The intrinsic value of bitcoin

Metcalfe's law is the concept that a networks' value is proportional to the square of the number of users. This is pretty well known in the cryptocurrency world. For example, this paper goes into great detail, providing a robust argument for the case of bitcoin's price in particular being heavily related to the Metcalfe value.  I have recreated the formulas below and updated the estimations provided with recent data.

I had to tweak a few of the formulas to make them work, but eventually I found the parameters that produced estimates similar to the paper.  I sourced all data from blockchain.info.

It should be noted that this will likely be an underestimate of the Metcalfe value, since it cannot take into account off chain transactions.

NB All logs are natural logs.

(1) log(Metcalfe) = A * log (transaction pairs) / gompertz sigmoid

Where:
  A = 0.945
  transaction pairs = number of wallets * 
                     (number of wallets - 1) / 2
  gompertz sigmoid = coins in circulation * 
                     log (21000000/ coins in circulation)/1000000

So the expected bitcoin value from this is 

(2) bitcoin value = exp(A * log (transaction pairs) / gompertz sigmoid)

I explored this relationship in STATA 14.2. As can be observed in the figure below, the price spikes in late 2017 (of which occurred mostly after the authors original paper was published) were not supported by the Metcalfe value. However what is important to note is that the price appears to have retreated no further than the Metcalfe value, adding further weight to the idea that the intrinsic value of the network is established in the Metcalfe law. 

It should also be noted that the authors of the above paper indicated that the increase in the value of the bitcoin price compared to the Metcalfe value in 2014 was likely due to some form of market manipulation. I will perform the same tests they did on the recent spikes (but it would appear at least graphically to be a similar situation)

Figure 1: Metcalfe value vs actual price in USD. Spikes always revert to the Metcalfe value. 

Focusing now on the last little while, we can really see the price "bounce" off the Metcalfe line.

Figure 2: Metcalfe value is a strong support line for the bitcoin price

Post hoc ergo propter hoc - Just because something happened after the fact, does not necessarily mean it happened because of that fact. As I stated above, Metcalfe's law is reasonably well known within the cryptocurrency sector. It may be that there are entities that have already calculated these figures, and decided because the value is approaching the Metcalfe value, it is time to "go long". However, in reality I think that whilst there may be a few cases of this, there probably is not enough of it to actually influence the price so dramatically.

What this analysis does do is provide some evidence to the testimony that there is some intrinsic relationship between the bitcoin price, the number of users and the number of coins and the Metcalfe value.

Exploring this, I ran a multiple linear regression for the log(Metcalfe value) against the number of wallets and number of coins (again all info from blockchain.info)

. regress logmetcalfe  nobitcoins nowallets

      Source |       SS           df       MS      Number of obs   =     1,260
-------------+----------------------------------   F(2, 1257)      >  99999.00
       Model |  5449.40689         2  2724.70344   Prob > F        =    0.0000
    Residual |  6.33766904     1,257  .005041901   R-squared       =    0.9988
-------------+----------------------------------   Adj R-squared   =    0.9988
       Total |  5455.74456     1,259   4.3333952   Root MSE        =    .07101

------------------------------------------------------------------------------
  logmetcalf |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  nobitcoins |   5.21e-07   1.39e-09   375.43   0.000     5.18e-07    5.23e-07
   nowallets |   1.08e-07   4.81e-10   223.91   0.000     1.07e-07    1.09e-07
       _cons |  -2.662352   .0165411  -160.95   0.000    -2.694803   -2.629901
------------------------------------------------------------------------------

Figure 3: Correlation of linear prediction and natural log of Metcalfe value

This is a reasonably sound correlation and not entirely unexpected given the construction of the Metcalfe value from the independent variables.

Some diagnostic plots:

Figure 4: A quantile plot of the standardised residuals indicates that the residuals are likely to be non normal. However this is also unlikely to be a problem for the purposes of this model.

Figure 5: The histogram reveals the true nature of the issue - the residuals are bidistributed (there is more than one distribution within the residuals). Again, this is likely a non issue for our purposes.

Figure 6: Whilst there are some high leverage points, there aren't many with high residual as well. If I were to take this further, I would investigate the points above the 0.006 leverage line and above the 0.004 residual square line, however i don't think this is necessary in this case.

Figure 7: The residuals appear to be reasonably scattered evenly around zero for each of the fitted values. 

A simple estimation of the Metcalfe value is then:

(3)  Metcalfe ≈ exp(-2.662352 + 1.08e-07*N + 5.21e-07*C)

Where N is the number of wallets and C is the circulating supply.

Next time, I will try to work out the circumstances of when bitcoin's Metcalfe value will be equal to $1M.


Thanks for reading crypto addicts.
Týr