Expected Value for Bitcoin in the Long Term

[Jason]

I wondered if it might be possible to come up with a mathematical model for the value of bitcoin in order to compute an expected value over time.  The goal isn't necessarily to predict exactly what the future value of bitcoin will be, but rather to gather some insight into different investment strategies ranging from selling everything now to holding indefinitely.

As with all models, I'm going to start simple and make some big assumptions which some of you will disagree with.

Assumption #1:  Bitcoin will have a relatively large growth rate, which is going to decline over time as Bitcoin achieves greater market penetration and eventually reaches some sort of value equilibrium.

Assumption #2:  There are risk factors which could cause Bitcoin to fail.  These risk factors are roughly constant with time.  They include things such as the discovery of fatal flaws in the protocol, or multinational efforts to outlaw it and shut it down.

I will model the first assumption with the equation (1 + R*exp(-d*(n-1)))^n where R is the initial growth rate, d is the rate at which the growth declines, and n is the number of growth periods.

I will model the second assumption with the equation S^n where S is the probability of survival in each growth period and should be < 1.

This leads to the equation V = (S^n)((1 + R*exp(-d*(n-1)))^n).

Letting the growth rate period be one year and plugging some vaguely plausible numbers into the equation leads to something I can actually solve (S=0.9, R=9, d=0.5 -- the R=9 reflects a potential 10x increase in value relative to the US dollar this year)

V = (0.9^n)((1 + 9*exp(-0.5*(n-1)))^n)

This is what a plot of the expected value looks like for year 1 to year 4:



As you can see from the above plot (or by maximizing the equation), the highest expected value occurs at n = 3.34 years with an expected value approximately 60 times greater than the initial value.  Beyond that time the expected value begins to decline.

If the assumptions are valid in the general sense (e.g. the shapes of the curves are approximately correct), then this model could be useful for making some rough estimates about the future value of one's bitcoin portfolio, with the caveat that the constants be updated to reflect changing conditions in the market, or the model itself be updated to reflect additional insights.  Some attempt to actually fit the parameters into recent price data would also be very useful.

In creating this model, I have made no attempt to predict the short-term behavior of bitcoin prices, so this would be of little or no use for high frequency trading purposes.  My purpose is only to attempt to predict the long-term behavior of the market.  I make no claims of success, but just wanted to use this simple model to stimulate further discussion on the subject by those with an interest in the subject.

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[bb113]

In my opinion it makes more sense to model different scenarios rather than try to include the chance of failure into the value calculation. I get that its an attempt to incorporate that risk into the calculations, but really seems non-intuitive to me. If I understand correctly, the chart you show doesn't represent any outcome you are expecting in reality, the downslope to the right is actually supposed to be due to a chance of 1 btc=0 usd.

[MagicBit15]

It would just be similar to game theory and using stock projections. The rate it's going now I mean over a hundred by next year I think will be easy I am thinking maybe over 200 honestly!

I would really like to see in 5 years what is going on especially after the next halving.

[Canaanite]

You cannot try and put Bitcoin expected growth into a formula yet, Too many unkown parameters and not enough historical information
If you want to estimate future prices in case of success you should compare it to other similar coins of companies

we do know that the value of the all bitcoins is around 500million usd and we have volume trades published at MTgox
I would try and compare it to Amazon 2012 revenues (61billion usd), Paypal volume 2012 (145 billion) or if you want to see a really high number, choose a mid coin like the Swiss franc and compare the monthly forex volume to the one of the Bitcoin at Mtgox.
That can give you nice estimations but it really means nothing

amazon - 61/0.5=122; 122*60 = 7,320$
paypal - 145/0.5=290; 290*60= 17,400$

Ohad

[nwbitcoin]

This is an exercise in futility!

You can't predict the value of anything unless you have at least some information on what is going to cause that growth.

What we have at the moment is a very rough technical infrastructure with no legal or political backing. 

The more I look at bitcoin, the more I see a similar path to Linux developing.  Its going to take a numbers of years before it gains momentum, and then, suddenly, every is using it, because they didn't realise they were. 

Everyone wants to see the numbers go to the sky, and to some extent they will, but without a time frame, and some idea of real growth drivers, your graph just looks very pretty!

