Valuation of Cyber Catastrophe Bonds and Their Role in Portfolio Efficiency: An Analysis of Model Selection and Investment Implications

Valuation of Cyber Catastrophe Bonds and Their Role in Portfolio Efficiency: An Analysis of Model Selection and Investment Implications | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Valuation of Cyber Catastrophe Bonds and Their Role in Portfolio Efficiency: An Analysis of Model Selection and Investment Implications Niklas Alexander Anders This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7923044/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study evaluates and compares five pricing models for cyber catastrophe (cat) bonds—the Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach, and Copula-POT Model—to identify the most effective method for loss prediction and portfolio optimization under Modern Portfolio Theory (MPT). Using CISSM data, the Copula-POT Model shows the highest predictive accuracy and robustness, making it the preferred framework despite a zero trigger probability due to limited extreme value data. Integrating cyber cat bonds priced by this model into an MPT-optimized portfolio improves diversification and risk-adjusted returns, outperforming traditional high-yield bonds. The study highlights challenges including data scarcity, parameter sensitivity, and model uncertainty, and proposes hybrid modeling and data enrichment as directions for future research. Overall, these findings emphasize the potential of cyber cat bonds as an innovative asset class and an effective tool for cyber risk transfer in capital markets. Cyber cat bonds Copula-POT Model Modern Portfolio Theory (MPT) portfolio optimization tail risk pricing models Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 1. Introduction In recent years, technological advancements have become deeply integrated into global business, transforming industries into the digital economy and reshaping corporate value through intangible assets such as data, algorithms, intellectual property, and organizational expertise. The rapid development of artificial intelligence (AI) and automation has accelerated innovation and efficiency while simultaneously increasing exposure to cyber risk. Cyber threats, including data breaches, ransomware attacks, and system failures, pose significant operational and financial challenges. Global cybercrime costs are projected to reach $ 10.5 trillion annually by 2025 from $ 3 trillion in 2015 (Cybersecurity Ventures 2020) and may rise to $ 15.63 trillion by 2029 (Statista 2025 ), reflecting an approximate 15% compound annual growth rate. In the United States alone, cybercrime losses grew by 22% in 2023 and surged by 33% in 2024 (FBI 2023, 7; FBI 2024, 4), averaging nearly 27% growth during this period. As the frequency and severity of cyber incidents increase, the need for robust mitigation mechanisms has become urgent, fueling the rapid growth of the cyber risk insurance market. Despite this, the global premium volume for cyber insurance remains low, standing at $ 15.3 billion in 2024 (Munich Re 2025), highlighting a substantial coverage gap and exposing financial vulnerabilities. Innovative solutions such as cyber catastrophe bonds, or cyber cat bonds, have emerged as instruments to transfer cyber risk to capital markets. Over the past 15 years, capital market participation in bearing insurance risks has grown (Braun et al. 2023 , 684). Nevertheless, despite $ 18.5 billion in cat bond issuance in the first half of 2025, cyber risk accounted for only 1.4% of the total $ 56.1 billion in risk capital outstandings (Artemis.bm 2025), indicating significant untapped potential. Unlike traditional cat bonds covering natural disasters, cyber cat bonds aim to cover financial losses from high-impact cyber events, yet they face substantial challenges. These include high risk premiums, uncertainties in cyber risk modeling, lack of historical loss data, extreme tail risks, and limited familiarity among market participants (Woods and Wolff 2023, 14; Kolesnikov et al. 2022, 1). The inherent volatility of cyber risks, coupled with the potential for interconnected losses such as widespread ransomware attacks or supply chain disruptions, complicates accurate assessment and pricing. Despite these obstacles, the strong returns of traditional cat bonds over the past decade suggest that cyber cat bonds could become attractive capital instruments if modeling and market challenges are addressed (Willard 2025 ). Existing research proposes various pricing models, but comparative evaluations of predictive validity and real-world applicability remain scarce. This study seeks to systematically assess multiple models to determine effective strategies for valuing cyber cat bonds and to explore implications for portfolio management. Specifically, it evaluates and compares pricing models, identifies the most predictive model based on accuracy, robustness, and relevance, and applies the results to Modern Portfolio Theory to optimize risk-adjusted returns. The investigation also examines the potential of cyber cat bonds to enhance portfolio performance through diversification, owing to their low correlation with traditional assets (S&P Global Ratings 2025; Mouelhi 2021 , 82), and explores investment opportunities arising from market inefficiencies, such as pricing discrepancies. The study contributes to bridging theoretical developments in cyber cat bond modeling with practical portfolio applications. By comparing pricing models and integrating their outputs into portfolio optimization techniques, it provides a comprehensive understanding of cyber cat bonds as instruments for both risk transfer and investment, offering valuable insights for investors, insurers, and the growing cyber risk insurance market. 2. Literature Review 2.1 Cyber Risk and Cyber Catastrophe Bonds The rise of the digital economy has elevated cyber risk as a critical concern for organizations worldwide, encompassing potential adverse outcomes from cyberattacks, system vulnerabilities, and human errors that compromise confidentiality, integrity, or availability of information systems. Threats such as data breaches, ransomware attacks, distributed denial-of-service (DDoS) attacks, and phishing present dynamic challenges (Carter 2018 , 30–31). Unlike traditional insurable risks, cyber risk is broad and rapidly evolving due to technological advancements like AI and machine learning (Chimamiwa 2024 , 71). The accelerated digital transformation, particularly during the COVID-19 pandemic, has expanded vulnerabilities, providing new opportunities for cybercriminals (Chigada and Madzinga 2021, 1). Cyber incidents can generate systemic losses, spreading across companies, sectors, and even global relationships, causing financial, operational, and reputational damage (Mastroeni et al. 2023, 15). Data breaches now cost an average of $ 4.88 million, a 10% increase year-over-year (IBM 2024 ). The cyber risk insurance market has grown to mitigate these financial consequences, yet remains underinsured, with a global premium volume of $ 15.3 billion projected to reach $ 16.3 billion in 2025 (Munich Re 2025), far below the expected $ 10.5 trillion annual cost of cybercrime by 2025, rising to $ 15.63 trillion by 2029 (Munich Re 2025; Cybersecurity Ventures 2020; Statista 2025 ). Catastrophe bonds, traditionally used to transfer natural disaster risks to capital markets, offer a model for cyber risk transfer, allowing insurers to shift extreme loss exposures to investors in exchange for high risk-adjusted returns (Cummins and Weiss 2009, 1). Cyber cat bonds adapt this structure to the unique characteristics of cyber risks. Global issuance reached $ 533.75 million in 2024, representing only 1.4% of total cat bond risk capital (Artemis.bm 2025). Challenges remain, including difficulties in modeling cyber risks, lack of historical data, information asymmetry between insurers and investors, and trigger design limitations (Orlando et al. 2018 , 2; Skeoch and Ioannidis 2023, 25; Cummins and Weiss 2009, 42). Table 1 : Comparison between Trigger Types of Cyber Cat Bonds illustrates the differences between parametric and indemnity triggers and highlights the trade-offs between accuracy and payout speed that must be considered in the design of cyber cat bonds. Pricing models often rely on statistical and machine learning approaches but are constrained by data scarcity and potential biases (Dubois et al. 2022 , 47). Despite these issues, cyber cat bonds offer growing potential for risk management and investment, particularly as the cyber landscape evolves (S&P Global Ratings 2025). Table 1 Comparison between Trigger Types of Cyber Cat Bonds Trigger Type Advantages Limitations Cyber-Specific Challenges Parametric Fast payout, low basis risk for defined events May not fully reflected actual losses Higher basis risk due to unpredictable attack vectors Indemnity Directly tied to insured losses Validation delays, moral hazard Difficult validation in opaque Incidents; potential for asymmetric information 2.2. Pricing Models for Cyber Catastrophe Bonds This section focuses on the pricing solutions of cyber cat bonds and conducts a comprehensive review and analysis of the pricing framework from Kolesnikov et al. (2022), Domfeh et al. ( 2022 ), Lane ( 2000 ), Li and Mamon (2023), and Tang et al. ( 2023 ). A systematic comparison of these models is conducted to understand their approaches, underlying assumptions, strengths, and limitations, and this comparison offers strong and direct evidence on the theoretical basis of the present study, which also provides a theoretical foundation for subsequent empirical studies. Kolesnikov et al. (2022) propose a pricing model based on loss distribution that focuses on modeling the expected loss and risk premium of cyber cat bonds. Using a cyber events database published on the public domain, they estimate parameters for the cyber loss distribution and calculate bond prices, yields, and other features by numerical simulation. The cyber losses are assumed to share a heavy-tailed distribution (e.g., Pareto distribution) to account for the extreme nature of cyber event losses. Monte Carlo simulations are employed by the model to generate a high number of loss scenarios and to simulate the expected loss and the likelihood of the bond being triggered based on these scenarios. Kolesnikov et al. emphasized that, due to the diversity of types of cyber loss, the model should account for the joint distribution of various types of loss (such as direct economic loss and reputational losses) for a more realistic characterization of the risk exposure of the bond. The strength of the model lies in its capacity to adapt to different types of cyber events and triggering conditions. But it does have its drawbacks. The model is highly sensitive to the parameters of the loss distribution. The limited historical data make the parameter estimation more uncertain. Moreover, the computation cost of Monte Carlo simulation is very high, and it may constrain the practical usage of Monte Carlo simulation. Nonetheless, this model was selected due to the ability to handle the heavy-tailed nature that is prevalent in cyber losses, hence filling the lack of historical data that is related to extraordinary events; additionally, it is among the few models that are specifically focused on cyber bonds and offers an intuitive approach, making it even more relevant and providing a foundational tool to value within this emergent market. Domfeh et al. ( 2022 ) propose a unified Bayesian framework for pricing cat bond derivatives and that is specifically designed for handling the associated uncertainty involved in catastrophic risks. This model utilized a Dirichlet Prior-Hierarchical Bayesian Collective Risk Model (DP-HBCRM) to estimate the frequency and severity of insurance loss events. It is extended by adding a stochastic interest rate model to separate the risk premium, and it is thus capable of valuing cat bonds, including cyber cat bonds. The DP-HBCRM formulates the loss process as a compound Poisson model in which the number of events is a Poisson distribution with a rate parameter, loss values are governed by a distribution with heavy tails (e.g., lognormal), which accounts for the extremeness of cyber losses. Estimation of the parameter rests on Bayesian inference and a prior of the form of a Dirichlet distribution used for capturing event frequency. With this approach, the model can reflect a prior of current knowledge of event frequencies and use it to update the prior with observed data. This renders the model especially useful in situations where the history of data is limited, as is often the case in the context of cyber risk assessment. In addition, the model features a stochastic interest rate setting, usually a Vasiček process. Incorporation of the market action helps determine the risk premium such that realized bond prices reflect expected losses plus investors' views of risks. The strength of this framework is its applicability to consolidating heterogeneous data, namely industry reports on cyber events and public repositories and databases of incidents, to improve the prediction accuracy of loss. This flexibility is important in cyber risk, where little historical loss information exists. However, the model’s need of subjective prior distribution injects subjectivity into identifiability, which possibly would influence the parameter estimates’ stability. Furthermore, the computational cost of Bayesian inference, in particular with methods based in Markov Chain Monte Carlo (MCMC) simulations does not easily allow its practical use. Notwithstanding these limitations, this model was chosen for its effectiveness in uncertain environments, such as evolving cyber threats with limited prior data. It provides an organized and theoretically valid methodology for pricing cyber cat bonds, offering a platform for pricing activities in a new and data-limited market. Li and Mamon (2023) propose a new method to price network risk, rooted in signal processing. They treat network attack events as random signal sequences in their model. These sequences are modeled by Hidden Markov Model (HMM) and Expectation Maximization (EM) techniques. The model assumes periodic and trend behavior of network losses and it uses a non-homogeneous Regime-Switching Markov Model (RSMM) to detect both: first, the time at which network attacks happen, and second, the interval and duration of the attacks. The state transition modeling reflects three phases in the Cyber Kill Chain (CKC): firewall normal, firewall failure, and anti-phishing failure. The loss distribution is assumed to be of Doubly-Truncated Pareto Distribution and the discount rate is modeled by the Vasiček process. Based on the reference measure transformation and the filtering technique, it keeps the quadratic payment factor and stochastic discount factor both dynamic which is the desired form for the parameter estimation and premium evaluation in standard deviation and index premium principles, respectively. The model's innovation lies in its integration of signal processing techniques into the pricing of cyber risk, which is especially appropriate for including the time-dependent and dynamic nature of cyberattacks. This method naturally complements the traditional statistical techniques. Being explicit about cyber risks, it is also simultaneously a more suitable risk model for pricing cyber cat bonds. But the fact that it presupposes stability and cyclicality in network losses may not hold for cyber risk. Regardless, this approach was considered due to the emphasis on filtering noise out of incomplete cyber event reports, designed specifically for cyber risks, to complement predictive models. Its dynamic modeling suits the time-dependent nature of cyberattacks, offering a novel approach for cyber cat bond pricing, provided high-quality time series data becomes available. Lane ( 2000 ) proposes a general framework that aims to provide a systematic pricing methodology for risk transfer transactions. The model employs a regression analysis to estimate bond pricing parameters based on expected loss (EL), risk load, and market risk premium. It is centered on probability of first loss (PFL) and conditional expected loss (CEL) and decomposes the bond price into three components: Risk-Free Interest Rate (RFIR), Expected Loss (EL), and Risk Premium (RP). Lane emphasizes that pricing models have to take into account market dynamics, investor risk appetite and the asymmetric distribution of risk events so that the bond is attractive and saleable in the capital markets. While the model was originally created to estimate risks associated with natural disasters, such as earthquakes and hurricanes, its theoretical framework is quite general. This flexibility makes it suitable for use in cyber cat bonds. The simple and efficient structure can be conveniently applied in data-poor and immature market conditions, as in the cyber cat bond market at present. But its simplifying assumptions — such as risk premium being in a linear relationship to other, risk premium, itself – may not adequately capture cyber risk’s complex natures. These natures consist of systematic dependence, nonlinear properties of multi-event triggering mechanisms, and a very high level of tail risk. Moreover, the model has greater reliance on market data, and the scarce historical data of the cyber cat bond market further exacerbates parameter estimation uncertainty. Nevertheless, Lane's framework was selected to explore causal relationships between cyberattack variables and losses, enhancing interpretability in interconnected systems through empirical fitting of spreads. It can be considered a cornerstone of the price of cyber cat bonds. It breaks down the drivers of cyber risk to help investors and issuers understand the logic that underpins pricing cyber risk, which can be challenging in nascent markets. Tang et al. ( 2023 ) propose a pricing framework for multi-event triggered cat bonds, particularly for those associated with high-dependence structures. The model combines the Extreme Value Theory (EVT) Peaks-Over-Threshold (POT) methodology with a nested Archimedean Copula to describe the tail behavior of multi-trigger indicators and complex dependencies. It also analyzes the sensitivity of bond prices through Monte Carlo simulations. The model integrates the POT approach with the Generalized Pareto Distribution (GPD) to model tail risk and relies on the nested Archimedean Copula to estimate the nonlinear dependencies among indicators. This embedding layer can help the model effectively deal with the multiple event triggers as well. Tang et al. emphasize that the pricing model should reflect the asymmetric in the distribution of triggering indicators, as well as the dynamics of the frequency and the size of catastrophes and the risk appetite of market participants. This will help to improve the market attractiveness of bonds. While this model was not originally designed to address cyber risks, its methodology aligns well with the pricing of cyber cat bonds. This model's strength lies in its ability to model tail dependencies, particularly suited for cyber risks where multi-event triggers create correlated extremes, as the nested Archimedean copula captures asymmetric joint tails more effectively than independent assumptions in other models. It stands out in capturing joint distributions of rare events, offering superior accuracy in pricing high-impact, correlated losses and it is also why this model was chosen. However, the noted limitation in capturing systematic correlations suggests future research could integrate dynamic Copula structures to better reflect evolving cyber risk dependencies. Modern Portfolio Theory (MPT) provides a systematic framework to optimize the balance between risk and return in investment portfolios. Developed by Harry Markowitz in the 1950s, MPT enables investors to construct portfolios that maximize expected returns for a given risk level or minimize risk for a desired return through strategic asset selection. Its foundation, the Mean-Variance Model, formalizes portfolio risk using the standard deviation of returns and a covariance matrix capturing asset correlations, producing the Efficient Frontier of optimal risk-return combinations. Extensions such as Tobin’s Separation Theorem and Sharpe’s Capital Asset Pricing Model (CAPM) further refine the treatment of risk, introducing concepts like systematic risk and emphasizing diversification. In multi-asset portfolios, low correlation between assets reduces total risk, a principle particularly relevant for insurance-linked securities like cat bonds (Cummins and Weiss 2009, 439). Cyber cat bonds share the same return-risk structure as traditional cat bonds, but their risk stems from events such as data breaches or ransomware attacks, which exhibit volatile and systemic loss correlations (Kolesnikov et al. 2022, 2). Their low correlation with conventional assets makes them suitable for diversification, yet limited data and price uncertainty complicate risk assessment (Braun et al. 2023 , 1). While MPT assumes normally distributed returns and relies on historical data—assumptions challenged by the heavy-tailed, scarce data of cyber cat bonds (Mastroeni et al. 2022, 2; Woods and Wolff 2023, 3)—it still provides a structured framework to compute risk-adjusted returns, such as the Sharpe Ratio, and optimize portfolio efficiency. Integrating expected returns, risks, and correlations of cyber cat bonds allows investors to determine optimal allocations and evaluate their role in multi-asset portfolios alongside high-yield bonds. 2.3 Research Gaps As a new risk transfer tool, cyber cat bonds have gained significant attention in both theory and practice. However, this review uncovers persistent unresolved gaps that hinder their full development. It is evident from the literature that there are some academic and application issues that have not been solved, and different viewpoints on how to model and price cyber risks, such as the loss distribution approach from Kolesnikov et al. (2022), the Bayesian models from Domfeh et al. ( 2022 ), and the Copula-POT framework from Tang et al. ( 2023 ). Yet, most of these contributions focus only on refining and applying a single model, without undertaking a broader comparative analysis across multiple frameworks to gauge their predictive accuracy. This gap limits the ability to identify the most suitable pricing strategy and, in turn, undermines the precision of cyber cat bond valuations in practice. Moreover, previous research is limited in its application of pricing model results to the optimization of investment portfolios. Despite a widespread use of MPT in examining the diversification effects of cat bonds, the available literature about the investment implications of cyber cat bonds, particularly the specific influence of their returns, risks, and correlations on portfolio efficiency, remains disparate, comparing only isolated case studies rather than an integrated framework. Mastroeni et al. (2022) emphasize that the systematic nature of cyber risk combined with the lack of data increases the complexity of investment decisions. However, there exist few studies have unified pricing models with MPT to investigate comprehensive investment strategies or arbitrage positions of cyber cat bonds, including arbitrage positions that could exploit pricing inefficiencies in this nascent market. For example, the lack of historical and standardized loss data contributes to elevated spreads in the cyber cat bond market. Therefore, investors have opportunities to capitalize on pricing inefficiencies and potential misevaluations. Building on these gaps, the review establishes a foundation for pricing and investment strategies. Yet, no detailed systematic comparison and empirical validation exist. This study addresses these shortcomings by systematically comparing five different pricing models, determining the most predictive model based on accuracy, robustness and relevance. Then integrating these findings with MPT to access cyber cat bonds’ role in investment portfolios. This approach, as outlined in the methodology (Section 3 ), is designed to deliver actionable guidance for investors, including strategies to leverage market inefficiencies for arbitrage or diversification benefits. Thus, it advances both the theoretical understanding of cyber risk modeling and practical portfolio management in this evolving field. 3. Methodology 3.1. Data Collection The study utilizes a standardized cyber loss event dataset to ensure consistency and comparability across five pricing models, addressing the data scarcity challenge highlighted and enabling fair cross-model evaluation. The primary source is the Cyber Events Database maintained by the Center for International and Security Studies at Maryland (CISSM), which collects publicly available cyber events from 2014 to 2025. The database includes structured data with multiple columns such as event date, actor, organization, industry and event type. This database was selected as the most comprehensive free and publicly accessible resource from a reputable academic institution and has also been used by big organizations like European Central Bank or Bank of Japan. Paid databases with fuller loss data exist (e.g., Advisen), but their inaccessibility due to student research constraints is discussed. Data were extracted from the CISSM portal on April 15, 2025. The time window selected for the study is 2023 to 2024. It is because of the rapid evolution of cyber events, which makes the recent data more pertinent to the current situation. Although the CISSM dataset provides qualitative information on cyber events, it lacks the loss amount. Therefore, the first action taken was a screening of samples designed to bring out quantifiable losses suitable for modeling. I excluded events that are non-measurable or of zero loss, such as protest attacks or political-espionage. These non-measurables are of approximately 17% based on manual checking. This filtering enhances the dataset’s suitability for pricing models since events without loss amounts cannot be integrated into the pricing frameworks. However, this decision may cause a problem of selection bias that may skew the dataset towards greater-impact events. The following events were evaluated using a hybrid approach designed to maximize accuracy: (1) Primary method: Direct losses were sourced from SEC 8-K filings, company reports or credible news reports. (2) Secondary method: When direct data could not be obtained, category-specific proxies were used. For data breaches, loss amounts were estimated as the number of records stolen or compromised (from public sources like www.bleepingcomputer.com or www.therecord.media ) multiplied by an average loss per record based on the IBM data cost of in a breach report ( $ 165 per record in 2023 and $ 169 in 2024). Mega breaches were those with more than a million data records stolen. There is also an associated average cost of such an event in the IBM report. For DDoS attacks and service outages, loss amounts were calculated in terms of the length of downtime times the most recent annual revenue of the affected company, based on publicly available incident details. This multi-formation approach, aligning with industry standards, as similar methodologies are used in reports like Verizon's 2025 Data Breach Investigations Report (DBIR), which analyzes over 22,000 incidents, was designed to make the dataset solid and consistent. However, it may also underrepresent indirect losses, such as customer attrition, or simplify the appropriate the of certain financial effects (Braun et al. 2023 , 685). Despite these limitations, this unified dataset will be a strong foundation to test pricing models and estimate portfolio inputs, while the limitations give emphasis on cautious interpretation of the outcomes. 3.2. Overview of Pricing Models Given the theoretical foundation of cyber cat bond pricing models, the following sections present implementation details of the five pricing models with the standardized CISSM dataset to ensure consistency. These models include Loss Distribution Framework from Kolesnikov et al. (2022), the Unified Bayesian Framework from Domfeh et al. ( 2022 ), Signal-Processing Approach from Li & Mamon (2023), Regression Approach by Lane ( 2000 ) and Copula-COT Model from Tang et al. ( 2023 ) and represent diverse approaches to cyber loss prediction, from frequency-severity decomposition to tail dependence modeling. Each model uses the same cyber loss event dataset (2023–2024) for parameter estimation, assumptions and computational methodology adapted to allow for data constraints and comparisons across models. The implementations of each model are described in the subsections. Through systematic application of these models, I aim to determine the most accurate and reliable method for the valuation of cyber cat bond as described in the research objectives. 