DINAPOLI LEVELS

All oftoday's markets have something in common. Whether we are dealing with futures contracts, foreign exchange, equity markets, or even mutual funds. Speed and volatility have increased dramatically. What used to be a days range in the S&P in 1983 is now apparent in one bar on a five minute chart! Regardless of what you are trading, yesterday's methods are ill-equipped to handle today's challenges. The proper use of Leading Indicators address this issue.

GENERAL DISCUSSION:

Perhaps in the years to come you might use a portion of your trading profits to travel to some of the great cultural centers of present and past civilizations. If you do, you will find as I did, that Fibonacci relationships are intrinsic to the architecture of Athens, Rome, Amsterdam, Paris, Egypt, many areas of South America, and so on. There's an elemental resonance to the way these aesthetically pleasing shapes derive from these mathematical road maps.

You can observe a curious evolution of Fibonacci expansions played out in musical progressions, crystal formations, and even in the growth of rabbit populations. Whether in the DNA spiral, the preprogrammed construction of a bee hive honeycomb, or the inspirational great pyramid of Giza, Fibonacci relationships abound. The human body itself is a study in Fibonacci relationships. Recently, at one of my workshops I met a surgeon who had done his graduate thesis on facial reconstruction. The thrust of his study was to quantitatively relate post surgical success in appearance with how closely the reconstructed bones approximated the "Golden Mean." There is no denying that Fibonacci ratios and the numbers that generate these ratios are innate in all matter.

It takes no great leap of mental agility therefore, to expect that the combined activity of mankind will somehow follow these precepts. This is particularly apparent relative to the markets, since markets are linked so closely to the overriding human emotions of greed andfear.

A LITTLE HISTORY:

Leonardo was born the son ofGuilielmo Bonacci, a wealthy merchant in Pisa about 1170. In Italian, "figlio" means "son," hence "figlio Bonacci" which was shortened through the years to Fibonacci. Signior Leonardo Bonacci was a preeminent mathematician of his time. He's credited with the discovery of the numerical sequence and ratios that have come to be known as the Fibonacci series.

Discovering the Fibonacci series is sort of like discovering America. I'm sure the Indians

knew about it before Columbus did. Likewise the proportions defined by the

mathematical relationships so important to us as traders, have been around for quite some

time.

The Golden Mean or Golden Ratio 1.618, to 1 (or .618 to 1) which among other things closely approximates the cost of this book, has had many names. The Greeks designated the ratio by the letter "phi." Pacioli, a medieval mathematician, named it the "divine proportion." Kelper called it "one ofthejewels of geometry." Somewhere along the line, someone dubbed it the "ratio of the whirling squares." I'm glad that one didn't stick. Consider the implications for the title of this book: "The Practical Application of Whirling Squares to Investment Markets."

DERIVATION:

The Fibonacci number series has more interesting aspects to it than most of us can imagine or would wish to. While we might get dizzy considering the possibilities, it's a mathematician's hot fudge sundae, and then some. Consider the series as most of us know it. 1, 1, 2, 3, 5, 8, 13, 21 and so on to infinity. We arrive at the series by simply adding the last two numbers together, beginning with 1,1. The ratios come about from dividing the numbers in various ways. If we divided 13 by 21 for example, we get .619 while 21 divided by 13 = 1.615. If we skip a number and divide 8 by 21 we get .381. Conversely, 21 divided by 8 is 2.625. The higher we go in the number series, before dividing, the closer we come to achieving the exact numerical Fibonacci ratios. We never achieve this, however, since there are an infinite sequence of decimals stringing after it. This is known in mathematics as an irrational number.

One interesting aspect of the summation progression is that it doesn't matter where we start. We can take any two numbers, like 5 and 100. Soon we're dealing with the same series.

5, 100, 105, 205,310, 515, 825, 1340, 2165

While it is known that Mr. Fibonacci "discovered" the series after a trip to Egypt, I get a different image when I think of him. Imagine the son of Bonacci sitting under a tree some time in the 13th century, after consuming a great bowl of pasta. He'd likely run out of fingers and toes and had to go to his abacus when the lights suddenly went on. It must have been something like the way I felt when I began applying his discovery to the S&P...as in

WOW!!!

I could go on at great length about the poetry of Fibonacci relationships, but if I did, I'd never get to their practical application to the markets. If you want to pursue the subject, many books have been written discussing the more esoteric aspects 1 . They certainly do a better job of describing the mathematics involved than I have. Besides, my lack of reverence toward the subject is bound to upset some people. So, I will leave the poetry, the derivation, and the history of Fibonacci numbers and ratios to others. This book is about practical application of Fibonacci concepts to the market, and in that vein, I will bring us back to reality and repeat the following prohibition for those of you who have galloped ahead to this chapter.