We analyze dependence, tail behavior and multimodality of the conditional distribution of a loss random vector given that the aggregate loss equals an exogenously provided capital. This conditional distribution is a building block for calculating risk allocations such as the Euler capital allocation of Value-at-Risk. A level 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  are inherited from those of the underlying joint loss distribution. Among these properties, we find that multimodality of the conditional distribution is an important feature  related to the number of risky scenarios likely to occur in a stressed situation. Moreover, Euler allocation becomes less sound under multimodality than under unimodality. To overcome this issue, we propose a novel risk allocation called the maximum likelihood allocation (MLA), defined as the mode of the conditional distribution given the total capital. The process of estimating MLA turns out to be beneficial for detecting multimodality, evaluating the soundness of risk allocations , and constructing more flexible risk allocations based on multiple risky scenarios . Properties of the conditional distribution and MLA are demonstrated in numerical experiments. In particular, we observe that negative dependence among losses typically leads to multimodality, and thus to multiple risky scenarios and less sound risk allocations .
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 .
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