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Risks
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Multivariate Risks
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Multivariate Risks

A special issue of Risks (ISSN 2227-9091).

Deadline for manuscript submissions: closed (30 July 2022) | Viewed by 11689

Special Issue Editor

Prof. Dr. Anita Behme

E-Mail Website
Guest Editor
Institute for Mathematical Stochastics, Dresden University of Technology, 01069 Dresden, Germany
Interests: actuarial science; probability theory; risk theory; stochastic processes

Special Issue Information

Dear Colleagues,

Interest in multidimensional risk theory has grown substantially in recent years. The main reason for this is the fact that multidimensional risk models allow for a deeper insight into the interplay between different lines of business or different insurers. However, due to the various possible multidimensional ruin sets and due to the appearing dependencies, e.g., between claim severities or claim arrival times, the mathematical treatment of multidimensional risk models demands a higher level of complexity when compared to univariate risk models.

In this Special Issue, we welcome high-quality research papers addressing the various aspects of multidimensional risk theory.

You are therefore cordially invited to submit your latest results in the area of multivariate risk modeling, such as multivariate risk theory, optimal dividend problems, risk networks, multivariate heavy/light-tailed claims, and others.

Dr. Anita Behme
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Risks is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Non-life insurance
  • Life, health, and pension insurance
  • Risk theory
  • Multidimensional modeling
  • Dependent risks

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Published Papers (5 papers)

