Many people think the four year cycle has ended, $126,000 was
the top for Bitcoin, and we're only going down from here.
But what if I told you the math proves that the four year cycle
never really existed? Astrophysicist and mathematician
Steven Perroneau joins me again to discuss why Bitcoin
mathematically outperforms gold, why the volatility is fading,
how the four year having cycle narrative is false, and why
bubbles are actually spaced out by math and not by calendar
years. Steven reveals why the power law
and his new insights point to a new date for the next major
Bitcoin bubble. And it's further away than you
think. All right, let's dive in.
All right, Steven Perrina, welcome back to Bitcoin for
Millennials. Oh it's great to be here with
you again, bro. I'm a student here.
The audience here is a, is a student.
So yeah, I'll just listen to your new findings and and try to
ask clarifying or maybe even some noob questions probably.
And yeah, I'm, I'm excited to dive in and see what you've been
been focusing on. So let's just let's just go into
it. Great.
Well, I've been thinking a lot about bubbles all year, you
know, as to whether we're going to have one and how big it might
be and when it might be and looking at the history of
bubbles and seeing that they were fading, you know, and
wondering whether we're going to have bubbles in the future.
But in the last month, I've kind of made some progress,
significant progress, I believe. And I also want to talk about
relative to gold, because it's been a good year for gold and,
and less so for Bitcoin. And so I'm, I don't want to
sweep down under the rug. I want to dress that head on.
So on this first slide, this gentleman was the master of the
Royal Mint. His name you're familiar with,
it's Isaac Newton. And people, a lot of people
don't know that he spent just as much time, you know, down in
London as he did in Cambridge. He spent about 30 years as
master of the Royal Mint in London after his, you know, time
in, in Cambridge. So I'm actually going to talk
just a little bit about alchemy and energy because he was, he
was quite into alchemy, which, you know, at that time the
distinction between chemistry and alchemy was, was unclear,
shall we say. And then I'll get into, just
briefly remind you of the distinction between the way a
power law behaves and the way exponentials behave.
And then I want to get into a discussion of Bitcoin relative
to gold, and that will be the power law measured in gold
terms, in other words, Bitcoin measured in ounces.
I'll also look at the Kelly criterion.
If you have a 2 asset portfolio, how much would you put into
Bitcoin, how much into gold? And then I want to look at the
bubbles. And I have a thesis I'm
presenting the last few weeks that the bubbles are really
spaced in logarithmic time, not in linear time.
So I'm dismissing the idea of a four year cycle.
And then I will talk about Bitcoin versus the dollar, just
some of the latest regression. If we have time, I'll talk about
cities versus companies and also a cosmology analogy for Bitcoin.
And so people know Newton for calculus and from gravity.
And that was in the 1660s with calculus.
And he published Principia and laid out Newtonian dynamics in
1687. But then ten years later he, he
became master of the men and he held that job, you know, pretty
much until his death. So for three decades.
And he also, while at Cambridge, worked in the area of alchemy.
And you can make an analogy between alchemy and Bitcoin and,
and here it is in this table. Alchemy has this sort of raw
material that they want to transform into gold, and they
use some sort of Crucible or furnace, and then there's a
catalyst which they called the Philosopher's stone.
So maybe the idea was, could you turn lead or a base metal into
gold with some alchemy process? And so transmutation into gold
and their quest for eternal value.
And it was also tied in with the quest for immortality.
Well, Bitcoin has some analogy to all of this.
For US, electricity is the raw substance and the Crucible are
the ASIC miners. And of course on that, there's
the catalyst, which is the Nakamoto consensus.
And that is what enables the miner, which is just a piece of
hardware. It's true that the hardware is
optimized for running hashing, but running hashing is is not
enough to complete everything you need to run the full
Nakamoto consensus. So that's the catalyst that then
at the other end actually creates Bitcoin and then puts it
out, you know, for eternal value on hard capped and incorruptible
Ledger. And the idea of energy is
digital money has been around for 100 years.
And I think the earliest reference I can find is this
Nobel Prize winner in chemistry, not alchemy, but chemistry,
Frederick Saudi. And in 1921, just over 100 years
ago, he published a little pamphlet called Wealth, Virtual
Wealth and Debt. And he saw wealth is originating
from the transformation of energy.
And he noted that debt grows exponentially.
Real world wealth has growth and decay.
And he called for an energy currency.
And people are more familiar, I think, with Henry Ford's calls
for this, something like this in the 1920s.
Buckminster Fuller in 1969 talked about it.
And then the whole process that led to Bitcoin kind of started
with David Chalm and digital money in the 1980s.
And there are many, many contributors over more than 4
decades that, you know, led to the publication of the white
paper and end of 2008. Now a reminder that power laws
are not exponential and they're not compound annual growth rate.
We find, you know, in physics, we find power laws everywhere.
There are 4 fundamental forces. Two of them are kind of pure
power laws, gravity and electromagnetism.
It gets a little more complicated with general
relativity, but if you take Newtonian gravity, it's just a 1
/ r ^2 force, and electromagnetism is similar.
And then the other two forces are nuclear forces are strong
and the weak force and they have power law components as well as
well as some other components. That weak nuclear force has a
decay component because it governs decay.
And then the strong nuclear force has a very strong spring
component that whole confines the quarks.
And here's some other examples. The first one is just the
simplest kind of power law for a black hole that the radius is
proportional to the mass, which is not what you expect for the
Earth, right? But this is how it behaves for
black hole is the event horizon radius.
It's just a linear relationship. It's the kind of the simplest
kind of black hole. If you have two that are
spiraling around each other and eventually merge and create
gravitational waves, then you have a steep power law where the
time to the final merger increases very, you know, it's
very sensitive to the distance and so it's very long as the
distance is large. And a couple of others.
The last one is the the Higgs force and it shows both square
terms and the 4th power terms. So you can get rather steep
relationships in physics. And that's what we see with
Bitcoin. It's a steep power law.
Here are two graphs to show the distinction between exponentials
and power laws. I think that is very interesting
and kind of like attention or discussion type of of sense that
these ways to look at like traditional finance and and, and
you know, traditional assets is not necessarily the way you
should look at Bitcoin apparently, right?
Like that is why I'd say that the power laws of discovery is
like, OK, that this this Bitcoin thing is behaving very
differently than than traditional assets.
Like how, how, how people looked at that before.
