2020/05/06 by Takaaki Koike, Marius Hofert, Koike, Takaaki +1
Agricultural and Biological Sciences · Decision Sciences · Economics, Econometrics and Finance · #91B30 (Secondary) #91G70 (Primary) 91B82 #Agricultural risk and resilience #FOS: Economics and business #Financial Risk and Volatility Modeling #Risk Management (q-fin.RM) #Risk and Portfolio Optimization #msc:91B30 #msc:91B82 #msc:91G70 #q-fin.RM
paper · pdf · doi:10.48550/arxiv.2005.02950
41 pages, 4 figures, 4 tables
openalex publication_date 2020/05/06 · arxiv created 2020/11/18 · arxiv updated 2020/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the variability of a risk from the statistical viewpoint of multimodality of the conditional loss distribution given that the aggregate loss equals an exogenously provided capital. This conditional distribution serves as a building block for calculating risk allocations such as the Euler capital allocation of Value-at-Risk. A superlevel set of this conditional distribution can be interpreted as a set of severe and plausible stress scenarios the given capital is supposed to cover. We show that various distributional properties of this conditional distribution, such as modality, dependence and tail behavior, are inherited from those of the underlying joint loss distribution. Among these properties, we find that modality of the conditional distribution is an important feature in risk assessment related to the variety of risky scenarios likely to occur in a stressed situation. Under unimodality, we introduce a novel risk allocation method called maximum likelihood allocation (MLA), defined as the mode of the conditional distribution given the total capital. Under multimodality, a single vector of allocations can be less sound. To overcome this issue, we investigate the so-called multimodalty adjustment to increase the soundness of risk allocations. Properties of the conditional distribution, MLA and multimodality adjustment are demonstrated in numerical experiments. In particular, we observe that negative dependence among losses typically leads to multimodality, and thus a higher multimodality adjustment can be required.