Granularity Shock: A Small Perturbation Two-Factor Model

Osadchiy, Maksim (2025): Granularity Shock: A Small Perturbation Two-Factor Model.

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Credit portfolio model; Granularity adjustment; Value at Risk; Expected Shortfall
• Subjects: C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics; G - Financial Economics > G2 - Financial Institutions and Services > G21 - Banks ; Depository Institutions ; Micro Finance Institutions ; Mortgages; G - Financial Economics > G3 - Corporate Finance and Governance > G32 - Financing Policy ; Financial Risk and Risk Management ; Capital and Ownership Structure ; Value of Firms ; Goodwill

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