Bitcoin & Kaspa
Not your regular crypto
Disclaimer: nothing in this article is financial advice.
Someone that went by the alias Trolololo was the first to discover that the Bitcoin price follows a function similar to a power law, even though it was not fully recognised by him/her as such. Giovanni Santostasi was probably the first to really recognise the power law and it was further popularized by H.C.Burger, but strangely enough it never got the attention that other Bitcoin price models received such as the stock-to-flow model by PlanB. Only recently, the power law model popped up again on some popular Youtube channels by Andrei Jikh and Peter McCormack.
It seems the power law model is not understood well enough by the general public, because I think that if you do fully understand it should just completely blow your mind. Therefore, it’s worth another attempt to explain it as clearly as possible, starting with a gentle high-school level math introduction.
Additionally, all previous authors were missing some major observations with regards to the power law model, which I will discuss as well. And then there is Kaspa. Did it just solve the blockchain trilemma? Is this the only alt-coin that is not a shit-coin?
What is a power law?
A power law is simply a mathematical function that maps input values (x on the horizontal axis in fig.1) to output values (y on the vertical axis in fig.1) according to the form y=xᵏ, with k the power constant, i.e. for k=2 that means y=x²=x·x (orange line in fig.1), e.g. for x=20 then y=20²=20·20=400.
All three graphs above show the same three functions, but the left graph has linear axes, the middle graph has a linear x-axis and a logarithmic y-axis, and the right graph has both a logarithmic x- and y-axis. Note that y=x¹ (blue line in fig.1) is a linear function and thus no power law, but it was added to show what happens with a straight line in logarithmic graphs.
Figure 2 shows power laws of the form y=axᵏ with a the scaling factor set to 0.5, 1.0 and 2.0 respectively and k=2. As shown in the log-log graph, the scaling factor doesn’t change the slope, but it does change the intercept, i.e. the scaling factor is the value at which the function crosses the vertical line x=10⁰=1, since y=axᵏ=a·1ᵏ=a for x=1.
Logarithmic graphs
What logarithmic graphs are doing, is stretching space with small values and compressing space with large values. Log-log graphs are particularly interesting because power laws will always appear linear, which makes them ideal to discover if certain data, like the price of bitcoin, follows a power law.
Reading logarithmic graphs requires some practice. Each major tick represents a power of 10, with 10⁰=1, 10¹=10, 10²=100, 10³=1000 etc. Each minor tick represents a multiple of the major tick, i.e. the first line after 10⁰ marks 2·10⁰=2, next line is 3·10⁰=3 etc., up to 10·10⁰=10¹=10. Next minor tick marks 2·10¹=20, which seems like a discontinuity in the speed with which the values grow, but it isn’t because the distance in the graph from 10¹ to 2·10¹ is a lot bigger than the distance from 9·10⁰=9 to 10·10⁰=10¹=10. The tick marks for 11, 12, 13… are no longer drawn because these lines will get so close to each other that at some point they can no longer be properly drawn.
The problem with log-linear graphs
Nonetheless, in case of the Bitcoin price, having time on a logarithmic scale feels unnatural since time seems to progress linearly to us, which is probably one of the reasons that the log-linear rainbow chart became popular. Additionally, tools like trading view which are used for technical price analysis only support logarithmic scaling of the vertical axis, simply because there have never been any assets before that track a power law.
Apart from the bad fit for the example rainbow chart, there is a fundamental problem with these kind of graphs when the function that was used to draw it is not shown. There is no way to know how many parameters were used to fit the model to the data. A famous quote from John von Neumann: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk”, meaning no matter how complex the data, there is always a way to fit a model to it. On a log-linear graph the function can be arbitrarily complex and there is no way to see. Whenever new price data comes out, the model can be adapted to it and it will be difficult to detect (as long as only the graph is shown and not the underlying formula). Consequently, the creator can claim a valid model indefinitely. In contrast, while a straight line on a log-log graph can be slightly adjusted in terms of slope (and intercept), once the data no longer follows a power law it is no longer a straight line and the model breaks.