3.2.1. Implementation of Loss Distribution Framework The pricing model for cyber cat bonds proposed by Kolesnikov et al. (2022) uses a loss distribution framework to derive bond prices, yields, and trigger probabilities. This methodology incorporates a Pareto distribution to account for the heavy-tailed structure of cyber losses and employs Monte Carlo simulations to produce loss realizations. First, it begins by calibrating the cyber loss distributions, which are the key to this framework. Cyber event frequency is modeled with an exponential distribution for inter-event times, with the cumulative distribution function: $$\:P\left({\tau\:}_{k,r}\le\:y\right)=\:{F}_{k}\left(y\right)={F}_{k}\left(y;\theta\:\right)=1-{e}^{-\lambda\:y}$$ with \(\:{\tau\:}_{k,r}\) is the time interval between the incident \(\:r\) and \(\:r+1\) of type k, and \(\:\theta\:=\:\lambda\:\) represents the event rate and is estimated based on the average annual event frequency overserved in the database. Loss severity follows a log-normal distribution: $$\:P\left({\xi\:}_{k,r}\le\:x\right)=\:{G}_{k}\left(x\right)={G}_{k}\left(x;\lambda\:\right)=\:{\Phi\:}\left(\frac{\text{ln}x-\mu\:}{\sigma\:}\right)$$ where \(\:{\xi\:}_{k,r}\) is the loss amount, \(\:{\Phi\:}\) denotes the standard normal cumulative distribution function, and \(\:\sigma\:\) is the standard deviation. These distributions are controlled by vectors \(\:\theta\:\) and \(\:\lambda\:\) , confidence intervals \(\:\theta\:\in\:[{\theta\:}_{l};{\theta\:}_{u}]\) and \(\:\lambda\:\in\:[{\lambda\:}_{l};{\lambda\:}_{u}]\) to address parameter uncertainty caused by limited historical data. The model prices the bond using a standard formula for bond price: $$\:P=\sum\:_{i=1}^{6}C\bullet\:{e}^{-R\left(\frac{{d}_{i}}{365}\right)}+N\bullet\:{e}^{-R\left(\frac{d}{365}\right)}\:$$ where C is the coupon value, N is the notional value, R is the funding rate, \(\:{d}_{i}\) represents the coupon payment dates (in days), d is the maturity date, and N is the notional value. Coupon payments occur on days 182, 365, 547, 730, 912, and 1095, with the notional paid at maturity (day 1095). The model simulates cyber losses across the bond’s 3-year maturity with a comparison of total losses against coupon and notional trigger thresholds. Payments are made only if the actual losses are below these thresholds as measured with an indicator function during Monte Carlo simulations. The fair price is determined by maximizing over these intervals: $$\:Price={max}_{\theta\:\in\:\left[{\theta\:}_{l};{\theta\:}_{u}\right],\lambda\:\in\:\left[{\lambda\:}_{l};{\lambda\:}_{u}\right]}FairPrice({F}_{k}\left(\bullet\:\right)={F}_{k}\left(\bullet\:;\theta\:\right);{G}_{k}\left(\bullet\:\right)=\:{G}_{k}\left(\bullet\:;\lambda\:\right))$$ To calculate risk premium alternatives, I compute the coupon rate using the probability of loss approach: $$\:Coupon\:rate\:\left(\%\right)=LIBOR\:\left(\%\right)+PL\:\left(\%\right)$$ The predictive ability of the model is tested by comparing the simulated losses with actual losses from the CISSM dataset in terms of mean squared error and tail risk coverage, using 5000 Monte Carlo simulations as detailed in the paper (Kolesnikov et al. 2022). The assumption of exponential inter-event times implies constant event frequency and event-type independence, potentially failing to capture systemic correlations in cyber events. This assumption is adopted from Kolesnikov et al. (2022) to facilitate parameter estimation in a Poisson process framework. Despite potential mismatches with the CISSM dataset, it is retained for model testing purposes. Although the model is strong in handling different loss events and allows for modeling of tail risk, its dependence on limited historical data leads to parameter uncertainty and the simulation-based method is computationally demanding. This becomes particularly severe when targeting high precision (e.g., < 0.1% error) or handling complex scenarios, such as paths with thousands of time steps and sample sizes in the millions to billions, potentially requiring hours on standard hardware. Optimization via parallel computing is suggested for practical use. The performance results will be evaluated against other pricing models to test its suitability for valuing cyber cat bonds. 3.2.2. Implementation of Unified Bayesian Framework The Unified Bayesian Framework proposed by Domfeh et al. ( 2022 ), which was originally designed for natural cat bond, has the potential to be used to evaluate the pricing for cyber cat bonds. The process combines DP-HBCRM for cyber risk, a Cox-Ingersoll-Ross (CIS) model for interest rate risk, and a maximum entropy approach for risk-neutral pricing. The process begins with the calibration of the DP-HBCRM using cyber event data from the CISSM dataset, where the claim frequency and claim severity distributions are clustered by event type. Cyber event frequency ( \(\:{N}_{i,s}\) ) is modeled as a nonhomogeneous Poisson process, where for event type (i) and quarter (s) the number of events follows: $$\:{N}_{i,s}\:|{\:\lambda\:}_{i,s\:}\sim\:Poisson\left({\lambda\:}_{i,s}\right),\:\:{\lambda\:}_{i,s}>0$$ $$\:\text{log}\left({\lambda\:}_{i,s}\right)|{\alpha\:}_{i},\:{\beta\:}_{s},\:{X}_{s}=\:{\alpha\:}_{i}+\:{\beta\:}_{s}{X}_{s}$$ where \(\:{\:\lambda\:}_{i,s\:}\) is the claim intensity, \(\:{\alpha\:}_{i}\:\) is event type-specific intensity capture, \(\:{\beta\:}_{s}\) is the quarterly pattern and \(\:{X}_{s}\) is the seasonal indicator. The claim size ( \(\:{S}_{i}\) ) and loss severity ( \(\:{S}_{i,s}\) ) are modeled with an inverse gamma distribution: $$\:{X}_{i}|{\kappa\:}_{i},{\theta\:}_{i}\:\sim\:Inv.Gamma\left({\kappa\:}_{i},\:{\theta\:}_{i}\right),\:\:{\theta\:}_{i}>0,\:{\kappa\:}_{i}>0$$ $$\:{S}_{i,\:s}\:|\:{\kappa\:}_{i,s},{\theta\:}_{i}\:\sim\:Inv.Gamma\left({\kappa\:}_{i,s}\bullet\:{\kappa\:}_{i},\:{\theta\:}_{i}\right)$$ Dirichlet Process (DP) priors are specified for ( \(\:{\kappa\:}_{i},\:{\theta\:}_{i}\) ) and ( \(\:{\alpha\:}_{i}\) ), inducing clustering of event types based on shared loss characteristics: $$\:\left({\kappa\:}_{i},\:{\theta\:}_{i}\right)\:\sim\:DP\left({\gamma\:}_{1},\:{G}_{0}\left(\bullet\:\right)\right)$$ $$\:{\alpha\:}_{i}\:\sim\:DP({\gamma\:}_{2},\:{H}_{0}\left(\bullet\:\right))$$ $$\:{G}_{0}\left({\zeta\:}_{1},{\zeta\:}_{2},{\eta\:}_{1},{\eta\:}_{2}\right)=Gamma\left({\zeta\:}_{1},{\zeta\:}_{2}\right)\times\:Gamma({\eta\:}_{1},{\eta\:}_{2})$$ $$\:{H}_{0}({\psi\:}_{1},{\psi\:}_{2})=Gamma(\:{\psi\:}_{1},{\psi\:}_{2})$$ $$\:\text{w}\text{i}\text{t}\text{h}\:\text{h}\text{y}\text{p}\text{e}\text{r}\text{p}\text{a}\text{r}\text{a}\text{m}\text{e}\text{t}\text{e}\text{r}\text{s}\:{\zeta\:}_{1},{\zeta\:}_{2},{\eta\:}_{1},{\eta\:}_{2},{\psi\:}_{1},{\psi\:}_{2}\:\sim\:Gamma\left(\text{0.01,0.01}\right)\:\text{a}\text{n}\text{d}\:{\beta\:}_{s}\mathcal{\:}\sim\mathcal{\:}\mathcal{N}\left(\text{0,0.01}\right).$$ Posterior distributions were estimated MCMC simulations with 40,000 iterations and the first 10,000 iterations as burn-in. Then, a Bayesian CIR model is employed to model interest rates: $$\:{dr}_{t}=\left(\alpha\:-\:{\beta\:r}_{t}\right)dt+\:\sigma\:\sqrt{{r}_{t}}d{W}_{t}$$ Parameters ( \(\:\alpha\:,\:\beta\:,\:{\sigma\:}^{2}\) ) are inferred using MCMC with 15,000 iterations and a burn-in of the first 5,000. The bond price for a zero-coupon cyber cat bond is formulated as follows: $$\:\:\:\:{P}_{t}=\:\text{{\rm\:K}}\:{\mathbb{E}}^{\mathbb{Q}}\:\left[{e}^{-{\int\:}_{t}^{T}{r}_{s}ds}\bullet\:{V}_{T}\:\right|\:{\mathcal{F}}_{t}\:\:]\:\approx\:\:\sum\:_{i=1}^{N}\sum\:_{t=1}^{T}\left(\text{exp}\left(-\sum\:_{u=1}^{t}{r}_{u}^{\left(i\right)}\right)\bullet\:{V}_{t}^{\left(i\right)}\right){\pi\:}_{i}^{*}\:\:\:\:\:\:\:\:(1)$$ where \(\:{V}_{T}=K\bullet\:I\left({L}_{T}\le\:D\right)+\alpha\:\bullet\:K\bullet\:I({L}_{T}>D)\) , \(\:\text{{\rm\:K}}\) is the face value, D is the loss threshold and \(\:{\pi\:}_{i}^{*}\) is risk-neutral probabilities derived via maximum entropy. To adapt the model to the CISSM dataset, I used fixed and more conservative hyperparameters for the Dirichlet Process priors ( \(\:{\gamma\:}_{1}=10,\:{\gamma\:}_{2}=2,\:{\zeta\:}_{1}=15,\:{\zeta\:}_{2}=0.1,\:{\eta\:}_{1}=10,{\eta\:}_{2}=0.5,\:{\psi\:}_{1}=3,{\psi\:}_{2}=1\) ) to ensure stable posterior inference. I removed the seasonal component ( \(\:{\beta\:}_{s}{X}_{s}\) ) in the nonhomogeneous Poisson process, as the dataset showed limited evidence of quarterly patterns for cyber risks. It was also confirmed by a chi-square test on quarterly patterns (chi-square statistic: 6.4343, p-value: 0.092290), which fails to reject the null hypothesis of no seasonality. The MCMC iterations for the DP-HBCRM were reduced to 3,000 with 1,000 burn-in iterations to avoid the risk of overfitting, and because seasonal factors were removed in the nonhomogeneous Poisson process, which consequently reduced the complexity of the model. To ensure convergence, I did the Gelman-Rubin test, and the results showed the Gelman-Rubin statistics were below 1.1 for all parameters. This simplified model structure requires fewer iterations than the paper suggested to achieve a stable and better-performed posterior distribution. The model assumes independence of event frequency and severity and may ignore systemic cyber event dependencies across events such as those arising from correlated supply chain vulnerabilities (Mastroeni et al., 2023, p. 2). However, by incorporating prior information and classifying event types, this model can handle scarce historical data, but it is very computationally demanding. The runtime increases significantly as the iteration counts and chain number grow. It could extend processing to several hours on standard hardware, limiting scalability for high-precision applications. 3.2.3. Implementation of Signal-Processing Approach For pricing cyber cat bonds, I use the signal-processing method of Li and Mamon (2023). This framework represents the dynamics of cyberattacks using a nonhomogeneous Markov chain modulated by hidden Markov chain, and it offers a flexible framework for modelling CKC state change and pricing bonds under uncertainty. The implementation focuses on calibrating RSMM, estimating transition probabilities, pricing a zero-coupon cyber cat bond, and calculating risk premia. The process begins by constructing the RSMM to the CKC states: firewall working ( \(\:{f}_{1}\) ), firewall fail ( \(\:{f}_{2}\) ), and anti-phishing fail ( \(\:{f}_{3}\) ). The state process ( \(\:{y}_{k}\) ) evolves according to: $$\:{y}_{k+1}=B\left({z}_{n}\right){y}_{k}+\:{w}_{k+1}$$ 2 where \(\:B\left({z}_{n}\right)=\left({b}_{ij}\left({z}_{k}\right)\right)\) is the state-dependent transition matrix, \(\:{w}_{k+1}\) is a martingale increment with \(\:E\left[\:{w}_{k+1}\:\right|{\mathcal{F}}_{k}]=0\) , and \(\:{\mathcal{F}}_{k}\) is the filtration generated by \(\:{\mathcal{F}}_{k}^{y}\) and the HMM chain \(\:{\mathcal{F}}_{k}^{z}\) . Eq. ( 2 ) is then revised by the authors, to accommodate certain established results of homogeneous HMM with a discrete range: $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{c}_{ji}\left({y}_{k}\right)\:{|}_{{y}_{k}={f}_{l}}\::=P\left({y}_{k+1}=\:{f}_{j}|{y}_{k}={f}_{l},\:{z}_{k}={e}_{i}\right)=\:{b}_{jl}\left({z}_{k}\right){|}_{{z}_{k}={e}_{i}}$$ 3 Combine (2) and (3) is equivalent to: $$\:{y}_{k+1}=C\left({y}_{k}\right){z}_{k}+\:{w}_{k+1}$$ 4 where \(\:C=\left({c}_{ji}\left({y}_{k}\right)\right)\) . Eq. ( 4 ) is a one-step delay model, which is reasonable because \(\:{y}_{k+1}\) may not react to z immediately. Second, the transition probabilities are determined considering recursive filters: $$\:{p}_{k}=\:\prod\:diag\left({d}_{k}\right){p}_{K-1}$$ $$\:\gamma\:\left({\mathcal{J}}_{k}^{j,r}{z}_{k}\right)=\:\prod\:diag\left({d}_{k}\right)\gamma\:\left({\mathcal{J}}_{k-1}^{j,r}{z}_{k-1}\right)+{d}_{k}^{\left(r\right)}{\pi\:}_{jr}{e}_{j}$$ $$\:\gamma\:\left({\mathcal{O}}_{k}^{r}{z}_{k}\right)=\:\prod\:diag\left({d}_{k}\right)\gamma\:\left({\mathcal{O}}_{k-1}^{r}{z}_{k-1}\right)+{d}_{k}^{\left(r\right)}{\pi\:}_{r}$$ \(\:\gamma\:\left({\mathcal{T}}_{k}^{s,r}\left({y}_{k},{f}_{i}\right){z}_{k}\right)=\prod\:diag\left({d}_{k}\right)\gamma\:\left({\mathcal{T}}_{k-1}^{s,r}\left({y}_{k-1},{f}_{i}\right){z}_{k-1}\right)\) \(\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+m{c}_{sr}\left({f}_{i}\right){\pi\:}_{r}\:\) $$\:\gamma\:({\mathcal{T}}_{k}^{r}\left({f}_{i}\right){z}_{k}=\:=\prod\:diag\left({d}_{k}\right)\gamma\:\left({\mathcal{T}}_{k-1}^{r}\left({f}_{i}\right){z}_{k-1}\right)+{d}_{k}^{\left(r\right)}{\pi\:}_{r}$$ where \(\:{d}_{k}\) represents observation probabilities, \(\:{\mathcal{J}}_{k}^{j,r}\) is the number of jumps from \(\:{e}_{r}\) to state \(\:{e}_{j}\) in time k, \(\:{\mathcal{O}}_{k}^{r}\) is the amount of time that the Markov chain z spent in state \(\:{e}_{r}\) up to k, \(\:{\mathcal{T}}_{k}^{s,r}\left({y}_{k},{f}_{i}\right)\) counts the number of times up to k that y is in state \(\:{f}_{s}\) , \(\:{\mathcal{T}}_{k}^{s,r}\left({f}_{i}\right)\) counts the number of times up to k, and \(\:\prod\:\) governs \(\:B\left({z}_{n}\right)\) transitions. They use change of measure, as in this joint implementation \(\:{\stackrel{-}{{\Lambda\:}}}_{k}=\:{\prod\:}_{l=1}^{k}\:{\stackrel{-}{\lambda\:}}_{l}\) , and do their calculations under a measure ( \(\:\stackrel{\sim}{P}\) ) in which the \(\:B\left({z}_{n}\right)\) are independent with uniform distribution. Then, estimate the optimal parameters using the EM algorithm and filtering technique: $$\:{\widehat{\pi\:}}_{jr}=\:\frac{\gamma\:\left({\mathcal{J}}_{k}^{j,r}\right)}{\gamma\:\left({\mathcal{O}}_{k}^{r}\right)}$$ $$\:{\widehat{c}}_{sr}\left({f}_{i}\right)=\:\frac{\gamma\:\left({\mathcal{T}}_{k}^{s,r}\left({y}_{k},{f}_{i}\right)\right)}{\gamma\:\left({\mathcal{T}}_{k}^{r}\left({f}_{i}\right)\right)}$$ Finally, loss severities are modeled with a doubly truncated Pareto distribution and transform this loss on dollar terms by proportinality. The bond price is calculated under a risk-neutral measure ( \(\:\mathbb{Q}\) ): $$\:{P}_{t}=\:{\mathbb{E}}^{\mathbb{Q}}\left[{e}^{-{\int\:}_{t}^{T}{r}_{s}{d}_{s}}\bullet\:K\left({F}_{T}\left(D\right)+a\left(1-{F}_{T}\left(D\right)\right)\right)\:\right|{\mathcal{F}}_{t}]$$ where \(\:{F}_{T}\left(D\right)=\text{P}\text{r}[{L}_{T}\le\:D]\) and the Vasiček model is used to generate interest path and to calculate the discount factor. The model is based on Markov state transitions, independent breach severities, and a stationary interest rate process. Loss and interest rate scenarios are simulated by Monte Carlo (10000 iterations) with 5,000 iterations used for the EM convergence. Regarding this approach, the RSMM captures state-dependent transitions, and hence improves pricing accuracy. However, the Monte Carlo simulation is computationally heavy, and thus efficient algorithms are necessary. To adapt the model to the dataset, where all events are state 3 (anti-phishing fail) based on CKC states, I implemented the simplified focus only on loss severity modeling using the doubly truncated Pareto distribution. This CKC transitions ineffective, limiting demonstration of the model's theoretical strength in dynamic state modeling—future datasets with diverse states could fully exploit this. The loss amounts were aggregated by industry and date to align with the structure of the dataset. Moreover, the shape parameter of the Pareto distribution was estimated via maximum likelihood estimation (MLE), with a stability check enforcing a minimum shape of 1.1 to prevent infinite expected losses. For loss perditions, I used directly the expected loss from the Pareto distribution instead of Monte Carlo simulations. 3.2.4. Implementation of Regression Approach The regression approach proposed by Lane ( 2000 ), which offers a general model for pricing ILS, including cyber cat bonds, by using a regression-based technique to estimate risk premia. This section describes the methodology of applying Lane’s regression model to cyber cat bond pricing and is consistent with the study’s objective of estimating loss prediction accuracy using the CISSM dataset. The regression method splits the pricing of cyber cat bond into three parts: the RFIR, EL and RP, that is, the spread over LIBOR of the bond. At the heart of the model are two risk metrics – PFL and CEL – that address the frequency and severity of cyber events, respectively. PFL is the probability of cumulative losses ( \(\:{L}_{T}\) ) exceeding the threshold ( \(\:D\) ): $$\:PFL=\text{P}\text{r}[{L}_{T}>D]$$ CEL is, in turn, the expected loss given that a loss occurs ( \(\:{L}_{T}>D\) ): $$\:CEL=E\left[{L}_{T}\right|\:{L}_{T}>D]/K$$ where K is the face value, \(\:{L}_{T}={\sum\:}_{i=1}^{{N}_{T}}{L}_{i}\) , \(\:{N}_{T}\) is the number of cyberattacks and \(\:{L}_{i}\) is the loss per attack. Here, I will test different distribution modes for \(\:{L}_{T}\) and \(\:{N}_{T}\) , and show the result in section 4 . The risk premium is expressed as a function of expected excess return (EER), PFL and CEL via the power-law form of Cobb-Douglas production function: $$\:EER=\:\gamma\:\times\:{\left(PFL\right)}^{\alpha\:}\times\:{\left(CEL\right)}^{\beta\:}$$ Here, \(\:\gamma\:=0.5551\) , \(\:\alpha\:=0.4946\) , and \(\:\beta\:=0.5741\) are sample values taken from the 1999 ILS market which are based on a historical trade-off between frequency and severity. The EL is calculated as the product of PFL and CEL: $$\:EL=PFL\:\times\:\:CEL$$ The full premium, which determines the bond’s spread over LIBOR, is the sum of EL and EER, adjusted for a 365-day count convention: $$\:Full\:Premium=EL+EER$$ The bond spread over LIBOR (S) is as follows: $$\:S=RFIR+EL+EER=LIBOR+Full\:Premium$$ To adapt the model to the dataset, I first grouped all events by industry, geography and classification of cyber risk. This step was adopted to emulate Lane's empirical approach. Trigger amounts ( \(\:D\) ) are established based on cyber cat bond data in the past two years (2023–2024, Artemis.bm, 2025), with values set at $ 576.49 million for the United States and $ 539.74 million for other countries. To address the scarcity of the historical data and the limited dataset, PFL is estimated as \(\:({N}_{Triggered}+1)/({N}_{events}+2)\) , with \(\:{N}_{Triggered}\) denoting the number of cyber events for which the loss amounts exceed the trigger amounts, and \(\:{N}_{events}\) denoting the number of cyber events in the groups. To avoid the case \(\:{N}_{Triggered}=0\) , Laplace smoothing is incorporated. CEL is calculated as the mean loss exceeding \(\:D\) : $$\:\left\{\begin{array}{c}\frac{\frac{\sum\:{L}_{Triggered}}{{N}_{Triggered}}}{\text{max}{L}_{Triggered}},if{\:N}_{Triggered}\ne\:0\:\\\:\sum\:{L}_{i}*\frac{0.01}{\text{max}{L}_{Triggered}},if{\:N}_{Triggered}=0\end{array}\right.$$ where \(\:{L}_{Triggered}\) denotes the loss amount of cyber events exceeding the trigger amount and \(\:{L}_{i}\) denotes the loss amount of cyber events. Subsequently, the computed PFL and CEL values, derived from the grouped data, are assigned to each cyber event. Departing from Lane’s original use of fixed parameters ( \(\:\gamma\:=0.5551\) , \(\:\alpha\:=0.4946\) , and \(\:\beta\:=0.5741\) ) derived from 1999 ILS data, this study uses a data-driven approach. An ordinary least squares (OLS) regression on log-transformed variables is employed to fit the parameters of the Cobb-Douglas production function, as detailed below: $$\:\text{log}\left(1+Loss\:Amount\right)=log\gamma\:+\alpha\:\bullet\:\text{log}\left(PEL\right)+\:\beta\:\bullet\:\text{log}\left(CEL\right)+\:ϵ$$ where \(\:ϵ\) represents the error term. For loss predictions, I split the dataset into a 70:30 ratio for training and predictive validation, respectively, with 5-fold cross-validation to enhance robustness. The model does not allow for asymmetric cyber risk loss distributions and for PFL and CEL for the risk profile. It also assumes the model parameters fitted to the 1999 ILS are also suitable for cyber risks, although the loss dynamic may differ, i.e. the systemic correlation or the multi-event trigger. The assumption of a linear relation between lay of risk and risk measures may not account for the full range of non-linearity in cyber events. However, its understandable form may help to make the model more approachable for the new cyber cat bond market where we have limited data. Yet its assumptions require careful consideration, particularly with respect to the portability of ILS parameters and the ability of PFL and CEL to fully address systemic cyber risk. These issues will be further addressed in the empirical analysis to allow me to assess the model’s predictive power in the more general comparative context. 3.2.5. Implementation of Copula-POT Model To achieve the goal of valuating cyber cat bonds and assessing their role in portfolio efficiency, I apply the Copula-POT model by Tang et al. ( 2023 ) to price cyber cat bonds using a database of cyber events with loss amounts attached to them. The methodology combines loss prediction accuracy, tail risk prediction, and contribution to portfolio enhancement to provide a solid base for model selection and application to MPT. The first part is to characterize the marginal distribution of the indicators of cyber losses using POT method with a Beta-generalized Pareto (GP) model introduced in Tang et al. ( 2023 ). Due to the presence of heavy tails on the losses, the database is decomposed into bulk and tail parts. The threshold \(\:{u}_{i}\) of each cyber loss indicator \(\:{X}_{i}\) (e.g., financial loss, number of affected records) is decided by inspecting the mean residual life plot using the following equation: $$\:{e}_{n}\left(u\right)=\:\frac{1}{{n}_{u}}\sum\:_{i=1}^{{n}_{u}}({x}_{\left(i\right)}-u)$$ The threshold-excesses ( \(\:{X}_{i}-{u}_{i}|{X}_{i}>{u}_{i}\) ) are modeled using the GPD: $$\:G\left(y\right)=1-(1+\frac{{\xi\:}_{y}}{\sigma\:}{)}^{\raisebox{1ex}{$-1$}\!\left/\:\!\raisebox{-1ex}{$\xi\:$}\right.}\:,\:\:y>0$$ where \(\:\xi\:\) and \(\:\sigma\:\) are the shape and scale parameters, respectively. Non-exceedances are modeled using a Beta distribution: $$\:{Beta}_{\alpha\:,\beta\:}\left(y\right)=\:\frac{1}{B\left(\alpha\:,\beta\:\right)}{y}^{\alpha\:-1}(1-y{)}^{\beta\:-1}\:,\:\:\:\:\:\:\:\:\:0<y<1$$ Maximum likelihood estimation is applied to estimate \(\:\xi\:,\:\sigma\:,\:\alpha\:,\:\beta\:\) , ensuring the distribution \(\:{F}_{i}\left(x\right)\) captures both normal and extreme losses as follows: $$\:{F}_{i}\left(x\right)=\:\left\{\begin{array}{c}1-\frac{{n}_{{u}_{i}}}{n}{\stackrel{-}{G}}_{{\xi\:}_{i},{\sigma\:}_{i}}\left(x-{u}_{i}\right),\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:x>{u}_{i}\\\:\frac{1}{B({\alpha\:}_{i},{\beta\:}_{i})}{\int\:}_{0}^{1-(x-{m}_{i})/({u}_{i}-{m}_{i})}{t}^{{\alpha\:}_{i}-1}(1-t{)}^{{\beta\:}_{i}-1}dt,\:\:\:\:x\le\:{u}_{i}\end{array}\right.$$ where \(\:\frac{{n}_{{u}_{i}}}{n}\) is the exceedance proportion and \(\:{m}_{i}\) is the sample minimum. The dependence structure between the multiple cyber loss indicators is then modeled by means of a nested Archimedean copula: $$\:C\left({u}_{1},\:\cdots\:,\:{u}_{m}\right)=\:{C}_{outer}({C}_{inner}\left({u}_{1},\cdots\:,\:{u}_{k};{\theta\:}_{1}\right),\:{u}_{k+1},\cdots\:,\:{u}_{m};{\theta\:}_{2})$$ with parameters \(\:{\theta\:}_{1}\) and \(\:{\theta\:}_{2}\) are estimated via maximum likelihood to capture the larger dependence among subsets of indicators, which will lead to more precise modeling of joint loss events. The cyber loss indicators are converted into uniform variates using \(\:{\stackrel{\sim}{u}}_{ij}={F}_{i}\left({x}_{ij}\right)\) , and a nested Frank copula is chosen because it fits best (Tang et al., 2023 ). Pricing of the cyber cat bond is then performed through Monte Carlo simulation as following equation: $$\:{P}_{t}=\:\sum\:_{s=t+1}^{T}\mathbb{E}\left\{{C}_{t,s}\right\}p\left(t,s\right)+\mathbb{\:}\mathbb{E}\left\{{F}_{t,T}\right\}p(t,T)$$ The coupon and principal payments are determined by the retention proportions \(\:{\alpha\:}_{t},\:{\beta\:}_{t},\:{\gamma\:}_{t}\) (as specified in above equations), acknowledging that they cut into loss indicators if they exceed their attachment points. The discount factor \(\:p(t,T)\) is calculated using the CIR model: \(\:p\left(t,T\right)=A(t,T){e}^{-B\left(t,T\right)r\left(t\right)}\) . The frequency of cyber events per year is calculated using an ARMA process in order to predict event intensity along the bond’s life. The CISSM dataset is preprocessed by selecting key loss indicators: Loss amount, geography and event type impact. The latter is calculated as a weighted impact from event type (e.g., data breach, DDOS). Thresholds ( \(\:{u}_{i}\) ) for these indicators are set with the 98th percentile instead of the mean residual life plot to focus on tail risks, ensuring sufficient exceedances (at least 10) for robust estimation. The Gumbel-Gumbel copula is chosen for its emphasis on upper tail dependence and because it showed the best result compared to other combinations of copula. For loss predictions, I first split the dataset randomly into a 70:30 ratio for training and testing sets. Secondly, Monte Carlo simulations were conducted with 200,000 iterations to predict losses. Finally, the loss prediction ability was evaluated by comparing simulated losses with actual losses from the test set. 3.3. Model Evaluation In order to identify the most appropriate model for pricing cyber cat bonds in my study, I review five pricing models mentioned above. The assessment is conducted on three points of aspects of loss prediction ability measured by Mean Squared Error (MSE), pricing efficiency measured by Sharpe ratio, and tail risk prediction accuracy measured by Conditional Value at Risk (CVaR). The models are tested against the CISSM dataset to see how well they can reproduce the CISSM data using statistical approaches, including goodness of fit testing and error metrics. The first and the most critical criterion in the study, the loss prediction ability, measures how accurately each model predicts cyber losses. MSE is computed by simulating losses against a validation subset of the dataset and a low MSE indicates high predictive accuracy. To handle the right-skewed distribution of loss amount data, using the original scale to calculate MSE directly would result in the evaluation results being overly influenced by extreme values. I apply a log1p transformation (natural logarithm of the value plus one, i.e., \(\:\text{l}\text{n}(1+x)\) ) to both actual and predicted losses before calculating the MSE, resulting in a standardized log MSE. This transformation is chosen because it effectively handles zero or small values without causing undefined logarithms, stabilizing the metric for skewed data. The log MSE is defined as: $$\:log\:MSE=\:\frac{1}{n}\sum\:_{i=1}^{n}[\text{l}\text{o}\text{g}(1+{\widehat{L}}_{i})-\text{l}\text{o}\text{g}(1+{L}_{i}{\left)\right]}^{2}$$ where \(\:{\widehat{L}}_{i}\) represents the predicted loss, \(\:{L}_{i}\) is the actual loss, and \(\:n\) is the number of observations. I also use the Kolmogorov-Smirnov (KS) goodness-of-fit test to assess the validity of loss distributions by comparing the empirical distribution of historical losses with the cumulative distribution functions of simulated losses, with p-value of 0.05 as the threshold to indicate an adequate fit. For each model, I do loss predictions with Monte Carlo simulations, adjusted to each model’s specific methodology. This entails fitting each model to the data, simulating losses over a range of scenarios and comparing their predictions with historical losses to gauge their predictive accuracy. This loss prediction capability forms the foundation for model evaluation, but to ensure practical applicability in cyber cat bond valuation, the next criterion assesses how well the model's pricing mechanism translates these predictions into efficient risk-return profiles. The second criterion, pricing efficiency, evaluates the effectiveness of each cyber cat bond pricing model in generating attractive returns relative to the associated risks, considering the diverse pricing methodologies across models. I quantify this with the Sharpe ratio, which is: $$\:Sharpe\:Ratio=\:\frac{\mathbb{E}[{R}_{p}-{R}_{f}]}{{\sigma\:}_{p}}$$ where \(\:{R}_{p}\) is the cyber cat bond return, \(\:{R}_{f}\) is the risk-free rate, and \(\:{\sigma\:}_{p}\) is the standard deviation of the bond’s returns, reflecting its risk or volatility. To do this, I employ a Monte Carlo approach to simulate 10,000 times the returns of a 2-year-zero-coupon cyber cat bond with a fixed face value in the way each model prescribes them. The bond prices are the result of a price computation in a risk-neutral framework, incorporating the risk premia (spread) of 11.3%, which represents the average spread of existing cyber cat bonds in 2023–2024. The loss trigger is set at $ 558.11 million, based on the average trigger amount of existing cyber cat bonds during the same period. The risk-free rate ( \(\:{R}_{f}\) ) is set to 4.00%, which is the average yield of 10-year U.S. Treasury notes also from the same period. The Sharpe ratio is then calculated to compare the pricing efficiency of each model, enabling an assessment of how effectively each pricing mechanism balances the high-risk, high-return nature of cyber cat bonds under varying loss scenarios. To assess robustness, sensitivity analysis is conducted by varying trigger levels, ensuring the model's pricing remains stable across realistic market conditions. Building on pricing efficiency, which previews investment implications, the third criterion focuses on the models' ability to handle extreme events, which is crucial for the tail-heavy nature of cyber risks. The third criterion, tail risk prediction accuracy, examines the degree to which the models capture the extreme cyber loss events, which is particularly important because cyber risks are naturally skewed. I measure it via the Conditional Value at Risk (CVaR) at the 99% confidence level as: $$\:{CVaR}_{\alpha\:}=\mathbb{\:}\mathbb{E}\left[L\:\right|\:L\ge\:Va{R}_{\alpha\:}]$$ where \(\:Va{R}_{\alpha\:}\) is the Value at Risk at confidence level \(\:\alpha\:=0.99\) . I calculate CVaR by simulating loss distributions using 10,000 Monte Carlo paths to generate robust tail estimates. For each model, I match the loss distribution for its tail and calculate the expected loss given that the VaR has been exceeded. This allows testing how well each model captures extreme cyber events. Computationally, I build the models in Python with packages like NumPy, SciPy, and pandas. The simulations of each model are calibrated to their theoretical framework, using preprocessing that scales the data and makes it consistent across models. For example, the Copula-POT must estimate the copula parameter while in a regression model, the nonlinear least squares are adopted. I monitor convergence and check results via diagnostics, like presenting the standard deviations for MSE, or checking the p-values of KS test. To integrate these criteria and select the most suitable model, a composite score is calculated as follows: Composite Score = 0.6 × (Normalized Loss Prediction Score) + 0.2 × (Normalized Sharpe Ratio) + 0.2 × (Normalized CVaR Accuracy). Normalization scales each metric to [0,1] across models, with higher values indicating better performance (e.g., lower log MSE yields a higher score). The weighting gives 60% to the Loss Prediction Score, where the loss prediction ability is referred to as the most critical criterion in the study for the model selection due to the high influence on precise risk measurement and bond pricing. The remaining 20% each for Sharpe Ratio and CVaR Accuracy reflects their supportive roles in evaluating pricing efficiency and tail risk management, ensuring a balanced yet prioritized evaluation. Monte Carlo simulations and statistical tests ensure robustness, providing a foundation for model selection that optimally trades off predictive accuracy, portfolio optimization, and tail risk estimation—driving the design of cyber risk transfer instruments. 