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<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Utility of classical insurance risk models for measuring the risks of cyber incidents | Japanese Journal of Statistics and Data Science </title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="Utility of classical insurance risk models for measuring the risks of cyber incidents"/> <meta name="twitter:description" content="Japanese Journal of Statistics and Data Science - We demonstrate that the classical insurance risk models yield significant advantages in the context of cyber risk analysis. This model exhibits..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig1_HTML.png"/> <meta name="journal_id" content="42081"/> <meta name="dc.title" content="Utility of classical insurance risk models for measuring the risks of cyber incidents"/> <meta name="dc.source" content="Japanese Journal of Statistics and Data Science 2024 7:2"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2024-09-24"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2024 The Author(s)"/> <meta name="dc.rights" content="2024 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="We demonstrate that the classical insurance risk models yield significant advantages in the context of cyber risk analysis. This model exhibits commendable attributes in terms of both computational efficiency and predictive capabilities. Utilizing several compound point risk models, we derive the conditional Value-at-Risk and Tail Value-at-Risk predictions for the cumulative breach size within specified time intervals. To verify the reliability of our method, we conduct backtesting exercises, comparing our predictions with actual breach sizes."/> <meta name="prism.issn" content="2520-8764"/> <meta name="prism.publicationName" content="Japanese Journal of Statistics and Data Science"/> <meta name="prism.publicationDate" content="2024-09-24"/> <meta name="prism.volume" content="7"/> <meta name="prism.number" content="2"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1059"/> <meta name="prism.endingPage" content="1084"/> <meta name="prism.copyright" content="2024 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s42081-024-00273-y"/> <meta name="prism.doi" content="doi:10.1007/s42081-024-00273-y"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s42081-024-00273-y.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s42081-024-00273-y"/> <meta name="citation_journal_title" content="Japanese Journal of Statistics and Data Science"/> <meta name="citation_journal_abbrev" content="Jpn J Stat Data Sci"/> <meta name="citation_publisher" content="Springer Nature Singapore"/> <meta name="citation_issn" content="2520-8764"/> <meta name="citation_title" content="Utility of classical insurance risk models for measuring the risks of cyber incidents"/> <meta name="citation_volume" content="7"/> <meta name="citation_issue" content="2"/> <meta name="citation_publication_date" content="2024/11"/> <meta name="citation_online_date" content="2024/09/24"/> <meta name="citation_firstpage" content="1059"/> <meta name="citation_lastpage" content="1084"/> <meta name="citation_article_type" content="Original Paper"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1007/s42081-024-00273-y"/> <meta name="DOI" content="10.1007/s42081-024-00273-y"/> <meta name="size" content="651093"/> <meta name="citation_doi" content="10.1007/s42081-024-00273-y"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1007/s42081-024-00273-y&api_key="/> <meta name="description" content="We demonstrate that the classical insurance risk models yield significant advantages in the context of cyber risk analysis. This model exhibits commendable"/> <meta name="dc.creator" content="Shimizu, Yasutaka"/> <meta name="dc.creator" content="Takagami, Yutaro"/> <meta name="dc.subject" content="Statistical Theory and Methods"/> <meta name="dc.subject" content="Statistics and Computing/Statistics Programs"/> <meta name="dc.subject" content="Statistics for Business, Management, Economics, Finance, Insurance"/> <meta name="dc.subject" content="Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences"/> <meta name="dc.subject" content="Statistics for Life Sciences, Medicine, Health Sciences"/> <meta name="dc.subject" content="Statistics for Social Sciences, Humanities, Law"/> <meta name="citation_reference" content="citation_journal_title=European Actuarial Journal; citation_title=Modeling and pricing cyber insurance: Idiosyncratic, systematic, and systemic risks; citation_author=K Awiszus, T Knispel, I Penner, G Svindland, A Voß, S Weber; citation_volume=13; citation_issue=1; citation_publication_date=2023; citation_pages=1-53; citation_doi=10.1007/s13385-023-00341-9; citation_id=CR1"/> <meta name="citation_reference" content="Bank for International Settlements. 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This model exhibits commendable attributes in terms of both computational efficiency and predictive capabilities. Utilizing several compound point risk models, we derive the conditional Value-at-Risk and Tail Value-at-Risk predictions for the cumulative breach size within specified time intervals. To verify the reliability of our method, we conduct backtesting exercises, comparing our predictions with actual breach sizes."/> <meta property="og:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig1_HTML.png"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link 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aria-hidden="true"> <div class="c-context-bar__container u-container"> <div class="c-context-bar__title"> Utility of classical insurance risk models for measuring the risks of cyber incidents </div> <div data-test="inCoD" data-track-context="sticky banner"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s42081-024-00273-y.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> </div> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Yasutaka-Shimizu-Aff1" data-author-popup="auth-Yasutaka-Shimizu-Aff1" data-author-search="Shimizu, Yasutaka" data-corresp-id="c1">Yasutaka Shimizu<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0003-3479-1149"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0003-3479-1149</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Yutaro-Takagami-Aff2" data-author-popup="auth-Yutaro-Takagami-Aff2" data-author-search="Takagami, Yutaro">Yutaro Takagami</a><sup class="u-js-hide"><a href="#Aff2">2</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>461 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s42081-024-00273-y/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>We demonstrate that the classical insurance risk models yield significant advantages in the context of cyber risk analysis. This model exhibits commendable attributes in terms of both computational efficiency and predictive capabilities. Utilizing several compound point risk models, we derive the conditional Value-at-Risk and Tail Value-at-Risk predictions for the cumulative breach size within specified time intervals. To verify the reliability of our method, we conduct backtesting exercises, comparing our predictions with actual breach sizes.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w92h120/springer-static/cover-hires/book/978-3-031-69561-2?as=webp" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/978-3-031-69561-2_7?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/978-3-031-69561-2_7">Cyber Risk and Cyber Insurance </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Chapter</span> <span class="c-article-meta-recommendations__date">© 2025</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w92h120/springer-static/cover-hires/book/978-3-030-99638-3?as=webp" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/978-3-030-99638-3_23?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/978-3-030-99638-3_23">Cyber Risk: Estimates for Malicious and Negligent Breaches Distributions </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Chapter</span> <span class="c-article-meta-recommendations__date">© 2022</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w92h120/springer-static/cover-hires/book/978-3-031-64273-9?as=webp" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/978-3-031-64273-9_43?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/978-3-031-64273-9_43">Challenges in Cyber Risk Insurance </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Chapter</span> <span class="c-article-meta-recommendations__date">© 2024</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1743399493, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=42081" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>In recent decades, the escalation of cyber incidents has paralleled the rapid advancement of Information Technology. Consequently, certain non-life insurance companies have introduced cyber-risk insurance, underscoring the importance of accurately assessing the risks associated with cyber incidents. Cyber risk analysis is a globally prominent and dynamically evolving field, with numerous researchers contributing to its discourse.</p><p>Some recent and notable studies include the works of Farkas et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Farkas, S., Lopez, O., & Thomas, M. (2021). Cyber claim analysis using generalized Pareto regression trees with applications to insurance. Insurance: Mathematics and Economics, 98, 92–105." href="/article/10.1007/s42081-024-00273-y#ref-CR10" id="ref-link-section-d432018368e383">2021</a>), Woods and Böhme (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Woods, D. W., & Böhme, R. (2021). SoK: Quantifying cyber risk. In 2021 IEEE symposium on security and privacy (pp. 211–228)." href="/article/10.1007/s42081-024-00273-y#ref-CR22" id="ref-link-section-d432018368e386">2021</a>), Eling et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2022" title="Eling, M., Elvedi, M., & Falco, G. (2022). The economic impact of extreme cyber risk scenarios. North American Actuarial Journal, 27, 1–15." href="/article/10.1007/s42081-024-00273-y#ref-CR8" id="ref-link-section-d432018368e389">2022</a>), Peters et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Peters, G. W., Malavasi, M., Sofronov, G., Shevchenko, P. V., Trück, S., & Jang, J. (2023). Cyber loss model risk translates to premium mispricing and risk sensitivity. The Geneva Papers on Risk and Insurance-Issues and Practice, 48(2), 372–433." href="/article/10.1007/s42081-024-00273-y#ref-CR16" id="ref-link-section-d432018368e392">2023</a>), and many more as referenced in these papers provide valuable insights and references for further exploration of the topic. Research focusing on the mathematical and technical aspects of cyber risk quantification and predictive distribution has been conducted relatively long. For example, Maillart and Sornette (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2010" title="Maillart, T., & Sornette, D. (2010). Heavy-tailed distribution of cyber risks. The European Physical Journal B, 75, 357–364." href="/article/10.1007/s42081-024-00273-y#ref-CR13" id="ref-link-section-d432018368e395">2010</a>) claim that the breach size distribution of cyber incidents seems heavy-tailed. Peng et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2016" title="Peng, C., Xu, M., Xu, S., & Hu, T. (2016). Modeling and predicting extreme cyber attack rates via marked point processes. Journal of Applied Statistics, 44(14), 2534–2563." href="/article/10.1007/s42081-024-00273-y#ref-CR15" id="ref-link-section-d432018368e399">2016</a>) discussed predicting cyber attack rates using marked point processes. Xu et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2018" title="Xu, M., Schweitzer, K. M., Bateman, R. B., & Xu, S. (2018). Modeling and predicting cyber hacking breaches. IEEE Transactions on Information Forensics and Security, 13, 2856–2871." href="/article/10.1007/s42081-024-00273-y#ref-CR23" id="ref-link-section-d432018368e402">2018</a>) employed ACD (Autoregressive Conditional Duration) and ARMA-GARCH models to characterize the frequency and magnitude of cyber incidents, respectively. Sun et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2021" title="Sun, H., Xu, M., & Zhao, P. (2021). Modeling malicious hacking data breach risks. North American Actuarial Journal, 25(4), 484–502." href="/article/10.1007/s42081-024-00273-y#ref-CR21" id="ref-link-section-d432018368e405">2021</a>) categorized cyber incident data into business sectors and leveraged copulas to forecast cyber risks at the organizational level. These papers delve into complex modeling, statistical methods, and risk assessment techniques to better understand cyber risk. In recent years, increasing tools such as machine learning and AI has attracted further attention to computational demands. For example, Zhan et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2015" title="Zhan, Z., Xu, M., & Xu, S. (2015). Predicting cyber attack rates with extreme values. IEEE Transactions on Information Forensics and Security, 10(8), 1666–1677." href="/article/10.1007/s42081-024-00273-y#ref-CR24" id="ref-link-section-d432018368e408">2015</a>) harnessed machine learning techniques for forecasting incident frequency, integrating extremal theory and time-series analysis to enhance predictive accuracy.