The Bitcoin power law
Figure 3 shows the Bitcoin price together with its lower and upper trend lines on a linear, log-linear and log-log graph. The legend shows the power law functions that define the lower and upper trend lines (with x replaced by year-2009 for ease of use; 2009 being the year where Bitcoin was created), e.g. according to this model the price for February 2024 (year≅2024.083) should be between $27k and $223k (fill the year into the formulas and you can calculate this yourself). At the time of writing bitcoin is trading at $43k.
Whenever the price hits the lower trend line (which it does a lot), it seems there are no more sellers left and the remaining holders are all long-term holders (or so called diamond hands/HODL-ers). This is why the lower trend line can be seen as the ‘true’ adoption trend line. Whenever the price gets close to the upper trend line (which it only does at the peaks of the 4-year halving cycle) there are no more buyers/speculators left and there is only one direction left for the price to go.
Interestingly, the upper and lower trend lines converge. Google (when following the link hover over the second intersection) can show us that these trend lines intersect around July 2138. That is very close to the year 2140 at which the last Bitcoin is mined and Bitcoin supply inflation becomes truely 0. Is that a coincidence? Before we try to answer that, let’s look at Kaspa first.
Kaspa — Bitcoin’s little brother
As far as I know Kaspa is the only other crypto whose price follows a power law. Kaspa is very similar to Bitcoin (Proof of Work, fair launch end of 2021), but its unique selling point is a transaction speed limit which (ultimately) will be approximately three orders of magnitude higher, which is achieved by not using a blockchain like Bitcoin does, but by using a blockDAG (Directed Acyclic Graph).
The paper Phantom GhostDAG — A scalable generalization of Nakamoto Consensus explains the main ideas behind Kaspa and is worth a read if you’re a bit technically oriented. Interestingly, as the title already suggests, Kaspa is actually a generalization of Bitcoin. It introduces a concept called the maximum k-cluster that is used to mark a set of well connected blocks in the graph for which the size of the anticone of each of those blocks (i.e. blocks outside of the history and future of those blocks) is provably limited to k. When k is set to 0, Kaspa’s blockDAG basically collapses to Bitcoin’s blockchain, where no parallel blocks are allowed, and therefore can only be operated securely under the assumption of slow block creation (i.e. only 1 block per 10 minutes, instead of 1 (and soon 10) block(s) per second for Kaspa).
To keep node storage requirements low, the trade-off is that transaction history is kept only for 3 days for regular nodes (instead of up till the genesis block like Bitcoin does), while only keeping the full UTXO database next to it. While not needed for proper operation, but for the sake of traceability and prove of fairness there is a collection of archival nodes that keep the full blockDAG and transaction history till genesis.
Kaspa is an ancient Aramaic word for silver, which is how its meant to be used; not as a direct competitor to Bitcoin — synonymous to a digital store of value — but as a means of exchange, like silver was in the past.
The Kaspa power law
Figure 4 shows the Kaspa price with its power law trend lines on a linear, log-linear and log-log graph, similar to figure 3 for Bitcoin. Usually, alt-coins move in tandem to Bitcoin, and since Bitcoin has been in a bear market for the past two years or so, it’s quite surprising that Kaspa has been making these huge gains.
Figure 5 shows that according to the trend lines Kaspa’s price will reach $1 halfway 2026 earliest and end of 2027 latest, and it will reach $10 in 2034 earliest and 2036 latest. At the time of writing one Kaspa trades at $0.10. Note that Kaspa ultimately will have a supply of 28.7 billion coins (of which 78% is already mined), so multiplying its price by 1000 gives a rough estimate on where it is in its price development compared to Bitcoin (Bitcoin was priced at $100 halfway 2013).
Kaspa’s trend lines also converge. Google (when following the link hover over the second intersection) can show us that Kaspa’s trend lines intersect around April 2059. Again, that is very close to the moment the last Kaspa is mined, which is around Oktober 2057, as can be seen when looking at the emission schedule (drag the right slider all the way to the right). Coincidence, or not?
