Dollar cost averaging Bitcoin - can we do better?

[hd49728]

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.

It is natural that in bull market, models can give over estimation while in bear market, models can give underestimation and sometimes there are wild periods that make good models seem to be broken, and invalid.

Your discovery is interesting but it can be explained, with higher price, bigger market cap, it's understandable that later market cycles and future cycles will bring smaller ROIs than past cycles.

This chart of Mayer Multiple reflects this fact too.
https://charts.bitbo.io/mayermultiple/

[MaxMueller]

Just now i noticed that i missed last month of discussion.

@d5000: the approach is the same like in early S2F-Models, which finally brokes to my mind when it left the path of log regression. But there is a common missunderstanding, log regression just fits a line to a bunch of data points. As every day close price adds a new data point the regression curve is outdated every single day, but of course a few days does not make a huge different. But the last S2F model was not updated for years and thats why was far beside track.
Regarding the function you could choose whatever function you like. With 5000 data points you could choose y=ax^5000 + bx^4999 + ... to hit every data point of the past precisly, but having 100% chance to miss the next future data point because it would be extremely overfitted. Therefore you want to keep your function as simple as possible and log regression is a simple fit for any kind of exponential growth and simple to calculate as well.
Regarding the time horizon you want to stay with history data points >> investment period. To find a good exit point during the current bullrun i guess it would be sufficient to only take the last 8 years (i agree that 4 years is too less due to covid influence). But for long term investment period (hodl) you wanna take all available data points even knowing that leads to an too optimistical result. But it does not really matters as your allocation rebalances with kelly, if you choose weekly or monthly rebalancing you have some kind of DCA withrawal plan.
The discussion ln above trend vs. +days ahead is by the way just a question of presentation, the underlying results do not differ at all.

@virginorange: could you please update the following grafic

What kind of valuation would be realistic 12 months from now?



The orange trend line shows for a given historical bitcoin valuation: "How did the Bitcoin price change after 12 months during the last bull market?"
The red trend line shows, for a given historical bitcoin valuation: "How did the bitcoin price change after 12 months during the last bear market?"
The green line shows the current valuation of bitcoin.

I would be curious were the yellow dots of the last 6 months have occured and what conclusions we can take for my interpretation here.

[virginorange]

...
I would be curious were the yellow dots of the last 6 months have occured and what conclusions we can take for my interpretation here.

I have updated the chart to include the most recent data points until 16th of Oct. 2024:



The most recent data points confirm that low current valuation leads to high returns within the next 12 months.

We can also see that our blue data points (2023) look a bit better than our green data points (2022). For a given valuation we got a higher yield in 2023 than in 2022. Can we confirm our spiral?



Separating the year 2023 by quater doesn't look like we are moving to the top right. Which is consistent with the poor Bitcoin performance the last couple of months. However we are also not lower than our bear market trend line.

What would happen, if the current Bitcoin price would freeze for the next 12 months?



If Bitcoin does not break out higher, we would remain on our current bear market line.
If we get a decent Bitcoin price performance in the next 12 months we will continue our spiral.
If Bitcoin price drops significantly, this could indicate model break down. However I consider model break down a very unlikely future scenario given the low Bitcoin volatility and the good regulatory adoption in the US (options, MSTR, ETFs).

