Granularity Shock: A Small Perturbation Two-Factor Model

The paper presents a small perturbation two-factor model designed to capture granularity risk, extending the Vasicek Asymptotic Single Risk Factor (ASRF) portfolio loss model. By applying the Lyapunov Central Limit Theorem, we demonstrate that, for small values of the Herfindahl-Hirschman Index (HHI), granularity risk, conditional on market risk, is proportional to a standard normal random variable. Instead of studying the behavior of a heterogeneous portfolio, we examine the behavior of a homogeneous portfolio subjected to a small perturbation induced by granularity risk. We introduce the Vasicek-Herfindahl portfolio loss distribution, which extends the Vasicek portfolio loss distribution for heterogeneous portfolios with low HHI values. Utilizing the Vasicek-Herfindahl distribution, we derive closed-form granularity adjustments for the probability density function and cumulative distribution function of portfolio loss, as well as for Value at Risk (VaR) and Expected Shortfall (ES). We compare the primary results of our approach with established findings and validate them through Monte Carlo simulations.