And what would life be like without it?
by Harry Edwards
The causes of the drunkard’s walk are market noise, randomness and murky market dynamics. But how would you define it mathematically? Take a look at the diagram below. It’s not typically obvious which points peak — then start a decline or bottom and then start and advance with market prices. Smoothing markets introduces time lags — while traditional representations leaves the points labeled as a, b and c camouflaged. In this article — I’m going to show you the solution and the limitless potential of it.
The above diagram illustration of Rolle’s Theorem was offered by Michel Rolle in 1691. At the peak, the first derivative between the two distinct points A and B is zero. But the peaks and valleys in market prices are confusing. Can’t see point c. You can’t repeatedly determine points a, b and c without distorting those points — which smoothing does by introducing a time lag. Life with the drunkard’s walk means attempting to determine point c — which is useful for optimal stopping — is an unsolved problem. Given the high variability nature of market dynamics — it’s no wonder most people have difficulty finding a simple market theory that works consistently enough for them. The simplest and most powerful solution has remained hidden due to the drunkard’s walk.
Time Value of Money Theorem (TVM)
The Time Value of Money dates at least to Martín de Azpilcueta (1491–1586) and is the central concept in finance theory. The concept allows the valuation of a likely stream of income and/or cash flows — both even and variable in nature between specific points in time. Variable sources can also be considered such as interest, projects cash flows and so forth.
Liquid market assets have fluctuating values. The key is to be able to dynamically value information streams so that money can be extracted via a compounding process without complex assumptions. This requires a unique solution that determines when the time lines should start, stop and when a shift of position should occur. This is the case because to know whether money is compounding for — rather than against you — requires a timeline so that the two points defined as a and b in Diagram 1 exist. Point b always unfolds sequentially in time — and positive compounding occurs naturally between points a and c. I will summarize a theorem and process that makes it possible to optimize TVM via the identification of the crucial points on timelines including market information streams.
Optimal Accumulation Theorem (OAT)
After point c in Diagram 1 dimensioning returns will occur — followed by negative returns. The challenge is to be able to classify system dynamics so that as little time as possible is spent attempting to accumulate resources after point c. Our theorem states: “In system dynamics which vary from favorable to unfavorable as defined by point a to b — there is some point called d — beyond c which presents an unacceptable risk.”
In order to optimize the rate of asset accumulation — the identification of point d is required to sequentially convert market information into variable returns streams. While the returns will contain both positive or negative results — classification for the information also enables strategy to limit the negative return streams. With fixed rate compounding — what is called a periodic “accumulator factor” is static and so is the conversion frequency of the compounding period.
The optimal accumulation theorem is an umbrella theorem which covers risk /reward conflicts in human and animal domains. While the scope of this article does not permit its discussion in more detail — readers will grasp the details in the examples that follow using Marginal Value Theorem and Information Foraging which are related concepts. MVT says a marginal rate forager should leave the current path once the marginal rate decreases pass point c in the diagram 1.
A typical illustration is competitive apple picking by humans. On each new apple tree, the number of apples picked per minute is high but decreases fast with time. Continuing to pick apples until the last few apples can be shown to be quantitatively suboptimal by the MVT. In OAT, the competition can take place on protected patches involving human conflicts that can have an extended range of position or negative outcome. It’s a completely general framework applicable where system dynamics as sequential patches can be mapped by partitioning as shown in Diagram 1.
Accumulation Rate Optimization (ARO)
Instead of a fixed accumulation factor — Accumulation Rate Optimization (ARO) uses a variable accumulation factor in an optimal compounding process. Point d as well as the classification of points between a to b are used to maximize performance by decreasing the time spent after the accumulator factor passes point c. Of course point c remains unknown until after it has occurred. An exit in the area of d is practical without prediction or probability estimates. The solution to drunkard’s walk makes discovering c and d simple classification problem. Optimizing positive and negatives returns is therefore a matter of maximizing time in partitions with a positive accumlation factor and minimizing time in those where it is negative.
The MVT implies that a forager should leave the current patch once the marginal rate decreases pass point c in the diagram 1 (which is similar to an MTV diagram). In ORA, optimizing the rate asset accumulation — is an optimal compounding process in which the patches resemble Diagram 1 — where a to b represent the timeline. The best accumulation factor for optimal compounding is the same advantage formula used in blackjack — as it defines mathematical expectation.
Hence, the advantage is able to confirm the correct mapping from a to b. How? Positive and negative return streams appear in their appropriate location in each patch is shaped as redundant curve. Each point exist within partition that makes it easier to determine point d. Additionally, each market patch is self-organized so that both the timeline and time spent in each is optimized by an actual — rather than estimated advantage.
What would life — be like — without the drunkard’s walk?
Imagine your financial life changed by a solution that enables you to use an optimal compounding process. Instead of debating or protesting about the inequalities in the financial industry — you could level the playing field. What makes our solution potential limitless is the ease which those without finance or math background — can do exactly that. How? Perceive how the concept works — then as our affiliate — learn to use it to control your financial destiny by helping a world full of people looking to do the same.