The Limits of Granularity Adjustments

Abstract

We provide a rigorous proof of granularity adjustment (GA) formulas to evaluate loss distributions and risk measures (value-at-risk) in the case of heterogenous portfolios, multiple systemic factors and random recoveries. As a significant improvement with respect to the literature, we detail all the technical conditions of validity and provide an upper bound of the remainder term at a finite distance. Moreover, we deal explicitly with the case of general loss distributions, possibly with masses. For some simple portfolio models, we prove empirically that the granularity adjustments do not always improve the infinitely granular first-order approximations. This stresses the importance of checking our conditions of regularity before relying of such techniques. And smoothing the underlying loss distributions through random recoveries or exposures improves the GA performances in general.

Multifactor granularity adjustments for market and counterparty risks

“…Under certain technical conditions and when n tends to the infinity, it can be proved that the cdf of L n is arbitrarily close to the cdf of µ(X), plus the function T n,∞ (•): see Gordy (2004) or Fermanian (2014), among others. Therefore, in this case, the portfolio value-at-risk can be approximated by a so-called granularity adjustment formula, i.e.…”

“…We are interested in checking whether the GA approximations are suffering from such a feature. Indeed, in Fermanian (2014), it has been noticed that fat-tailed loss distributions can disturb GA approximations.…”

Robust and consistent estimation of generators in credit risk

“…• Although more sophisticated methods are available for calculation of VaR and ES (see [Fer14]), we calculate the risk charges using Monte Carlo. This is sufficient here since the portfolios are small relative to a typical bank portfolio, therefore we can obtain accurate estimates using a reasonable number of simulations.…”