Bitcoin Weibull S-curve vs. Power Law update

Simpler is More Beautiful

What about S-curves?

People often wonder if Bitcoin is following an S-curve of technology adoption. For now all the evidence is that it adheres to a power law and the foundational understanding for that is its adherence to Metcalfe’s law that says a network’s value grows as the square of the number of nodes. If the number of nodes also grows as a power law, the result is Bitcoin’s price, it’s measure of value, growing as a power law. The number of nodes, addresses holding Bitcoin, is indeed growing in that fashion, nearly as the cube of the age of Bitcoin.

The net result is a power law with an index between 5 and 6. Price climbs as something steeper than the 5th power of time, or on a log-log graph, a straight line with a steep slope of between 5 and 6.

The general motivation around thinking about an S-curve is that eventually Bitcoin might inhale hundreds of trillions of dollars of value from bonds, equities, and real  estate. And that over time it will run up against the limits of available fiat and fiat-priced assets that are being redirected toward the world’s first scientifically precise monetary technology, that is also the best store of value around since its inception in 2009.

A logistic regression form of S-curve is a poor choice for an asset that has grown nearly 8 factors of 10 in value in just 15.7 years, a factor close to 100 million.

There is a better choice for an S-curve function that applies across a broad dynamic range; it is the Weibull cumulative distribution function (CDF).  It looks like this: f = 1 - exp [-(t/c)^k] where f is the fraction of ultimate asymptotic value, c is the characteristic time scale, and k is known as the shape factor. When t = c, true for all k, a knee in the curve at (1 - 1/e) or 63.2% of asymptotic value is reached.

There is a trick that allows you to perform a linear regression with this Weibull CDF function. Rearrange and take a double logarithm and you end up with a relation defining a variable y:

y = ln [ -ln (1-f)] = k [ln (t) - ln (c) ]

You now have a linear expression of y vs. ln (t) and can regress with ordinary least squares, k the shape parameter is the slope, and -k ln (c) is the intercept, so you can recover c, the characteristic time, as well.

The Weibull CDF has the wonderful property for our purposes that in early years, before it reaches the knee of the curve, the function expands to a power law, namely (t/c)^k = C * t^k to first order.

Comparison of Power Law and Weibull S-Curve for Bitcoin

I performed Weibull CDF regressions for two cases: $1 million maximum value of Bitcoin, and $10 million maximum value. I used block monthly data since 2011 (block months are 4375 blocks in duration). Both have excellent R² values of 0.95 and nearly identical shape parameters of 5.41 and 5.42, respectively.

The characteristic time scales differ, for the first case it is 27.2 years and in the second case it is 41.8 years to reach the higher valued knee in the curve. Bitcoin is currently 16.4 block years of age, so the $1 million maximum case would reach a knee in just under 11 years, the year 2035. And the best fit trend price would be (1 - 1/e) x $1 million or $632 K. For the $10 million case the knee would be reached in 25 years from now, the year 2049, at a price of $6.3 million.

In a recent interview with Robin Seyr I suggested a $9 million price target for 2049; that is based on a power law extrapolation.

For a straight power law fit we get the same R² value of 0.95, and a slope 5.41, identical to the shape parameter (slope of y vs ln t) of the $10 million case. (If we had performed the regression in calendar years the slope would be around 5.7 since the number of calendar years is lower at every point; block years have proceeded somewhat faster but are quite close to a Gregorian year recently).

Figure 1. Weibull CDF regression for $10 million asymptotic Bitcoin case. The best fit shape parameter is 5.41 and the R² value is 0.95.

In the figure I have plotted the Weibull regression line for y against the natural log of the age of Bitcoin, for the $10 million case. It’s a straight line of slope (shape parameter) 5.41. The definition of y is a double natural log of [1 - price/(ultimate price)] = ln [-ln (1 - price/$10 million)] for this case.

Now the Weibull CDF has three parameters: shape, timescale, and maximum value whereas the power law has only two: a constant and a power law index (the Weibull shape parameter is the analog to the power law). Considering that in its early years the Weibull CFD behaves just as a power law, and applying Occam’s razor that a simpler model is to be preferred we choose the power law.

The strong statistical similarity between the power law fit and the $10 million result is suggestive of a power law that can persist for another two decades and that implies a price over 100 x greater than today can be reached in due time.

Think of that as in 2024 dollars since as long as there are $ there will be inflation and thus presumably no ultimate price for Bitcoin with its absolutely finite supply. What we are trying to understand here is when Bitcoin’s price advance would drop to the inflation rate, for the next two decades it is much, much higher than M2 monetary supply inflation (around 7% long term).

When would the Weibull CDF depart from a power law?

Figure 2. Comparison of Weibull CDF and power law for shape parameter (power law index) of 5.4. They diverge at around x = 0.8 since the power law has no asymptotic endpoint.

In the above figure 2, I have plotted the Weibull CDF curve in a semi-log plot and for a 5.4 shape factor. And also, as the dashed line, we see a power law of 5.4 index, as appropriate if the fits are based on the block year calendar. The two curves track very closely until age 0.8 x c, where c is the time scale to the knee, that depends on the asymptotic value for a given shape parameter.

For the $1 million value case it is age 0.8 x 27.2, or age 22 years, under 6 years from now. For the $10 million asymptotic value case, the separation of the curves is at 0.8 of 41.8 years  = 33 block years or about 17 years from now.

Comparison of Power Law and Weibull for Bitcoin priced in Gold ounces

But if you are worried about the inflation correction we can also do the regression in gold ounce terms. Gold generally holds its value in real terms (but not more than that) over long time frames. Today the ratio is about 23 ounces for Bitcoin priced in gold.

Here the regression is in calendar years rather than block years. The power law regression results have a power law index of 5.46 and R² = 0.94.

For the Weibull CDF we assume a ‘good delivery’ 400 ounce gold bar which is also an ultimate value around $1 million. The linear regression of  y  against  the natural log of age yields the following results: slope 5.48 and R² = 0.94, very similar to the power law regression.

The critical time for the knee of the curve has c = 23.5 years or 8 years from now when BTC/Au would reach 0.632  x 400 = 253 ounces, or almost 8 kilograms of gold for one Bitcoin.

So for a gold comparison we get a similar slope and similar R² as against the USD. And between the power law and Weibull for BTC/Au we see essentially the same slope and R² .

Again, we see that when evaluating Bitcoin’s price rise relative to gold there is no need to go beyond the power law.

Note also that if and when Bitcoin is worth 250 ounces of gold, that would correspond to over 5 billion ounces market cap, and the gold total supply is 6 billion ounces. Bitcoin will be close to ‘flipping’ gold by that time as the premier commodity asset and should also be worth much more in market cap than any one of the top global companies.

Figure 3. Weibull CDF regression for 400 ounces “good delivery” gold bar asymptotic Bitcoin price case. The best fit shape parameter is 5.48 and the R² value is 0.94.

Occam’s Razor

Occam’s razor says if two models perform similarly, with similar statical robustness, the simpler one, the one with fewer parameters, is to be preferred.

Thus we prefer the power law model with two parameters and its robust performance over a decade and a half to the Weibull CDF S-curve models with three parameters. Indeed the Weibull models give the same results for slope and R², there is no reason to prefer them as long as the power law is performing well.

We also prefer the power law model to Saylor’s Bitcoin24 scenarios of which there are three. And each one of those requires four parameters, and uses a stair step function rather than a continuous function introducing even more unnecessary complexity.

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