# Granularity Adjustment

## Definition

The **Granularity Adjustement** (GA) for the ASRF model is an approximation formula for calculating the capital needed to cover the risk arising from the potential default of large borrowers.

## Context

The granularity adjustment is an extension of the ASRF model which forms the theoretical basis of the Internal Ratings-Based (IRB) approaches of Basel II/Basel III. Through this adjustment, single Name Concentration is integrated into the ASRF model thereby making capital requirements more risk sensitive.

The ASRF model assumes that portfolios are fully diversified with respect to individual borrowers, so that risk capital depends only on Systematic Risk. Hence, the IRB formula omits the contribution of the residual Idiosyncratic Risk to the required capital.

A granularity adjustment that incorporates name concentration in the IRB model was already included in the Second Consultative Paper of Basel II and was later refined by the work of Martin and Wilde ^{[1]} and further in Gordy and Lütkebohmert ^{[2]}.

Given a portfolio of N borrowers, Gordy and Lütkebohmert developed a simplified formula for an add-on to the capital for unexpected loss (UL capital) in a single-factor model. The simplified formula follows from the ‘full’ granularity adjustment of Gordy and Lütkebohmert if quadratic terms are dropped. An alternative interpretation is to assume that any idiosyncratic risk in recovery rates that is still explicitly captured by the ‘full’ adjustment formula is eliminated by diversification.

## Issues and Challenges

- The simplicity of the granularity adjustment formula comes at the price of a potential model error. The reason is that the granularity adjustment, unlike the IRB model, was developed in a CreditRisk+ setting, and the CreditRisk+ model differs in the tail of the loss distribution from the IRB model. For this reason, the formula comes close to, but is not fully consistent, with the IRB model.
- More broadly the approximation is only valid within a narrow and fairly simplistic view of credit portfolios