Dollar cost averaging Bitcoin - can we do better?

[Turbartuluk]

I always support pure DCA, which is buying whenever you set, be it a fixed time, which is what I do, more or less beginning of the month when salary comes, or a fixed amount, which is what I consider my small earnings on forum, sig campaign, which is fixed at $$.

Sure, you can view trends and predict and time, but that asks for additional management. First is to monitor market, and this seems easy, but anyone who traded knows the temptation is to get smart and then start to measure, and try to perform better. This alone is time consuming and possibly gets into emotion. It's very easy to get dragged in and get absorbed believe me. Even not to mention distraction of altcoins!!

Second, less of issue but still management, is to set aside cash to buy, and to use it up in low price. That's also very hard to do with discipline.

I noticed that there are different understandings for DCA. Often it is used as synonym for simple savings plan (DCA on timeline) but this is not really DCA for me. The show off strength is when you set up DCA on the price axis, e.g. limit orders in a grid each -5%. If you set up some package of backloaded limit orders you can be sure that you will buy BTC at a very good average price as long as BTC price don't fall off the bottom of your grid.

Of course you can combine both possibilities of DCA, like this:

• monthly 1000$ saving plan from bank account to crypto exchange
• weekly 250$ saving plan USD into BTC, consisting of:
• 100$ instant buy BTC
• 3x50$ limit Buy orders @ current BTC price -5%/-10%/-15%

 

If you choose such an approach then the model of @virginorange could be great to adjust instant buy vs. limit orders on the one hand and max drawdown respectively positions of limit orders on the other hand.

Let me give you an idea for such a DCA savings plan for accumulation between trend line (0.0) and bottom line (-1.0) with a stack of 5 limit orders (simplified regarding of the previous bear market):

Example 1: Date 01.04.2022
BTC Trend line price (TL): ~26.500€
BTC Bottom line price (TL-1.0): ~10.000€
Current BTC Price (cP): ~42.000€ or +0.45 above Trend line
Conclusion: 0% Instant Buy, 5x 20% each in limit orders from 26.5k€ (0.0), to 20.5k€ (-0.25), 16k€ (-0.5), 12.5k€ (-0.75) down to 10k€ (-1.0)

Example 2: Date 01.07.2022
BTC Trend line price (TL): ~30.000€
BTC Bottom line price (TL-1.0): ~10.800€
Current BTC Price (cP): ~30.000€ or +0.0 above Trend line
Conclusion: 0% Instant Buy, 5x 20% each in limit orders from 30k€ (0.0), to 22.8k€ (-0.25), 18k€ (-0.5), 13.8k€ (-0.75) down to 10.8k€ (-1.0)

Example 3: Date 01.10.2022
BTC Trend line price (TL): ~32.500€
BTC Bottom line price (TL-1.0): ~12.000€
Current BTC Price (cP): ~20.000€ or -0.5 below Trend line
Conclusion: 50% Instant Buy, 5x 10% each in limit orders from 18k€ (-0.6), to 16k€ (-0.7), 14.5k€ (-0.8 ), 13.0k€ (-0.9) down to 12k€ (-1.0)

Example 4: Date 01.01.2023
BTC Trend line price (TL): ~36.500€
BTC Bottom line price (TL-1.0): ~13.500€
Current BTC Price (cP): ~15.500€ or -0.85 below Trend line
Conclusion: 85% Instant Buy, 5x 3% each in limit orders from 15.5k€ (-0.85) down to 13.5k€ (-1.0)

Do not take the calculation too serious, as it shall only demonstrate the principle.
Roughly estimated the average BTC buy price with this strategy would have been ~20k€ with 70-80% of FIAT swapped. To come up to 100% FIAT swapped you could also adjust your limit orders to increasing trend or bottom line or cancel limit orders and swap cash to BTC via market order if you believe you already passed the bottom (regarding to cycle position). If you are really brave you could set up the lowest 1-2 orders as 5x margin orders, to be sure to swap 100% of FIAT without adjustment and take a leverage slightly above 1.0x.
Either way you would have hit the ground quite well.

Obviously you could do the same as a withdrawal plan to sell your BTC in the area of [+0.0 .. +1.0] during the bull market.
In addition you can use the cycle position to switch beetween both plans (e.g. ~2y each) or just switch them on/off (e.g. ~1y each, separated by ~1y break). No emotions in here, easy to calculate, easy to automate and due to exponential nature of calculation the DCA would already be backloaded by nature.
  

The highes recent risk value was 531 at Dec. 31th of 2022. This was the highest risk value since 551 on Sep. 11th 2016, which was 6.3 years earlier. The absolute risk value has not important meaning for our interpretation. However our risk value in comparison to previous risk values is important for our interpretation.

@virginorange: I noticed that hightest recent risk values seems to be some of the best entry points. A future black swan event leading to very high risk values could also be a perfekt "buy the dip" opportunity.
I am not sure if it might be a good idea to cover a possible model failure outside the BTC allocation (e.g. one-way rebalancing to lower risk assets), so that you gain 100% model trust insight the BTC allocation. This might be necessary to prevent uncertainty and emotional stress in such a situation.

[slaman29]

I noticed that there are different understandings for DCA. Often it is used as synonym for simple savings plan (DCA on timeline) but this is not really DCA for me. The show off strength is when you set up DCA on the price axis, e.g. limit orders in a grid each -5%. If you set up some package of backloaded limit orders you can be sure that you will buy BTC at a very good average price as long as BTC price don't fall off the bottom of your grid.

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If you choose such an approach then the model of @virginorange could be great to adjust instant buy vs. limit orders on the one hand and max drawdown respectively positions of limit orders on the other hand.

I understand what you're saying, believe me. And I really really like what @virginorange proposed, and what you proposed.

But I still think that the majority of people actually don't have the time or expertise to properly take advantage of setting up advanced DCA. And as I mentioned, even if they did, they will expose themselves to the emotional and discipline factors that can easily suck people in. I speak from experience here

Back in forex and gold trading days this was also the case. Especially with currencies like in Asia where you actually get more value saving in forex USD or EUR especially. The stats of average users show they still gain guaranteed value in gold and forex with simple DCA. Savings plan, without committing more time to gaining and implementing expertise.

