  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.
