An Analytical Review of Cyber Risk Management for Insurers

Executive Summary

This briefing document synthesizes a review of the mathematical models and quantitative tools used by insurance companies for cyber risk management. The analysis reveals that cyber risk presents a unique and complex challenge due to several inherent factors: a lack of reliable historical data, its dynamic and evolving nature, the strategic intent of threat actors, and the high degree of interconnectivity that creates systemic and accumulation risks.

Key takeaways include:

• Data Scarcity and Complexity: The reluctance of firms to disclose incidents and the rapid evolution of threats severely hamper the development of robust loss models. Cyber risk is characterized by heavy‑tailed loss distributions, where rare but catastrophic events can cause massive economic damage.
• Modeling Interdependence: Standard insurance models that assume independence between risks are inadequate. A single vulnerability can trigger cascading failures across sectors, necessitating advanced techniques like copula models (including elliptical, Archimedean, and vine copulas) to accurately capture the complex dependency structures and avoid underestimating capital requirements.
• Regulatory Framework: European regulators, notably EIOPA, are pushing for greater resilience. Insurers are now required to integrate cyber scenarios into stress tests, addressing both their own operational vulnerabilities and the underwriting risks from their cyber insurance portfolios. Accurately calculating the Solvency Capital Requirement (SCR) is a central challenge.
• Pricing and Product Innovation: Traditional pricing principles are being adapted to the unique characteristics of cyber risk. Models now incorporate Value‑at‑Risk (VaR), Tail Value‑at‑Risk (TVaR), and higher‑order statistical moments (skewness, kurtosis) to better account for extreme events. Innovations like Bonus‑Malus systems and incident‑specific policies aim to mitigate moral hazard and information asymmetry, which remain significant barriers to market efficiency.
• Persistent Challenges: Despite advancements, the market is hindered by information asymmetry, adverse selection, and moral hazard. Standardizing policy terms and improving data‑sharing initiatives are critical for the sustainable growth of the cyber insurance market. Future research must focus on adaptive pricing, hybrid models combining statistical and engineering approaches, and new reinsurance structures like cyber catastrophe bonds.

The Evolving Landscape and Unique Characteristics of Cyber Risk

The management and transfer of cyber risk are complicated by a unique combination of factors that distinguish it from traditional insurable risks. These challenges form the foundational context for the development of quantitative management tools.

• Lack of Reliable Data: A primary obstacle is the scarcity of robust historical data on cyber incidents. This is exacerbated by the reluctance of companies to disclose breaches due to fears of reputational damage, making it difficult to build accurate loss models (Zängerle and Schiereck 2023).
• Dynamic and Evolving Nature: Cyber risk is inherently dynamic, constantly evolving with technological innovation, regulatory changes, and the pace of digital transformation. This creates a continuous stream of new vulnerabilities (Georgescu 2021; Smith and Lostri 2020). The rise of AI, smart cities, and insurtech tools both creates new efficiencies and introduces new risks (Ratnawat 2025; Zraqou et al. 2025).
• Intentional and Strategic Threats: Unlike many traditional risks, cyber incidents are often deliberate acts carried out by strategic adversaries motivated by economic or political gain. These actors are supported by a mature cybercrime ecosystem that provides tools and services on demand (Abou El Houda 2024; Munk 2022).
• Interconnectivity and Accumulation Risk: The high degree of interconnectivity between IT systems creates significant interdependence and accumulation risks. A single vulnerability can lead to cascading effects across networks and industries (Kröger 2008; Maglaras et al. 2018). This interconnectedness also generates negative externalities, where underinvestment in security by one entity compromises the resilience of the entire network.
• Information Asymmetry: Challenges such as adverse selection and moral hazard hinder the effectiveness of cyber insurance solutions. Insurers struggle to accurately assess the security posture of potential policyholders, and insurance coverage may disincentivize security investments (Franke 2017; Pal et al. 2025).
• Dual Exposure for Insurers: Insurance companies face a twofold vulnerability. They are directly exposed as users of IT systems and are indirectly exposed through the underwriting risk associated with their cyber insurance products (European Insurance and Occupational Pensions Authority 2023).
• Non‑Affirmative Risk: A significant concern is "non‑affirmative" or "silent" cyber risk, where traditional insurance policies (e.g., property, liability) may inadvertently provide coverage for cyber events without explicitly pricing for them. This can lead to unexpected loss accumulations.

