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.
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.