A Little Math and a Bitcoin Forecast
Anyone who seriously studies Bitcoin, asks why it is so successful, and what comes next will hardly get past the stock-to-flow theory of PlanB@100trillionUSD .
If you don’t know this theory yet, you should take a closer look at it here: PlanBTC.com.
This theory explains in simple mathematical formulas the intrinsic value of Bitcoin. Bitcoin is viewed here as a scarce commodity that is highly sought after similar to gold, follows the laws of the market, but serves little purpose other than a monetary investment.
This theory answers a question I had been asking myself for quite some time. I had come across a chart in an article about the progression of bitcoin value over the years. This chart’s special thing was that it did not start from 2014 as usual, but from the first block in 2009 to the present time . The graph was equipped with a logarithmic Y-axis for the value.
Suppose you have such a chart in front of you. In that case, you will immediately recognize that the value progression follows a curve, which looks as if it were logarithmic itself (in fact, it’s not logarithmic, it will end up nearly linear in this format).
There is almost always a mathematical formula behind it with such a regular shape, using which you can extrapolate future values. You only have to find it and justify it. Precisely this was made possible by the S2F theory.
If you look at the calculations of the S2F theory, these values magically match the actual Bitcoin prices achieved over the entire period of its trading on the exchanges. It has also been repeatedly adjusted and extended to increase its accuracy.
However, one should keep in mind that these logarithmic graphs look more accurate than they are in reality. Even the slightest deviations can lead to considerable differences in price. If you are not familiar with logarithmic charts, you should be very careful.
There are also cyclical phases in the Bitcoin course, where the values do not match the calculated values. Deviations of up to 10 times the value are not uncommon here.
All in all, however, there is probably no other theory that can make reliable predictions over so many orders of magnitude.
The newest Bitcoin equation from PlanB is of the following form:
$/BTC = e ^(3.21 * ln(SF) — 1.6)
[equation by PlanB@100trillionUSD]
In contrast to PlanB, I consider this formula as a long term prediction only.He changed the calculation of the SF-value to become more accurate. Therefore he now uses a longer time span for the SF calculation instead of daily values. This leads to ‘ramps’ in the chart instead of the previously displayed steps. A better overall prediction is the result, but he loses his argument that the bitcoin values immediately followed the new SF-value at the beginning of 2013. In my opinion, however, this argument is still valid.
I propose a different approach to predicting the bitcoin price’s short-term progress in the growth phase after the halving.
My goal is to develop an understanding of the Bitcoin price course and check new approaches to explain the price fluctuations and the prediction.
The more one knows about the Bitcoin system; the more confident one becomes in dealing with it.
The Growth Phase
Interestingly, the intrinsic value of Bitcoin does not change steadily but seems to run in leaps. However, these leaps can be explained very easily by the halvings, which immediately reduce the amount of newly generated Bitcoins and strongly influence the value in the S2F calculation.
The question arises here, why Bitcoin does not change its value instantly after a halving and switches to the new intrinsic value. If you look at the following months after a halving, not much seems to change in the course. However, you have to look closer to understand these changes.
In PlanB’s charts, there is also a self-similarity (thus fractal structure) that seems to repeat itself in different orders of magnitude every four years.
This was the starting point for my calculations. Indeed it had to be possible to compare the price trends from 2013 to the end of 2017 with today’s price trends, starting in 2017 and ending in 2021.
If you look at the price trend and take the lowest value as a reference point, you can divide it into a growth phase and a recession phase. Since we are currently in the growth phase according to this classification, I have concentrated on this phase for the time being. I will certainly analyze the recession phase later.
So I got all the historical bitcoin values since 2014  and looked at the phase in which bitcoin’s value increased. Then I looked for the beginning of the new growth phase from 2018 and compared these values with those from 4 years ago. With a factor of 17.5 for the old values and a shift of exactly 1399 days, both value trends could be superimposed well. The 1399 days correspond precisely to the time distance of the two halvings, which took place in the respective period.
The result astonished me myself. The similarities of both progressions are visible. Although the bitcoin value in 2019 was enormously exaggerated almost over the entire year. And as if it were natural, both progressions unite again, as if they knew the way to the common goal.
However, I would like to point out that the courses’ comparison is not meant to equate them. I am only interested in finding out the similarities.
A Bitcoin Formula
You can recognize two different phases in the above graph. The phase with the flatter course before the halving and the phase afterward, in which the values increase rapidly.
