Unified Approach to Portfolio Optimization using the `Gain Probability Density Function’ and Applications

ArXiv ID: 2512.11649 “View on arXiv”

Authors: Jean-Patrick Mascomère, Jérémie Messud, Yagnik Chatterjee, Isabel Barros Garcia

Abstract

This article proposes a unified framework for portfolio optimization (PO), recognizing an object called the `gain probability density function (PDF)’ as the fundamental object of the problem from which any objective function could be derived. The gain PDF has the advantage of being 1-dimensional for any given portfolio and thus is easy to visualize and interpret. The framework allows us to naturally incorporate all existing approaches (Markowitz, CVaR-deviation, higher moments…) and represents an interesting basis to develop new approaches. It leads us to propose a method to directly match a target PDF defined by the portfolio manager, giving them maximal control on the PO problem and moving beyond approaches that focus only on expected return and risk. As an example, we develop an application involving a new objective function to control high profits, to be applied after a conventional PO (including expected return and risk criteria) and thus leading to sub-optimality w.r.t. the conventional objective function. We then propose a methodology to quantify a cost associated with this optimality deviation in a common budget unit, providing a meaningful information to portfolio managers. Numerical experiments considering portfolios with energy-producing assets illustrate our approach. The framework is flexible and can be applied to other sectors (financial assets, etc).

Keywords: Portfolio Optimization, Gain Probability Density Function (PDF), Risk Decomposition, Energy Assets, High Moments

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper presents a sophisticated, unified theoretical framework using probability density functions and generalizes existing risk measures, requiring advanced mathematics for derivation and application. It includes numerical experiments with energy assets and proposes specific optimization algorithms (projected gradient) and cost quantification methods, indicating a significant data and implementation focus.

  flowchart TD
    A["Research Goal: Unified PO Framework via Gain PDF"] --> B["Data: Energy Assets Dataset"]
    B --> C["Methodology: Compute Gain PDF for Portfolio"]
    C --> D["Compute: High Moment Objective & Cost of Deviation"]
    D --> E["Outcome: New Objective Function for High Profits"]
    E --> F["Outcome: Quantified Cost of Optimality Deviation"]
    F --> G["Outcome: Unified Framework for PO & Risk Decomposition"]