The disappointment of missing out, or making wrong calls, it leads to other psychological/emotional elements that ruin the whole idea of DCA in the first place Which is to simplify for the average user and ensure they don't miss out on great gains just because they don't understand TA.

[virginorange]

TL;DR: I present a second strategy to outperform buy and hold. Bitcoin has excellent performance when stocks have low volatility. Bitcoin has poor performance when stocks have high volatility. Only holding Bitcoin during low-volatility-times gives you higher returns with lower drawdowns than buy and hold. However this strategy is still inferior to timing the Bitcoin cycle (buy trend-40%, sell trend+750 days). Combining cycle and volatility information does not improve results vs. only timing the cycle, but can give you additional clarity about our position in the cycle.

How would you use our indicator (Bitcoin outperforming during periods of low volatility) to modify buy and hold or DCA?


Above average volatility in the S&P 500 stock index is associated with poor Bitcoin returns in a stable way.

evidence #1






Chart description:
- blue line = Bitcoin price at low S&P volatility
- orange line = Bitcoin price during high S&P volatility
- green circle = expected result: high S&P volatility leads to poor bitcoin performance
- red circle = unexpected result: good bitcoin performance despite high S&P volatility
- yellow circle = no significant bitcoin price movement during high S&P volatility

Interpretation: As you can see, most of our cycles are green, so high S&P volatility almost always gives us poor Bitcion returns. There are a few periods where our bitcoin volatility indicator "does not work" and bitcoin performs well despite above average S&P volatility. I like the stability of the results. No matter how we measure short-term (30 days or 90 days) and long-term (360 days or 500 days) volatility, our volatility indicator works reliably.

evidence #2



Chart description: We can plot the relative short-term to long-term volatility of the S&P on the x-axis. We can then plot the expected daily price return on Bitcoin on the y-axis. To average out the noise, I have calculated 500 day averages of 250 data points with worse higher volatility and 250 data points with lower volatility.

Interpretation: High S&P volatility gives you negative bitcoin returns the following day. Low volatility gives you positive bitcoin returns the next day. The positive Bitcoin returns are quite stable between -0.7 (very low volatility) and -0.1 (moderately low volatility). It is also stable no matter how we measure our volatility (30d/90d/360d/500d).

Why do we like stable results? Stable results in the past are more likely to be stable in the future and are more likely to provide outperformance than a model based on unstable results.

An example of unstable results is plotting bitcoin's next-day price returns against bitcoin's own volatility.



Chart description: Same like last chart, except the x-axis representing Bitcoin's own volatility instead of the S&P's volatility.

Interpretation: The 30-day results are the opposite of the 90-day results. We also cross the x-axis (which separates positive and negative returns) several times. Both show us low stability when estimating bitcoin retrurns from bitcoin volatility. Estimating bitcoin returns from S&P volatility is much more fruitful than estimating bitcoin returns from bitcoin volatility.

Choosing the volatility measure an the cut-off: We have seen that there is little difference in the choice (30 days or 90 days and 360 days and 500 days) to measure our relative S&P volatility. I don't like to trade, so for similar out-performance I prefer the model with the least number of trading signals, which would be 90 days volatility versus 500 days volatility. Because of evidence #2, I use a cut-off point of -0.1. This means that our signal becomes bullish when the S&P 90-day volatility is slightly below the 500-day volatility:



Chart description: see evidence #1

Interpretation: We can see that most signals look decent, but we have one very harmful signal (red cycle) in recent years. At the end of 2018, we had a period of very positive bitcoin performance despite high S&P volatility.

evidence #3
How good is our volatility signal?



Chart description: I draw a line starting on the first day after the low volatility period begins. Each line is described with its first day. The line stops at the end of the low volatility period. The x-axis represents the number of days in low volatility market conditions. The y-axis show how much Bitcoin price increased sind day 0. Green lines represent very good Bitcoin performances. Yellow lines represent ok performances. The black line represents the average of all other lines.

Interpretation: For most of the low volatility periods we can see very decent returns for the bitcoin price. The worst period gives us a return of around 0, while the best period gives us a return of around 2 (=638%). Periods of high volatility generally show poor returns, with some exceptions showing positive returns. I can conclude that trading bitcoin according to the S&P volatility looks very promising. However, some risk remains. We are quite safe buying bitcoin when stocks enter a period of low volatility.



Chart description: Same like above, but we are looking at the performance for the high volatility periods.

Interpretation: Selling bitcoin when we enter a high volatility period for stocks is a bit riskier than buying at the beginning of low volatility periods. Sometimes the price of bitcoin still performs well and we have to buy back more expensive.

Now we have established 3 pieces of evidence to show that Bitcoin performs much better during periods of low S&P volatility than during periods of high S&P volatility. However our volatility signal is not perfect.



How damaging are our few bad signals? Let's do a backtest.



Chart description:
We compare two strategies:
1. (black line): We buy 10€ of Bitcoin on 01/01/2005 and hold it until today.
2. (brown line): We start with 10€. We hold bitcoin during periods of low volatility and cash during periods of high volatility.

Interpretation: We can see that we start with some underperformance (brown line is blowing black line) which turns into outperformance later on. Perhaps Bitcoin is now trading with a higher correlation to traditional financial markets than in the past.
I would say that the observed outperformance of our strategy alone is a bit weak, as we can also observe historical periods of underperformance. Together with the significantly lower drawdown of our strategy (-53% vs. -83% for buy and hold), the strategy is still okay. If you dislike drawdowns, the significant reduction in maximum drawdown could make this strategy still interesting despite the mixed outperformance track record.


Comparison with cycle timing



Chart description:
We add one more strategy (yellow line): Buy Bitcoin at trend*e⁻0.5 and sell Bitcoin at trend+750 days.

Interpretation: When we back-test our valuation trigger for trading Bitcoin in the OP, we get even better results (yellow line). We can see significant outperformance, hardly any underperformance and only a slightly worse drawdown.



Results: Both strategies, trading by volatility (strategy 2) and trading by valuation (strategy 3), outperform buy and hold (strategy 1) before tax with higher average returns and lower drawdowns. However, strategy 3, as explained in the OP, performs much better.

