The Power Laws of Bitcoin: How can an S-curve be a power law?
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The document presents a comprehensive analysis of Bitcoin's price modeling through various frameworks, including power laws and S-curve models. It discusses the implications of the Lindy effect on Bitcoin's longevity and forecasts substantial price increases based on historical data. Additionally, it highlights the comparative analysis with gold and emphasizes the concepts of scarcity, security, and the utility of Bitcoin as a reserve asset.
![LINDY EFFECT
(LONG RUNNING SHOWS/TECH LAST LONGER)
Can apply to persistent technology
E[T-t] = p*t , where T-t is time until expiry, p is Lindy
proportion
If use annual % price gain as a p estimator and taking
geometric mean from 2011-2024, then
p = 2.4 roughly
So, if Bitcoin has survived 15+ years, it should have
another 36 years ahead (and once it reaches age 50
years, another 120 etc.)
Taleb’s anti-fragility
1921-2018 5](https://image.slidesharecdn.com/powerlawsofbitcoinsperrenod-240322030951-a847ac59/85/The-Power-Laws-of-Bitcoin-How-can-an-S-curve-be-a-power-law-5-320.jpg)
![SOBERING PART
BUT MORE “PHYSICAL” THAN S2F
Power law index varied fair amount, but settled down by
year 8
Expected gain declines with time, gradual
fl
attening (even
before S-curve considered)
Expected gain fair value [(B+1)/B]^k, or 39% for the next
block year and 37% year after
Standard deviation is, in log terms, .315 (.253) or a
multiplicative factor of 2.07 (1.79) for all data (last 4 years)
Power
law
slope
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Block years
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5
k slope
Evolution of the slope parameter, oscillated in early years but has settled down to
somewhat over k = 5.
8](https://image.slidesharecdn.com/powerlawsofbitcoinsperrenod-240322030951-a847ac59/85/The-Power-Laws-of-Bitcoin-How-can-an-S-curve-be-a-power-law-8-320.jpg)
![TECHNOLOGY S-CURVE
WEIBULL CUMULATIVE DISTRIBUTION FUNCTION
Weibull cdf: f = 1 -exp [-(t/c)^k]
Normalized to an asymptotic value
Three parameters:
Characteristic time scale, c
Scale factor k
Asymptotic value
Regression requires a double log of both sides
10](https://image.slidesharecdn.com/powerlawsofbitcoinsperrenod-240322030951-a847ac59/85/The-Power-Laws-of-Bitcoin-How-can-an-S-curve-be-a-power-law-10-320.jpg)
![WEIBULL CDF
Ln (1-f) = [-(t/c)^k]; ln (-ln (1-f)) = k ln (t/c) ; substitute for LHS: y = k ln (t) - k ln (c)
So for y = ax + b; a := k and x := ln (t) and b: - k ln (c) . And can linearly regress (StatPlus)
So we are regressing a double log function of the market cap (relative to its asymptotic value) vs. the
log of time
We run a series of models, varying asymptotic market cap when
fi
tting to data and then determine
best
fi
t for c, k for each model
Characteristic time = time to 1 - 1/e fraction of asymptotic MC (for any k)
Scale parameter k
11](https://image.slidesharecdn.com/powerlawsofbitcoinsperrenod-240322030951-a847ac59/85/The-Power-Laws-of-Bitcoin-How-can-an-S-curve-be-a-power-law-11-320.jpg)
![WEIBULL CDF REGRESSION (MARKET CAP)
BLOCKYEARS [2, 15.75]
Weibull cdf: f = 1 -exp [-(t/c)^k]
For early t << c, f = 1 - [ 1 - (t/c)^k ] = (t/c)^k and it’s just a power law of k! (Market cap ~ t^6)
So as long as a power law holds we are not close to the knee at age c; knee = 1 - 1/e = 63% of
asymptotic value
And that means the longer it holds, then the asymptotic market cap can be signi
fi
cantly higher
Year of Knee of S-curve and Best
fi
t price at Knee
ASYMPTOTIC MC K SHAPE C TIMESCALE R2 F YEAR PRICE
POWER LAW 5.986 2x 1.93 years 0.954 3374 2025 88K
$3 T 6.051 18.34 0.953 3304 2026 96K
$10 T 6.004 22.67 0.953 3358 2030 310K
$30 T 5.992 27.32 0.954 3369 2035 950K
$100 T 5.988 33.40 0.954 3373 2041 3.1M
12](https://image.slidesharecdn.com/powerlawsofbitcoinsperrenod-240322030951-a847ac59/85/The-Power-Laws-of-Bitcoin-How-can-an-S-curve-be-a-power-law-12-320.jpg)
The Power Laws of Bitcoin: How can an S-curve be a power law?
