A tale of two allocations: Risk capital contributions versus risk contributions in the tail

Nawaf Mohammed; Edward Furman

Home/Archive/arXiv

arXiv · 2609.36177

A tale of two allocations: Risk capital contributions versus risk contributions in the tail

Nawaf Mohammed·Edward Furman

View original source ↗Read source document ↗Export metadata ↓

Abstract

We compare two natural proportional notions of a risk component's contribution to the aggregate tail risk of a collection of risks: the fraction of aggregate tail risk capital allocated to the component under Conditional Tail Expectation (CTE), and the component's expected realized share of aggregate risk under Geometric Tail Expectation (GTE). The resulting proportional allocations generally differ. For arbitrary random vectors-allowing atoms, signed risks, and any tail domain--we establish when the allocations agree, determine their ordering when they do not, and characterize their asymptotic separation. Both proportional allocations are weighted averages of the conditional risk share of a component given the aggregate: the proportional CTE allocation weights tail scenarios by severity, whereas the proportional GTE allocation weights them uniformly. Their difference is therefore a normalized tail covariance. The proportional CTE and GTE allocations agree throughout a tail exactly when the conditional risk share is constant there; under a mild unimodality condition, dominance is characterized by its monotonicity. Among independent exponential dispersion models with heterogeneous natural parameters, exact agreement is possible only for the scaled Poisson family; under comonotonicity, it is equivalent to proportional quantile functions. In the extreme tail, the limiting relationship between the proportional CTE and GTE allocations is governed by the ratio of Expected Shortfall to Value-at-Risk: boundedness ensures common limits, convergence to one forces the allocations to merge, and divergence can cause their limits to separate.

Explore related subjects

q-fin.RM

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Nawaf Mohammed, Edward Furman. 2026-09-28. A tale of two allocations: Risk capital contributions versus risk contributions in the tail. https://arxiv.org/abs/2609.36177

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A multi-view contrastive learning framework for spatial embeddings in risk modelling

Incorporating spatial information, particularly when related to climate, weather, and demographic factors, is crucial for improving underwriting precision and enhancing risk management in insurance. However, spatial data are often unstructured, high-dimensional, and difficult to integrate into predictive models. Embedding methods are needed to convert spatial data into meaningful representations for modelling tasks. We propose a novel multi-view contrastive learning framework for generating spatial embeddings that combine information from multiple spatial data sources. To train the model, we construct a spatial dataset that merges satellite imagery and OpenStreetMap features across Europe. The framework aligns these spatial views with coordinate-based encodings, producing low-dimensional embeddings that capture both spatial structure and contextual similarity. Once trained, the model generates embeddings directly from latitude-longitude pairs, enabling any dataset with coordinates to be enriched with meaningful spatial features without requiring access to the original spatial inputs. In a case study on French real estate prices, we compare models trained on raw coordinates against those using our spatial embeddings as inputs. The embeddings consistently improve predictive accuracy across generalised linear, additive, and boosting models, while providing post-hoc explainable spatial effects and demonstrating generalisation of the fitted spatial effects to regions without training observations. A second case study on flood claim counts across Belgian postal codes confirms that the embeddings improve territorial risk classification in an insurance context.

q-fin.RM↗

arXiv2026

Robust distortion risk metrics with Wasserstein and Unimodal Constraints

We establish sharp upper and lower bounds for distortion risk metrics under distributional uncertainty. The uncertainty sets are characterized by four key features of the underlying distribution: mean, variance, unimodality, and Wasserstein distance to a reference distribution. We first examine a broad class of distortion functions, assuming only finite variation and imposing neither continuity nor monotonicity. This class includes important examples such as the Gini deviation, the mean-median deviation, and inter-quantile differences. When the uncertainty set is defined by a fixed mean, variance, and Wasserstein distance, we derive the worst- and best-case values of the distortion risk metric and identify the corresponding extremal distributions. We then impose an additional unimodality constraint. In this case, for absolutely continuous distortion functions, we again characterize the worst- and best-case values and explicitly determine the optimal distributions attaining these bounds. Moreover, an algorithm is developed to compute sharp upper bounds and the worst-case quantiles. We apply our results to a robust portfolio optimization problem of interest.

q-fin.RM↗

arXiv2026

Causal Discovery via Simultaneous DAG Recovery Using the Angles Space of Directional Dependence Measures

Most causal discovery algorithms utilizing a Directed Acyclic Graph (DAG) framework execute sequentially (e.g. constraint-based models) or iteratively (e.g. score-based or functional causal models). In contrast, we develop a new method, Angles-based Directional Dependence (ADD), that executes over the entire DAG space simultaneously, based on only two matrix estimations. We apply dual orderings on any (positive definite) directional dependence measure for all pairwise relationships, identify statistically significant directional dependence in the (positive definite) angles space, and then enforce acyclicality to make proper causal interpretations. Potential benefits of the approach include increased coherence, due to simultaneous edge-calling within a positive definite space, increased power, due to the ability to use any directional dependence measure and thus, opportunistically adapt to different or varying data conditions, and increased speed and scalability, due to the need for only two matrix estimations, and two (fast) simulations to define empirical confidence bounds under independence (regardless of the size of the DAG space). We conduct a preliminary empirical study evaluating direct adjacency under nonlinear, asymmetric, heavy-tailed data conditions. The promising results indicate applications to feature selection in quantitative finance, and justify and encourage a more extensive follow-up study to benchmark against competing algorithms to more fully test the above-mentioned potential benefits of ADD.

q-fin.RM↗