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The Performance of Cryptocurrencies Under the Rigors of Mathematics - An Overvalued Bitcoin is Detrimental to Ether

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February 27, 2018 | 

Joanna Newman |  0 Comments| 

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Cryptocurrencies are an entirely new type of financial asset. Now, an increasing number of analysis tools are now beginning to make themselves known in favor of the entire planet’s lust for this new form of asset. Some of these tools are simply recreations of the tools used by other industries that share similarities with Blockchain technology, namely: Metcalfe’s Law.

The Principles of Metcalfe’s Law


Robert Metcalfe is the founder and creator of the Ethernet protocol. In the 80s, he stipulated a law that claims that a telecommunication network’s worth is directly proportionate to the number of users squared (n2). In other words, the more users on a network, the more the network’s value will increase. Its mathematical application is based on the number of nodes and the value is in relation to the number of potential links within a network.



For a network of nodes (n), the number of links is equal to n(n-1)/2 or n2/2.

Metcalfe’s Law finds its home “in IT networks such as the internet, social networks and the World Wide Web. Often used in combination with other mathematical tools, Metcalfe’s Law is the most popular network analysis calculation.

The Blockchain is the network on which cryptocurrencies are based. It consists of nodes (devices that have a wallet address) and is an exchange system that relies on transaction links between users. Therefore, Blockchain technology is the perfect subject for analysis using Metcalfe’s Law and its derivatives. At least, that’s the belief of Ken Alarabi in his book Digital Blockchain Networks Appear to be Following Metcalfe’s Law.

Bitcoin and Ethereum’s Worth According to Metcalfe’s Law


In his book, Alarabi analyses the two biggest Blockchains in terms of their market caps. He compares the Metcalfe Index of both Bitcoin and Ethereum based on their number of active nodes and transactions in relation to their coin’s current value.

From here on, we will refer to M as the Metcalfe Index and V as the volume of daily transactions on the Blockchain. In combination with the Metcalfe Index, we will also take into consideration other Indexes such as the Sardoff Index (S) and the Zipf Index (Z).



















   N      N1.5   2*N1.5100*        N*ln (N)
Sardoff’s Law (S)Metcalfe’s Law (M)    M1 M2  M3   Zipf’s Law

N = Number of daily transactions

S = Current value of given currency

Bitcoin Analysis


The following is a calculation of the Pearson correlation of 6 difference indexes with the price of Bitcoin.

In the first line of the first table, you will see the Pearson correlation index (r) of Bitcoin’s price in relation to each of the formulas from above.

For reference, ADR is for the indexes in which only addresses were used, USD will be the indexes for which only the price was taken into consideration and TX will be the indexes for which both the address and the price are used for the calculation.

The second line of the table shows the Pearson Index calibrated and averaged for a period of 30 days. And, the third line is the same index, but over a period of 90 days. The last two lines on the chart are the correlation index calculations for the natural logarithm (ln) of the USD’s value.



It’s easy to see the perfect correlation between the 6 calculated indexes and the price of Bitcoin, especially in the lines that describe the natural logarithm. One can even deduce that the value of r(ln) is invariable since the difference is so small.

In the following table, the same calculations were made using the number of active addresses.



Note that the correlation between the number of address with the price is made clear by all the indexes calculated with the natural logarithm.

To illustrate the that Bitcoin’s price follows Metcalfe’s Law, Alarabi superimposed graphs that pair the USD’s correlation to the M2 Index. As can be seen, the two graphs are nearly identical.


Ethereum Analysis


Again, we are looking at the Pearson Index in correlation with Ethereum’s value in USD. As opposed to Bitcoin’s single address per wallet, Ethereum relies on a system that uses multiple addresses. To replace the number of addresses, the number of daily transactions is used instead. Like in Bitcoin’s analysis, days where the value was at $0, were not included.



There is a strong correlation between the 6 indexes when calculated with Ethereum, even for lines that did not use the natural logarithm. The difference between this and Bitcoin’s result could be explained by the sheer amount of data that’s been recorded for Bitcoin. For Bitcoin’s there has been 8 years’ worth of data, whereas Ethereum has only been around for 3 years.

As with Bitcoin, it is the graphs correlated with the USD prices and the Metcalfe Index, during a 30-day period, that offers the most similarities.



Once the correlation between their prices and the number of daily transactions has been established, we must then ask ourselves; what should the market cap of these currencies be based on these numbers?

To answer this question, we will look at a new index (NVT), developed by Dimitri Kalichkin, and which has already proven itself in several market studies. Called the NVT or Network Value-to-Transactions, it can be obtained by calculating the ratio of daily capitalization in relation to the Metcalfe Index over a period of 30 days.

Bitcoin’s NVT


The following figure shows the evolution of Bitcoin price in relation to its NVT.

The orange outlines Bitcoin’s real price, whereas the blue is the price that it should be based on the number of transactions.

We can then conclude that on December 23rd, 2015, Bitcoin had been undervalued by about 25% and on December 25th, 2017, it was undervalued by 100%.


Ethereum’s NVT


The following shows the evolution of Ethereum’s price according to its NVT Index.

Again, the orange shows Ethereum’s real price and the blue shows the price that it should have according to the number of transactions that occurred on that day.

Here, we can conclude that Ethereum has been undervalued since the 1st of May 2017. July 1st, 2017 saw Ethereum undervalued by 75% and on January 1st, 2018, it was nearly 150% undervalued.



From an analysis perspective, it is plain to see how the Metcalfe Index is a good forecasting tool that provides good insight into a cryptocurrency’s history. And, when combined with the NVT, Ethereum is given proper justice since it has suffered from Bitcoin’s popularity and still not did score much lower than it.

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Total Market Cap The Total Market Capitalization (Market Cap) is an indicator that measures the size of all the cryptocurrencies.It’s the total market value of all the cryptocurrencies' circulating supply: so it’s the total value of all the coins that have been mined.

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