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

TL;DR

This paper introduces a unified framework for portfolio optimization centered on the gain probability density function, enabling flexible, interpretable, and customizable approaches, including direct target PDF matching and deviation cost quantification.

Contribution

It presents a novel gain PDF-based framework that unifies existing methods and allows direct control over portfolio objectives through target PDF matching.

Findings

Framework effectively incorporates existing approaches like Markowitz and CVaR.

Method enables direct targeting of desired profit distributions.

Numerical experiments demonstrate practical applicability to energy asset portfolios.

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).