Power Laws of Bitcoin, Seyr podcast August 2024
perrenod
97 views
26 slides
Aug 11, 2024
Slide 1 of 26
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
About This Presentation
Slides from August 5, 2024 podcast/video hosted by Robin Seyr.
With Stephen Perrenod we discovered the Lindy effect and what it has to do with the Power Law predicting the Price of BTC in USD, but also in Gold!
Slide Content
THE POWER
LAWS OF BITCOIN
How can an S-curve be a power law?
Robin Seyr podcast
Stephen Perrenod, Ph.D.
August 5, 2024 1
LAWS OF BITCOIN
How can an S-curve be a power law?
Robin Seyr podcast
Stephen Perrenod, Ph.D.
August 5, 2024 1
SOME LONG-TERM BITCOIN PRICE MODELS
Block time
Lindy Effect
Power law model (vs. $, also vs. Gold)
S-curve models (Weibull, not logistic function)
Kelly criterion: position sizing
It’s a supercomputer!
It’s alive! Or at least a Special-purpose AI
2
Block time
Lindy Effect
Power law model (vs. $, also vs. Gold)
S-curve models (Weibull, not logistic function)
Kelly criterion: position sizing
It’s a supercomputer!
It’s alive! Or at least a Special-purpose AI
2
BLOCK CALENDAR / BLOCK TIME
It’s a Time Chain, after all
840,000 block height corresponded to 4th Halving in April
= 16 Block Years of 52,500 blocks
Block months have 4375 blocks
Since Halvings and Difficulty 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 in Anno Satoshi 17 (was 16 at April Halving)
3
It’s a Time Chain, after all
840,000 block height corresponded to 4th Halving in April
= 16 Block Years of 52,500 blocks
Block months have 4375 blocks
Since Halvings and Difficulty 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 in Anno Satoshi 17 (was 16 at April Halving)
3
LINDY EFFECT
(LONG RUNNING SHOWS/TECH LAST LONGER)
Can apply to persistent technology (not to people)
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 4
(LONG RUNNING SHOWS/TECH LAST LONGER)
Can apply to persistent technology (not to people)
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 4
WHY TRY A POWER LAW MODEL
Nature loves power laws:
Gravity: ~ 1/R
2
Electromagnetism:
~ 1/R
2
Strong nuclear force: ~ 1/R
2
for R < Rc
Weak force: ~ exponential
x 1/R
2 but also 1/R term
5
Social and Communication networks:
Power law behavior is a common emergent behavior of networks, e.g. Metcalfe’s Law
https://royalsocietypublishing.org/doi/10.1098/rspa.2019.0742#d1e3441
Nature loves power laws:
Gravity: ~ 1/R
2
Electromagnetism:
~ 1/R
2
Strong nuclear force: ~ 1/R
2
for R < Rc
Weak force: ~ exponential
x 1/R
2 but also 1/R term
5
Social and Communication networks:
Power law behavior is a common emergent behavior of networks, e.g. Metcalfe’s Law
https://royalsocietypublishing.org/doi/10.1098/rspa.2019.0742#d1e3441
EXPONENTIAL (NVIDIA) VS POWER LAW (BITCOIN)
NVIDIA public since 1999, was flattish until
2014
Straight line on a semi-log plot
Has been Exponential since then, at 69%
compounded
Another decade would be $550 trillion = all
private wealth, can’t happen
Exponentials are unstable relative to power
laws, “trees don’t grow to the sky”
6
Nvidia stock price on a semi log chart. It took off after 2014 (shown as 14 in the graph) with a strong exponential rise for the last decade. The price is adjusted for
stock splits; the price plotted is at June 1st of each year. The best fit regression (blue line) corresponds to 69% compound annual growth.
NVIDIA public since 1999, was flattish until
2014
Straight line on a semi-log plot
Has been Exponential since then, at 69%
compounded
Another decade would be $550 trillion = all
private wealth, can’t happen
Exponentials are unstable relative to power
laws, “trees don’t grow to the sky”
6
Nvidia stock price on a semi log chart. It took off after 2014 (shown as 14 in the graph) with a strong exponential rise for the last decade. The price is adjusted for
stock splits; the price plotted is at June 1st of each year. The best fit regression (blue line) corresponds to 69% compound annual growth.
