Bitcoin is Highly Convex

Jensen 'ratio' of 4

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.

Current Power Law best fit

This is a current best fit power law relation for Bitcoin’s price over the past 13 plus years:

P = $38,788 * (B/15)^5.41 .

It’s a steep power law, better than the 5th power of time elapsed since Bitcoin was first created. Here B is the number of block-years elapsed, each of 52,500 blocks duration. The best fit for all monthly data points since B =2 through the latest point at B = 15.5 has an R^2 of 0.94 and an F-statistic of 2535, excellent for 163 data points.

I have normalized to B = 15 in the expression above, we are currently at B =15.5 block years, that is we are half a year away from the next Halving (halfining for Hal Finney fans) at B = 16 (arriving in April 2024). The current model fair value is $46,309, so we are still well below fair value at around $35,000.

Of course the volatility is high. It should be considered in logarithmic or ratio terms, not in linear terms. The standard deviation is 0.319 in the log10 of price, a factor of over two. During the last four years it has decreased to 0.253, a factor of 1.79 up or down as the one sigma variation. So a +/- one sigma range is currently [$25871, $82893].

One must be prepared for large volatility. But that also means high upside and Bitcoin skews toward that, toward the right tail. We could drop $9K or we could go up by $44K and remain within a single standard deviation from the power law model.

In a Medium article in August, 2020 https://medium.com/the-capital/bitcoins-positive-skew-e3f6db6cd8ac I showed that Bitcoin’s skewness for a power law model (call a Lindy tech adoption model in that article) was 0.64; it is similar today with a value of 0.59.

I wrote then as follows:

Jensen’s Inequality Shows Convexity

Positive convexity essentially means there is more upside than downside in an investment, and that its response to volatility is favorable. One might expect that Bitcoin itself has positive convexity, since if you buy it straight, not on margin, then the downside is limited to the cost, but the upside might be much more than the initial price, given it is a new technology with a strong long-term trend, and provided you have sufficient patience (HODLing).

The Jensen inequality takes the form:

E[F(x)] > F[E(x)] .

This means the expected value E of the model F built on input parameter x is greater than the model’s output for the expected or average value of the input parameter, E(x).

At that time what I call the Jensen ratio, the degree of inequality expressed as the ratio of the left hand side of the equation to the right hand side, was 3.84.

It’s slightly larger now, the situation has only improved. Just to clarify, the right hand side has an expectation value for the input parameter, which in our case is block time, and that is simply the mean block time for the series, which is 8.75 (since we started the regression at B = 2). So we take the model price at that time, which was $2107.

And the left hand side is the mean of the function evaluated at each monthly block time considered, which is the expectation value at that time. The left hand side is much larger because we have substantially higher prices at later times, Bitcoin’s steep long-term ascent is very far from a random walk. The mean of all such model prices is $8394.

Clearly the LHS is much larger than the RHS, the power law function is highly convex, and as long as it remains a good statistical fit as it is today, then Bitcoin is behaving as a highly convex asset.

The Jensen ratio today is 3.98 very similar but higher than it was three years ago. We like Bitcoin’s convexity, the upside that it presents.