Can combine two good strategies (2 and 3) to get an even better trading strategy?
We know that buying Bitcoin cheap gives us outperformance (strategy 3). We also know that Buying bitcoin during a period of low volatility gives us outperformance (strategy 2).



Buying bitcoin cheap (green box) and entering a period of low volatility (green line) could be an excellent investment, as all green lines end up much higher than they started (marked in green), while the less useful green lines (marked in red) are not included.

This idea sounds good, but waiting for (a) valuation and (b) volatility to turn favorable (strategy 4) gives us a lower return (101% p.a.) than only waiting for (a) valuation to turn favorable (121%).

We are missing out on returns by sitting in cash while waiting for volatility to come down. While waiting for volatility to come down may not be a good quantitative strategy, it could still be useful from a psychological perspective. If you didn't buy enough Bitcoin during the bear market, because you were scared, but the volatility didn't turn favorable, it could give you the confirmation to buy. At this point, we are still very early in the bear market.

The second combination possibility would be strategy 5:



Strategy 3: We are selling at an overvaluation (Bitcoin price = Bitcoin price trend + 750 days). However, the historical top was around +1000 days or even +1250 days. I still use the 750 days cut-off point in my strategy 3 because we only know the exact top in hindsight. I want to allow for some margin of error to not miss the top in case the next top is lower.
Strategy 5 modifies strategy 3:  We could wait for bitcoin to reach trend + 750 days overvaluation and only sell when the volatility trigger turns red (= entering a period of high volatility).



This strategy 5 gives us an even better return (131% p.a.) than pure cycle timing (121%) with similar drawdowns. However, despite the higher average return, I still prefer pure cycle timing (strategy 3) because we sell a bit before the peak rather than a bit after the peak. Since in real life we will need some time to sell, I would rather start too early than too late.


Valuation leads to action, volatility is confirmation
We buy when Bitcoin is 40% below trend. Entering the next period of low volatility in equities is likely to mark the end of the bottom. We sell when Bitcoin price is 750 days ahead of trend. The subsequent entry of a period of high volatility marks the end of the top.

Let me give you an idea for such a DCA savings plan for accumulation between trend line (0.0) and bottom line (-1.0) with a stack of 5 limit orders (simplified regarding of the previous bear market):

I also thought about how to use our position in the cycle to modify my DCA strategy.

Background: I changed my model for calculation the optimal allocation into Bitcoin from adjusted risk parity (=old model) to maximum allocation after taking into account liquidity constraints (= new model). As a result I have to sell a lot of stocks (e.g. 15.000 €) to buy more Bitcoin. However I'm already exposed to Bitcoin since end of 2021.

I create 3 baskets:
- very cheap basket (trend * e^-0.5): 5.000€
- cheap basket (below trend): 5.000€
- ok basket (trend to trend + 150 days): 5.000€

Investment decision triggers:

❶ If Bitcoin price is below trend*e^-0.5, I will invest all my remaining funds into Bitcoin. This would be 15.000€ as of today.

❷ If Bitcoin price is above trend*e^-0.5, but below trend, I will invest 1000€ from cheap basket and 1000€ from ok basket every month.

❸ If Bitcoin price is above trend, but below trend +150 days, I will invest 1000€ from ok basket every month.

❹ If Bitcoin price is above trend + 150 days, but below trend + 750 days, I will DCA surplusses into Bitcoin, but not sell any extra shares.

❺ If Bitcoin price reaches trend +750 days, I will sell Bitcoin. I will sell 3x my monthly net salary each month until we fall below trend + 750 days. All the selling will go 50% to the very cheap basket and 50% to the cheap basket.

❻ On the way down from trend + 750 days to trend I will put my DCA into my 3 baskets and only buy Bitcoin, when Basket rules allow for it.



Model maintenance triggers:

My DCA management can be easily implemented in excel and does not cause much of a headache or costs a lot of time. However I'm very dependent on my model to continue to describe the Bitcoin price behavior. Therefore I have to check, if my model is still correct.

We assume Bitcoin price is moving around a upwards trend.



Sometimes Bitcoin price is supported by the macroeconomic environment



❶ Maintenance trigger "new top found":

in theory the trigger looks like this:


    1st trigger:
        Bitcoin price = trend +750 days
    2nd trigger:
        [Bitcoin price 10% below all time high] and [S&P 90d volatility > S&P 500d volatility] or
        [Bitcoin price 50% below all time high]

I will then check, if the slope is still slightly below 6.


A significantly lower slope would require a new hedonistic optimization to get a new bitcoin allocation (e.g. 50% instead of 66%).

I will also check how far above trend the Bitcoin price went, to set a new trading trigger for the next cycle (e.g. 500 days overvaluation instead of 750 days or switiching form days overvaluation to ln overvaluation).

❷ Maintenance trigger "new top missed":



    trigger:
        no euphoria since 1621 or 1677 days





❸ Maintenance trigger "new bottom found":



    1st trigger:
        Bitcoin price = 40% below trend
    2nd trigger:
        [Bitcoin price 10% above 12 months low] and [S&P 90d volatility < 90% S&P 500d volatility] or
        [Bitcoin price 50% above 12 months low]

I will then check, if the slope is still slightly below 6. Also I will check the picture of undervaluation and volatility to re-calibrate our trading triggers and our maintenance triggers.

❹ Maintenance trigger "bottom broken":




    trigger:
        crushing with high volatility through the floor or
        gradually lower and lower bottoms

[bitcoin_mining]

The approach you have outlined in investing is very similar to auto investing. That's why I mentioned auto investment because in case of auto investment you have to keep enough money in the wallet and go to auto investment option to confirm how often the bot will invest money from your main account. In this case, if asked to invest a certain amount of money in a row, then the bot will do so. But investing in this method is probably more about relying on yourself than on bots.

[virginorange]

TL;DR: If you have a lump sum to invest into Bitcoin, buying now gives you more Bitcoin than DCA over the next couple of months. However most people should DCA, because it is mentally easier, or it fits better to your liquidity profile (monthly salary) or you enter the bitcoin space during overvaluation.