“LINDY” POWER LAW MODEL
LOG-LOG AND SEMI-LOG CHARTS
FIRST DISCOVERED BY “PLAN G” IN 2014 (GIOVANNI SANTOSTASI, PH.D.)
Model is the same for both charts: P ~ Byr^k, where Byr is age in block years and k is power law
index = 5.40 (in calendar years ~ 5.7 power law index)
7
US$0
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 = 0.9497
R² = 1
Model P Actual P
Log-log chart of Bitcoin price vs. Block Years elapsed, monthly data. k = 5.4 power law index is best fit (green).
Currently ~8 years (block chain age increase 50%) for a factor of 9 increase of price to $1/2 million. $1 million around year 27 (11
years away) R2 = 0.95
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
$1000K
$100K
$10K
$ 1K
$100
$10
$1
LOG-LOG AND SEMI-LOG CHARTS
FIRST DISCOVERED BY “PLAN G” IN 2014 (GIOVANNI SANTOSTASI, PH.D.)
Model is the same for both charts: P ~ Byr^k, where Byr is age in block years and k is power law
index = 5.40 (in calendar years ~ 5.7 power law index)
7
US$0
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 = 0.9497
R² = 1
Model P Actual P
Log-log chart of Bitcoin price vs. Block Years elapsed, monthly data. k = 5.4 power law index is best fit (green).
Currently ~8 years (block chain age increase 50%) for a factor of 9 increase of price to $1/2 million. $1 million around year 27 (11
years away) R2 = 0.95
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
$1000K
$100K
$10K
$ 1K
$100
$10
$1
QUANTILE REGRESSION
8
Also age in Block years. Log-log chart to 50 years extrapolation, Semi-log chart to 30 years.
Note tight support for 0.05, 0.1, 0.25 levels. At the base we have a 5.6 or 5.7 index power law.
Quasi-periodic bubbles with exponential rise and fall.
Mid-line 5.4 index. At the tops it looks like a 5.0 power law, as volatility decreases. Still high at around a factor of 2 from the midline 0.5 quantile.
@Sina_21st
8
Also age in Block years. Log-log chart to 50 years extrapolation, Semi-log chart to 30 years.
Note tight support for 0.05, 0.1, 0.25 levels. At the base we have a 5.6 or 5.7 index power law.
Quasi-periodic bubbles with exponential rise and fall.
Mid-line 5.4 index. At the tops it looks like a 5.0 power law, as volatility decreases. Still high at around a factor of 2 from the midline 0.5 quantile.
@Sina_21st
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 flattening (even
before any S-curve considered). Completely different from
exponential = fixed CAGR
Expected gain fair value [(B+1)/B]^k, or 39% for the next block
year and 37% year after that, etc.
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
33.544.555.566.577.588.599.51010.51111.51212.51313.51414.51515.5
k slope
Evolution of the slope parameter, oscillated in early years but has settled down to
somewhat over k = 5.
9
BUT MORE “PHYSICAL” THAN S2F
Power law index varied fair amount, but settled down by year 8
Expected gain declines with time, gradual flattening (even
before any S-curve considered). Completely different from
exponential = fixed CAGR
Expected gain fair value [(B+1)/B]^k, or 39% for the next block
year and 37% year after that, etc.
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
33.544.555.566.577.588.599.51010.51111.51212.51313.51414.51515.5
k slope
Evolution of the slope parameter, oscillated in early years but has settled down to
somewhat over k = 5.