Price channel convergence
Convergence of these trend lines should not come as a surprise. When an asset class matures and its market cap grows it usually becomes less volatile, which narrows the price channel. Additionally, Bitcoin has halving cycles, i.e. every four years the reward on Bitcoin mining is cut in half. This causes a supply squeeze, because all of a sudden the amount of Bitcoin that is sold by miners on the market is cut in half, which causes the price to skyrocket. Nevertheless, this supply squeeze effect is slowly diminishing with every cycle, because the ratio between the newly created Bitcoin and the existing supply is slowly dropping towards 0. You cannot expect that in 2036, when Bitcoin’s yearly supply inflation rate drops from ~0.2% to ~0.1% and there is only just ~0.8% of the Bitcoin total supply left to be mined in the remaining 104 years, that this event is going to cause a significant supply squeeze, let alone when the yearly inflation drops from ~0.02% to 0.01% in 2048 when there is ~0.1% supply left to be mined. At that point, the Bitcoin halving cycle has no significant impact on the price of Bitcoin any longer.
As a matter of fact, to see where Bitcoin is headed in terms of volatility within its price channel, we can look a bit at Kaspa. Kaspa has no 4-year halving cycle. Kaspa’s supply halves every year, but does that by reducing the mining reward gradually every month. So Kaspa’s mining reward reduction is not causing significant supply squeezes. It’s price action is mainly driven by adoption (the slope of the lower trend line) and it’s volatility inside the price channel simply by human emotions, i.e. fear and greed, and of course significant news events. These price drivers are obviously all present for Bitcoin as well.
At the time of writing (February 2024) Kaspa’s trading range is bounded by $0.05 on the low end and $0.17 on the high end, meaning its price channel width as a percentage of the lower price trend is 100·0.17/0.05=340%, while for Bitcoin we get 100·223/27=826%. Given that Bitcoin’s market cap is currently over 300 times as high as Kaspa’s market cap and given that Bitcoin is about 4 halving cycles old and Kaspa just ~1.5 halving cycle, based on these figures alone you would probably assume the opposite. That’s why it’s reasonable to assume that Bitcoin’s supply squeeze effect caused by its 4-year halving cycle is the main driver for its volatility, which is also supported by the looks of its price history.
Nonetheless, Bitcoin’s price channel width is not decreasing as rapidly as you would assume based on its quickly dropping supply inflation rate. Additionally, the upper trend line of the Bitcoin price is not a very good fit, especially looking at the price during the last halving cycle.
Figure 6 shows the Bitcoin price on a log-log graph with an upper trend line that nicely fits the price peaks of the halving cycles, while still showing the original power law trend line faded to the background. Not wanting to disturb Von Neumann too much, I simply chose to increase the initial slope to a power of 5.5 and linearly dropping it to 4.28 over time till the end of 2030, thereby adding just one parameter, but adding time to the power makes it a decreasing exponential function and no longer a power law; hence the curved line on the log-log graph. While this trend line probably better fits to the idea of the diminishing halving cycle supply squeeze effect, we’re obviously running into an increased risk of overfitting. With this trend line the peak of the next halving cycle will be around $150k somewhere halfway 2025. Only time will tell.
When will the power law model break?
Every model like this eventually breaks. So, how and when?
First of all, a power law is just a mathematical formula and it grows indefinitely without caring about the limited number of people on earth or the limited number of assets/dollars in the world. Looking at the technology adoption life cycle shown in figure 7, people seem to agree that for Bitcoin we’re starting to get closer to the end of the early adopters phase. Although according to this source, currently just 5% of the worldwide population owns some Bitcoin, even though that is not very evenly distributed. Countries like Nigeria (13%) are front-runners, for obvious reasons.
If we’re indeed moving into the early majority phase, and the number of people using Bitcoin is the limiting factor in Bitcoin’s price growth then the slope (or power) of the lower trend line (i.e. the adoption trend line) will start dropping in the near future, because when running up the Bitcoin adoption life cycle bell curve from left to right the acceleration (second order derivative) drops to 0 at around 16%, meaning the steepness (i.e. velocity or first order derivative) no longer increases, but starts to decrease.
