Summary for Binge-Reading: (#0-#13) My "ABC-K6 Model" that naturally mimics the ETF ProShares UltraPro QQQ (Ticker: TQQQ) by adopting biased probabilities
Since last year, I have been working through trial and error to create a simulation model that naturally mimics the ProShares UltraPro QQQ (hereinafter TQQQ), an ETF that manages the NASDAQ-100 index with 3x leverage.
As an outsider to financial engineering, I started by using "biased dice" as a tool, choosing an approach to create replicas of time-series data with various characteristics using only simple arithmetic within the scope of finite, measurable, and countable computational complexity. Eventually, I arrived at my ABC-K6 model (hereinafter K6).
As I repeatedly fine-tuned K6 and updated it to "Ver. K6αA," I began to feel its potential as an experiential simulator—a "full-length mirror" or something close to a driving simulator—for observing my own emotional tolerance in advance when operating TQQQ.
The content I considered while building K6 has been serialized in note articles #0 through #13. This time, I have compressed them and summarized them so they can be read all at once.
In the first half, especially up to #7, I focused on the thought process leading up to the construction of K6. In the second half, I have organized the tools that make up the K6 model and the parameters of the arithmetic formulas.
Those interested in "why it became this kind of model" might want to read from the first half, while those interested in the "concrete structure of the model" might prefer to start from the second half.
My model is not a simulation that accurately predicts a single point in the future. I believe it is useful as an aid for observing one's emotional tolerance according to one's lifestyle and stage, and for thinking about countermeasures to bring unexpected crashes within the scope of expectations based on "time axis" and "crash depth," while replicating a large amount of time-series data of various valuation amounts that are probabilistically possible in the future.
★(Note)
My articles are not intended as recommendations for specific financial products or investment advice.
Furthermore, my model is a thought experiment constructed based on my own subjectivity and has not been academically verified.
#0 (Introduction) Humans and Dice: An Entrance to the Probability Drifting Through Civilization and Society
The starting point of this series was a sense of discomfort: "Is the financial market purely random?"
It seems to me that human society and civilization have advanced into the future with a "bias" greater than mere coincidence. The idea that probability containing that bias might be seeping into the stock market was the starting point of my idea.
Do people roll fair dice in the stock market? Or rather, are people continuing to roll "rigged dice" that weave in civilization, technology, beliefs, and the will for the future?
I decided to call the resulting tilt in probability "Civilization Probability" or social probability.
To me, the charts of the NASDAQ-100 and its 3x leveraged ETF, TQQQ, do not look like mere price trajectories, but like the ridgeline of a mountain range that civilization has climbed. If the market were completely random, that mountain should not have a shape.
This note is neither a mathematical paper nor investment advice. It is a story of accumulating small thought experiments to observe the slight tilt of probability distorted by civilization and to explore its meaning.
#1 Probability Bias and Drifting Time
I have come to redefine asset management not as the "rise and fall of valuation," but as the conversion of time. "Time is Money" is a well-known phrase, but I feel we have not paid enough attention to the equivalence of Money is Time that lies behind it.
From my perspective, a crash in investment assets is not "a decrease in valuation," but an extension of the time it takes to reach the future. Loss is a debt of time, and recovery is its repayment. I perceive that increasing assets is a phenomenon where time is compressed, and decreasing assets is a phenomenon where time expands.
Time does not disappear. It is simply converted into another form called money—that is how I perceive it. Time has been feared because it has no form. But even if invisible, it is certainly stored within assets.
From this perspective, investment is not a game of courage, but an engineering task of how to design the resource called time.
Also, I decided to measure the time of the financial market not by calendars or clocks, but by "the number of times the dice of price movement is rolled = number of steps." By doing so, I was able to make it easier to handle with my original simulation model.
While respecting existing financial theory, I begin a journey to find a mathematical model that naturally mimics the TQQQ, where price movements run wild, on my own two feet, with primitive arithmetic and dice with a slight bias.
My motto is "measurable, finite, countable." Even in the fog, count what can be counted—that is my way.
Welcome.To the world of Money is Time. From here, the journey around Civilization Probability begins.
#2 My "Style" of Viewing the Financial Market
I have organized my "style"—the attitude with which I observe the financial market, form hypotheses, and attempt to incorporate them into a mathematical model.
I value an attitude of facing a small number of samples and errors head-on rather than drawing conclusions all at once from massive data. Like the t-distribution shown by Gosset of Guinness, the fewer the samples, the more uncertainty is held. I feel that attitude is important when dealing with the financial market. Therefore, my basic principle is "measurable, finite, countable." Count what can be counted, and leave the errors without intentionally erasing them.