And, and the, the foundational explanation for that is that
Bitcoin is first and foremost on network.
It's a technology and it manifests as a network and it's
actually a meta network. There are many different sort of
complementary networks. Just for example, there's a
network of minors and there's a network of full nodes and then
you have exchanges, you know, and so forth.
And so you get network effects. And what we know is from
communication networks, and this is true for the Internet
certainly, that you get these kind of square effects where the
value of the network grows with the square as the network scales
to more and more users. And it was true for the fellow
telephone system. And it it was true during the
history of, of the Internet. It followed apparel law and it
eventually saturated. So it's, you know, it's
saturated eventually at 5 billion or whatever.
Yeah. And really, Bitcoin is the
Internet plus money, right? And it's so this power law that
it scales is rather similar. But we also find the power laws
in the things like the hash rate.
So it's not just the price, it's the address count.
It's the price, it's the hash rate.
And some other attributes show this kind of parallel behavior.
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That's gemini.com/brom. So it's exponentials have a time
scale and you can say it's the time scale to grow by a factor
of E, the base of the natural algorithm, or you can say it's
the time scale to double. And for people that know the
rule of 72 in finance, you know if you get 8% compounded, it
takes nine years to double and that'll happen again and again,
it'll keep doubling like that. And so if you plotted log
linear, which is the one in the center of the slide and the
orange line for an exponential, it's just a straight line on the
log linear. So you'll see a lot of charts,
you know, you'll see in finance with stock charts, you'll see
linear linear charts or log linear charts.
You never see log log charts. You never see ones that are
logarithmic in time in, in Treadfi.
Now a power law will start out and look more steep than the
exponential and eventually will be less steep.
Now if you look in the right hand side, it's log log.
And in a log log plot a parallel straight and there's no
characteristic time scale. You have to know what the origin
time is that's relevant or the origin position if it's like
gravity or something. But there's no fixed doubling
time. And I'll show you later how the
doubling time changes and how the growth rate changes.
And so the power law can be more steep than the exponential
depending on the what the actual values of parameters are and
eventually less steep. So in these examples, they're
actually tangential at the same price point, but then they have
very different behaviors in the past and in the future.
OK, let's get into Bitcoin versus gold.
And I'll give you a very simple example that shows that gold
does not follow a power law on time.
And, you know, people would not be, would not naturally think
that since gold's been around for 5000 years.
And we, we, we're sort of comfortable with the idea of
what the gold prices were. You know, we have a rough, most
people have a rough idea what it was 10 years ago and 20 years
ago. And this just shows annual
prices for 15 years and, you know, in the first of the year.
And then instead of age, I've got log 10 of age and I've got
log 10 of the Bitcoin price in dollars.
And then the next column has log 10 of gold in dollars.
And then the final column has the ratio.
So it's Bitcoin measured in gold oz, but it's the log of that
quantity. So if you take the slope, you
know, comparing the third column to the second column, that slope
is 5.84. And that's the Bitcoin power
law. It's about 5.8 slope if you take
the last column and calculate the slope relative to log age.
So log of Bitcoin measured in gold oz versus log age.
Just do this in a spreadsheet. Slope is 5.5 S One reason to
test against gold is well, people sometimes say well what
about inflation? You know, the dollar inflates,
the money supply goes up about 7% on average and this
eliminates most of that objection because the gold
supply increases about 1 1/2 to 2% per year.
So most of the inflation objection is eliminated.
If if you make put gold into the denominator and you still see a
power law and you look at the R-squared, for Bitcoin dollar
it's .96 and for Bitcoin gold is .94.
So it's very, very high in both cases.
Now if you try to do log gold, power log against log age, you
know, you get a slope, you have to, you're going to measure
something. But the R-squared is very low,
it's .28. So it's, it's not a, it's not a
good fit. And in fact, it fits well with a
7.4% compound annual growth. And and as a reminder, the
R-squared is the yeah sorry amount that measures measures
how statistically correct. It's the quality of the fit.
It's the correlation coefficient squared of the.
Data you're you're trying to correlate, right?
Right. This is just ordinarily squared.
And so it's looking at how you know how well the two series
align and it's very high. You know, 9495 is a very high
correlation. It's a rare thing to see.
Yeah, 1 is the highest 2/2, right?
Basically 100% it would be 1, right?
That would be like a perfect perfect correlation basically.
Right. OK.
And so gold, your point here is that gold versus dollars there
there is no way to predictably well to to predict.
You can predict on long periods. On very long periods, it keeps
up with inflation, right? Yeah, roughly.
So it kind of keeps up with the money supply.
The money supply grows at 7%, and gold has been growing at
about 7%. And some years it'll do much
better and some it won't do, you know, and and it's got a lot of
volatility, not as high as Bitcoin, but over 50 years, you
know, you see the same sort of thing.
It basically keeps up with the inflation.
I mean, you know, to first order.
Obviously gold supply is also increased.
Some Bitcoin has work to do to prove itself.
Too. Yeah, exactly.
Yeah. Yeah, right.
That that you're invested in gold.
Yeah, I agree. OK.
This is just a graph of of what I showed before essentially.
So you can just take these annual points and you can see
the Bitcoin measured in gold. This is again, it's a log log
chart. So what we're seeing on the X
axis is the log age. So where it says 1.0, that's 10
years of age. So that's beginning of 2019,
right? And now we're a little over 1.2
in, in log 10 terms. And so that, you know, first
thing to notice is the points get closer and closer, right?
Because a year is getting shorter and shorter on a log
chart, right? And actually, if you're going to
do the regression right, you would have points that are
equally spaced in log terms. And I've gone back and done that
for, for Bitcoin dollar and it doesn't change the results.
We've got enough data that we're determining it well, but this
just gives you a feel and the, the band is sort of the, the
uncertainty. And you can see, you know, the
points mostly lie within the band.
That's the one Sigma ±1 standard deviation.
So Bitcoin is a power law in gold.
It's a little bit less than you see in dollars.
It's about 5.45 point 5, depending on what data you use.
This is just the correlations between Bitcoin and stocks and
gold, and it shows that Bitcoin has been more correlated with
tech stocks than it has with gold.
But the first thing I want to call your attention to is the
chart in the upper left. And it goes back to 2011, a
little bit earlier than 2011. Bitcoin went up in value by 100
/ 100,000 times. Gold went up in value by less
than three times. You know, you just don't hear
the gold bars talk about this too much, right?
