Latest update:

**2014-11-22**. See below for history.

**The spreadsheet extrapolation.** Many people plot log price versus time, find the best fitting line, and extrapolate it. This is what they get:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1.615e+00 4.340e-02 -37.2 <2e-16 ***

day 5.618e-03 4.729e-05 118.8 <2e-16 ***

R-squared: 0.8995

R2 = 90%! This model MUST be correct. Add in the bias -- if this model is correct, we will be making 0.56% per day forever, compounded. Who doesn't like that.

What others point out in response is that the trend cannot go on forever. Although, even if log price vs. time is an S-shaped curve, we really don't know where we are in that curve. If we are near the beginning of that curve, then the linear extrapolation is fine. Or are we further along?

Even if we are near the beginning of the S-curve, there is another issue. OLS assumes that the residuals are independent of each other, which, in a time series, is clearly and completely false. And we should talk about confidence intervals as well, not just a point estimate.

**A better model.** Here is a very basic model, the assumptions for which are actually not violated. I call it the Basic Long-Term Model (BLTM).

diff log price ~ Normal(mu, sigma)

The difference in the log price, which is approximately the daily percent return, has a Normal distribution with mean mu and standard deviation sigma. Am I saying that this is the "true" correct model? No. This is a useful simplification and an improvement on the log price chart extrapolation. It's easy to perform these calculations, easy to explain, and easy to understand.

The model does not take into account the fact that the model parameters -- mu and sigma -- cound change over time. That is simply outside the scope of this basic model. I am using all available data to estimate the parameters.

**Predictions.** Here are the parameter estimates and the prediction of this simplistic model.

n.data from to mu sigma z.stat

1 1,579 2010-07-17 2014-11-22 0.00561 0.0602 0.0932

n.fut date p_5 p_25 p_50 p_75 p_95

1 0 2014-11-22 NA NA 352 NA NA

2 1 2014-11-23 321 340 354 369 391

3 7 2014-11-29 282 329 366 408 476

4 30 2014-12-22 241 333 417 522 720

5 39 2014-12-31 234 339 438 567 820

6 61 2015-01-22 225 359 496 685 1,090

7 91 2015-02-21 222 394 587 874 1,550

8 122 2015-03-24 224 438 698 1,110 2,170

9 152 2015-04-23 230 489 826 1,400 2,970

10 183 2015-05-24 239 550 983 1,760 4,050

11 213 2015-06-23 250 619 1,160 2,190 5,430

12 244 2015-07-24 263 700 1,380 2,740 7,300

13 274 2015-08-23 278 791 1,640 3,390 9,670

14 304 2015-09-22 294 895 1,940 4,200 12,800

15 335 2015-10-23 314 1,020 2,310 5,230 17,000

16 365 2015-11-22 335 1,150 2,730 6,460 22,300

17 404 2015-12-31 365 1,360 3,400 8,480 31,600

18 731 2016-11-22 835 5,640 21,300 80,300 543,000

* In agreement with the log price chart extrapolation, overall, bitcoin has been making 0.56% per day.

* According to the median scenario (p_50), we will hit 1,000 again sometime in early summer 2015.

* According to the pessimistic scenario (p_5), we won't get back to 1,000 even in two years.

* According to the optimistic scenario (p_95), we will break 10,000 by early fall 2015.

* Just for fun -- projected price on 2014-12-31 is 438.

**Break-even point.** If you buy today and thing go bad, how long will you have to hold to get your fiat back?

n.fut date p_5 p_25 p_50 p_75 p_95

1 388 2015-12-15 352 1,270 3,110 7,580 27,400

388 days.

Comments?

**History.***

2014-05-08*

2014-06-26*

2014-11-19