With crypto adoption rates increasing, the future value of this asset class has become the subject of speculation. People who predict, house people predict. Some economists like Nouriel Roubini think the price will return to zero within five years, while John McAfee shocked the world by claiming a price of $1 million per Bitcoin by the end of 2020. Others have made predictions. guess is in this wide range.

Overall, the price of Bitcoin has increased very rapidly since its inception in 2009 and has also fluctuated wildly. Periods of rapid increases and booms seem to encourage people like McAfee to make very optimistic predictions about future prices, while downtrends have led some economists to predict a drop to zero. We look at Bitcoin’s full price history and find that Bitcoin’s price evolution can be understood as moving in a corridor defined by two time-based power-laws. time. Although the idea of modeling Bitcoin price using a power-law rule is not new, in this article, we will explain it in detail so that readers can better understand the problem.

Moreover, we will see that the price corridor can be divided into two bands, one is located at the lower end of the price predictions and quite thin, the other is much larger and located at the higher predictions. Bitcoin price spends equal amount of time in both bands. This implies that increases and decreases are likely to continue to occur. The predictions above seem very broad, but accurate enough to contradict the predictions of some others. This price pattern will also help identify the pros to enter or leave the market.

The price will actually develop approximately as stated in this article. In fact, it’s more likely that these predictions will be too low instead of too high: I believe Bitcoin has more upside potential than downside when it comes to large exogenous shocks. But the article will not attempt to make any predictions regarding large exogenous shocks. Instead, we will assume that everything continues “as usual”.

**Different views on the price**

The most interesting and amazing aspect of the Bitcoin price is that it has gone through many large orders within a few years. The first example of a publicly listed price I could find was $0.05 per Bitcoin on the Mt Gox exchange, on 7/17/2010, but before that date many Bitcoins were trading for the price. much lower, such as on May 22, 2010, when Laszlo Hanyecz paid 10,000 BTC for two pizzas, which equates to a price of just $0.0025 (0.25 cents) per Bitcoin. At the time of writing, the price of one Bitcoin hovered around $10,873, about 4 million times what Laszlo Hanyecz offered at the time.

Going through a lot of large orders is unusual for a financial instrument, and actually looking at the Bitcoin price chart over time can be a bit confusing (if prices are expressed on a linear scale). Below is a chart of the Bitcoin price starting from July 17, 2010 to the time of writing. Similar price lots can be found at any website that lists Bitcoin prices.

Any price swings close to current levels are huge when compared to past prices, even showing that past prices seem meaningless. However, in order to get a long-term price trend, all past price levels are important. The reason for the above effect is that the linear scale is quite inconvenient to use for anything that goes through a lot of commands. Using a logarithmic scale instead of a linear scale is more useful. Logarithmic scale for evenly spaced intervals, for example 0.01 to 0.1 or 1000 to 10000. Looking at it this way, the overall picture of Bitcoin’s price evolution becomes clearer:

What becomes clear is that the growth rate of the Bitcoin price appears to be slowing down. The price went from $0.10 to $1 – a factor of 10 – in just a few months. The next rally reached a slower multiplier of 10.

In the chart above, price (y-axis) has been scaled logarithmically, but not time (x-axis). Let’s see what happens when the x-axis is also scaled logarithmically to form a log-log plot:

**Linear regression**

Since this data looks very linear, we try to use linear regression on it. This idea is not new, e.g. I found a post on reddit that used to do exactly this.

Visually, this model works very well. It performed well on the way back to the first price listed by the exchanges. Interestingly, the reddit post was written about 1 year ago and the results are still surprisingly similar. In addition, the coefficient of determination is high at 0.93139763, which gives another indication that the model fits well. We can look at how the coefficient of determination evolves over time. Surprisingly, the model tends to fit the data better as time goes on:

The x-axis represents the number of data points (days) used for the linear regression model, while the y-axis represents a measure of goodness of fit. Bitcoin price is more and more in line with power-law.

If we move the fit slightly lower (but without changing the slope) we will find a support line that looks to be working well. Except for one case in 2010, the price has never breached this line.