3.4. Portfolio Optimization Building on the selected pricing model from section 3.3 , which demonstrates superior loss prediction, pricing efficiency, and tail risk accuracy via the composite score, this section applies its outputs to a modified MPT framework. This approach evaluates the investment application value of cyber cat bonds by assessing their impact on risk-adjusted returns and diversification benefits, aligning with the study's objectives of bridging theoretical valuation to practical portfolio management. To evaluate this, I use a modified MPT approach. With a fixed asset allocation, I can access the impact of cyber cat bonds on risk-adjusted return, as measured by the Sharpe Ratio. This approach avoids calculating the efficient frontier, as different pricing models for cyber cat bonds could lead to inconsistent portfolio allocations, and it will complicate cross-model comparisons. Instead, I use a fixed allocations that allow for focused analysis of diversification benefits, reflecting real-world investor perspectives who seek to understand how cyber cat bonds enhance standard portfolios without requiring dynamic rebalancing. While this does not fully "maximize" returns as in standard MPT (Markowitz, 1952 ), it provides practical insights into incremental improvements, particularly for institutional investors. The baseline portfolio of 60/40 allocation (60% stock index, 40% bond index) is designed th reflect a realistic investment scenario. It comprises 40% U.S. stock index (S&P 500), 20% worldwide stock index (All-World ex-US), 30% U.S. investment-grade bonds, and 10% international bonds. This composition mirrors common allocation in balanced funds and institutional portfolios. Moreover, combining US and international exposure can capture the benefits of international diversification and eventually improve the Sharpe Ratio (Pham et al., 2025, p.38). For these assets, I estimate expected returns and volatilities from historical data (2019–2024, 5 years), with correlations computed using pairwise return series. The portfolio also includes a 2-year zero-coupon cyber cat bond (face value $ 100 million). The cyber cat bond is designed to repay its initial investment plus a risk premium (11.3%) at maturity unless catastrophe losses exceed the trigger amount, which is set at $ 558.11 million on per-occurrence basis. The indemnity trigger and per-occurrence basis are following the market preference, with 70% of cyber cat bonds using indemnity triggers. Since they offer the most direct linkage to policyholder losses and result in lower basis risk compared to parametric or index-based trigger variants. And the trigger of $ 558.11 million is adopted to ensure market relevance, based on average trigger amounts from existing cyber cat bonds (Artemis.bm 2025). Parameters for the cyber cat bond — expected return, volatility and correlations — are generated by the pricing model chosen, from which loss distributions and risk-adjusted returns are simulated. The expected return is the risk-neutral expected payoff — discounted at the risk-free rate and with an allowance for a risk premium from the pricing model. Volatility comes from the standard deviation of simulated bond returns, which captures loss trigger uncertainty. Correlations between the cyber bonds and traditional assets are still not clear and due to the lack of historical data on cyber cat bonds. Based on a report of S&P Global Ratings 2025, which estimates correlations with stocks at 0.1–0.24 (potentially rising above 0.3 during large-scale cyber events) and with bonds below 0.2, a conservative baseline correlation of 0.25 is assumed. Using MPT, I calculate the Sharpe Ratio defined as: $$\:Sharpe\:Ratio=\:\frac{\mathbb{E}\left[{R}_{p}\right]-{R}_{f}}{{\sigma\:}_{p}}$$ where \(\:\mathbb{E}\left[{R}_{p}\right]=\:\sum\:_{i=1}^{n}{w}_{i}\mathbb{E}\left[{R}_{i}\right]\) is the portfolio expected return, \(\:{w}_{i}\) is the weight of asset \(\:i\) , \(\:\mathbb{E}\left[{R}_{i}\right]\) is the expected return of asset \(\:i\) , \(\:{R}_{f}\) is the risk-free rate, and \(\:{\sigma\:}_{p}\) is the portfolio volatility, defined as: $$\:{\sigma\:}_{p}=\:\sqrt{{\sum\:}_{i=1}^{n}{\sum\:}_{j=1}^{n}{w}_{i}{w}_{j}{\sigma\:}_{i}{\sigma\:}_{j}{\rho\:}_{ij}}$$ with \(\:{\sigma\:}_{i}\:and\:{\sigma\:}_{j}\) are the volatilities of assets \(\:i\) and \(\:j\) , and \(\:{\rho\:}_{ij}\) is the correlation between their returns. To ensure robustness, I conduct sensitivity analysis by testing alternative portfolio allocations with equity/fixed income ratio of 70/30 and 50/50, using the same cyber cat bond proportions and trigger amounts. Additionally, I test different correlation assumptions (0.1, 0.4) and different trigger amounts ( $ 300 million, 800 million) to address uncertainty due to limited historical data. This analysis examines whether cyber cat bonds consistently improve the Sharpe Ratio across different risk profiles and market conditions, offering insights for investors with varying risk preferences. I also discuss the potential for cyber cat bonds to replace HY bonds, evaluating their investment value in a fixed-income portfolio. Using ICE BofA High Yield Index as a benchmark (with parameters estimated from 2019–2025 historical data), I compare the Sharpe Ratio for three portfolios: Benchmark (100% HY bonds), 30% cyber cat bonds + 70% HY bonds and 70% cyber cat bonds + 30% HY bonds. The same methods and assumptions for cyber cat bonds are used. By comparing these metrics, I analyze whether cyber cat bonds enhance risk-adjusted returns and therefore could be an alternative to HY bonds. To ensure robustness, I also conduct sensitivity analysis by testing different trigger amounts ( $ 300 million, 800 million) and different correlation assumptions (0.1, 0.4). Reflecting on this approach, expanding MPT to include cyber cat bonds emphasizes how they can improve portfolio diversification by offering an exceptionally diluted risk concept. But the reliance on model-based inputs adds uncertainty, especially when it comes to estimating a correlation that may not be well-anchored in scarce empirical evidence. This framework achieves a balance between analytical rigor and real-world applicability, providing investors with a clear perspective on the value of cyber cat bonds in managing cyber risk within a globally diversified portfolio. 4. Results and Analysis 4.1 Pricing Model Performance To determine the most suitable pricing models for cyber cat bonds, this section evaluates five pricing models – Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach and Copula-POT Model – based on three key criteria: loss prediction ability (log MSE), pricing efficiency (Sharpe ratio), and tail risk prediction accuracy (via \(\:{CVaR}_{0.99}\) relative standard error), as detailed in section 3.3 . I selected the log MSE as the metric to compare each model’s ability to predict losses. A lower log MSE and standard deviation of log MSE (log MSE SD) indicate higher predictive accuracy and higher stability, respectively. Additionally, K-Statistic and associated p-value from Monte Carlo simulations calibrated to the CISSM dataset are used to access the statistical significance of the results. Table 3 shows the summary of the empirical results. Table 3 Results of Loss Prediction Ability Model Log MSE Log MSE SD K-Statistic p-value Loss Distribution Framework 0.8514 0.0048 0.0081 0.5235 Unified Bayesian Framework 1.7538 0.018 0.2666 0.0000 Signal-Processing Approach 9.7226 0.4147 0.4722 0.0000 Regression Approach 3.8699 0.0978 0.4984 0.0000 Copula-POT Model 0.0211 0.0001 0.2238 0.0000 The Copula-POT Model achieves the highest loss prediction ability with the lowest log MSE (0.0211) and an exceptionally low log MSE SD (0.0001). This indicates both high accuracy and stability, aligning with the capacity to model the heavy-tailed distribution. Its reliance on copula functions and EVT effectively captures tail dependencies. However, its KS p-value of 0.0000 suggests a statistical bias, which requires further investigation of its distributional assumptions, possibly through refined copula parameter selection or increasing simulation iterations. The Loss Distribution Framework follows with a log MSE of 0.8514 and a relatively low standard deviation (0.0048) and is the only model to pass the KS test (p-value: 0.5235), showing adequate performance and a strong fit to the historical loss distribution. Though its higher MSE compared to Copula-POT suggests less precision for extreme tails. The Unified Bayesian Framework also performs well with a log MSE of 1.7538, but its higher standard deviation (0.018) suggests less consistent prediction, possibly due to reliance on prior assumptions that may not fully reflect the volatile nature of cyber data. In contrast, the Regression Approach and Signal-Processing Approach show not only poor predictive accuracy but also high variability. Moreover, they both fail the KS test (p-value: 0.0000), indicating significant deviations from the observed distribution. I selected Sharpe Ratio to evaluate the risk-adjusted returns of a 2-year zero-coupon cyber cat bond with a trigger of $ 558.11 million, a risk premium of 11.3%, and a risk-free rate of 4.00%. A higher Sharpe Ratio indicates better return relative to risk, reflecting the model's ability to generate efficient bond valuations amid market discrepancies. The trigger probability derived from Monte Carlo simulations (10,000 paths) calibrated to the CISSM dataset, represents the likelihood of a cyber cat bond being triggered. A lower probability indicates lower risk for investors. Table 4 presents the empirical results. The Loss Distribution Framework achieves the highest pricing efficiency with a Sharpe Ratio of 1.8230 and a low trigger probability of 0.50%, indicating a strong balance between risk and return. Combined this performance with its reliable loss prediction ability makes it suitable for investors seeking stable returns with minimal trigger risk and aligns with the study's emphasis on diversification benefits. Table 4 Results of Pricing Efficiency Model Sharpe Ratio Trigger probability Loss Distribution Framework 1.8230 0.50% Unified Bayesian Framework 0.4907 3.76% Signal-Processing Approach 0.0000 0.00% Regression Approach 0.0000 0.00% Copula-POT Model 0.0000 0.00% The Unified Bayesian Framework follows with a moderate Sharpe Ratio of 0.4907 but a higher trigger probability of 3.76%, suggesting less efficiency due to the higher trigger risk. This performance may be attributed to a higher standard deviation of loss prediction (Log MSE SD = 0.018). In contrast, the Signal-Processing Approach, Regression Approach and Copula-POT Model all yield a Sharpe Ratio with 0.0000 and a trigger probability of 0.00%, implying no bonds were triggered during the simulation and thus negligible returns. For the Signal-Processing Approach and Regression Approach, this aligns with their poor loss prediction performance, rendering them unsuitable for practical valuation. Notably, the Copula-POT Model has the most accurate and stable loss prediction ability, yet its Sharpe Ratio and trigger probability are both zero. It suggests an overly conservative pricing strategy, possibly due to data limitations, such as an insufficient number of extreme values for loss amount in CISSM dataset. This prevents the GPD from learning a sufficiently heavy tail, making it difficult to simulate extreme losses. The result highlights the Loss Distribution Framework as the most efficient model for pricing cyber cat bond, balancing high returns with low risk. The Copula-POT Model’s conservatism, despite its loss predication accuracy, warrants further investigation into its pricing assumption. I selected CVaR at the 99% confidence level to evaluate tail risk prediction accuracy, which measures the expected loss in the worst 1% of scenarios. A lower value indicates a better ability to manage extreme losses. VaR at the 99% confidence level and the relative standard error of CVaR, derived from Monte Carlo simulations (10,000 paths) calibrated to the CISSM dataset, provide additional insights into the models’ performance in capturing tail risk and prediction stability. Table 5 presents the empirical results. The Copula-POT Model shows the strongest overall performance with a low \(\:{CVaR}_{0.99}\) and the lowest relative standard error (0.02%), reflecting exceptional stability and accuracy in capturing tail risk. The low SE aligns with its superior loss prediction ability reinforcing its robustness despite a conservative pricing strategy. Table 5 Results of Tail Risk Prediction Accuracy Model \(\:{CVaR}_{0.99}\) ( $ M) \(\:{VaR}_{0.99}\) ( $ M) CVaR Relative SE Loss Distribution Framework 21,728.2538 14,090.5604 0.27% Unified Bayesian Framework 37,743.7876 25,817.1887 0.68% Signal-Processing Approach 1.7444 0.3061 1.24% Regression Approach 3.8023 3.1217 0.50% Copula-POT Model 7,250.2630 5,439.1907 0.02% The Loss Distribution Framework follows with a \(\:{CVaR}_{0.99}\) of $ 21,728.2538 million and a low relative standard error of 0.27%, indicating reliable tail risk estimates, consistent with its good loss prediction ability and high pricing efficiency. The Regression Approach achieves a moderate \(\:{CVaR}_{0.99}\) of $ 3.8023 million but a higher relative standard error (0.50%), suggesting less stable tail risk predictions, possibly due to its simplistic linear assumptions. The Unified Bayesian Framework performs poorly, with the highest \(\:{CVaR}_{0.99}\) and high relative standard error (0.68%), indicating weak tail risk prediction. It is possibly due to parameter selection in the DP-HBCRM framework, which may overestimate losses given the right-skewed nature of the distribution. Surprisingly, the Signal-Processing Approach exhibits the lowest \(\:{CVaR}_{0.99}\) . However, it also has high relative standard error (1.24%) and poor performance in loss prediction and investment efficiency. This suggests that these results may stem from model failure or unrealistic estimates. This subsection synthesizes the findings to evaluate the overall performance of five selected pricing models across three key criteria: loss prediction ability, pricing efficiency, and tail risk prediction accuracy. By integrating the results, I aim to identify the most suitable models for pricing cyber cat bonds and select candidates for portfolio optimization, addressing the research objectives outlined. The Loss Distribution Framework exhibits balanced performance across all criteria. It has a low Log MSE of 0.8514 and Log MSE SD of 0.0048, indicating reliable loss prediction. Moreover, it is the only model that passes the KS test (p-value = 0.5235), meaning a strong fit to historical loss distribution. It also achieves the highest pricing efficiency (Sharpe Ratio = 0.9542) showing its ability to balance risk and return. Furthermore, its \(\:{CVaR}_{0.99}\) of $ 21,728.2538 million and low relative standard error of 0.27% confirm the robust tail risk prediction, making it a versatile choice for investors navigating the evolving cyber risk market. The Copula-POT Model excels in loss predictive accuracy but is limited by its pricing efficiency. It shows the lowest Log MSE (0.0210) and Log MSE SD (0.0001), demonstrating superior loss prediction. Though its p-value of 0.0000 suggests distributional bias requiring further parameter refinement. Its \(\:{CVaR}_{0.99}\) of $ 7,250.2630 million and relative standard error of 0.02% highlight exceptional tail risk prediction, likely due to the POT model, which utilized EVT and large-scale simulations via Monte Carlo simulations. However, its Sharpe Ratio of 0.0000 indicates a conservative pricing strategy. It may be related to the high trigger amount or insufficient extreme value data in the CISMM dataset and will be examined during sensitivity analysis. The Unified Bayesian Framework shows mixed performance. It has moderate loss prediction ability with Log MSE of 1.7538 and Log MSE SD of 0.018, yet with p-value of 0.0000 suggests poor distribution fit. It also has second-highest pricing efficiency. However, its high trigger risk (trigger probability = 3.75%), which is higher than the average of 1.83% for cyber cat bonds (Artemis.bm 2025), limits its appeal. Furthermore, the second highest relative standard error of \(\:{CVaR}_{0.99}\) (0.68%) reflects weak tail risk prediction, likely due to DP-HBCRM parameter overestimation of right-skewed tails.The Signal-Processing Approach and Regression Approach underperform across all criteria. They both have high Log MSE and Log MSE SD among five models and zero Sharpe Ratio and trigger probability. Even though the Signal-Processing Approach achieves the lowest \(\:{CVaR}_{0.99}\) of $ 1.7444 million, its high relative standard error of 1.24% indicates a potential for methodological unsuitability for cyber cat bonds. It is possibly due to the inadequacy of the CISMM dataset for the model and will be further explored. The Regression Approach demonstrates moderate in tail risk prediction; however, the overall results remain weak. This may be attributed to its simplistic linear or parametric assumptions and will be investigated. To compare the relative performance, a weighted scoring system, as mentioned in section 3.3 , is applied: Composite Score = 0.6 × (Normalized Loss Prediction Score) + 0.2 × (Normalized Sharpe Ratio) + 0.2 × (Normalized CVaR Accuracy. Scores range from 1.0 (best) to 0.2 (worst) per criterion, with lower log MSE and relative SE, and higher Sharpe Ratio, receiving higher scores. Figure 5 shows the visualized results in a bar chart. Based on these findings, I select Loss Distribution Framework, Copula-POT Model and Unified Bayesian Framework for portfolio optimization in 4.3. The Loss Distribution Framework and the Copula-POT Model achieve the highest composite score, indicating balanced performance. The Loss Distribution Framework has a balanced performance with the highest pricing efficiency, showing the potential to be the most appropriate pricing model for cyber cat bonds and providing stable and risk-adjusted returns for cyber cat bond investors. The Copula-POT Model exhibits superior loss prediction and tail risk accuracy. However, its Sharpe Ratio of 0.0000 requires further investigation, though it remains a promising candidate for a reliable pricing model for cyber cat bonds. The Unified Bayesian Framework, ranked third among five models, shows moderate investment efficiency and dynamic updating potential. The Signal-Processing Approach and Regression Approach are excluded due to their consistent underperformance and low composite scores. 4.2 Portfolio Optimization Results This section presents the results of adding cyber cat bonds into a diversified investment portfolio and a HY Index, using a modified MPT approach. The analysis evaluates three selected pricing models—Loss Distribution Framework, Copula-POT Model, and Unified Bayesian Framework—based on their ability to enhance risk-adjusted returns, as measured by the Sharpe Ratio. I integrate cyber cat bonds into a baseline 60/40 portfolio and a HY benchmark; therefore, it is possible to access the practical value of these pricing models from an investor’s perspective, including diversification and optimization of portfolios. Results are presented with a cyber cat bond allocation of 5% and 10% for the baseline portfolio (reflecting conservative institutional strategies) and 30% and 70% in the HY portfolio (simulating alternative asset replacement scenarios), following sensitivity analyses to test robustness across different assumptions. The baseline portfolio, designed to reflect a realistic balanced fund, yields an annualized return of 6.18%, an annualized volatility of 12.42%, and a Sharpe Ratio of 0.176. This performance serves as a benchmark for assessing the impact of adding cyber cat bonds, which are modeled as 2-year zero-coupon bonds with a face value of $ 100 million, a risk premium of 11.3%, and a per-occurrence indemnity trigger of $ 558.11 million, consistent with market standards. Tables 6 and 7 show the results for cyber cat bond allocations of 5% and 10%, respectively. They reveal a clear improvement over the baseline portfolio, indicating that cyber cat bonds enhance risk-adjusted returns, likely due to their low correlation with traditional assets (assumed at 0.25) and attractive risk premia. Table 6 Results of Portfolio Optimization with 5% Cyber Cat Bond Model Sharpe Ratio Ann. Expected Return Ann. Volatility Loss Distribution Framework 0.352 7.00% 8.54% Unified Bayesian Framework 0.323 6.83% 8.78% Copula-POT Model 0.353 6.97% 8.40% Table 7 Results of portfolio optimization with 10% cyber cat bond Model Sharpe Ratio Ann. Expected Return Ann. Volatility Loss Distribution Framework 0.441 7.81% 8.64% Unified Bayesian Framework 0.378 7.47% 9.18% Copula-POT Model 0.447 7.74% 8.36% The Copula-POT Model shows the highest Sharpe Ratio at both 5% (0.353) and 10% (0.447) allocations, paired with the lowest annualized volatility. However, this outcome is attributed to zero trigger probabilities, suggesting that its conservative pricing strategy may limit standalone performance but becomes advantageous in a diversified portfolio. This discrepancy warrants further investigation, which I will discuss in section 5 . The Loss Distribution Framework perform robustly, with a Sharpe Ratio of 0.352 (5%) and 0.441 (10%), and annualized volatility of 8.54% and 8.64%. The slight increase in volatility at the 10% allocation shows a trade-off between higher returns and higher risk. Nonetheless, its balanced performace across all criteria makes it a reliable choice for investor. The Unified Bayesian Framework has the weakest performance with a Sharpe Ratio of 0.323 (5%) and 0.378 (10%), and the highest annualized volatility. Its relatively higher trigger probability likely causes the increasing portfolio risk and then reduces its risk-adjusted returns compared to the other models. The benchmark, represented by the ICE BofA High Yield Index, yields an annualized return of 4.11%, an annualized volatility of 9.69%, and a Sharpe Ratio of 0.0116, offering a starting point to evaluate the potential of cyber cat bonds. Table 8 and Table 9 present the results for portfolios mixing cyber cat bonds with HY bonds, with 2 different allocations, using the same pricing models and assumptions as in the previous subsection. The results show that all models achieve higher Sharpe Ratios than the benchmark across both allocations. Expect to the Unified Bayesian Framework of 70% allocation, other cases all exhibit lower annualized volatility with higher annualized returns and Sharpe Ratio. This suggests cyber cat bonds, especially when priced with the Copula-POT Model and Loss Distribution Framework, can significantly enhance risk-adjusted returns, making them a compelling alternative to HY bonds. Table 8 Sharpe Ratio for 30% Cyber Cat Bonds + 70% HY Bonds Model Sharpe Ratio Ann. Expected Return Ann. Volatility Loss Distribution Framework 0.4905 7.70% 7.54% Unified Bayesian Framework 0.2800 6.68% 9.56% Copula-POT Model 0.5116 7.47% 6.78% Table 9 Sharpe Ratio for 70% Cyber Cat Bonds + 30% HY Bonds Model Sharpe Ratio Ann. Expected Return Ann. Volatility Loss Distribution Framework 1.3909 12.48% 6.10% Unified Bayesian Framework 0.4591 10.10% 13.29% Copula-POT Model 2.7348 11.95% 2.91% To access the robustness of these results, I also conducted univariate sensitivity analyses by varying key assumptions individually, such as cyber cat bond allocations (5% vs. 10%) and portfolio allocations (50/50, 60/40, 70/30) for the baseline portfolio, trigger amounts ( $ 300 million, $ 558.11 million, $ 800 million) and correlation coefficients (0.1, 0.25, 0.4). For baseline portfolio, these analyses build on the 10% allocation results from Table 7 , and for HY bonds, these analyses build on the 70% cyber cat bonds + 30% HY bonds results from Table 9 . Figure 6 shows the trigger amount analysis and Table 10 shows the trigger probabilities and corresponding Sharpe Ratio (single bond). Table 10 indicates an inverse relationship between trigger amounts and trigger probabilities—higher triggers reduce the likelihood of bond activation. Table 10 Trigger Probability and Sharpe Ratio of Individual Bond by Trigger Amount Model Trigger: $ 300M Trigger: $ 588.11M Trigger: $ 800M Loss Distribution Framework 1.10% (1.192) 0.5% (1.923) 0.24% (2.665) Unified Bayesian Framework 3.98% (0.483) 3.76% (0.490) 3.64% (0.498) Copula-POT Model 0.01% (0.000) 0.00% (0.000) 0.00% (0.000) The Copula-POT Model maintains the highest and stable Sharpe Ratio of 0.447 across all trigger levels. This phenomenon may be attributed to the fact that the trigger probabilities are equal to zero, consequently, minimizing expected losses and leading to the identical bond prices. The Loss Distribution Framework shows moderate sensitivity to trigger amounts, with Sharpe Ratios increasing from 0.43 at $ 300 million to 0.447 at $ 800 million. This behavior is likely due to its pricing model for zero-coupon bonds, where the price is determined only by the discounted present value of the face value, unaffected by trigger amounts. Therefore, higher trigger amounts (lower trigger probabilities) yield to higher Sharpe Ratio. In contrast, the Unified Bayesian Framework exhibits a flat Sharpe Ratio of 0.379 − 0.378, suggesting robust results and limited sensitivity even with different trigger probabilities. This could also result from its bond pricing mechanism. A higher trigger amount reduces the probability of the bond being triggered, thereby lowering expected losses and stabilizing returns. Overall, the Sharpe Ratio of the Loss Distribution Framework varies with trigger amounts; however, the results are very similar to Table 6 , irrespective of the selected trigger amount. The Copula-POT Model demonstrates the best performance, while the Loss Distribution Framework follows. The Unified Bayesian Framework performs the worst. Figure 7 shows the correlation coefficient analysis, highlighting the diversification benefits of cyber cat bonds. The Copula-POT Model holds a consistent Sharpe Ratio of 0.447 across correlation levels. The Loss Distribution Framework and the Unified Bayesian Framework both exhibit the same trend: lower correlations lead to higher Sharpe Ratios. However, while the Loss Distribution Framework maintains a relatively steady performance across different correlation levels, the Unified Bayesian Framework shows greater sensitivity to higher correlations, likely due to its higher volatility. The overall results suggest that lower correlations significantly enhance Sharpe Ratio, and they remain consistent with the results from Table 6 . Notably, at a correlation of 0.1, the Loss Distribution Framework slightly outperforms the Copula-POT Model, possibly due to favorable simulation conditions. Figure 8 shows the portfolio allocation analysis. These results indicate that Sharpe Ratio tends to increase as the equity proportion increases. Moreover, they also remain identical ranking as Table 7 , indicating the result is not affected by portfolio allocation. For the HY portfolio, Figs. 9 and 10 present the similar sensitivity analyses for trigger amounts and correlations. The Copula-POT Model maintains outperformance and stability across all assumptions. The Loss Distribution Framework follows with moderate results; however, it demonstrates the sensitivity to both trigger amounts and correlation coefficient. The Unified Bayesian Framework perform the worst as Table 9 presents across all assumptions, but it shows the relative stable Sharpe Ratio. It is evident that, irrespective of the assumption applied, the 70% cyber cat bonds + 30% HY bonds allocation exhibits superior performance in comparison to its benchmark (100% HY bonds). The sensitivity analyses reinforce the results from Table 7 and Table 9 that the Copula-POT Model and Loss Distribution Framework outperform the Unified Bayesian Framework, offering reliable performance for cyber cat bond integration into diversified portfolios and HY bonds, and cyber cat bonds exhibit significant potential to serve as an alternative to HY bonds while offering better risk-adjusted returns. Furthermore, the practical significance lies in guiding investors toward higher trigger amounts (e.g., $ 800 million) and lower correlations (e.g., 0.1) to maximize returns. Regarding model robustness the Copula-POT Model demonstrates high stability across all parameters, suggesting minimal need for structural changes, though its zero trigger probability may reflect data limitations. The Loss Distribution Framework shows moderate sensitivity to trigger amounts and correlation coefficients in both portfolios. The Unified Bayesian Framework is sensitive to correlation coefficients in the baseline portfolio, highlighting a need to reduce model volatility. These insights indicate the importance of precise historical loss data and correlation assumptions to strengthen model reliability. However, the analysis is constrained by several limitations. The assumed correlation ranges (0.1–0.4) are based on sparse empirical evidence from sources like S&P Global Ratings (2025), which could potentially underrepresent real-world dynamics. Additionally, the univariate approach may overlook interaction effects between parameters, such as the combined impact of trigger amounts and correlations, which could alter outcomes. This section consolidates the key insights derived. Firstly, incorporating cyber cat bonds into the baseline portfolio enhances risk-adjusted returns of the portfolio, likely due to low correlation between cyber cat bonds and traditional assets and diversification benefits. Secondly, replacing a proportion of HY bonds with cyber cat bonds also enhances risk-adjusted returns. Therefore, cyber cat bonds have great potential to become an alternative to HY bonds. Additionally, the sensitivity analysis confirms the robustness of these findings across different assumptions such as trigger amounts and correlation coefficients. These results emphasize the prospective value of cyber cat bonds as a distinct asset class within the modified framework of MPT, thereby establishing a basis for improved investment strategies, although challenges due to data availability and model accuracy are to be noted. 