When will bitcoin be worth 1 million dollars?

This is the million dollar question.  When will bitcoin be worth 1 million dollars. Will John McAfee have to eat his own dick? In this post I (building heavily upon the work of others) established that bitcoins' value is solidly supported by the Metcalfe value. I gave the simple estimation for the Metcalfe value of

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

Using a little algebra, first we set the Metcalfe value to $1000000, and then solve for the number of bitcoins in circulation (which happily reduces to a function of the number of wallets).

The function is calculated as:

(2) No. bitcoins ≈ 0.207294*(1.52573*10^8 - no. wallets)

Substituting (2) back into (1) and solving for no. wallets, will give us a function that describes the number of users required on the network to support a value of $1000000.

(3) 1000000 ≈ exp(-2.662352 +                1.08e-07*N +                5.21e-07*0.207294*               (1.52573*10^8 -  N)) N ≈ 2.75803*10^8

What this indicates is that there will need to be around two hundred and seventy-five million wallets to support a Metcalfe value of $1000000. This does not seem impossible, nor even remotely unreasonable. 

For example, displayed in the figure below is the current number of wallet users from blockchain.info:

Figure 1: Current wallet numbers. around a 10x increase will put the bitcoin Metcalfe value at $1000,000

A 10x (11.36 to be precise) increase in wallet users will result in a Metcalfe value of $1000000, and as has been established previously, the Metcalfe value is a strong support line for the bitcoin value. 

Now that we have a target established, we can work on trying to magic 8 ball the number of users. The chart in figure 1 looks vaguely linear, so I will perform a linear regression on the number of users against date. All data is sourced from blockchain.info.

number of months	year	normalised  number of wallets	4th root normalised number of wallets
2	2011	2436	7.02537193
12	2012	77232	16.6705267
12	2013	962069	31.3185434
12	2014	2723272	40.6230713
12	2015	5428667	48.2695556
12	2016	10961809	57.5400928
12	2017	21506448	68.0992254
4	2018	72724815	92.346546

number
of months

year

normalised
 number of wallets

4th root
normalised number of wallets

2

2011

2436

7.02537193

12

2012

77232

16.6705267

12

2013

962069

31.3185434

12

2014

2723272

40.6230713

12

2015

5428667

48.2695556

12

2016

10961809

57.5400928

12

2017

21506448

68.0992254

4

2018

72724815

92.346546

Table 1: Wallet data has been normalised against the number of months available for a given year. 

The maximum number of wallets was taken for each year. Upon inspection, a 4th root transformation 

target has been identified for the regression.

. regress throotnormalisednumberofwallets year

      Source |       SS           df       MS      Number of obs   =         8
-------------+----------------------------------   F(1, 6)         =    250.14
       Model |  5267.39179         1  5267.39179   Prob > F        =    0.0000
    Residual |  126.344853         6  21.0574755   R-squared       =    0.9766
-------------+----------------------------------   Adj R-squared   =    0.9727
       Total |  5393.73665         7  770.533807   Root MSE        =    4.5888

------------------------------------------------------------------------------
throotnorm~s |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        year |   11.19884   .7080738    15.82   0.000     9.466249    12.93144
       _cons |  -22514.83   1426.416   -15.78   0.000    -26005.15   -19024.52
------------------------------------------------------------------------------

This gives us the regression equation for the number of wallets based on year.  

The equation is therefore:

   (4) Number of Wallets = (11.19884 * year - 22514.83)^4

So how many wallets will there be in 2020? Well if the equation in 3 is to be believed, there will be around 130 million. 130 million wallets substituting back into (3) we have a bitcoin Metcalfe value of around $1000000. Looks like McAfee may have to go hungry.

Next time: Remember this?

Figure 2: Metcalfe value vs bitcoin price.

Well I am going to apply some tests about these peaks to see if they are statistically significantly different to the Metcalfe value, and then explore what that means if they are.

Thanks for reading crypto addicts.
Týr

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