If you are going to make a nice graph, you need to build in a SWOT analysis to give you a far more realistic background which will help create a very useful guide of everyone on the future direction of bitcoin.

[segfault]

I'd hope we could project a very rough growth curve given the set inflation rate and available information (difficulty, network peers, transaction volume etc). We don't have an active forward market so getting a real liquid future value is out, but we should at least be able to project with some reasonable confidence levels against the major currencies.

Thoughts?

[qwk]

As with all models, I'm going to start simple and make some big assumptions which some of you will disagree with.

I will disagree with both.

Assumption #1:  Bitcoin will have a relatively large growth rate, which is going to decline over time as Bitcoin achieves greater market penetration and eventually reaches some sort of value equilibrium.

The growth rate can not be expected to simply decline. I expect a slow and steady, almost linear growth rate at the very beginning. After passing a certain tipping point, i'd expect a huge growth to adapt to the "masses" which would eventually fade into a steadily declining growth as you predict it.
So: linear -> exponential -> degressive? Something like that...

Assumption #2:  There are risk factors which could cause Bitcoin to fail.  These risk factors are roughly constant with time.  They include things such as the discovery of fatal flaws in the protocol, or multinational efforts to outlaw it and shut it down.

The risk factors are definitely not constant with time.
The discovery of fatal flaws in the protocol becomes more and more unlikely over time.
The same goes for the ability of national or even multinational efforts to outlaw it.

[Jason]

Thank you for the feedback I've received so far.  I'm no professional when it comes to creating models, although I have taken an introductory course in the subject as I found it interesting.

In my opinion it makes more sense to model different scenarios rather than try to include the chance of failure into the value calculation. I get that its an attempt to incorporate that risk into the calculations, but really seems non-intuitive to me. If I understand correctly, the chart you show doesn't represent any outcome you are expecting in reality, the downslope to the right is actually supposed to be due to a chance of 1 btc=0 usd.

Yes, the chart is a bit non-intuitive in that it doesn't represent any actual outcome, but rather the expected output.  Perhaps an analogy would help.  If someone is selling lottery tickets for 1 BTC each, and the lottery prize is 1 million BTC, then a rational actor would base his decision on whether to buy a lottery ticket based on the expected value of a lottery ticket.  If there were 500'000 tickets sold, then the probability p of a ticket being the winning ticket is p=1/500'000.  So the expected value is now V = p * 1 million BTC or 2 BTC.  So in this case, it is rational to buy lottery tickets since they have an expected value greater than their cost.

The chart is my attempt at doing something similar with bitcoins.  The question I'm attempting to answer is essentially this:  If I have 1 BTC in my hand right now, how much can I expect it to be worth at some future date.  There are two pieces of the model.  One models future growth with an exponentially decaying growth rate.  The second models some inherent risks of the new technology.

You cannot try and put Bitcoin expected growth into a formula yet, Too many unkown parameters and not enough historical information
If you want to estimate future prices in case of success you should compare it to other similar coins of companies

I think you are suggesting modeling bitcoin prices based on stock prices?  There is already a model that does this and it's called a biased random walk.  You start out at some initial value v0 and then move up or down with probability p0 or p1 respectively.  If p0 != p1 then it's biased.  This type of model is often used with stock prices and in fact doing a linear regression on stock prices in order to calculate the best fitting line through the stock prices is one example of this.  However, from what little I know about computational investing, most stock models would fail to capture the dynamics we have seen in action in the bitcoin market.  This was my motivation for coming up with the different model with growth based on a decaying exponential.  To be sure it is a gross simplification, and can and should be refined further.  I believe there is value in starting simple and evolving complexity.

This is an exercise in futility!

You can't predict the value of anything unless you have at least some information on what is going to cause that growth.

In the field of modeling, many models are based on knowledge of physical laws which regulate a system.  In such cases, one can develop some confidence about the basis for ones models.  However, there are other cases where there is little or no knowledge of the physical laws which regulate a system.  And in such cases, mathematical models can still be created based on observing the data and coming up with reasonable equations to model the data.  There is no doubt that many of these types of models turn out to be failures, but many of them also turn out to be quite accurate and useful.  For example, Schelling's Segregation Model or the Standing Ovation Model are simple models that make gross assumptions about individual behaviors that turn out to be quite accurate when applied on an aggregate basis.  A more complex yet highly illustrative model that lacks a physical basis for reality and yet is well regarded is the Solow Growth Model.