5. Discussion The findings from section 4 provide key insights into the valuation of cyber cat bond pricing models and their implications for portfolio optimization. The comparative analysis of five models—the Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach, and Copula-POT Model—highlights their respective strengths and limitations. The Loss Distribution Framework aligns well with historical data, passing the KS test, and performs robustly across loss prediction, pricing efficiency, and tail risk criteria. The Unified Bayesian Framework benefits from incorporating prior knowledge, enhancing robustness under data scarcity, though improper priors can introduce bias. The Signal-Processing and Regression Approaches underperform due to limited variability and insufficient data, yielding poor predictive accuracy. The Copula-POT Model excels in loss prediction and tail risk estimation, achieving the highest composite scores, though its zero trigger probability limits individual bond Sharpe Ratios, likely due to dataset constraints. Nevertheless, in diversified portfolios, this conservatism enhances risk-adjusted performance under the modified MPT framework (FBI 2023, 7; FBI 2024, 4). Portfolio optimization results demonstrate the practical benefits of integrating cyber cat bonds. Adding 10% of cyber cat bonds priced via the Copula-POT Model increases the Sharpe Ratio from 0.176 to 0.447 in a baseline portfolio, while substituting 30% of HY bonds raises it from 0.0116 to 0.5116. These improvements stem from low correlation with traditional assets, offering diversification and superior risk-adjusted returns compared to HY bonds. Sensitivity analyses confirm the robustness of these results across varying trigger amounts and correlations, emphasizing the strategic advantage of cyber cat bonds in institutional portfolios. Challenges primarily relate to data limitations. The CISSM dataset lacks detailed loss amounts and extreme event observations, constraining model accuracy, especially for extreme-value methods like the Copula-POT Model. Manual data supplementation may introduce inconsistencies, while parameter selection, log transformations, and model assumptions create additional uncertainty. Moreover, cyber risks are inherently volatile and rapidly evolving, making historical data less predictive for future losses. These limitations underscore the need for caution in interpreting results but do not diminish the observed diversification and portfolio benefits. Future research should expand datasets, utilize machine learning to impute missing data, and develop hybrid models combining the Copula-POT and Loss Distribution Framework strengths. Interdisciplinary approaches could investigate investor behavior to improve adoption of cyber cat bonds. Empirical validation through recent deals will help refine model accuracy and assess real-world portfolio benefits. Advancing these directions will strengthen cyber cat bonds’ role as a viable alternative to HY bonds and enhance their integration into capital markets, addressing both predictive and portfolio optimization objectives (FBI 2023, 7; FBI 2024, 4). 6. Conclusion This study has explored the pricing models for cyber cat bonds and their integration into investment portfolios, addressing a critical gap in the implementation of cyber risk management with modern financial strategies. The key findings reveal that among the five pricing models evaluated—the Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach, and Copula-POT Model—the Copula-POT Model demonstrates superior and robust performance in loss prediction capability. It achieved the highest scores in accuracy and stability due to its effective capture of dependencies via Copula functions and extreme value modeling with POT, making it ideal for handling the volatile nature of cyber losses. The Loss Distribution Framework followed with the same composite score as the Copula-POT Model, excelling in pricing efficiency with a Sharpe Ratio of 0.9542 and robust distribution fit, as confirmed by its successful KS test results. Other models, such as the Unified Bayesian Framework and Regression Approach, showed limitations in predictive ability due to parameter sensitivities and incomplete data, while the Signal-Processing Approach underperformed owing to uniform event states in the CISSM dataset. In portfolio optimization, incorporating cyber cat bonds under a modified MPT framework enhances diversification, offering a distinct risk-return profile that outperforms the balanced portfolio and traditional HY bonds in stability across varying market conditions, as shown in the comparative and sensitivity analyses. These contributions advance the literature by providing a robust framework for pricing cyber cat bonds and empirical evidence of their role as an alternative asset class, addressing three primary objectives outlined and filling research gaps identified regarding model comparisons and practical applications. Empirically, the study assessed and compared the performance of multiple pricing models using a standardized CISSM dataset supplemented with public loss amount data, systematically evaluating their predictive validity and real-world applicability through metrics like accuracy, robustness, and KS tests. Moreover, the identification of the highest-performing model for loss prediction was also achieved. Theoretically, the application of these results to MPT maximized risk-adjusted returns, demonstrating through baseline optimizations and comparisons that adding cyber cat bonds into a balanced portfolio can enhance the risk-adjusted return, and they can serve as a viable substitute for HY bonds, providing investors with diversification benefits. Practically, the portfolio optimization results demonstrate tangible benefits, such as elevating the baseline portfolio's Sharpe Ratio from 0.176 to 0.447 with a 10% allocation priced via Copula-POT Model, and outperforming HY bonds with a Sharpe Ratio of 2.7348 in a 30% HY bonds and 70% cyber cat bonds allocation. Sensitivity analyses confirm robustness across trigger amounts, correlation and allocation, emphasizing the models' resilience under varying assumptions. While limitations such as data inconsistencies and model complexities tempered the results, the objectives were met by offering actionable insights that bridge theoretical modeling with real-world investment implications. In final remarks, cyber cat bonds may represent a transformative tool in investment portfolios, particularly in a time when cyber risks are projected to cost the global economy $ 10.5 trillion annually by 2025 (Cybersecurity Ventures 2020). Their inclusion not only mitigates systemic risks through capital market transfer but also enhances investment efficiency, as evidenced by the 154% Sharpe Ratio improvement in optimized portfolios. As markets evolve, with cyber cat bonds comprising only 1.4% of the $ 56.1 billion cat bond market (Artemis.bm 2025), their untapped potential underscores the need for greater adoption. This study demonstrated that the Copula-POT Model is the most suitable choice among five selected pricing models for valuing cyber cat bonds and cyber cat bonds offer diversification beyond traditional assets, appealing to institutional investors seeking HY alternatives with low market linkage. However, unlocking their full potential requires resolution of data and modeling issues as posited in future research initiatives. In summary, this study affirms the strategic role of cyber cat bonds in encouraging innovative and resilient risk management and consequently greater and safer acceptance in the financial world. Declarations Author Contribution I have no statement References Anderson, David. 2018. Desmond Higham, and Yuyuan Sun. Computational complexity analysis for Monte Carlo approximations of classically scaled population processes. arXiv preprint arXiv:1512.01588v3 [math.NA]. https://arxiv.org/abs/1512.01588v3 . Accessed September 24, 2025. Artemis.bm. 2025. Catastrophe bonds & ILS outstanding by risk or peril. https://www.artemis.bm/dashboard/cat-bonds-ils-by-risk-or-peril/ . Accessed September 24, 2025. Artemis.bm. 2025. Catastrophe bonds & ILS Market Dashboard. https://www.arte- mis.bm/dashboard/ . Accessed September 24, 2025. Artemis.bm. 2025. Catastrophe Bond & Insurance-Linked Securities Deal Directory. https://www.artemis.bm/deal-directory/ . Accessed September 24, 2025. Bodie, Zvi, and Alex Kane, Alan Marcus. 2014. Investments . 10th ed. New York: McGraw Hill Higher Education. Braun, Alexander, Martin Eling, and Christian Jaenicke. 2023. Cyber insurance-linked se- curities. ASTIN Bulletin 53(3):684–705. 10.1017/asb.2023.22 Carter, Steve. 2018. Cyber-Catastrophe Insurance-Linked Securities . on Smart Ledgers. Long Finance. Chigada, Joel Rujeko Madzinga. 2021. Cyberattacks and threats during COVID-19: A systematic literature review. SA Journal of Information Management 23. 10.4102/sajim.v23i1.1277 Chimamiwa, George. 2024. Managing cyber risks in the face of AI- and ML-Driven Ad- versarial Attacks. 71–79. 10.70301/CONF.SBS-JABR.2024.1/1.6 Cummins, J. David. 2013. and Pauline Barrieu. Innovations in Insurance Markets: Hybrid and Securitized Risk-Transfer Solutions. 10.1007/978-1-4614-0155-1_20 Cummins, J. David, Philippe Trainar. 2009. Securitization, Insurance, and Reinsurance. The Journal of Risk and Insurance 76(3):463–492. 10.1111/j.1539-6975.2009.01313.x Cummins, J., Daivd, and Mary A. Weiss. 2009. Convergence of Insurance and Financial Markets: Hybrid and Securitized Risk-Transfer Solutions. Journal of Risk and Insurance 76(3):493–545. 10.2139/ssrn.1260399 CyberCube. 2023. Digital Ties and Natural Divides: Correlation and Diversification in Cyber Catastrophe Bonds. https://insights.cybcube.com/correlation-and-diversification-in- cyber-catastrophe-bonds . Accessed September 24, 2025. Cybersecurity Ventures. 2020. Cybercrime To Cost The World $ 10.5 Trillion Annually By 2025. https://cybersecurityventures.com/hackerpocalypse-cybercrime-report-2016/ . Ac- cessed September 24, 2025. Demers-Bélanger, and Kim, Can Sin Lai. 2020. Diversification benefits of cat bonds: An in‐depth examination. Financial Markets Institutions & Instruments 29(5):165–228. 10.1111/fmii.12132 Domfeh, Daniel, Anirban Chatterjee, and Matthew Dixon. 2022. A Unified Bayesian Framework for Pricing Catastrophe Bond Derivatives. arXiv preprint arXiv:2205.04520. https://arxiv.org/abs/2205.04520 . Accessed September 24, 2025. Drobetz, Wolfgang, and Henning Schröder, Lars Tegtmeier. 2019. The Role of CAT Bonds in an International Multi-Asset Portfolio: Diversifier, Hedge, or Safe Haven? SSRN Elec- tronic Journal . 10.2139/ssrn.3359277 Dubois, Elisabeth V., F. Omer, and Keskin. 2022. and Unal Tatar. Cyber Risk Modeling Meth- ods and Data Sets: A Systematic Interdisciplinary Literature Review for Actuaries. SOA Research Institute. Federal Bureau of Investigation. 2023. Internet Crime Complaint Center Report. https://www.ic3.gov/Media/PDF/AnnualReport/2023_IC3Report.pdf . Accessed September 24, 2025. Federal Bureau of Investigation. 2024. Internet Crime Complaint Center Report. https://www.ic3.gov/Media/PDF/AnnualReport/2024_IC3Report.pdf . Accessed September 24, 2025. FRED. 2025. ICE BofA US High Yield Index Effective Yield. https://fred.stlou- isfed.org/series/BAMLH0A0HYM2EY . Accessed September 24, 2025. FRED. 2025. Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity, Quoted on an Investment Basis. https://fred.stlouisfed.org/series/DGS10 . Accessed Sep- tember 24, 2025. Götze, Thomas, and Marc Gürtler, Eileen Witowski. 2020. Improving CAT bond pricing models via machine learning. Journal of Asset Management 21(5):428–446. 10.1057/s41260-020-00167-0 Harry, Charles, Nancy Gallagher. 2018. Classifying Cyber Events. Journal of Infor- mation Warfare 17(3):17–31. Hofer, Lorenz, Patrizio Gardoni, and Michele A. Zanini. 2019. Risk-Based CAT Bond Pricing Considering Parameter Uncertainties. Sustainable and Resilient Infrastructure 6(5–6):315–331. 10.1080/23789689.2019.1667116 IBM. 2024. Cost of a Data Breach Report. https://www.ibm.com/reports/data-breach . Ac- cessed September 24, 2025. ICE. 2025. Index Platform. https://indices.ice.com . Accessed September 24, 2025. Kish, Richard. 2023. Catastrophe (CAT) bonds: risk offsets with diversification and high returns. Financial Services Review 25(3):303–329. 10.61190/fsr.v25i3.3281 Kshetri, Nir. 2020. The evolution of cyber-insurance industry and market: An institutional analysis. Telecommunications Policy 44 (8). 102007. 10.1016 /j.telpol.2020.102007 Kolesnikov, Oleg, Aleksei Markov, Dmitry Smagulov, and Sergejs Solovjovs. 2022. Cyber. Loss Distribution Fitting. 2022. A General Framework towards Cyber Bonds and Their Pricing Models. International Journal of Mathematics and Mathematical Sciences 1–20. 10.1155/2022/7689828 Lane, Marton. 2000. Pricing Risk Transfer Transactions. ASTIN Bulletin 30(2):259–293. 10.2143/AST.30.2.504635 Li, Yifei, Richard Mamon. 2023. The Price Tag of Cyber Risk: A Signal-Processing Approach. Ieee Access : Practical Innovations, Open Solutions 11:44294–44318. 10.1109/ACCESS.2023.3272572 Markowitz, Harry. 1952. Portfolio Selection. The Journal of Finance 7(1):77–91. 10.2307/2975974 Mastroeni, Loretta, and Alessandro Mazzoccoli, Maurizio Naldi. 2023. Cyber Insurance Premium Setting for Multi-Site Companies under Risk Correlation. Risks 11(10):167. 10.3390/risks11100167 Mouelhi, Chaker. 2021. The Relationship Between Cat Bond Market and Other Financial Asset Markets: Evidence from Cointegration Tests. European Journal of Business and Management Research 6(2):78–85. 10.24018/ejbmr.2021.6.2.790 Munich Re. 2025. Cyber Insurance: Risks and Trends 2025. https://www.munichre.com/en/insights/cyber/cyber-insurance-risks-and-trends-2025.html . Accessed September 24, 2025. Niehaus, Greg. 2002. The Allocation of Catastrophe Risk. Journal of Banking and Fi- nance 26(2–3):585–596. 10.1016/S0378-4266(01)00235-7 Orlando, Albina, Maria Francesca Carfora, Fabio Martinelli, and Francesco Mercaldo. 2018. and Art- siom Yautsiukhin. Cyber Risk Management: A New Challenge for Actuarial Mathe- matics: MAF 2018. 10.1007/978-3-319-89824-7_36 Pain, David. 2024. Catalysing Cyber Risk Transfer to Capital Markets: Catastrophe bonds and beyond. Geneva Association . https://www.genevaassociation.org/publica- tion/cyber/catalysing-cyber-risk-transfer-capital-markets-catastrophe-bonds-and-beyond . Accessed September 24, 2025. Peters, Gareth, Pavel Shevchenko, Ruben Cohen, and Diane Maurice. 2017. Understand- ing Cyber Risk and Cyber Insurance. SSRN Electronic Journal . 10.2139/ssrn.3065635 Pham, Nhan, and Bing Cui. 2025. and Umar Ruthbah. The Performance of the 60/40 Portfolio: A Historical Perspective. CFA Institute. https://rpc.cfainstitute.org/research/re- ports/2025/performance-of-the-60-40-portfolio . Accessed September 24, 2025. Polacek, Andy. 2018. Catastrophe bonds: A primer and retrospective. Chicago Fed Letter . https://www.chicagofed.org/publications/chicago-fed-letter/2018/405 . Accessed September 24, 2025. S&P Global Rating. 2025. Cyber Risk Insights: Cyber Catastrophe Bonds Offer Greater Scope for Risk Mitigation. https://www.spglobal.com/ratings/en/regulatory/article/250317-cyber-risk-insights-cyber- catastrophe-bonds-offer-greater-scope-for-risk-mitigation-s13438912 . Accessed September 24, 2025. Sharpe, William. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Con- ditions of Risk. The Journal of Finance 19(3):425–442. 10.2307/2977928 Skeoch, Hannah. 2023. and Christos Ioannidis. The barriers to sustainable risk transfer in the cyber-insurance market. arXiv preprint arXiv:2303.02061. https://arxiv.org/abs/2303.02061 . Accessed September 24, 2025. Statista. 2025. Estimated Cost of Cybercrime Worldwide 2018–2029. https://www.sta- tista.com/forecasts/1280009/cost-cybercrime-worldwide . Accessed September 24, 2025. Strupczewski, Grzegorz. 2021. Defining Cyber Risk. Safety Science 135:105143. 10.1016/j.ssci.2020.105143 Tang, Yuchen, Chang Wen, Chenggang Ling, and Yujun Zhang. 2023. Pricing Multi- Event-Triggered Catastrophe Bonds Based on a Copula–POT Model. Risks 11(8):151. 10.3390/risks11080151 Tobin, James. 1958. Liquidity Preference as Behavior Towards Risk. The Review of Eco- nomic Studies 25(2):65–86. 10.2307/2296205 Trottier, Denis-Alexandre, Van Son Lai, and Anne-Sophie Charest. 2017. CAT Bond Spreads via HARA Utility and Nonparametric Tests. Journal of Fixed Income 28(2):52–69. 10.3905/jfi.2018.1.062 Vaugirard, Victor E. 2003. Pricing Catastrophe Bonds by an Arbitrage Approach. Quar- terly Review of Economics and Finance 43(1):119–132. 10.1016/S1062- Verizon. 2025. 2025 Data Breach Investigations Report. https://www.verizon.com/busi- ness/resources/reports/dbir/ . Accessed September 24, 2025. Willard, John. 2025. Cyber ILS market poised for growth, but must overcome key risk challenges: S&P. Artemis.bm . https://www.artemis.bm/news/cyber-ils-market-poised-for- growth-but-must-overcome-key-risk-challenges-sp/ . Accessed September 24, 2025. Woods, Daniel, Jessica Wolff. 2023. A History of Cyber Risk Transfer. SSRN Elec- tronic Journal . 10.2139/ssrn.4493171 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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1","display":"","copyAsset":false,"role":"figure","size":49577,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 5: Standardized Performance of Pricing Models Across Criteria\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7923044/v1/4e35c90e1c2c307daebe0860.png"},{"id":96060493,"identity":"802aa6ec-7b02-4d64-89f7-9d52c7909b6e","added_by":"auto","created_at":"2025-11-17 08:34:01","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":44252,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 6: Sensitivity Analysis of Sharpe Ratio by Trigger Amount for Baseline Portfolio\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7923044/v1/8c07014303792e172139e95c.png"},{"id":96060502,"identity":"90e11d44-28dd-4615-8ac2-81582a54b6a5","added_by":"auto","created_at":"2025-11-17 08:34:01","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":46922,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 7: Sensitivity Analysis of Sharpe Ratio by Correlation Coefficients for Baseline Portfolio\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7923044/v1/6b0209b1b7da2417ffb45b85.png"},{"id":96060498,"identity":"efc8b0df-d7be-4a72-b0ce-2e14acaeb7d5","added_by":"auto","created_at":"2025-11-17 08:34:01","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":42535,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 8: Sensitivity Analysis of Sharpe Ratio by Portfolio Allocations for Baseline Portfolio\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7923044/v1/48b2518bebd95c2249a0d06f.png"},{"id":96060496,"identity":"fd60a378-e164-4ca1-b8cf-e4ce94e20801","added_by":"auto","created_at":"2025-11-17 08:34:01","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":38509,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 9: Sensitivity Analysis of Sharpe Ratio by Trigger Amount for HY Portfolio\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7923044/v1/c176d5d48fd65020f5750e99.png"},{"id":96246333,"identity":"cf0ed3f0-1ce3-4ddb-9c67-65f5fc93bccc","added_by":"auto","created_at":"2025-11-19 07:25:27","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":41004,"visible":true,"origin":"","legend":"\u003cp\u003eFigure 10: Sensitivity Analysis of Sharpe Ratio by Correlation Coefficients for HY Portfolio\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-7923044/v1/d6ce3406bbe944b7442f9b41.png"},{"id":96362768,"identity":"537fdd5d-059d-4b54-b554-00dc7d5f18c9","added_by":"auto","created_at":"2025-11-20 09:48:46","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1638923,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7923044/v1/ebab22e8-5f37-4ae7-9502-2427c031dace.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Valuation of Cyber Catastrophe Bonds and Their Role in Portfolio Efficiency: An Analysis of Model Selection and Investment Implications ","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn recent years, technological advancements have become deeply integrated into global business, transforming industries into the digital economy and reshaping corporate value through intangible assets such as data, algorithms, intellectual property, and organizational expertise. The rapid development of artificial intelligence (AI) and automation has accelerated innovation and efficiency while simultaneously increasing exposure to cyber risk. Cyber threats, including data breaches, ransomware attacks, and system failures, pose significant operational and financial challenges. Global cybercrime costs are projected to reach \u003cspan\u003e$\u003c/span\u003e10.5 trillion annually by 2025 from \u003cspan\u003e$\u003c/span\u003e3 trillion in 2015 (Cybersecurity Ventures 2020) and may rise to \u003cspan\u003e$\u003c/span\u003e15.63 trillion by 2029 (Statista \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2025\u003c/span\u003e), reflecting an approximate 15% compound annual growth rate. In the United States alone, cybercrime losses grew by 22% in 2023 and surged by 33% in 2024 (FBI 2023, 7; FBI 2024, 4), averaging nearly 27% growth during this period. As the frequency and severity of cyber incidents increase, the need for robust mitigation mechanisms has become urgent, fueling the rapid growth of the cyber risk insurance market. Despite this, the global premium volume for cyber insurance remains low, standing at \u003cspan\u003e$\u003c/span\u003e15.3\u0026nbsp;billion in 2024 (Munich Re 2025), highlighting a substantial coverage gap and exposing financial vulnerabilities.\u003c/p\u003e\u003cp\u003eInnovative solutions such as cyber catastrophe bonds, or cyber cat bonds, have emerged as instruments to transfer cyber risk to capital markets. Over the past 15 years, capital market participation in bearing insurance risks has grown (Braun et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2023\u003c/span\u003e, 684). Nevertheless, despite \u003cspan\u003e$\u003c/span\u003e18.5\u0026nbsp;billion in cat bond issuance in the first half of 2025, cyber risk accounted for only 1.4% of the total \u003cspan\u003e$\u003c/span\u003e56.1\u0026nbsp;billion in risk capital outstandings (Artemis.bm 2025), indicating significant untapped potential. Unlike traditional cat bonds covering natural disasters, cyber cat bonds aim to cover financial losses from high-impact cyber events, yet they face substantial challenges. These include high risk premiums, uncertainties in cyber risk modeling, lack of historical loss data, extreme tail risks, and limited familiarity among market participants (Woods and Wolff 2023, 14; Kolesnikov et al. 2022, 1). The inherent volatility of cyber risks, coupled with the potential for interconnected losses such as widespread ransomware attacks or supply chain disruptions, complicates accurate assessment and pricing.\u003c/p\u003e\u003cp\u003eDespite these obstacles, the strong returns of traditional cat bonds over the past decade suggest that cyber cat bonds could become attractive capital instruments if modeling and market challenges are addressed (Willard \u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). Existing research proposes various pricing models, but comparative evaluations of predictive validity and real-world applicability remain scarce. This study seeks to systematically assess multiple models to determine effective strategies for valuing cyber cat bonds and to explore implications for portfolio management. Specifically, it evaluates and compares pricing models, identifies the most predictive model based on accuracy, robustness, and relevance, and applies the results to Modern Portfolio Theory to optimize risk-adjusted returns. The investigation also examines the potential of cyber cat bonds to enhance portfolio performance through diversification, owing to their low correlation with traditional assets (S\u0026amp;P Global Ratings 2025; Mouelhi \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2021\u003c/span\u003e, 82), and explores investment opportunities arising from market inefficiencies, such as pricing discrepancies.\u003c/p\u003e\u003cp\u003eThe study contributes to bridging theoretical developments in cyber cat bond modeling with practical portfolio applications. By comparing pricing models and integrating their outputs into portfolio optimization techniques, it provides a comprehensive understanding of cyber cat bonds as instruments for both risk transfer and investment, offering valuable insights for investors, insurers, and the growing cyber risk insurance market.\u003c/p\u003e"},{"header":"2. Literature Review","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1 Cyber Risk and Cyber Catastrophe Bonds\u003c/h2\u003e\u003cp\u003eThe rise of the digital economy has elevated cyber risk as a critical concern for organizations worldwide, encompassing potential adverse outcomes from cyberattacks, system vulnerabilities, and human errors that compromise confidentiality, integrity, or availability of information systems. Threats such as data breaches, ransomware attacks, distributed denial-of-service (DDoS) attacks, and phishing present dynamic challenges (Carter \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2018\u003c/span\u003e, 30\u0026ndash;31). Unlike traditional insurable risks, cyber risk is broad and rapidly evolving due to technological advancements like AI and machine learning (Chimamiwa \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2024\u003c/span\u003e, 71). The accelerated digital transformation, particularly during the COVID-19 pandemic, has expanded vulnerabilities, providing new opportunities for cybercriminals (Chigada and Madzinga 2021, 1). Cyber incidents can generate systemic losses, spreading across companies, sectors, and even global relationships, causing financial, operational, and reputational damage (Mastroeni et al. 2023, 15). Data breaches now cost an average of \u003cspan\u003e$\u003c/span\u003e4.88\u0026nbsp;million, a 10% increase year-over-year (IBM \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2024\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eThe cyber risk insurance market has grown to mitigate these financial consequences, yet remains underinsured, with a global premium volume of \u003cspan\u003e$\u003c/span\u003e15.3\u0026nbsp;billion projected to reach \u003cspan\u003e$\u003c/span\u003e16.3\u0026nbsp;billion in 2025 (Munich Re 2025), far below the expected \u003cspan\u003e$\u003c/span\u003e10.5 trillion annual cost of cybercrime by 2025, rising to \u003cspan\u003e$\u003c/span\u003e15.63 trillion by 2029 (Munich Re 2025; Cybersecurity Ventures 2020; Statista \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). Catastrophe bonds, traditionally used to transfer natural disaster risks to capital markets, offer a model for cyber risk transfer, allowing insurers to shift extreme loss exposures to investors in exchange for high risk-adjusted returns (Cummins and Weiss 2009, 1). Cyber cat bonds adapt this structure to the unique characteristics of cyber risks. Global issuance reached \u003cspan\u003e$\u003c/span\u003e533.75\u0026nbsp;million in 2024, representing only 1.4% of total cat bond risk capital (Artemis.bm 2025).\u003c/p\u003e\u003cp\u003eChallenges remain, including difficulties in modeling cyber risks, lack of historical data, information asymmetry between insurers and investors, and trigger design limitations (Orlando et al. \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2018\u003c/span\u003e, 2; Skeoch and Ioannidis 2023, 25; Cummins and Weiss 2009, 42). Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e: Comparison between Trigger Types of Cyber Cat Bonds illustrates the differences between parametric and indemnity triggers and highlights the trade-offs between accuracy and payout speed that must be considered in the design of cyber cat bonds. Pricing models often rely on statistical and machine learning approaches but are constrained by data scarcity and potential biases (Dubois et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2022\u003c/span\u003e, 47). Despite these issues, cyber cat bonds offer growing potential for risk management and investment, particularly as the cyber landscape evolves (S\u0026amp;P Global Ratings 2025).\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eComparison between Trigger Types of Cyber Cat Bonds\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eTrigger Type\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eAdvantages\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eLimitations\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eCyber-Specific Challenges\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eParametric\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eFast payout, low basis risk for defined events\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMay not fully reflected actual losses\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003eHigher basis risk due to\u003c/p\u003e\u003cp\u003eunpredictable attack vectors\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eIndemnity\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eDirectly tied to insured losses\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eValidation delays, moral hazard\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003eDifficult validation in opaque\u003c/p\u003e\u003cp\u003eIncidents; potential for asymmetric information\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2. Pricing Models for Cyber Catastrophe Bonds\u003c/h2\u003e\u003cp\u003eThis section focuses on the pricing solutions of cyber cat bonds and conducts a comprehensive review and analysis of the pricing framework from Kolesnikov et al. (2022), Domfeh et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), Lane (\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2000\u003c/span\u003e), Li and Mamon (2023), and Tang et al. (\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). A systematic comparison of these models is conducted to understand their approaches, underlying assumptions, strengths, and limitations, and this comparison offers strong and direct evidence on the theoretical basis of the present study, which also provides a theoretical foundation for subsequent empirical studies.\u003c/p\u003e\u003cp\u003eKolesnikov et al. (2022) propose a pricing model based on loss distribution that focuses on modeling the expected loss and risk premium of cyber cat bonds. Using a cyber events database published on the public domain, they estimate parameters for the cyber loss distribution and calculate bond prices, yields, and other features by numerical simulation. The cyber losses are assumed to share a heavy-tailed distribution (e.g., Pareto distribution) to account for the extreme nature of cyber event losses. Monte Carlo simulations are employed by the model to generate a high number of loss scenarios and to simulate the expected loss and the likelihood of the bond being triggered based on these scenarios. Kolesnikov et al. emphasized that, due to the diversity of types of cyber loss, the model should account for the joint distribution of various types of loss (such as direct economic loss and reputational losses) for a more\u0026ensp;realistic characterization of the risk exposure of the bond. The strength of the model lies in its capacity to adapt to different types of cyber events and triggering conditions. But it does have its drawbacks. The model is highly sensitive to the parameters of the loss distribution. The limited historical data make the parameter estimation more uncertain. Moreover, the computation cost of Monte Carlo simulation is very high, and it may constrain the practical usage of Monte Carlo simulation. Nonetheless, this model was selected due to the ability to handle the heavy-tailed nature that is prevalent in cyber losses, hence filling the lack of historical data that is related to extraordinary events; additionally, it is among the few models that are specifically focused on cyber bonds and offers an intuitive approach, making it even more relevant and providing a foundational tool to value within this emergent market.\u003c/p\u003e\u003cp\u003eDomfeh et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) propose a unified Bayesian framework for pricing cat bond derivatives and that is specifically designed for handling the associated uncertainty involved in catastrophic risks. This model utilized a Dirichlet Prior-Hierarchical Bayesian Collective Risk Model (DP-HBCRM) to estimate\u0026ensp;the frequency and severity of insurance loss events. It is extended by adding a stochastic interest rate model to separate the risk premium, and it is thus capable of valuing cat bonds, including cyber cat bonds. The DP-HBCRM formulates the loss process as a compound Poisson model in which the number of events is a Poisson distribution with a rate parameter, loss values are governed by a distribution with heavy tails (e.g., lognormal), which accounts for the extremeness of cyber losses. Estimation of the parameter rests on Bayesian inference and a prior of the form of a Dirichlet distribution used for capturing event frequency. With this approach, the model can reflect a prior of current knowledge of event frequencies and use it to update the prior with observed data. This renders the model especially useful in situations where the history of data is limited, as is often the case in the context of cyber risk assessment. In addition, the model features a stochastic interest rate setting, usually a Vasiček process. Incorporation of the market action helps determine the risk premium such that realized bond prices reflect expected losses plus investors' views of risks. The strength\u0026ensp;of this framework is its applicability to consolidating heterogeneous data, namely industry reports on cyber events and public repositories and databases of incidents, to improve the prediction accuracy of loss. This flexibility is important\u0026ensp;in cyber risk, where little historical loss information exists. However, the model\u0026rsquo;s need of\u0026ensp;subjective prior distribution injects subjectivity into identifiability, which possibly would influence the parameter estimates\u0026rsquo; stability. Furthermore, the computational cost of Bayesian inference, in particular with methods based in Markov Chain\u0026ensp;Monte Carlo (MCMC) simulations does not easily allow its practical use. Notwithstanding these limitations, this model was chosen for its effectiveness in uncertain environments, such as evolving cyber threats with limited prior data. It provides an organized and theoretically valid methodology for pricing cyber cat bonds, offering a platform for pricing activities in a new and data-limited market.\u003c/p\u003e\u003cp\u003eLi and Mamon (2023) propose a new method to price network risk, rooted in signal processing. They treat network attack events as random signal sequences in their model. These sequences are modeled by Hidden Markov Model (HMM) and Expectation Maximization (EM) techniques. The model assumes periodic and trend behavior of network losses and\u0026ensp;it uses a non-homogeneous Regime-Switching Markov Model (RSMM) to detect both: first, the time at which network attacks happen, and second, the interval and duration of the attacks. The state transition modeling reflects three phases in the\u0026ensp;Cyber Kill Chain (CKC): firewall normal, firewall failure, and anti-phishing failure. The loss distribution is assumed to be of Doubly-Truncated Pareto Distribution and the discount rate is modeled by the Vasiček\u0026ensp;process. Based on the reference measure transformation and the filtering technique, it keeps the quadratic payment factor and stochastic discount factor both dynamic which\u0026ensp;is the desired form for the parameter estimation and premium evaluation in standard deviation and index premium principles, respectively. The model's innovation lies in its integration of signal processing techniques into the pricing of cyber risk, which is especially appropriate for including the time-dependent and dynamic nature of cyberattacks. This method naturally complements the traditional statistical techniques. Being explicit about cyber risks, it is also simultaneously a more suitable risk model for pricing cyber cat bonds. But the fact that it presupposes stability and cyclicality in network losses may not hold for cyber risk. Regardless, this approach was considered due to the emphasis on filtering noise out of incomplete cyber event reports, designed specifically for cyber risks, to complement predictive models. Its dynamic modeling suits the time-dependent nature of cyberattacks, offering a novel approach for cyber cat bond pricing, provided high-quality time series data becomes available.\u003c/p\u003e\u003cp\u003eLane (\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2000\u003c/span\u003e) proposes a general framework that aims to provide a systematic pricing methodology for risk transfer transactions. The model employs a regression analysis to estimate bond pricing parameters based on expected loss (EL), risk load, and market risk premium. It is centered on probability of first loss (PFL) and conditional expected loss (CEL) and decomposes the bond price into three components: Risk-Free Interest Rate (RFIR), Expected Loss (EL), and Risk Premium (RP). Lane emphasizes that pricing models have to take into account market dynamics, investor risk appetite and the asymmetric distribution of risk events so that the bond is attractive and saleable in the capital markets. While the model was originally created to estimate risks associated with natural disasters, such as earthquakes and hurricanes, its theoretical framework is quite general. This flexibility makes it suitable for use in\u0026ensp;cyber cat bonds. The simple and efficient structure can be conveniently applied in data-poor and immature market\u0026ensp;conditions, as in the cyber cat bond market at present. But its simplifying assumptions \u0026mdash; such as risk premium being\u0026ensp;in a linear relationship to other, risk premium, itself \u0026ndash; may not adequately capture cyber risk\u0026rsquo;s complex natures. These natures consist of systematic dependence, nonlinear properties of multi-event triggering mechanisms, and a very high level of\u0026ensp;tail risk. Moreover, the model has greater reliance on market data, and the scarce historical data of the cyber cat bond market further exacerbates parameter estimation uncertainty. Nevertheless, Lane's framework was selected to explore causal relationships between cyberattack variables and losses, enhancing interpretability in interconnected systems through empirical fitting of spreads. It can be considered a cornerstone of the price of cyber cat bonds. It breaks down the drivers of cyber risk to help investors and issuers understand the logic that underpins pricing cyber risk,\u0026ensp;which can be challenging in nascent markets.\u003c/p\u003e\u003cp\u003eTang et al. (\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) propose a pricing framework for multi-event triggered cat bonds, particularly for those associated with high-dependence structures. The model combines the Extreme Value Theory (EVT) Peaks-Over-Threshold (POT) methodology with a nested Archimedean Copula to describe the tail behavior of multi-trigger indicators and complex dependencies. It also analyzes the sensitivity of bond prices through Monte Carlo simulations. The model integrates the POT approach with the Generalized Pareto Distribution (GPD) to model tail risk and relies on the nested Archimedean Copula to estimate the nonlinear dependencies among indicators. This embedding layer can help the model effectively deal with the\u0026ensp;multiple event triggers as well. Tang et al. emphasize that the pricing model should reflect the asymmetric in the distribution of triggering indicators, as well as the dynamics of the frequency and the size of catastrophes and the risk appetite of market participants. This will help to improve the market attractiveness of bonds. While this model was not originally designed to address cyber risks, its methodology aligns well with the pricing of cyber cat bonds. This model's strength lies in its ability to model tail dependencies, particularly suited for cyber risks where multi-event triggers create correlated extremes, as the nested Archimedean copula captures asymmetric joint tails more effectively than independent assumptions in other models. It stands out in capturing joint distributions of rare events, offering superior accuracy in pricing high-impact, correlated losses and it is also why this model was chosen. However, the noted limitation in capturing systematic correlations suggests future research could integrate dynamic Copula structures to better reflect evolving cyber risk dependencies.\u003c/p\u003e\u003cp\u003eModern Portfolio Theory (MPT) provides a systematic framework to optimize the balance between risk and return in investment portfolios. Developed by Harry Markowitz in the 1950s, MPT enables investors to construct portfolios that maximize expected returns for a given risk level or minimize risk for a desired return through strategic asset selection. Its foundation, the Mean-Variance Model, formalizes portfolio risk using the standard deviation of returns and a covariance matrix capturing asset correlations, producing the Efficient Frontier of optimal risk-return combinations. Extensions such as Tobin\u0026rsquo;s Separation Theorem and Sharpe\u0026rsquo;s Capital Asset Pricing Model (CAPM) further refine the treatment of risk, introducing concepts like systematic risk and emphasizing diversification. In multi-asset portfolios, low correlation between assets reduces total risk, a principle particularly relevant for insurance-linked securities like cat bonds (Cummins and Weiss 2009, 439). Cyber cat bonds share the same return-risk structure as traditional cat bonds, but their risk stems from events such as data breaches or ransomware attacks, which exhibit volatile and systemic loss correlations (Kolesnikov et al. 2022, 2). Their low correlation with conventional assets makes them suitable for diversification, yet limited data and price uncertainty complicate risk assessment (Braun et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2023\u003c/span\u003e, 1). While MPT assumes normally distributed returns and relies on historical data\u0026mdash;assumptions challenged by the heavy-tailed, scarce data of cyber cat bonds (Mastroeni et al. 2022, 2; Woods and Wolff 2023, 3)\u0026mdash;it still provides a structured framework to compute risk-adjusted returns, such as the Sharpe Ratio, and optimize portfolio efficiency. Integrating expected returns, risks, and correlations of cyber cat bonds allows investors to determine optimal allocations and evaluate their role in multi-asset portfolios alongside high-yield bonds.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e2.3 Research Gaps\u003c/h2\u003e\u003cp\u003eAs\u0026ensp;a new risk transfer tool, cyber cat bonds have gained significant attention in both theory and practice. However, this review uncovers persistent unresolved gaps that hinder their full development. It is evident from the literature that there are some academic and application issues that have not been solved, and different viewpoints on how to model and price cyber risks, such as the loss distribution approach from Kolesnikov et al. (2022), the Bayesian models from Domfeh et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), and the Copula-POT framework from Tang et al. (\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Yet, most of these contributions focus only on refining and applying a single model, without undertaking a broader comparative analysis across multiple frameworks to gauge their predictive accuracy. This gap limits the ability to identify the most suitable pricing strategy and, in turn, undermines the precision of cyber cat bond valuations in practice.\u003c/p\u003e\u003cp\u003eMoreover, previous research is limited in its application of pricing model results to the optimization of investment portfolios. Despite a widespread use of MPT in examining the diversification effects of cat bonds, the available literature about the investment implications of cyber cat bonds, particularly the specific influence of their returns, risks, and correlations on portfolio efficiency, remains disparate, comparing only isolated case studies rather than an integrated framework. Mastroeni et al. (2022) emphasize that the systematic nature of cyber risk combined with the lack of data increases the complexity of investment decisions. However, there exist few studies have unified pricing models with MPT to investigate comprehensive investment strategies or arbitrage positions of cyber cat bonds, including arbitrage positions that could exploit pricing inefficiencies in this nascent market. For example, the lack of historical and standardized loss data contributes to elevated spreads in the cyber cat bond market. Therefore, investors have opportunities to capitalize on pricing inefficiencies and potential misevaluations.\u003c/p\u003e\u003cp\u003eBuilding on these gaps, the review establishes a foundation for pricing and investment strategies. Yet, no detailed systematic comparison and empirical validation exist. This study addresses these shortcomings by systematically comparing five different pricing models, determining the most predictive model based on accuracy, robustness and relevance. Then integrating these findings with MPT to access cyber cat bonds\u0026rsquo; role in investment portfolios. This approach, as outlined in the methodology (Section \u003cspan refid=\"Sec6\" class=\"InternalRef\"\u003e3\u003c/span\u003e), is designed to deliver actionable guidance for investors, including strategies to leverage market inefficiencies for arbitrage or diversification benefits. Thus, it advances both the theoretical understanding of cyber risk modeling and practical portfolio management in this evolving field.\u003c/p\u003e\u003c/div\u003e"},{"header":"3. Methodology","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e3.1. Data Collection\u003c/h2\u003e\u003cp\u003eThe study utilizes a standardized cyber loss event dataset to ensure consistency and comparability across five pricing models, addressing the data scarcity challenge highlighted and enabling fair cross-model evaluation. The primary source is the Cyber Events Database maintained by the Center for International and Security Studies at Maryland (CISSM), which collects publicly available cyber events from 2014 to 2025. The database includes structured data with multiple columns such as event date, actor, organization, industry and event type. This database was selected as the most comprehensive free and publicly accessible resource from a reputable academic institution and has also been used by big organizations like European Central Bank or Bank of Japan. Paid databases with fuller loss data exist (e.g., Advisen), but their inaccessibility due to student research constraints is discussed. Data were extracted from the CISSM portal on April 15, 2025.\u003c/p\u003e\u003cp\u003eThe time window selected for the study is 2023 to 2024. It is because of the rapid evolution of cyber events, which makes the recent data more pertinent to the current situation. Although the CISSM dataset provides qualitative information on cyber events, it lacks the loss amount. Therefore, the first action taken was a screening of samples designed to bring out quantifiable losses suitable for modeling. I excluded events that are non-measurable or of zero loss, such as protest attacks or political-espionage. These non-measurables are of approximately 17% based on manual checking. This filtering enhances the dataset\u0026rsquo;s suitability for pricing models since events without loss amounts cannot be integrated into the pricing frameworks. However, this decision may cause a problem of selection bias that may skew the dataset towards greater-impact events.\u003c/p\u003e\u003cp\u003eThe following events were evaluated using a hybrid approach designed to maximize accuracy:\u003c/p\u003e\u003cp\u003e(1) Primary method: Direct losses were sourced from SEC 8-K filings, company reports or credible news reports.\u003c/p\u003e\u003cp\u003e(2) Secondary method: When direct data could not be obtained, category-specific proxies were used. For data breaches, loss amounts were estimated as the number of records stolen or compromised (from public sources like \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e\u003ca href=\"http://www.bleepingcomputer.com\" target=\"_blank\"\u003ewww.bleepingcomputer.com\u003c/a\u003e\u003c/span\u003e\u003cspan address=\"http://www.bleepingcomputer.com\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e\u003ca href=\"http://www.bleepingcomputer.com\" target=\"_blank\"\u003ewww.therecord.media\u003c/a\u003e\u003c/span\u003e\u003cspan address=\"http://www.therecord.media\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e) multiplied by an average loss per record based on the IBM data cost of in a breach report (\u003cspan\u003e$\u003c/span\u003e165 per record in 2023 and \u003cspan\u003e$\u003c/span\u003e169 in 2024). Mega breaches were those with\u0026ensp;more than a million data records stolen. There is also an associated average cost\u0026ensp;of such an event in the IBM report. For DDoS attacks and service outages, loss amounts were calculated in terms of the length of downtime times the most recent annual revenue of the affected company, based on publicly available incident details.\u003c/p\u003e\u003cp\u003eThis multi-formation approach, aligning with industry standards, as similar methodologies are used in reports like Verizon's 2025 Data Breach Investigations Report (DBIR), which analyzes over 22,000 incidents, was designed to make the dataset solid and consistent. However, it may also underrepresent indirect losses, such as customer attrition, or simplify the appropriate the of certain financial effects (Braun et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2023\u003c/span\u003e, 685). Despite these limitations, this unified dataset will be a strong foundation to test pricing models and estimate portfolio inputs, while the limitations give emphasis on cautious interpretation of the outcomes.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e3.2. Overview of Pricing Models\u003c/h2\u003e\u003cp\u003eGiven the theoretical foundation of cyber cat bond pricing models, the following sections present implementation details of the five pricing models with the standardized CISSM dataset to ensure consistency. These models include Loss Distribution Framework from Kolesnikov et al. (2022), the Unified Bayesian Framework from Domfeh et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2022\u003c/span\u003e),\u0026ensp;Signal-Processing Approach from Li \u0026amp; Mamon (2023), Regression Approach by Lane (\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2000\u003c/span\u003e) and Copula-COT Model from Tang et al. (\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) and represent diverse approaches to cyber loss prediction, from frequency-severity decomposition to tail dependence modeling. Each model uses the same cyber loss event dataset (2023\u0026ndash;2024) for parameter estimation, assumptions and computational methodology adapted to allow for data constraints and comparisons across models. The implementations of each model are described in the subsections. Through systematic application of these models, I aim to determine the most accurate and reliable method for the valuation of cyber cat bond as described in the research objectives.\u003c/p\u003e\u003cdiv id=\"Sec9\" class=\"Section3\"\u003e\u003ch2\u003e3.2.1. Implementation of Loss Distribution Framework\u003c/h2\u003e\u003cp\u003eThe pricing model for cyber cat bonds proposed by Kolesnikov et al. (2022) uses a loss distribution framework to derive bond prices, yields, and trigger probabilities. This methodology incorporates a Pareto distribution to account for the heavy-tailed structure of\u0026ensp;cyber losses and employs Monte Carlo simulations to produce loss realizations.\u003c/p\u003e\u003cp\u003eFirst, it begins by calibrating the cyber loss distributions, which are the key to this framework. Cyber event frequency is modeled with an exponential distribution for inter-event times, with the cumulative distribution function:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:P\\left({\\tau\\:}_{k,r}\\le\\:y\\right)=\\:{F}_{k}\\left(y\\right)={F}_{k}\\left(y;\\theta\\:\\right)=1-{e}^{-\\lambda\\:y}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewith \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\tau\\:}_{k,r}\\)\u003c/span\u003e\u003c/span\u003e is the time interval between the incident \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r+1\\)\u003c/span\u003e\u003c/span\u003e of type k, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\theta\\:=\\:\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e represents the event rate and is estimated based on the average annual event frequency overserved in the database. Loss severity follows a log-normal distribution:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:P\\left({\\xi\\:}_{k,r}\\le\\:x\\right)=\\:{G}_{k}\\left(x\\right)={G}_{k}\\left(x;\\lambda\\:\\right)=\\:{\\Phi\\:}\\left(\\frac{\\text{ln}x-\\mu\\:}{\\sigma\\:}\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\xi\\:}_{k,r}\\)\u003c/span\u003e\u003c/span\u003e is the loss amount, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Phi\\:}\\)\u003c/span\u003e\u003c/span\u003e denotes the standard normal cumulative distribution function, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\sigma\\:\\)\u003c/span\u003e\u003c/span\u003e is the standard deviation. These distributions are controlled by vectors\u0026ensp;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\theta\\:\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e, confidence intervals \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\theta\\:\\in\\:[{\\theta\\:}_{l};{\\theta\\:}_{u}]\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\lambda\\:\\in\\:[{\\lambda\\:}_{l};{\\lambda\\:}_{u}]\\)\u003c/span\u003e\u003c/span\u003e to address parameter uncertainty caused by limited historical data.\u003c/p\u003e\u003cp\u003eThe model prices the bond using a standard formula for bond price:\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:P=\\sum\\:_{i=1}^{6}C\\bullet\\:{e}^{-R\\left(\\frac{{d}_{i}}{365}\\right)}+N\\bullet\\:{e}^{-R\\left(\\frac{d}{365}\\right)}\\:$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere C is the coupon value, N is the notional value, R is the funding rate, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{d}_{i}\\)\u003c/span\u003e\u003c/span\u003e represents the coupon payment dates (in days), d is the maturity date, and N is the notional value. Coupon payments occur on days 182, 365, 547, 730, 912, and 1095, with the notional paid at maturity (day 1095). The model simulates cyber losses across the bond\u0026rsquo;s 3-year maturity with a comparison of total losses against coupon and notional trigger thresholds. Payments are made only if the actual losses are below these thresholds as measured with an indicator function during Monte Carlo simulations.\u003c/p\u003e\u003cp\u003eThe fair price is determined by maximizing over these intervals:\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:Price={max}_{\\theta\\:\\in\\:\\left[{\\theta\\:}_{l};{\\theta\\:}_{u}\\right],\\lambda\\:\\in\\:\\left[{\\lambda\\:}_{l};{\\lambda\\:}_{u}\\right]}FairPrice({F}_{k}\\left(\\bullet\\:\\right)={F}_{k}\\left(\\bullet\\:;\\theta\\:\\right);{G}_{k}\\left(\\bullet\\:\\right)=\\:{G}_{k}\\left(\\bullet\\:;\\lambda\\:\\right))$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTo calculate risk premium alternatives, I compute the coupon rate using the probability of loss approach:\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:Coupon\\:rate\\:\\left(\\%\\right)=LIBOR\\:\\left(\\%\\right)+PL\\:\\left(\\%\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe predictive ability of the model is tested by comparing the simulated losses with actual losses from the CISSM dataset in terms of mean squared error and tail risk coverage, using 5000 Monte Carlo simulations as detailed in the paper (Kolesnikov et al. 2022). The assumption of exponential inter-event times implies constant event frequency and event-type independence, potentially failing to capture systemic correlations in cyber events. This assumption is adopted from Kolesnikov et al. (2022) to facilitate parameter estimation in a Poisson process framework. Despite potential mismatches with the CISSM dataset, it is retained for model testing purposes. Although the model is strong in handling different loss events and allows for modeling of tail risk, its dependence on limited historical data leads to parameter uncertainty and the simulation-based method is computationally demanding. This becomes particularly severe when targeting high precision (e.g., \u0026lt;\u0026thinsp;0.1% error) or handling complex scenarios, such as paths with thousands of time steps and sample sizes in the millions to billions, potentially requiring hours on standard hardware. Optimization via parallel computing is suggested for practical use. The performance results will be evaluated against other pricing models to test\u0026ensp;its suitability for valuing cyber cat bonds.