Mining hardware resale value vs projected return

[Altoidnerd]

I wrote an entry level type editorial on the important question of whether to keep a piece of mining equipment or whether to sell it.  I would like feedback from the forum if you guys don't mind.  I don't think the conclusions will be particularly shocking to anyone here, but perhaps at least entertaining.

http://altoidnerd.wordpress.com/2013/11/07/when-should-i-sell-my-bitcoin-mining-hardware-bitcoin-mining-hardware-resale-value-vs-projected-return/

Please point out any stylistic problems or factual mistakes....whatever you can do.

I do plan to expand the analysis quite a bit, which may be more useful or interesting here.  There's a lot you can do with the general idea of the geometric approximation to the total BTC value intrinsic in a miner.

Thank you

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[optimator]

Great article. It's well written and easy to understand.

Just a quick question... You take the estimated btc for the current 2 week period and divide it by the "common ratio". Why not the percent increase? (y2 - y1) / y1

Cheers!

[Altoidnerd]

Great article. It's well written and easy to understand.

Just a quick question... You take the estimated btc for the current 2 week period and divide it by the "common ratio". Why not the percent increase? (y2 - y1) / y1

Cheers!

Thanks for asking.  The answer is a bit mathematical but I'll see if I can explain that supposing you have not seen a geometric series before.

I divide it by I(1 - common ratio) because of a mathematical result that coincides with my model, which I chose for its simplicity.

I say let us call this value of the 1st 2-week blocks total earnings... I dont know, how about B for bitcoin.  Ok so

total earnings in first two week period = B

Then I postulate it is sufficient to say therefore the total earnings in the following two weeks will be a certain fraction r of last times:

total earnings in the 2nd fortnight = r * B [/center

(Note that a fortnight is 14 days.  Bitcoin has given us reason to finally use this hilarious word).

I have furthermore supposed that the ratio of earnings between to fortnights is always r.  This is equivalent to saying that the difficulty rises by the same fraction each time it increases.  If you look at this chart:

http://blockchain.info/charts/difficulty

you can obtain "r" by dividing the difficulty between steps.  Is the ratio, r, between adjacent steps actually always the same?  Not really.  But it is loosely constant within each 4 or 5 steps - and 4 or 5 steps is 10 weeks which is actually about the useful life of a miner.

So then

3rd fortnight's earnings =  r * (second fortnights earnings) = r*( r*B ) =  r² *B

and the 4th fortnight yields r^3 * B...and the nth fornight r^(n-1) * B which looks like

total bitcoins ever = 1st fortnight's + 2nd fortnights + 3rd fortnight's + ... = B + r * B + r²*B + r³*B + r⁴*B + ... on and on forever

and if you look at the last line of the proof given here

http://en.wikipedia.org/wiki/Geometric_series#Formula

you will perhaps understand why I have said take the first fortnight's earnings and divide by (1-r).

[optimator]

Wow! What a great, detailed explanation.  I understand the calculation of the sum of a geometric series, but I guess I'm still confused about the denominator.

if, d1=390 & d2=510, then r= d1/d2= .76

and your denominator is 1-r = .24

This is equivalent to saying that the difficulty rises by the same fraction each time it increases.

So are you saying difficulty rises by 24%? That's the part I don't get.

390 * (1+24%) = 484. This isn't 510.

The increase was (510-390)/390 = 30.7%

I'm sure I'm mixing something up.

[Altoidnerd]

Wow! What a great, detailed explanation.  I understand the calculation of the sum of a geometric series, but I guess I'm still confused about the denominator.

if, d1=390 & d2=510, then r= d1/d2= .76

and your denominator is 1-r = .24

This is equivalent to saying that the difficulty rises by the same fraction each time it increases.

So are you saying difficulty rises by 24%? That's the part I don't get.

390 * (1+24%) = 484. This isn't 510.

The increase was (510-390)/390 = 30.7%

I'm sure I'm mixing something up.

Hmmm the common ratio itself, r represents the fractional decrease in mining profits with each diff change¹

That is why you compare the two difficulties.  So if the common ratio is .76, that means you will mine 24% less this fortnight than last fortnight yes.  BUT  don't make the leap with the 1/(1-r) thing.  The 1/(1-r) term keeps tracks of the infinite sum, so I wouldn't bother even thinking of that as some fractional change.  

Proof:

 A = 1 + r + r² + r³ + r⁴ +  ...
    = 1 + r ( 1 + r + r² + r³ + r⁴ +  ...)
    = 1 + r A

A = 1 + r A    thus

A = 1/ (1 - r)

You can see that there's a little infinity thing happening there so you do not want to worry about 1/(1-r) being some sort of fraction.

¹ Note that your actual mining profits will not exactly feel this type of sharp dropoff...the actual way that works is more complicated but I don't think it is worth it to count the effects of such things.  This model works alright despite its simplicity.