I'd hope we could project a very rough growth curve given the set inflation rate and available information (difficulty, network peers, transaction volume etc). We don't have an active forward market so getting a real liquid future value is out, but we should at least be able to project with some reasonable confidence levels against the major currencies.

Since we don't have (and probably never will have) an understanding of the many forces driving the price of Bitcoin, I don't see any choice but to make some gross assumptions based on intuition and then see how well models based on these assumptions fit the existing data.  One really good example of this can be seen here:  http://en.wikipedia.org/wiki/Neoclassical_growth_model Of course they do have the advantage of being able to tune their model based on how well it fits measured economic growth in the real world.  We lack historical data in the same domain as Bitcoin against which we could tune our models.

The growth rate can not be expected to simply decline. I expect a slow and steady, almost linear growth rate at the very beginning. After passing a certain tipping point, i'd expect a huge growth to adapt to the "masses" which would eventually fade into a steadily declining growth as you predict it.
So: linear -> exponential -> degressive? Something like that...

Actually, I don't disagree with this.  I left off the linear growth rate because I believe we are now in the exponential part of the growth.  For the purposes of predicting the future expected value, fitting the past data in the linear growth phase is not necessary as far as I can see.

The risk factors are definitely not constant with time.
The discovery of fatal flaws in the protocol becomes more and more unlikely over time.
The same goes for the ability of national or even multinational efforts to outlaw it.

Your points are well taken, and again I have no significant disagreement.  However, I still believe that as a first order approximation, modeling the risk factors as being constant is not too bad.  Consider:

• In my opinion, the risk of multinational efforts to outlaw bitcoin have not peaked yet.  It will have to grow considerably more before it is seen as a clear and present danger to TPTB.
• New threats to bitcoin will constantly arise as new technology is invented which could pose a threat (quantum cryptography has already been discussed and while it wouldn't necessarily kill bitcoin, it would require some major changes to be made to the protocol)

Factor these into the equation, and add the caveat that the risk factors are only approximately constant with time.  This suggests that this portion of the model may be reasonable for the next few years (with appropriate parameterization), but will become increasingly unreliable beyond that as we are unable to foresee what threats may or may not exist in the future.  However, if you have a different model in mind, please share it as I would certainly be interested in it.

[mestar]

You took some function with 3 parameters, then you took some random parameter values, and drew that. 

You didn't even try to fit the real data or anything.  Why do you think Bitcoin will grow one year?  Perhaps the growth period is over.  Perhaps, it will grow for the next 10 years.

Also, your function has no similarities to the actual price charts.  I don't think we can learn anything from all that.

[Jason]

You took some function with 3 parameters, then you took some random parameter values, and drew that. 

You didn't even try to fit the real data or anything.  Why do you think Bitcoin will grow one year?  Perhaps the growth period is over.  Perhaps, it will grow for the next 10 years.

Also, your function has no similarities to the actual price charts.  I don't think we can learn anything from all that.

I'm sorry if you can't learn anything from what I've written.  Your comments suggest that you don't know the difference between expected value and predicted value, so I would start there if I were you.

[porcupine87]

I agree with mestar. We can't know. If we could know, we would only deploy right decisions. In a free market information about the future is all. The more information you have the richer you can get. Maybe you have an advantage over others with you formula. Maybe the probability your assumption is right is higher than the assumptions and conclusions of others. Maybe just 1-2% better.

But what you can derive something form the cource of bitcoin to USD. The expectations rise and so the probability that Bitcoin can get a real thing in the future. Prices are the articlation of collective intelligence. Unfortunately it's not perfect, but better than other forecasts (see prediction markets).

Otherwise you can't derive higher future from price rises in the past. As well as you can't predict the bext bend based on the view into the driving mirror.

[mestar]

Your comments suggest that you don't know the difference between expected value and predicted value, so I would start there if I were you.

I see you want straight away to an "attack the person" mode. No need to do that.

You took your R, or "initial growth" as 9, and your d, "grow decline" as 0.5.  Why?

