The Effect of a BCH Price Rise on the BTC Blockrate

Epoch 252: February 6 - 19

Below is a formula for calculating the BTC blockrate for any BCH price (vs BTC) during the current difficulty period which ends around February 19th.

The Formula for Calculating the BTC Blockrate is:

BTC blocks/hr = 7.02 - 5.94 * BCH price / (1 + BTC fees per block/12.5)

Some sample values are provided for illustration:

If BTC fees did not change from their level of around 1 BTC/block at period start:

BCH price   BTC blocks/hr (if fees are 1 BTC per block)
  0.13             6.3
  0.20             5.9
  0.30             5.4
  0.40             4.8
  1.00             1.5
  1.27             0.0

And what might happen for various levels of BTC fees:

BCH price    Blocks(1BTC fees)  Blocks(5BTC fees)  Blocks(10BTC fees)
  0.13             6.3                6.5                6.6
  0.20             5.9                6.2                6.4
  0.30             5.4                5.7                6.0
  0.40             4.8                5.3                5.7
  1.00             1.5                2.8                3.7
  1.27             0.0                1.6                2.8

The formula is derived as follows.

We have a reasonable idea of the equilibrium mining state at the beginning of the current BTC difficulty period. We will take these as the starting values:

BTC price: 1
BTC blockrate: 6.3
BTC hash share: 89.8%
BTC fees per block: 1 BTC
BCH price: 0.13
BCH blockrate: 6
BCH hash share: 10.2%
BCH fees per block: 0

BCH difficulty adjusts to maintain 6 BCH blocks/hr. BCH vs BTC mining profitability is not considered explicitly by the Difficulty Adjustment Algorithm (DAA). However, assuming a sufficient number of miners switch to the more profitable coin, the DAA has the effect of ensuring BCH and BTC are, over time, equally profitable to mine and it does this by adjusting (in effect) BCH difficulty in line with the change in the ratio of BCH block reward to BTC block reward. For example, if the BCH block reward doubled vs BTC's, BCH difficulty and hashrate would also double so that it again becomes equally profitable to mine BCH or BTC.

We assume zero BCH fees, making the BCH block reward 12.5 BCH. The BTC block reward is 12.5 + fees. Expressing the BCH reward in BTC and dividing throughout by 12.5 gives a block reward ratio of:

BCH price / (1 + BTC fees/12.5)      

As said above, BCH’s hash share changes in proportion to the change in the BCH vs BTC block reward, so BCH’s hash share is given by:

BCH hash share = BCH starting share * (new BCH reward/new BTC reward) / (starting BCH reward/starting BTC reward)

BCH's starting hash share was 10.2%; BCH's starting price was 0.13; BTC's starting fees per block were 1 BTC, so the above equation becomes:

BCH new hash share = 10.2 * (BCH price / (1 + BTC fees/12.5)) / (0.13 / (1 + 1/12.5))
BCH new hash share = 84.7 *  BCH price / (1 + BTC fees/12.5)

BTC's new hash share percentage is:

BTC new hash share = 100 - BCH new hash share
BTC new hash share = 100 - 84.7 * BCH price / (1 + BTC fees/12.5)

BTC's starting blockrate was 6.3, so BTC's new blockrate will be:

BTC blocks/hr = 6.3 * BTC new hash share / BTC starting hash share

Substituting and putting in BTC's starting hash share of 89.8 gives:

BTC blocks/hr = 6.3 * (100 - 84.7 * BCH price / (1 + BTC fees/12.5)) / 89.8

Giving us the Formula for Calculating the BTC Blockrate:

BTC blocks/hr = 7.02 - 5.94 * BCH price / (1 + BTC fees/12.5)

The formula cannot give the exact BTC blockrate as there are unknowns: the overall hashrate; miners may not always mine the more profitable coin; a large BTC price fall could make mining unprofitable.