Tuesday, November 14, 2017

Power laws and cryptocurrencies

The Power Law is used to describe phenomena where large occurrences are rare but small ones are quite common. For example, there are few billionaires while most people make only a modest income; there are few large cities but many small towns; there are few very frequent words but many rare words.

Mathematically, Power Laws are of interest because of what is known as "scale invariance", as well as the fact that there is no well-defined average value. Furthermore, Power Laws are considered to be universal — you can read about this in Wikipedia. One of the more obvious places that we might expect to find them is in the exchange rates of currencies (their "worth") — there will be a few of great worth (the "major currencies") and lots of lesser worth.

For example, I recently read the headline: Bitcoin isn't "too expensive", says BTCC boss Bobby Lee. He was defending the price of the digital currency Bitcoin, which has increased in value more than 600 percent this year, claiming that this is not evidence of a financial bubble, but instead is evidence that the currency is proving its utility in the digital world. Obviously, I cannot let this claim pass without turning a quantitative eye upon it.


Bitcoin is the original cryptocurrency, established in 2009, just after the financial crash of that time. It is a digital currency, which by design has no central bank or regulatory authority supporting it. The coins don’t exist in a tangible form, but instead exist solely in a digital "wallet". Nevertheless, they can still be exchanged and used in transactions, just as with any fiat currency.

Bitcoin is based on a technology now referred to as the blockchain, which seriously has the potential to redefine future economic and legal transactions. Indeed, it is the blockchain idea that has proven to be of interest to financial and legal institutions, not the currency itself (which is just an example of using the blockchain). Blockchain is a distributed digital database, where every transaction is broadcast over the net and stored publicly, making it immutable as well as transparent. Compared to traditional financial and legal systems, this provides increased security, higher efficiency, greater error resistance, and reduced transaction costs. You can read about it in The ultimate 3500-word guide in plain English to understand Blockchain.

Bitcoin was launched for around $US0.005 (ie. half a cent). It was pretty much ignored for 4 years, but it has increased greatly in popularity over the past 4 years. Its exchange rate first exploded to a peak in late 2013, followed by a slow decline of nearly 90% (associated with the collapse of the Mt Gox digital currency exchange). It has achieved near-manic popularity in the past year, as shown in the first graph.

From CoinGecko
Bitcoin exchange rate with the US dollar

So, we now have headlines like this: Bitcoin just surged over $4000 and is near biggest financial crash in 400 years. The reference is to to what is known as Tulip mania, in the Netherlands in 1636-1637, where the tulip bulb prices quickly went from 1 guilder to 60, exploded to 1,000 or more, and then crashed. This is the context within which Bobby Lee made his claim (quoted above) that the current Bitcoin price is not too high.

The important point for our purposes here is that Bitcoin has spawned a host of imitators. So, there are now, or have been, more than 1,000 cryptocurrencies in existence. Many of them are intended as genuine digital currencies, each one addressing one or more of the perceived limitations of the original Bitcoin (such as its inability to scale up to a large number of transactions, or to process transactions faster). Indeed, we may see Bitcoin as a proof of concept and/or pilot study for digital currencies.

Most of the so-called altcoins, however, are not intended as general-use currencies at all. Instead, they form a totally new mode of fundraising for start-up companies, which now sell custom cryptocurrencies in order to raise investment. That is, instead of issuing shares as an IPO (initial public offer) they have an ICO (initial coin offer), thus bypassing the traditional venture capital processes. There is is a whole new world of digital finance emerging (see Cryptocurrency mania fuels hype and fear at venture firms).


In order to assess the comparative price of Bitcoin to the altcoins, I need the exchange rate of the current crop of cryptocurrencies. I took the CoinGecko rates at 14:25 UTC on 11 November 2017 (they change by the minute!). There were 735 coins listed, of which I took the top 100 exchange rates in US dollars. I then ignored the data for the Bit20 coin, which is actually related to an index fund, and thus has a price that is unrelated to the other currencies.

The next graph shows the currencies listed in the rank order of their value. This should illustrate a special case of the Power Law that is known as Zipf's Law, which refers to the "size" of each event relative to it's rank order of size. The standard way to evaluate the Zipf pattern is to plot the data with both axes of the graph converted to logarithms, under which circumstances the data should form a straight line.

As you can see, the exchange rates do fit Zipf's Law very well. In particular, Bitcoin, which is the #1 ranked coin, is not over-priced relative to the other coins. Note that this does not address the question as to whether all of the coins are over-priced or not. That would be a separate question, about the intrinsic value of cryptocurrencies.

Note that the top 25 ranked coins do not fit the Power Law as well as do the remaining 75 coins. So, we might also look at these top coins separately. This is shown in the next graph.

These 25 coins also fit Zipf's Law very well, but the power exponent is clearly smaller than for the remaining coins. In this case, Bitcoin fits the Power Law even better than before. Like it or not, relative to the other coins, Bitcoin is, indeed, not "too expensive".

Very few of the coins appear to be be over-priced (ie. far above the line), but a few of them might be considered under-priced (ie. far below the line). In particular, the #4 ranked coin is the SegWit2x [Futures]. This coin represents a controversial suggestion to split off from Bitcoin. It has not received a great deal of support from the Bitcoin community, and the proposed split was officially suspended only a few days ago. Whether it will go ahead eventually is unclear. The #5 ranked coin is Dash, which is often touted as a currency much more like cash, in the sense that the users can remain almost completely anonymous (which is actually a bit tricky with Bitcoin).

In the world of currency exchange, the big three pieces of information about each currency are (i) the Price of each coin, (ii) the Market Capitalization, which is the total coin supply multiplied by the coin price, and (iii) the Liquidity, which refers to how easy it is to buy and sell coins without causing a change in their price (it is used to measure the market share, market maturity and market acceptance). We could summarize this information for each coin by using a phylogenetic network.

So, I took the information as supplied by CoinGecko (see above) in US dollars, and log-transformed the numbers (economic worth is usually considered to be log-normally distributed). I then calculated the manhattan distances pairwise between the currencies, and plotted this using a NeighborNet graph, as shown in the final figure. The 10 top-price currencies have their full name shown, while the remainder are labeled with their exchange abbreviation. As usual, coins that have similar financial characteristics are near each other in the network; and the further apart the coins are in the network then the more different are their characteristics.

There are basically four neighborhoods in the graph, representing four different types of coins. Those coins at the top-right of the network all have a high Price, Capitalization and Liquidity. These are the coins that currently dominate the market. Moving leftwards from there in the graph, the Price, Capitalization and Liquidity all decrease, so that the coins in the middle of the network have low values of all three criteria. The coins at the top-left of the network have a relatively high Price but still have a low Capitalization and Liquidity. Those coins isolated at the bottom of the network currently have no Market Capitalization at all, even though they are available for trading and thus have a Price (this includes the SegWit2x Futures).


So, should you invest your hard-earned savings in cryptocurrencies? Plenty of people are doing so. For example, Coinbase, the largest cryptocurrency exchange in the USA, reportedly now has 12 million customers.

The general consensus seems to be "yes" to investment only if you like a bit of a gamble, because you may win big, but otherwise the answer is currently "no". The attributes that currently make cryptocurrencies such a speculative investment, such as their big price swings, their volatility and unpredictability, and their potentially lucrative payoffs, actually make them pretty useless as currencies. If you are looking for a long-term investment, then you probably need to find an altcoin that is either useful as a transaction medium, or provides an innovative application of the blockchain technology.

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