</p><p>On the other hand, these approaches may incur substantial computational costs, and the opacity of these AI models can introduce challenges in their interpretability. One of these challenges is the “black box” nature of many machine learning algorithms, making it difficult to interpret and understand the inner workings of these models. Furthermore, in the field of cyber risk analysis, there is a growing interest in combining machine learning methods with statistical (theoretical) methods. These hybrid methods are often referred to as “gray methods.” The need to strike a balance between methods’ transparency and effectiveness is an ongoing concern in this area of research.</p><p>In this paper, we dare to shed light on the classical model again to recognize the usefulness of a simple model. The model we employed adheres to classical actuarial practices. It offers simplicity, ease of comprehension, and computational efficiency, which are advantageous in practical applications. It is also powerful enough to predict risk quantification in the future. We shall demonstrate the efficacy of classically employed risk models, particularly those involving composite point processes, in achieving substantial risk reduction without incurring substantial computational expenses. Even when Monte Carlo simulations are necessary, the model’s straightforward nature and explicitly computable structure make it a valuable tool for efficiently assessing and managing cyber risks. This aligns with the actuarial principle of using well-understood and computationally manageable risk analysis and management models.</p><p>In our cyber risk analysis, we will adopt a quite simple compound risk model, a classic paradigm in insurance risk assessment: Let <i>N</i> be a random variable representing the frequency of cyber incidents occurring within a defined period, and <span class="mathjax-tex">$U_{i}$</span> be the breach size of the <i>i</i>th incident involving information leakage, with the common distribution <span class="mathjax-tex">$F_{U}$</span>. Then the total amount of breaches, say <i>S</i>, is given by</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} S=\sum _{i=1}^{N}U_{i};\qquad U_{i}\overset{i.i.d.}{\sim }F_{U}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (1.1) </div></div><p>While it may be considered a straightforward model, it boasts a reasonable degree of expressiveness and is supported by various distribution approximations, making it possible for easy statistical inference. As in Awiszus et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Awiszus, K., Knispel, T., Penner, I., Svindland, G., Voß, A., & Weber, S. (2023). Modeling and pricing cyber insurance: Idiosyncratic, systematic, and systemic risks. European Actuarial Journal, 13(1), 1–53." href="/article/10.1007/s42081-024-00273-y#ref-CR1" id="ref-link-section-d432018368e576">2023</a>) and Dacorogna and Kratz (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Dacorogna, M., & Kratz, M. (2023). Managing cyber risk, a science in the making. Scandinavian Actuarial Journal, 2023(10), 1000–1021." href="/article/10.1007/s42081-024-00273-y#ref-CR6" id="ref-link-section-d432018368e579">2023</a>), the classification of cyber risks is complex, and such a classical <i>frequency-severity approach</i> may have limitations. However, we usually employ this model within a single period but adapt it to encompass multi-period risks, which is the novelty of our paper, and still propose statistical inference for point processes; see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec3">3</a>. We revisit this classic risk model, highlighting its potential and demonstrating that it can effectively predict cyber risks with ingenuity even within its classical framework.</p><p>Nonetheless, constructing a more detailed model needs an elaborate examination of actual data. Thus, before introducing a specific model, we shall look at the dataset employed in this paper. We use an open dataset given by Privacy Rights Clearinghouse (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Privacy Rights Clearinghouse. (2023). Retrieved from 
 https://www.privacyrights.org/data-breaches
 
 " href="/article/10.1007/s42081-024-00273-y#ref-CR18" id="ref-link-section-d432018368e591">2023</a>). It includes information about cyber incidents in the United States, with a public date, breach size, type of breach, and business field, among others. To get the model insight, let us review the data. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig1">1</a> presents a distribution of breach sizes for each incident recorded between 2005 and 2020, exhibiting a pronounced long (heavy) right tail. In this paper, as in Maillart and Sornette (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2010" title="Maillart, T., & Sornette, D. (2010). Heavy-tailed distribution of cyber risks. The European Physical Journal B, 75, 357–364." href="/article/10.1007/s42081-024-00273-y#ref-CR13" id="ref-link-section-d432018368e597">2010</a>), we assume that the tail function <span class="mathjax-tex">$\overline{F}_U(x):=1-F_U(x)$</span> conforms to the concept of being ‘regularly varying’ with the index <span class="mathjax-tex">$-\kappa $</span>, where <span class="mathjax-tex">$\kappa \ge 0$</span> is a key parameter:</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \lim _{x\rightarrow \infty }\dfrac{\overline{F}_U(tx)}{\overline{F}_U(x)}=t^{-\kappa }\quad \hbox { for all}\ t>0, \end{aligned}$$</span></div></div><p>which is denoted as</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \overline{F}_U \in {\mathscr {R}}_{-\kappa }. \end{aligned}$$</span></div></div><p>Furthermore, we should note that this breach data encompasses numerous instances of zero values (i.e., 0-inflated), signifying the occurrence of cyberattacks without resulting in any information leakage. Since our primary interest lies in evaluating the actual damage risk stemming from cyberattacks, our focus is on assessing tail risk, involving the calculation of ’Value-at-Risk’ (VaR) and ’Tail Value-at-Risk’ (TVaR), both of which are commonly established tail risk metrics. Given that these metrics are reliant on the tail properties of the distribution, we employ the ’Peaks-Over-Threshold’ method from extreme value theory and model the tail distribution using the ’generalized Pareto distribution (GPD),’ as outlined in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar6">A.1</a>. This approach serves as the standard procedure for dealing with data characterized by heavy tails; see, e.g., Embrechts et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2003" title="Embrechts, P., Klüppelberg, C., & Mikosch, T. (2003). Modeling extremal events for insurance and finance. Springer." href="/article/10.1007/s42081-024-00273-y#ref-CR9" id="ref-link-section-d432018368e915">2003</a>) or Resnick (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2008" title="Resnick, S. I. (2008). Extreme values, regular variation and point processes. Springer." href="/article/10.1007/s42081-024-00273-y#ref-CR19" id="ref-link-section-d432018368e918">2008</a>), among others.</p><p>Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig2">2</a> illustrates the time series of frequencies spanning the years 2005–2020, suggesting that the frequency of cyber incidents should be addressed through the modeling of a stochastic (point) process, as opposed to being represented by a single random variable <i>N</i> as previously discussed. We assert that modeling this dynamic time series of frequencies is pivotal to our analysis. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec3">3</a>, we will introduce and elaborate upon several stochastic processes to address this modeling challenge.</p><p>Based on the observations presented in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig3">3</a>, it is evident that breach sizes have exhibited a significant increase since 2016. While the precise cause of this trend remains uncertain, it is likely influenced by a legislative amendment in 2015, which mandated the reporting of cyber attack data in the United States. Consequently, some of the leakage data recorded before 2016 may have been consolidated and reported in 2016, thus affecting the accuracy and continuity of the breach size time-series data. Furthermore, it’s important to note that the data available for 2005 and beyond 2019 is notably limited, as numerous breaches during these periods have yet to be formally recorded. As a result, we have chosen to focus our data analysis on the period from 2006 to 2018.</p><p>Considering these factors, we have divided the data into two distinct cases: Case 1, our primary dataset, excludes data recorded after 2016 due to the observed shift in frequency tendencies, as previously discussed. Case 2 encompasses all available data and serves as a reference, as summarized in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab1">1</a>.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-1"><figure><figcaption class="c-article-table__figcaption"><b id="Tab1" data-test="table-caption">Table 1 Usage of data</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/1" aria-label="Full size table 1"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Considering this dataset’s distinctive attributes, we introduce specific models in the subsequent section. The outcomes of our data analysis employing these dedicated models are presented in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec8">4</a>, culminating with the paper’s conclusions in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec13">5</a>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/1" rel="nofollow"><picture><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="417"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Breach size of single incident</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/2" rel="nofollow"><picture><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="570"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Frequency (monthly)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/3" rel="nofollow"><picture><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="622"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Breach size (monthly)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="Multi-period compound risk models"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Multi-period compound risk models</h2><div class="c-article-section__content" id="Sec2-content"><p>As described in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec1">1</a>, we expand upon the single-period model (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ1">1.1</a>) to formulate a multi-period framework. In this multi-period analysis, we partition the observation period into distinct segments, presupposing that the cumulative breach count within each period has a potentially different compound risk model. On the other hand, when examining the breach size distribution, as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig1">1</a>, we presume a common heavy-tailed distribution spanning all the observation periods:</p><ul class="u-list-style-bullet"> <li> <p>Let <span class="mathjax-tex">$N_{k}$</span> be a random variable describing the frequency of breaches in <i>k</i>th period, satisfying that there exists some <span class="mathjax-tex">${\epsilon }>0$</span> such that </p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{r=0}^{\infty }(1+{\epsilon })^r {{\mathbb {P}}}(N_k=r)<\infty , \end{aligned}$$</span></div></div><p> which corresponds to the condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ11">A.1</a>) in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar12">A.4</a>.