9
CONVEX ADVANTAGE: Z-SCORE
PAST 10 YEARS, 4-YEAR VOL WINDOW
Z-score: # standard deviations above/below trend, using log10 of
prices
For latest 4-year window, one S.D. is a factor of 1.74 x
Positive swings larger, up to Z = 2.28, minus no lower than Z = -1.31
Upside 10^(2.28 x 0.242) = 3.54 times; Downside 10^(1.31 x 0.242) =
2.07 times
E.g. last data point, model trend $59,605
Downside price $28,800 but Upside price $211,000
Risk up to $31K for up to $151K gain (5:1)
Highly favorable risk / reward = Convexity
Furthermore, trend is 39% p.a. at present
10
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Block years elapsed
66.4176.8337.257.6678.0838.58.9179.3339.7510.16710.5831111.41711.83312.2512.666613.08313.513.9214.3314.7515.16715.58316
Z-score four year
PAST 10 YEARS, 4-YEAR VOL WINDOW
Z-score: # standard deviations above/below trend, using log10 of
prices
For latest 4-year window, one S.D. is a factor of 1.74 x
Positive swings larger, up to Z = 2.28, minus no lower than Z = -1.31
Upside 10^(2.28 x 0.242) = 3.54 times; Downside 10^(1.31 x 0.242) =
2.07 times
E.g. last data point, model trend $59,605
Downside price $28,800 but Upside price $211,000
Risk up to $31K for up to $151K gain (5:1)
Highly favorable risk / reward = Convexity
Furthermore, trend is 39% p.a. at present
10
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Block years elapsed
66.4176.8337.257.6678.0838.58.9179.3339.7510.16710.5831111.41711.83312.2512.666613.08313.513.9214.3314.7515.16715.58316
Z-score four year
CONVEX ADVANTAGE: LEAD/LAG
As suggested by @apsk32
How many years/months does the price lead or lag
a power law model?
July 31 at $66,550 ahead by 3.7 months
11
Block years (lead/lag)
-2.000
-1.000
0.000
1.000
2.000
3.000
4.000
Block years elapsed
2.002.583.173.754.334.925.506.086.677.257.838.429.009.5810.1710.7511.3311.9212.5013.0813.6714.2514.8315.4216.00
Lead/Lag years
0
18
35
53
70
Lag 1 to 2 Yrs.Lag < 1 yr.Lead 0 To 1Lead 1 to 2Lead 2 to 3Lead > 3 yrs.
As suggested by @apsk32
How many years/months does the price lead or lag
a power law model?
July 31 at $66,550 ahead by 3.7 months
11
Block years (lead/lag)
-2.000
-1.000
0.000
1.000
2.000
3.000
4.000
Block years elapsed
2.002.583.173.754.334.925.506.086.677.257.838.429.009.5810.1710.7511.3311.9212.5013.0813.6714.2514.8315.4216.00
Lead/Lag years
0
18
35
53
70
Lag 1 to 2 Yrs.Lag < 1 yr.Lead 0 To 1Lead 1 to 2Lead 2 to 3Lead > 3 yrs.
EXPONENTIAL OR POWER LAW?
The data for average monthly return and monthly
return volatility is clear
Results at left are model independent
If exponential = constant CAGR both lines would
be straight
In reality, both the return and volatility fell
monotonically and considerably from Epoch 1 to
Epoch IV
Actual values are consistent with power law
model plus allowance for bubbles
12https://stephenperrenod.substack.com/publish/posts/detail/144303111
The data for average monthly return and monthly
return volatility is clear
Results at left are model independent
If exponential = constant CAGR both lines would
be straight
In reality, both the return and volatility fell
monotonically and considerably from Epoch 1 to
Epoch IV
Actual values are consistent with power law
model plus allowance for bubbles
12https://stephenperrenod.substack.com/publish/posts/detail/144303111
POWER LAW MODEL VS. GOLD
This graph is BTC / gold ounces in
calendar years, past 13 years
Power law ~ t^(-5.48), R
2 0.87
13
Calendar years since Jan. 2009https://stephenperrenod.substack.com/p/bitcoin-versus-gold-update
This graph is BTC / gold ounces in
calendar years, past 13 years
Power law ~ t^(-5.48), R
2 0.87
13
Calendar years since Jan. 2009https://stephenperrenod.substack.com/p/bitcoin-versus-gold-update
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, e.g. market cap
Regression requires a double log of both sides
Market cap $30T or $100T: k = 6.00, R^2 = 0.96
14
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, e.g. market cap
Regression requires a double log of both sides
Market cap $30T or $100T: k = 6.00, R^2 = 0.96