Why not 30 for R, or 3.  Why not 0.01 for d, or 1? 

Basically, you made up those numbers, then your model spit out that the maximum "value" for bitcoin will be in 3.5 years.   That 3.5 is as good as your input parameters.  Garbage in, garbage out.

[sgbett]

I have some coins. When the dollar value of 1 BTC rises by a multiple. I convert a percentage of my remaining BTC into fiat.

If BTC collapses you get a consolation of having amassed a bit of fiat.
If BTC takes over the world you didn't make the mistake of selling out early.

Choose values depending on what you want in each scenario. On paper, its pretty easy to not lose. Gotta pick your values and stick to the plan though

[Jason]

I have some coins. When the dollar value of 1 BTC rises by a multiple. I convert a percentage of my remaining BTC into fiat.

If BTC collapses you get a consolation of having amassed a bit of fiat.
If BTC takes over the world you didn't make the mistake of selling out early.

Choose values depending on what you want in each scenario. On paper, its pretty easy to not lose. Gotta pick your values and stick to the plan though

This is exactly the sort of strategy I'm using myself.  The trick, as you say, is picking your values.

The point of my attempt at a model was to act as an aid in picking some reasonable values, rather than just picking them arbitrarily.

[Jason]

You took your R, or "initial growth" as 9, and your d, "grow decline" as 0.5.  Why?

Why not 30 for R, or 3.  Why not 0.01 for d, or 1? 

Basically, you made up those numbers, then your model spit out that the maximum "value" for bitcoin will be in 3.5 years.   That 3.5 is as good as your input parameters.  Garbage in, garbage out.

Please do me the courtesy of at least reading what I wrote before asking questions which I have already answered.

This leads to the equation V = (S^n)((1 + R*exp(-d*(n-1)))^n).

The above is the general equation for my model.  If you want to call it, "garbage" then do so by pointing out what is wrong with the model, not with an arbitrary set of parameters.

Letting the growth rate period be one year and plugging some vaguely plausible numbers into the equation leads to something I can actually solve (S=0.9, R=9, d=0.5 -- the R=9 reflects a potential 10x increase in value relative to the US dollar this year)

V = (0.9^n)((1 + 9*exp(-0.5*(n-1)))^n)

The above values were used solely for illustrative purposes (e.g. to show you the shape of the curve).  It is obvious that by changing the values of the parameters you can change the peak and the maximal value to whatever you desire.

Also note that a couple of paragraphs later I wrote:

Some attempt to actually fit the parameters into recent price data would also be very useful.

So if you want a set of parameters that are not "garbage," then why not go fit them into the model yourself and then see for yourself what results?

[mestar]

The above is the general equation for my model.  If you want to call it, "garbage" then do so by pointing out what is wrong with the model, not with an arbitrary set of parameters.

Well, nothing is wrong with the model.

 V = (S^n)((1 + R*exp(-d*(n-1)))^n)

You could rewrite that as

( S * (1 + R*exp(-d*(n-1))))^n
( S + S*R*exp(-d*(n-1)) )^n

So, your parameters S and d are both slowing the growth, just one does it lineary, and the other does it exponentially with time.  Also, the influence of "d", and with it R, dies very quickly, and your model goes to S^n for large n, as long as "d" is not zero.   

My guess is that you could replace S and d with just one parameter.

Looking at this:
http://bitcoincharts.com/charts/mtgoxUSD#tgTzm1g10zm2g25zv

Any model would have problems in November 2011 if you look only at data up to that moment.

Another guess is that a simple quadratic polinomial model on a log chart would have all the power of your model.

By all means, fit the actual data and post the results.

[Jason]

V = (S^n)((1 + R*exp(-d*(n-1)))^n)

You could rewrite that as

( S * (1 + R*exp(-d*(n-1))))^n
( S + S*R*exp(-d*(n-1)) )^n

I wrote the equation as I did so that the two components of it would be separated out in an obvious fashion.  Namely, the growth factor and the risk factor.  You can attempt to simplify it if you wish, but I don't see the point.

So, your parameters S and d are both slowing the growth, just one does it lineary, and the other does it exponentially with time.  Also, the influence of "d", and with it R, dies very quickly, and your model goes to S^n for large n, as long as "d" is not zero.   