My approach is inductive, not deductive. First observe, intuit, and set a hypothesis; the mathematical formula comes later. Just as Mendel hypothesized invisible genes, I tentatively place the "tilt" that repeatedly appears in the market as a hypothesis. I want to verify if that hypothesis functions as a map.
If the market were completely random, there are many words and systems in front of me that I cannot explain.
The empirical rule that "assets are likely to increase if you spend time," evolving stock indices, capitalism incorporating human choices, and the accumulation phenomenon where capital calls for capital.
To me, they seem to suggest the possibility that a slight probability bias is established in the market.
I do not want to deny the random walk. Rather, while respecting it, I want to gently place another probability created by civilization and human will next to it. Not as a rebellion, but as a counterpoint.
The time civilization has walked is not just noise. I will quietly and carefully incorporate that belief into a measurable, finite, and countable model. From here, I begin the journey of the hypothesis called "Civilization Probability" in earnest.
Digression: Commentary on the "Inverse Calculation Problem of Financial Statistics"
How is my ABC-K6 model (hereinafter K6) created? In this digression, I have intuitively organized the basic methods of mathematical model construction.
The starting point of K6 was to intentionally step outside the usual financial engineering approach of "estimating the future from past time-series data." This is because I felt there was a structural limit to estimating a population with only one time series observable in the actual market as the sole sample.
So I switched to the idea of thinking backwards from given financial statistics (CAGR, MaxDD, standard deviation, etc.) rather than the time series itself.
" Instead of relying on past data, I should create a new "time-series population of another world" that satisfies the statistics"
I call this virtual set a time-series (population) replica.
In K6, I do not handle financial statistics as they are, but decompose them into measurable, finite, and countable parts. It is "geometrization of financial statistics"—breaking down complex statistics into parts that can be reconstructed using only biased dice and arithmetic.
By reassembling these parts, I can generate countless population replicas that satisfy the same statistics but have completely different time-series paths. Moreover, they tend to naturally gather near the median, and extreme distortions can be expressed by moving the parameters.
K6 is not a model that predicts the future at a single point. It is a device for looking at a "set of possible futures" that have the same properties, leaving behind the curse of "past = 1."Observing the risk structure of the financial market from the outside with only biased dice and arithmetic—
"These are the outlines of how I build K6, and the contours of the world this model shows me."
#3 My Leveraged NASDAQ-100 TQQQ Simulation Model Adopting Biased Probabilities: The Probability of Rigged Coin Flips
I will conduct a simple thought experiment regarding my discomfort with "Is the financial market a random walk?"
If the market were truly governed by fair random probabilities, it should be modelable as a simple coin flip (50:50). However, a world where assets continue to grow in the long term cannot be established with fair coin flips. In a fair coin-flip game, many market participants would not increase their assets and would eventually exit.
But in reality, investors have increased, the stock market has expanded, and assets have grown over the long term. This fact suggests the possibility that the "coins" being flipped in the financial market are not completely fair. Just a slight bias—that alone quietly but decisively changes the long-term results.
However, it is impossible to explain the extreme behavior of leveraged products like TQQQ with just one rigged coin. So I advanced my thinking and thought about flipping "multiple rigged coins" at the same time.
When you flip two coins, results like:
・Both heads
・Heads and tails
・Both tails
are born, and there, not a single coincidence, but a combination and chain of coincidences appear. If heads continue, optimism strengthens; if tails continue, caution grows—I felt that this "chain of luck" is closer to the reality of the financial market.
Thus, in this chapter, I intuitively derive the structure of:
・Market growth that cannot be explained by fair randomness
・A chain of "good luck" and "bad luck" born from the overlap of multiple coincidences
・The invisible probability bias that slowly tilts the world behind it
.
This idea of "multiple rigged coins" later developed into the biased dice used in the K6 model, and became the entrance for treating the "runaway" of leveraged products like TQQQ as a chain of coincidences.
#4 My Leveraged NASDAQ-100 TQQQ Simulation Model Adopting Biased Probabilities: A World Governed by Good and Bad Luck
I redefine the price movements of the stock market as something that is neither "completely random" nor "clear intent."
In my hypothesis, stock market price movements are like a world governed by the outcome of dice. I think that those outcomes create "good luck" and "bad luck," and by chaining them together, the rhythm of the market is formed.
Also, even a market that looks random at first glance can sometimes appear to have a bias like "gravity" that tilts slightly toward good luck when viewed over a long period.