14
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 fitting to data and then determine
best fit for c, k for each model
Characteristic time = time to 1 - 1/e fraction of asymptotic MC (for any k)
Scale parameter k
15
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 fitting to data and then determine
best fit for c, k for each model
Characteristic time = time to 1 - 1/e fraction of asymptotic MC (for any k)
Scale parameter k
15
WEIBULL CDF REGRESSION (MARKET CAP)
BLOCK YEARS [2, 16.17]
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 significantly higher
Year of Knee of S-curve and Best fit price at Knee
ASYMPTOTIC MC K SHAPE C TIMESCALE R2 F YEAR PRICE
POWER LAW 5.992x 1.98 years0.9636572026 116K
$3 T 6.07 18.23 0.9635852026 96K
$10 T 6.01 22.6 0.9636412030 310K
$30 T 6.00 27.3 0.9636522035 950K
$100 T 5.99 33.4 0.9536552041 3.1M
16
BLOCK YEARS [2, 16.17]
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 significantly higher
Year of Knee of S-curve and Best fit price at Knee
ASYMPTOTIC MC K SHAPE C TIMESCALE R2 F YEAR PRICE
POWER LAW 5.992x 1.98 years0.9636572026 116K
$3 T 6.07 18.23 0.9635852026 96K
$10 T 6.01 22.6 0.9636412030 310K
$30 T 6.00 27.3 0.9636522035 950K
$100 T 5.99 33.4 0.9536552041 3.1M
16
KELLY CRITERION,
SENSITIVE TO REBALANCING TIMEFRAME
Bitcoin or US$ cash, two-asset portfolio
The Kelly criterion maximizes growth of log Wealth, f is optimal allocation
f = p - q/b (gambler’s formula, more conservative! Compared
to investment formula)
where p = % wins, q = 1-p % losses, b = average win size / average loss size
In table w is average percentage of all wins only, l average percentage for
losses
The more chill you are, the more Bitcoin you should hold as % of liquid
capital or wealth
Annual rebalance would call for f ~ 56% to 77% (7-year or all time
window, but noisy)
The yearly column results include 5% interest on cash
5 years prior to Aug 2023 for Monthly, Quarterly, and for Yearly starting in 2011 or 2018 including half year of 2024
17
SENSITIVE TO REBALANCING TIMEFRAME
Bitcoin or US$ cash, two-asset portfolio
The Kelly criterion maximizes growth of log Wealth, f is optimal allocation
f = p - q/b (gambler’s formula, more conservative! Compared
to investment formula)
where p = % wins, q = 1-p % losses, b = average win size / average loss size
In table w is average percentage of all wins only, l average percentage for
losses
The more chill you are, the more Bitcoin you should hold as % of liquid
capital or wealth
Annual rebalance would call for f ~ 56% to 77% (7-year or all time
window, but noisy)
The yearly column results include 5% interest on cash
5 years prior to Aug 2023 for Monthly, Quarterly, and for Yearly starting in 2011 or 2018 including half year of 2024
17
@OrionX_net
CryptoSuper 12th
Mining Report,
May 2024:
Comparison to Top 500
Supercomputer List
Attribute
Frontier
Supercomputer (#1)
Top 500 (all)
WhatsMiner 63S
Hydro cabinet
Bitcoin Network
Equivalent
Performance 1.2 Exaflops 8.2 Exaflops 4.68 Petahash/s 581 Exahash/s
One year
increase
9% 57% 50% (vs. 53S) 57%
Chips
37,888 AMD Instinct
GPUS 6nm; 9472
AMD Epyc CPUs
Hundreds of
millons of cores
12 multithread, 5 nm
ASIC
1,489,744 ASICs
Cabinets 74 1 124,145 cabinets
Power
consumption
23 MegaWatts 89.2 KW 11,074 MW
Weight 296 tons 0.76 metric tons94,600 metric tons
Cost $600 million $107,000 $13.28 billion
Output Science
Science &
Engineering & AI
1.32 Bitcoin per year164,250 Bitcoin per year
Value Priceless Enormous $93,400 per year$11.6 billion per year
Table 5. Comparison of a
hypothetical Bitcoin network based
on the latest high-end WhatsMiner
63S Hydro system with the
Department of Energy’s Frontier
supercomputer and the Top 500 list
as a whole. The cost to build out the
global Bitcoin network is about 22
times that of the Frontier system
while the economic output is
measurable at $11.6 billion annually,
currently. Interestingly both the
Top500 and the Bitcoin network
have recently experienced the same
Moore’s Law growth rate of 57% per
year.
