Existence, uniqueness, and convergence rates of the hierarchical risk‑parity fixed point

Establish existence, uniqueness, and convergence rates for the hierarchical risk‑parity fixed point that equalizes risk contributions at both the sector and within‑sector levels under realistic block covariance structures (i.e., asset‑level covariance matrices partitioned into sector blocks). Derive sufficient conditions—such as block diagonal dominance or restricted eigenvalue bounds—under which the two‑level risk‑parity fixed‑point updates converge to a unique solution.

Background

The paper introduces a two‑level Bayesian Hierarchical Risk Parity (BHRP) framework in which final asset weights are constructed from sector weights and within‑sector weights. The covariance matrix is partitioned into sector blocks, enabling risk‑parity conditions to be enforced both across sectors and within each sector.

While the authors provide fixed‑point update procedures and algorithmic characterizations for achieving equal risk contributions at both levels, they explicitly note that formal theoretical guarantees—existence, uniqueness, and convergence rates—have not been derived for realistic (non‑ideal) block structures. They call for sufficient conditions, such as block diagonal dominance or restricted eigenvalue bounds, to be identified.