They kind of like to look at the most recent year when they've
had a good year. But if you look at the the vast
sweep of Bitcoin history, there's just been no contest.
Bitcoin has just been displacing, displacing,
displacing. In terms of relative terms, it's
gone up over 30,000 * 40,000 times in relative to gold.
Now it's about 20 oz. And at one time it was like a
thousandth of an ounce. The slide below that shows that
this is the Grainger causality test and it shows do Bitcoin.
Prices or gold prices lead, and in fact it's got both curves and
the blue curves is testing whether Bitcoin leads gold.
And here you want to be low on the slide.
You don't want to be high on the slide.
This P value is just what's the probability that you can produce
this with random numbers? And so the Bitcoin gold blue
line, the numbers being high means that there's no
correlation or causality here of Bitcoin prices leading gold
prices. But on the other hand, the gold
colored does gold leading Bitcoin by anywhere from 2 to 12
weeks. They're all below the 5% level
and that's where it becomes statistically significant.
So gold prices do lead Bitcoin and it can range from a few
weeks up to 1/4 kind of a thing. So that's kind of something
we've had going for us. It gives us a little hope that
we'll maybe see some shift in the next year, better
performance for Bitcoin. And the slide on the right shows
the rolling 52 week correlation of Bitcoin and gold.
And you can see sometimes it's positive and sometimes it's
negative and it moves around a lot.
It's hard to conclude very much about that.
It's an unstable kind of relationship.
And this year they've been, you know, not not too well
correlated. OK.
This shows a more complete Bitcoin, Bitcoin versus gold
regression. And it shows both ordinary least
squares tests and quantile regression tests that are done
at 5 different levels of point 1.3 up to .9.
And those are the dashed lines. And then the solid line in the
middle is the ordinary least squares test.
So if you look at the line that's green dashed, it's close
to the solid line. That's the median in the
quantile regression, and it turns out they both give a
parallel index of 5.3 and the uncertainty there is small just
point O2 and R-squared is high, it's .94, but the trend values
would be much higher than we see today.
They should be around in the high 40s or low 50s oz based on
that. Be double almost.
It should be. Actually we're down at about 20
oz right now. So it should be because gold
keeps hitting new highs. I think it just hit 4400.
So we're around 20 oz and we should be above 40 just to be at
the power launch, long term power launch trend.
So we can say well, yeah, OK, we're down below the 10% level.
It's a little bit concerning. It's certainly weak performance.
And if that continues, you would start to measure a lot somewhat
lower power law index. And in fact we've done that over
the last couple of years. It's it's lowered a little bit,
but it doesn't break the idea of a power law.
And so that the assumption would be we'd get some restoration and
it would, you know, revert at some point.
We've actually modelled how this behaves and there is this kind
of springiness to it. And I've done this in both gold
terms and in dollar terms where you get restoration back, you
know, from a bubble or from a through if you get restoration
back, yeah. I have a question about this
because I've, I, I think what is, I'd say difficult to
understand. You just correctly said, you
know, Bitcoin still has to prove itself right.
I I think that is more like conceptually rather than price
wise. I mean they're kind of
intertwined, right? Like if you understand that
Bitcoin is superior, has superior characteristics over
gold. That's that's one.
But like acting on that is 2 right?
So I think there's, there's just a big learning curve there for
for a lot of people, right? And then you based on that, it's
either people buy Bitcoin or they buy gold, right?
Other people don't understand that, you know, just like gold,
you shouldn't really trade or leverage Bitcoin, although you
can do that 24/7, 365, right? That's why we also see a lot of
leverage and you know, increased volatility, etcetera.
So I think that's also like another part of just the
understanding of of Bitcoin and how how people handle that.
So there's kind of like this qualitative understanding
learning curve. And then when you look at like
the, the, the, the, the, the power law rights for me, the
essence of the power law is, you know, apparently there's a
predictable way to look at Bitcoins adoption in, in several
different dimensions, including price, like you said, hash rate
notes, etcetera, right? Like that, That is the discovery
and the, and the, and the, and the connected theory.
So when I look at this slide, you know, the conclusion
basically is Bitcoin is undervalued or miss or
mispriced, whatever definition you want to you want to use,
right, based on the power law that up until this point has
proven itself with the also this high, high R-squared, right?
So I think that is what together creates this conclusion of
Bitcoin. Bitcoin is currently
underpriced. How can other people think true
that in in ways of yeah, kind of like judging the math versus the
sentiment versus the perception versus the, you know, I don't
know, like the understanding of of of of the market.
Does that make sense? Yeah, absolutely.
And you know, the market is right in the short term and the
math is correct and long term. You know, I think Buffett talked
about voting machine and weighing machine, right, which
is actually goes back to Benjamin Graham.
So the the voting machine is the short term, it's the market.
Now the weighing machine is the math, the long term.
What is the fundamental for Bitcoin?
The thing to keep in mind, of course, is the volatility is
very large. So the volatility is sitting
there in logarithm space and everybody gets all excited
because Bitcoin moves $5000. But that's noise. 1 standard
deviation is .2 in log 10 and .2 in log 10 is a factor of 1.58, a
multiplicative factor of 1.58. So if fair value were at 100,000
and now it's higher than that, but if it were at $100,000, then
to move by 1 standard deviation you'd have to get to $158,000 to
move to 1 standard deviation. On the downside, you know you'd
be below 70,000. So that is a normal.
It's normal to be in the 70,000 to 150,000 kind of range.
This is another way of looking at the persistence of Bitcoin.
It's called the Hearst exponent. I'm not going to go through the
method. I think I did it with on your
show half a year ago. What the method does is it
changes the window size of the data that it's looking at, and
it measures the full range, the full range of variation in the
data, and also measures the statistical uncertainty, the
standard deviation. And it looks at the ratio of
those two. And it finds as you grow the
window size, that actually that ratio has a power law behavior.
And normally this Hurst exponent is between zero and one.
And if you have a random process, it's going to be 1/2.
And if you have something that's reverting to mean, it's going to
be near 0. I took 10 years of the SPY,
which is the SP500, you know, ETF and it has a value of 0.18.
So that indicates strong mean reversion for Bitcoin.
And you have to do this with the residuals.