To me, it looks like a quiet tilt between coincidence and necessity created by civilization and human behavior.
The market is a "biased random terrain" governed by good and bad luck.
Understanding that terrain also means reinterpreting the meaning of time and money.
#5 My Leveraged NASDAQ-100 TQQQ Simulation Model Adopting Biased Probabilities: "Learning from the Past" and "Joy and Sorrow"
In asset management, I do not subscribe to the attitude of "applying past data to the future as it is," nor to the attitude of stopping thinking by saying "the future cannot be read."
Past data is not the answer, but merely small "pebbles" to be picked up. That is what I think.
By reconnecting the fluctuations of those minute numbers and probabilities, the structure connecting luck and time becomes visible.
What is important to me is not the reproduction of the past, but the "extraction of structure."
Also, the joy and sorrow and emotions of market participants look like noise that shakes prices. However, I feel that behind that, a simple probability structure called a "chain of good luck and bad luck" is quietly at work.
That is why I focus not on the emotions themselves, but on the "outcome of the biased dice" behind them.
The market is not just a wave of emotions, but a probability terrain formed as a result of people continuing to bet on the future within civilization.
I perceive the slight tilt born in that terrain as the trajectory of "Civilization Probability," which is my hypothesis.
#6 My Leveraged NASDAQ-100 TQQQ Simulation Model Adopting Biased Probabilities: "Prosperity Must Decline = Does the Declined Must Prosper?" And, "To be Better than Today."
I redefine the rise and fall of the market not as mere results, but as a "probabilistic pendulum motion" where prosperity must decline and the declined must prosper appear alternately.
The market constantly changes its appearance like the flow of a river; it sometimes sinks and sometimes floats again.
"Prosperity must decline" is certainly true. But at the same time, I feel that "the declined must prosper" also exists as a principle of the stock market.
People and civilization have an impulse to To be better than today—to be even a little better than today.
As a result of that impulse accumulating, the probability of the world is not a completely fifty-fifty random, but is distorted in a form slightly tilted toward the future. That is how I hypothetically perceive it.
So even if bad luck exists, it might not be just pure coincidence.
Isn't the history itself, where humanity has stood up even after falling and rebuilt even after losing, proof that a "coin that is slightly easier to land on heads" is embedded in this world?
What I want to capture is the bias of probability that looks like coincidence, born from the breathing of that civilization—the mathematics of destiny.
#7 My Leveraged NASDAQ-100 TQQQ Simulation Model Adopting Biased Probabilities: Biased Luck and Chains of Coincidence, and Arithmetic Formulas to Replicate the Future
I assume that human society and financial markets are not driven by "perfectly fair chance," but by dice that are slightly tilted toward good fortune.
Both good and bad fortune are matters of chance. However, when they chain together and are compounded by "events of different qualities," the direction of life and the market changes significantly.
The keys to shaping the future are,
1. Bias of good/bad fortune
2. Chain of chance
3. Qualitative hierarchy of luck
I believe these are the three.
I have expressed this idea with the following arithmetic formula.
Future = Now × (Bias) × (Chain) × (Quality)
I used multiplication instead of addition because real life and markets move synergistically. The future is not an extension of the past; it always begins from "now."
If the probability is tilted slightly forward, the scenery will change over time—that is what I believe.
This formula is not magic that guarantees the future.
It is simply a map for perceiving the future not as "total darkness," but as a probability with a direction.
Whether to give meaning to those numbers and accept responsibility for them is ultimately left to us, the users.
Digression: My financial model, the "ABC-K6 Model": What can be seen and what cannot be seen through this arithmetic formula
Here, I will pause to organize what my arithmetic model, ABC-K6, can and cannot show.
What K6 outputs is not the correct answer or a predicted value for the future.
It is a collection of countless valuation time-series replicas generated probabilistically under identical conditions.
It is not "a single point in the future," buta device for observing a set of multiple future paths that could have occurredis what it is.
From these replicas, one can analyze distributions such as CAGR, maximum DD (crash), and recovery periods, but this is merely an act of observing properties within the probability structure of K6, not a judgment of the real market.
Mathematical formulas are inorganic, and meaning and judgment are always left to the observer.
The greatest feature of K6 is that all internal calculation histories for each time series are preserved.
Therefore, the depth of DD and the recovery process can be observed as "paths with a story," andit functions like an experiential simulator (e.g., a driving simulator) to help one become aware of emotional fluctuations and tolerance in asset management in advance.
