Is Bitcoin Displacing Gold?

A Technology S-curve Analysis

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. 

The general concept is that new technologies displace old ones, and that Bitcoin is a monetary technology replacing gold, another pure monetary asset, but old tech.

Bitcoin at nearly $900 billion market cap has reached an 8% market share relative to all gold in the world ($11 trillion). Typically a new technology even at that early level of maturity starts to grab all of the incremental available value at the expense of the old tech (think smart phones vs. earlier mobile phones).

A Weibull Technology S-Curve

This article presents the technology S-curve analysis using a Weibull cumulative distribution function for Bitcoin’s price growth relative to gold.

The function is widely used in failure analysis, and is described in MIT lecture notes here. In this case, gold is possibly failing relative to Bitcoin’s displacement of it as a long term store-of-value. It could lose its monetary premium while retaining a smaller jewelry and industrial value.

The Weibull function looks like this:

f = 1 - exp [- (t/c)^k ],

where f is the fraction of an ultimate market capitalization relative to gold’s as a function of time, t. The characteristic timescale is c, and k is a power law index. One assumes the ultimate market cap and then runs a regression analysis to determine c and k.

So for example one might assume an asymptotic value of Bitcoin priced in gold as, say 1000 ounces ($1.8 million at today’s gold price), and the function tells you when you reach 300 ounces, etc.

The regression analysis is done by a transformation that takes the natural log twice in succession:

(t/c)^k = - ln (1-f)

followed by

k ln(t/c) = k ln (t) - k ln (c) = ln (- ln(1-f)) = y

This allows a linear regression of the form y = a x +b by identifying y with the logarithmically transformed function of f, a = k, and ln (t) as x.

The procedure is described in more detail in this article:

https://medium.com/the-capital/bitcoins-long-term-value-e077e212264d

There are a couple of interesting facts about the Weibull (S-curve) function. One is that at early times it reduces to a power law f ~ (t/c)^k. And the second is that when t = c, at the characteristic time, for all k, the f value is saturating and has reached 1 - 1/e = 0.632.

Regression Results

I have run regressions with monthly Bitcoin and gold prices over a 101 month period from early 2013 through August 2021. We are evaluating f = price of Bitcoin in gold ounces, with the starting time for the models set to January 2009 when Bitcoin was introduced.

In addition to the S-curve models, we show a power law model (the early stages of the S-curve resemble a power law), and an exponential with time model.

The power law model shows us that the index is around 5, that is Bitcoin’s price in gold terms has been growing by the 5th power of time since it was introduced. An exponential model also fits the historical data well with a short time scale of less than 4 years for a factor of 10 price increase. However, it is very divergent going forward since it represents a factor of 1000 price increase in just 11 years.

Table 1: This table summarizes parameters and statistical measures for two S-curve models, representing government-held gold and all gold, respectively. Also included are power law and exponential models for Bitcoin priced in ounces of gold. All models fit the 8 plus years of monthly historical data well, with R^2 of 0.83 or better and F-test values in the neighborhood of 500.

For the S-curve I ran two models, the third one in the table is a 60 ounces of gold (almost 2 kilograms) ultimate value, corresponding to about $2 trillion value representing all gold held by governments around the world. The second model has an asymptotic value of 10 kilograms, roughly $11 trillion at current gold prices, representing all the gold in the world.

So the interpretation is that if Bitcoin ultimately reaches the market cap of all government gold then the characteristic time scale to reach 63% of that is about 17 years. Add that to 2009 and you have the year 2026.

Here is the historical regression from StatPlus in the transformed y variable versus ln of calendar year since 2009, for a 60 ounces of gold target. Despite the transformation we can see clearly the three peaks roughly associated with the Bitcoin Halvings.

Figure 1. This figure shows the regression of y against the ln of calendar years elapsed since 2009. y is a double logarithmic transform of the fraction of the asymptotic price, expressed as 60 gold ounces (roughly $2 trillion market cap).

Forecasted Fair Values, Bitcoin in gold ounces

Currently Bitcoin is at 26 ounces, around 44% of the asymptotic value and corresponding to a y-value of -0.55 of the graph above. Again, y = 0 corresponds to 1 - 1/e or .632 as the value of f. To reach a value y = 1 requires f around 0.95, and y =2 corresponds to over 99.9% saturation. The model assumption breaks if Bitcoin exceeds 60 ounces.

If Bitcoin is going to equal the market cap of all gold and saturate, then the time scale is 24 years, and the 63% point (6.3 kilograms) would be reached around 2033 with that S-curve model.

Here is a forecast through the end of the decade for the fair value of Bitcoin in gold terms for each of the models.

Table 2: Bitcoin fair value forecasts in gold ounces for four models. Three of the models range from about 2 to about 5 kilograms by the end of the decade; the exponential model goes wild, predicting ten 400 ounces good delivery bars.

Figure 2. Graph of the predicted fair values shown in Table 2. All models are flattening out at a few kilograms by the end of the decade, apart from the exponential model. Only the two S-curve models are not mathematically divergent.

A different, less technical yet longer, version of this article is also published here: https://cryptoassets0417.medium.com/is-bitcoin-displacing-gold-f12d14b681e8?sk=7d97f83677a4b5e03069f6b8b5090971