9.15 A fair price for an ASIC miner

26th April 2013

Previous posts in series

0. Introduction

I was recently asked by a couple of bitcointalk.org forum members, Rampion and zedicus, what I thought would be a reasonable BTC price for Avalon batch #4, if the delivery date is sometime in July. This is a very interesting question - how do you decide what a reasonable price is, given that the timeframe in which scant profits can be earned is short? It's a question that applies equally to all ASIC manufacturers, and having an index of a maximum price limit vs estimated date of arrival allows one to make decisions not just about how much you should pay for an Avalon batch 4, but allows you to compare prices between manufacturers.

At the moment, pricing schema has become somewhat guess and giggle. Initially, ASICs were priced in  US$, which allowed a simpler comparison between devices but made estimating a return on investment difficult. Recently Avalon made the interesting choice of pricing batch 3 devices in BTC. This makes a kind of sense now and for the foreseeable future while the difficulty is low enough that electricity and manufacturing costs are a negligible compared to profits. 

The exact method used was: 

"The price of each unit is the current mining difficulty which at the time of writing, just got readjusted to about 6,695,826. We take that number and multiply it by two ( predicting the network speed will double. ) and calculate the return in a thirty day window, which is about 75 bitcoins. See this site for more details."

So the price was what they estimated the thirty day earnings to be once the batch 3 miners were delivered - about 1.12 BTC/Ghps. It's an arbitrary choice, but since Avalon appeared to be the only manufacturer delivering they had no problems selling all their devices at this price. Was it a fair price?

In contrast, the Butterfly Labs Bitforce SC Single, rated at 50Ghps, is currently priced at US$2,499 (16.44 BTC)  or 0.34 BTC/Ghps. The ASICMiner auction for their 10Ghps blades went as high as 76 BTC or 7.6 BTC/Ghps. Clearly, there's much confusion about what a suitable price for an ASIC miner is at a given time.

If ASIC miners' prices are based on their ability to earn, we would expect the BTC price per Ghps to reduce as difficulty increases. Certainly if you wish to purchase an ASIC miner and are using an estimate of the bitcoins it earns in a given time period to decide on a suitable price, the date the device arrives will be the largest influence on how much you should be willing to spend.

This method of pricing has limits - if difficulty increases faster than the exchange rate, then at some point a BTC price based on what a miner can earn is not going to cover manufacture costs, and may not be affordable to miners once electricity costs are significant. For the moment retail prices seem to be headed toward btc pricing, so that's (partially) what this post is about.

1. Some basics
If the network hashrate is constant, your expected btc earnings are easy to accurately determine:

btc per unit time = hashrate / 2^32 / difficulty * bitcoin reward

cumulative btc = hashrate / 2^32 / difficulty * bitcoin reward  * time units

If the network hashrate is changing, it's a little harder to figure out how much you should earn. I've seen estimates of earnings based on exponential decay for times when the hashrate increases linearly but that is an incorrect model. When hashrate increases linearly, reward per second can be estimated as:

btc per unit time = hashrate / (at + b) * blocks per unit time * bitcoin reward

where a = hashrate at onset of linear network hashrate increase
      b = increase in hashrate per unit time

      t = time since onset of linear network hashrate increase

If we integrate this from time t = t0 (some time after the onset of linear network hashrate increase) to time t = t1, then we can estimate the total earnings of a miner with hashrate H in the epoch t0 to t1:

H / (at + b) * r * B = 1/(t + b/a) * r * B * H / a 

Indefinite integral 1/(t + b/a) dt = log(t + b/a) + constant

btc per epoch = (log(t + b/a) - log(b/a)) * r * B * H / a 

              = log(t * (a/b) + 1) * r * B * H / a 

where H = miner hashrate per unit time
      r = block rate per unit time

      t = seconds since onset of linear network hashrate increase

The decrease in earnings per unit time is proportional to 1 / t, and the cumulative earnings are logarithmic. This means:

1. If you start mining just at the onset of linear network hashrate increase, most of your earnings will occur earlier rather than later.
2. The longer you start mining after the onset of linear network hashrate increase, the closer your accumulated earnings approach that of a miner in a network with a constant hashrate.
3. There is no short term cumulative earning asymptote as there would be if the decay in earnings per time period was exponential. A maximum will eventually be reached, but it will depend on the exchange rate, the cost of electricity and the bitcoin mining difficulty.