And normally what people do is they just take the kind of log
variation day-to-day because they're assuming a kind of
compound annual growth process. But for Bitcoin, if you take the
residuals after taking out the power law, then you find that
this Hearst exponent is .9. And doesn't matter whether you
do it in gold or dollars, daily or weekly, it gets stronger.
If you do it daily, it's very, very high.
It's like nothing one sees. So this is a measure of very
long term persistence. It's another way of saying that
Bitcoin has embedded alpha, if you will.
What? What does that mean?
It means long term outperformance.
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OK, now I want to look at the Kelly criterion for optimal log
wealth. Shannon is the guy who figured
out entropy for information and John Kelly was one of his
associates at Bell Labs in the early 50s.
He he published his his criterion and it's derived from
the entropy of information criteria, but it's applied to a
kind of betting process or investing process and.
This last time right like the criterion is, is a way to to
calculate your your portfolio allocation over a certain amount
of of time. So I'm right and I'm going to
show you some update and comparison.
Bitcoin versus gold, if you had a 2 asset portfolio and also a
different method that would include stocks also.
And you could do this with master against Bitcoin if you
want to, but I'm not going to show that today.
And this is just think about what's minted, what's mined and
what's a printed, right? Bitcoins minted.
Gold is mined because it's not really mined.
We don't, we don't dig into the ground we minted.
Yeah, these are 10 years of annual returns with 16 different
asset classes, half of which are debt kind of classes.
So they don't have huge returns. So I'm going to throw those out
in the next chart. But Bitcoin from 2015 to 2024
won 8 out of 10 years. It was #1 in these amongst these
16 asset classes. And you can see gold bounced
around. And of course, Bitcoin was the
biggest loser also too, too, when it had the the lowest
return. The gold is in dark green, so
you can see where the gold was. But now I, I took out the bond
once because they don't return a lot and I so I've got 8 asset
classes here, Bitcoin, broad tech, large and small caps, real
estate, gold, commodities, international equities.
And I apply this Kelly criterion.
Now the way it works is you look at not how many years did it
come in first place, but you look at how many years did it
out did it outperform. So the first thing you can do is
just cast it outperform cash and you can do that with or without
a risk free rate for the cash. And for Bitcoin it was 8 times
out of 10. So this probability, which is
the first term in the Kelly fraction F is 0.8.
Then you have to subtract out your losses, but you divide your
losses by the edge that you have.
So you actually get 2 edges in this formula.
The first edges if you win more than half of the time.
And that's the P and the Q, right?
The P is the win percentage, the Q is the loss percentage.
P + Q is 1, but the B is the ratio.
If your wins are bigger than your losses, then that's a
second advantage. And so if if your wins and
losses were equal, you'd say P -, Q.
So if you had a processor where you won 70% of the time and you
lost 30% of the time, P -, Q would be 40% and your optimal
portfolio allocation would be 40%.
That would be your your the maximizing the growth of your
log wealth. But you can allocate more if
your wins outperform your losses.
Well, for Bitcoin, they do very substantially.
In fact, the average win was close to 300%.
The average loss was about 70%. So this B ratio is 44.2.
That means the F is 75%. So it would tell you that in a 2
asset portfolio with dollars, you can allocate 75% to Bitcoin.
Now in the next two columns I make adjustment for earning AT
bill rate and I set it to fours or risk free rate.
It's still 75% in the case of Bitcoin.
If you go through this for technology, you had a high win
rate in recent years and it comes out to be 6065 percent,
66% depending on whether you apply the risk free rate.
Small caps didn't work out at all once you've applied the risk
free rate. Large caps were OK because
they're not that different from broad tech lately.
And after the risk free rate it would be a 40% maximum
allocation in A2 asset portfolio with dollars.
Real estate was a loser once you risk free rate, and gold did
well. But once you subtract the risk
free rate, you have a problem. Because look, gold had an
average win of 13%, but it had an average loss of 8%.
If you subtract 4% from this, it's down to 9.
And if you subtract 4% from, this is -12 right?
Yeah. So even.
Clue include this right, because the risk free rate is of the 4%
of bonds is, is is like a like a classic place to park your money
basically. So that that's, that's yeah.
So that's kind of a, an assured return of 4% that you use as a
let's call it an existing alternative, right?
It's not necessarily the a good alternative, but it exists if
you want to take air quotes less risk.
You know, that's another debate if, if T-bills are risky, but
you know, nominally there's a 4% return.
And you actually have to allocate some to cash because it
it OK, you take the best asset, Bitcoin, you shouldn't be
allocating more than 75% according to Kelly.
So the rest should be in cash, just T bill waiting for an
opportunity when the price pulls back.
And the idea is that you rebalance, right?
And the rebalance window is very, very important.
So this is for an annual rebalance window.
And for Bitcoin, the longer your window is, the higher the Kelly
fraction. Actually, if you use four years,
you'd have a Kelly fraction of 100% because every four year
period has been a winning period.
Yeah, but the problem with doing that is you'd only have four
data points, so your statistics are poor.
So the rebalancing is in bit in between, right?
So if, if the, if the, if the, if the price goes down, you go
below your 75% total allocation in your portfolio and you can
use the part of the 25% cash or well the cash you had to to to
buy more Bitcoin to get back to that allocation, right.
OK. That's that's the idea, yeah.
OK. Now Bitcoin, you know, we've
talked about a little bit about how the return actually drops
with time, and we had some crazy years in the beginning, right,
that we don't expect to see again.
If we just go back to that slide for a moment, this third one,
14149%, yeah, we don't expect to see that again.
Here's another one, 300% Yeah, we might see that.
So I've also calculated with five years only just to mitigate
or see how strong the effect was.
But it's great still. I mean, the B drops to 2.4, but
the the computed F is now 71%. It didn't drop that much at all.
Now if you were to pick just three assets, Bitcoin, broad
tech in gold and look at their F values and then normalize
everything, it would say 52% Bitcoin, 42% broad tech, 6%
gold. Over 10 years, right?
That's good to add. Over those last.
May. Yeah, exactly.
Yeah. OK, OK.
What about this year? This year we we have to adjust
for and we say, OK, we didn't get another.
You know we're down this year, right?
So instead of being 8 out of 10, we're 8 out of 11.
So that drops us to 73% win rate.
But interestingly, the B value goes up because the loss here is
based on a totally two years and they were very big losses,
whereas this year we've got a loss, it's very small.
We're down like 5% or something for the year.