We can also try to do a linear regression on only 3 highs reached in 2011, 2013 and 2017. Interestingly, this fit works very well. All three data points are close to the support line.

Market highs also seem to follow the power-law rule. If the next market top also follows this rule, it will be above the support line. The power-law slope is 5.02927337, while the fit across all data produces a slightly larger slope of 5.84509376. As such, the Bitcoin bull market could be contained relative to the overall trendline. This is perhaps to be expected: as the market matures and the order book gets deeper, people should expect less volatility.

We now have two power rules between where Bitcoin price moves: a lower support line and a higher line defined by three market tops.

Now, notice which data point fits the model best. We will use random sample consensus (RANSAC), which is an iterative form to remove outliers: First, linear regression is performed on all data points. Then, the data point that is less suitable for the dataset is deleted and the linear regression is performed again. We will stop when 50% of the data points have been deleted. This chart shows the results:

The data points have been selected and highlighted by RANSAC in this graph. There seem to be two sets of data points: Those that have been selected by RANSAC are very close to the model fit. In the group of data points not selected by RANSAC, the values are mostly on the model fit. In fact, some of them are much higher than the right model. These data points occur mainly in bull markets.

We found that the fit using all data and the RANSAC results had very similar slopes, but slightly different biases. This is because bull market prices have been largely excluded by RANSAC.

**Model prediction**

The pattern predicts the price will move between the red support line and the blue top line. The purple/RANSAC fit line identifies the center of the “normal mode”. The two previous halvings as well as the estimated future halving have been marked by black vertical lines.

We can divide this corridor into two bands, one corresponding to the “normal” mode and the other corresponding to the “bullish” mode. So far, the price has spent half of the time in the lower “normal mode” band and the rest of the time in the higher “bullish mode” band.

**Explain**

The exponential rule model predicts a continuous, but slow growth of the Bitcoin price. It also predicts a decrease, but still large volatility in the future. The model predicts that the price will not reach $100,000 by 2021 and will not be lower than $100,000 by 2028. It predicts that the price will not reach $1,000,000 before 2028 but will not be below that level after 2037 either. predict prices will continue to rise, albeit at a slower rate.

These predictions seem a bit ‘dovish’ compared to others. According to this model, McAfee’s famous prediction is overly optimistic.

In fact, the rather wide price corridor combined with the slowing growth rate means that unlucky investors will have to wait longer before their initial investment is safely recovered. For example, investors who bought Bitcoin at the top of the bubble in 2011 only had to wait about 2 years until 2013 for the Bitcoin price to recover permanently. However, investors who bought at the height of the 2013 bubble had to wait about four years, until 2017, before the price recovered to that price and formed above it. The model predicts that the price point reaching the top of the bubble in 2017 may not happen until the end of 2023, about six years later.

Up until now, each 4-year halving has had a bubble that generates prices that exceed the next period bubble. Due to the slow growth of the above points and the width of the corridor, this is not guaranteed to continue in the future. For example, the model allows the following scenario:

The price is around $150,000 in early 2022, which is the next 4 year period and the fourth 4 year period. The price is lower than $150,000 until mid-2028, i.e. in the 6th 4-year period. Such a scenario would give Bitcoin detractors an opportunity to criticize, but otherwise it is not what it is. worrisome, as long as one is prepared.

Why does Bitcoin follow an exponential rule and should we expect it to continue? Observation shows that Bitcoin follows a recognized power rule for a particular purpose. In addition, there are other factors besides time that will affect the price of Bitcoin, such as its scarcity. However, Bitcoin scarcity is programmatic so also time based. It is by no means, therefore, that a simple time-based model will continue to hold true in the future. The power-rule fit that works better and better measured in log-log plots is an indication that this can indeed hold.

**Conclusion**

In this article, we presented a simple time-based equation to model the Bitcoin price. It’s worth noting that the equations are both 1. simple and 2. use time as the only variable, but work very well over long periods of time.

This model does not attempt to predict a bull market that seems to happen periodically. However, the bull market is expected to be within the corridor defined by this pattern.