</p> </li> <li> <p>Let <span class="mathjax-tex">$U^{(k)}:=\{U_{1}^{(k)}, U_{2}^{(k)},..., U_{N_{k}}^{(k)}\}$</span> be the sets of breach sizes of each incident occurring in <i>k</i>th period with <span class="mathjax-tex">$U_{i}^{(k)}\overset{i.i.d}{\sim }F_{U}$</span>, and assume that there exists a constant <span class="mathjax-tex">$\kappa >1$</span> such that </p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \overline{F}_U \in {\mathscr {R}}_{-\kappa }. \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.1) </div></div> </li> <li> <p>For <span class="mathjax-tex">$t\in {\mathbb {N}}$</span>, let <span class="mathjax-tex">${\mathscr {F}}_{0}$</span> and <span class="mathjax-tex">${\mathscr {F}}_{t}$</span> be a <span class="mathjax-tex">$\sigma $</span>-field such that <span class="mathjax-tex">${\mathscr {F}}_{0}:=\{\emptyset ,\Omega \}$</span> and, by induction, </p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathscr {F}}_{t}:={\mathscr {F}}_{t-1}\vee \sigma (N_{t};U^{(t)}),\quad t=1,2,\dots . \end{aligned}$$</span></div></div> </li> </ul><p>Then, the total amount of breaches in the <i>k</i>th period is given by</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} S_k = \sum _{i=1}^{N_k} U_i^{(k)}, \quad k = 1,2,\dots , \end{aligned}$$</span></div></div><p>and we are interested in the following <i>conditional (Tail-) Value-at-Risk</i>:</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\left\{ \begin{array}{ll} & VaR_{\alpha }^{(t)}(S_{k}):=\inf \{x\ge 0\ |\ {{\mathbb {P}}}(S_{k}\le x|{\mathscr {F}}_{t})\ge \alpha \}\\ & TVaR_{\alpha }^{(t)}(S_{k}):=\dfrac{1}{1-\alpha }\displaystyle \int _{\alpha }^{1}VaR_{u}^{(t)}(S_{k})\,du \end{array}\right. },\quad k>t. \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.2) </div></div><p>To approximate these risk measures in heavy-tailed situations, the following result by, e.g., Biagini and Ulmer (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2009" title="Biagini, F., & Ulmer, S. (2009). Asymptotics for operational risk quantified with expected shortfall. ASTIN Bulletin, 39, 735–752." href="/article/10.1007/s42081-024-00273-y#ref-CR4" id="ref-link-section-d432018368e2230">2009</a>), Theorem 2.5 is useful. More detailed asymptotic estimates are given in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar12">A.4</a> in Appendix. See also Böcker and Klüppelberg (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2005" title="Böcker, K., & Klüppelberg, C. (2005) Operational VaR: A closed-form solution. RISK Magazine, December, pp. 90–93." href="/article/10.1007/s42081-024-00273-y#ref-CR5" id="ref-link-section-d432018368e2236">2005</a>), Peters et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2013" title="Peters, G. W., Targino, R. S., & Shevchenko, P. V. (2013). Understanding operational risk capital approximations: First and second orders. Journal of Governance and Regulation, 2, 58–78." href="/article/10.1007/s42081-024-00273-y#ref-CR17" id="ref-link-section-d432018368e2240">2013</a>) and references therein.</p> <h3 class="c-article__sub-heading" id="FPar1">Theorem 2.1</h3> <p>(Biagini and Ulmer <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2009" title="Biagini, F., & Ulmer, S. (2009). Asymptotics for operational risk quantified with expected shortfall. ASTIN Bulletin, 39, 735–752." href="/article/10.1007/s42081-024-00273-y#ref-CR4" id="ref-link-section-d432018368e2250">2009</a>) Suppose that the index in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ2">2.1</a>) satisfies <span class="mathjax-tex">$\kappa >1$</span>. Then, as <span class="mathjax-tex">$\alpha \rightarrow 1$</span>, it holds that</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} VaR^{(t)}_{\alpha }(S_k)&\sim VaR_{\beta ^{(t)}_k}(U); \\ TVaR^{(t)}_{\alpha }(S_k)&\sim \frac{\kappa }{\kappa -1}VaR_{\beta ^{(t)}_k}(U) \end{aligned}$$</span></div></div><p>where</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \beta ^{(t)}_k:=1-\frac{(1-\alpha )}{{\mathbb {E}}[N_k|{\mathscr {F}}_t]}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.3) </div></div> <p>Let us explore an further approximation for <span class="mathjax-tex">$VaR_{\beta ^{(t)}_k}(U)$</span>, which facilitates the explicit computation of <span class="mathjax-tex">$VaR^{(t)}_{\alpha }(S_k)$</span> and <span class="mathjax-tex">$TVaR^{(t)}_{\alpha }(S_k)$</span>. Given that <span class="mathjax-tex">$\beta ^{(t)}_k \rightarrow 1\ a.s.$</span> as <span class="mathjax-tex">$\alpha \rightarrow 1$</span> for any values of <i>k</i> and <i>t</i>, we can approximate<span class="mathjax-tex">$VaR_{\beta ^{(t)}_k}(U)$</span> as <span class="mathjax-tex">$\beta ^{(t)}_k\rightarrow 1$</span> through the conventional arguments inherent to extreme value theory, as outlined below: it follows for any <span class="mathjax-tex">$u\in {\mathbb {R}}$</span> that</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F_{U}(x)&= \overline{F}_{U}(u)F_{U}(x-u|u)+F_{U}(u), \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$F_{U}(x-u|u):= [F_{U}(x)-F_{U}(u)]/\overline{F}_{U}(u)$</span>. When <span class="mathjax-tex">$x = VaR_{\beta _k^{(t)}}(U)$</span> and <span class="mathjax-tex">$u>0$</span> is “large enough”, Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar6">A.1</a> gives the following approximation of <span class="mathjax-tex">$VaR_{\beta _k^{(t)}}(U)$</span> by replacing <span class="mathjax-tex">$F_{U}(x-u|u)$</span> with a generalized Pareto distribution (GPD) <span class="mathjax-tex">$G_{\xi ,\sigma }(x-u)$</span>:</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} VaR_{\beta _k^{(t)}}(U)&= F_U^{-1}(\beta ^{(t)}_k) \nonumber \\&\sim u+G_{\xi ,\sigma }^{-1}\left(1-\frac{1-\beta _k^{(t)}}{\overline{F}_{U}(u)}\right) =u+\frac{\sigma }{\xi }\left\{ \left( \frac{\overline{F}_{U}(u)}{1-\beta _k^{(t)}}\right) ^{\xi }-1\right\} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.4) </div></div><p>See Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar12">A.4</a> for details of the validation of this formula. The value for <i>u</i> is determined using the Peaks-Over-Threshold method, a standard approach within this context. The estimation of the parameters <span class="mathjax-tex">$\xi $</span> and <span class="mathjax-tex">$\sigma $</span> from the data is elaborated upon in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec8">4</a>.</p> <h3 class="c-article__sub-heading" id="FPar2">Remark 2.2</h3> <p>Thus, the fact that the conditional (Tail) VaR can be written explicitly is a major advantage in numerical calculations. Since the risk measures we need to compute are <span class="mathjax-tex">${\mathscr {F}}_t$</span>-conditional random quantities, their prediction requires Monte Carlo calculations based on their distribution. With complex models, even a single computation of a risk measure requires a Monte Carlo calculation, which must be repeated many times to examine its distribution. Our method eliminates the initial Monte Carlo calculation; see also Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar3">3.1</a>.</p> </div></div></section><section data-title="Specific models for frequencies"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3"><span class="c-article-section__title-number">3 </span>Specific models for frequencies</h2><div class="c-article-section__content" id="Sec3-content"><h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">3.1 </span>Negative binomial model</h3><p>We assume that the distribution of <span class="mathjax-tex">$N_k\ (k=1,2,\dots )$</span> will change according to the period. However, assuming a single distribution for each time period typically provides only a limited amount of data for estimating that distribution. To address this limitation, we further divide each period into several sub-periods, such that for a given integer <span class="mathjax-tex">$m\in {\mathbb {N}}$</span>, we express the frequency <span class="mathjax-tex">$N_k$</span> as a sum of individual sub-period frequencies:</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} N_k := N_{k,1} + N_{k,2} + \dots + N_{k,m}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.1) </div></div><p>where <i>m</i> is the number of the sub-period and <span class="mathjax-tex">$N_{k,j}\ (j=1,\dots ,m)$</span> is the number of breaches in the sub-period <span class="mathjax-tex">$k_j$</span>. We make the following assumptions to guide our analysis: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">[NB1]</span> <p><span class="mathjax-tex">$N_{k,j}(j=1,...,m)$</span> are i.i.d. random variables, each of which follows a geometric distribution <span class="mathjax-tex">$N_{k,j}\overset{i.i.d}{\sim }Ge(p_k)$</span>, where the parameter <span class="mathjax-tex">$p_k$</span> is constant during the <i>k</i>th period: for <span class="mathjax-tex">$p_k\in (0,1)$</span>, </p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathbb {P}}}(N_{k,j} = r) = (1-p_k)p_k^r,\quad r=1,2,.... \end{aligned}$$</span></div></div> </li> </ol><p>Then the distribution of <span class="mathjax-tex">$N_k$</span>, which is the i.i.d. sum of geometric variables becomes the negative binomial distribution <span class="mathjax-tex">$N_{k}\sim NBin(m, p_k)$</span>:</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathbb {P}}}(N_k = r) = {m + r - 1 \atopwithdelims ()r}(1 - p_k)^m p_k^r,\quad r=1,2,\dots . \end{aligned}$$</span></div></div><p>Note that</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathbb {E}}[N_k] = \frac{mp_k}{1-p_k}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.2) </div></div><p>Moreover, under this assumption, the condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ11">A.1</a>) in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar12">A.4</a> is obvious since <span class="mathjax-tex">$p_k<1$</span>.</p><p>We further assume a time series model for <span class="mathjax-tex">$\{p_k\}_{k=1,2,\dots }$</span>: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">[NB2]</span> <p>The value of the parameter <span class="mathjax-tex">$p_{k}$</span> changes stochastically according to <i>k</i>, and the logit transformation of <span class="mathjax-tex">$p_{k}$</span>, say <span class="mathjax-tex">$\text{ logit }p_k:=\log p_k/(1-p_k)$</span> follows an ARIMA(<i>p</i>, <i>d</i>, <i>q</i>) process: </p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text{ logit }\,p_{k}-\text{ logit }\,p_{k-d}=c+{\epsilon }_{k}+\sum _{i=1}^{p}\,\phi _{i}\text{ logit }\,p_{k-i}+\sum _{i=1}^{q}\theta _{i}{\epsilon }_{k-i}, \end{aligned}$$</span></div></div><p> where <span class="mathjax-tex">${\epsilon }_{k}\overset{i.i.d}{\sim }{\mathscr {N}}(0,\sigma ^{2})$</span>, <span class="mathjax-tex">$c\in {\mathbb {R}}$</span>, and <span class="mathjax-tex">$\sigma >0$</span>. <span class="mathjax-tex">$\theta _{i}$</span> and <span class="mathjax-tex">$\phi _{i}$</span> are the regression coefficients.