MYE Yellow Paper – Version Q4 2025

September 29, 2025

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📄 Download the full MYE Yellow Paper (PDF)

Abstract

This algorithm forms the technical core of the Messiah Yield Engine (MYE), which will be deployed in a phased rollout.

This paper describes the mathematical properties of the initial algorithm designed to manage a small, diverse pool of treasury assets.

The system is architected for continuous, on-the-fly improvement, and thus the underlying model is subject to iterative enhancement as more research is conducted and more financial data becomes available.

The core idea is grounded in the academic literature of modern portfolio theory. It builds upon Markowitz’s mean-variance optimization for optimal asset allocation, extends it by incorporating higher-moment measures (skewness, kurtosis) to account for non-normal return distributions, and finally hardens it with distributionally robust optimization techniques to ensure performance is maintained under model uncertainty and adverse market regimes.

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Base Algorithm and Distributionally Robust Gamma Algorithm

Enhancements via Higher-Order Portfolio Optimization

This section provides a concise mathematical summary of the Base Algorithm and the Distributionally Robust (DR) Gamma Algorithm, along with their extensions using higher-order portfolio optimization.
These enhancements incorporate non-Gaussian characteristics of asset returns, particularly relevant for cryptocurrency markets.

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0.1 Base Algorithm (Nominal End-to-End Learning)

The Base Algorithm unifies prediction and optimization within an end-to-end framework, where the objective is to minimize a task-specific loss tied to out-of-sample portfolio performance, rather than isolated prediction errors.

Mathematical Formulation

Prediction Layer:
A parameterized function

$$g_θ : ℝ^m → ℝ^n$$

generates point estimates of asset returns from input features ( x_t ), yielding

$$\hat{y}_t = g_θ(x_t)$$

Deviation Risk Measure:
For a portfolio ( z ∈ 𝒵 ), the risk is quantified via a deviation function ( R ) (e.g., ( R(X) = X^2 ) for variance) applied to historical prediction errors ( ε = {ε_j} ):

$$f_ε(z, q) = \min_c \sum_{j=1}^{T} q_j \, R(ε_j^T z - c)$$

with ( q ) denoting the nominal uniform distribution (( q_j = 1/T )).

Optimization Layer (Nominal):
The optimal portfolio is obtained by solving:

$$z_t^* = \arg\min_{z ∈ 𝒵} \Big[ f_ε(z, q) - γ \, \hat{y}_t^T z \Big]$$

where ( γ ) represents the risk appetite parameter.

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Enhancement via Higher-Order Moments

The deviation function ( R(X) ) may be extended beyond quadratic variance (( X^2 )) to account for higher-order moments, such as skewness and kurtosis. For instance:

Skewness Adjustment

$$R(X) = X^2 - η_3 X^3$$

with ( η_3 ) mitigating negative skewness (asymmetric downside risks).

Kurtosis Adjustment

$$R(X) = X^2 + η_4 X^4$$

with ( η_4 ) addressing heavy-tailed distributions (extreme events).

Such generalizations enable the Base Algorithm to optimize portfolios aligned with utility functions that reflect the non-Gaussian dynamics prevalent in cryptocurrency returns.

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0.2 Distributionally Robust (DR) Gamma Algorithm

The DR Gamma Algorithm augments the Base Algorithm by incorporating distributional ambiguity, replacing the nominal distribution ( q ) with a worst-case distribution from a ϕ-divergence-bounded ambiguity set ( 𝒫(δ) ).

Mathematical Formulation

Ambiguity Set: Constructed via a ϕ-divergence (e.g., Hellinger distance):

$$𝒫(δ) = \{\, p ∈ Δ_T : I_ϕ(p, q) ≤ δ \,\}$$

where ( Δ_T ) is the probability simplex over ( T ) outcomes, and ( δ ) governs the robustness level.

DR Minimax Problem:
Portfolio selection proceeds as:

$$z_t^* = \arg\min_{z ∈ 𝒵} \max_{p ∈ 𝒫(δ)} \Big[ f_ε(z, p) - γ \, \hat{y}_t^T z \Big]$$

Dual Reformulation (Tractable Form):
Convex duality yields an equivalent minimization:

$$z_t^* = \arg\min_{z ∈ 𝒵,\, λ≥0,\, ξ,\, c} \left[ ξ + δλ + \frac{λ}{T} \sum_{j=1}^{T} ϕ^*\!\left( \frac{R(ε_j^T z - c) - ξ}{λ} \right) - γ \, \hat{y}_t^T z \right]$$

where ( ϕ^* ) is the convex conjugate of the ϕ-divergence kernel, ensuring computational tractability.

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Enhancement via Higher-Order Moments

The aforementioned extensions to ( R(X) ) (incorporating skewness and kurtosis) integrate seamlessly into the DR framework, rendering the robustness sensitive to non-Gaussian tail risks and asymmetries.
Consequently, the adversarial distribution ( p ) in the worst-case analysis exploits not only variance but also higher moments, yielding portfolios resilient to extreme prediction discrepancies and cryptocurrency-specific market regimes.

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0.3 Summary of Enhancements with Higher-Order Portfolio Optimization

To transcend mean-variance limitations, both algorithms admit enhancements through higher-order considerations:

1. Generalization of ( R(X) ):
   Substitute the quadratic form ( R(X) = X^2 ) with a polynomial utility incorporating elevated moments:

   $$R(X) = X^2 + η_4 X^4$$

   where ( η_4 > 0 ) addresses excess kurtosis (tail risks).

   The third moment (skewness) has been explicitly excluded from this formulation to maintain the convexity properties essential for computational tractability.

   This preserves the convex optimization structure while capturing the critical non-Gaussian effects of heavy-tailed distributions.

2. Integration into End-to-End Learning:
   The coefficient ( η_4 ) is optimized jointly with ( θ, γ, ) and ( δ ) through backpropagation across the optimization layer, guided by a task loss (e.g., Sharpe ratio).

   Incorporation of the third moment would require more advanced non-convex solver approaches, which are currently under development for future iterations.

3. DR with Higher Moments:
   Within the DR setup, the worst-case distribution ( p ) adversarially targets these augmented moments, enhancing protection against extreme deviations and tail events.

   The current convex formulation ensures this protection remains computationally feasible in production environments.

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These extensions preserve the foundational end-to-end learning paradigm while accommodating the higher-moment features indispensable for cryptocurrency portfolio management.
The architecture maintains explicit convexity to ensure tractable optimization while providing a pathway for future incorporation of skewness through advanced solver methodologies.

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References

1. Amos, B., & Kolter, J. Z. (2017). OptNet: Differentiable Optimization as a Layer in Neural Networks. Proceedings of the 34th International Conference on Machine Learning (ICML), 136–145.
2. Kuhn, D., Esfahani, P. M., Nguyen, V. A., & Shafieezadeh-Abadeh, S. (2019). Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning. arXiv preprint arXiv:1908.08729.
3. Martellini, L., & Ziemann, V. (2010). Improved Estimates of Higher-Order Comoments and Implications for Portfolio Selection. The Review of Financial Studies, 23(4), 1467–1502.