CryptoSuper 12th
Mining Report,
May 2024:
Comparison to Top 500
Supercomputer List
Attribute
Frontier
Supercomputer (#1)
Top 500 (all)
WhatsMiner 63S
Hydro cabinet
Bitcoin Network
Equivalent
Performance 1.2 Exaflops 8.2 Exaflops 4.68 Petahash/s 581 Exahash/s
One year
increase
9% 57% 50% (vs. 53S) 57%
Chips
37,888 AMD Instinct
GPUS 6nm; 9472
AMD Epyc CPUs
Hundreds of
millons of cores
12 multithread, 5 nm
ASIC
1,489,744 ASICs
Cabinets 74 1 124,145 cabinets
Power
consumption
23 MegaWatts 89.2 KW 11,074 MW
Weight 296 tons 0.76 metric tons94,600 metric tons
Cost $600 million $107,000 $13.28 billion
Output Science
Science &
Engineering & AI
1.32 Bitcoin per year164,250 Bitcoin per year
Value Priceless Enormous $93,400 per year$11.6 billion per year
Table 5. Comparison of a
hypothetical Bitcoin network based
on the latest high-end WhatsMiner
63S Hydro system with the
Department of Energy’s Frontier
supercomputer and the Top 500 list
as a whole. The cost to build out the
global Bitcoin network is about 22
times that of the Frontier system
while the economic output is
measurable at $11.6 billion annually,
currently. Interestingly both the
Top500 and the Bitcoin network
have recently experienced the same
Moore’s Law growth rate of 57% per
year.
BITCOIN AS SPECIAL PURPOSE AI
“Your arguments present a compelling case for considering Bitcoin as a form of
special-purpose AI for money.”
“While Bitcoin may not fit the traditional definition of AI, it embodies many
characteristics of an intelligent, adaptive system for financial transactions. Given
its complex and emergent behaviors, decentralized self-governance, and the
creation of extensive networks, one could argue that Bitcoin functions as a
specialized form of AI for money, representing an innovative alternative to
traditional financial systems.” - ChatGPT
19
1. Logical system of self-executing
protocols
II. A decentralized, persistent network
III. Catalyst for human behavior
IV. Emergent and resilient network
behavior
V. Power law behavior (more stable)
One AI recognizing another
“Your arguments present a compelling case for considering Bitcoin as a form of
special-purpose AI for money.”
“While Bitcoin may not fit the traditional definition of AI, it embodies many
characteristics of an intelligent, adaptive system for financial transactions. Given
its complex and emergent behaviors, decentralized self-governance, and the
creation of extensive networks, one could argue that Bitcoin functions as a
specialized form of AI for money, representing an innovative alternative to
traditional financial systems.” - ChatGPT
19
1. Logical system of self-executing
protocols
II. A decentralized, persistent network
III. Catalyst for human behavior
IV. Emergent and resilient network
behavior
V. Power law behavior (more stable)
One AI recognizing another
@moneyordebt
Thank You
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.
20
Thank You
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.