</p> </li> </ol><p>To compute the approximated (T)VaR, we require the prediction of the conditional expectation</p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathbb {E}}[N_k|{\mathscr {F}}_t],\quad k > t \end{aligned}$$</span></div></div><p>based on observations <span class="mathjax-tex">$(N_{k,1},N_{k,2},\dots , N_{k,m})_{k=1,2,\dots ,t}$</span>. Given that <i>m</i> observations <span class="mathjax-tex">$N_{k,j}\,(j=1,\dots ,m)$</span> are independently and identically distributed samples from a geometric distribution with parameter <span class="mathjax-tex">$p_k$</span>, the log-likelihood is expressed as</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} L_n(p_k):= \sum _{j=1}^m \log (1-p_k)p_k^{N_{k,j}}, \end{aligned}$$</span></div></div><p>and the MLE of <span class="mathjax-tex">$p_k$</span> up to time <i>t</i> is computed as</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widehat{p}_k:= 1 - \frac{1}{1 + N_k/m},\quad k=1,\dots ,t. \end{aligned}$$</span></div></div><p>If <span class="mathjax-tex">$\widehat{p}_k\,(k=1,\dots t)$</span> provides a reliable estimate of <span class="mathjax-tex">$p_k$</span>, we can assume that these estimated values approximately satisfy the ARIMA process described in [NB2]. Consequently, we can proceed to estimate the parameters <span class="mathjax-tex">$(p,d,q; c,\phi _i,\vartheta _i,\sigma )$</span> based on the sequence <span class="mathjax-tex">$\{\widehat{p}_k\}{k=1,\dots t}$</span>, resulting in <span class="mathjax-tex">$(\widehat{p},\widehat{d},\widehat{q}; \widehat{c},\widehat{\phi }_i,\widehat{\vartheta }_i,\widehat{\sigma })$</span>.</p><p>Subsequently, the logit of <span class="mathjax-tex">$p_k^{(t)}:=p_k|_{{\mathscr {F}}_t}$</span>, representing the ’future’ parameter conditional on <span class="mathjax-tex">${\mathscr {F}}_t$</span>, can be predicted through the estimated ARIMA<span class="mathjax-tex">$(\widehat{p},\widehat{d},\widehat{q})$</span> model, as follows:</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text{ logit }\,p_k^{(t)}= \widehat{c} + \text{ logit }\,p^{(t)}_{k-\widehat{d}} + {\epsilon }_{k}+\sum _{i=1}^{\widehat{p}}\widehat{\phi }_{i}\text{ logit }\,p_{k-i}^{(t)}+\sum _{i=1}^{\widehat{q}}\widehat{\theta }_{i}{\epsilon }_{k-i},\quad {\epsilon }_k \sim {\mathscr {N}}(0,\widehat{\sigma }^{2}). \end{aligned}$$</span></div></div><p>Generating a sample of <span class="mathjax-tex">$p^{(t)}_k$</span> as well as the expression (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ7">3.2</a>), we have the predictor</p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathbb {E}}[N_k|{\mathscr {F}}_t]\approx \frac{m p^{(t)}_k}{1-p^{(t)}_k},\quad k>t, \end{aligned}$$</span></div></div><p>and the predictor of <span class="mathjax-tex">$\beta ^{(t)}_k$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ4">2.3</a>) is given by</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widehat{\beta }^{(t)}_k:=1-\frac{(1-\alpha )(1-p^{(t)}_k)}{mp_k^{(t)}}. \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar3">Remark 3.1</h3> <p>Since <span class="mathjax-tex">$VaR_{\widehat{\beta }^{(t)}_k }(U)$</span> is a random variable via <span class="mathjax-tex">$p_k^{(t)}$</span> that follows ARIMA model, we must estimate its distribution to predict <span class="mathjax-tex">$VaR_{\widehat{\beta }^{(t)}_k }(U)$</span>. This involves generating random samples of <span class="mathjax-tex">$VaR_{\widehat{\beta }^{(t)}_k }(U)$</span>. By repeating the aforementioned procedure, for example, <i>B</i> times, and obtaining predictor values <span class="mathjax-tex">$\widehat{\beta }^{(t)}_{k,1},\widehat{\beta }^{(t)}_{k,2}, \dots ,\widehat{\beta }^{(t)}_{k,B}$</span>, we accumulate a set of <i>B</i> samples of <span class="mathjax-tex">$VaR_\beta (U)$</span>:</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widehat{{\varvec{V}}}:=\left\rbrace VaR_{\widehat{\beta }^{(t)}_{k,1} }(U), VaR_{\widehat{\beta }^{(t)}_{k,2} }(U), \dots , VaR_{\widehat{\beta }^{(t)}_{k,B} }(U) \right\lbrace . \end{aligned}$$</span></div></div><p>Consequently, a predictor for <span class="mathjax-tex">$VaR_{\widehat{\beta }^{(t)}_k }(U)$</span> can be approximated as</p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} VaR_{\widehat{\beta }^{(t)}_k }(U) \approx \textrm{mean}(\widehat{{\varvec{V}}}) = \frac{1}{B}\sum _{j=1}^B VaR_{\widehat{\beta }^{(t)}_{k,j}}(U), \end{aligned}$$</span></div></div><p>and each <span class="mathjax-tex">$VaR_{\widehat{\beta }^{(t)}_{k,j}}(U)$</span> is calculated according to the formula in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ5">2.4</a>); see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec8">4</a> for the practical procedure. Moreover, the <span class="mathjax-tex">$\alpha $</span>-confidence interval is given by</p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} [\widehat{{\mathbb {V}}}_{(1-\alpha )/2}, \widehat{{\mathbb {V}}}^{(1-\alpha )/2}],\quad \alpha \in (0,1), \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">${\mathbb {V}}_{(1-\alpha )/2}$</span> and <span class="mathjax-tex">${\mathbb {V}}^{(1-\alpha )/2}$</span> are the lower and upper <span class="mathjax-tex">$(1-\alpha )/2$</span>-empirical quantile for <span class="mathjax-tex">$\widehat{{\varvec{V}}}$</span>, respectively.</p> <p>In this procedure, if <span class="mathjax-tex">$VaR_{\widehat{\beta }^{(t)}_{k,j}}(U)$</span> had to be calculated again by Monte Carlo, it would be a significant computational cost. However, in our simple model approach, this can be written in explicit form, which significantly reduces the amount of computation.</p> <h3 class="c-article__sub-heading" id="Sec5"><span class="c-article-section__title-number">3.2 </span>Compound Poisson model</h3><p>The second candidate for <i>N</i> is the compound Poisson process with stochastic intensity. We maintain the structure of Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ6">3.1</a>) for <span class="mathjax-tex">$N_k$</span> but alter the distribution of <span class="mathjax-tex">$N_{k,j}$</span> to follow the Poisson distribution. We make the following assumptions: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">[CP1]</span> <p><span class="mathjax-tex">$N_{k,j} (j=1,...,m)$</span> are independent and identically distributed random variables, each of which conforms to a Poisson distribution, denoted as <span class="mathjax-tex">$N_{k,j}\overset{i.i.d}{\sim }Po(\Lambda _k/m)$</span>. Here, the parameter <span class="mathjax-tex">$\Lambda _k$</span> remains constant throughout the <span class="mathjax-tex">$k^{th}$</span> period, with <span class="mathjax-tex">$\Lambda _k > 0$</span>: for <span class="mathjax-tex">$\Lambda _k > 0$</span>, </p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathbb {P}}}(N_{k,j} = r) = e^{-\Lambda _k/m}\frac{(\Lambda _k/m)^\ell }{\ell !},\quad \ell =0,1,2,\dots . \end{aligned}$$</span></div></div> </li> <li> <span class="u-custom-list-number">[CP2]</span> <p>The value of the parameter <span class="mathjax-tex">$\Lambda _{k}$</span> changes depending on <i>k</i>. In particular, log transfomation of <span class="mathjax-tex">$\Lambda _{k}(:=\displaystyle \log \Lambda _{k})$</span> follows ARIMA(<i>p</i>, <i>d</i>, <i>q</i>) process, i.e. </p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \log \Lambda _{k}-\log \Lambda _{k-d}=c+{\epsilon }_{k}+\sum _{i=1}^{p}\phi _{i}\log \Lambda _{k-i}+\sum _{i=1}^{q}\theta _{i}{\epsilon }_{k-i}, \end{aligned}$$</span></div></div><p> where <span class="mathjax-tex">${\epsilon }_{k}\overset{i.i.d}{\sim }{\mathscr {N}}(0,\sigma ^{2})$</span>.</p> </li> </ol><p>Under the condition [CP1], it is evident that the condition in Eq. (<a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar8">A.2</a>) from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar12">A.4</a> holds, given that <span class="mathjax-tex">$N_k\sim Po(\Lambda _k)$</span>.</p><p>We follow the same procedure as the previous section to predict <span class="mathjax-tex">${\mathbb {E}}[N_k|{\mathscr {F}}_t]\ (k>t)$</span>. To commence, we estimate each <span class="mathjax-tex">$\Lambda _k/m\ (k=1,2,\dots ,t)$</span> based on observations <span class="mathjax-tex">$(N_{k,1},N_{k,2},\dots , N_{k,m})_{k=1,2,\dots ,t}$</span> through the MLE:</p><div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{\widehat{\Lambda }_k}{m} = \frac{N_{k,1} + \dots + N_{k,m}}{m} = \frac{N_k}{m}\quad \Leftrightarrow \quad \widehat{\Lambda }_k = N_k. \end{aligned}$$</span></div></div><p>Next, if the <span class="mathjax-tex">$\widehat{\Lambda }_k$</span> estimates the true <span class="mathjax-tex">$\Lambda _k$</span> well, then we can regards that <span class="mathjax-tex">$\{\log \widehat{\Lambda }_k\}_{k\in {\mathbb {N}}}$</span> follows the ARIMA(<i>p</i>, <i>d</i>, <i>q</i>), and that <span class="mathjax-tex">$\Lambda _k^{(t)} = \Lambda _k|_{{\mathscr {F}}_t}$</span> is predicted by</p><div id="Equ38" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \log \Lambda _{k}^{(t)}-\log \Lambda _{k-\widehat{d}}^{(t)} =\widehat{c}+{\epsilon }_{k}+\sum _{i=1}^{\widehat{p}}\widehat{\phi }_{i}\log \Lambda _{k-i}^{(t)}+\sum _{i=1}^{\widehat{q}}\widehat{\theta }_{i}{\epsilon }_{k-i},\quad {\epsilon }_k\sim {\sim }{\mathscr {N}}(0,\widehat{\sigma }^{2}), \end{aligned}$$</span></div></div><p>where all the unknown parameters are estimated from <span class="mathjax-tex">$\{\log \widehat{\Lambda }_k\}_{k=1,2,\dots t}$</span>.</p><p>Since <span class="mathjax-tex">${\mathbb {E}}[N_k]=\Lambda _k$</span>, we approximate <span class="mathjax-tex">${\mathbb {E}}[N_k|{\mathscr {F}}_t]$</span> as follows:</p><div id="Equ39" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathbb {E}}[N_k|{\mathscr {F}}_t] \approx \Lambda _k^{(t)},\quad k>t, \end{aligned}$$</span></div></div><p>and the predictor of <span class="mathjax-tex">$\beta _k^{(t)}$</span> is given by</p><div id="Equ40" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widehat{\beta }_k^{(t)} = 1 - \frac{1-\alpha }{\Lambda _k^{(t)}}. \end{aligned}$$</span></div></div><p>Then the <span class="mathjax-tex">$VaR_{\widehat{\beta }_k^{(t)}}$</span> is predicted by the same procedure as in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar3">3.1</a>.</p><h3 class="c-article__sub-heading" id="Sec6"><span class="c-article-section__title-number">3.3 </span>Hawkes processes</h3><p>The third candidate for <i>N</i> is represented by a <i>Hawkes process</i> <span class="mathjax-tex">$\widetilde{N}=(\widetilde{N}t){t\ge 0}$</span>, which is a point process characterized by stochastic intensity: for given <span class="mathjax-tex">${\mathscr {F}}_t$</span>,</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \lambda _t=\mu +\sum _{t_{i}<t}g(t-t_{i}), \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.3) </div></div><p>where <span class="mathjax-tex">$\mu \ge 0$</span>, <i>g</i> is a <i>kernel function</i>, and <span class="mathjax-tex">$t_{i}$</span> is the <span class="mathjax-tex">$i_{th}$</span> incident time.</p><p>We suppose that <span class="mathjax-tex">$\widetilde{N}_k$</span> is the number of incidents up to time <i>k</i>. Then our <span class="mathjax-tex">$N_k$</span>, which is the number of incident in the <span class="mathjax-tex">$k^{th}$</span> period, is given by</p><div id="Equ41" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} N_k = \widetilde{N}_{k} - \widetilde{N}_{k-1}. \end{aligned}$$</span></div></div><p>Given that obtaining the expectation <span class="mathjax-tex">${\mathbb {E}}[\widetilde{N}_k]$</span> is typically challenging, we make an additional assumption concerning the kernel function <i>g</i>, which assumes the form</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} g_\vartheta (x)=\alpha \beta \exp (-\beta x),\quad x\in {\mathbb {R}}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.4) </div></div><p>with a parameter <span class="mathjax-tex">$\vartheta =(\alpha ,\beta ) \in {\mathbb {R}}_{+}^2$</span>. Importantly, we assume that the parameter <span class="mathjax-tex">$\vartheta $</span> remains constant across periods (<i>k</i>), whereas in Sects. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec4">3.1</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec5">3.2</a>, we had assumed period-dependent <span class="mathjax-tex">$\vartheta $</span> values.</p><p>According to Lesage et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2020" title="Lesage, L., Deaconu, M., Lejay, A., Meira, A. J., Nichil, G. & State, R. (2020). Hawkes processes framework with a Gamma density as excitation function: application to natural disasters for insurance. Retrieved from 
