**OVERVIEW**

The 'Bitcoin Power Law Bands' indicator is a set of three DXY price trendlines and two price bands for BTC , indicating overall long-term trend, support and resistance levels as well as oversold and overbought conditions. The magnitude and growth of the middle (Center) line is determined by double logarithmic (log-log) regression on the entire DXY price history of BTC . The upper (Resistance) and lower (Support) lines follow the same trajectory but multiplied by respective (fixed) factors. These two lines indicate levels where the price of BTC is expected to meet strong long-term resistance or receive strong long-term support. The two bands between the three lines are price levels where BTC may be considered overbought or oversold.

All parameters and visuals may be customized by the user as needed.

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**CONCEPTS**

**Long-term models**

Long-term price models have many challenges, the most significant of which is getting the growth curve right overall. No one can predict how a certain market, asset class, or financial instrument will unfold over several decades. In the case of BTC , price history is very limited and extremely volatile, and this further complicates the situation. Fortunately for us, a few smart people already had some bright ideas that seem to have stood the test of time.

**Power law**

The so-called power law is the

*only*long-term BTC price model that has a chance of survival for the years ahead. The idea behind the power law is very simple: over time, the rapid (exponential) initial growth cannot possibly be sustained (see The seduction of the exponential curve for a fun take on this). Year-on-year returns, therefore, must decrease over time, which leads us to the concept of diminishing returns and the power law. In this context, the power law translates to linear growth on a chart with both its axes scaled logarithmically. This is called the log-log chart (as opposed to the semilog chart you see above, on which only one of the axes - price - is logarithmic).

**Log-log regression**

When both price and time are scaled logarithmically, the power law leads to a linear relationship between them. This in turn allows us to apply linear regression techniques, which will find the best-fitting straight line to the data points in question. The result of performing this log-log regression (i.e. linear regression on a log-log scaled dataset) is two parameters: slope (m) and intercept (b). These parameters fully describe the relationship between price and time as follows: log(P) = m * log(T) + b, where P is price and T is time. Price is measured in DXY , and Time is counted as the number of days elapsed since BTC 's genesis block.

**DPC model**

The final piece of our puzzle is the Dynamic Power Cycle (DPC) price model of BTC . DPC is a long-term cyclic model that uses the power law as its foundation, to which a periodic component stemming from the block subsidy halving cycle is applied dynamically. The regression parameters of this model are re-calculated daily to ensure longevity. For the 'Bitcoin Power Law Bands' indicator, the slope and intercept parameters were calculated on publication date (March 6, 2022). The slope of the Resistance Line is the same as that of the Center Line; its intercept was determined by fitting the line onto the Nov 2021 cycle peak. The slope of the Support Line is the same as that of the Center Line; its intercept was determined by fitting the line onto the Dec 2018 trough of the previous cycle. Please see the

**Limitations**section below on the implications of a static model.

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**FEATURES**

**Inputs**

• Parameters

• Center Intercept (b) and Slope (m): These log-log regression parameters control the behavior of the grey line in the middle

• Resistance Intercept (b) and Slope (m): These log-log regression parameters control the behavior of the red line at the top

• Support Intercept (b) and Slope (m): These log-log regression parameters control the behavior of the green line at the bottom

• Controls

• Plot Line Fill: N/A

• Plot Opportunity Label: Controls the display of current price level relative to the Center, Resistance and Support Lines

**Style**

• Visuals

• Center: Control, color, opacity, thickness, price line control and line style of the Center Line

• Resistance: Control, color, opacity, thickness, price line control and line style of the Resistance Line

• Support: Control, color, opacity, thickness, price line control and line style of the Support Line

• Plots Background: Control, color and opacity of the Upper Band

• Plots Background: Control, color and opacity of the Lower Band

• Labels: N/A

• Output

• Labels on price scale: Controls the display of current Center, Resistance and Support Line values on the price scale

• Values in status line: Controls the display of current Center, Resistance and Support Line values in the indicator's status line

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**HOW TO USE**

The indicator includes three price lines:

• The grey Center Line in the middle shows the overall long-term BTC USD price trend

• The red Resistance Line at the top is an indication of where the BTC USD price is expected to meet strong long-term resistance

• The green Support Line at the bottom is an indication of where the BTC USD price is expected to receive strong long-term support

These lines envelope two price bands:

• The red Upper Band between the Center and Resistance Lines is an area where BTC is considered overbought (i.e. too expensive)

• The green Lower Band between the Support and Center Lines is an area where BTC is considered oversold (i.e. too cheap)

The power law model assumes that the price of BTC will fluctuate around the Center Line, by meeting resistance at the Resistance Line and finding support at the Support Line. When the current price is well below the Center Line (i.e. well into the green Lower Band), BTC is considered too cheap (oversold). When the current price is well above the Center Line (i.e. well into the red Upper Band), BTC is considered too expensive (overbought). This idea alone is not sufficient for profitable trading, but, when combined with other factors, it could guide the user's decision-making process in the right direction.

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**LIMITATIONS**

The indicator is based on a static model, and for this reason it will gradually lose its usefulness. The Center Line is the most durable of the three lines since the long-term growth trend of BTC seems to deviate little from the power law. However, how far price extends above and below this line will change with every halving cycle (as can be seen for past cycles). Periodic updates will be needed to keep the indicator relevant. The user is invited to adjust the slope and intercept parameters manually between two updates of the indicator.

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**RAMBLINGS**

The 'Bitcoin Power Law Bands' indicator is a useful tool for users wishing to place BTC in a macro context. As described above, the price level relative to the three lines is a rough indication of whether BTC is over- or undervalued. Users wishing to gain more insight into BTC price trends may follow the author's periodic updates of the DPC model (contact information below).

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**NOTES**

The author regularly posts on Twitter using the @DeFi_initiate handle.

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**THANKS**

Many thanks to the following individuals, who - one way or another - made the 'Bitcoin Power Law Bands' indicator possible:

• TradingView user 'capriole_charles', whose open-source 'Bitcoin Power Law Corridor' script was the basis for this indicator

• Harold Christopher Burger, whose Bitcoin’s natural long-term power-law corridor of growth article (2019) was the basis for the 'Bitcoin Power Law Corridor' script

• Bitcoin Forum user "Trololo", who posted the original power law model at Logarithmic (non-linear) regression - Bitcoin estimated value (2014)