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec10\" class=\"Section3\"\u003e\u003ch2\u003e3.2.2. Implementation of Unified Bayesian Framework\u003c/h2\u003e\u003cp\u003eThe Unified Bayesian Framework proposed by Domfeh et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), which was originally designed for natural cat bond, has the potential to be used to evaluate the pricing for cyber cat bonds. The process combines DP-HBCRM for cyber risk, a Cox-Ingersoll-Ross (CIS) model for interest rate risk, and a maximum entropy approach for risk-neutral pricing.\u003c/p\u003e\u003cp\u003eThe process begins with the calibration of the DP-HBCRM using cyber event data from the CISSM dataset, where the claim frequency and claim severity distributions are clustered by event type. Cyber event frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{N}_{i,s}\\)\u003c/span\u003e\u003c/span\u003e) is modeled as a nonhomogeneous Poisson process, where for event type (i) and quarter (s) the number of events follows:\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$$\\:{N}_{i,s}\\:|{\\:\\lambda\\:}_{i,s\\:}\\sim\\:Poisson\\left({\\lambda\\:}_{i,s}\\right),\\:\\:{\\lambda\\:}_{i,s}\u0026gt;0$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e\n$$\\:\\text{log}\\left({\\lambda\\:}_{i,s}\\right)|{\\alpha\\:}_{i},\\:{\\beta\\:}_{s},\\:{X}_{s}=\\:{\\alpha\\:}_{i}+\\:{\\beta\\:}_{s}{X}_{s}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\:\\lambda\\:}_{i,s\\:}\\)\u003c/span\u003e\u003c/span\u003eis the claim intensity, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}_{i}\\:\\)\u003c/span\u003e\u003c/span\u003e is event type-specific intensity capture,\u0026ensp;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{s}\\)\u003c/span\u003e\u003c/span\u003e is the quarterly pattern and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{s}\\)\u003c/span\u003e\u003c/span\u003e is the seasonal indicator. The claim size (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{S}_{i}\\)\u003c/span\u003e\u003c/span\u003e) and loss severity (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{S}_{i,s}\\)\u003c/span\u003e\u003c/span\u003e) are modeled with an inverse gamma distribution:\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equh\" name=\"EquationSource\"\u003e\n$$\\:{X}_{i}|{\\kappa\\:}_{i},{\\theta\\:}_{i}\\:\\sim\\:Inv.Gamma\\left({\\kappa\\:}_{i},\\:{\\theta\\:}_{i}\\right),\\:\\:{\\theta\\:}_{i}\u0026gt;0,\\:{\\kappa\\:}_{i}\u0026gt;0$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equi\" name=\"EquationSource\"\u003e\n$$\\:{S}_{i,\\:s}\\:|\\:{\\kappa\\:}_{i,s},{\\theta\\:}_{i}\\:\\sim\\:Inv.Gamma\\left({\\kappa\\:}_{i,s}\\bullet\\:{\\kappa\\:}_{i},\\:{\\theta\\:}_{i}\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eDirichlet Process (DP) priors are specified for (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\kappa\\:}_{i},\\:{\\theta\\:}_{i}\\)\u003c/span\u003e\u003c/span\u003e) and (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}_{i}\\)\u003c/span\u003e\u003c/span\u003e), inducing clustering of event types based on shared loss characteristics:\u003cdiv id=\"Equj\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equj\" name=\"EquationSource\"\u003e\n$$\\:\\left({\\kappa\\:}_{i},\\:{\\theta\\:}_{i}\\right)\\:\\sim\\:DP\\left({\\gamma\\:}_{1},\\:{G}_{0}\\left(\\bullet\\:\\right)\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equk\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equk\" name=\"EquationSource\"\u003e\n$$\\:{\\alpha\\:}_{i}\\:\\sim\\:DP({\\gamma\\:}_{2},\\:{H}_{0}\\left(\\bullet\\:\\right))$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equl\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equl\" name=\"EquationSource\"\u003e\n$$\\:{G}_{0}\\left({\\zeta\\:}_{1},{\\zeta\\:}_{2},{\\eta\\:}_{1},{\\eta\\:}_{2}\\right)=Gamma\\left({\\zeta\\:}_{1},{\\zeta\\:}_{2}\\right)\\times\\:Gamma({\\eta\\:}_{1},{\\eta\\:}_{2})$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equm\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equm\" name=\"EquationSource\"\u003e\n$$\\:{H}_{0}({\\psi\\:}_{1},{\\psi\\:}_{2})=Gamma(\\:{\\psi\\:}_{1},{\\psi\\:}_{2})$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equn\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equn\" name=\"EquationSource\"\u003e\n$$\\:\\text{w}\\text{i}\\text{t}\\text{h}\\:\\text{h}\\text{y}\\text{p}\\text{e}\\text{r}\\text{p}\\text{a}\\text{r}\\text{a}\\text{m}\\text{e}\\text{t}\\text{e}\\text{r}\\text{s}\\:{\\zeta\\:}_{1},{\\zeta\\:}_{2},{\\eta\\:}_{1},{\\eta\\:}_{2},{\\psi\\:}_{1},{\\psi\\:}_{2}\\:\\sim\\:Gamma\\left(\\text{0.01,0.01}\\right)\\:\\text{a}\\text{n}\\text{d}\\:{\\beta\\:}_{s}\\mathcal{\\:}\\sim\\mathcal{\\:}\\mathcal{N}\\left(\\text{0,0.01}\\right).$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ePosterior distributions were estimated MCMC simulations with 40,000 iterations and the first 10,000 iterations as burn-in.\u003c/p\u003e\u003cp\u003eThen, a Bayesian CIR model is employed to model interest rates:\u003cdiv id=\"Equo\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equo\" name=\"EquationSource\"\u003e\n$$\\:{dr}_{t}=\\left(\\alpha\\:-\\:{\\beta\\:r}_{t}\\right)dt+\\:\\sigma\\:\\sqrt{{r}_{t}}d{W}_{t}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eParameters (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:,\\:\\beta\\:,\\:{\\sigma\\:}^{2}\\)\u003c/span\u003e\u003c/span\u003e) are inferred using MCMC with 15,000 iterations and a burn-in of the first 5,000. The bond price for a zero-coupon cyber cat bond is formulated as follows:\u003cdiv id=\"Equp\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equp\" name=\"EquationSource\"\u003e\n$$\\:\\:\\:\\:{P}_{t}=\\:\\text{{\\rm\\:K}}\\:{\\mathbb{E}}^{\\mathbb{Q}}\\:\\left[{e}^{-{\\int\\:}_{t}^{T}{r}_{s}ds}\\bullet\\:{V}_{T}\\:\\right|\\:{\\mathcal{F}}_{t}\\:\\:]\\:\\approx\\:\\:\\sum\\:_{i=1}^{N}\\sum\\:_{t=1}^{T}\\left(\\text{exp}\\left(-\\sum\\:_{u=1}^{t}{r}_{u}^{\\left(i\\right)}\\right)\\bullet\\:{V}_{t}^{\\left(i\\right)}\\right){\\pi\\:}_{i}^{*}\\:\\:\\:\\:\\:\\:\\:\\:(1)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{V}_{T}=K\\bullet\\:I\\left({L}_{T}\\le\\:D\\right)+\\alpha\\:\\bullet\\:K\\bullet\\:I({L}_{T}\u0026gt;D)\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{{\\rm\\:K}}\\)\u003c/span\u003e\u003c/span\u003e is the face value, D is the loss threshold and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\pi\\:}_{i}^{*}\\)\u003c/span\u003e\u003c/span\u003e is risk-neutral probabilities derived via maximum entropy.\u003c/p\u003e\u003cp\u003eTo adapt the model to the CISSM dataset, I used fixed and more conservative hyperparameters for the Dirichlet Process priors (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\gamma\\:}_{1}=10,\\:{\\gamma\\:}_{2}=2,\\:{\\zeta\\:}_{1}=15,\\:{\\zeta\\:}_{2}=0.1,\\:{\\eta\\:}_{1}=10,{\\eta\\:}_{2}=0.5,\\:{\\psi\\:}_{1}=3,{\\psi\\:}_{2}=1\\)\u003c/span\u003e\u003c/span\u003e) to ensure stable posterior inference. I removed the seasonal component (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\beta\\:}_{s}{X}_{s}\\)\u003c/span\u003e\u003c/span\u003e) in the nonhomogeneous Poisson process, as the dataset showed limited evidence of quarterly patterns for cyber risks. It was also confirmed by a chi-square test on quarterly patterns (chi-square statistic: 6.4343, p-value: 0.092290), which fails to reject the null hypothesis of no seasonality. The MCMC iterations for the DP-HBCRM were reduced to 3,000 with 1,000 burn-in iterations to avoid the risk of overfitting, and because seasonal factors were removed in the nonhomogeneous Poisson process, which consequently reduced the complexity of the model. To ensure convergence, I did the Gelman-Rubin test, and the results showed the Gelman-Rubin statistics were below 1.1 for all parameters. This simplified model structure requires fewer iterations than the paper suggested to achieve a stable and better-performed posterior distribution.\u003c/p\u003e\u003cp\u003eThe model assumes independence of event frequency and severity and may ignore systemic cyber event dependencies\u0026ensp;across events such as those arising from correlated supply chain vulnerabilities (Mastroeni et al., 2023, p. 2). However, by incorporating prior information and classifying event types, this model can handle scarce historical data, but it is very computationally demanding. The runtime increases significantly as the iteration counts and chain number grow. It could extend processing to several hours on standard hardware, limiting scalability for high-precision applications.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec11\" class=\"Section3\"\u003e\u003ch2\u003e3.2.3. Implementation of Signal-Processing Approach\u003c/h2\u003e\u003cp\u003eFor pricing cyber cat bonds, I use the signal-processing method of Li and Mamon (2023). This framework represents the dynamics of cyberattacks using a nonhomogeneous Markov chain modulated by hidden Markov chain, and it offers a flexible framework for modelling CKC state change and pricing bonds under uncertainty. The implementation focuses on calibrating RSMM, estimating transition probabilities, pricing a zero-coupon cyber cat bond, and calculating risk premia.\u003c/p\u003e\u003cp\u003eThe process begins by constructing the RSMM to the CKC states: firewall working (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{1}\\)\u003c/span\u003e\u003c/span\u003e), firewall fail (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{2}\\)\u003c/span\u003e\u003c/span\u003e), and anti-phishing fail (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{3}\\)\u003c/span\u003e\u003c/span\u003e). The state process (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}_{k}\\)\u003c/span\u003e\u003c/span\u003e) evolves according to:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{y}_{k+1}=B\\left({z}_{n}\\right){y}_{k}+\\:{w}_{k+1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:B\\left({z}_{n}\\right)=\\left({b}_{ij}\\left({z}_{k}\\right)\\right)\\)\u003c/span\u003e\u003c/span\u003e is the state-dependent transition matrix, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{w}_{k+1}\\)\u003c/span\u003e\u003c/span\u003e is a martingale increment with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:E\\left[\\:{w}_{k+1}\\:\\right|{\\mathcal{F}}_{k}]=0\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathcal{F}}_{k}\\)\u003c/span\u003e\u003c/span\u003e is the filtration generated by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathcal{F}}_{k}^{y}\\)\u003c/span\u003e\u003c/span\u003e and the HMM chain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathcal{F}}_{k}^{z}\\)\u003c/span\u003e\u003c/span\u003e. Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e2\u003c/span\u003e) is then revised by the authors, to accommodate certain established results of homogeneous HMM with a discrete range:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:{c}_{ji}\\left({y}_{k}\\right)\\:{|}_{{y}_{k}={f}_{l}}\\::=P\\left({y}_{k+1}=\\:{f}_{j}|{y}_{k}={f}_{l},\\:{z}_{k}={e}_{i}\\right)=\\:{b}_{jl}\\left({z}_{k}\\right){|}_{{z}_{k}={e}_{i}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eCombine (2) and (3) is equivalent to:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{y}_{k+1}=C\\left({y}_{k}\\right){z}_{k}+\\:{w}_{k+1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:C=\\left({c}_{ji}\\left({y}_{k}\\right)\\right)\\)\u003c/span\u003e\u003c/span\u003e. Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e4\u003c/span\u003e) is a one-step delay model, which is reasonable because \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}_{k+1}\\)\u003c/span\u003e\u003c/span\u003e may not react to z immediately.\u003c/p\u003e\u003cp\u003eSecond, the transition probabilities are\u0026ensp;determined considering recursive filters:\u003cdiv id=\"Equq\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equq\" name=\"EquationSource\"\u003e\n$$\\:{p}_{k}=\\:\\prod\\:diag\\left({d}_{k}\\right){p}_{K-1}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equr\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equr\" name=\"EquationSource\"\u003e\n$$\\:\\gamma\\:\\left({\\mathcal{J}}_{k}^{j,r}{z}_{k}\\right)=\\:\\prod\\:diag\\left({d}_{k}\\right)\\gamma\\:\\left({\\mathcal{J}}_{k-1}^{j,r}{z}_{k-1}\\right)+{d}_{k}^{\\left(r\\right)}\u0026lt;{p}_{k-1},{e}_{r}\u0026gt;{\\pi\\:}_{jr}{e}_{j}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equs\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equs\" name=\"EquationSource\"\u003e\n$$\\:\\gamma\\:\\left({\\mathcal{O}}_{k}^{r}{z}_{k}\\right)=\\:\\prod\\:diag\\left({d}_{k}\\right)\\gamma\\:\\left({\\mathcal{O}}_{k-1}^{r}{z}_{k-1}\\right)+{d}_{k}^{\\left(r\\right)}\u0026lt;{p}_{k-1},{e}_{r}\u0026gt;{\\pi\\:}_{r}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\gamma\\:\\left({\\mathcal{T}}_{k}^{s,r}\\left({y}_{k},{f}_{i}\\right){z}_{k}\\right)=\\prod\\:diag\\left({d}_{k}\\right)\\gamma\\:\\left({\\mathcal{T}}_{k-1}^{s,r}\\left({y}_{k-1},{f}_{i}\\right){z}_{k-1}\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:+m\u0026lt;{p}_{k-1},{e}_{r}\u0026gt;\u0026lt;{y}_{k},{f}_{s}\u0026gt;\u0026lt;{y}_{k-1},{f}_{i}\u0026gt;{c}_{sr}\\left({f}_{i}\\right){\\pi\\:}_{r}\\:\\)\u003c/span\u003e\u003c/span\u003e\u003cdiv id=\"Equt\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equt\" name=\"EquationSource\"\u003e\n$$\\:\\gamma\\:({\\mathcal{T}}_{k}^{r}\\left({f}_{i}\\right){z}_{k}=\\:=\\prod\\:diag\\left({d}_{k}\\right)\\gamma\\:\\left({\\mathcal{T}}_{k-1}^{r}\\left({f}_{i}\\right){z}_{k-1}\\right)+{d}_{k}^{\\left(r\\right)}\u0026lt;{p}_{k-1},{e}_{r}\u0026gt;\u0026lt;{y}_{k-1},{f}_{i}\u0026gt;{\\pi\\:}_{r}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{d}_{k}\\)\u003c/span\u003e\u003c/span\u003e represents observation probabilities, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathcal{J}}_{k}^{j,r}\\)\u003c/span\u003e\u003c/span\u003e is the number of jumps from \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{e}_{r}\\)\u003c/span\u003e\u003c/span\u003e to state \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{e}_{j}\\)\u003c/span\u003e\u003c/span\u003e in time k, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathcal{O}}_{k}^{r}\\)\u003c/span\u003e\u003c/span\u003e is the amount of time that the Markov chain z spent in state \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{e}_{r}\\)\u003c/span\u003e\u003c/span\u003e up to k, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathcal{T}}_{k}^{s,r}\\left({y}_{k},{f}_{i}\\right)\\)\u003c/span\u003e\u003c/span\u003e counts the number of times up to k that y is in state \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{s}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathcal{T}}_{k}^{s,r}\\left({f}_{i}\\right)\\)\u003c/span\u003e\u003c/span\u003e counts the number of times up to k, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\prod\\:\\)\u003c/span\u003e\u003c/span\u003e governs \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:B\\left({z}_{n}\\right)\\)\u003c/span\u003e\u003c/span\u003e transitions. They use change of measure, as in this joint implementation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\stackrel{-}{{\\Lambda\\:}}}_{k}=\\:{\\prod\\:}_{l=1}^{k}\\:{\\stackrel{-}{\\lambda\\:}}_{l}\\)\u003c/span\u003e\u003c/span\u003e, and do their calculations under a measure (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\stackrel{\\sim}{P}\\)\u003c/span\u003e\u003c/span\u003e) in which the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:B\\left({z}_{n}\\right)\\)\u003c/span\u003e\u003c/span\u003e are independent with uniform distribution.\u003c/p\u003e\u003cp\u003eThen, estimate the optimal parameters using the EM algorithm and filtering technique:\u003cdiv id=\"Equu\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equu\" name=\"EquationSource\"\u003e\n$$\\:{\\widehat{\\pi\\:}}_{jr}=\\:\\frac{\\gamma\\:\\left({\\mathcal{J}}_{k}^{j,r}\\right)}{\\gamma\\:\\left({\\mathcal{O}}_{k}^{r}\\right)}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equv\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equv\" name=\"EquationSource\"\u003e\n$$\\:{\\widehat{c}}_{sr}\\left({f}_{i}\\right)=\\:\\frac{\\gamma\\:\\left({\\mathcal{T}}_{k}^{s,r}\\left({y}_{k},{f}_{i}\\right)\\right)}{\\gamma\\:\\left({\\mathcal{T}}_{k}^{r}\\left({f}_{i}\\right)\\right)}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eFinally, loss severities are modeled with a doubly truncated Pareto distribution and transform this loss on dollar terms by proportinality. The bond price is calculated under a risk-neutral measure (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbb{Q}\\)\u003c/span\u003e\u003c/span\u003e):\u003cdiv id=\"Equw\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equw\" name=\"EquationSource\"\u003e\n$$\\:{P}_{t}=\\:{\\mathbb{E}}^{\\mathbb{Q}}\\left[{e}^{-{\\int\\:}_{t}^{T}{r}_{s}{d}_{s}}\\bullet\\:K\\left({F}_{T}\\left(D\\right)+a\\left(1-{F}_{T}\\left(D\\right)\\right)\\right)\\:\\right|{\\mathcal{F}}_{t}]$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{F}_{T}\\left(D\\right)=\\text{P}\\text{r}[{L}_{T}\\le\\:D]\\)\u003c/span\u003e\u003c/span\u003e and the Vasiček model is used to generate interest path and to calculate the discount factor.\u003c/p\u003e\u003cp\u003eThe model is based on Markov state transitions, independent breach\u0026ensp;severities, and a stationary interest rate process. Loss and interest rate scenarios are simulated by\u0026ensp;Monte Carlo (10000 iterations) with 5,000 iterations used for the EM convergence. Regarding this approach, the RSMM captures state-dependent transitions, and hence improves pricing accuracy. However, the Monte Carlo simulation is computationally heavy, and thus efficient algorithms are necessary.\u003c/p\u003e\u003cp\u003eTo adapt the model to the dataset, where all events are state 3 (anti-phishing fail) based on CKC states, I implemented the simplified focus only on loss severity modeling using the doubly truncated Pareto distribution. This CKC transitions ineffective, limiting demonstration of the model's theoretical strength in dynamic state modeling\u0026mdash;future datasets with diverse states could fully exploit this. The loss amounts were aggregated by industry and date to align with the structure of the dataset. Moreover, the shape parameter of the Pareto distribution was estimated via maximum likelihood estimation (MLE), with a stability check enforcing a minimum shape of 1.1 to prevent infinite expected losses. For loss perditions, I used directly the expected loss from the Pareto distribution instead of Monte Carlo simulations.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec12\" class=\"Section3\"\u003e\u003ch2\u003e3.2.4. Implementation of Regression Approach\u003c/h2\u003e\u003cp\u003eThe regression approach proposed by Lane (\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2000\u003c/span\u003e), which offers a general model for pricing ILS, including cyber cat bonds, by using a regression-based technique to estimate risk premia. This section describes the methodology of applying Lane\u0026rsquo;s regression model to cyber cat bond pricing and is consistent with the study\u0026rsquo;s objective of estimating loss prediction accuracy using the CISSM dataset.\u003c/p\u003e\u003cp\u003eThe regression method splits the pricing of cyber cat bond into three\u0026ensp;parts: the RFIR, EL and RP, that is, the spread over LIBOR of the bond. At the heart of the model are two risk metrics \u0026ndash; PFL and CEL \u0026ndash; that address the frequency and severity of cyber events, respectively. PFL is the probability of cumulative losses (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{T}\\)\u003c/span\u003e\u003c/span\u003e) exceeding the threshold (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:D\\)\u003c/span\u003e\u003c/span\u003e):\u003cdiv id=\"Equx\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equx\" name=\"EquationSource\"\u003e\n$$\\:PFL=\\text{P}\\text{r}[{L}_{T}\u0026gt;D]$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eCEL is, in turn, the expected loss given that a loss occurs (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{T}\u0026gt;D\\)\u003c/span\u003e\u003c/span\u003e):\u003cdiv id=\"Equy\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equy\" name=\"EquationSource\"\u003e\n$$\\:CEL=E\\left[{L}_{T}\\right|\\:{L}_{T}\u0026gt;D]/K$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere K is the face value, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{T}={\\sum\\:}_{i=1}^{{N}_{T}}{L}_{i}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{N}_{T}\\)\u003c/span\u003e\u003c/span\u003e is the number of cyberattacks and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the loss per attack. Here, I will test different distribution modes for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{T}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{N}_{T}\\)\u003c/span\u003e\u003c/span\u003e, and show the result\u0026ensp;in section \u003cspan refid=\"Sec16\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eThe risk premium is expressed as a\u0026ensp;function of expected excess return (EER), PFL and CEL via the power-law form of Cobb-Douglas production function:\u003cdiv id=\"Equz\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equz\" name=\"EquationSource\"\u003e\n$$\\:EER=\\:\\gamma\\:\\times\\:{\\left(PFL\\right)}^{\\alpha\\:}\\times\\:{\\left(CEL\\right)}^{\\beta\\:}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\gamma\\:=0.5551\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:=0.4946\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\beta\\:=0.5741\\)\u003c/span\u003e\u003c/span\u003e are\u0026ensp;sample values taken from the 1999 ILS market which are based on a historical trade-off between frequency and severity. The EL is calculated as the product of PFL and CEL:\u003cdiv id=\"Equaa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equaa\" name=\"EquationSource\"\u003e\n$$\\:EL=PFL\\:\\times\\:\\:CEL$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe full premium, which determines the bond\u0026rsquo;s spread over LIBOR, is the sum of EL and EER, adjusted for a 365-day count convention:\u003cdiv id=\"Equab\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equab\" name=\"EquationSource\"\u003e\n$$\\:Full\\:Premium=EL+EER$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe bond spread over LIBOR (S) is as follows:\u003cdiv id=\"Equac\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equac\" name=\"EquationSource\"\u003e\n$$\\:S=RFIR+EL+EER=LIBOR+Full\\:Premium$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTo adapt the model to the dataset, I first grouped all events by industry, geography and classification of cyber risk. This step was adopted to emulate Lane's empirical approach. Trigger amounts (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:D\\)\u003c/span\u003e\u003c/span\u003e) are established based on cyber cat bond data in the past two years (2023\u0026ndash;2024, Artemis.bm, 2025), with values set at \u003cspan\u003e$\u003c/span\u003e576.49\u0026nbsp;million for the United States and \u003cspan\u003e$\u003c/span\u003e539.74\u0026nbsp;million for other countries. To address the scarcity of the historical data and the limited dataset, PFL is estimated as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:({N}_{Triggered}+1)/({N}_{events}+2)\\)\u003c/span\u003e\u003c/span\u003e, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{N}_{Triggered}\\)\u003c/span\u003e\u003c/span\u003e denoting the number of cyber events for which the loss amounts exceed the trigger amounts, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{N}_{events}\\)\u003c/span\u003e\u003c/span\u003e denoting the number of cyber events in the groups. To avoid the case \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{N}_{Triggered}=0\\)\u003c/span\u003e\u003c/span\u003e, Laplace smoothing is incorporated. CEL is calculated as the mean loss exceeding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:D\\)\u003c/span\u003e\u003c/span\u003e:\u003cdiv id=\"Equad\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equad\" name=\"EquationSource\"\u003e\n$$\\:\\left\\{\\begin{array}{c}\\frac{\\frac{\\sum\\:{L}_{Triggered}}{{N}_{Triggered}}}{\\text{max}{L}_{Triggered}},if{\\:N}_{Triggered}\\ne\\:0\\:\\\\\\:\\sum\\:{L}_{i}*\\frac{0.01}{\\text{max}{L}_{Triggered}},if{\\:N}_{Triggered}=0\\end{array}\\right.$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{Triggered}\\)\u003c/span\u003e\u003c/span\u003e denotes the loss amount of cyber events exceeding the trigger amount and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{i}\\)\u003c/span\u003e\u003c/span\u003e denotes the loss amount of cyber events. Subsequently, the computed PFL and CEL values, derived from the grouped data, are assigned to each cyber event.\u003c/p\u003e\u003cp\u003eDeparting from Lane\u0026rsquo;s original use of fixed parameters (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\gamma\\:=0.5551\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:=0.4946\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\beta\\:=0.5741\\)\u003c/span\u003e\u003c/span\u003e) derived from 1999 ILS data, this study uses a data-driven approach. An ordinary least squares (OLS) regression on log-transformed variables is employed to fit the parameters of the Cobb-Douglas production function, as detailed below:\u003cdiv id=\"Equae\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equae\" name=\"EquationSource\"\u003e\n$$\\:\\text{log}\\left(1+Loss\\:Amount\\right)=log\\gamma\\:+\\alpha\\:\\bullet\\:\\text{log}\\left(PEL\\right)+\\:\\beta\\:\\bullet\\:\\text{log}\\left(CEL\\right)+\\:ϵ$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:ϵ\\)\u003c/span\u003e\u003c/span\u003e represents the error term. For loss predictions, I split the dataset into a 70:30 ratio for training and predictive validation, respectively, with 5-fold cross-validation to enhance robustness.\u003c/p\u003e\u003cp\u003eThe model does not allow for asymmetric cyber risk loss distributions and for PFL and CEL for the risk profile. It also assumes the model parameters fitted to the 1999 ILS are also suitable for cyber risks, although the loss dynamic may differ, i.e. the\u0026ensp;systemic correlation or the multi-event trigger. The assumption of a linear relation between lay of risk and risk measures may not account for the full range of non-linearity in cyber events. However, its understandable form may help to make the model more approachable for the new cyber cat bond market where we have limited data. Yet its assumptions require careful consideration, particularly with respect to the portability of ILS parameters and the ability of PFL and CEL to fully address systemic cyber risk. These issues will be further addressed in the empirical analysis to allow me to\u0026ensp;assess the model\u0026rsquo;s predictive power in the more general comparative context.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec13\" class=\"Section3\"\u003e\u003ch2\u003e3.2.5. Implementation of Copula-POT Model\u003c/h2\u003e\u003cp\u003eTo achieve the goal of valuating cyber cat bonds and assessing their role in portfolio efficiency, I apply the Copula-POT model by Tang et al. (\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) to price cyber cat bonds using a database of cyber events with loss amounts attached to them. The methodology combines loss prediction accuracy, tail risk prediction, and contribution to portfolio enhancement to provide a solid base for model selection and application to MPT.\u003c/p\u003e\u003cp\u003eThe first part is to characterize the marginal distribution of the indicators of cyber losses using POT method with a Beta-generalized Pareto (GP) model introduced in Tang et al. (\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Due to the presence of heavy tails on the losses, the database is decomposed into bulk and tail parts. The threshold \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{u}_{i}\\)\u003c/span\u003e\u003c/span\u003e of each cyber loss indicator \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{i}\\)\u003c/span\u003e\u003c/span\u003e (e.g., financial loss, number of affected records) is decided by inspecting the mean residual life plot using the following equation:\u003cdiv id=\"Equaf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equaf\" name=\"EquationSource\"\u003e\n$$\\:{e}_{n}\\left(u\\right)=\\:\\frac{1}{{n}_{u}}\\sum\\:_{i=1}^{{n}_{u}}({x}_{\\left(i\\right)}-u)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe threshold-excesses (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{X}_{i}-{u}_{i}|{X}_{i}\u0026gt;{u}_{i}\\)\u003c/span\u003e\u003c/span\u003e) are modeled using the GPD:\u003cdiv id=\"Equag\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equag\" name=\"EquationSource\"\u003e\n$$\\:G\\left(y\\right)=1-(1+\\frac{{\\xi\\:}_{y}}{\\sigma\\:}{)}^{\\raisebox{1ex}{$-1$}\\!\\left/\\:\\!\\raisebox{-1ex}{$\\xi\\:$}\\right.