\( \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\textm}[1]{\textsf{#1}} \newcommand{\textnormal}[1]{\textsf{#1}} \def\T{{\mkern-2mu\raise-1mu\mathsf{T}}} \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\E}{{\rm I\kern-.2em E}} \newcommand{\w}{\bm{w}} % bold w \newcommand{\bmu}{\bm{\mu}} % bold mu \newcommand{\bSigma}{\bm{\Sigma}} % bold mu \newcommand{\bigO}{O} %\mathcal{O} \renewcommand{\d}[1]{\operatorname{d}\!{#1}} \)

12.5 Summary

Graphs provide a convenient and compact way to represent big data, exposing the underlying structure and existing patterns that may otherwise go unnoticed. In the context of portfolio design, the following key takeaway points can be identified.

  • In finance, graphs can represent the relationship among assets: each asset is a node and their pairwise relationships are represented by edges of different strength.

  • Financial graphs can be learned automatically from collected data (see Chapter 5). The recommended graph learning formulations are the heavy-tailed Markov random field (12.3) or, if a clustered graph is desired, the \(k\)-component version (12.4).

  • Hierarchical clustering methods can be used to partition the assets into clusters with different levels of detail from the graph information.

  • Graph information of assets should be taken into account in the portfolio formulation. Further work needs to be done, but some notable examples include: the hierarchical \(1/N\) portfolio, the hierarchical risk parity portfolio, and the hierarchical equal risk contribution portfolio.