Background: I came to the conclusion that my Bitcoin allocation is too low. As a result I have to sell shares to buy Bitcoin. So my question is, do I go all in now? Or do I cost average into Bitcoin over several months?
The first and last time I invested significant money (relative to my net worth) into Bitcoin was Nov. 2021 very close to the peak. Now after several years I broke even (not inflation adjusted). This time I want to make a better decision.

I compared a lump sum investment vs. a DCA investment spread out over 2 to 12 months for every day since 16.07.2010.



The longer you spread out your Bitcoin investment the less Bitcoin you get. If you split your initinal investment into 2 equal monthly investments, you get 3% less Bitcoin on average. Spreading your investments out over 12 months makes gives you 25% less Bitcoin on average. The decision to DCA into Bitcoin is around 1/3 of the time better than a lump sum investment (32% for 12 months DCA).



If we only take data from 2015, DCA makes you miss out on less Bitcoin (18% instead of 25% for 12 months), while DCA remains a better decision than lump sum investment 31% of days.

This makes sence, since Bitcoin has a positive price trend vs. cash on average. The longer we wait to buy Bitcoin the higher Bitcoin's price trend increased. Waiting to buy Bitcoin is always a very dangerous game.


So when does DCA outperform?



We can see most of the time the curves are blow zero meaning most days DCA underperforms, but we have some time frames DCA looks like a better idea.



The more wie spread out our investment into Bitcoin, the more extreme our results become (bigger wins and bigger losses). DCA outperforms lump sum investment, when we start close to a peak, but underperforms otherwise.



We can plot our DCA outperformance vs. our position in the Bitcoin cycle. For undervaluation and moderate overvaluation it  lump sum investment was better than DCA. For significant overvaluation the longer we spread out our Bitcoin buys in time the more successful we were in avoiding buying at the peak.



If we only look at data since 2015 the effect is smaller, but the decision is the same. For undervaluation and moderate overvaluation better buy Bitcoin as soon as possible (= lump sum, no DCA).

Currently we are at 0 overvaluation.



Since 2015 we can observe 12 month DCA buying 21% less bitcoin than lump sum investments. However on the time frame between 2 and 3 months DCA performed roughly as well as a lump sum investment.

Since DCA makes us buy Bitcoin later than buying at the first possibility, I would expect I'm missing out on the Bitcoin price trend with the proportion of cash not invested yet. This expectation is quantified in the column "expected DCA underperformance". We can see, the observed and the expected underperformance roughly match.

Since Bitcoin price currently is roughly equal to trend, I would expect to miss out on the Bitcoin price trend growth, if I would DCA instead of investing everything now. However the trend 2024-2025 is lower than the trend 2015-2024. Therefore the expected underperformance would between 2% (2 months) and 16% (12 months)


Conclusion: DCA can be useful, if the monthly Bitcoin buys matches your monthly liquidity surplus (salary minus food and rent). However if you already have some liquidity, which has to be invested into Bitcoin to reach optimal Bitcoin allocation, it is most cases optimal to buy Bitcoin as soon as liquidity is available. Spreading your buys out over several months is only optimal, if Bitcoin is significantly above trend. This this case, we should just wait until Bitcoin gets closer to trend again.

[Turbartuluk]

~

Regarding your S&P Volatility analysis i like the idea to use it as a psychological support in times of uncertainty, but i guess i would never use S&P related indicators for BTC trading.
The reason is simple, S&P cycles are much longer than BTC cycles and i would not mix them.

Regarding BTC we had some full cycles since 2010 and a lot of data Points relating to all cycle phases.
Regarding S&P we only have one big bull market since 2010 only interupted by some "minor" corrections.

In simple words: continuously printing money as hell (low vola) is good for S&P and BTC. Major global crises resulting in corrections (high vola) is bad for S&P and BTC. Thats my stronly shortened interpretation of causality behind those correlations presented. That could be completly different in an S&P bear market.
Despite that i guess low S&P vola can be interpreted as low risk of fundamental changes in world order. I guess therefore it is still a good "soft" indicator to confirm the gut feeling.  
I also guess there are some opportunities to use this as an indicator across the board, e.g. for rebalancing different assets.

~

Regarding your DCA Analysis i would partly agree. Within a strong bullish trend overlaid by cyclical fluctuations it seems logical to me that a simple savings plan is mostly weeker compared to lump sum investment. BUT: I don't think that is the question!

The question for me is: Can we improve the cycle based entry points by using DCA as a money management tool (not as an indicator)?
Using time based DCA: Is it better to buy @ Trend price -40% / ln(-0.5) all at once or starting the saving plan for X month from than.
Using price based DCA: Is it better to buy @ Trend price -40% / ln(-0.5) all at once or split basket to buy @ -0.5, -0.6, -0.7, -0.8, -0.9 and -1.0 (and eventually miss the dip with some orders an have some FIAT left)?  

[virginorange]

S&P volatility is a useful indicator for Bitcoin, but not for trading

Regarding your S&P Volatility analysis i like the idea to use it as a psychological support in times of uncertainty, but i guess i would never use S&P related indicators for BTC trading. The reason is simple, S&P cycles are much longer than BTC cycles and i would not mix them.

I agree, S&P volatility is not very helpful for trading as shown in my backtest. It outperforms buy and hold, but underperforms cycle valuation trading.

However I will add the volatility indicator to give me a rule based confirmation that the top or the bottom is in:



What I like about my bottom / top confirmation indicator:
- rule based
- no wrong signals up to now
- top confirmation is quite close to the real top
- bottom confirmation is still reasonable close to the bottom as a model maintenance trigger
- very interesting is that during the double tops 2013 and 2022 our indicator only shows us the 2nd top

What I dislike about my bottom / top confirmation indicator:
- The trigger is not useful for trading since the Bitcoin price has moved quite a bit when my indicator calls the top or bottom.



for trading I will stick to valuation

Regarding BTC we had some full cycles since 2010 and a lot of data Points relating to all cycle phases.
Regarding S&P we only have one big bull market since 2010 only interupted by some "minor" corrections.

The problem is not the lack of S&P-volatility signals for Bitcoin, we had more than 10 signals since 2015, ...



