Flat Betting Model: Analysis of a Linear Position-Sizing Strategy

The Flat Model represents a deterministic position-sizing strategy in which the absolute magnitude of each transaction remains constant regardless of the current portfolio balance. Formally, if we denote the position size as S, then S(t) = S₀ for all t ∈ [0, T], where T is the observation horizon. This strategy constitutes a special case of linear capital management with a zero adaptation coefficient. The absence of feedback between iteration outcomes and subsequent position sizes renders the model fully stationary in terms of stochastic process theory. It is precisely this stationarity that ensures a predictable variance profile across any arbitrary interval.

The mathematical proof of Flat Model stability is grounded in the Central Limit Theorem (CLT). Let X₁, X₂, ..., Xₙ be a sequence of independent, identically distributed random variables representing the outcome of each iteration. The sum Sₙ = Σ Xᵢ as n → ∞ is asymptotically normally distributed with parameters (nμ, nσ²), where μ is the expected value of a single outcome and σ² is its variance. Critically, the linear growth of variance is proportional to √n rather than n, implying that relative volatility diminishes as sample size increases. This property formally demonstrates that the coefficient of variation CV = σ/μ converges to zero as the number of iterations grows without bound.

Variance analysis of the Flat Model reveals a fundamental distinction from progressive systems. The total portfolio variance over n iterations is computed as Var(Sₙ) = n · S₀² · σ²ₓ, where σ²ₓ is the normalized variance of a single outcome. Unlike geometric progressions, where variance grows exponentially, linear growth ensures a controllable risk profile. The standard deviation of the result after N iterations is SD(N) = S₀ · σₓ · √N, which enables the construction of confidence intervals with high precision. The practical significance of this property lies in the ability to accurately forecast the Maximum Drawdown (MDD) at a specified confidence level.

Expected Value (EV) convergence in the Flat Model is governed by the Law of Large Numbers. The sample mean Ŝₙ = Sₙ/n converges in probability to μ as n → ∞, with convergence rate determined by Chebyshev's inequality: P(|Ŝₙ - μ| ≥ ε) ≤ σ²/(nε²). In practice, this implies that achieving a relative precision of δ = 1% with a standard deviation σ = 0.5 requires at least n = σ²/δ² = 2,500 iterations. Notably, the Flat Model provides the fastest EV convergence among all strategies with a fixed upper bound on position size. This property establishes it as the benchmark strategy for comparative analysis of more sophisticated capital management systems.

Practical implementation of the Flat Model in automated systems requires addressing several engineering challenges. First, the optimal position size S₀ must be determined as a function of initial capital C₀ and the acceptable risk level R: S₀ = C₀ · R / (z_α · σₓ · √N_max), where z_α is the quantile of the normal distribution and N_max is the planned number of iterations. Second, the system must implement a monitoring mechanism for detecting deviations of actual results from the theoretical distribution using the χ² test or the Kolmogorov-Smirnov test. Third, the system architecture requires a State Manager module that ensures atomicity of each transaction and provides correct logging for subsequent statistical analysis. Deployment within a microservices architecture enables horizontal scaling of computational resources when processing parallel data streams.