Yes, we average those 3. Instead of it being a 70%
average, it's like a 40 or 45% average.
So then the ratio with the winds, the winds are still the
same and now the B value goes up.
So the F value doesn't drop that much, but it does drop to about
.6868% allocation. That's a 2 asset portfolio
against dollars. And if you just say, well, just
the past four years, then you do get a little lower value, you
get about 60%. OK.
I think this is a good noob question.
You said the wins, the the wins are still high, right?
And I think a classic like investment finance disclaimer is
right like past performance is not indicative of the of the
future, right. So I think a good noob question
would be why are you using still the 292% of of the past?
You know, like how, how can you take that towards the future, if
that makes sense? That is sounds like a noob
question. So it's a good question, you
know what I mean? Like how?
Why can't you still use that number?
First of all, you've got statistics of small numbers, so
you kind of have to live with what you've got, right?
And over time they roll off. But if you don't like that, look
at the last row. That's just five years.
And so they're the win. The average win is 155%.
So it's only half as large, right?
And the average loss is 64%. And then if I fold in, you know,
that's a ratio of 2.41. But if I fold in the latest 5%
loss, that ratio actually jumps up to 4.4 because it gets weight
weighted in with these other two numbers that gave you this one.
Actually that's one number. They gave you this one.
So the 64% against average with the -5%.
And then this ratio goes up. So it does drop from .71 down to
.59. But it's not an enormous drop.
But it gives you a feeling and you can, you know, try these.
And then, but I would say the one thing is people should not
go out beyond the edge because the curve kind of rises like
this and falls sort of steeply if if you get out over your skis
on this. It's basically quantifying kind
of like the risk appetite for a person who holds Bitcoin
depending on the time frame that they look at Bitcoin, right?
So if you look at it, if your outlook is five years, you know
it's you're, you're probably towards the 59%, right?
If your outlook is 10 years, then you know you could take the
risk again air quotes to like 75% allocation if if that is the
frame of reference that you constantly want to use, not only
looking back but but also looking looking forward, right?
Right. OK.
And this actually shows how the Kelly fraction develops for two
cases. 1 is where you have annual window and the other is
you have quarterly window and it's a little bit longer series.
It goes back like 14 years in the quarterly case and about 12
years in the other case. And yeah, I mean sort of 1213
years and you can see that in the beginning it jumps up to
100% and it drops down and it moves up again depending on when
the bubbles hit etcetera. But it settles down and for the
annual it ends up essentially at at around 75%.
This does not subtract out the risk free rate, I don't think.
Yeah, I don't think it does. And then the other one settles
at around 50%. So if you're using quarterly
rebalancing, you are going to have a lower allocation than if
you use annual rebalancing. Yeah, yeah, yeah.
Because you're just more likely to win if you've looked at a
full year. Yeah, this is illustrating my
point, right. Just depending on how and when
you want to allocate, right? Or like kind of like rebalance
your risk, so to say. It's funny because this shows
that huddling, right? Just huddling is, is, is is
giving you a better result also, right.
And that, that I, I find one of the most fascinating things
about this, like the math backs up the meme.
So just huddle and you don't have to try to like time, time
the market and, and, and constantly do this, this, this
reallocation. But again, like that is
connected to what I said before. It just depends on your
understanding and your own conceptualization of what
Bitcoin is by itself. And then also compared to all
the other things that you, you could invest in, right?
So I, I, you know, there's, there's different pieces of
understanding and, and, and weight also that eventually help
a person create their own investment thesis and allocation
and and risk appetite. But I, I think it's very
interesting that the the math connects back to, to the meme.
So huddling is yeah, preferred that sense.
Yeah. This is a different technique
and it gives somewhat similar results.
It's called MVSK and it looks, it's kind of like sharp ratios,
but it adds skew and kurtosis, which is the S and the K and you
do the same thing. You take out the parallel trend
and then look at the residuals. And this is showing the
percentage in a three asset portfolio actually with both
gold and stocks. And even then, you're above 50%
in most cases. And what you have on these two
axes is the skew and the kurtosis weight.
So if you look at the one for the beta, that's how much weight
you want to put on the skew, you know, which is the moment.
This is envy is mean and variance, right?
So these are the moments and the skew is the next moment.
And then how much you want to weight the kurtosis.
And that's the Lambda. So if you just say 1:00 and
1:00, you're going to be 58% even in a three asset portfolio
with with golden stocks. OK, But I want to move on
because I want to get to the log periodic stuff.
That's just Buffett saying don't over diversify.
You know, if you know what you're doing, you can huddle.
OK, this is Bitcoin against the dollar.
It's a log, log chart slope 5.75.9 depending on whether you
use ordinary least squares or whether you use quantile.
There's just two parameters, you know, there's a constant in
front of everything. And then the K is the power law
index, which is very similar for the two techniques.
And then you can see the projections there for December
of next year. If you're on the mean or the
median for the QR, you'd be around 160,000 a year from now.
Yeah. And maybe maybe that's not this
slide, we probably have another slide.
But what you sent me upfront is that you know, when you when you
talk about the existence of of bubbles and basically put aside
the four year cycle frame and you just look at the at the
math, right? You say your timing suggests
like a very strong 2026 in the in the second-half, like leading
towards a peak more like mid 2027 and then something similar
for gold. But now because we're under the
the power line trend, basically what you're saying, what the
math shows you is that that we are, yeah, looking towards a
catch up back to to the trend and then and then back above,
right. Is that correct?
Right. And, and to set up that
discussion, note first that the first bubble was in 2011, the
next one was 2013. So that gap was a little over 2
years, nothing like a four year gap.
Then 2013 to 2017. Yes, you had four years, 2017 to
2021. It actually was less than four
years. It was more like 3.3 because the
left shoulder in April is higher relative to the regression line
than the right shoulders. Yeah, that was more like a
double top also. Right, double top.
OK, how to calculate the growth? The formula is you take the
ratio of ages, you raise it to the K power and you subtract 1
and that gives you the percentage growth.
So if you want to look at one year is t + 1 / t raised to the
K power -1 And if you do what's called a Taylor series
expansion, you kind of only need the 1st 2 terms and first term
is just K / t. So if K right now is 5.8 and our
age is 17, that gives you 34.6% point 346.
And then there's a second order term that goes as a square of
that K / T, it gives you another 6%.