Also, K6 assumes only "financial products whose time is not cut off midway." Therefore, while it is compatible with index funds and leveraged ETFs, it is fundamentally incompatible with margin trading or CFDs that involve settlement deadlines, margin calls, and loss cuts.
I have concluded that even with the same "3x leverage," products that continue over time and contracts that end midway are essentially different things.
K6 is not a model that guarantees high accuracy or safety in asset management.
The time-series replicas generated by K6 include deep DDs and long periods of stagnation at various timings and levels. Even so, the future does not end midway, and diverse recovery scenarios can be observed to the end—in that one respect, leveraged ETFs can be subjects that can be handled by the K6 model.
From #8 onwards, I will dissect in order how this "biased probability" is embedded into the arithmetic formula and rises as future time-series replicas.
#8: A fair lottery system that draws biased luck: "BES³"
In this chapter, I will introduce how I embedded the "bias of good/bad fortune" which is the foundation of the K6 model, into the formula.
I perceive the market and life not as completely random, but as a world where there exists a "50+α% good fortune" that tilts slightly in a direction where the future becomes better.
I believe that the "α%" originates from my hypothesis of "civilization probability"—rooted in the instinct of "To be better than today" that has allowed humanity to rebuild even after falling and to accumulate civilization.
To reproduce this bias, I introduced a bias-embedded lottery mechanism.
It is a mechanism that provides a lottery device (a die as a metaphor) that has bias as a design philosophy, rather than just leaving it to luck, giving the probability a slight asymmetry.
However, in the market, there are differences in the quality of price movements, such as gentle rises, stagnation, declines like storms, and Big Events that change lives.
Therefore, I did not fix the bias to one type, but prepared multiple dice with different personalities (bias distributions) and adopted a nested structure where which one is applied is itself decided by a fair lottery.
I named this mechanism of "continuously choosing a lottery mechanism with bias through a fair lottery" BES³ (Bias Embedded Stochastic Selection System).
From the next time, I will break down the basic formula
Future = [Now] × [Bias of good/bad fortune] × [Chain of chance] × [Qualitative hierarchy of luck]
and introduce step-by-step which coefficients the lottery results of BES³ determine.
As a result, the arithmetic formula of the ABC-K6 model changes its appearance to a multiplication structure with more symbols:
Future = {Now × j1 × (α × β × γ) × j2 × δ} × ε
.
It starts to look mysterious, but my intention is to keep it within the range of "arithmetic that can be calculated by hand." It is an arithmetic formula that intuitively visualizes the mechanism by which bias, chains, and quality shape the future.
#9 Explanation of "α" in the arithmetic formula of my leveraged Nasdaq TQQQ simulation model, the ABC-K6 model, which adopts biased probabilities
In this chapter, we will receive the results from the biased lottery system introduced previously,BES³ and explain the "arithmetic formula (the brain)" of the K6 model that actually calculates future valuation.
I have named this arithmetic structure the MSM³ (Markov–Semi-Markov Modulated Multiplicative Process).
The basic form of MSM³ is the multiplication formula:
Future = [Now] × [Bias of Good/Bad Luck] × [Chain of Coincidence] × [Qualitative Hierarchy of Luck]
, which is broken down within the K6 model as follows:
Future = {Now × j1 × (α × β × γ) × j2 × δ} × ε
Although the number of symbols increases, the content is entirely an arithmetic formula consisting only of multiplication.
First, let's introduce the fundamental parameter, "Bias of Good/Bad Luck = α."
In K6, the price movement of TQQQ over a certain period is simplified into three states: "Zero Growth, Positive Growth, Negative Growth," and each is determined by the lottery results of BES³.
The first coefficient is α, which is a multiplier representing "the basic growth direction and strength for that period."
α is first determined by a high-level lottery using two fair dice to decide "Zero Growth (α=1)" first; only if it is not zero growth, two biased dice (A/B/C set) are rolled to determine it.
If the sum of the biased dice is even, it is positive growth; if odd, it is negative growth, and the growth rate is calculated as
○ Positive Growth
△α = 1 + (Average of Sum / n)
○ Negative Growth
▼ α = 1 - (Average of Sum / n)
(n is an arbitrary parameter to adjust the magnitude of the growth rate).
As described above, α in the arithmetic formula is derived from the "results of a set of dice with biased probabilities."
Supplement to #9: How does the bias of the parity of the sum change when using a "set of two" dice with biased probabilities?
In this supplement, I will clarify the points that are intuitively difficult to understand regarding the rule explained in #9: "using two biased dice as a pair."
