**“How fast will Bitcoin grow further?” - This should be the main question for most crypto investors. Every day, many people try to give an answer to it, but the high volatility of Bitcoin and the hype cycles make such point forecasts are almost useless. However, using standard regression methods that take into account the uncertainty in the price dynamics of bitcoin, one can predict range price values, which will be much more likely to coincide with reality. **

**In this article I will introduce a simple, but effective, change to the commonly used power model and try using the resulting model to predict some**

*ranges*probable values - for the date the bitcoin price reaches key levels and average annual profitability. The results can be informative and useful for making investment decisions.## Methods

### Pricing model

The history of the price of bitcoin already has about 10years. 3319 daily data points in my source (.csv file) - this is more than enough to apply basic mathematical modeling, and the choice of affordable price models is great. Here I will mention two of them:

- An excellent model based on scarcity (defined by the ratio of stocks to growth) from PlanB
- DaveTheWave Models Based on Two-Parameter Logarithmic Regression

Although there are fundamental differences between the models of the two authors, the following equation is applied in both for the input data array:

</p>or, in another entry:

</p>A and B in these equations are constants,defining a linear function between log (x) and log (y), and are obtained by minimizing the least summed squares of the distances between the model and the realized observations. The linear relationship between the logarithms of two entities is a property of the law of power dependence. Power-law dependencies in price models for bitcoin fundamentally have an absolute meaning, as PlanB clearly explained and demonstrated based on a gradually increasing deficit of bitcoin.

### Adding a time shift parameter

Although my first samples are quite satisfactoryreconstructed the published diagrams of the above models, and I could stop there, I was able to improve the fit by adding the third parameter S to the equation:

</p>or, in another entry:

</p>Parameter A is the slope of the line between log (x + S) andlog (y), B is the intersection point of this line (the intersection of the Y axis), and S is responsible for the singularity point of the input logarithmic function. S shifts the reference point, so that instead of assuming that the first available price data coincides with the reference point, or that the reference point coincides with the Bitcoin genesis block, we provide the model with the ability to independently determine the reference point.

### Fitting method

When using a two-parameter powerit is enough for the model to perform a linear fit between log (price) and log (time). With the introduction of the third parameter, it becomes necessary to use the nonlinear fitting method. It can be performed using the Levenberg-Marquardt algorithm, applicable with any parametric nonlinear equations. I used the LabVIEW platform for this, but the same can be done on other platforms.

## results

### Fit

An equation in which y = the price of bitcoin at the end of September 1, 2019, and x = the number of days from the earliest point in the historical price data of bitcoin (July 17, 2010), will look like this:

</p>or, in another entry:

</p>R² (coefficient of determination) is 0.935, whichmeans that the smoothed curve defined by three parameters is 93.5% of the total variance in the history of the price of bitcoin. Not bad. In the graph below, this fit is shown by a green line (the timeline is linear, prices are logarithmic).

Please note that 10 ^ (- 13.43) is a numbersmall, and (x + 312) ^ 4,860 is a time-varying large number. By multiplying these numbers, the magic of creating the green line above is achieved. Let's make sure the model is working by applying this formula to the values on September 2, 2019, 3334 days after July 17, 2010, our first point of bitcoin price data:

This is the price *models* for September 2. *Actual* the price at the end of the day was $ 9941.97 (source). Thus, according to the model, bitcoin at that time was overrated by 31%. For Bitcoin, this is nothing special.

### Estimated Time Range

Making sure that the model is satisfactory andplausibly corresponds to changes in the price of bitcoin over time, I moved on to setting the interval around the price of the model, as wide as necessary, so that it includes highs and lows throughout the entire available history of the price of bitcoin. Having tried several methods for this, I noticed that bitcoin price highs were formed at times approximately 7 times higher than the simulated price (the red line in the graph above), and lows - about 0.4 times lower than the modeled price (blue line ) Using 7 and 0.4 as constant coefficients, I got a price range for each day. On the graph, it seems that this band of bands expands over time, but this is an optical illusion: the length of the vertical segment between the red and blue lines anywhere in the graph will be the same. Now that we have a range of price values, we can easily get the time range by projecting this band of the price range onto the time axis (using the inverse non-linear equation). In the diagram below, the result obtained by this method is clearly presented for the levels of 100 thousand and 1 million dollars.

This method returns the following estimated dates and time ranges for Bitcoin to reach iconic price levels of $ 100,000 and $ 1 million:

- Date of achievement of 100 thousand $: August 15, 2026 (from January 11, 2021 to February 20, 2030)
- Date of achievement of $ 1 million: November 23, 2036 (from November 30, 2027 to July 20, 2042)

These ranges are much more informative than anypoint forecasts, because they are generated by strictly rational methods, based on extrapolating all our knowledge of the price of bitcoin, and using the already well-established power model.

### Not earlier and not later

Since this model estimates the maximum range of bitcoin price values for each point in time, it also gives the following fairly accurate clues:

- Until January 11, 2021, the price of bitcoin is unlikely to reach $ 100 thousand.
- Until November 30, 2027, the price of bitcoin is unlikely to reach $ 1 million.
- After February 20, 2030, the price of Bitcoin, most likely, will no longer drop below $ 100 thousand.
- After July 20, 2042, the price of bitcoin will most likely no longer fall below $ 1 million.

### Estimated Annual Growth Range

The model can also be used to estimate the average annual return (CAGR) of investments made today in bitcoin as the price follows these steps:

- CAGR upon reaching $ 100K: 40% (25–455%)
- CAGR upon reaching $ 1 million: 31% (23–76%)

The estimated results for $ 1 million tell us thateven with the least sharp growth within a given range, the average annual return on investment in bitcoin will be 23% for a period of 22.9 years. Under the scenario of maximum growth, the average annual return will be 76% for a period of 8.3 years, while with a simulated growth, the average annual return will be 31% for a period of 17.2 years.

### Reference point for bitcoin and model validity

The parameter S is in the best agreement withwith actual data at a value of 312. My historical data on the bitcoin price starts from July 17, 2010 (x = 0), so S = 312 corresponds to September 8, 2009 (July 17, 2010 minus 312 days) as the starting date, selected model. This is a few months after the date of the formation of the genesis block of Bitcoin (January 3, 2009), which is used by others in their price models as a constant (a completely natural choice). In the first months, the number of transactions and addresses was very low, so it can hardly be said that bitcoin pricing began immediately from the moment of genesis. The model was free to choose any start date, even before the genesis block. The fact that she chose a date after the genesis and is so close to it reinforces the hypothesis that bitcoin price data do follow a power-type relation, which, of course, is very difficult to prove and such a conclusion cannot be made only on the basis of a linear form on the log scale (price) -log (time). In other words, the fact that the model of the three parameters so accurately selects the start date increases the credibility of this model.

## findings

These calculated ranges are obtained byprojecting the full currently available data on the price of bitcoin into the future using a three-parameter power model that sets labels for orientation in future changes in the price of bitcoin. Moreover, in the most conservative scenario, the estimated average annual yield in dollar terms is 23% for 23 years (quite easy to remember ;-), this seems like a good investment. Since the indicators of the model are updated with each new daily point of data on the price of Bitcoin, the estimated dates and growth rates will gradually change over time. But if Bitcoin continues to behave the same as in the past almost ten years, these ranges should remain fairly stable.

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