After a few trials, it seemed clear. The phase before the halving corresponds to constant exponential growth. In a logarithmic plot, this results in a straight line (light blue in the graph). Here, the value of Bitcoin roughly doubles every year. You can see the parallel course of the values to this straight line in both progressions (red and blue), even if they often move very far away from this straight line.
To force the phase after the halving into a mathematical formula was much more complicated. Exponential growth is not enough here. Only a triple exponential growth brought the desired success.
To understand such growth, one can imagine an interest rate. A single exponential growth corresponds to a constant interest rate, while a multiple exponential growth is an exponentially increasing interest rate.
The general equation is of the following form:
$/BTC = $/BTC(at halving) • e^(ln(a) t • (ln(b) t² + 1))
a is the growth factor at halving (1.07 per month in 2020)
b is the growth factor of the growth
t is the time since halving
Parameter a is the constant growth factor before the halving, and b is related to the change of scarcity and the market conditions. So it’s probably related to the new S2F value, the trading volume, and the euphoria on the market.
So the formula for the growth after the halving is easily explainable by the initial growth that already existed before halving.
If you change the market by creating a scarcity of Bitcoins, you create the second exponential growth function, which ‘adds’ to the existing one. Without the initial growth, you would have created a standard, exponential growth, but now you have a double exponential growth.
With a little trial and error, I adjusted the parameters to create the green curve in the graph. It’s impressive how well the real bitcoin values held to this formula in 2017.
So the Bitcoin formula for the actual growth phase is:
USD/BTC ≈ 9100 • e^(0.81t • (2.35t²+1))
USD/BTC ≈ 9100 • e^(0.81t + 1.9t³)
where t = years since halving 10.05.2020
Everyone is invited to check this formula for themselves. Its amazing!
I had initially used the largest value of 12/17/2017 (around $20000 at the time) as the target for my formula. However, this did not work. The values’ slope just before the highest value was too big, and the values didn’t fit into any formula. I realized that the ‘true’ Bitcoin peak was reached only a month later and that the higher value before was simply a temporary exaggeration.
The short-term exaggerations are nothing unusual. For Professor Didier Sornette of the University of Zurich , a growth stronger than exponential is usually an unmistakable sign of a financial bubble .
Such a growth phase is also called hyper-exponential growth. In this phase, you will see many ‘baby bubbles’ and mini crashes.
These exaggerations and crashes are oscillating faster and faster. This leads to the bursting of the bubble and thus to a substantial loss in value.
See also the following article: The Financial Bubble Experiment (german language, see the last paragraph)
The growing euphoria of the investors characterizes this phase. Interestingly, it begins directly after the halving, and most of the time, the values are not higher than expected.
But if the Bitcoin values become more significant than the predicted S2F value, it is, of course, a bubble that will burst sometime. And the value then returns to normal after a certain time.
This time is fortunately minimal with Bitcoin due to the 4-year cycles. However, as PlanB has already mentioned, the value can temporarily drop by 80% 
I was unsure if this formula with all parameters would be valid for the current bitcoin value or if there is a difference. Therefore, I tried to create a second set of parameters that correspond to the current bitcoin curve. This is the dashed curve in the chart below. However, I still have too few data points to confirm this progression. But I will, of course, adjust the calculation over time and keep the chart up to date.
If you look at the bitcoin values in my chart, you can see that PlanB’s prediction with his newer S2FX model  should not be so wrong. He assumes there a target value of $288,000 for the bitcoin. My calculation at least shows the possibility that such values could be reached effortlessly.
There are certainly many questions about this calculation. Still, the most crucial question for many other Bitcoin enthusiasts and me is at what time exactly the maximum value will be reached.
According to S2F theory, the intrinsic bitcoin value is around 85 to 100,000 $ by the end of next year. But how high may continue the Bitcoin rise until the next panic on the market forces the value down? Every investor should be aware that the probability of a crash above a certain amount becomes bigger and bigger day by day.
There is a good article on how to get information about these value peeks here: 📈 5 crypto-native indicators to enrich your market cycle analysis 
If one should miss this crash, only patient waiting for the next growth phase will help. It will probably start in 2023.
Or else one waits for my analysis of the Bitcoin recession and behaves accordingly. However, followers of the strict EMH theory  are unlikely to show much interest here…
 S2F Theory by PlanB@100trillionUSD
 historical bitcoin chart
 Yahoo Finance Data
 Li Lin and Didier Sornette, Diagnostics of Rational Expectation Financial Bubbles with Stochastic Mean-Reverting Termination Times
 Didier Sornette
 S2FX model by PlanB@100trillionUSD
 Felipe G., crypto indicators
 PlanB@100trillionUSD, Efficient Market Hypothesis and S2F model
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