# Fibonacci Ratios: No One Knows Why They Work, but They Do

A number sequence known as the Fibonacci series was proposed as an answer to the question, “If we start with one pair of rabbits, how many pairs of rabbits will there be one year later if we assume that every month each pair reproduces and adds a new pair to the group?”

This number series has been written about since the 13th century, when it was identified by mathematician Leonardo Fibonacci, and is widely used by traders in the 21st century.

The answer to the original question is that the number of rabbits in any given month is equal to the sum of the number of rabbits in the two previous months. If we start with one pair, in the second month, there is still one pair as they have not yet delivered any baby rabbits. In the third month, there are two pairs, in the fourth month, there are three pairs, and in the fifth month, there are five pairs of rabbits. In the twelfth month, there would be 144 pairs of rabbits.

The Fibonacci series is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc…

Three ratios are commonly associated with this series:

1. The ratio of a number to the next higher number in the series is about 0.618 for numbers above 5. For example, 34/55 is equal to 0.6182.

2. The ratio of a number to the next lower number in the series is about 1.618, the inverse of 0.618. For example, 55/34 equals 1.6182.

3. The ratio of numbers that are separated by one additional number (such as 5 and 13) is about 2.618, or its inverse 0.382. For example, 89/233 equals 0.382 and 233/89 equals 2.618.

Traders believe that these ratios are often found in price data. Fibonacci ratios are applied to individual stocks and futures contracts, as well as indices, and can be used in any time frame.

As an example of how the Fibonacci ratios are applied, after a significant price decline, traders generally look for at least a short rebound in price, and many traders will expect the price rise to stop, at least temporarily, at a Fibonacci ratio related to the decline. The Fibonacci ratio may also prove to be a long-term resistance level. This is shown in the chart of the Nasdaq 100 index below. After the Nasdaq 100 crashed in 2000, the 38.2% Fibonacci ratio served as resistance and held back further advances for 12 years. Once the 38.2% level was decisively broken in the spring of 2012, traders looked at the 61.8% retracement as the next resistance level. The market moved easily through that level in 2013, and it should now be expected to serve as support.

Fibonacci ratios are also used to define upside price targets and support levels, and can help identify times when a stock or index is likely to turn.