In K6, a set of dice with biased probabilities (A/B/C) is selected by BES³, and the "parity of the sum of the results" determines whether it is positive growth (even) or negative growth (odd).
What is important here is that the bias in the parity of a single die does not simply intensify when used as a pair.
For example, if the parity ratio per die is
A: 70:30, B: 60:40, C: 50:50
, the parity of the sum of the two-set is
A set: 58:42, B set: 52:48, C set: 50:50
, which shows that while the bias remains, it is inevitably mitigated. Intuitively, one might feel that "if you use two 7:3 dice, even numbers will appear more often." However, in reality, only "even + even" or "odd + odd" create an even number, and mixing parity results in an odd number, creating a structure where the bias is neutralized.
K6 intentionally utilizes this property to create a state where:
"There is a bias, but it is not extreme"
"Ease of winning/losing does not become fixed"
"Fluctuations are likely to connect naturally"
This is because if parity judgment were done with only one die, the bias would be too strong, making the entire model unstable.
In other words, the α design of K6 is a refinement to ensure that "
only fluctuations with a directional tendency remain
."
#10: Explanation of "β" in the arithmetic formula of my Leveraged Nasdaq TQQQ simulation model, the ABC-K6 model, which adopts biased probabilities
In this chapter, I will introduce the parameter chain of coincidence, parameter β (beta), which quantifies the concept in the K6 model.
While α, discussed up to the previous chapter, determines the direction and magnitude of fluctuation—whether it "went up or down that week"—β does not represent the direction of price movement itself, but rather plays the role of amplifying the fact that "the same direction continued."
The K6 arithmetic formula has the structure
Future = [Now] × [Bias] × [Chain] × [Quality]
↓
Future = {Now × j1 × (α × β × γ) × j2 × δ} × ε
, in which the division of labor is:
・α: Determination of rise, fall, or zero growth
・β: Bonus/penalty due to continuation in the same direction
β is triggered only when α continues in the same direction. In K6, the influence of β increases according to the "chain length in the same direction," and the chain ends when an α in the opposite direction appears.
However, as an important design, we adopt the rule that "
zero growth (0α) does not break the chain". This models the intuition that "even if it pauses for a moment, it does not necessarily mean the trend has changed."
As an example of a chain, when positive growth (△α) chains for 2 or 3 times,
(▼α)(△α1)(△α2)(▼α)
(▼α)(△α1)(0α)(△α2)(▼α)
(▼α)(△α1)(△α2)(0α)(▼α)
Even if 0α is sandwiched, the chain continues if it is in the same direction.
A negative chain simply reverses △ and ▼.
The calculation of β is defined as follows, with the chain length as m (m≥2):
・Positive chain: β = K^(m−1)
・Negative chain: β = L^(m−1)
The reason the exponent is (m−1) is that the first growth is already included in α, and β amplifies "only the chain effect from the second time onwards."
What β represents is not the emotion of "being lucky/unlucky," but the non-linear amplification effect caused by the continuation of coincidence
. Whether a trend is extending or a decline is unstoppable, what determines the visceral impact is not the "correctness of the direction" but "whether the same direction continued"—K6 incorporates this intuition as β.
#11: Explanation of "γ" and "δ" in the arithmetic formula of my Leveraged Nasdaq TQQQ simulation model, the ABC-K6 model, which adopts biased probabilities
In this chapter, I will explain the parameters "
Future = [Now] × [Bias] × [Chain] × [Qualitative Hierarchy of Luck]
↓
Future = {Now × j1 × (α × β × γ) × j2 × δ} × ε
," specifically Qualitative Hierarchy of Luck parameters γ (gamma) and δ (delta), which have different properties from the α (direction and magnitude) and β (chain in the same direction) that we have covered so far.
In K6, "quality of luck" is not treated as a monolith but is intentionally separated into two types. Intuitively, it is organized as follows:
・γ: Luck that occurs within the flow (triggered in conjunction with the α chain)
・δ: Luck that interrupts from outside the flow (triggered independently of the α chain)
First,
γ is a parameter that is triggered the "moment" the α chain, which had been continuing in the same direction, breaks.While β exponentially amplifies the momentum of the α chain, γ acts as a "brake and cushion" that works at the boundary where the chain ends.There are two types of γ.