- 1. THE POWER LAWS OF BITCOIN How can an S-curve be a power law? Stephen Perrenod, Ph.D. 1
- 2. ABOUT ME Early start and fi nish in Finance Older than Michael Saylor Younger than fellow ΠΛΦ brother Feynman Astrophysics and then High performance computing industry for a few decades and then Technology consulting Worked for Fred’s college roommate OrionX.net , Analyst and Partner 2
- 3. SOME LONG-TERM BITCOIN PRICE MODELS Block time Lindy Effect Power law model (also vs. Gold) S-curve models Kelly criterion position sizing What about utility? 3
- 4. BLOCK CALENDAR / BLOCKTIME It’s a Time Chain 840,000 block height corresponds to 4th Halving in April = 16 BlockYears of 52,500 blocks Block months have 4375 blocks Since Halvings and Dif fi culty adjustments happen on block boundaries, best fundamental time system for regressions Just as the Gregorian calendar has earthly, lunar, solar rhythms, so does Satoshi’s calendar Already Anno Satoshi 16 prior to halving 4
- 5. LINDY EFFECT (LONG RUNNING SHOWS/TECH LAST LONGER) Can apply to persistent technology E[T-t] = p*t , where T-t is time until expiry, p is Lindy proportion If use annual % price gain as a p estimator and taking geometric mean from 2011-2024, then p = 2.4 roughly So, if Bitcoin has survived 15+ years, it should have another 36 years ahead (and once it reaches age 50 years, another 120 etc.) Taleb’s anti-fragility 1921-2018 5
- 6. WHYTRY A POWER LAW MODEL Nature loves power laws: Gravity: ~ 1/R2 Electromagnetism: ~ 1/R2 Strong nuclear force: ~ 1/R2 for R < Rc Weak force: ~ exponential x 1/R2 but also 1/R term 6
- 7. LINDY POWER LAW MODEL SEMI-LOG AND LOG-LOG CHARTS Model is the same for both P ~ B^k, where B is age in block years and k is power law index = 5.40 US$0 US$1 US$10 US$100 US$1,000 US$10,000 US$100,000 US$1,000,000 1 10 100 R² = 0.943 R² = 1 Model Price Actual Price Log-log chart of Bitcoin price vs. Block Years elapsed, monthly data. k = 5.4 power law index is best fi t (green). Currently ~8 years (block chain age increase 53%) for a factor of 10 increase of price. $1 million around year 27 (11 plus years away) Log10 BTC Price USD -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Block Years Elapsed 2 3.25 4.5 5.75 7 8.25 9.5 10.75 12 13.25 14.5 15.75 Log10 Price Model Log P Model - 1 SD Model + 1 SD 7
- 8. SOBERING PART BUT MORE “PHYSICAL” THAN S2F Power law index varied fair amount, but settled down by year 8 Expected gain declines with time, gradual fl attening (even before S-curve considered) Expected gain fair value [(B+1)/B]^k, or 39% for the next block year and 37% year after Standard deviation is, in log terms, .315 (.253) or a multiplicative factor of 2.07 (1.79) for all data (last 4 years) Power law slope 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Block years 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 k slope Evolution of the slope parameter, oscillated in early years but has settled down to somewhat over k = 5. 8
- 9. POWER LAW MODELVS. GOLD This is BTC / gold ounces in calendar years, past 11 years Exponential 69% cagr, R2 .85 Power law ~ t^4.7, R2 .87 Gold ounces 0.01 0.10 1.00 10.00 100.00 Years since Jan. 2009 4.25 4.67 5.08 5.50 5.92 6.33 6.75 7.17 7.58 8.00 8.42 8.83 9.25 9.67 10.08 10.50 10.92 11.33 11.75 12.17 12.58 13.00 13.41 13.83 14.25 14.67 15.08 BTC/Gold oz. Model Exponential Model Power Law 9
- 10. TECHNOLOGY S-CURVE WEIBULL CUMULATIVE DISTRIBUTION FUNCTION Weibull cdf: f = 1 -exp [-(t/c)^k] Normalized to an asymptotic value Three parameters: Characteristic time scale, c Scale factor k Asymptotic value Regression requires a double log of both sides 10