}\\:,\\:\\:y\u0026gt;0$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\xi\\:\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\sigma\\:\\)\u003c/span\u003e\u003c/span\u003e are the shape and scale parameters, respectively. Non-exceedances are modeled using a Beta distribution:\u003cdiv id=\"Equah\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equah\" name=\"EquationSource\"\u003e\n$$\\:{Beta}_{\\alpha\\:,\\beta\\:}\\left(y\\right)=\\:\\frac{1}{B\\left(\\alpha\\:,\\beta\\:\\right)}{y}^{\\alpha\\:-1}(1-y{)}^{\\beta\\:-1}\\:,\\:\\:\\:\\:\\:\\:\\:\\:\\:0\u0026lt;y\u0026lt;1$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eMaximum likelihood estimation is applied to estimate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\xi\\:,\\:\\sigma\\:,\\:\\alpha\\:,\\:\\beta\\:\\)\u003c/span\u003e\u003c/span\u003e, ensuring the distribution \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{F}_{i}\\left(x\\right)\\)\u003c/span\u003e\u003c/span\u003e captures both normal and extreme losses as follows:\u003cdiv id=\"Equai\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equai\" name=\"EquationSource\"\u003e\n$$\\:{F}_{i}\\left(x\\right)=\\:\\left\\{\\begin{array}{c}1-\\frac{{n}_{{u}_{i}}}{n}{\\stackrel{-}{G}}_{{\\xi\\:}_{i},{\\sigma\\:}_{i}}\\left(x-{u}_{i}\\right),\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:x\u0026gt;{u}_{i}\\\\\\:\\frac{1}{B({\\alpha\\:}_{i},{\\beta\\:}_{i})}{\\int\\:}_{0}^{1-(x-{m}_{i})/({u}_{i}-{m}_{i})}{t}^{{\\alpha\\:}_{i}-1}(1-t{)}^{{\\beta\\:}_{i}-1}dt,\\:\\:\\:\\:x\\le\\:{u}_{i}\\end{array}\\right.$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{{n}_{{u}_{i}}}{n}\\)\u003c/span\u003e\u003c/span\u003e is the exceedance proportion and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{m}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the sample minimum.\u003c/p\u003e\u003cp\u003eThe dependence structure between the multiple cyber loss indicators is then modeled by means of a nested Archimedean copula:\u003cdiv id=\"Equaj\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equaj\" name=\"EquationSource\"\u003e\n$$\\:C\\left({u}_{1},\\:\\cdots\\:,\\:{u}_{m}\\right)=\\:{C}_{outer}({C}_{inner}\\left({u}_{1},\\cdots\\:,\\:{u}_{k};{\\theta\\:}_{1}\\right),\\:{u}_{k+1},\\cdots\\:,\\:{u}_{m};{\\theta\\:}_{2})$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewith parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{1}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\theta\\:}_{2}\\)\u003c/span\u003e\u003c/span\u003e are estimated via maximum likelihood to capture the larger dependence among subsets of indicators, which will lead to more precise modeling of joint loss events. The cyber loss indicators are converted into uniform variates using \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\stackrel{\\sim}{u}}_{ij}={F}_{i}\\left({x}_{ij}\\right)\\)\u003c/span\u003e\u003c/span\u003e, and a nested Frank copula is chosen because it fits best (Tang et al., \u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e\u003cp\u003ePricing of the cyber cat bond is then performed through Monte Carlo simulation as following equation:\u003cdiv id=\"Equak\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equak\" name=\"EquationSource\"\u003e\n$$\\:{P}_{t}=\\:\\sum\\:_{s=t+1}^{T}\\mathbb{E}\\left\\{{C}_{t,s}\\right\\}p\\left(t,s\\right)+\\mathbb{\\:}\\mathbb{E}\\left\\{{F}_{t,T}\\right\\}p(t,T)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe coupon and principal payments are determined by the retention proportions \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}_{t},\\:{\\beta\\:}_{t},\\:{\\gamma\\:}_{t}\\)\u003c/span\u003e\u003c/span\u003e (as specified in above equations), acknowledging that they cut into loss indicators if they exceed their attachment points. The discount factor \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:p(t,T)\\)\u003c/span\u003e\u003c/span\u003e is calculated using the CIR model: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:p\\left(t,T\\right)=A(t,T){e}^{-B\\left(t,T\\right)r\\left(t\\right)}\\)\u003c/span\u003e\u003c/span\u003e. The frequency of cyber events per year is calculated using an ARMA process in order to predict event intensity along the bond\u0026rsquo;s life.\u003c/p\u003e\u003cp\u003eThe CISSM dataset is preprocessed by selecting key loss indicators: Loss amount, geography and event type impact. The latter is calculated as a weighted impact from event type (e.g., data breach, DDOS). Thresholds (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{u}_{i}\\)\u003c/span\u003e\u003c/span\u003e) for these indicators are set with the 98th percentile instead of the mean residual life plot to focus on tail risks, ensuring sufficient exceedances (at least 10) for robust estimation. The Gumbel-Gumbel copula is chosen for its emphasis on upper tail dependence and because it showed the best result compared to other combinations of copula. For loss predictions, I first split the dataset randomly into a 70:30 ratio for training and testing sets. Secondly, Monte Carlo simulations were conducted with 200,000 iterations to predict losses. Finally, the loss prediction ability was evaluated by comparing simulated losses with actual losses from the test set.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\u003ch2\u003e3.3. Model Evaluation\u003c/h2\u003e\u003cp\u003eIn order to identify the most appropriate model for pricing cyber cat bonds in my study, I review five pricing models mentioned above. The assessment is conducted on three points of aspects of loss prediction ability measured by Mean Squared\u0026ensp;Error (MSE), pricing efficiency measured by Sharpe ratio, and tail risk prediction accuracy measured by Conditional Value at Risk (CVaR). The models are tested against the CISSM dataset to see how well they can reproduce the CISSM data using statistical approaches, including goodness of fit testing and error metrics.\u003c/p\u003e\u003cp\u003eThe first and the most critical criterion in the study, the loss prediction ability, measures\u0026ensp;how accurately each model predicts cyber losses. MSE is computed by simulating losses against a validation subset of the dataset and a low MSE indicates high predictive accuracy. To handle the right-skewed distribution of loss amount data, using the original scale to calculate MSE directly would result in the evaluation results being overly influenced by extreme values. I apply a log1p transformation (natural logarithm of the value plus one, i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{l}\\text{n}(1+x)\\)\u003c/span\u003e\u003c/span\u003e) to both actual and predicted losses before calculating the MSE, resulting in a standardized log MSE. This transformation is chosen because it effectively handles zero or small values without causing undefined logarithms, stabilizing the metric for skewed data. The log MSE is defined as:\u003cdiv id=\"Equal\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equal\" name=\"EquationSource\"\u003e\n$$\\:log\\:MSE=\\:\\frac{1}{n}\\sum\\:_{i=1}^{n}[\\text{l}\\text{o}\\text{g}(1+{\\widehat{L}}_{i})-\\text{l}\\text{o}\\text{g}(1+{L}_{i}{\\left)\\right]}^{2}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{L}}_{i}\\)\u003c/span\u003e\u003c/span\u003e represents the predicted loss, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{L}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the actual loss, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:n\\)\u003c/span\u003e\u003c/span\u003e is the number of observations. I also use the Kolmogorov-Smirnov (KS) goodness-of-fit test to assess the validity of loss distributions by comparing the empirical distribution of historical losses with the cumulative distribution functions of simulated losses, with p-value of 0.05 as the threshold to indicate an adequate fit. For each model, I do loss predictions with Monte Carlo simulations, adjusted to each model\u0026rsquo;s specific methodology. This entails fitting each model to the data, simulating losses over a range of scenarios and comparing their predictions with historical losses to gauge their predictive accuracy.\u003c/p\u003e\u003cp\u003eThis loss prediction capability forms the foundation for model evaluation, but to ensure practical applicability in cyber cat bond valuation, the next criterion assesses how well the model's pricing mechanism translates these predictions into efficient risk-return profiles. The second criterion, pricing efficiency, evaluates the effectiveness of each cyber cat bond pricing model in generating attractive returns relative to the associated risks, considering the diverse pricing methodologies across models. I quantify this with the Sharpe ratio, which is:\u003cdiv id=\"Equam\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equam\" name=\"EquationSource\"\u003e\n$$\\:Sharpe\\:Ratio=\\:\\frac{\\mathbb{E}[{R}_{p}-{R}_{f}]}{{\\sigma\\:}_{p}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{R}_{p}\\)\u003c/span\u003e\u003c/span\u003e is the cyber cat bond return, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{R}_{f}\\)\u003c/span\u003e\u003c/span\u003e is the risk-free rate, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{p}\\)\u003c/span\u003e\u003c/span\u003e is the standard deviation of the bond\u0026rsquo;s returns, reflecting its risk or volatility. To do this, I employ a Monte Carlo approach to simulate 10,000 times the returns of a 2-year-zero-coupon cyber cat bond with a fixed face value in the way each model prescribes them. The bond prices are the result of a price computation in a risk-neutral framework, incorporating the risk premia (spread) of 11.3%, which represents the average spread of existing cyber cat bonds in 2023\u0026ndash;2024. The loss trigger is set at \u003cspan\u003e$\u003c/span\u003e558.11\u0026nbsp;million, based on the average trigger amount of existing cyber cat bonds during the same period. The risk-free rate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{R}_{f}\\)\u003c/span\u003e\u003c/span\u003e) is set to 4.00%, which is the average yield of 10-year U.S. Treasury notes also from the same period. The Sharpe ratio is then calculated to compare the pricing efficiency of each model, enabling an assessment of how effectively each pricing mechanism balances the high-risk, high-return nature of cyber cat bonds under varying loss scenarios. To assess robustness, sensitivity analysis is conducted by varying trigger levels, ensuring the model's pricing remains stable across realistic market conditions.\u003c/p\u003e\u003cp\u003eBuilding on pricing efficiency, which previews investment implications, the third criterion focuses on the models' ability to handle extreme events, which is crucial for the tail-heavy nature of cyber risks. The third criterion, tail risk prediction accuracy, examines the degree to which the models capture the extreme cyber\u0026ensp;loss events, which is particularly important because cyber risks are naturally skewed. I\u0026ensp;measure it via the Conditional Value at Risk (CVaR) at the 99% confidence level as:\u003cdiv id=\"Equan\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equan\" name=\"EquationSource\"\u003e\n$$\\:{CVaR}_{\\alpha\\:}=\\mathbb{\\:}\\mathbb{E}\\left[L\\:\\right|\\:L\\ge\\:Va{R}_{\\alpha\\:}]$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:Va{R}_{\\alpha\\:}\\)\u003c/span\u003e\u003c/span\u003e is the Value at Risk at confidence level \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:=0.99\\)\u003c/span\u003e\u003c/span\u003e. I calculate CVaR by simulating loss distributions using 10,000 Monte Carlo paths to generate\u0026ensp;robust tail estimates. For each model, I match the loss\u0026ensp;distribution for its tail and calculate the expected loss given that the VaR has been exceeded. This allows testing how well each model captures extreme cyber events.\u003c/p\u003e\u003cp\u003eComputationally,\u0026ensp;I build the models in Python with packages like NumPy, SciPy, and pandas. The simulations of each model are calibrated\u0026ensp;to their theoretical framework, using preprocessing that scales the data and makes it consistent across models. For example, the Copula-POT must estimate the copula\u0026ensp;parameter while in a regression model, the nonlinear least squares are adopted. I monitor convergence and check results via diagnostics, like presenting the standard deviations for MSE, or checking the p-values of KS test.\u003c/p\u003e\u003cp\u003eTo integrate these criteria and select the most suitable model, a composite score is calculated as follows: Composite Score\u0026thinsp;=\u0026thinsp;0.6 \u0026times; (Normalized Loss Prediction Score)\u0026thinsp;+\u0026thinsp;0.2 \u0026times; (Normalized Sharpe Ratio)\u0026thinsp;+\u0026thinsp;0.2 \u0026times; (Normalized CVaR Accuracy). Normalization scales each metric to [0,1] across models, with higher values indicating better performance (e.g., lower log MSE yields a higher score). The weighting gives 60% to the Loss Prediction Score, where the loss prediction ability is referred to as the most critical criterion in the study for the model selection due to the high influence on precise risk measurement and bond pricing. The remaining 20% each for Sharpe Ratio and CVaR Accuracy reflects their supportive roles in evaluating pricing efficiency and tail risk management, ensuring a balanced yet prioritized evaluation. Monte Carlo simulations and statistical tests ensure robustness, providing a foundation for model selection that optimally trades off predictive accuracy, portfolio optimization, and tail risk estimation\u0026mdash;driving the design of cyber risk transfer instruments.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\u003ch2\u003e3.4. Portfolio Optimization\u003c/h2\u003e\u003cp\u003eBuilding on the selected pricing model from section \u003cspan refid=\"Sec14\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e, which demonstrates superior loss prediction, pricing efficiency, and tail risk accuracy via the composite score, this section applies its outputs to a modified MPT framework. This approach evaluates the investment application value of cyber cat bonds by assessing their impact on risk-adjusted returns and diversification benefits, aligning with the study's objectives of bridging theoretical valuation to practical portfolio management.\u003c/p\u003e\u003cp\u003eTo evaluate this, I use a modified MPT approach. With a fixed asset allocation, I can access the impact of cyber cat bonds on risk-adjusted return, as measured by the Sharpe Ratio. This approach avoids calculating the efficient frontier, as different pricing models for cyber cat bonds could lead to inconsistent portfolio allocations, and it will complicate cross-model comparisons. Instead, I use a fixed allocations that allow for focused analysis of diversification benefits, reflecting real-world investor perspectives who seek to understand how cyber cat bonds enhance standard portfolios without requiring dynamic rebalancing. While this does not fully \"maximize\" returns as in standard MPT (Markowitz, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e1952\u003c/span\u003e), it provides practical insights into incremental improvements, particularly for institutional investors.\u003c/p\u003e\u003cp\u003eThe baseline portfolio of 60/40 allocation (60% stock index, 40% bond index) is designed th reflect a realistic investment scenario. It comprises 40% U.S. stock index (S\u0026amp;P 500), 20% worldwide stock index (All-World ex-US), 30% U.S. investment-grade bonds, and 10% international bonds. This composition mirrors common allocation in balanced funds and institutional portfolios. Moreover, combining US and international exposure can capture the benefits of international diversification and eventually improve the Sharpe Ratio (Pham et al., 2025, p.38). For these assets, I estimate expected returns and volatilities from historical data (2019\u0026ndash;2024, 5 years), with correlations computed using pairwise return series.\u003c/p\u003e\u003cp\u003eThe portfolio also includes a 2-year zero-coupon cyber cat bond (face value \u003cspan\u003e$\u003c/span\u003e100\u0026nbsp;million). The cyber cat bond is designed to repay its initial investment plus a risk premium (11.3%) at maturity unless catastrophe losses exceed the trigger amount, which is set at \u003cspan\u003e$\u003c/span\u003e558.11\u0026nbsp;million on per-occurrence basis. The indemnity trigger and per-occurrence basis are following the market preference, with 70% of cyber cat bonds using indemnity triggers. Since they offer the most direct linkage to policyholder losses and result in lower\u0026ensp;basis risk compared to parametric or index-based trigger variants. And the trigger of \u003cspan\u003e$\u003c/span\u003e558.11\u0026nbsp;million is adopted to ensure market relevance, based on average trigger amounts from existing cyber cat bonds (Artemis.bm 2025). Parameters for the cyber cat bond \u0026mdash; expected return, volatility and correlations \u0026mdash; are generated by the pricing model chosen, from which loss distributions and\u0026ensp;risk-adjusted returns are simulated. The expected return is the risk-neutral expected payoff \u0026mdash; discounted at the risk-free rate and with an allowance for a risk premium from the pricing model. Volatility comes from the standard deviation of simulated bond returns, which captures loss trigger uncertainty.\u003c/p\u003e\u003cp\u003eCorrelations between the cyber bonds and traditional assets are still not clear and due to the lack of historical data on cyber cat bonds. Based on a report of S\u0026amp;P Global Ratings 2025, which estimates correlations with stocks at 0.1\u0026ndash;0.24 (potentially rising above 0.3 during large-scale cyber events) and with bonds below 0.2, a conservative baseline correlation of 0.25 is assumed.\u003c/p\u003e\u003cp\u003eUsing MPT, I calculate the Sharpe Ratio defined as:\u003cdiv id=\"Equao\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equao\" name=\"EquationSource\"\u003e\n$$\\:Sharpe\\:Ratio=\\:\\frac{\\mathbb{E}\\left[{R}_{p}\\right]-{R}_{f}}{{\\sigma\\:}_{p}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbb{E}\\left[{R}_{p}\\right]=\\:\\sum\\:_{i=1}^{n}{w}_{i}\\mathbb{E}\\left[{R}_{i}\\right]\\)\u003c/span\u003e\u003c/span\u003e is the portfolio expected return, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{w}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the weight of asset \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:i\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbb{E}\\left[{R}_{i}\\right]\\)\u003c/span\u003e\u003c/span\u003e is the expected return of asset \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:i\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{R}_{f}\\)\u003c/span\u003e\u003c/span\u003e is the risk-free rate, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{p}\\)\u003c/span\u003e\u003c/span\u003e is the portfolio volatility, defined as:\u003cdiv id=\"Equap\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equap\" name=\"EquationSource\"\u003e\n$$\\:{\\sigma\\:}_{p}=\\:\\sqrt{{\\sum\\:}_{i=1}^{n}{\\sum\\:}_{j=1}^{n}{w}_{i}{w}_{j}{\\sigma\\:}_{i}{\\sigma\\:}_{j}{\\rho\\:}_{ij}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewith \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{i}\\:and\\:{\\sigma\\:}_{j}\\)\u003c/span\u003e\u003c/span\u003e are the volatilities of assets \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:i\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:j\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\rho\\:}_{ij}\\)\u003c/span\u003e\u003c/span\u003e is the correlation between their returns.\u003c/p\u003e\u003cp\u003eTo ensure robustness, I conduct sensitivity analysis by testing alternative portfolio allocations with equity/fixed income ratio of 70/30 and 50/50, using the same cyber cat bond proportions and trigger amounts. Additionally, I test different correlation assumptions (0.1, 0.4) and different trigger amounts (\u003cspan\u003e$\u003c/span\u003e300\u0026nbsp;million, 800\u0026nbsp;million) to address uncertainty due to limited historical data. This analysis examines whether cyber cat bonds consistently improve the Sharpe Ratio across different risk profiles and market conditions, offering insights for investors with varying risk preferences.\u003c/p\u003e\u003cp\u003eI also discuss the potential for cyber cat bonds to replace HY bonds, evaluating their investment value in a fixed-income portfolio. Using ICE BofA High Yield Index as a benchmark (with parameters estimated from 2019\u0026ndash;2025 historical data), I compare the Sharpe Ratio for three portfolios: Benchmark (100% HY bonds), 30% cyber cat bonds\u0026thinsp;+\u0026thinsp;70% HY bonds and 70% cyber cat bonds\u0026thinsp;+\u0026thinsp;30% HY bonds. The same methods and assumptions for cyber cat bonds are used. By comparing these metrics, I analyze whether cyber cat bonds enhance risk-adjusted returns and therefore could be an alternative to HY bonds. To ensure robustness, I also conduct sensitivity analysis by testing different trigger amounts (\u003cspan\u003e$\u003c/span\u003e300\u0026nbsp;million, 800\u0026nbsp;million) and different correlation assumptions (0.1, 0.4).\u003c/p\u003e\u003cp\u003eReflecting on this approach, expanding MPT to include cyber cat bonds emphasizes how they can improve portfolio diversification by offering an exceptionally diluted risk concept. But the reliance on model-based inputs adds uncertainty, especially when it comes to estimating a correlation that may not be well-anchored in scarce empirical evidence. This framework achieves a balance between analytical rigor and real-world applicability, providing investors with a clear perspective on the value of cyber cat bonds in managing cyber risk within a globally diversified portfolio.\u003c/p\u003e\u003c/div\u003e"},{"header":"4. Results and Analysis","content":"\u003cdiv id=\"Sec17\" class=\"Section2\"\u003e\u003ch2\u003e4.1 Pricing Model Performance\u003c/h2\u003e\u003cp\u003eTo determine the most suitable pricing models for cyber cat bonds, this section evaluates five pricing models \u0026ndash; Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach and Copula-POT Model \u0026ndash; based on three key criteria: loss prediction ability (log MSE), pricing efficiency (Sharpe ratio), and tail risk prediction accuracy (via \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e relative standard error), as detailed in section \u003cspan refid=\"Sec14\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eI selected the log MSE as the metric to compare each model\u0026rsquo;s ability to predict losses. A lower log MSE and standard deviation of log MSE (log MSE SD) indicate higher predictive accuracy and higher stability, respectively. Additionally, K-Statistic and associated p-value from Monte Carlo simulations calibrated to the CISSM dataset are used to access the statistical significance of the results. Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the summary of the empirical results.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eResults of Loss Prediction Ability\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"5\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eLog MSE\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eLog MSE SD\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eK-Statistic\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003ep-value\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.8514\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0048\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0081\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.5235\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.7538\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.018\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.2666\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSignal-Processing Approach\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e9.7226\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.4147\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.4722\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRegression Approach\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e3.8699\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0978\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.4984\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.0211\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0001\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.2238\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe Copula-POT Model achieves the highest loss prediction ability with the lowest log MSE (0.0211) and an exceptionally low log MSE SD (0.0001). This indicates both high accuracy and stability, aligning with the capacity to model the heavy-tailed distribution. Its reliance on copula functions and EVT effectively captures tail dependencies. However, its KS p-value of 0.0000 suggests a statistical bias, which requires further investigation of its distributional assumptions, possibly through refined copula parameter selection or increasing simulation iterations. The Loss Distribution Framework follows with a log MSE of 0.8514 and a relatively low standard deviation (0.0048) and is the only model to pass the KS test (p-value: 0.5235), showing adequate performance and a strong fit to the historical loss distribution. Though its higher MSE compared to Copula-POT suggests less precision for extreme tails. The Unified Bayesian Framework also performs well with a log MSE of 1.7538, but its higher standard deviation (0.018) suggests less consistent prediction, possibly due to reliance on prior assumptions that may not fully reflect the volatile nature of cyber data. In contrast, the Regression Approach and Signal-Processing Approach show not only poor predictive accuracy but also high variability. Moreover, they both fail the KS test (p-value: 0.0000), indicating significant deviations from the observed distribution.\u003c/p\u003e\u003cp\u003eI selected Sharpe Ratio to evaluate the risk-adjusted returns of a 2-year zero-coupon cyber cat bond with a trigger of \u003cspan\u003e$\u003c/span\u003e558.11\u0026nbsp;million, a risk premium of 11.3%, and a risk-free rate of 4.00%. A higher Sharpe Ratio indicates better return relative to risk, reflecting the model's ability to generate efficient bond valuations amid market discrepancies. The trigger probability derived from Monte Carlo simulations (10,000 paths) calibrated to the CISSM dataset, represents the likelihood of a cyber cat bond being triggered. A lower probability indicates lower risk for investors. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e4\u003c/span\u003e presents the empirical results.\u003c/p\u003e\u003cp\u003eThe Loss Distribution Framework achieves the highest pricing efficiency with a Sharpe Ratio of 1.8230 and a low trigger probability of 0.50%, indicating a strong balance between risk and return. Combined this performance with its reliable loss prediction ability makes it suitable for investors seeking stable returns with minimal trigger risk and aligns with the study's emphasis on diversification benefits.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eResults of Pricing Efficiency\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSharpe Ratio\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eTrigger probability\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.8230\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.50%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.4907\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e3.76%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSignal-Processing Approach\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.00%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRegression Approach\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.00%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.0000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.00%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe Unified Bayesian Framework follows with a moderate Sharpe Ratio of 0.4907 but a higher trigger probability of 3.76%, suggesting less efficiency due to the higher trigger risk. This performance may be attributed to a higher standard deviation of loss prediction (Log MSE SD\u0026thinsp;=\u0026thinsp;0.018). In contrast, the Signal-Processing Approach, Regression Approach and Copula-POT Model all yield a Sharpe Ratio with 0.0000 and a trigger probability of 0.00%, implying no bonds were triggered during the simulation and thus negligible returns. For the Signal-Processing Approach and Regression Approach, this aligns with their poor loss prediction performance, rendering them unsuitable for practical valuation. Notably, the Copula-POT Model has the most accurate and stable loss prediction ability, yet its Sharpe Ratio and trigger probability are both zero. It suggests an overly conservative pricing strategy, possibly due to data limitations, such as an insufficient number of extreme values for loss amount in CISSM dataset. This prevents the GPD from learning a sufficiently heavy tail, making it difficult to simulate extreme losses. The result highlights the Loss Distribution Framework as the most efficient model for pricing cyber cat bond, balancing high returns with low risk. The Copula-POT Model\u0026rsquo;s conservatism, despite its loss predication accuracy, warrants further investigation into its pricing assumption.\u003c/p\u003e\u003cp\u003eI selected CVaR at the 99% confidence level to evaluate tail risk prediction accuracy, which measures the expected loss in the worst 1% of scenarios. A lower value indicates a better ability to manage extreme losses. VaR at the 99% confidence level and the relative standard error of CVaR, derived from Monte Carlo simulations (10,000 paths) calibrated to the CISSM dataset, provide additional insights into the models\u0026rsquo; performance in capturing tail risk and prediction stability. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e5\u003c/span\u003e presents the empirical results. The Copula-POT Model shows the strongest overall performance with a low \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e and the lowest relative standard error (0.02%), reflecting exceptional stability and accuracy in capturing tail risk. The low SE aligns with its superior loss prediction ability reinforcing its robustness despite a conservative pricing strategy.