Quote from: d5000 on September 26, 2024, 10:17:36 PM

I think we can agree that we're searching for a formula responding to the question: "how can I time the market if the price behaves a bit like in the past and Bitcoin does not fail."

The formula can not only help us to time the market. We can also decide if Bitcoin is experiencing historical usual price volatility (stay calm and Hodl) or if Bitcoin breaks historical price patterns (indication that Bitcoin might fail). Third we could see a downward shift in Bitcoin's price trend growth, which would lower the optimal allocation of life savings into Bitcoin according the the Kelly criterion or my hedonistic optimization.

Quote from: d5000 on September 26, 2024, 10:17:36 PM

it would make sense to concede more weight to the later price data than to the earlier movements

In my previous posts I estimated the trend function with all available data (2010-2024, black). However in the past we didn't have all price data, e.g. on 02.02.2020, we only had data until 01.02.2020 to estimate the Bitcoin price trend. So the green line shows a fair Bitcoin price with a trend function estimated with all the data until one day before the current day (2010-previous day, green).

We can also limit our data set to estimate our trend function based on the last X years of price data (last 4 years = red, last 8 years = dark yellow). Estimating with more years is vulnerable missing a recent breaking of our trend. Estimating with less years makes us vulnerable confusing cycle an trend. During euphoria we estimate a higher Bitcoin price trend growth than during the bear markets.

I think basing our Bitcoin price trend function on only the last four years would be too short. 2017-2018 the red line was a poor indicator of the Bitoin price trend. The green line would have been much better than the red line. I would therefore take the last 8 years to get a reasonable stable trend line.

We were already discussing how stable our Bitcoin price trend slope is. Is it lower than 5.83? How stable is the slope? Does it change if we look only on recent data? The results are shown in the 2nd chart.

Calculating or trend Bitcoin price trend growth based on all data until the day before (green line) shows quite a stable trend. Taking only the recent 4 years into consideration the trend moves between 1.9 and 9. However we have seen that 4 years is too short. Taking the last 8 years we see a slope that is declining but still more than 5, which still translate in a high annual yield of 36% (instead of 41% for 5.83, or 12% for 1.9).

In the 3rd chart we can see if Bitcoin is over- oder undervalued vs. trend, depending when we calculate the trend and how many recent years we take into account for our trend calculation. The picture stays roughly the same no matter with time horizon we take to base our trend estimation on.

Quote from: d5000 on September 26, 2024, 10:17:36 PM

For this reason, I'd like to ask you a particular question: what's your reason to use the natural logarithm (base e) and not another function? Did it "just" seem to fit with the "straight line", or is it based on another theory, e.g. some kind of "natural" exponential growth?

The natural logarithm for time and price fits the data well. Exponential growth functions would suggests much higher prices by now. Maybe there is some kind of natural growth in the nature of Bitcoin price, Bitcoin adoption, address usage, etc. However verbally you can make up plausible reasons for everything and its opposite.

Quote from: d5000 on September 26, 2024, 10:17:36 PM

I've re-read the first page and while you discussed an alternative with Tubartuluk based on "days ahead", I've not seen the answer to this particular question. So if you want you can explain this here

During the Bitcoin bullmarket we saw price spikes carrying the Bitcoin price significantly higher than trend. However those price spikes become smaller and smaller as time goes on.

By calculating the bull market price deviation not with e^X but with days ahead, we get more stable price spikes, making it easier to decide when to sell.