- 11. WEIBULL CDF Ln (1-f) = [-(t/c)^k]; ln (-ln (1-f)) = k ln (t/c) ; substitute for LHS: y = k ln (t) - k ln (c) So for y = ax + b; a := k and x := ln (t) and b: - k ln (c) . And can linearly regress (StatPlus) So we are regressing a double log function of the market cap (relative to its asymptotic value) vs. the log of time We run a series of models, varying asymptotic market cap when fi tting to data and then determine best fi t for c, k for each model Characteristic time = time to 1 - 1/e fraction of asymptotic MC (for any k) Scale parameter k 11
- 12. WEIBULL CDF REGRESSION (MARKET CAP) BLOCKYEARS [2, 15.75] Weibull cdf: f = 1 -exp [-(t/c)^k] For early t << c, f = 1 - [ 1 - (t/c)^k ] = (t/c)^k and it’s just a power law of k! (Market cap ~ t^6) So as long as a power law holds we are not close to the knee at age c; knee = 1 - 1/e = 63% of asymptotic value And that means the longer it holds, then the asymptotic market cap can be signi fi cantly higher Year of Knee of S-curve and Best fi t price at Knee ASYMPTOTIC MC K SHAPE C TIMESCALE R2 F YEAR PRICE POWER LAW 5.986 2x 1.93 years 0.954 3374 2025 88K $3 T 6.051 18.34 0.953 3304 2026 96K $10 T 6.004 22.67 0.953 3358 2030 310K $30 T 5.992 27.32 0.954 3369 2035 950K $100 T 5.988 33.40 0.954 3373 2041 3.1M 12
- 13. KELLY CRITERION, SENSITIVETO REBALANCINGTIMEFRAME Bitcoin or US$ cash, two-asset portfolio The Kelly criterion maximizes growth of log Wealth, f is optimal allocation f = p - q/b where p = % wins, q = 1-p % losses, b = average win size / average loss size In table W is sum of all percentage wins The more chill you are, the more Bitcoin you should hold as % of liquid capital or wealth Annual rebalance would call for f ~ 44% to 70% (5-years or all time window, but noisy) 5 years prior to Aug 2023 13
- 14. SCARCITY, SECURITY > UTILITY Use dissipates (decreases) value The faster we spend fi at, the more prices will rise and the fi at devalues Fiat is designed for utility fi rst Bitcoin is designed for security and scarcity needed for a long-term Store of Value Ends up being a hard asset, a reserve asset with fi rst layer utility appropriate to a reserve currency, not to retail spending Because security keeps rising, and scarcity keeps rising Tick tock, next block : Lindy Effect 14
- 15. @moneyordebt ThankYou This is not investment advice. Bitcoin is highly volatile. Past performance of back-tested models is no assurance of future performance. Only invest what you can afford to lose. You must decide how much of your investment capital you are willing to risk with Bitcoin. No warranties are expressed or implied. Money has become information. Bitcoin is energy securely encapsulated as information. Electrons to eternal bits. 15
- 16. Appendix 16
- 17. MODELING BITCOIN VALUE:THREE METHODS EACH A POWER LAW: PRICE - STOCK2FLOW ~ $70K IN 3/2023 PRICE - DIFFICULTY ~ $71K IN 3/2023 PRICE -TIME ~ $37K IN 3/2023 AND K = 5.42 (MID MARCH 2023 ACTUAL ABOUT $25K) December 2019 inThe Dark Side on Medium 17
- 18. PRICE DOUBLINGTIME WITH POWER LAW K = 5.4 18 0 1 2 3 4 5 Block Years 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Doubling time, block years Doubling time, for Block Year to the 5.4 Power Law model Block year Calendar Year Doubling time, block years CAGR 4 2012 0.55 234% 5 2013 0.68 168% 6 2014 0.82 130% 7 2015 0.96 106% 8 2016 1.10 89% 9 2017 1.23 77% 10 2018 1.37 67% 11 2019 1.51 60% 12 2020 1.64 54% 13 2021 1.78 49% 14 2022 1.92 45% 15 2023 2.05 42% 16 2024 2.19 39% 17 2024 2.33 36% 18 2024 2.47 34% 19 2024 2.60 32% 20 2024 2.74 30% 21 2024 2.88 29% 22 2024 3.01 27% 23 2024 3.15 26% 24 2024 3.29 25% 25 2024 3.42 24% 26 2024 3.56 23% 27 2024 3.70 22% 28 2024 3.83 21% 29 2024 3.97 20% 30 2024 4.11 19% Double ((B+x)/B)^5.4 = 2 x = B* (2^(1/5.4)-1)