Conclusion:
❶ The relationship (cheap Bitcoin = good returns) remains in place. In combination with Bitcoin currently being moderately cheap vs. trend should make it a decent investment for the next 12 months.
❷ We have no evidence that the relationship between Bitcoin valuation and 12m-return is breaking to the downside (below the lower bear market line), which would still yield a positive 12m-return given the current valuation. Bitcoin is currently 18% below trend. Even the bear market line indicates a e^0.1 = 10% return for the next 12 months (68.000€ per Bitcoin).
❸ We have some evidence that Bitcoin yield is higher than the current bear market line, but certainly lower than the previous bull market line. We have not enough data to pinpoint the expected yield for our current valuation, but it should be between e^0.5 = 64% = 100k per Bitcoin and e^1.0 = 171% = 170k per Bitcoin, which would be between last bull market and last bear market line. 100k per Bitcoin would be trend by then. 170k would be e^0.5 or around 500 days overvaluation. 500 days of overvaluation could be also a good moment to sell Bitcoin for traders or people needing cash in the near future.
❹ Bitcoin price has high probability to outperform gold, the S&P and the cost of capital (my mortgage) in the next years and thus should be heavily included for our long term savings. However vor short and medium term liquidity management we should not forget cash and Gold/S&P.


How robust is our Bitcoin trend price estimation?
My trend price calculations assume that the trend price function form [liniar relationship between ln(time) and ln(price)] and the parameters of the function should be quite stable over time. The function already assumes Bitcoin price growth is decreasing, but it could be possible that Bitcoin price growth is recently decreasing quicker than historically observed. This would not impact our decision to hold Bitcoin for the long term, but it would have a minor impact on the proportion of total current and future life savings we allocate to Bitcoin. It would have a major impact, if we want to trade the cycle.

I estimated the Bitcoin trend function for all of Bitcoins history as well as for the last 4, 5, ..., 13 years.



Bitcon's recent price performance (last 4-8 years) seems to be on the weak side. The expected trend price growth of Bitcoin also based on recent history far exceeds other assetclasses. However if we base our trend evaluation on recent years, Bitcoin could be slightly more expensive than trend. This information would not impact our decision to buy Bitcoin as of today, but we could overestimate Bitcoin's price at the next peak and thus miss our exit (if we plan to exit this cycle).


How stable are my Bitcoin price estimations?

We can't directly observe the Bitcoin price trend, but we can estimate a Bitcoin price trend. If we take a long history, we can estimate the Bitcoin price trend without being confused by the Bitcoin price cycle. If we take a shorter history, we can see a breaking in the Bitcoin trend earlier, but the cycle confuses our results more. We have a trade off between long history (to remove the cycle) and a short history (to see a change in trend).

I estimated the Bitcoin price trend on:
(i) the basis of all data up to now
(ii) all data known point in time
(iii) last 8 years
(iv) last 6 years
(v) last 4 years

I can then calculate the difference between trend and current price to estimate the level of over- or undervaluation of Bitcoin as shown in the blue line in the chart "BTC_vs_trend".

I can also calculate realized forward return for the next:
(a) 24 months
(b) 12 months

The chart on the right side shows the relationship between the Bitcoin's valuation (calculated as described in i to v) vs. Trend and forward return (a or b). The green line shows Bitcoin's the current valuation versus trend for a given trend model.













So what kind of forward return can we expect right now?
Very decent but not overwhelming returns, with some possible downside.



The rather weak Bitcoin price performance in the recent 4 years indicates gives us much lower Bitcoin price estimates for models based on the most recent price data. This results in some variance of our model outputs. However most estimates for the Bitcoin price in EUR are positive. On average over all models we could expect 15% price return in 1 year and 45% return in 2 years.

Please be aware that those are point in time measurements and during the next 2 years there will be likely higher and lower Bitcoinprices than indicated as well.


TL;DR

What I found:
❶ Sufficient evidence of higher trend price growth before 2013.
❷ Not enough evidence to divide the time series further.
❸ I revise my fair Bitcoin trend price estimation down to 63k EUR.


Consequences for…:
… cycle trading: Some decrease in trend price, slightly over valued instead of slightly undervalued.
… long term Bitcoin allocation: Some decrease in trend price growth, but no different allocation


Why do we need to monitor the Bitcoin trend function?
❶ Finding the optimal Bitcoin allocation: The intrinsic properties of Bitcoin make a 20% Bitcoin allocation prudent. Since Bitcoin also has by far the best expected return of all asset classes, the optimal Bitcoin allocation would be 66% of total life savings. The Bitcoin allocation is limited by liquidity management (due to volatility) as well as diversification requirements (due to the risk of Bitcoin failing calculated based on the lindy effect).
❷ Manage deviations from dollar cost averaging: If we want to buy or sell a significant amount of Bitcoin, we would like to know if Bitcoin is currently over- or under valuated vs. trend.


