Signed network models for dimensionality reduction of portfolio optimization

TL;DR

This paper develops a time-series-based signed network model for dimensionality reduction in portfolio optimization, grounded in Markowitz's portfolio theory and extended to incorporate higher-order moments of asset return distributions, and introduces a combinatorial interpretation of higher-order moments.

Abstract

In this paper, we develop a time-series-based signed network model for dimensionality reduction in portfolio optimization, grounded in Markowitz's portfolio theory and extended to incorporate higher-order moments of asset return distributions. Unlike traditional correlation-based approaches, we construct a complete signed graph for each trading day within a specified time window, where the sign of an edge between a pair of assets is determined by the relative behavior of their log returns with respect to their mean returns. Within this framework, we introduce a combinatorial interpretation of higher-order moments, showing that maximizing skewness and minimizing kurtosis correspond to maximizing balanced triangles and balanced 4-cliques with specific signed edge configurations respectively. We establish that the latter leads to an NP-hard combinatorial optimization problem, while the former is naturally guaranteed by the structural properties of the signed graph model. Based on this interpretation, we propose a dimensionality reduction method using a combinatorial formulation of the mean-variance optimization problem through a combinatorial hedge score metric for assets. The proposed framework is validated through extensive backtesting on 199 S\&P 500 assets over a 16-year period (2006 - 2021), demonstrating the effectiveness of reduced asset universes for portfolio construction using both Markowitz optimization and equally weighted strategy.

Figures (5)

• Figure 1: (a) Threshold function harary2002signed for signed network formation. $c_{ij}$ denotes the covariance or correlation strength for the assets $i$ and $j$, (b) The formation of positive and negative edges in $G^s_t(\boldsymbol{\mu,R_N})$, $(c)$ Signed triangles in signed graphs, green and red colored edges represent the positive and negative edges respectively.
• Figure 2: The possible $4$-cliques in $G^s_t(\boldsymbol{\mu}, R_N)$. Edges in green and red indicate positive and negative edges respectively.
• Figure 3: Annual Return: (a) $K=20$, (b) $K=30$, (c) $K=40$, (d) $K=50.$
• Figure 4: Annual Volatility: (a) $K=20$, (b) $K=30$, (c) $K=40$, (d) $K=50.$
• Figure 5: Sharpe ratio: (a) $K=20$, (b) $K=30$, (c) $K=40$, (d) $K=50.$

Theorems & Definitions (2)

• proof
• proof