So if K is 5.87, you're getting about 41% as the projection for
the next year for the main trend, how much you would expect
the main trend to grow? So that's kind of the fair value
price on trend. Yeah, there's a trick that you
can do. If they use the OLS index,
that's 5.68. If you increase the age by 50%,
you're raising 1.5 to the 5.68 power.
That's exactly 1010.00, just by coincidence.
So increase the age by 50%, you expect 10 times the price.
So if we're age 17 now you add 8 1/2 years, you're at 2034.5
because we're at the beginning of 2026 and you expect over
1,000,000 on the power log trend.
Because right now the power log trend is about it's a little
over 110,000. And then another 12.75 years
after that, you'd expect to be a 10 million or above 1 trend and
that would be early 2047. Yeah.
And on trend would also be. Is that the same as fair value
then? Right, because it's?
Yeah, that's my fair value quote UN quote.
Yeah, exactly. So the what does it say 2034
like half, Yeah. So I I think the estimates are
kind of between 2032 and 2035 depending on like it could hit a
million before, right, if it's in this above trend, right.
But basically you're saying within 10 years we are at a
million on the on the trend. Doesn't mean it could be a bit
over, doesn't mean it could be a bit below.
But just like you now calculated I think around what was the the
fair value now it's it's like over just over 100, right.
That would be the fair value at that at that point.
Yeah, right. And this sorry if you go back.
Yeah. And then because this kind of
shows the speed that you talked about before, right?
2047 the, the, the, the 10 million.
So although when you look at the power law and you, you see the
diminishing returns in percentages, the, the absolute
nominal amounts are are at at at a totally different level,
right? Like so the.
Drops kind of a couple of percent as you go from age 17 to
age 18, that kind of a thing. Yeah, but I mean more the the.
I don't know if the correct word is exponential, but like just
the going from the one to 10 so you see it takes longer.
Yes. Right from 8 1/2 to 1275.
But it's still predictable. That's the whole point of the of
the of the power law. And it's still rapid.
This just shows the stability of the power law over time.
It started out noisy, you know, it shot up.
There's the big bubbles in 11 and 2013.
And so it shot up above 10. It's just if you measured the
power law at any given time and you only had the date up till
then, what would you see? But by 2016, it dropped out to
six. And then since then it's been
relatively stable around, you know, 5.65.7, 5.8 kinds of
numbers. Right now it's 5.68 for the
ordinarily squared. And this band around here shows
the uncertainty which has gotten smaller and smaller.
This shows how the volatility has come down.
Sorry, that's what I remember from our last conversation is
that really struck me that that that is also why I say like, OK,
this is not an opinion, right? It's not an opinion.
Like sometimes you see people to say, oh, the power law is an
opinion. But this actually shows almost
like a constant in, in the level of certainty slash slash
uncertainty, however you want to define it, right?
But I think that actually is extremely interesting.
Like this is not a random, random formula or calculation
that that you're following. Apparently it is something that
stays predictable within a certain range, right?
Like there's the. Yeah, it's, you know, to go into
the physics of it a bit and the physics and networks, if the
networks have structure and they develop like fat nodes, you
know, where you've got some fat wallets or you've got some
exchanges and so forth. And they have preferential
connectivity that tends to give rise to parallel behavior.
And they, it's stabilizing the system actually so that it
doesn't behave in an exponential manner because exponentials run
away on the upside, but then they crash on the downside.
So it's exponential can be in both directions, right?
And this is like a system that's operating kind of at the edge of
stability, but it's self regulating.
And you can model it as having this kind of spring like motion
where you get liquidity waves come in, it moves up, you get a
bubble, but you get liquidity absorption and you get
restoration back to the power law.
And you could model it with dynamics and include a spring
like term to restore. OK.
This just shows though that the volatility has come down and not
for the points that are below the line.
If you split into like you can split into two bell curves, 1 is
for the bubbles and one is for the core behavior.
But essentially, and what you find is the volatility has been
flat, the green line for their core behavior, but the bubbles,
it's been coming down. So it, you know, it was a factor
of 3 and it's cut in half to a factor of 1 and a half, 1.6.
This shows the same kind of thing.
It's basically come down as a reciprocal of time.
And this is now looking at the bubbles, setting a threshold and
saying you have to be above a certain threshold of .2 in the
log 10, which is the current 1 standard deviation and you need
to have at least a 10 day duration.
And when you do that, you see a bunch of tiny bubbles and anti
bubbles, you know, both, both directions, but they're only
four that really stand out and they're the ones we all know in
2011, 2013, 27, 142021. If you take out the power law,
you get the lower chart and you can approximate the energy by
drawing these triangles. And the the triangle height and
width is just measuring what's above the green dashed line.
So it's only if it went above the threshold that it's
represented by the triangle. And you see the other things are
just noise pretty much, although there are fewer and fewer even
of the noisy ones. And the energy has gone down in
the bubbles and the distance between them is about 2 1/2
years, four years and about 3.3. And we've had no bubbles since
mid 2022. So I believe the four year cycle
is invalidated by two things. One is 2011, which everyone
likes to ignore, and then that we've had no bubble in 2025 S.
Translation would be like like bubbles up and down are noise
right? Then the signal is basically a a
a constant growth. You don't.
There is one. Direction you don't get big anti
bubbles. In other words what you get is
you get bubble motion up and then restore to the power law
and below the power law but not as far.
So if you say your threshold is is .2 in log space, you don't
get below that very much. In other words, the right tails
a lot bigger than the left tail, right.
You can fit a bubble with this large periodic power law that's
been popularized by Cernad for a variety of phenomena, including
financial markets. This was a projection I did in
July and it said we could get a peak in in mid-september.
It turned out to be early October.
And so that was one case. When it's right, it's pretty
tricky to use this thing because it has seven parameters.
So you need to always be rechecking and tested on
different scales. But it's it's this long periodic
parallel that's was the motivator for how you fit the
whole series of bubbles. And actually Giovanni did this
back in 2019. And so we only had the 2011,
2013, 2017 large bubbles. And he fits something like this,
but it's a slightly different formula because with the one
that I showed you before, it works up to a critical time.
And then the idea is that there's a crash.