My guess is that you could replace S and d with just one parameter.

You are getting distracted with the equation itself and losing sight of the model it is representing.  The intent here is to model two separate factors:

• 1.  The risk of systemic failure (brought on by a multitude of possibilities) modeled by S^n
• 2.  The growth, modeled by (1+R*exp(-d*(n-1)))^n

If you were to attempt to fit the model to real data, you would only use the equation from #2 above, not from #1, since #1 converts the formula from a long-term predicted growth model to a long-term expected value model.

Looking at this:
http://bitcoincharts.com/charts/mtgoxUSD#tgTzm1g10zm2g25zv

Any model would have problems in November 2011 if you look only at data up to that moment.

If I had any interest in modeling the short-term future value of bitcoin, then I would choose a completely different model more suitable for such things (such as a biased random walk -- though I would not choose to use that model myself for reasons which are way outside the scope of this thread).  Again, I think you misunderstand what I am trying to do here, which is to compute a future expected value.

Let me go back to my lottery example which I used earlier.

If you are holding a lottery ticket in your hand and the lottery pays 1 million BTC, then what is the expected value of that lottery ticket?  It depends on how many lottery tickets have been issued.  If 100'000 tickets have been issued, the you can compute the expected value to be 10 BTC.  That is the average value of a lottery ticket.  999'999 tickets will be worth 0 BTC, and 1 will be worth 1 million BTC.

Similarly, I'm not exactly trying to predict how much a Bitcoin will be worth in 1 year, or 2.34 years, or whatever.  I'm trying to get an idea for the expected value of 1 BTC in my hand today based on long-term growth rates and risk factors.  My model is factoring risk into the equation and thus it is not predicting the actual value of BTC, but rather first predicting the actual value of BTC based on the growth portion of the equation, and then discounting the value based on risk factors.

The purpose of this is to develop a rational strategy for selling ones bitcoins so as to maximize the value returned.

Another guess is that a simple quadratic polinomial model on a log chart would have all the power of your model.

By all means, fit the actual data and post the results.

Don't take this the wrong way, but once again you are showing that you do not understand what I am attempting to do.

Anyway, I have obviously made a mistake in bringing up this subject here and I will not repeat it again in the future.

[tickets]

Bitcoin is just a (virtual) commodity which functions as a medium of reliable value exchange.

Its price is set by fundamentals, by supply/demand, and by speculation.

Why not try to conjecture about future price targets by trying to infer a valuation, based on potential market etc.
This should yield at least an order-of-magnitude kind of number which would be better than nothing.

Using very rough numbers, if $1B worth of transactions are made using 10M coins, the price per coin should be somewhere
around $100/BTC, right?    We already have example market sizes and growth rates from Silk Road and 2 main gambling sites -- it would be an interesting
exercise to project that growth rate forward for those markets. 

The question of whether the current price is 'reasonable' could then be addressed.  Equities trade at some multiple of current earnings (commonly 15x) in anticipation of future growth...   

Has anyone tried to put together long-term targets using that kind of reasoning?  If not, I'll put up a blog post doing just that, of course for constructive  critique..

[jubalix]

This is an exercise in futility!

You can't predict the value of anything unless you have at least some information on what is going to cause that growth.

What we have at the moment is a very rough technical infrastructure with no legal or political backing. 

The more I look at bitcoin, the more I see a similar path to Linux developing.  Its going to take a numbers of years before it gains momentum, and then, suddenly, every is using it, because they didn't realise they were. 

Everyone wants to see the numbers go to the sky, and to some extent they will, but without a time frame, and some idea of real growth drivers, your graph just looks very pretty!

If you are going to make a nice graph, you need to build in a SWOT analysis to give you a far more realistic background which will help create a very useful guide of everyone on the future direction of bitcoin.

 no legal or political backing. 
<<

this is what make it strong

[Wekkel]

Has anyone tried to put together long-term targets using that kind of reasoning?  If not, I'll put up a blog post doing just that, of course for constructive  critique..

Hess, as a matter of fact it has been done quite a few times. Just looking at market cap and the size of online business, $10,000-$100,000 is not that unrealistic. But that requires quite some work on Bitcoin software to iron out the flaws.