According to Allianz Commercial (2025), cyber risk is the top concern for companies, a position solidified by the interdependence of risks and an increase in attacks fueled by AI technologies.

A Brief History of Cyber Risk Quantification

Early models for cyber risk analysis focused on assessing organizational vulnerability and quantifying potential losses. These foundational concepts have evolved into more sophisticated stochastic and dynamic approaches.

Vulnerability Functions

The seminal work by Gordon and Loeb (2002) introduced the vulnerability function, which expresses the probability of an information compromise as a function of investment in security measures. A key insight from their model is that it is never economically rational to invest more than 37% of the expected loss value to protect an information asset. Subsequent researchers have proposed alternative vulnerability models to capture different aspects of security investment effectiveness.

Frequency‑Severity Models

A common approach to modeling cyber attacks is the frequency‑severity framework. Research indicates that the severity of losses from cyber incidents follows a heavy‑tailed distribution, meaning that while most events cause minor damage, a small number of extreme events can lead to catastrophic losses.

Maillart and Sornette (2010) analyzed a dataset of cyber incidents and found that economic damages follow a power‑law distribution: P(X > x) ∼ x⁻α. They estimated the exponent α ≈ 1.7, which implies a theoretically infinite variance and a highly unstable mean. This finding underscores the inadequacy of traditional risk assessment methods based on means and standard deviations for modeling cyber risk.

Other studies have used techniques such as K‑means clustering, logistic regression, Fourier series expansions, and differential equations to model incident severity, predict attack likelihood, and quantify system resilience (Peters et al. 2017; Sobchuk et al. 2023; Weisman et al. 2025).

Cyber Insurance: Regulation, Interdependence, and Products

The growing cyber insurance market has attracted significant regulatory attention, with a focus on managing the systemic nature of the risk and refining insurance products to address market inefficiencies.

Managing Interdependence in Insurance Regulation

The European Insurance and Occupational Pensions Authority (EIOPA) has been instrumental in shaping the regulatory landscape.

• EIOPA Surveys (2018, 2019): Early investigations revealed an embryonic market heavily reliant on qualitative methods for pricing and risk assessment. Key limitations identified were a lack of claims data, the need to quantify non‑affirmative risks, and significant variability in how insurers calculate their Solvency Capital Requirements (SCR).
• EIOPA Framework (2023): In line with the Digital Operational Resilience Act (DORA), EIOPA defined a framework for assessing insurer resilience. This requires insurers to conduct stress tests for adverse cyber scenarios, considering both their own cyber resilience and the cyber underwriting risk from their portfolios.

The Solvency II framework allows SCR to be calculated via a standard formula or an internal model. A critical flaw in standard approaches is the "fallacy of independence"‑treating risks as independent leads to a dangerous underestimation of aggregate risk.

To address this, copulas have become a crucial tool for modeling the dependence between cyber events. Sklar's theorem provides the theoretical basis for separating the marginal distributions of individual risks from their joint dependence structure.

• Elliptical Copulas (Gaussian, t‑copula): Model symmetric, linear dependence structures. The t‑copula is particularly useful as it can capture tail dependence, reflecting the reality that extreme events are often correlated.
• Archimedean Copulas (Gumbel, Clayton, Frank): Offer more flexibility to model asymmetric dependencies, such as left‑tail (Clayton) or right‑tail (Gumbel) dependence.
• Vine Copulas: Decompose a multivariate dependence structure into a series of bivariate copulas, allowing for highly flexible and accurate modeling of complex relationships between different types of cyber breaches (Carannante et al. 2023).

Cyber Insurance Products and Market Challenges

Despite growing demand, the expansion of the cyber insurance market is limited by several structural barriers and market failures.