... the problem might be a possible change in the relationship between S&P volatility and Bitcoin during a long term S&P bear market. One example for a change in relationship would be the gold price. During the bond bull market gold moved inverse to real interest rates, but in the bond bear market during last few years gold decoupled from real interest rates. So relationships can change.
I would guess a liquidity or a risk event pushing the S&P volatility above average would definitely be bad for Bitcoin price in the short run. Especially since Bitcoin and legacy financial markets are connected via. ETFs and Futures.

The question for me is: Can we improve the cycle based entry points by using DCA as a money management tool (not as an indicator)?
Using time based DCA: Is it better to buy @ Trend price -40% / ln(-0.5) all at once or starting the saving plan for X month from than.

Is it better to buy @ Trend price -40% (= trend*e^-0.5) all at once. Starting the saving plan for X month from than yields you an underperformance for all X. For X = 12 you get an underperformance of e⁻0.4 = 33%

DCA costs performance, but may be necessary to keep your bank happy

Using price based DCA: Is it better to buy @ Trend price -40% / ln(-0.5) all at once or split basket to buy @ -0.5, -0.6, -0.7, -0.8, -0.9 and -1.0 (and eventually miss the dip with some orders and have some FIAT left)?  

Missing the dip and being partly invested in cash will yield you a significant underperformance over a whole cycle. We could reduce the underperformance by canceling the -1.0 after a certain time and placing a market order at -0.2 instead. On average you will loose with the -1.0 bet. Not only are you more likely to go from -0.5 to 0.0 than to -1.0, but additionally you have a positive trend.

Nevertheless I'm interested in this question as well to place the orders around a peak and a bottom. Not so much in order to get further outperformance, but form an operational stand point to not overburden my bank accounts with inflows triggering AML procedures in the bank. I can prove all my buys and all taxes are paid. However it is still a hassle I would like to avoid. Also some banks might dislike Bitcoin, despite buying Bitcoin is legal.

As Bitcoin prices increase, it is possible I can only sell 3% of my Bitcoin peer to peer. A centralized exchange CEX would make additional 10% sales possible. If I sell at the best possible moment, I could only sell 3% (incl. CEX 13%) of my holdings. Spreading it out over 4 months would make it possible trade 12% (incl. CEX 22%).

[Turbartuluk]

Is it better to buy @ Trend price -40% (= trend*e^-0.5) all at once. Starting the saving plan for X month from than yields you an underperformance for all X. For X = 12 you get an underperformance of e⁻0.4 = 33%

DCA costs performance, but may be necessary to keep your bank happy

You have to keep in mind your derivation of Cut-Off points. Based on past data you selected a "good" cut-off point for lump sum investment. Comparing DCA with this "good" lump sum investment might be kind of recursive and results might be prone to overfitting.
Your conclusion is true for past Data, but i guess you can not expect it to be true in general, at least it was partly false during the last cycle. Even if there is some build in safety factor there is still some risk left that an undervaluation of -0.5 will not be reached in the future anymore (e.g. this Scenario)

DCA can be used to spread this risk and be sure that you are at least partly invested to profit "somehow".


Regarding the performace side there might be second perspective...
If you want to invest a fixed amount of money i guess you are right: A price based DCA is likely designed in a way that you dont fall off the grid, therefore you will have some open orders left if everything turnes out as expected and you will "loose" some performance. The benefit on the other hand is, that the strategy is more robust to misjudgment of Cut-Off Points, like i said before.

Looking Back the last 10y the correct portfolio weighting of BTC would have been 100% or even above. Obviously that was not the strategy from most of us. Choosing a lower weighting results in more confidence in the investment strategy.
You can use DCA in the same manner if you keep your weighting flexible. Choosing a backloaded DCA e.g. with orders down to ln(-1.0) will result in a "good" average buy price, as long you dont fall of the grid (roughly right Magnitude of cut-off value). For the invested money DCA seems to improve the strategy. The "underperformance" due to liquidity left over is meanwhile only based on missed profits or untaken opportunities. At least for me i can say that in my point of view the "cost" of unused liquidity is just inflation and not missed profits as they happen every day. Therefore it does not really hurt me to be less invested than i could.

Therefore the perspective might change with an approach of "flexible weighting".  

[virginorange]

DCA can be used to spread this risk and be sure that you are at least partly invested to profit "somehow".

You are correct. I was conservative with my cut-off points, but a significant change in the Bitcoin price behavior could make me miss the top or the bottom. Therefore I won't trade 100% of my Bitcoin and I will DCA (mainly because of banking restrictions but also to reduce risk).


Adjusting our strategy (sell at trend+750 days, buy at trend*e^-0.5) for bank account limitations.

Problem:
Peter has 1 Bitcoin. He feels comfortable trading 0.5 Bitcoin while holding the remaining 0.5 Bitcoin. The optimum strategy would be sell 0.5 Bitcoin at trend+750 days for cash and use this cash to buy back more Bitcoin at trend*e^-0.5. However selling 0.5 Bitcoin at trend+750 days would lead to an cash inflow into his bank account of 65.000 EUR far exceedingd his salary cash inflow of 2k. This will trigger anti money laundering actions from his bank. The bank might file a report to the government, block the cash inflow or require additional information from Peter about the source of the funds. Peter always remained within the law and paid his taxes in full. However he would still like to avoid the hassle of filing a lot of paper works.
He therefore doesn't want his cash inflow from selling Bitcoin to exceed 4k a month (2x his salary). As a result he can't sell his 0.5 Bitcoin for 65.000 EUR at once but would need several months to sell his Bitcoin. However the Bitcoin peak only lasts a short time. So either (i) Peter won't be able to sell his 0.5 Bitcoin in full or (ii) he will have sell his Bitcoin over several months for a lower price above trend or mixture of i and ii.

Solution:
4k a month is 1k a week, which is 3.2% of 0.5 Bitcoin at trend price of 62k. For a 3.2% weekly cash turnover bank account limit the optimal time to start selling is roughly at trend+200 days (currently 76k EUR) and the optimum time to start buying back is at trend*e^-0.3 (currently 46k EUR).