Bitcoinmodeler thinks the Bitcoin price trend broke 2018
https://bitcoinmodeler.substack.com/
Bitcoinmodeler’s Bitcoin price trend function depends on square root of time, while I used ln of time. His approach confirms the original model in this thread with a current annual Bitcoin price trend of slightly north of 40%. I think it is not so important which of both function forms we choose to model the bitcoin price trend, since both function forms give very similar results in the short term and in the long term both function forms will break down anyway.
Bitcoinmodeler argues we have a break in the Bitcoin price trend around 2018 with a higher price trend growth before 2018 and a lower price trend growth after 2019. He argues that after 2018 the network growth (as measured with on chain addresses) has slown down and thus the price trend also has slown down. I think it is certainly possible that the Bitcoin price trend and Bitcoin network adoption has slowed down. However, I think we have difficulties measuring the network effects (on chain, software, MSTR, ETFs, Options, regulatory approval, Lightning, Fedimint) and relying only on chain addresses gives you signals with a low time lag, but on chain addresses are not sufficient to measure adoption.
Bitcoinmodeler estimates a lower Bitcoin trend price of 63k USD for yearend 2024.


I’m certain the Bitcoin price trend will break one day, since all models will ultimately break. I’m curious, if I can find a different price trend for different periods in the Bitcoin price data. I don’t think however the number of Bitcoin addresses still accurately fully describe the Bitcoin network effect


Searching for a break of the Bitcoin price trend

Splitting our data
If we had to split our data set into two parts, at which date should we split it? It would make sense to split our data set at a date, in which the first part and the second part have a vastly different trend functions (e.g. the slope of our function should differ in the 1st and in the 2nd period).

I split the data set 16.07.2010 until 18.10.2024 into all possible 2 parts:

16.07.2010 until 17.07.2010 and 17.07.2010 until 18.10.2024
16.07.2010 until 18.07.2010 and 18.07.2010 until 18.10.2024
…
16.07.2010 until 17.10.2024 and 17.10.2024 until 18.10.2024

The blue line at 16.07.2011 is at 12. The orange line is at 6. If we split the data on 16.07.2011, we use 1 year (16.07.2010 until 16.07.2011) to estimate the Bitcoin trend function and we use 16.07.2011 until 18.10.2024 to estimate the slope of a trend function for the 2nd period.



If we calculate our slope based on too few data points the slope of our trend line is influenced too much by the cycle. Let’s say we would split our data set in before and after 01.01.2024. In this case our Bitcoin price trend slope would capture the cyclical Bitcoin price recovery since beginning of this year giving us a growth rate, which is too high. If we would split the data set in Q1-2021 we would get a trend growth rate, which is too low.

Due to lack of data the cycle interferes with the trend estimation. The blue line is not very reliable for dates before 2013. The orange line is not very reliable for dates after 2020. The green box marks more reliable estimation for the slope of the Bitcoin trend function.



No matter where we split our data set, the slope for the trend function calculated on the basis of the more recent history is lower than for Bitcon’s early history (orange line below blue line, grey line lower than zero). The blue line always captures the high price growth rate of before 2013 and thus is always higher than the orange line.

We can also see lower lows and lower highs.

19.03.2014: -1.21
26.10.2017: -1.46
21.06.2019 -1,46
01.01.2020: -1.73

This would indicate that my estimated trend function may bit slightly too optimistic. How bad is it?

For each day in history we can estimate a Bitcoin trend price function. Based on this function we can estimate Bitcoin’s trend price in EUR today.