 https://hal.inria.fr/hal-03040090
 
 " href="/article/10.1007/s42081-024-00273-y#ref-CR12" id="ref-link-section-d432018368e10995">2020</a>), the expectation of the Hawkes process with the kernel (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ9">3.4</a>) is written as</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathbb {E}}[\widetilde{N}_{t}]=\frac{\mu }{1-\alpha }t-\frac{\mu \alpha }{\beta (1-\alpha )^{2}}\left[1-\exp \{-(1-\alpha )\beta t\}\right], \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.5) </div></div><p>although it is generally hard to find the explicit expression of the expectation of a Hawkes process.</p><p>Next, we check the condition (<a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar8">A.2</a>) in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar12">A.4</a>. According to Daley and Vere-Jones (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2003" title="Daley, D. J., & Vere-Jones, D. (2003). An introduction to the theory of point processes-volume I: Elementary theory and methods (2nd ed.). Springer." href="/article/10.1007/s42081-024-00273-y#ref-CR7" id="ref-link-section-d432018368e11160">2003</a>),</p><div id="Equ42" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathbb {P}}}(\widetilde{N}_k - \widetilde{N}_{k-1}=r)={\mathbb {E}}\Biggl [\exp \left( -\int _{k-1}^{k}\lambda _{s}\,ds\right) \frac{(\int _{k-1}^{k}\lambda _{s}\,ds)^{r}}{r!}\Biggr ]. \end{aligned}$$</span></div></div><p>Hence it follows for any <span class="mathjax-tex">${\epsilon }>0$</span> that</p><div id="Equ43" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{r=0}^\infty (1+{\epsilon })^r {{\mathbb {P}}}(N_k=r)&= \sum _{r=0}^\infty (1+{\epsilon })^r {{\mathbb {P}}}(\widetilde{N}_k - \widetilde{N}_{k-1}=r)\\&= \sum _{r=0}^{\infty }(1+{\epsilon })^{r}\cdot {\mathbb {E}}\Biggl [\exp \left( -\int _{k-1}^{k}\lambda _{s}ds\right) \dfrac{(\int _{k-1}^{k}\lambda _{s}ds)^{r}}{r!}\Biggr ]\\&= {\mathbb {E}}\Biggl [\sum _{r=0}^{\infty }\exp \left( -\int _{k-1}^k \lambda _{s}ds\right) \dfrac{\{(1+{\epsilon })\int _{k-1}^{k}\lambda _{s}ds\}^{r}}{r!}\Biggr ]\\&= {\mathbb {E}}\Biggl [\exp \left( {\epsilon }\int _{k-1}^{t}\lambda _{s}ds\right) \sum _{r=0}^{\infty }\exp \left\{ -(1+{\epsilon })\int _{k-1}^{k}\lambda _{s}ds\right\} \\&\quad \times \dfrac{\{(1+{\epsilon })\int _{k-1}^{k}\lambda _{s}ds\}^{r}}{r!}\Biggr ]\\&= {\mathbb {E}}\Biggl [\exp \left( {\epsilon }\int _{k-1}^{k}\lambda _{s}ds\right) \Biggr ] < \infty . \end{aligned}$$</span></div></div><p>To estimate the parameters <span class="mathjax-tex">$\vartheta = (\mu ,\alpha ,\beta )$</span>, we require knowledge of the incident times <span class="mathjax-tex">$t_i$</span>, which are not available in the PRC dataset [18]. Only the date of each incident is provided. Consequently, we resort to a hypothetical stochastic generation of incident times and substitute these with random numbers to construct an estimator for the parameters. This process is iterated multiple times, and the estimated values of <span class="mathjax-tex">$\widehat{\vartheta }$</span> are computed by averaging these estimators.</p><p>Given that numerous incident times occur within a single day, an approximation in which the times are assumed to be uniformly distributed throughout a day is generally acceptable. The averaging process will help mitigate any errors. In practical terms, we follow these steps: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>Generate the quasi-occurrence times <span class="mathjax-tex">$t_i < t$</span> uniformly within a daily scale, denoted as <span class="mathjax-tex">$\tau _1,\dots , \tau _{N_k}$</span>.</p> </li> <li> <span class="u-custom-list-number">2.</span> <p>Estimate the <span class="mathjax-tex">$\vartheta =(\mu ,\alpha ,\beta )$</span> by the maximum likelihood method, where the likelihood function is given by </p><div id="Equ44" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} L(\mu ,\alpha ,\beta ):= \sum _{j=1}^{N_k} \log \lambda _{\tau _j} - \int _0^t \lambda _s\,ds \end{aligned}$$</span></div></div> </li> <li> <span class="u-custom-list-number">3.</span> <p>Iterate this procedure <i>B</i> times, and compute the MLE <span class="mathjax-tex">$\widehat{\vartheta }^{t,j} = (\widehat{\mu }^{(t,j)}, \widehat{\alpha }^{(t,j)}, \widehat{\beta }^{(t,j)})$</span> in the <span class="mathjax-tex">$j^{th}$</span> step <span class="mathjax-tex">$(j=1,2,\dots ,B)$</span>. Then, aggregate these individual estimates as follows: </p><div id="Equ45" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widehat{\vartheta }^{(t)} = \frac{1}{B}\sum _{j=1}^B \widehat{\vartheta }^{(t,j)}. \end{aligned}$$</span></div></div><p> This approach allows us to estimate the parameters <span class="mathjax-tex">$\vartheta $</span> with repeated sampling and averaging for enhanced accuracy.</p> </li> </ol><p>From the expression (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ10">3.5</a>), we have the approximation</p><div id="Equ46" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathbb {E}}[\widetilde{N}_k|{\mathscr {F}}_t] \approx \frac{\widehat{\mu }^{(t)}}{1-\widehat{\alpha }^{(t)}}k-\frac{\widehat{\mu }^{(t)}\widehat{\alpha }^{(t)}}{\widehat{\beta }^{(t)}(1-\widehat{\alpha }^{(t)})^{2}}\left\rbrace 1-\exp \left[-(1-\widehat{\alpha }^{(t)})\widehat{\beta }^{(t)}k\right]\right\lbrace =: \Pi _k^{(t)},\quad k>t. \end{aligned}$$</span></div></div><p>Since <span class="mathjax-tex">$N_k = \widetilde{N}_k - \widetilde{N}_{k-1}$</span>, the predictor of <span class="mathjax-tex">$\beta _k^{(t)}$</span> is given by</p><div id="Equ47" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widehat{\beta }_k^{(t)} = 1 - \frac{1-\alpha }{\Pi _k^{(t)} - \Pi _{k-1}^{(t)}}. \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar4">Remark 3.2</h3> <p>It’s important to note that in this particular model, the predictor for <span class="mathjax-tex">$\widehat{\beta }_k^{(t)}$</span> in the future is not subject to randomness. This is because the model assumes that the parameter values remain constant. As a result, there is no need to follow the procedure outlined in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar3">3.1</a>. The value of <span class="mathjax-tex">$VaR_{\widehat{\beta }_k^{(t)}}$</span> is solely determined by the estimated value of <span class="mathjax-tex">$\widehat{\beta }_k^{(t)}$</span>.</p> <h3 class="c-article__sub-heading" id="Sec7"><span class="c-article-section__title-number">3.4 </span>Approximation and estimation of <span class="mathjax-tex">$F_U$</span> </h3><p>Across all the models described above, we maintain the assumption:</p><div id="Equ48" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \overline{F}_U \in {\mathscr {R}}_{-\kappa },\quad \kappa >1. \end{aligned}$$</span></div></div><p>which enables us to apply Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar6">A.1</a>, and for ‘large <span class="mathjax-tex">$u>0$</span>’, we can approximate</p><div id="Equ49" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F_U(x|u) \approx G_{\xi ,\sigma }(x), \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\xi \ge 0$</span> and <span class="mathjax-tex">$\sigma >0$</span> represent the parameters. Determining a ‘suitable’ value for <span class="mathjax-tex">$u>0$</span> is crucial. The <i>Peaks-Over-Threshold (POD) method</i> is a widely recognized approach for selecting a threshold <span class="mathjax-tex">$u>0$</span>. We make this determination visually by utilizing the <i>mean excess (ME)-plot</i>. Further details can be found in Embrechts et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2003" title="Embrechts, P., Klüppelberg, C., & Mikosch, T. (2003). Modeling extremal events for insurance and finance. Springer." href="/article/10.1007/s42081-024-00273-y#ref-CR9" id="ref-link-section-d432018368e13877">2003</a>), Section 6.5.</p><p>As an example, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig4">4</a> displays the ME-plot for Case 2 (2006–2016); refer to Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab1">1</a>. We can opt for a threshold such as <span class="mathjax-tex">$u=6.6\times 10^6\ (\mathrm {6.6e+06})$</span>. Subsequently, we estimate the parameters <span class="mathjax-tex">$\xi $</span> and <span class="mathjax-tex">$\sigma $</span> using data that exceeds this threshold <i>u</i>, employing the maximum likelihood method as outlined in Embrechts et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2003" title="Embrechts, P., Klüppelberg, C., & Mikosch, T. (2003). Modeling extremal events for insurance and finance. Springer." href="/article/10.1007/s42081-024-00273-y#ref-CR9" id="ref-link-section-d432018368e13984">2003</a>), Section 6.5.1. These estimated values are denoted as <span class="mathjax-tex">$\widehat{\xi }{u}^{(t)}$</span> and <span class="mathjax-tex">$\widehat{\sigma }{u}^{(t)}$</span>. It’s important to note that <span class="mathjax-tex">$t=2013$</span> in Case 1 and <span class="mathjax-tex">$t=2016$</span> in Case 2. The estimated values are presented in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab2">2</a>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Fig. 4"><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/4" rel="nofollow"><picture><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig4_HTML.png" alt="figure 4" loading="lazy" width="685" height="372"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>Mean excess plot (Case 2: 2006–2016)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-2"><figure><figcaption class="c-article-table__figcaption"><b id="Tab2" data-test="table-caption">Table 2 Estimated values of <span class="mathjax-tex">$\xi $</span> and <span class="mathjax-tex">$\sigma $</span> (MLE) with the threshold <i>u</i></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/2" aria-label="Full size table 2"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Furthermore, based on these estimated values, we conducted the <i>Kolmogorov–Smirnov (KS) goodness-of-fit test</i> for the estimated probability density function: <span class="mathjax-tex">$G_{\widehat{\xi }{u}^{(t)}, \widehat{\sigma }{u}^{(t)}}$</span>. The KS test statistics are provided in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab3">3</a>, and we found that the hypothesis stating the distribution of <span class="mathjax-tex">$U_i>u$</span> follows a Generalized Pareto Distribution (GPD) was not rejected at the 5% significance level in both Cases 1 and 2. For your reference, these density functions are depicted in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig5">5</a>.