And we kind of saw that, right? And in October, in this case,
you just run time forward and so you get these waves going
forward. And so he matched the 2011,
2013, 2017, not necessarily the peaks, but when they occurred
and his projection that the next one would be at age 18, actually
17.96. And he's got it right there.
He's got the X value, it's the log of number of days, and it
works out to be 18.0 or 17.96. Now I've now got six years more
of data, so I've got a somewhat more precise reading on this,
but the spirit is the same and the idea that it's long
periodic, that the spacing interval grows by roughly a
factor of 2 and we the age went from 2.45 to 4.9 to 9.
And so you can see that the bubbles are getting further and
further apart. Yeah, this is interesting,
right? Because this is what people also
talk about with regard to gold that I I don't have the the the
periods on top of mind, but there was basically was that the
introduction of the gold ETFs. I think that was it that that
they also talked about like gold basically had a seven-year kind
of like a bull bull run. I found it interesting that
there is a certain behavior that that you now show with Bitcoin
also that it's kind of like the, the volatility dampens and the,
the space between the, the, the bubbles gets longer, which in my
mind would translate to a good thing that that means like
Bitcoin is actually an established or becoming an
established asset. That is a, that is a like in, in
a serious consideration for for people to, to, to put in their
portfolio, right. Like that, that whole argument
of, oh, it's volatile, etcetera. That well that's basically this
graph I think showing that that that is a non argument
basically. Over time, I mean, you know.
Exactly you. You, you buy Bitcoin at the
volatility you can handle, right?
And so higher tiers of capital come in and their institutional
capital. There are two things that they
expect that they understand the volatility and can manage it.
And in general they would like lower volatility.
And the second is that Bitcoin is expensive enough that if they
buy some that it makes a difference to their portfolio.
And This is why Apple hasn't bought a lot of Bitcoin yet
because they cannot change their life by buying Bitcoin.
Sailor could change his life because he had a small company
by going all in. But as Bitcoin goes to higher
and higher value, it attracts higher tiers of capital.
And this is actually a dynamic behind this long periodic
behavior. You can model it with a simpler
kind of equation that has five terms and it's got this large
spacing parameter, which is Lambda, and it relates to the
angular frequency, which is in this cosine term.
It's cosine of a Omega, which is an angular frequency.
And then there's a log of the time and there's a phase term.
So that's essentially what we're modeling to time when when the
peaks happen. So in order to do that and
extract that Lambda value, I did a 4A transform of the data.
First I determine the residuals. I did this for both gold and
dollar, but this chart is Bitcoin measured against gold
because I thought, well, you know, I don't know for fun,
let's try it against gold 1st. And so these are just the
residuals. The power law is taken out.
I do the 4A transform and I find that this fundamental parameter
Lambda has a value of two point O 7.
It's a 5.3 power law as I showed you before.
And you can see the residuals are the peaks and the bubbles
are quite well defined and sharp.
And then the red lines are the fundamental behavior with this
parameter, Lambda equals 2 point O 7.
And they you can see they line up right with the peaks and
there's a face term and it's not too hard to match the peaks.
But once you've lined those up, where's the next peak?
Well, it's out here way on the right.
It's at age 18.4. So that's May twenty 27th and
that's when you'd expect the next fundamental peak.
So that explains 2011, 2013, 2017.
Well, what about 2021? People that play music know that
you know you have C and then you have the next octave is a
harmonic and you can have subharmonics.
So you could have the lower octave and those are slower, so
the first harmonics faster. And you see three, they actually
happen in this case, they happen in between the red ones.
They happen logarithmically in between.
So there's like the square root of this two point O 7.
So there are three blue vertical lines marked and one of them
lines up with the 2021 bubble and the other two don't work
that well. They sort of kind of fit against
local peaks, but they're not very large amplitude.
And then the one sub harmonic is only one because it's slower
appears and it does line up with a little bubble roughly that
happened in in the 2019 time frame.
So I I suggest that this is a better explanation of what's
going on than a four year cycle. I think yeah, it's interesting
because some some people would argue that's kind of like the
the the 2021 bull market, you know, that people would expect
when they follow the four year cycle was kind of like a muted
thing kind of went up crazy. But it was it was also a weird
period with with COVID and and stuff like that, right.
So I think that's a general thing that people say like, oh,
that was kind of like a muted bull market.
So that's kind of what I'm recognizing here.
And on the on the other side, what people kind of like say now
is, OK, maybe 2025 was almost like a bear, bear market year,
right? Like that there hasn't even been
a bull type market let's let's call it, you know, a right a
rising market that I don't think bull and bear kind of applies to
to this. But I think that's what the
graph does show, right. There's like this very long
almost yeah, like a like a chop solidation type behavior here
that I think a lot of people discuss.
But you're showing that this is actually part of the
mathematical behavior in a sense that is not a logical to to see
this, this kind of kind of what how I would translate this in in
my head, right. So people create certain
narratives to talk about what is happening.
Whether that's correct, correct description or not, I don't, I
don't know. But I think it's interesting
that the math does kind of like match how people talk about the
the market that we saw like 2021 or like what is happening now.
And when I look at this, I think that then there's some
interesting times to come, right?
Like the comment 12 1618 months? 18 months, yeah.
Yeah. Now, this value of 2.0 is
characteristic of a lot of other physical processes.
If you look at earthquakes, a typical land is 2 to 2.4.
Rock fractures it's around 2. Fluid turbulence cascades around
2, tremors and volcanoes, and even the financial bubbles.
If you look at the tail distributions even in stocks,
you can find these in ecology and in statistical physics.
They all kind of around 2, or you know, not too far removed
from 2 for this parameter. I also did this with a technique
called wavelets and it found several wavelets, but they were
all between 2.0 and 2.1, so they were consistent.
Broad idea is that you're seeing the capital inflow come in
discrete hierarchy tiers and it takes longer and longer for
Bitcoin to have its price go up by a factor of 10 with the power
law. So it takes longer and longer to
get these big new entrants of capital, whether it be the
treasury companies or the ETFs or earlier on hedge funds.
And so when the price gets to a certain level, it kind of opens
it up for the hedge funds to come in or later for the
treasury companies come in. And that helps drive, you know,
a bubble. And people kind of jump on when
they see the price momentum as well.
But eventually that's over, plays his hand and it comes
back, you know, to blow the power law and then catches up to
the power law. When you say this is such a
clean result, how, how surprised are you by the fact that Bitcoin
again apparently behaves so predictably or like according to
almost natural law, right? Like I'd like the power law you
you see in nature and that you know, my conclusion here is that
this is not a random thing, right?