Nonetheless, I think it’s more reasonable to think that the Bitcoin share in global financial assets is the limiting factor of its growth, because this is what Bitcoin is competing against. The total amount of global financial assets is estimated at around $1000 trillion according to a report by McKinsey, excluding the $500 trillion of non-financial assets in the real economy. Bitcoin’s market cap is currently about $850 billion, so this represents a 100·0.85/1000=0.085% share in global financial assets.
Of course it’s not so easy to determine what share of global financial assets Bitcoin can absorb, but I think we can agree that it’s nowhere close to saturation. With the recent Bitcoin ETF approval institutional investments will start to flow into Bitcoin. Let’s assume Bitcoin can reach the market cap of gold (~$13.7 trillion), which means a single bitcoin will ultimately be worth ~$650k, which it will have reached around 2035 according to the current lower (adoption) trend line. Hypothetically, when Bitcoin absorbs all global financial assets a single bitcoin will be worth $47.6 million, which means that the power law model will definitely break before the year 2064. All-in-all, it’s reasonable to assume that somewhere in the next 10 or 20 years the adoption of Bitcoin will start to slow down.
Of course other scenarios are possible as well. If we enter some nightmare hyper-inflationary Weimar type of scenario the power law model will break to the upside. A deflationary collapse might cause the model to break early to the downside, even though that is quite unlikely because it’s the only scenario that central banks around the world can willingly avoid. Then of course, the success of Bitcoin is not unavoidable. A critical technical failure could occur, or governments all around the world could unite and try to ban Bitcoin. Even though Bitcoin poses a threat to the power of governments, the latter is probably not a high probability outcome given how divided governments around the world are on way more important topics like climate change and poverty. The resilience of the Bitcoin network was tested before when China banned Bitcoin mining, but that hardly made a dent.
For Kaspa it’s slightly different because it’s operating in the shadow of Bitcoin, which makes it grow at a much slower pace. As Google showed us before, at the intersection of the lower and upper trend lines one kaspa reaches $122 in 2059, which corresponds with a market cap of 122·28.7·10⁹=$3.5 trillion (about 3 times the market cap of physical silver), which is still only a 100·3.5/1000=0.35% share in global financial assets. Additionally, Kaspa is still heavily under development, which suggests that changes to its organic network growth characteristics are still possible.
Use as a real currency
Bitcoin itself is less of a threat to governments then Kaspa is in the long run, because Bitcoin will never be used as a currency itself due to its limit on transaction speed (max. ~7 transactions per second). That’s why Bitcoin needs second layer scaling solutions like the Lightning Network. But even Lightning has its limitations. For Lightning to work, a channel needs to be opened after which off-chain transactions can occur. To settle, again an on-chain transaction is needed.
Only to on-board 8 billion people on Lightning, at 7 people per second that will take 8·10⁹/7·3600·24·365~=36 years, and then nobody settled any transactions yet and Bitcoin was not used for anything other than on-boarding people to Lightning. It is suggested to use federations of trusted groups of people that together share settlements, but that concept causes centralisation of power again. Slowly at first, but faster once people forget about the need for self custody and start trusting bigger and bigger federations with lower overhead costs (since the Bitcoin transaction fee will not always stay this low), which goes against the original idea of Bitcoin.
Kaspa, once up to speed, can do several 1000s of transactions per second, close to what centralised payment services like Visa or Mastercard can handle. Adding a scaling layer to such a fast settlement base layer really will enable the original Satoshi Nakamoto vision of a peer-to-peer electronic cash system. According to the current Kaspa adoption trend, this will be a vision for the very long term. Unless it breaks early to the upside of course. Let’s hope it does.
Future year-over-year price growth
Figure 8 shows the year-over-year growth of the lower trend line of Bitcoin and Kaspa, so not considering price fluctuations within the channel. It gives a rough comparison on the relative growth of adoption between Bitcoin and Kaspa for the coming years. Investing in Kaspa right now is a bit like investing in Bitcoin a few years ago. From 2033 onwards the lower bound price appreciation of Bitcoin overtakes Kaspa again if the models stay accurate. If so, in 2033 both assets will still yield a decent year-over-year return on investment of around 28%; not quite so easily achievable with traditional assets.