So, if the network hashrate increases linearly ( it is unlikely to increase at a polynomial rate for any length of time, unlike the exchange rate recently ) then there is no simple way that I could think of to determine a "fair price". If there is a way to do this, please post in the comments.

Since I couldn't determine a method to determine a "fair price", I decided it would be better to present expected earnings figures in a format that will allow prospective ASIC purchasers make the decision for themselves, regardless of whether they are purchasing retail, wholesale or second hand.

2. BTC per Ghps cumulative earnings threshold

In the chart below I provide the following data, based on my assumptions from the last post in this series:

1. The days required (tile colour) for a miner to earn more than a minimum earnings threshold in btc per Ghps (y axis), depending on the deployment date (x axis). 
2. The cumulative earnings in btc per Ghps over 60 days minus the minimum earnings threshold in btc per Ghps (net profit, tile colour and y axis, respectively), depending on the the deployment date (x axis). 
3. Similar as 2. but expressing the cumulative earnings in 60 days minus the minimum earnings threshold as a percentage of the minimum earnings threshold - a percentage return.

The dates and the 60 day limit in the charts are a limit determined by my confidence in the model. I'm still not confident to model the network hashrate past August until I have a better idea of which and when manufacturers devices will be deployed. Note that I've used labelled contours instead of a colourbar to indicate values.

Notice that the BFL Single SC at $2500 / 50 Ghps (currently 19.2 btc / 50 Ghps, or 0.38 btc / Ghps) is quite profitable, right up to the end of May. It would need about twenty days mining to pay for itself, and once the 0.38btc / Ghps threshold is reached will return 200% of the purchase cost within the next seventy days. Based on this estimate, if I could be guaranteed delivery by the end of May I'd purchase a BFL Single SC.

Dataset is here.

3. Conclusions

• If the network hashrate increases linearly, cumulative earnings increase logarithmically.
• The earlier mining is deployed after the onset of the long term linear hashrate increase, the more is earned earlier compared to later. The later mining is deployed after the onset of the long term linear hashrate increase, the more the cumulative earnings approximate that of a miner in a network with a constant hashrate.
• Since there's no simple short term cumulative earning maximum, I haven't been able to comply with the request Rampion and zedicus made - I hope the charts above are sufficient.

If you're asking me what I would choose, I'd be using the 100% line in the bottom plot to determine what I'd pay per Ghps for a mining device priced in BTC, so that I'd expect to have paid for the mining device and cleared the same amount again within sixty days - but only if I had a reasonable idea of when I'd be able to deploy it.

4. But wait! A present for Rampion and zedicus
Since zedicus was kind enough to donate to the blog, I decided to answer their actual question a little more closely:

"... what would you believe to be a reasonable pricing for Avalon batch #4, if official shipping date is "early June" and therefore most likely delivery date in July."

Well, I've explained why I can't provide an estimate for a reasonable retail miner price - I simply don't know how to derive such an estimate (although Meni Rosenfeld probably could).  What I've addressed in the next plot is the request for a July deployment schedule - actually I've covered May, June and July, and a ninety day cumulative earnings limit instead of a sixty day limit.

This of course takes the estimates well past my limit of confidence in the model, so take the results with a grain of salt. I'll post an update of the chart above whenever new devices priced in btc are announced, so you'll have an estimate closer to the date.

Dataset (or at least as much of it as Bitbin will allow me to paste) is here.

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