Taking into account the price data 2011 and after to estimate our trend function, Bitcoin trend price today would be almost 70.000 EUR. Taking into account all data after 01.01.2017 would give us a trend Bitcoin price of 55.000 EUR. Data after 2020 I would disregard as of today, since they are too much influenced by the cycle.

19.03.2014: Estimated function a = 5,690932543, b = -38,17162062, today’s Bitcoin price = 66.925€
26.10.2017: Estimated function a = 4,318053405, b = -26,54126982, today’s Bitcoin price = 51.672€
21.06.2019 Estimated function a = 4,49448015, b = -28,020024, today’s Bitcoin price = 54.271€
01.01.2020: Estimated function a = 4,21023669, b = -25,591115, today’s Bitcoin price = 52.530€

05.01.2016: Estimated function a = 5,83140729, b = -39,332817, today’s Bitcoin price = 70.730€
15.11.2018: Estimated function a = 5,23391209, b = -34,322047, today’s Bitcoin price = 60.063€

Between 2013 and 2020 I don’t think we have a certain date after which Bitcoin’s price trend breaks lower.




If Bitcoin’s price trend would have broken in 2018 as argued by Bitcoinmodeler, I would expect the blue line flat before 2018 and falling after 2018. For the orange line I would expect a falling line until 2018 and a stable line after 2018 (until we get distorted by the cycle around 2020).

The high blue line before 2013 in combination with the blue line being above the orange line until the very recent past (thus heavily distorted by the cycle) indicates a higher growth period for the Bitcoin price before 2013. Thus, my first split is data before and after 01.01.2013.

Should we split the data after 01.01.2013 one more time?

After removing the years before 2013 from the data set we get the following chart:



End of 2018 the orange and the blue line have the same hight. If we would separate our data set into 2013-2018 and 2018-2024 both trend functions would be identical. Therefore a split does not make sense.

We have significant differences between the blue and the orange line at the for early or late dates (yellow box), however I would consider those data points too biased by the cycle, due to the short remaining history. Focusing on the green box:



We can see a deviation between the blue and the orange line after the end of 2018. This could indicate a lower trend growth. However, I would guess it is down do cycle interference. End of 2018 we had a cycle top. Starting from a cycle top to now (which is not a top), would underestimate the trend.

Starting from the a cycle top distorts the data especially if we don’t have a lot of data e.g. less than one cycle.



As the orange line becomes unreliable for high dates (because it only looks at data after the date to estimate the trend function) the blue line becomes unreliable for
low dates (because it only looks at data before this date but after 2013). I would consider the slope calculations before end of 2018 to be too biased by the cycle, but after 2018 the results become pretty. Incorporating more data after 2013 does not give us a declining slope and the slope is quite stable at 5,2 or 5.4.

We can see, a 2nd split is not necessary. However due to the first split (before and after 2013) the slope after 01.01.2013 is now 5.4 while the slope calculated with the full dataset including the data before 2013 gives us a slope of 5.8.

So my best guess would be a slope of 5.4, which would be a fair Bitcoin trend price of 63k EUR.



A reasonable conservative estimate for the slope could be 4.5 (orange line summer 2019), which would cover roughly one cycle:



And lead to a conservative fair Bitcoin trend price of 54k EUR, which would indicate slight overvaluation as of today.

[MaxMueller]

~

Thank you for the update. It's another great post worth a lot merit (merits i don't have and you don't care).


To break it down for me and others:

If we build up our trend line model only from data points of the last 4-8 years, the trend price curve is more flatten, trend prices are lower and with current price we are already in an area of overvaluation. We may gain another e^0,5 = +65% from current price 70k -> 115k until reaching cycle top, which might be relativly close (3-6 month).

If we build up our trend line model in comparison from all available data points, the trend price curve is steeper, trend prices are highter and with current price we have a decent undervaluation. We may gain another e^1 = +170% from current price 70k -> 190k until reaching cycle top, which leads to a peak much later (maybe Q3-Q4 2025).