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-3"><figure><figcaption class="c-article-table__figcaption"><b id="Tab3" data-test="table-caption">Table 3 KS-test</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/3" aria-label="Full size table 3"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-5" data-title="Fig. 5"><figure><figcaption><b id="Fig5" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 5</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/5" rel="nofollow"><picture><img aria-describedby="Fig5" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig5_HTML.png" alt="figure 5" loading="lazy" width="685" height="257"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-5-desc"><p>Estimated density of GPD with data; Case 1 (left) and Case 2 (right)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/5" data-track-dest="link:Figure5 Full size image" aria-label="Full size image figure 5" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="Data analysis: prediction of tail risks"><div class="c-article-section" id="Sec8-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec8"><span class="c-article-section__title-number">4 </span>Data analysis: prediction of tail risks</h2><div class="c-article-section__content" id="Sec8-content"><p>Utilizing each of the models previously outlined, namely NB (negative binomial), CP (compound Poisson), and HK (Hawkes process), we estimate the conditional (Tail) Value-at-Risk as described in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar3">3.1</a>.</p><p>In the NB and CP models, we set <span class="mathjax-tex">$m=7$</span> to represent a week, where <span class="mathjax-tex">$N_{kj}$</span> signifies the number of incidents on the <i>j</i>th day of the <i>k</i>th week. Under each model, we generate 1000 samples of <span class="mathjax-tex">$VaR_{\beta _k^{(t)}}(U)$</span> and <span class="mathjax-tex">$TVaR_{\beta _k^{(t)}}(U)$</span>, calculate the mean, and determine the 95% confidence interval. Subsequently, we compare these results with the test data in Cases 1 and 2, respectively.</p><h3 class="c-article__sub-heading" id="Sec9"><span class="c-article-section__title-number">4.1 </span>Negative Binomial model</h3><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab4">4</a> provides the results of estimating the ARIMA process for <span class="mathjax-tex">$p_k$</span> as assumed in [NB2]. We select the values of (<i>p</i>, <i>d</i>, <i>q</i>) using Akaike’s Information Criteria (AIC) through maximum likelihood estimation (MLE).</p><p>We estimate the 99% and 99.9% (Tail) VaR and present the results in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig6">6</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig7">7</a> alongside the testing data for backtesting.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-4"><figure><figcaption class="c-article-table__figcaption"><b id="Tab4" data-test="table-caption">Table 4 Estimation of ARIMA process for <span class="mathjax-tex">$p_k$</span> in [NB2]</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/4" aria-label="Full size table 4"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-6" data-title="Fig. 6"><figure><figcaption><b id="Fig6" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 6</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/6" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig6_HTML.png?as=webp"><img aria-describedby="Fig6" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig6_HTML.png" alt="figure 6" loading="lazy" width="685" height="423"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-6-desc"><p>NB model, Case 1: 99% and 99.9% (T)VaR with breaches in 2014–2015</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/6" data-track-dest="link:Figure6 Full size image" aria-label="Full size image figure 6" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-7" data-title="Fig. 7"><figure><figcaption><b id="Fig7" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 7</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/7" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig7_HTML.png?as=webp"><img aria-describedby="Fig7" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig7_HTML.png" alt="figure 7" loading="lazy" width="685" height="440"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-7-desc"><p>NB model, Case 2: 99% and 99.9% (T)VaR with breaches in 2017–2018</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/7" data-track-dest="link:Figure7 Full size image" aria-label="Full size image figure 7" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec10"><span class="c-article-section__title-number">4.2 </span>Compound Poisson model</h3><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab5">5</a> is estimating the result of the ARIMA process for <span class="mathjax-tex">$\Lambda _k$</span> assumed in [CP2]. AIC also selects the parameter (<i>p</i>, <i>d</i>, <i>q</i>). We show 99% and 99.9% (Tail) VaR in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig8">8</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig9">9</a>.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-5"><figure><figcaption class="c-article-table__figcaption"><b id="Tab5" data-test="table-caption">Table 5 Estimation of ARIMA process for <span class="mathjax-tex">$\Lambda _k$</span> in [CP2]</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/5" aria-label="Full size table 5"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-8" data-title="Fig. 8"><figure><figcaption><b id="Fig8" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 8</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/8" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig8_HTML.png?as=webp"><img aria-describedby="Fig8" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig8_HTML.png" alt="figure 8" loading="lazy" width="685" height="423"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-8-desc"><p>CP model, Case 1: 99% and 99.9% (T)VaR with breaches in 2014–2015</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/8" data-track-dest="link:Figure8 Full size image" aria-label="Full size image figure 8" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-9" data-title="Fig. 9"><figure><figcaption><b id="Fig9" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 9</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/9" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig9_HTML.png?as=webp"><img aria-describedby="Fig9" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig9_HTML.png" alt="figure 9" loading="lazy" width="685" height="440"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-9-desc"><p>CP model, Case 2: 99% and 99.9% (T)VaR with breaches in 2017–2018</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/9" data-track-dest="link:Figure9 Full size image" aria-label="Full size image figure 9" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec11"><span class="c-article-section__title-number">4.3 </span>Hawkes Process</h3><p>We present the backtesting results for the HK models in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig10">10</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig11">11</a>. As mentioned at the end of Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s42081-024-00273-y#Sec6">3.3</a>, the values of <span class="mathjax-tex">$VaR_{\widehat{\beta }_k^{(t)}}(U)$</span> <span class="mathjax-tex">$(k=1,2,\dots )$</span> are computed deterministically based on the estimated values of <span class="mathjax-tex">$\widehat{\beta }_k^{(t)}$</span>. Consequently, we cannot provide confidence intervals for the VaR as in the other models; see also Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar4">3.2</a>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-10" data-title="Fig. 10"><figure><figcaption><b id="Fig10" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/10" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig10_HTML.png?as=webp"><img aria-describedby="Fig10" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig10_HTML.png" alt="figure 10" loading="lazy" width="685" height="423"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-10-desc"><p>HK model, Case 2: 99% and 99.9% (T)VaR with breaches in 2014–2015</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/10" data-track-dest="link:Figure10 Full size image" aria-label="Full size image figure 10" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-11" data-title="Fig. 11"><figure><figcaption><b id="Fig11" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 11</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/11" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig11_HTML.png?as=webp"><img aria-describedby="Fig11" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42081-024-00273-y/MediaObjects/42081_2024_273_Fig11_HTML.png" alt="figure 11" loading="lazy" width="685" height="423"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-11-desc"><p>HK model, Case 2: 99% and 99.9% (T)VaR with breaches in 2017–2018</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s42081-024-00273-y/figures/11" data-track-dest="link:Figure11 Full size image" aria-label="Full size image figure 11" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec12"><span class="c-article-section__title-number">4.4 </span>Back testing the models</h3><p>We provide the backtesting results in Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab6">6</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab7">7</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab8">8</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab9">9</a>, where:</p><ul class="u-list-style-bullet"> <li> <p>‘95%-lower’ represents the rate at which the actual breaches are less than the 95%-lower bound of the confidence interval.</p> </li> <li> <p>‘95%-upper’ indicates the rate at which the actual breaches are less than the 95%-upper bound of the confidence interval.</p> </li> <li> <p>‘Mean’ represents the rate at which the actual breaches are less than the mean of the Monte Carlo samples of (T)VaR. This can be considered as the risk reserve for the insurer of the cyber risks.</p> </li> </ul><p>From a theoretical perspective, these rates (especially ‘Mean’) are expected to be close to 99% in VaR and even higher in TVaR because TVaR is a more conservative risk measure than VaR.</p><p>The results show that the 99%-VaR behaves as the theory suggests, and TVaR is more conservative, which appears to be sufficient for risk management. In Case 1, each rate is around 99% for VaR, while in Case 2, they are slightly underestimated but not far from 99%. This outcome is reasonable considering the trend changes since 2016, as discussed in the Introduction.</p><p>Figures <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig6">6</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig7">7</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig8">8</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig9">9</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig10">10</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s42081-024-00273-y#Fig11">11</a> illustrate that the results with NB and CP are similar, making it challenging to determine which is superior. The HK model yields slightly inferior results compared to the other two, even though it is often used to analyze cyber risks. Hence, the NB and CP models are sufficiently suitable for practical risk management, and there may be no compelling reason to opt for the HK model, which involves more complex estimation and modeling processes.