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Research

20 pages, 457 KiB  
Article
Sharp Probability Tail Estimates for Portfolio Credit Risk
by Jeffrey F. Collamore, Hasitha de Silva and Anand N. Vidyashankar
Risks 2022, 10(12), 239; https://doi.org/10.3390/risks10120239 - 14 Dec 2022
Viewed by 2130
Abstract
Portfolio credit risk is often concerned with the tail distribution of the total loss, defined to be the sum of default losses incurred from a collection of individual loans made out to the obligors. The default for an individual loan occurs when the [...] Read more.
Portfolio credit risk is often concerned with the tail distribution of the total loss, defined to be the sum of default losses incurred from a collection of individual loans made out to the obligors. The default for an individual loan occurs when the assets of a company (or individual) fall below a certain threshold. These assets are typically modeled according to a factor model, thereby introducing a strong dependence both among the individual loans, and potentially also among the multivariate vector of common factors. In this paper, we derive sharp tail asymptotics under two regimes: (i) a large loss regime, where the total number of defaults increases asymptotically to infinity; and (ii) a small default regime, where the loss threshold for an individual loan is allowed to tend asymptotically to negative infinity. Extending beyond the well-studied Gaussian distributional assumptions, we establish that—while the thresholds in the large loss regime are characterized by idiosyncratic factors specific to the individual loans—the rate of decay is governed by the common factors. Conversely, in the small default regime, we establish that the tail of the loss distribution is governed by systemic factors. We also discuss estimates for Value-at-Risk, and observe that our results may be extended to cases where the number of factors diverges to infinity. Full article
(This article belongs to the Special Issue Multivariate Risks)
15 pages, 446 KiB  
Article
Multi-Variate Risk Measures under Wasserstein Barycenter
by M. Andrea Arias-Serna, Jean Michel Loubes and Francisco J. Caro-Lopera
Risks 2022, 10(9), 180; https://doi.org/10.3390/risks10090180 - 7 Sep 2022
Viewed by 2003
Abstract
When the uni-variate risk measure analysis is generalized into the multi-variate setting, many complex theoretical and applied problems arise, and therefore the mathematical models used for risk quantification usually present model risk. As a result, regulators have started to require that the internal [...] Read more.
When the uni-variate risk measure analysis is generalized into the multi-variate setting, many complex theoretical and applied problems arise, and therefore the mathematical models used for risk quantification usually present model risk. As a result, regulators have started to require that the internal models used by financial institutions are more precise. For this task, we propose a novel multi-variate risk measure, based on the notion of the Wasserstein barycenter. The proposed approach robustly characterizes the company’s exposure, filtering the partial information available from individual sources into an aggregate risk measure, providing an easily computable estimation of the total risk incurred. The new approach allows effective computation of Wasserstein barycenter risk measures in any location–scatter family, including the Gaussian case. In such cases, the Wasserstein barycenter Value-at-Risk belongs to the same family, thus it is characterized just by its mean and deviation. It is important to highlight that the proposed risk measure is expressed in closed analytic forms which facilitate its use in day-to-day risk management. The performance of the new multi-variate risk measures is illustrated in United States market indices of high volatility during the global financial crisis (2008) and during the COVID-19 pandemic situation, showing that the proposed approach provides the best forecasts of risk measures not only for “normal periods”, but also for periods of high volatility. Full article
(This article belongs to the Special Issue Multivariate Risks)
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16 pages, 773 KiB  
Article
Diffusion Approximations of the Ruin Probability for the Insurer–Reinsurer Model Driven by a Renewal Process
by Krzysztof Burnecki, Marek A. Teuerle and Aleksandra Wilkowska
Risks 2022, 10(6), 129; https://doi.org/10.3390/risks10060129 - 17 Jun 2022
Cited by 1 | Viewed by 1951
Abstract
We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume [...] Read more.
We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume that the flow of claims is governed by a general renewal process. A simple ruin probability formula for the model is known only in infinite time for the special case of the Poisson process and exponentially distributed claims. Therefore, there is a need for simple analytical approximations. In the literature, in the infinite-time case, for the Poisson process, a De Vylder-type approximation has already been introduced. The idea of the diffusion approximation presented here is based on the weak convergence of stochastic processes, which enables one to replace the original risk process with a Brownian motion with drift. By applying this idea to the insurer–reinsurer model, we obtain simple ruin probability approximations for both finite and infinite time. We check the usefulness of the approximations by studying several claim amount distributions and comparing the results with the De Vylder-type approximation and Monte Carlo simulations. All the results show that the proposed approximations are promising and often yield small relative errors. Full article
(This article belongs to the Special Issue Multivariate Risks)
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23 pages, 3568 KiB  
Article
Optimal Dividends for a Two-Dimensional Risk Model with Simultaneous Ruin of Both Branches
by Philipp Lukas Strietzel and Henriette Elisabeth Heinrich
Risks 2022, 10(6), 116; https://doi.org/10.3390/risks10060116 - 2 Jun 2022
Cited by 1 | Viewed by 2096
Abstract
We consider the optimal dividend problem in the so-called degenerate bivariate risk model under the assumption that the surplus of one branch may become negative. More specific, we solve the stochastic control problem of maximizing discounted dividends until simultaneous ruin of both branches [...] Read more.
We consider the optimal dividend problem in the so-called degenerate bivariate risk model under the assumption that the surplus of one branch may become negative. More specific, we solve the stochastic control problem of maximizing discounted dividends until simultaneous ruin of both branches of an insurance company by showing that the optimal value function satisfies a certain Hamilton–Jacobi–Bellman (HJB) equation. Further, we prove that the optimal value function is the smallest viscosity solution of said HJB equation, satisfying certain growth conditions. Under some additional assumptions, we show that the optimal strategy lies within a certain subclass of all admissible strategies and reduce the two-dimensional control problem to a one-dimensional one. The results are illustrated by a numerical example and Monte Carlo simulated value functions. Full article
(This article belongs to the Special Issue Multivariate Risks)
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13 pages, 458 KiB  
Article
EM Estimation for the Bivariate Mixed Exponential Regression Model
by Zezhun Chen, Angelos Dassios and George Tzougas
Risks 2022, 10(5), 105; https://doi.org/10.3390/risks10050105 - 17 May 2022
Viewed by 2586
Abstract
In this paper, we present a new family of bivariate mixed exponential regression models for taking into account the positive correlation between the cost of claims from motor third party liability bodily injury and property damage in a versatile manner. Furthermore, we demonstrate [...] Read more.
In this paper, we present a new family of bivariate mixed exponential regression models for taking into account the positive correlation between the cost of claims from motor third party liability bodily injury and property damage in a versatile manner. Furthermore, we demonstrate how maximum likelihood estimation of the model parameters can be achieved via a novel Expectation-Maximization algorithm. The implementation of two members of this family, namely the bivariate Pareto or, Exponential-Inverse Gamma, and bivariate Exponential-Inverse Gaussian regression models is illustrated by a real data application which involves fitting motor insurance data from a European motor insurance company. Full article
(This article belongs to the Special Issue Multivariate Risks)
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