Of course, outside and in between there are even more possible scenarios. The question is, if it is possible to estimate the probability of those different scenarios somehow?!


@virginorange: choosing different values for trend line and price vs. trend calculation reminds me somehow on monte-carlo-simulation. Even if it is a completely different aproach it migt be a good addition and valuable examination of results.

What i have in mind is to calculate the rate of change = (close-open)/open for each day since the beginning. You could than randomly pick 365 values (either from all data or only from last 4, 6, 8 years) and multiply all those rate of changes to the current price. Repeat it a hundred times to get a hundred different price devellopments for the upcoming 365 days. my supposition is, that with those calculated prices for today +365 days (which might have some kind of Gaussian distribution) you could evaluate if the different trend curves are too optimistic/pessimistic, depending on were the trendline cuts the distribution?

What do you think about that approach?

 

[dansus021]

Well this is good insight Mr. virginorange Welldone. 

To be honest Im still playing with Cex trade robot that basically its like DollarCostAveraging but you can still sell at the high and buy at low but all of that is automatically but there is a catch and that is the bot is gonna take the stop loss if the position is turn down a lot which is good but for long term holder like most of people they didn't really care much about couple of down.

But this is really good insight frenn it basically you do dca when the price is down but take profit a little or don't add position if the bitcoin price is at high clap for you

[virginorange]

Of course, outside and in between there are even more possible scenarios. The question is, if it is possible to estimate the probability of those different scenarios somehow?!

@virginorange: choosing different values for trend line and price vs. trend calculation reminds me somehow on monte-carlo-simulation. Even if it is a completely different aproach it migt be a good addition and valuable examination of results.

What i have in mind is to calculate the rate of change = (close-open)/open for each day since the beginning. You could than randomly pick 365 values (either from all data or only from last 4, 6, 8 years) and multiply all those rate of changes to the current price. Repeat it a hundred times to get a hundred different price devellopments for the upcoming 365 days. my supposition is, that with those calculated prices for today +365 days (which might have some kind of Gaussian distribution) you could evaluate if the different trend curves are too optimistic/pessimistic, depending on were the trendline cuts the distribution?

What do you think about that approach?

 

We want the Monte Carlo simulation to give us a probability distribution of possible future bitcoin prices.

Are randomly chosen daily yields really a good prediction?

1.) Yield & volatility are decreasing: Bitcoin's volatility and expected return decrease over time. So randomly picking a daily return would give us some daily returns from the early years of bitcoin with a volatility and expected return that exceeds bitcoin's price movement today. We could solve this by adjusting the return and volatility of the earlier return data downwards.

Let's assume a current expected daily bitcoin yield of 0.1% and a volatility of 60.
A historical data point would have a bitcoin yield of 0.3% and a volatility of 120.
So we adjust the historical data point yield_adjusted = yield_observed * 60/120 -0.05%.

2.) Mean reversion around the trend: Undervaluation gives you better future performance than overvaluation.

3.) Auto-correlation of valatility: Volatile days are, on average, followed by other volatile days. Therefore, during periods of above-average volatility, we would expect to see more volatile daily returns in the future.

4.) Missing paths: The bitcoin price has a >0% probability of going (close to) zero. This is not included in the historical data set and would therefore be missed by the Monte Carlo simulation. The risk of bitcoin dying within the next X years I would derive from Lindy.

We could learn more about the nature of Bitcoin's volatility (decreasing trend, occastional spikes and autocorrelation). Create a normalized dataset of Bitcoin's daily yield. Estimate volatility paths and subsequently estimate returns.

I would guess it would be simpler and more reliable (less assumptions needed) to derive Bitcoin price probabilities from options e.g. form Deribit.

[DenBlad]

great analysis and DCA advice. I think it works in these terms very well if you look at the long term rate