The limits of granularity adjustments

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 systematic 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 for finite portfolio sizes. 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 some conditions of regularity before relying on such techniques. Smoothing the underlying loss distributions through random recoveries or exposures improves the GA performances in general.