A Better Mining Calculator

The current mining calculators are linear. By our observations, the key drivers to projecting mining profitability exhibit exponential behavior. A better model is needed.

I’m not claiming our model is perfect – but it’s better than a linear projection.

The Simulation Model

Premise

So if you use any of the other popular mining calculators, you punch in your miner details and it tells you how much you’ll make today given your contribution to the global hash rate. Then in multiplies it by 7 to get a weekly projection. By 30 for a month. By 365ish for a year. Those pictures start to look real rosy, especially during a paradigm shift when a more efficient piece of hardware is introduced and the calculator says to you, “I know there’s no such thing as a get-rich-quick scheme but this one’s for real!”. This isn’t to say that mining isn’t profitable. It is. Wildly. But a better method is needed to more accurately project what a miner will make such that one can make more accurate plans.

The premise of the model is that there are three primary driving variables to the amount of value produced by a miner: hash rate, the global hash power, and exchange rate.

Hash rate is an independent variable. It’s supplied by the user and represents the individual hash contribution (per second) that the miner contributes to the global hash rate.

The global hash rate is the sum total of all miners focused on finding a block for a given network at a given time; the total sum of all of the individual hash rate contributions.

The exchange rate (hereafter “fx rate”) is the ratio of the cryptocurrency the miner produces to another variable in which value is more relatable to the user. Fx rate is obviously not required to determine value, but to the uninitiated it would be akin to attempting the mental math required to convert kilometers to miles in which the ratio swings wildly on any given day.

So global hash rate isn’t static. It’s growing – there are always new miners for sale. Miners have been getting faster contributing more and more hash power to the global network. So as more machines are added to the network, the hash rate grows. This needs to be modeled.

The fx rate is also changing. Now I’m the first to tell you that the answer to whether or not the fx rate will go up or down is “yes… probably”. But there is information to be gleaned from how its behaved in the past, and how it behaves with respect to other things that may or may not be related.

Deciding what is related to what, or which variable is the cause or the effect – that’s not for me to determine. Instead we model the correlation – which as they say does not imply causation – between global hash rate growth and fx appreciation/depreciation.

The Math

The Simulation Model doesn’t presume to know anything about either of the variables. Global hash rate and exchange rate are modeled as simple geometric Brownian motions. These models follow a differential stochastic process known as a Weiner process: There’s a drift or at trend component, and a random component governed by statistics like the standard deviation.

We grab hash rate history and exchange rate history and collect statistics using an exponentially-weighted moving average model. This method has the benefit of weighting more recent events more heavily and letting the effect of more distant data points fade into obscurity. This allows us to collect information on the mean which we use as the drift, and standard deviation which is represented by sigma in the model above.

Finally, we need to generate the random variables used in the simulation. To do so, we need to generate a value for each variable that makes sense if it were to happen together. You need to generate values that form the normal curve (image right) that was the input you gave it, with information on the distribution (sigma), the lean toward positive or negative (skewness), or the girth of the tails (kurtosis).

To perform this process multi-dimensionally, we use a create Gaussian Copulas, or uniform multivariate probability distributions (image left). Imagine taking one of the two-dimensional normal distributions and spinning it like a dreidel in three dimensions. Then recognize that each dimension might be shaped differently, like in the image above right.

This allows us to create scenarios in which both variables exist in a space that are occur within the probability space defined by the time series of co-movements.

In short, we grab the time series to determine how much the global hash rate growth varies (volatility), the direction it’s been headed lately (the mean), and how things tend to move together.

To simplify these calculations, we had to make some sweeping assumptions.

As we built this model over the past two weeks in the second half of September 2017. This was a particularly interesting time to perform this analysis. It yielded some interesting results.

The Simulation Model Results

The variance is too high, for a myriad of reasons.

First and foremost is recent history. We’re observing the beginnings of new paradigms of all three currencies under observation: Dash ASICs were introduced days ago, Litecoin a few months ago – still within reach of our ewma’s 72-day lookback period, and Bitcoin just forked contentiously allowing the same miner to target two different currencies: Bitcoin or Bitcoin Cash resulting changes of 50% or more to the global hash rate as the hash rate swings between BCH and BTC. Of course the global hash rate will explode up the J-curve to infinity when the standard deviation is built off of observations where the hash rate swings 50% a day.

We’ve enabled use of this model, but we expect until things calm down (if they ever calm down), this model may produce some wonky results.

The Batch Model

During an internal discussion, we realized we had a pretty good idea of how much the hash rate grows. While creating this model, we accurately projected the hash rate growth of the dash network. Bitmain even felt obligated to clarify that they don’t’ disclose batch sizes conspicuously close to the release of our projection – something I’m not sure they’ve had to do before, indicating we were likely on the right track.

So if that’s the case, we know what the hash rate is. Roughly 4000 units every 3-4 weeks. Why bother projecting it? We’ll simulate fx rate independently and grow the hash linearly.

Try it out.

Sweeping Assumptions

Hash Rate

-continuous

Price

-IR parity

-fx arbitrage

-

Payments

The next thing to consider would be the pool size and the probability that the pool finds x blocks per day. To simplify the model, we assume that no one pool has advantages over another. You know, nobody is gaining an advantage through ASIC boost, mining empty blocks, or just generally severing the elegant symbiosis between capitalistic motivations and benefit of the network. Therefore, the long run equilibrium state is that pool distributions trend toward the hash reward being evenly distributed weighted on hash power.

This makes the math tremendously easier to work with, as we can allocate the daily rewards of the entire network by individual hash rate contribution. Now it’s continuous instead of discrete.

Perhaps more concerning…

Things we could incorporate but willfully chose to omit:

The obvious next step is to fix our time series. We need to improve our data source such that hash rate is imputed from block time directly observed from each respective blockchain. You’d then want to run some sort of smoothing algorithm to reduce the aforementioned volatility problem.

Once block time granularity is achieved, you’d want to project per block. You’d therefore need to add the independent random variable of block time, known officially as luck. It’s possible Miner #247 finds the solution with a lucky guess five whole minutes ahead of schedule. This would allow you to run paths with a time-step interval of a block.

You’d then capture difficulty adjustments. Every 2048 blocks you’d adjust the difficulty at that snapshot, resulting in more accurate network behavior.

For profitability sake, there’s a few other considerations including:

Transaction fees (on the network)

Transaction fees when converting to USD

Bitcoin halving

Other currency FX rates, for those of you living outside the US. I seem to recall that you can find this data from the Federal Reserve Bank of St. Louis. Note that this information would need to be simulated as well and added to the covariance matrix.

You could include interest rates if you wanted to present value the investment. You can find interest rates (or ones literally close enough for government work, as they say) from ISDA who has to make them public for Credit Default Swap standardization models. Thank the credit crisis.