Like this is not a random product that this introduced
into a market where, well, some people might see the value and
buy it and other people's don't, other people don't, right?
Like there's not really, well, there's a surprising amount of
predictability, let's call it that, right?
And, and I think that by itself is something that is very hard
to, I almost want to say create, right?
Like like with just random new companies going on the market
with their IPO and stuff like that.
They are, they are way less predictable than Bitcoin, right?
So how, how, how, how surprised are you by that, right?
Or how special is that? Maybe that's a better question,
right? Like why?
Why should you pay attention when you're seeing stuff like
this? It is special.
Bitcoin belongs to a set with one member.
It's unique, you know, there are thousands and thousands of
companies that trade on public stock exchanges.
There's no crypto that's like Bitcoin.
They don't belong to the same set that Bitcoin belongs to.
And you know, I, I think, you know, we know why it is a
definitional scientific standard for money rooted in energy and
cryptography and built on a network that has achieved, you
know, it's past the level of critical success and it's built
this structure. And really what this is saying
is that it has this structure that's self reinforcing is a
word for it. It's called autopoietic.
It's self reinforcing. It has to live in a larger
environment, but cells are autopoietic.
They have to depend on organs, but they're self organizing.
They have their own rules of self organization and then they
absorb energy and resources from the environment.
Bitcoin is autopoietic. It has its own rules of its
organization, but it it absorbs Fiat into the environment and it
incentivizes humans to go out and mine it, mint it, and to run
full nodes right and to participate in the ecosystem
because of what it does. And it incentivizes new tiers of
capital to come in as it grows its price in this parallel
manner. It's very interesting.
I, I, I googled it, I never, I never heard this term, but it
says the core idea is self creation.
So the system's processes generate its own components and
organization. It's self maintenance.
It defines and preserves its own boundaries.
That's interesting, right? Because that is, that is, you
know, if you, if you think about Bitcoin as this like 3D cube
with the 21 million units and like all the energy is basically
captured in that. But also the protocol has
boundaries, right? There's just a set amount of
rules and that's just it like the rules live in this, in this
system. And then the third one is
autonomy itself producing itself sustaining, not dependent on
external creation, right? And I think it's so fascinating
that a lot of these things come together.
Just the idea of a protocol enabling something that is
digitally scarce, I think it's one thing, but also the amount
of like, like taking physical energy, which is high entropy,
turning it into a low entropy, basically immutable database
that the longer it lives and the more data it stores, it becomes
more trustworthy. So more people are gravitated
towards that, right? They mine and they have wallets
and, and, and, you know, more addresses and stuff.
And so it's, it's all these elements to together that kind
of fuel this reinforcing loop that is, do you have any outside
of human body that, that that's the example I, I saw, right.
Are there any like other systems that are that are like that?
Because it's pretty wild if you can compare Bitcoin to like how
a human cell behaves. Sure, galaxies are like that is
self organizing, self-sustaining, but they also
interact. They interact with their
neighbors or they have, you know, as they form gas comes in,
they do suffer collisions and they'll reorganize when they
have collisions. And, and they show this kind of
same kind of capability that they have a power law in terms
of the continuous behavior of the spiral arms, but they also
have this log periodic behavior and some of the details of the
spacing of those or when you have branching and and those.
So that's another example. It's wild.
Bitcoin is at that level. I just think that's mind blowing
that that again, apparently this is this is so.
Yeah. All right.
Now if you want to know when the peaks happen, you can do this
weird numerology. You could just take this Lambda
and add it to 2009. That gives you your first peak.
You can take Lambda squared, that's your next peak Lambda
cube 2017 Lambda to the 4th power 2027 and then the one
after that would be 2047. Wow.
And the value spacing the, you know, the price in the market
cap spacing is Lambda to the K power and we know K is about 5.7
and Lambda's a little over 2 and it's turned out to be 60 times.
So very roughly you going to see 60 times increase in market cap
as you go from one to the next to the next to the next.
Maybe eventually you saturate before 2087, right?
And you become, you know, hyper Bitcoin happens right?
If we keep going by that time the 2047 takes you to as we said
before something of order 10 million or so, which is over 200
trillion and that'll put it in the roughly won't be quite as
large as the future in two of the US dollar, but it'll be in
the in the same class of that. So huddling is going to be
harder. Hardly is going to be hard,
yeah, but you can afford to sell a small amount and huddle the
rest. No, I I agree, but I think it's
just interesting that the whole idea of your own proof of work
is going to be tested the longer you you, you own this asset,
right? Like can you that that is why I
again, I also love this stuff. It's kind of like if, if, if you
follow the math, that's quite reassuring in a sense that you
don't like, you're just your ego and like your own perception of
adoption or the speed or, you know, however that's mixed with
your wants and needs and, and desires.
You can kind of like put that aside and and and and follow the
math also as a as a guide in in holding and also understanding
not only the direction, but of like the size where this is
going, right. So yeah, I think it's very
interesting. Actually I thought I would close
with this slide. This is the comparison of the
four year cycle to the long periodic bubbles.
So does bubble years down here 2011 through a future
possibility in 2027 and noting there was none in 2025.
And so four year model ignores 2011.
It says yes to 20/13/2017. It was correct for 20/21.
It failed for this year. The long periodic is correct for
20/11/2013 2017. It says that 2021 is the first
harmonic of that same fundamental mode explains why
there was nothing in 2025 and then we have expected in 2027.
So we're right five times so far as opposed to being wrong two
times. And the other thing is you can
measure the RMS error and that's, you know, how wrong is
your prediction relative to when the thing actually happened.
And the case of the four year model, your average error is
over one year, it's like 1.2 years, but for the long periodic
you're only off by half a year. So it's about 3 times better in
terms of the error. Fascinating man like thanks,
thanks so much for for taking me true.
Like I just said, I, I think this for me as a, as a, as a non
mathematician, I think it's very helpful to see that there's like
practical guidance in the, in the math, right?
I think it's amazing you're doing this work and, and I
really also appreciate your time just sharing that here.
I'm going to link to, you know, your socials and stuff like that
so people can can follow you. And yeah, I hope we can we can
do this again, like do some updates, see, see where we're
at, right? Yeah.
I want to thank you for your time.
I really appreciate it. It's been my pleasure, enjoyed
being on again. Likewise, thank you, thank you.
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