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

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).

💡 Research Summary

The paper introduces a unified framework for portfolio optimization (PO) that places the gain probability density function (gain PDF) at the core of the decision‑making process. Starting from the classical Markowitz formulation, the authors argue that traditional approaches rely on a limited set of scalar statistics—typically expected return and a risk measure such as variance, Value‑at‑Risk (VaR) or Conditional‑VaR (CVaR). While these metrics capture only a single aspect of the underlying distribution, they become inadequate when the gain function is non‑linear (e.g., return on investment) or when the asset returns deviate from Gaussian assumptions.

The authors first formalize the statistical setting: a set of N assets, each described by L stochastic features collected in a matrix Y. Scenarios s = 1,…,S are assumed i.i.d., providing a data‑generating process for the random vector Y. A portfolio weight vector P satisfies the usual budget and non‑negativity constraints, possibly augmented by linear limits on individual assets. The gain function g(P, Z) maps a realized scenario Z to a scalar profit (or ROI). By applying g to the random matrix Y, the scalar random variable G = g(P, Y) is obtained, whose probability density with respect to Lebesgue measure is denoted σ(P, u). This σ, the gain PDF, is one‑dimensional for any portfolio and contains the full probabilistic information needed for optimization.

All conventional objective functions can be expressed as functionals of σ. For example, the classic Markowitz objective becomes
gain(σ) = ∫ u σ(P, u) du,
risk_Mark(σ) = ∫ u² σ(P, u) du − gain(σ)².
Similarly, CVaR‑deviation is written as risk_Cdev_β(σ) = CVaR_β(σ) − gain(σ) with CVaR_β(σ) = min_α