Finding the optimal valuations for buying and selling Bitcoin

We have learned that we could outperform buy and hold by buying Bitcoin cheap in the cycle and selling Bitcoin expensive in the cycle. Best time to buy Bitcoin was at trend*e^-0.5. Best time to sell Bitcoin was trend+750 days. How do the two limits (-0.5 and 750) change as a function of how limited our bank account turn over is relative to the value of our Bitcoin we are willing to cycle-trade.

We can be sure the limits (-0.5 and 750) will move closer to trend (e.g. -0.3 and +300) especially for severely restricted bank accounts, because we can't buy or sell all Bitcoin at the optimal moment. We have to choose a less optimal moment or not trade at all.

Backtests

We can calculate our out performance for multiple scnearios. For each bank account restriction (buy/sell X% of our tradable Bitcoin per week), we ceck for different buy and sell triggers the outperformance. The goal is to find the combinations of triggers that gives us the optimal outperformance. For the optimal combination (marked in bold) a graphical output of our backtest is shown.



Even if our bank account is severly restrictive (0.1% weekly trading) the optimum limits only go to (-0.25 and +180)





For 1%-3% trading restriction our limits are around (-0.3 and +200).




For a restriction of 5% our lower limit (-0.48) is almost at the optimum level without any restrictions (-0.5), but the sell limit is still much lower (250 instead of 750). The reason for this is, that the tops are short, but the bottoms take many days longer. For our trades to complete we therefore need to ease the restriction for selling more than the restriciton for buying.



Even if our bank account is not very restrictive and allows us to trade 10% a week our sell trigger has to be eased quite a bit from 750 to 300.


Why not use all your Bitcoin for cycle-timing?

Timing the cycle outperforms buy and hold by a wide margin, so why not use sell all our Bitcoin high and buy back low?

Safety: Do we feel comfortable owning zero Bitcoin?  Selling all your Bitcoin and leaving all that money in a bank may be risiky depending on your juristiction (Libanon, Egypt, Nigeria) and your relationship to the people in power (your are known as a critic of a powerfull corrupt politician in your contry).

Privacy: You don't want to be targeted by robbers, who may hurt you and your family members to get you to transfer all your Bitcoin to them. So you don't want to spread the information how much Bitcoin you own around too much. Buying Bitcoin once, especially without KYC, checking your balances with your own node or over TOR, leaves less traces. Buying and selling all your Bitcoin every couple of years on a centralized exchange, leaves a lot of data for hacker and robbers to use.

Non-fungibility: Maybe you think some UTXOs could have lower regulatory risk, so you might only sell UTXOs with higher regulatory risk and keep UTXOs with lower regulatory risk.

-> We have to earmark which UTXOs and how many Bitcoin are we willing to sell.


Steps to perform for finding the optimal buy and the sell levels:

Estimating the maximum fiat turnover:
- How many and which bank accounts can you use?
- How much turn over each bank account can stomach each month?

Estimating maximum Bitcoin trading:
- Earmarking UTXOs for selling: which UTXO in which order would I like to sell?
- [Bitcoin earmarked for potential selling] * [Bitcoin trend price]

[Bank limit in %] = [maximum fiat turnover] / ( [Bitcoin earmarked for potential selling] * [Bitcoin trend price] )

Backtesting the optimal buy and sell limit for your [Bank limit in %]

[Turbartuluk]

...for identifiing peak you could use 2 different indicators: BTC change ln-price +1 (~250% within 1year) which equals red 2 or ln-price +0,5-1 above trend which equals red 3/4[/li][/list]

~

@virginrange:
I re-read the thread focussing on your volatility analysis. Not as a hint that the model is broken, but as a kind of confirmation that the peak is incoming.
Maybe you could redo your calculations of "Bitcoin change ln-price vs. trend" for 30d / 90d and Vola30d + Vola90d for daily ln-returns for data points price_ahead > +750 days, to maybe see some volatility levels as selling confirmation?!

I guess that would be much more precise than BTC change ln-price +1 within 1year. 

[d5000]

In another subforum I have referred to this (excellent) thread recently, so maybe it's time to refresh it a bit!

First I'd like to ask @virginorange: do you still think the original formula holds? I repeat it here:

e^[ 5,8323 * ln(Date - 03.Jan.2009) - 39,296 ]

For those who want a quick & dirty command line script, in the Spanish subforum I've published a little script to calculate the value for each day, in Python:

Code:

import sys
import datetime
import math

rawdate = sys.argv[1]

# fórmula @virginorange: e^[ 5,8323 * ln(Date - 03.Jan.2009) - 39,296 ]
genesis = datetime.date.fromisoformat("2009-01-03")
date = datetime.date.fromisoformat(rawdate)
delta_since_genesis = date - genesis
days_since_genesis = delta_since_genesis.days
price = math.e ** (5.8323 * math.log(days_since_genesis) - 39.296)

print(price)

I personally consider it a tiny bit too optimistic, because I believe that the price in early 2024 may have been an "overshoot". I have experimented a bit with the formula and got that the following values would be currently quite close:

e^[ 5,82 * ln(Date - 03.Jan.2009) - 39,4 ]

Which gives me the current price of ~ € 58358.60 and is thus very close to the actual price. (Of course we don't know if we're currently above or below the trend but I think my point becomes clear ...).

I'm however not good at math, so it's not completely clear for me how I'd adjust it to get lower values in the future preserving the approximate curve until now (or if this is even possible with that formula).

[virginorange]

First I'd like to ask @virginorange: do you still think the original formula holds? I repeat it here:

e^[ 5,8323 * ln(Date - 03.Jan.2009) - 39,296 ]

The formula is estimated to minimize the square difference between the daily bitcoin prices and the trend (ls-estimation). However since we work in ln-time earlier data points carry more weight.



As a result the formula is pretty sticky and does not change a lot with recent data points. One day this will be the downfall of the trend model since the model will show a change in trend very late.

To check if the trend has already broken down I estimated the slope of different sub sections of our curve.




We don't see a different slope before and after 2017. Both intervals have a slope of 5.7.
However dividing our curve into segments bottom to bottom or top to top gave us much less stable results.

I personally consider it a tiny bit too optimistic, because I believe that the price in early 2024 may have been an "overshoot". I have experimented a bit with the formula and got that the following values would be currently quite close:

e^[ 5,82 * ln(Date - 03.Jan.2009) - 39,4 ]

Which gives me the current price of ~ € 58358.60 and is thus very close to the actual price. (Of course we don't know if we're currently above or below the trend but I think my point becomes clear ...).