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-6"><figure><figcaption class="c-article-table__figcaption"><b id="Tab6" data-test="table-caption">Table 6 Empirical test for <span class="mathjax-tex">$VaR_{0.99}$</span></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/6" aria-label="Full size table 6"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-7"><figure><figcaption class="c-article-table__figcaption"><b id="Tab7" data-test="table-caption">Table 7 Empirical test for <span class="mathjax-tex">$VaR_{0.999}$</span></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/7" aria-label="Full size table 7"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-8"><figure><figcaption class="c-article-table__figcaption"><b id="Tab8" data-test="table-caption">Table 8 Empirical test for <span class="mathjax-tex">$TVaR_{0.99}$</span></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/8" aria-label="Full size table 8"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-9"><figure><figcaption class="c-article-table__figcaption"><b id="Tab9" data-test="table-caption">Table 9 Empirical test for <span class="mathjax-tex">$TVaR_{0.999}$</span></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/9" aria-label="Full size table 9"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-10"><figure><figcaption class="c-article-table__figcaption"><b id="Tab10" data-test="table-caption">Table 10 <i>p</i>-values for binomial backtesting of <span class="mathjax-tex">$VaR_{0.99}$</span>. Bold letters are the results where <span class="mathjax-tex">$H_0$</span> is rejected at a significance level of 10%</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/10" aria-label="Full size table 10"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-11"><figure><figcaption class="c-article-table__figcaption"><b id="Tab11" data-test="table-caption">Table 11 <i>p</i>-values for binomial backtesting of <span class="mathjax-tex">$VaR_{0.999}$</span>. Bold letters are the results where <span class="mathjax-tex">$H_0$</span> is rejected at a significance level of 10%</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1007/s42081-024-00273-y/tables/11" aria-label="Full size table 11"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The above empirical backtesting may need to be revised: backtesting of VaR and TVaR is described in detail in Bayer and Dimitriadis (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2022" title="Bayer, S., & Dimitriadis, T. (2022). Regression-based expected shortfall backtesting. Journal of Financial Econometrics, 20(3), 437–471." href="/article/10.1007/s42081-024-00273-y#ref-CR3" id="ref-link-section-d432018368e17740">2022</a>) and Nolde and Ziegel (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Nolde, N., & Ziegel, F. (2017). Elicitability and backtesting: Perspectives for banking regulation. The Annals of Applied Statistics, 11(4), 1833–1874." href="/article/10.1007/s42081-024-00273-y#ref-CR14" id="ref-link-section-d432018368e17743">2017</a>), and R packages are available. However, they did not work well on our dataset. The exact reasons are unclear, but in particular, during the computational process, irregular matrices appeared and the errors could not be removed until the end. This may be due to the peculiarities of our data. As mentioned in the Introduction, our data are not partly recorded as correct time stamps. This is a drawback of this open data, and hence, the distribution of the data is rather biased.</p><p>Therefore, as a standard backtest for VaR, we conducted a backtest using the binomial distribution mentioned in the Basel documents (Bank for International Settlements <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2013" title="Bank for International Settlements. (2013). Consultative document: Fundamental review of the trading book: A revised marked risk framework. Retrieved from 
 http://www.bis.org/publ/bcbs265.pdf
 
 ." href="/article/10.1007/s42081-024-00273-y#ref-CR2" id="ref-link-section-d432018368e17750">2013</a>) (Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab10">10</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab11">11</a>). This method serves our purpose here, as it is a method that can be described in the framework of the Nolde and Ziegel (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Nolde, N., & Ziegel, F. (2017). Elicitability and backtesting: Perspectives for banking regulation. The Annals of Applied Statistics, 11(4), 1833–1874." href="/article/10.1007/s42081-024-00273-y#ref-CR14" id="ref-link-section-d432018368e17759">2017</a>) backtest and can be used without any distributional assumptions on the loss data (see Nolde and Ziegel <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2017" title="Nolde, N., & Ziegel, F. (2017). Elicitability and backtesting: Perspectives for banking regulation. The Annals of Applied Statistics, 11(4), 1833–1874." href="/article/10.1007/s42081-024-00273-y#ref-CR14" id="ref-link-section-d432018368e17762">2017</a>, Example 1). The null hypothesis in our test is the following:</p><blockquote class="c-blockquote"><div class="c-blockquote__body"> <p><span class="mathjax-tex">$H_0$</span>: The sequence of our predicted <span class="mathjax-tex">$\{VaR_{\alpha }^{(t)}\}_{t\in {\mathbb {N}}}$</span> is <i>conditionally calibrated</i> (certainly the value-at-risk with level <span class="mathjax-tex">$\alpha $</span>).</p> </div></blockquote><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab10">10</a> shows that the <i>p</i>-value for the HK model is relatively small, e.g., <span class="mathjax-tex">$H_0$</span> is rejected at a significance level of 5% for the HK-Mean, Case 2, but is at a level of stable acceptance for both NB and CP (in the Mean). Also, from Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1007/s42081-024-00273-y#Tab11">11</a>, the performance of HK is somewhat inferior to the other two models, as is also the case at the 99.9% level. It is safe to say that very classical models such as NB and CP are also adequate in practical terms, at least in our dataset.</p> <h3 class="c-article__sub-heading" id="FPar5">Remark 4.1</h3> <p>All existing methods for backtesting against TVaR rely on the distribution of the loss data and/or the asymptotic variance of the test statistic. In particular, it was difficult to find a suitable method for TVaR backtesting, as the loss data in this case was heavy-tailed, and even the existence of variance was doubtful. Due to those limitations, only the above-mentioned empirical results are identified here.</p> </div></div></section><section data-title="Conclusion and future works"><div class="c-article-section" id="Sec13-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec13"><span class="c-article-section__title-number">5 </span>Conclusion and future works</h2><div class="c-article-section__content" id="Sec13-content"><p>We extend the classical (single-period) insurance risk model to a multi-period framework for more effective cyber risk assessment. By evaluating the performance of different models, including the negative binomial model, Poisson process, and Hawkes process, you provide valuable insights into their ability to predict VaR and TVaR in future periods.</p><p>Our data analysis revealed that both the negative binomial and Poisson models effectively predict VaR and TVaR for cyber risks. However, there was no significant difference in their performance, suggesting that either of these models can be used effectively for risk assessment. Surprisingly, the Hawkes model, which is commonly used for predicting cyber risks, did not exhibit superior performance in this specific dataset.</p><p>Our study demonstrates that a classical and simple model can effectively manage cyber risks. The explicit calculations and low computational costs make this approach practical and accessible. Moreover, the straightforward statistical procedures involved in this model make it a valuable tool for cyber risk assessment. This research emphasizes the importance of using models that are not only effective but also easy to implement in practice.</p><p>On the other hand, recent survey studies related to cyber risk and insurance, such as Awiszus et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Awiszus, K., Knispel, T., Penner, I., Svindland, G., Voß, A., & Weber, S. (2023). Modeling and pricing cyber insurance: Idiosyncratic, systematic, and systemic risks. European Actuarial Journal, 13(1), 1–53." href="/article/10.1007/s42081-024-00273-y#ref-CR1" id="ref-link-section-d432018368e17932">2023</a>) and Dacorogna and Kratz (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Dacorogna, M., & Kratz, M. (2023). Managing cyber risk, a science in the making. Scandinavian Actuarial Journal, 2023(10), 1000–1021." href="/article/10.1007/s42081-024-00273-y#ref-CR6" id="ref-link-section-d432018368e17935">2023</a>), have pointed out that while emphasizing the importance and utility of the classical actuarial approach, it is difficult to address the complexity of cyber risk data using only the classical frequency-severity approach. This suggests that there is still room for development in the direct application of our classical model. However, it should not be forgotten that the explicit expressiveness of the simple classical model has computational advantages, which cannot be ignored in practice. Furthermore, in the above studies, the use of a single-period model is assumed for frequency modeling. Our novelty lies in extending this to a multi-period model and addressing its statistical inference. Awiszus et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2023" title="Awiszus, K., Knispel, T., Penner, I., Svindland, G., Voß, A., & Weber, S. (2023). Modeling and pricing cyber insurance: Idiosyncratic, systematic, and systemic risks. European Actuarial Journal, 13(1), 1–53." href="/article/10.1007/s42081-024-00273-y#ref-CR1" id="ref-link-section-d432018368e17938">2023</a>) mention using a Cox process for frequency modeling, but this approach does not yield explicit expressions for VaR or TVaR. Our approach emphasizes explicitness. Making our multi-period model the standard benchmark model and incorporating the characteristics of cyber risk may serve as a trigger for more complex modeling.</p><p>Despite using the PRC data [18], the largest dataset available to our knowledge, several issues still need to be solved. First, many cyber incidents have likely yet to be disclosed. Open databases for cyber attacks may help address this. Secondly, inaccuracies in the incident dates are a concern. The reported dates in PRC are not necessarily when the breaches occurred but when they were made public. As a result, some incidents may be reported long after they occurred, akin to the Incurred But Not Reported (IBNR) concept in insurance. This issue needs further attention in future research.</p><p>While we used all data without categorization in our data analysis, the trends may differ depending on the type of breaches. For instance, some incidents, such as hacking or insider breaches, may be malicious, while others could result from negligence, like administrative errors. Additionally, the trends may vary based on business sectors such as companies, educational institutions, and medical facilities. Consequently, future analyses should ideally be based on more finely categorized and detailed data. Unfortunately, such data are not readily available as open-source, and developing a comprehensive database is still an ongoing challenge. The cyber risk analysis field would greatly benefit from establishing more extensive and categorized datasets for improved insights and risk management. </p></div></div></section> </div> <section data-title="Data Availability"><div class="c-article-section" id="data-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="data-availability">Data Availability</h2><div class="c-article-section__content" id="data-availability-content"> <p>All the data we used in this paper are available on Privacy Rights Clearinghouse (2023) website.</p> </div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ul class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item"><p class="c-article-references__text" id="ref-CR1">Awiszus, K., Knispel, T., Penner, I., Svindland, G., Voß, A., & Weber, S. 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Also, the authors extend their sincere appreciation to the anonymous reviewers for their insightful comments that have contributed to enhancing the quality of this paper.