Granularity Shock: A Small Perturbation Two-Factor Model

Maksim Osadchiy

MPRA Paper from University Library of Munich, Germany

Abstract: This paper proposes a small perturbation two-factor model designed to capture granularity risk, extending the classical Vasicek Asymptotic Single Risk Factor (ASRF) portfolio loss model. By applying the Lyapunov Central Limit Theorem, we demonstrate that, for small Herfindahl-Hirschman Index (HHI) values, granularity risk – conditional on market risk – is approximately proportional to a standard normal random variable. Instead of analyzing heterogeneous portfolios directly, we focus on a homogeneous portfolio subject to a small perturbation induced by granularity risk. We propose the Vasicek-Herfindahl portfolio loss distribution, which extends the Vasicek portfolio loss distribution to account for portfolio concentration. Utilizing this distribution, we derive closed-form granularity adjustments for the probability density function (PDF) and cumulative distribution function (CDF) of portfolio loss, as well as for Value at Risk (VaR) and Expected Shortfall (ES). We compare our primary results with existing findings and validate them through Monte Carlo simulations.

Keywords: Credit portfolio model; Granularity adjustment; Value at Risk; Expected Shortfall (search for similar items in EconPapers)
JEL-codes: C46 G21 G32 (search for similar items in EconPapers)
Date: 2025-03-31, Revised 2025-06-15
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