Paper: arXiv 2302.10573

Authors: Andries Steenkamp

Abstract

We consider the multi-objective mean-variance-skewness-kurtosis (MVSK) problem in portfolio selection, with and without shorting and leverage. Additionally, we define a sparse variant of MVSK where feasible portfolios have supports contained in a chosen class of sets. To find the MVSK problem’s Pareto front, we linearly scalarize the four objectives of MVSK into a scalar-valued degree four polynomial $F_λ$ depending on some hyper-parameter $λ\in Δ^4$. As one of our main results, we identify a set of hyper-parameters for which $F_λ$ is convex over the probability simplex (or over the cube). By exploiting the convexity and neatness of the scalarization, we can compute part of the Pareto front. We compute an optimizer of the scalarization $F_λ$ for each $λ$ in a grid sampling of $Δ^4$. To see each optimizer’s quality, we plot scaled portfolio objective values against hyper-parameters. Doing so, we reveal a sub-set of optimizers that provide a superior trade-off among the four objectives in MVSK.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical and mathematical contribution by establishing conditions for the convexity of degree-four polynomial scalarizations in higher-moment portfolio optimization (MVSK). However, it focuses primarily on the optimization methodology and synthetic grid sampling rather than rigorous empirical backtesting against real-world market datasets.

Research Flowchart

  flowchart TD
    A[Goal: Solve Multi-Objective MVSK Portfolio Optimization] --> B[Inputs: Portfolio Constraints & Hyper-parameter Grid]
    B --> C[Method: Linear Scalarization into Degree-4 Polynomial F_λ]
    C --> D[Identify Hyper-parameter Set Ensuring Convexity of F_λ]
    D --> E[Compute Optimizers via Convex Optimization over Grid]
    E --> F[Plot Scaled Objective Values vs. Hyper-parameters]
    F --> G[Outcomes: Compute Pareto Front & Identify Superior Trade-off Portfolios]