Where do Fibonacci Ratios come from (retracement levels) There are Fibonacci ratios that traders use in stock markets. These percentages are 23.6%, 38.2%, 61.8%.
I know 38 and 62 equal 100, but where does the 23.6% come from? What is the full sequence of these ratios?
 A: The Golden ratio is $\phi = \frac{\sqrt{5}+1}{2} \approx 1.618$ and the Fibonacci numbers $1,1,2,3,5,8,13,\ldots$ are $\mathrm{Fib}_n=\dfrac{\phi^n-\left(\frac{-1}{\phi}\right)^n}{\phi+\frac{1}{\phi}}$.
Then

*

*$\dfrac{1}{\phi}=\phi-1\approx 0.618$

*$\dfrac{1}{\phi^2}=2-\phi\approx 0.382$

*$\dfrac{1}{\phi^3} = 2\phi-3\approx 0.236$
though why this should be used in stock market technical analysis is not much more credible than astrology.
A: The Fibonacci sequence $1,1,2,3,5,8,\ldots$ is generated by $F_0 = F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$ with the well known property
$$\lim_{n \to \infty} \frac{F_n}{F_{n+1}} = \frac{1}{\phi} \approx 0.618,$$
where $\phi \approx 1.618 $ is the golden ratio.
The retracements are
$$\lim_{n \to \infty} \frac{F_n}{F_{n+1}}= \frac{1}{\phi} \approx 0.618 \\ \lim_{n \to \infty} \frac{F_n}{F_{n+2}}=   \lim_{n \to \infty} \frac{F_n}{F_{n+1}}\frac{F_{n+1}}{F_{n+2}}= \frac{1}{\phi}\frac{1}{\phi}\approx 0.382, \\ \lim_{n \to \infty} \frac{F_n}{F_{n+3}} = \lim_{n \to \infty} \frac{F_n}{F_{n+1}}\frac{F_{n+1}}{F_{n+2}}\frac{F_{n+2}}{F_{n+3}}= \frac{1}{\phi}\frac{1}{\phi}\frac{1}{\phi} \approx 0.236$$
Each successive retracement level is obtained by dividing the preceeding level by $\phi = 1.618...$ or, equivalently, as roughly $61.8\%$ of the preceeding level.
