=*=~-~~~~*~~~~~-~==--=--*===

13 MONTH CHANCE UITH MONTH P FWMA .8 .9 1.0 1.1 1.2 1.3 1.4 f031=IPII==tPrSrlPI’OIPI- -IIIIPlllllltlllllll=~~=~~~~~~=~-~=~~~~~~*==~~=~~*~*----

9 14.1 17.1 16.5 17.1 16.9 17.2 18.4 10 17.2 16.6 16.3 14.8 15.7 16.5 17.1’ 11 30.5 29.3 16.9 16.3 16.1 17.0 16.3 12 28.6 29.3 29.3 14.2 14.1 14.1 13.6 13 22.9 25.8 26.5 29.1 13.6 14.4 14.1 14 19.6 23.7 23.7 25.6 26.7 26.1 13.8

Ill~tltlll=lllllllll==~~=**=~~~*~~~*~**-~~*~~~.~~~**~~~-----~.-*.*--~~~.---===--

14 MONTH CHANGE WITH MONTH P FWMA .8 .9 1.0 1.1 1.2 1.3 1.4 lI*~llllI*IIlIIrIIII~=*~*====~~*~~*~~*=~.~~.~~.=.==*~~*~-~*~~=~~*~--.~--==~~~~-.

9 16.1 16.5 17.6 17.6 17.6 19.0 19.0 10 14.8 16.1 15.8 16.4 16.5 16.5 17.3 11 14.2 13.4 14.5 14.0 14.2 17.0 17.0 12 12.6 12.4 11.2 13.1 13.6 15.6 16.2 13 11.2 12.9 6.7 6.3 11.2 11.2 13.7 14 11.2 12.9 12.9 6.7 6.1 5.7 6.4

I~llll~lll~l~lt~l~ll~=*~=~=~=.*~=~.~=~~.~~*.~=*.~=~.=~---=--===-=.**.~~.--~--**.

MTAJcuEwALWlXER 1988/1989 83

TFtBlF 18 11 month Zchangt smoothed by 12 month FWMA with P- .9

********************************

I*****28**APR 1924 l * * 54.90%

2****128**AUG 1932 l ** 12.11% 3****160**APR 1935 l ** 71.69% 4****200**AUG 1938 l ** 14.81% 5+***219*+MAR 1940 l ** 14.81% 6****232+*APR 1941 l ** 14.81% 7****249**SEP 1942 l ** 14.81% 8**+*308**AUG 1947 l ** 57.70% 9+***33A**OCT 1949 l ** 57.70%

10****385-*FE8 1954 l *+ 74.14% 11**+*439**JUL 19si l ** 16.29% 12****~63**~AN 1361 l * * 36.79%

13****49J*~FE8 1963 l * * 36.79%

14****54~~*FE8 1967 l ** 10.97% 15+***595**5EP 1970 l * * 40.33%

15**+*633-*FE8 1975 l *. 18.51% 17****676**APR 1978 l ** 26.78% 19****7~B+*AUG 1982 l * * 0.00%

19****757*+JAN 1985 4.4 0.00% *******I-- --**P*******************

TFIBLF 12 12 month Xchange smoothed by 12 month FWMA with P= 1.3

********************************

l**++*29**MAY 1924 l * * 54.35%

2***+128**AUG 1932 l +* 12.11% 3+*++161*+MAY 1335 l ** 42.00% 4+***199**JUL 1938 +** 13.24% 6****2Z0**APR 1940 l ** 13.24% 6****23j**MAY 1941 +*+ 13.24% 7****250**OCT 1342 l * * 13.24%

8++**30B**AUG 1917 l ** 60.62% 9+***315**WAR 1948 l ** 60.62%

10****333**SEP 1919 l ** 60.62% l1****386**FE8 1954 l ** 74.14% 12****4jA*+F~8 155B ++* 27.59% 13**+*438**JUN 13SB l ** 27.53% 14****469**JAN 1361 l ** 43.61% 15+***493**JAN 1363 +++ 43.61% 16****54i**FEB 1367 l +* 10.87% 17****S95*+SEP 1370 l +* 40.33%

1B****638**FEB 1975 l *4 18.51% 13****676**APR 1579 l *+ 26.78% 21****7;7**JAN 20****779**SEP 1535 1592 l * * 0.00::

l * * 0.00%

***.****.****************==***=*

TF\BLF 11 11 month %change smoothed by 12 month FWMA with P- 1

********************************

1**++*28**APR 1924 l * * 54.90%

2*+**1I!8**AUC 1932 l ** 12.11% 3**+*160**APR 1935 l ** 71.69% 4***+199*+JUL 1938 l * * 9.59%

5+***219**MAR 1940 ++* 9.59% 6**+*232**APR 1941 l * * 9.59%

7++**249+*SEP 1942 l * * 9.59%

8****308**AUG 1947 l ** 57.70% 9++**33A**OCT 1949 l *. 57.70%

10****386*~FES 1954 l *+ 74.14% 11***+439**JUL 1958 l ** 16.29% 1:****463*-OEC 1550 l +* Ai.A0%

lj**+*434**FE8 lj6j l +* 42.40% l4*+*+542**FEB 1567 l ** 10.87% 15***~635*+SEP !370 l *+ 40.33%

16***+63B**FEB lS?!5 l ** 18.51% l7****6?6**APR 1378 +** 26.78% 13****7iB**AUG 1332 l * * 0.00%

19****757*+JAN !335 l * * 0.80%

********************************

TFSBLE 13 11 month Xchangt smoothed by 13 month FWMA with P- 1.3

l*+*+*tB*+APR 1924 l * * 54.90%

2**+*1:9**AUG 1332 l ** lZ.ll% 3****163**APR 1935 .** 71.69% 4+***199*+JUL 1938 l * * 9.59%

S*+**219**MAR 1940 l * * 9.59x

6****~32**ApR 1941 l * * 9.59%

7****249**SEP 1942 l * * 9.59%

3+***307**JUL 1947 l +* 55.62% 9****jj4**OCT 1949 l ** 53.62%

10+eoejj;+.FEB 1954 l * * 74.14% 11**+*4~9**JL’L 1958 l ** 16.29% I?*-+*463**OEC 1969 .*. 42.40% !j++a+AgA++FEB 1153 et+ 42.40X l$*.**~~~*.~~~ :3S7 l ** 10.37% lSr**~SSS**SEp 1970 l * * AO. 23:!

16**+*637**JAN 1975 l * * 23.30%

17***-676**AFR 1378 l .+ 26.78% ?B+***723+*AUG 1982 l * * 0.00%

19****757**JAN 1935 l * * 0.00%

********************************

84 MTAJCURNALWINTEIR 1988/1989

Journal of Technical Analysis (JOTA). Issue 31 (1988, Winter)

Economy & Finance