You optimize the formula to our recent history. I optimized the formula over the total price history of Bitcoin. Optimizing for only a shot time span gives you unstable results depending on where we are in the bull/bear-cycle.

Our model will break down one day. Therefore I have implemented model maintenance triggers. We should question our model, when the Bitcoin price goes lower than  trend*e⁻1 or when we don't reach a euphoria phase for a long time (more than 1600-1700 days after the last euphoria).

Model maintenance triggers:

My DCA management can be easily implemented in excel and does not cause much of a headache or costs a lot of time. However I'm very dependent on my model to continue to describe the Bitcoin price behavior. Therefore I have to check, if my model is still correct.

...

❶ Maintenance trigger "new top found":

in theory the trigger looks like this:


    1st trigger:
        Bitcoin price = trend +750 days
    2nd trigger:
        [Bitcoin price 10% below all time high] and [S&P 90d volatility > S&P 500d volatility] or
        [Bitcoin price 50% below all time high]

I will then check, if the slope is still slightly below 6.


A significantly lower slope would require a new hedonistic optimization to get a new bitcoin allocation (e.g. 50% instead of 66%).

I will also check how far above trend the Bitcoin price went, to set a new trading trigger for the next cycle (e.g. 500 days overvaluation instead of 750 days or switiching form days overvaluation to ln overvaluation).

❷ Maintenance trigger "new top missed":



    trigger:
        no euphoria since 1621 or 1677 days





❸ Maintenance trigger "new bottom found":



    1st trigger:
        Bitcoin price = 40% below trend
    2nd trigger:
        [Bitcoin price 10% above 12 months low] and [S&P 90d volatility < 90% S&P 500d volatility] or
        [Bitcoin price 50% above 12 months low]

I will then check, if the slope is still slightly below 6. Also I will check the picture of undervaluation and volatility to re-calibrate our trading triggers and our maintenance triggers.

❹ Maintenance trigger "bottom broken":




    trigger:
        crushing with high volatility through the floor or
        gradually lower and lower bottoms

Currently we are around 25% blow trend, which does not seem to be unusual. Since Bitcoin volatility is low we are certainly not breaking our model for reason ❹. The new euphoria should arrive summer 2026 latest. Otherwise we would have to evaluate our model. So I can't confirm a model break down.

I'm however not good at math, so it's not completely clear for me how I'd adjust it to get lower values in the future preserving the approximate curve until now (or if this is even possible with that formula).

I used more sophisticated statistics programs during university and also one of my previous jobs, however for Bitcoin price trend estimation excel is enough. Excel offers a simple least squares regression. You only have to ln-transform the time values and the price values and excel can calculate a simple line for the trend. The results you can then transform back.

[d5000]

The formula is estimated to minimize the square difference between the daily bitcoin prices and the trend (ls-estimation). However since we work in ln-time earlier data points carry more weight.
As a result the formula is pretty sticky and does not change a lot with recent data points. One day this will be the downfall of the trend model since the model will show a change in trend very late.

Thanks! I already suspected something like that, because experimenting with values I found out that, if we maintain the same formula, it seems not to be possible to create a curve which still is close enough to our observed price history, but returns significantly lower results for years far in the future. Particularly from 2050 on I get values of more than 10 millions which are in my opinion extremely optimistic.

What I was thus wondering is if you can add an element to the formula to get slightly higher values for the early years, closer to the tops, and slightly lower ones for the more recent years. This would be a formula that is closer to the recent lows of the last 7-8 years, but in the first half would be closer to the highs in 2011 and 2013 for example.

So I continued to experiment, reading superficially some random website about log functions, and found the following formula:

e * 0.8^[ 5,83 * ln(Date - 03.Jan.2009) - 36.5 ]

Some values I get with this one:

- 2012-01-01: 28.02 € - close to the 2011 high
- 2015-01-01: 651.15 € - close to the 2014 high
- 2020-01-01: 10158.32 € - close to the 2019 high
- 2024-01-01: 41422.95 € - close to the real price in early 2024

Which is quite close to my intentions. Now on to the future:

- 2025-01-01: 55526.39 € - looks quite bearish, but it's possible for 2025 to be an euphoria phase so the real price can be much higher than that.
- 2026-01-01: 73064.35 € - could be close to the low of the next bear.
- 2030-01-01: 190316.34 € - feels already quite good.
- 2035-01-01: 500754.85 € - Gold flippening is close!
- 2040-01-01: 1110794.37 € - Looks already quite bullish, can we sustain that?
- 2050-01-01: 3942264.41 € - I think many would be happy with that result

In contrast the current formula gives for 2050: 19302683.81 € - Looks a bit "unbelievable".

This is only another example for such a formula, and it might be less elegant as it doesn't use the e constant. Perhaps it's too bearish, but I'm more comfortable with it ...

[virginorange]

Thanks! I already suspected something like that, because experimenting with values I found out that, if we maintain the same formula, it seems not to be possible to create a curve which still is close enough to our observed price history, but returns significantly lower results for years far in the future. Particularly from 2050 on I get values of more than 10 millions which are in my opinion extremely optimistic.

We have 15 years price history. An estimation for 2050 would be more than 25 years into the future from now, which is not prudent.

If we assume the Lindy effect there is a non significant risk of Bitcoin not surviving the next 25 years. However due to the high trend growth in Bitcoins price the Kelly criterion would still suggest to invest 66% of your current and future life savings now and rebalancing out of Bitcoin after 5-10 years to keep the allocation at 66%.

Ultimativ there will be a Bitcoin price after full adoption (which could be 50% of the market cap of gold = 9*10^12 USD or 20% of the marketcap of real estate, stocks, art and bonds = 30-40*10^12 USD), which will then only increase with global money supply (e.g. 8% a year). However I'm quite uncertain what market cap Bitcoin should have. The question is not if the model breaks down but when.

Even if the model would break down and Bitcoin trend growth would be only 25% per year instead of the estimated 40%, Bitcoin would still be an excellent investment.