• Barriers to Growth: The Organisation for Economic Co‑operation and Development (OECD) identified scarcity of historical data, regulatory uncertainty, and the difficulty of modeling systemic risk as major impediments (OECD 2017, 2020).
• Moral Hazard and Information Asymmetry: A core challenge is that insurance coverage can discourage policyholders from investing in security (moral hazard). Furthermore, insurers lack complete information about a client's security posture, leading to market inefficiencies (Shetty et al. 2010).
• Product and Pricing Heterogeneity: An analysis by Romanosky et al. (2019) of over 100 cyber insurance policies found significant heterogeneity in pricing. Many policies used a flat‑rate model based on revenue or asset value, ignoring actual security controls.
• Innovative Contract Structures:Researchers are proposing more dynamic and sophisticated contract models to overcome these limitations:
  • Marked Point Process: Zeller and Scherer (2022) proposed a model that represents cyber events as a sequence of marked points, capturing arrival time, severity, and the set of breached firms, allowing for dynamic pricing based on interdependent risk.
  • Bonus‑Malus System: Xiang et al. (2024) introduced a dynamic framework where premiums are updated based on claims history, using a Markov transition model to incentivize mitigation measures and reduce moral hazard.
  • Two‑Pillar Framework: Chong et al. (2025b) developed a framework combining cyber risk assessment (using frequency‑severity and cascade models) with cyber capital management to help firms optimally allocate capital between security investments, insurance, and reserves.

Cyber Reinsurance

Reinsurance is a critical tool for primary insurers to manage their exposure to large‑scale cyber events. Given that a single attack can affect thousands of policyholders simultaneously (e.g., an attack on a major cloud provider), the risk of accumulation is too great for a single insurer to bear. Reinsurance allows for the pooling and diversification of this risk across a broader capital base. However, reinsurers are cautious due to the same challenges facing primary insurers: lack of data, high uncertainty, and the potential for catastrophic losses. As an alternative, cyber catastrophe bonds (cat bonds) have been proposed to transfer cyber risk to capital markets, potentially offering a solution to capacity constraints (Mastroeni et al. 2022).

Mathematical Models for Insurance Pricing

Accurately pricing cyber insurance premiums is a central challenge. Various mathematical principles have been proposed, moving beyond simple expected value to account for the extreme volatility and dependencies inherent in cyber risk.

• Expected Value Premium Principle: The simplest model, where the pure premium is proportional to the expected loss: PP = (1 + λ)E[X]. This is considered unsuitable for cyber risk because it fails to account for dependencies and high variance (Lau et al. 2020).
• Value‑at‑Risk (VaR) and Tail Value‑at‑Risk (TVaR) Principles:These principles set the premium based on the tail of the loss distribution.
  • VaR Premium: PP = VaRα(X), setting the premium at a level where the probability of loss exceeding it is a small value α.
  • TVaR Premium: PP = TVaRα(X), which calculates the average loss in the tail beyond the VaR threshold, providing a more conservative premium.
• Mean‑Variance and Standard Deviation Principles:These models add a risk loading based on the variance or standard deviation of the loss distribution.
  • Mean‑Variance: PP = E[X] + (δ/2)V[X] (Lin et al. 2018).
  • Standard Deviation: PP = E[X] + δ√V[X] (Antonio et al. 2021).
• Fourth‑Order Statistics Principle: To better capture the shape of heavy‑tailed and asymmetric distributions, Naldi and Mazzoccoli (2018) proposed using higher‑order moments like skewness (S[X]) and kurtosis (K[X]): PP = E[X] + (δ/2)V[X] + (δ²/6)S[X]V^(3/2)[X] + (δ³/24)K[X]V²[X] This provides a more accurate representation of extreme risks but is more difficult to estimate reliably.

A summary of these fundamental principles is provided below.

Conclusion and Future Research Directions

The review demonstrates that while significant progress has been made in developing mathematical models for cyber risk, critical challenges persist. The lack of standardized data‑sharing practices continues to hinder the development of precise pricing models and limits market efficiency. Open questions remain regarding the accurate measurement of evolving threats, management of interconnected risks, and mitigation of information asymmetries.

Future research should focus on:

• Developing adaptive pricing systems and innovative actuarial models that can better capture the dynamic, systemic nature of cyber risk.
• Exploring hybrid models that integrate technical, engineering‑based insights (e.g., threat intelligence, network topology) with statistical and actuarial approaches.
• Investigating new forms of reinsurance structures and alternative risk transfer mechanisms like cyber cat bonds.
• Leveraging advanced data analytics and artificial intelligence to enhance risk prediction and underwriting processes.
• Designing regulatory frameworks that encourage information sharing and foster market resilience.

Closer collaboration between researchers, insurers, and policymakers is crucial to overcome these challenges and ensure the sustainable growth of the cyber insurance market as an effective risk management tool in an increasingly digital world.