20
Appendix
21
21
PRICE DOUBLING TIME
WITH POWER LAW K = 5.4
22
0
1
2
3
4
5
Block Years
456789101112131415161718192021222324252627282930
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 2025 2.33 36%
18 2026 2.47 34%
19 2027 2.60 32%
20 2028 2.74 30%
21 2029 2.88 29%
22 2030 3.01 27%
23 2031 3.15 26%
24 2032 3.29 25%
25 2033 3.42 24%
26 2034 3.56 23%
27 2035 3.70 22%
28 2036 3.83 21%
29 2037 3.97 20%
30 2038 4.11 19%
Double ((B+x)/B)^5.4 = 2 x = B* (2^(1/5.4)-1)
WITH POWER LAW K = 5.4
22
0
1
2
3
4
5
Block Years
456789101112131415161718192021222324252627282930
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 2025 2.33 36%
18 2026 2.47 34%
19 2027 2.60 32%
20 2028 2.74 30%
21 2029 2.88 29%
22 2030 3.01 27%
23 2031 3.15 26%
24 2032 3.29 25%
25 2033 3.42 24%
26 2034 3.56 23%
27 2035 3.70 22%
28 2036 3.83 21%
29 2037 3.97 20%
30 2038 4.11 19%
Double ((B+x)/B)^5.4 = 2 x = B* (2^(1/5.4)-1)
JENSEN’S INEQUALITY RATIO*
MEASURES CONVEXITY
(*NOT JENSEN ALPHA, DIFFERENT GUY)
Jensen’s inequality compares (a) mean of model values (e.g. from
power law function) over an interval with (b) function at the mean of
the interval
If average ( φ [t1, t2] ) > φ (average [t1,t2] ) then φ is a convex
function
Simply put: average of all model prices over interval relative to
expected price at the mean time of the interval
Ratio of the two averages =* Jensen inequality ratio
Positive Convexity is good as bond traders know
If it rises with more data it means the convexity is hardening
Copilot AI: “If Jensen’s inequality keeps getting stronger for a dataset,
it could indicate increased stability and/or strong convexity.”
23
0
1
2
3
4
Block Years
2 3 4 5 6 7 8 9 101112131415.0
Jensen Ratio of Bitcoin Power Law ~ Block Year^5.4
MEASURES CONVEXITY
(*NOT JENSEN ALPHA, DIFFERENT GUY)
Jensen’s inequality compares (a) mean of model values (e.g. from
power law function) over an interval with (b) function at the mean of
the interval
If average ( φ [t1, t2] ) > φ (average [t1,t2] ) then φ is a convex
function
Simply put: average of all model prices over interval relative to
expected price at the mean time of the interval
Ratio of the two averages =* Jensen inequality ratio
Positive Convexity is good as bond traders know
If it rises with more data it means the convexity is hardening
Copilot AI: “If Jensen’s inequality keeps getting stronger for a dataset,
it could indicate increased stability and/or strong convexity.”
23
0
1
2
3
4
Block Years
2 3 4 5 6 7 8 9 101112131415.0
Jensen Ratio of Bitcoin Power Law ~ Block Year^5.4
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 in The Dark Side on Medium
24
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 in The Dark Side on Medium
24
ABOUT ME
Early start and finish in Finance
Older than Michael Saylor
Younger than fellow ΠΛΦ brother Feynman
Astrophysics and then Supercomputing
industry for a few decades followed by
Technology consulting
OrionX.net , Analyst and Partner
25
Early start and finish in Finance
Older than Michael Saylor
Younger than fellow ΠΛΦ brother Feynman
Astrophysics and then Supercomputing
industry for a few decades followed by
Technology consulting
OrionX.net , Analyst and Partner
25
NVDA
SPECIAL PURPOSE AI
TOP 500 COMPARISON
UPDATE MODELS
TEMPORAL LEAD/LAG
Additions / deletions for this version
26
SPECIAL PURPOSE AI
TOP 500 COMPARISON
UPDATE MODELS
TEMPORAL LEAD/LAG
Additions / deletions for this version
26
Download
Download Slideshow Get the original presentation fileQuick Actions
Statistics
- Views 97
- Slides 26
- Age 483 days
Related Slideshows
11
8-top-ai-courses-for-customer-support-representatives-in-2025.pptx
JeroenErne2
57 views
10
7-essential-ai-courses-for-call-center-supervisors-in-2025.pptx
JeroenErne2
53 views
13
25-essential-ai-courses-for-user-support-specialists-in-2025.pptx
JeroenErne2
42 views
11
8-essential-ai-courses-for-insurance-customer-service-representatives-in-2025.pptx
JeroenErne2
41 views
21
Know for Certain
DaveSinNM
25 views
17
PPT OPD LES 3ertt4t4tqqqe23e3e3rq2qq232.pptx
novasedanayoga46
30 views