**Fibonacci Time Cycles**

Robert C. Miner proportions future time byFibonacci ratios. First, Minor applies Fibonacci Time-Cycle Ratios to the time duration of the latest completed price swing, using both trading days and calendar days. The most important Fibonacci ratios are: 0.382, 0.500, 0.618, 1.000, 1.618, 2.000, and 2.618.

Miner’s Alternative Time Projections are calculated as time ratios of the previous price swing in the same direction, that is, up swings are measured out as proportions of previous up swings, while down swings are measured out as proportions of previous down swings. Also, Alternative Time Projections may be derived from same-direction price swings earlier than the latest one.

Miner points out that there is a very high probability of trend change when both price and time ratios coincide.

Miner’s Trend Vibration^{TM} method is based on two directional movements early in a trend: the initial thrust and the initial corrective wave of that thrust. Together, these two movements are Elliott Waves one and two, and Miner calls the initial vibration. Fibonacci ratios of that initial vibration time projected forward coincide with subsequent turning dates, including the end point of the completed trend.

Of secondary importance are the day counts, using both trading days and calendar days. When one or more day counts is a number in the Fibonacci sequence, the probability of a directional trend change is heightened. The more hits on Fibonacci numbers, the greater the confirmation and power of that date.

As suggested by W.D. Gann, Miner also uses multiples of 30 (specifically, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, and 360), and multiples of 36 (specifically, 36, 72, 108, 144, 180, 216, 252, 288, 324, and 360) in his day counts. Anniversary dates of previous turning points in history also add value to his analysis of time.

Miner also uses Bollinger bands (which are also known as Standard Deviation Bands and Volatility Bands) to help identify and confirm time/price turning points. Two standard deviations above and below a moving average create a channel that encloses 95% of the price action. In relatively low volatility, sideways trading-range markets, such bands reliably indicate support and resistance. In trending markets, where the trend is strong and continuing, reactions against the trend often do not exceed the moving average mid-way between the upper and lower bands. In a bullish trend, price spends more of the time testing the upper band and the moving average. In a bearish trend, price spends more of the time testing the lower band and the moving average.

At the independently determined cyclical time of probable trend change, Miner has observed that price is often near one extreme band or the other. Then, to confirm the trend change, price moves quickly to the opposite band, in the direction of the new trend, showing a relatively high degree of absolute price velocity. Trend, Elliott Wave and Chart Pattern interpretation complement and complete Miner’s cycle analysis.

The time and price projection methods cited here are adapted with permission from Miner, Robert C., Dynamic Trading, Dynamic Traders Group Inc., 6336 N. Oracle, Suite 326-346, Tucson, AZ 85704. This recommended book offers practical guidelines for interpretation and a large number of actual trading examples. Miner also develops software to efficiently make calculations of the Fibonacci relationships, including time as well as price, in any market.

S*ource:* Colby, Robert. *The Encyclopedia of Technical Market Indicators*; (c) 2003.

**Fibonacci Number**

In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci (a contraction of filius Bonaccio, “son of Bonaccio”). Fibonacci’s 1202 book *Liber Abaci* introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.

The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself, yielding the sequence 0, 1, 1, 2, 3, 5, 8, etc. In mathematical terms, it is defined by the following recurrence relation:

That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers also denoted as *F _{n}*, for

*n*= 0, 1, 2, … ,20 are:

F_{0} |
F_{1} |
F_{2} |
F_{3} |
F_{4} |
F_{5} |
F_{6} |
F_{7} |
F_{8} |
F_{9} |
F_{10} |
F_{11} |
F_{12} |
F_{13} |
F_{14} |
F_{15} |
F_{16} |
F_{17} |
F_{18} |
F_{19} |
F_{20} |

0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 | 610 | 987 | 1597 | 2584 | 4181 | 6765 |

Every 3rd number of the sequence is even and more generally, every *kth* number of the sequence is a multiple of *F _{k}*. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property:

*gcd(F _{m},F_{n}) = F_{gcd(m,n)}*.

The sequence extended to negative index *n* satisfies *F _{n} = F_{n−1} + F_{n−2}* for

*all*integers

*n*, and

*F*:

_{−n}= (−1)^{n+1}F_{n}.., −8, 5, −3, 2, −1, 1, followed by the sequence above.

*Source: http://en.wikipedia.org/wiki/Fibonacci_number*

**Fibonacci Golden Ratio**

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

The closed-form expression (known as Binet’s formula, even though it was already known by Abraham de Moivre) for the Fibonacci sequence involves the golden ratio:

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence):

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

This identity allows any polynomial in φ to be reduced to a linear expression. For example:

*Source: htpp://en.wikipedia.org/wiki/Golden_Ratio#Relationship_to_Fibonacci_sequence*

The following slides were provided by Ralph Acampora, CMT and the New York Institute of Finance.

*Source: Information provided by the New York Institute of Finance (NYIF) and Ralph Acampora, CMT.*