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Department of Applied Mathematics, Waseda University, Shinjuku, Japan</p><p class="c-article-author-affiliation__authors-list">Yasutaka Shimizu</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Graduate School of Fundamental Science and Engineering, Waeda University, Shinjuku, Japan</p><p class="c-article-author-affiliation__authors-list">Yutaro Takagami</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Yasutaka-Shimizu-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Yasutaka Shimizu</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?sortBy=newestFirst&dc.creator=Yasutaka%20Shimizu" class="c-article-button" data-track="click" data-track-action="author link - 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There exists <span class="mathjax-tex">$\kappa >0$</span> such that <span class="mathjax-tex">$\overline{F}\in {\mathscr {R}}_{-\kappa }$</span> if and only if there exists a positive function <span class="mathjax-tex">$b(u)\rightarrow \infty $</span> as <span class="mathjax-tex">$u\rightarrow \infty $</span> such that</p><div id="Equ50" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \lim _{u\rightarrow \infty }\sup _{0<x<\infty }|F(x|u)-G_{\xi ,b(u)}(x)|=0, \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$F(x|u):={{\mathbb {P}}}(X-u\le x|X>u)$</span> and <span class="mathjax-tex">$\xi =1/\kappa $</span>, and <span class="mathjax-tex">$G_{\xi ,\sigma }$</span> is the <i>generalized Pareto distribution (GPD)</i> with the distribution function given by</p><div id="Equ51" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} G_{\xi ,\sigma }(x)= {\left\{ \begin{array}{ll} 1-\left( 1+\dfrac{\xi }{\sigma }x\right) ^{-1/\xi } & (\xi \ne 0)\\ 1-e^{-x/\sigma } & (\xi =0) \end{array}\right. }. \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar8">Proof</h3> <p>See Embrechts et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2003" title="Embrechts, P., Klüppelberg, C., & Mikosch, T. (2003). Modeling extremal events for insurance and finance. Springer." href="/article/10.1007/s42081-024-00273-y#ref-CR9" id="ref-link-section-d432018368e18642">2003</a>), Theorem 3.4.13. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar9">Lemma A.2</h3> <p>Let <i>S</i> is a compound risk model given in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ1">1.1</a>), and suppose that <i>N</i> satisfies</p><div id="Equ52" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{r=0}^{\infty }(1+{\epsilon })^{r}\cdot {{\mathbb {P}}}(N=r)<\infty \end{aligned}$$</span></div></div><p>for some <span class="mathjax-tex">${\epsilon }>0$</span>. Then, it holds that</p><div id="Equ53" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \overline{F}_{S}(x)\sim {\mathbb {E}}[N]\cdot \overline{F}_{U}(x),\quad x\rightarrow \infty . \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar10">Proof</h3> <p>See Embrechts et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2003" title="Embrechts, P., Klüppelberg, C., & Mikosch, T. (2003). Modeling extremal events for insurance and finance. Springer." href="/article/10.1007/s42081-024-00273-y#ref-CR9" id="ref-link-section-d432018368e18911">2003</a>), Theorem 1.3.9. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar11">Lemma A.3</h3> <p>For <span class="mathjax-tex">$\kappa >1$</span> and <span class="mathjax-tex">$\overline{F}_U \in {\mathscr {R}}_{-\kappa }$</span>, there exists a function <span class="mathjax-tex">$L(x) \in {\mathscr {R}}_0$</span> such that</p><div id="Equ54" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \int _x^\infty \overline{F}_U(y)\,dy \sim L(x) \frac{x^{1-\kappa }}{\kappa - 1},\quad x\rightarrow \infty . \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar12">Proof</h3> <p>See Grandell (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1997" title="Grandell, J. (1997). Mixed Poisson processes. Chapman & Hall." href="/article/10.1007/s42081-024-00273-y#ref-CR11" id="ref-link-section-d432018368e19181">1997</a>), p.181. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar13">Theorem A.4</h3> <p>Let <i>S</i> is a compound risk model given in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ1">1.1</a>), and suppose that <span class="mathjax-tex">$F_U$</span> is a (proper) distribution function with <span class="mathjax-tex">$F_U(x) <1$</span> for any <span class="mathjax-tex">$x\in {\mathbb {R}}$</span>, and that <span class="mathjax-tex">$\overline{F}_{U}\in {\mathscr {R}}_{-\kappa }$</span> where <span class="mathjax-tex">$\kappa >1$</span>. Moreover, suppose that there exists some <span class="mathjax-tex">${\epsilon }>0$</span> such that</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{r=0}^{\infty }(1+{\epsilon })^{r}\cdot {{\mathbb {P}}}(N=r)<\infty . \end{aligned}$$</span></div><div class="c-article-equation__number"> (A.1) </div></div><p>Then it follows for <span class="mathjax-tex">$\beta :=1-\dfrac{1-\alpha }{{\mathbb {E}}[N]}$</span> that</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\left\{ \begin{array}{ll} VaR_{\alpha }(S)\sim VaR_{\beta }(U)\\ TVaR_{\alpha }(S)\sim \dfrac{\kappa }{\kappa -1}VaR_{\beta }(U) \end{array}\right. },\ \alpha \rightarrow 1. \end{aligned}$$</span></div><div class="c-article-equation__number"> (A.2) </div></div><p>Furthermore, taking a sequence <span class="mathjax-tex">$u=u(\alpha ) \uparrow \infty $</span> such that</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \lim _{\alpha \uparrow 1}\frac{1-\alpha }{1-F_U(u(\alpha ))} \in (0,{\mathbb {E}}[N]), \end{aligned}$$</span></div><div class="c-article-equation__number"> (A.3) </div></div><p>it holds that</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} VaR_{\beta }(U)\sim u(\alpha )+\dfrac{\sigma }{\xi }\left\{ \left( \dfrac{\overline{F}_{U}(u(\alpha ))}{1-\beta }\right) ^{\xi }-1\right\} , \quad \alpha \rightarrow 1, \end{aligned}$$</span></div><div class="c-article-equation__number"> (A.4) </div></div><p>where <span class="mathjax-tex">$\xi = 1/\kappa $</span> and <span class="mathjax-tex">$\sigma $</span> is a function of <span class="mathjax-tex">$u=u(\alpha )$</span>.</p> <h3 class="c-article__sub-heading" id="FPar14">Remark A.5</h3> <p>As an example, taking <span class="mathjax-tex">$u=u(\alpha )$</span> such that, for a constant <span class="mathjax-tex">$\gamma \in (0,1)$</span>,</p><div id="Equ55" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} u(\alpha ) = F_U^{-1}\left(1 - \gamma ^{-1}\frac{1 - \alpha }{{\mathbb {E}}[N]}\right), \end{aligned}$$</span></div></div><p>then we have <span class="mathjax-tex">$u(\alpha )\rightarrow \infty $</span> as <span class="mathjax-tex">$\alpha \uparrow 1$</span>, and that</p><div id="Equ56" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \lim _{\alpha \uparrow 1} \frac{1-\alpha }{1-F_U(u(\alpha ))} = \gamma {\mathbb {E}}[N] \in (0,{\mathbb {E}}[N]), \end{aligned}$$</span></div></div><p>which satisfies (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ13">A.3</a>).</p> <h3 class="c-article__sub-heading" id="FPar15">Proof</h3> <p>The asymptotic equivalency (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ12">A.2</a>) is shown in Biagini and Ulmer (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2009" title="Biagini, F., & Ulmer, S. (2009). Asymptotics for operational risk quantified with expected shortfall. ASTIN Bulletin, 39, 735–752." href="/article/10.1007/s42081-024-00273-y#ref-CR4" id="ref-link-section-d432018368e20549">2009</a>), Theorem 2.5, so we omit the details. See also Shimizu (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2018" title="Shimizu, Y. (2018). Insurance mathematics with statistical methodologies. Kyoritsu Shuppan Co., Ltd." href="/article/10.1007/s42081-024-00273-y#ref-CR20" id="ref-link-section-d432018368e20552">2018</a>).</p> <p>As for the statement (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ14">A.4</a>), we note the following decomposition of <span class="mathjax-tex">$F_U$</span>: for any <span class="mathjax-tex">$u<x$</span>,</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F_U(x)&= \overline{F}_U(u) \frac{ \overline{F}_U(x) - \overline{F}_U(u) }{\overline{F}_U(u) } + F_U(u) \nonumber \\&= \overline{F}_U(u) \frac{{{\mathbb {P}}}(U\le x) - {{\mathbb {P}}}(U\le u)}{{{\mathbb {P}}}(U>u) } + F_U(u) \nonumber \\&= \overline{F}_U(u) F_U(x-u|u)+ F_U(u). \end{aligned}$$</span></div><div class="c-article-equation__number"> (A.5) </div></div><p>We see from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s42081-024-00273-y#FPar6">A.1</a> that there exists a function <span class="mathjax-tex">$\sigma =b(u)\uparrow \infty $</span> such that</p><div id="Equ57" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} r_u(x):= F_U(x-u|u) - G_{\xi ,\sigma }(x-u) \rightarrow 0,\quad u\rightarrow \infty \end{aligned}$$</span></div></div><p>uniformly in <span class="mathjax-tex">$x>u$</span> and that, for <span class="mathjax-tex">$x= VaR_\beta (U)$</span>, we have</p><div id="Equ58" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{G_{\xi ,\sigma }(VaR_\beta (U)-u)}{F_U(VaR_\beta (U)-u|u)} = 1 - \frac{1 - F_U(u)}{\beta - F_U(u)} r_u(VaR_\beta (U)), \end{aligned}$$</span></div></div><p>and</p><div id="Equ59" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} r_u(VaR_\beta (U)) \rightarrow 0, \end{aligned}$$</span></div></div><p>as <span class="mathjax-tex">$u\rightarrow \infty $</span> and <span class="mathjax-tex">$\beta \rightarrow 1$</span>. Here we note that, under (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ13">A.3</a>) with <span class="mathjax-tex">$u=u(\alpha )$</span>,</p><div id="Equ60" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 0< \frac{1 - F_U(u)}{\beta - F_U(u)} =\frac{1}{ 1 - \frac{1-\alpha }{1 - F_U(u)}\frac{1}{{\mathbb {E}}[N]} } = O(1)\quad (\alpha \rightarrow 1), \end{aligned}$$</span></div></div><p>which implies that</p><div id="Equ61" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{G_{\xi ,\sigma }(VaR_\beta (U)-u)}{F_U(VaR_\beta (U)-u|u)} = 1 -O(1)\cdot r_u(VaR_\beta (U)) \rightarrow 1 \end{aligned}$$</span></div></div><p>as <span class="mathjax-tex">$u=u(\alpha )$</span> and <span class="mathjax-tex">$\alpha \rightarrow 1$</span>. Therefore, we see that</p><div id="Equ62" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F_U(VaR_\beta (U)-u|u) \sim G_{\xi ,\sigma }(VaR_\beta (U)-u) \end{aligned}$$</span></div></div><p>under (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ13">A.3</a>). Substituting <i>x</i> with <span class="mathjax-tex">$VaR_\beta (U)$</span> in both sides of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ15">A.5</a>), we have that</p><div id="Equ63" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \beta \sim \overline{F}_U(u) G_{\xi ,\sigma }\left(VaR_\beta (U) - u\right)+ F_U(u), \quad \alpha \rightarrow 1, \end{aligned}$$</span></div></div><p>which concludes that</p><div id="Equ64" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} VaR_\beta (U) \sim u(\alpha ) + G^{-1}_{\xi ,\sigma }\left(1 - \frac{1-\beta }{\overline{F}_U(u)}\right) \sim u + \frac{\sigma }{\xi } \left\{ \left( \frac{\overline{F}_U(u)}{1-\beta } \right) ^\xi - 1\right\} ,\quad \alpha \rightarrow 1, \end{aligned}$$</span></div></div><p>with <span class="mathjax-tex">$u= u(\alpha )$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s42081-024-00273-y#Equ13">A.3</a>). <span class="mathjax-tex">$\square $</span></p> </div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Shimizu, Y., Takagami, Y. Utility of classical insurance risk models for measuring the risks of cyber incidents. <i>Jpn J Stat Data Sci</i> <b>7</b>, 1059–1084 (2024). https://doi.org/10.1007/s42081-024-00273-y</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s42081-024-00273-y?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-11-06">06 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