I estimated the slope of the curve again with current data. The slope decreases from 5.8394 to 5.823 with the full sample data suggesting a trend price of 71.5k EUR instead of 73k EUR. However, if we split our price data in half (before and after 14.11.2016) we only get a lower slope: e^(5.0743*ln(days)-32.942) suggesting a fair trend price of 59k EUR and lowering the trend price growth to 37%.

This formula based on the recent half of Bitcoin prices would not impact our Bitcoin allocation for holding much since the yield is still very high. However it would impact our decisions trying to trade the cycle. If we are too optimistic about the bitcoin price trend, we would miss to sell at the top and buy Bitcoin to early in the bear market.

- 2025-01-01: 55526.39 € - looks quite bearish, but it's possible for 2025 to be an euphoria phase so the real price can be much higher than that.
- 2026-01-01: 73064.35 € - could be close to the low of the next bear.
- 2030-01-01: 190316.34 € - feels already quite good.
- 2035-01-01: 500754.85 € - Gold flippening is close!
- 2040-01-01: 1110794.37 € - Looks already quite bullish, can we sustain that?
- 2050-01-01: 3942264.41 € - I think many would be happy with that result

I would use the trend formula for price projections of 3-8 years. For longer projections I would which market of savings Bitcoin will capture. Also for long term projections you have to estimate a probability of Bitcoin failing and significantly underperforming stocks/gold/realestate.

[Unity for Humanity]

According to experienced investors, the DCA investment method is an effective investment method during the week. Generally in this investment method every investor gets the opportunity to invest as desired. The investor has no pressure in this investment so he can invest any amount whenever he wants. The most positive aspect of this investment method is that any professional person can invest every day or every month with minimum amount of money and keep it consistent.

[sosruko]

Excellent Details. very informative
In my opinion, the current cycle is below the ATH by a significant amount, and I predict that BTC will peak between 63 and 73K .
I've read estimations that this bull run would continue until mid 2025, but november 2024 seems like a more realistic estimate.
You might be right in your analytical approach, but time will tell. but considering how much short form entertainment consumes the ordinary person, this is inevitable.
Although the market is unpredictable, my theory holds that the recent bad market we've had prevents us from having a price fall back to base level.

[tvplus006]

...Or we could sell some of our bitcoin close to the top and buy back close to the bottom. This would be a simple DCA strategy...

The DCA strategy does not involve selling at a high price and then buying at a low price. What you are writing about is the most common type of trading that investors follow, but this type of trading is not DCA.

[d5000]

We have 15 years price history. An estimation for 2050 would be more than 25 years into the future from now, which is not prudent.[...]

Ultimativ there will be a Bitcoin price after full adoption (which could be 50% of the market cap of gold = 9*10^12 USD or 20% of the marketcap of real estate, stocks, art and bonds = 30-40*10^12 USD), which will then only increase with global money supply (e.g. 8% a year). However I'm quite uncertain what market cap Bitcoin should have. The question is not if the model breaks down but when.

I agree here. So basically your model is valid for the "growth phase" in adoption, and when we approach saturation we have simply to accept it breaks. I also agree that it is difficult to predict how high the percentage of global savings could be in the "full adoption" scenario. And of course Bitcoin can fail.

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."

However, if we use a more conservative formula, then it is possible that we can predict the price growth rate a bit longer. We would still have a model backed by yesterday's data. However, in my opinion it would make sense to concede more weight to the later price data than to the earlier movements: The market is only slowly maturing, and thus the early movements until 2013-15 can be explained more by chance, short-term sentiments and external effects (e.g. MtGox bankruptcy) than the post-2015 (approx.) data which gives already more hints about the steepness of the adoption curve.

I'm interested on this because I would not like to see something in the next 2 years already similar to what happened to Stock-to-flow, which was corrected every year, and has now probably broken completely.

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?

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

I estimated the slope of the curve again with current data. The slope decreases from 5.8394 to 5.823 with the full sample data suggesting a trend price of 71.5k EUR instead of 73k EUR. However, if we split our price data in half (before and after 14.11.2016) we only get a lower slope: e^(5.0743*ln(days)-32.942) suggesting a fair trend price of 59k EUR and lowering the trend price growth to 37%.

Thanks for doing that "alternative" calculation! I'll look into that possibility too.

[tbct_mt2]

I agree here. So basically your model is valid for the "growth phase" in adoption, and when we approach saturation we have simply to accept it breaks. I also agree that it is difficult to predict how high the percentage of global savings could be in the "full adoption" scenario.

Majority of Americans aren’t confident in the safety and reliability of cryptocurrency.
The research on Americans but globally Bitcoin also has big open space for further adoption.

I agree with you, that everything will come to a point of "saturation" and Bitcoin adoption growth will hit the "saturation point" in future but based on the research result, I believe that we are still far from that point and in near future like next 2 or 3 market cycles, Bitcoin will continue its good growth when adoption continues to soar sharply.

And of course Bitcoin can fail.

Yes, Bitcoin can fail like others but risk of failure in 2024 is much smaller than in earlier years of Bitcoin history.

The bullish case for Bitcoin (Part 2).

Lindy effect works and Bitcoin is not excluded from Lindy effect.

Furthermore, the Lindy effect suggests that the longer Bitcoin remains in existence the greater society’s confidence that it will continue to exist long into the future. In other words, the societal trust of a new monetary good is asymptotic in nature, as is illustrated in the graph below:

See image in the medium article.

If Bitcoin exists for 20 years, there will be near-universal confidence that it will be available forever, much as people believe the Internet is a permanent feature of the modern world.

[virginorange]

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.

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.

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.

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.




How would early signs of the model breaking down look like?

I updated my chart. This time I didn't measure the price of Bitcoin in EUR, but in ounces of gold.



Bitcon trend price growth hit the lowest value ever on a horizon of 4 as well as 8 years. In roughly 100 days we will also have the longest time of Bitcoin saysing below trend (green line).

I would still estimate Bitcoin outperforming gold in the future for two reasons: (i) yellow line ist still at 4 (= 28% annual return over gold) and (ii) Bitcoin has more catching up to do to reach final adoption than gold. However Bitcoin will probably outperform Gold less than in the past.