・Rescue γ: Triggered when △α (rise) breaks ▼α (downward chain) → A bonus representing the feeling of being "saved" when a decline stops
・Resistance γ: Triggered when ▼α (decline) breaks △α (upward chain)
→ A penalty representing the feeling of "hitting a wall" when a rise stops
Both commonly suppress the runaway upside/downside caused by β and create a foundation for trend reversal.Next,
δ is completely different from γ, and is a "Big Event" parameter that appears probabilistically regardless of the flow of α and β.α and β flow and appears probabilistically regardless of the "Big Event" parameter.δ is introduced to represent discontinuous interruptions that "happen when they happen," such as earnings reports, economic indicators, monetary policy, geopolitical risks, and unexpected news.
By separating:・α = Basic direction of change and magnitude of change・β = Internal amplification due to the continuation of the chain
・γ = Boundary luck that occurs at the moment the chain breaks
・δ = Exogenous events that interrupt regardless of the flow
K6 has a structure that allows "momentum," "reversal," and "sudden events" to be incorporated into a mathematical formula simultaneously.
#12: Mathematical formula for my leveraged Nasdaq TQQQ simulation model using biased probabilities, the ABC-K6 model... Explanation of "ε" and "j1, j2"
In this chapter, I will explain the parameters ε, j1, and j2 that remained in the basic formula of the K6 model:
Future = {Now × j1 × (α × β × γ) × j2 × δ} × ε
First,
ε is a parameter that incorporates costs. It also includes "impairment costs" that occur over time, especially by holding TQQQ.While leveraged ETFs have explicit costs such as
expense ratios (e.g., 0.18% for QQQ vs. 0.82% for TQQQ), a more important "hidden cost" is volatility drag (volatility decay) derived from daily rebalancing.Volatility decay is a phenomenon where, when an index repeatedly moves up and down in a range-bound market, the "negative compounding effect" is amplified in leveraged ETFs, and the valuation naturally erodes.
For example, if an index moves -5% → +5.27%, the valuation of a non-leveraged ETF becomes 100.01% of the principal, recovering the principal.
However, with 2x leverage, the price movement becomes -10% → +10.54%, and the valuation becomes 99.49% of the principal, resulting in a loss of principal.
As this difference accumulates, a temporal wear specific to leveraged ETFs occurs, where "
a sideways market itself becomes a cost."In K6, I have incorporated this wear into ε as an impairment coefficient that is always active, creating a structure where the valuation gradually decreases due to ε even during periods where weekly change is zero (α=1).Next,
j1 and j2 are parameters for introducing inexplicable market fluctuations (irrational deviations), unlike "structural elements with design intent" such as α, β, γ, δ, and ε.j1 provides a "1 ± several %" fluctuation to the first half of the mathematical formula (mainly the part related to α), and j2 provides it additionally when a Big Event (δ) occurs, through independent dice rolls.
1 ± several % fluctuation.Although these are elements that are difficult to observe or define, by intentionally including them, I have given them the role of thinning out the discomfort of K6 "artificially mimicking TQQQ" and generating turbulent time-series replicas close to the real market.
# 13: My leveraged Nasdaq TQQQ simulation model using biased probabilities... Are time-series replicas generated with equal probability?
Are the time-series replicas generated by the ABC-K6 model probabilistically "equivalent"? I have such a simple question.
Since the K6 model is composed of a finite set of parameters and lottery probabilities, it is theoretically possible to create a "future catalog" that classifies all patterns of weekly changes by probability frequency.
In other words, each replica time series has a structure where an occurrence probability, "how likely it is to happen," can be defined.
However, when I actually generate 1000 replicas and arrange them in order of CAGR, I tend to have the illusion that "the top, median, and bottom are all equally likely to happen."
But K6 has history reference and also includes low-frequency elements such as Stall (no price movement) and Big Events.
Considering this structure, even if the number of generated replicas is equal, the ranking space is not necessarily probabilistically equal. Rather, I have come to feel that inequality is more natural.
In other words, there is a possibility that "how many appeared as CAGR ranking (visual distribution)" and "how naturally that future ranking is likely to occur (occurrence probability)" do not match.
Even in the current preliminary analysis, results are beginning to emerge that suggest a discrepancy between the rankings near the median and the ranking zones that are probabilistically dense. I feel that there are still mathematically unresolved issues in K6.
On the other hand, if you view the diverse time-series replicas generated by K6 as a "full-length mirror (experiential simulator)," I also think that being able to naturally experience the depth of a crash and the recovery period in various ranking zones is meaningful enough for measuring emotional tolerance.
This article about the K6 model, centered on TQQQ, still has more to come.
From the next time onwards, I intend to write about considerations useful for practical asset management based on the K6 model.