Quite often, when you read something about the markets, they talk about the Fibonacci retracements. When the name of Fibonacci appears, it's kind of interesting because we know the Fibonacci sequence
- 1,1,2,3,5,8,13,21,34,55, ...
in which a new element is the sum of the previous two. The asymptotic ratio of the neighbors approaches the Golden mean
- (sqrt(5)+1)/2 = 1.618034...
whose inverse happens to be the same number minus one
- (sqrt(5)-1)/2 = 0.618034...
Now it's useful to calculate the square root of both of these numbers, write 1-0.618=0.382, and add the numbers 1/2 and 1, and express everything as percentages:
The last one is the Golden mean squared. Why is it useful? Apparently, many traders use these numbers to predict the turning points of a price, see e.g.
Imagine that the price P of something kind of increases. If it returns a bit, it's a retracement. If the retracement is shallow, people believe that the increase is a trend. Now, if the retracement is bigger, it will probably stop and return to the increase when you return to 50% - to the arithmetic average of the last visible local minimum and the local maximum. Also, it's possible that the turning point is when you return by 38.2% of the difference between the last maximum and the previous minimum. Then you try to calculate all possible turning points by taking the "special" numbers above and multiplying them with the differences between various maxima and minima - and if they cluster, you get a likely turning point. Those with the Golden mean (without its square root) are "strong", I guess.
Do you believe this story? Well, even if it's nonsense, there are apparently many people who use it - and these people put a pressure so that their hypothetical turning points are turning points indeed. For example, euro's latest rally was from \$1.1986 in the summer to the maximum \$1.3664 last week. You can calculate that the 38.2% retracement is at \$1.3023 (very near the current rate) and the 50% retracement is at \$1.2825. If I understand well, the Fibonacci traders will think that one of these two values must be a turning point, and if these two points are broken, euro will have to drop to \$1.1986 again, assuming that the turning points are allowed to be repeated. ;-) If this is also broken (or if the exact rules don't allow it), you must probably make a 161.8% retracement and euro will have to drop to \$1.0949. ;-)
Although I believe that in the world of rational people without fairy-tales, the Fibonacci turning points would be uncorrelated with the real ones (and the "special" values are just some values that look sufficiently different from "small" variations), this fairy-tale sort of seems "self-consistent". For example, if A,B,C are the prices at three turning points and (C-A) = (B-A) * 0.618, then you can also calculate (B-A) = (C-A) * 1.618, and therefore various ways how you relate the points to one another are kind of equivalent, by the defining property of the Golden ratio. ;-) Your opinions about this new kind of science are welcome.