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eResults of Tail Risk Prediction Accuracy\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e (\u003cspan\u003e$\u003c/span\u003eM)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{VaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e (\u003cspan\u003e$\u003c/span\u003eM)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eCVaR Relative SE\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e21,728.2538\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e14,090.5604\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.27%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e37,743.7876\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e25,817.1887\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.68%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSignal-Processing Approach\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.7444\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.3061\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e1.24%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRegression Approach\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e3.8023\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e3.1217\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.50%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e7,250.2630\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e5,439.1907\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.02%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe Loss Distribution Framework follows with a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e of \u003cspan\u003e$\u003c/span\u003e21,728.2538\u0026nbsp;million and a low relative standard error of 0.27%, indicating reliable tail risk estimates, consistent with its good loss prediction ability and high pricing efficiency. The Regression Approach achieves a moderate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e of \u003cspan\u003e$\u003c/span\u003e3.8023\u0026nbsp;million but a higher relative standard error (0.50%), suggesting less stable tail risk predictions, possibly due to its simplistic linear assumptions. The Unified Bayesian Framework performs poorly, with the highest \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e and high relative standard error (0.68%), indicating weak tail risk prediction. It is possibly due to parameter selection in the DP-HBCRM framework, which may overestimate losses given the right-skewed nature of the distribution. Surprisingly, the Signal-Processing Approach exhibits the lowest \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e. However, it also has high relative standard error (1.24%) and poor performance in loss prediction and investment efficiency. This suggests that these results may stem from model failure or unrealistic estimates.\u003c/p\u003e\u003cp\u003eThis subsection synthesizes the findings to evaluate the overall performance of five selected pricing models across three key criteria: loss prediction ability, pricing efficiency, and tail risk prediction accuracy. By integrating the results, I aim to identify the most suitable models for pricing cyber cat bonds and select candidates for portfolio optimization, addressing the research objectives outlined. The \u003cb\u003eLoss Distribution Framework\u003c/b\u003e exhibits balanced performance across all criteria. It has a low Log MSE of 0.8514 and Log MSE SD of 0.0048, indicating reliable loss prediction. Moreover, it is the only model that passes the KS test (p-value\u0026thinsp;=\u0026thinsp;0.5235), meaning a strong fit to historical loss distribution. It also achieves the highest pricing efficiency (Sharpe Ratio\u0026thinsp;=\u0026thinsp;0.9542) showing its ability to balance risk and return. Furthermore, its \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e of \u003cspan\u003e$\u003c/span\u003e21,728.2538\u0026nbsp;million and low relative standard error of 0.27% confirm the robust tail risk prediction, making it a versatile choice for investors navigating the evolving cyber risk market. The \u003cb\u003eCopula-POT Model\u003c/b\u003e excels in loss predictive accuracy but is limited by its pricing efficiency. It shows the lowest Log MSE (0.0210) and Log MSE SD (0.0001), demonstrating superior loss prediction. Though its p-value of 0.0000 suggests distributional bias requiring further parameter refinement. Its \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e of \u003cspan\u003e$\u003c/span\u003e7,250.2630\u0026nbsp;million and relative standard error of 0.02% highlight exceptional tail risk prediction, likely due to the POT model, which utilized EVT and large-scale simulations via Monte Carlo simulations. However, its Sharpe Ratio of 0.0000 indicates a conservative pricing strategy. It may be related to the high trigger amount or insufficient extreme value data in the CISMM dataset and will be examined during sensitivity analysis.\u003c/p\u003e\u003cp\u003eThe \u003cb\u003eUnified Bayesian Framework\u003c/b\u003e shows mixed performance. It has moderate loss prediction ability with Log MSE of 1.7538 and Log MSE SD of 0.018, yet with p-value of 0.0000 suggests poor distribution fit. It also has second-highest pricing efficiency. However, its high trigger risk (trigger probability\u0026thinsp;=\u0026thinsp;3.75%), which is higher than the average of 1.83% for cyber cat bonds (Artemis.bm 2025), limits its appeal. Furthermore, the second highest relative standard error of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e (0.68%) reflects weak tail risk prediction, likely due to DP-HBCRM parameter overestimation of right-skewed tails.The \u003cb\u003eSignal-Processing Approach\u003c/b\u003e and \u003cb\u003eRegression Approach\u003c/b\u003e underperform across all criteria. They both have high Log MSE and Log MSE SD among five models and zero Sharpe Ratio and trigger probability. Even though the Signal-Processing Approach achieves the lowest \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{CVaR}_{0.99}\\)\u003c/span\u003e\u003c/span\u003e of \u003cspan\u003e$\u003c/span\u003e1.7444\u0026nbsp;million, its high relative standard error of 1.24% indicates a potential for methodological unsuitability for cyber cat bonds. It is possibly due to the inadequacy of the CISMM dataset for the model and will be further explored. The Regression Approach demonstrates moderate in tail risk prediction; however, the overall results remain weak. This may be attributed to its simplistic linear or parametric assumptions and will be investigated. To compare the relative performance, a weighted scoring system, as mentioned in section \u003cspan refid=\"Sec14\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e, is applied: Composite Score\u0026thinsp;=\u0026thinsp;0.6 \u0026times; (Normalized Loss Prediction Score)\u0026thinsp;+\u0026thinsp;0.2 \u0026times; (Normalized Sharpe Ratio)\u0026thinsp;+\u0026thinsp;0.2 \u0026times; (Normalized CVaR Accuracy. Scores range from 1.0 (best) to 0.2 (worst) per criterion, with lower log MSE and relative SE, and higher Sharpe Ratio, receiving higher scores. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e5\u003c/span\u003e shows the visualized results in a bar chart.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eBased on these findings, I select Loss Distribution Framework, Copula-POT Model and Unified Bayesian Framework for portfolio optimization in 4.3. The Loss Distribution Framework and the Copula-POT Model achieve the highest composite score, indicating balanced performance. The Loss Distribution Framework has a balanced performance with the highest pricing efficiency, showing the potential to be the most appropriate pricing model for cyber cat bonds and providing stable and risk-adjusted returns for cyber cat bond investors. The Copula-POT Model exhibits superior loss prediction and tail risk accuracy. However, its Sharpe Ratio of 0.0000 requires further investigation, though it remains a promising candidate for a reliable pricing model for cyber cat bonds. The Unified Bayesian Framework, ranked third among five models, shows moderate investment efficiency and dynamic updating potential. The Signal-Processing Approach and Regression Approach are excluded due to their consistent underperformance and low composite scores.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e\u003ch2\u003e4.2 Portfolio Optimization Results\u003c/h2\u003e\u003cp\u003eThis section presents the results of adding cyber cat bonds into a diversified investment portfolio and a HY Index, using a modified MPT approach. The analysis evaluates three selected pricing models\u0026mdash;Loss Distribution Framework, Copula-POT Model, and Unified Bayesian Framework\u0026mdash;based on their ability to enhance risk-adjusted returns, as measured by the Sharpe Ratio. I integrate cyber cat bonds into a baseline 60/40 portfolio and a HY benchmark; therefore, it is possible to access the practical value of these pricing models from an investor\u0026rsquo;s perspective, including diversification and optimization of portfolios. Results are presented with a cyber cat bond allocation of 5% and 10% for the baseline portfolio (reflecting conservative institutional strategies) and 30% and 70% in the HY portfolio (simulating alternative asset replacement scenarios), following sensitivity analyses to test robustness across different assumptions.\u003c/p\u003e\u003cp\u003eThe baseline portfolio, designed to reflect a realistic balanced fund, yields an annualized return of 6.18%, an annualized volatility of 12.42%, and a Sharpe Ratio of 0.176. This performance serves as a benchmark for assessing the impact of adding cyber cat bonds, which are modeled as 2-year zero-coupon bonds with a face value of \u003cspan\u003e$\u003c/span\u003e100\u0026nbsp;million, a risk premium of 11.3%, and a per-occurrence indemnity trigger of \u003cspan\u003e$\u003c/span\u003e558.11\u0026nbsp;million, consistent with market standards. Tables\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e6\u003c/span\u003e and \u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e7\u003c/span\u003e show the results for cyber cat bond allocations of 5% and 10%, respectively. They reveal a clear improvement over the baseline portfolio, indicating that cyber cat bonds enhance risk-adjusted returns, likely due to their low correlation with traditional assets (assumed at 0.25) and attractive risk premia.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eResults of Portfolio Optimization with 5% Cyber Cat Bond\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSharpe Ratio\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eAnn. Expected Return\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eAnn. Volatility\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.352\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e7.00%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e8.54%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.323\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e6.83%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e8.78%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.353\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e6.97%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e8.40%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eResults of portfolio optimization with 10% cyber cat bond\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSharpe Ratio\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eAnn. Expected Return\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eAnn. Volatility\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.441\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e7.81%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e8.64%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.378\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e7.47%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e9.18%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.447\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e7.74%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e8.36%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe Copula-POT Model shows the highest Sharpe Ratio at both 5% (0.353) and 10% (0.447) allocations, paired with the lowest annualized volatility. However, this outcome is attributed to zero trigger probabilities, suggesting that its conservative pricing strategy may limit standalone performance but becomes advantageous in a diversified portfolio. This discrepancy warrants further investigation, which I will discuss in section \u003cspan refid=\"Sec19\" class=\"InternalRef\"\u003e5\u003c/span\u003e. The Loss Distribution Framework perform robustly, with a Sharpe Ratio of 0.352 (5%) and 0.441 (10%), and annualized volatility of 8.54% and 8.64%. The slight increase in volatility at the 10% allocation shows a trade-off between higher returns and higher risk. Nonetheless, its balanced performace across all criteria makes it a reliable choice for investor. The Unified Bayesian Framework has the weakest performance with a Sharpe Ratio of 0.323 (5%) and 0.378 (10%), and the highest annualized volatility. Its relatively higher trigger probability likely causes the increasing portfolio risk and then reduces its risk-adjusted returns compared to the other models.\u003c/p\u003e\u003cp\u003eThe benchmark, represented by the ICE BofA High Yield Index, yields an annualized return of 4.11%, an annualized volatility of 9.69%, and a Sharpe Ratio of 0.0116, offering a starting point to evaluate the potential of cyber cat bonds. Table\u0026nbsp;\u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e8\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e9\u003c/span\u003e present the results for portfolios mixing cyber cat bonds with HY bonds, with 2 different allocations, using the same pricing models and assumptions as in the previous subsection. The results show that all models achieve higher Sharpe Ratios than the benchmark across both allocations. Expect to the Unified Bayesian Framework of 70% allocation, other cases all exhibit lower annualized volatility with higher annualized returns and Sharpe Ratio. This suggests cyber cat bonds, especially when priced with the Copula-POT Model and Loss Distribution Framework, can significantly enhance risk-adjusted returns, making them a compelling alternative to HY bonds.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab7\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 8\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eSharpe Ratio for 30% Cyber Cat Bonds\u0026thinsp;+\u0026thinsp;70% HY Bonds\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSharpe Ratio\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eAnn. Expected Return\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eAnn. Volatility\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.4905\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e7.70%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e7.54%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.2800\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e6.68%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e9.56%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.5116\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e7.47%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e6.78%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab8\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 9\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eSharpe Ratio for 70% Cyber Cat Bonds\u0026thinsp;+\u0026thinsp;30% HY Bonds\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSharpe Ratio\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eAnn. Expected Return\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eAnn. Volatility\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.3909\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e12.48%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e6.10%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.4591\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e10.10%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e13.29%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e2.7348\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e11.95%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e2.91%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTo access the robustness of these results, I also conducted univariate sensitivity analyses by varying key assumptions individually, such as cyber cat bond allocations (5% vs. 10%) and portfolio allocations (50/50, 60/40, 70/30) for the baseline portfolio, trigger amounts (\u003cspan\u003e$\u003c/span\u003e300\u0026nbsp;million, \u003cspan\u003e$\u003c/span\u003e558.11\u0026nbsp;million, \u003cspan\u003e$\u003c/span\u003e800\u0026nbsp;million) and correlation coefficients (0.1, 0.25, 0.4). For baseline portfolio, these analyses build on the 10% allocation results from Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e7\u003c/span\u003e, and for HY bonds, these analyses build on the 70% cyber cat bonds\u0026thinsp;+\u0026thinsp;30% HY bonds results from Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e9\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the trigger amount analysis and Table\u0026nbsp;\u003cspan refid=\"Tab9\" class=\"InternalRef\"\u003e10\u003c/span\u003e shows the trigger probabilities and corresponding Sharpe Ratio (single bond). Table\u0026nbsp;\u003cspan refid=\"Tab9\" class=\"InternalRef\"\u003e10\u003c/span\u003e indicates an inverse relationship between trigger amounts and trigger probabilities\u0026mdash;higher triggers reduce the likelihood of bond activation.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab9\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 10\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eTrigger Probability and Sharpe Ratio of Individual Bond by Trigger Amount\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eTrigger: \u003cspan\u003e$\u003c/span\u003e300M\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eTrigger: \u003cspan\u003e$\u003c/span\u003e588.11M\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eTrigger: \u003cspan\u003e$\u003c/span\u003e800M\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLoss Distribution Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.10% (1.192)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.5% (1.923)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.24% (2.665)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUnified Bayesian Framework\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e3.98% (0.483)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e3.76% (0.490)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e3.64% (0.498)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCopula-POT Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.01% (0.000)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.00% (0.000)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.00% (0.000)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe Copula-POT Model maintains the highest and stable Sharpe Ratio of 0.447 across all trigger levels. This phenomenon may be attributed to the fact that the trigger probabilities are equal to zero, consequently, minimizing expected losses and leading to the identical bond prices. The Loss Distribution Framework shows moderate sensitivity to trigger amounts, with Sharpe Ratios increasing from 0.43 at \u003cspan\u003e$\u003c/span\u003e300\u0026nbsp;million to 0.447 at \u003cspan\u003e$\u003c/span\u003e800\u0026nbsp;million. This behavior is likely due to its pricing model for zero-coupon bonds, where the price is determined only by the discounted present value of the face value, unaffected by trigger amounts. Therefore, higher trigger amounts (lower trigger probabilities) yield to higher Sharpe Ratio. In contrast, the Unified Bayesian Framework exhibits a flat Sharpe Ratio of 0.379\u0026thinsp;\u0026minus;\u0026thinsp;0.378, suggesting robust results and limited sensitivity even with different trigger probabilities. This could also result from its bond pricing mechanism. A higher trigger amount reduces the probability of the bond being triggered, thereby lowering expected losses and stabilizing returns. Overall, the Sharpe Ratio of the Loss Distribution Framework varies with trigger amounts; however, the results are very similar to Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e6\u003c/span\u003e, irrespective of the selected trigger amount. The Copula-POT Model demonstrates the best performance, while the Loss Distribution Framework follows. The Unified Bayesian Framework performs the worst. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e7\u003c/span\u003e shows the correlation coefficient analysis, highlighting the diversification benefits of cyber cat bonds. The Copula-POT Model holds a consistent Sharpe Ratio of 0.447 across correlation levels. The Loss Distribution Framework and the Unified Bayesian Framework both exhibit the same trend: lower correlations lead to higher Sharpe Ratios. However, while the Loss Distribution Framework maintains a relatively steady performance across different correlation levels, the Unified Bayesian Framework shows greater sensitivity to higher correlations, likely due to its higher volatility. The overall results suggest that lower correlations significantly enhance Sharpe Ratio, and they remain consistent with the results from Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Notably, at a correlation of 0.1, the Loss Distribution Framework slightly outperforms the Copula-POT Model, possibly due to favorable simulation conditions.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e8\u003c/span\u003e shows the portfolio allocation analysis. These results indicate that Sharpe Ratio tends to increase as the equity proportion increases. Moreover, they also remain identical ranking as Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e7\u003c/span\u003e, indicating the result is not affected by portfolio allocation.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFor the HY portfolio, Figs.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e9\u003c/span\u003e and \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e10\u003c/span\u003e present the similar sensitivity analyses for trigger amounts and correlations. The Copula-POT Model maintains outperformance and stability across all assumptions. The Loss Distribution Framework follows with moderate results; however, it demonstrates the sensitivity to both trigger amounts and correlation coefficient. The Unified Bayesian Framework perform the worst as Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e9\u003c/span\u003e presents across all assumptions, but it shows the relative stable Sharpe Ratio. It is evident that, irrespective of the assumption applied, the 70% cyber cat bonds\u0026thinsp;+\u0026thinsp;30% HY bonds allocation exhibits superior performance in comparison to its benchmark (100% HY bonds). The sensitivity analyses reinforce the results from Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e7\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e9\u003c/span\u003e that the Copula-POT Model and Loss Distribution Framework outperform the Unified Bayesian Framework, offering reliable performance for cyber cat bond integration into diversified portfolios and HY bonds, and cyber cat bonds exhibit significant potential to serve as an alternative to HY bonds while offering better risk-adjusted returns. Furthermore, the practical significance lies in guiding investors toward higher trigger amounts (e.g., \u003cspan\u003e$\u003c/span\u003e800\u0026nbsp;million) and lower correlations (e.g., 0.1) to maximize returns. Regarding model robustness the Copula-POT Model demonstrates high stability across all parameters, suggesting minimal need for structural changes, though its zero trigger probability may reflect data limitations. The Loss Distribution Framework shows moderate sensitivity to trigger amounts and correlation coefficients in both portfolios. The Unified Bayesian Framework is sensitive to correlation coefficients in the baseline portfolio, highlighting a need to reduce model volatility. These insights indicate the importance of precise historical loss data and correlation assumptions to strengthen model reliability. However, the analysis is constrained by several limitations. The assumed correlation ranges (0.1\u0026ndash;0.4) are based on sparse empirical evidence from sources like S\u0026amp;P Global Ratings (2025), which could potentially underrepresent real-world dynamics. Additionally, the univariate approach may overlook interaction effects between parameters, such as the combined impact of trigger amounts and correlations, which could alter outcomes.\u003c/p\u003e\u003cp\u003eThis section consolidates the key insights derived. Firstly, incorporating cyber cat bonds into the baseline portfolio enhances risk-adjusted returns of the portfolio, likely due to low correlation between cyber cat bonds and traditional assets and diversification benefits. Secondly, replacing a proportion of HY bonds with cyber cat bonds also enhances risk-adjusted returns. Therefore, cyber cat bonds have great potential to become an alternative to HY bonds. Additionally, the sensitivity analysis confirms the robustness of these findings across different assumptions such as trigger amounts and correlation coefficients. These results emphasize the prospective value of cyber cat bonds as a distinct asset class within the modified framework of MPT, thereby establishing a basis for improved investment strategies, although challenges due to data availability and model accuracy are to be noted.\u003c/p\u003e\u003c/div\u003e"},{"header":"5. Discussion","content":"\u003cp\u003eThe findings from section \u003cspan refid=\"Sec16\" class=\"InternalRef\"\u003e4\u003c/span\u003e provide key insights into the valuation of cyber cat bond pricing models and their implications for portfolio optimization. The comparative analysis of five models\u0026mdash;the Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach, and Copula-POT Model\u0026mdash;highlights their respective strengths and limitations. The Loss Distribution Framework aligns well with historical data, passing the KS test, and performs robustly across loss prediction, pricing efficiency, and tail risk criteria. The Unified Bayesian Framework benefits from incorporating prior knowledge, enhancing robustness under data scarcity, though improper priors can introduce bias. The Signal-Processing and Regression Approaches underperform due to limited variability and insufficient data, yielding poor predictive accuracy. The Copula-POT Model excels in loss prediction and tail risk estimation, achieving the highest composite scores, though its zero trigger probability limits individual bond Sharpe Ratios, likely due to dataset constraints. Nevertheless, in diversified portfolios, this conservatism enhances risk-adjusted performance under the modified MPT framework (FBI 2023, 7; FBI 2024, 4). Portfolio optimization results demonstrate the practical benefits of integrating cyber cat bonds. Adding 10% of cyber cat bonds priced via the Copula-POT Model increases the Sharpe Ratio from 0.176 to 0.447 in a baseline portfolio, while substituting 30% of HY bonds raises it from 0.0116 to 0.5116. These improvements stem from low correlation with traditional assets, offering diversification and superior risk-adjusted returns compared to HY bonds. Sensitivity analyses confirm the robustness of these results across varying trigger amounts and correlations, emphasizing the strategic advantage of cyber cat bonds in institutional portfolios. Challenges primarily relate to data limitations. The CISSM dataset lacks detailed loss amounts and extreme event observations, constraining model accuracy, especially for extreme-value methods like the Copula-POT Model. Manual data supplementation may introduce inconsistencies, while parameter selection, log transformations, and model assumptions create additional uncertainty. Moreover, cyber risks are inherently volatile and rapidly evolving, making historical data less predictive for future losses. These limitations underscore the need for caution in interpreting results but do not diminish the observed diversification and portfolio benefits. Future research should expand datasets, utilize machine learning to impute missing data, and develop hybrid models combining the Copula-POT and Loss Distribution Framework strengths. Interdisciplinary approaches could investigate investor behavior to improve adoption of cyber cat bonds. Empirical validation through recent deals will help refine model accuracy and assess real-world portfolio benefits. Advancing these directions will strengthen cyber cat bonds\u0026rsquo; role as a viable alternative to HY bonds and enhance their integration into capital markets, addressing both predictive and portfolio optimization objectives (FBI 2023, 7; FBI 2024, 4).\u003c/p\u003e"},{"header":"6. Conclusion","content":"\u003cp\u003eThis study has explored the pricing models for cyber cat bonds and their integration into investment portfolios, addressing a critical gap in the implementation of cyber risk management with modern financial strategies. The key findings reveal that among the five pricing models evaluated\u0026mdash;the Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach, and Copula-POT Model\u0026mdash;the Copula-POT Model demonstrates superior and robust performance in loss prediction capability. It achieved the highest scores in accuracy and stability due to its effective capture of dependencies via Copula functions and extreme value modeling with POT, making it ideal for handling the volatile nature of cyber losses. The Loss Distribution Framework followed with the same composite score as the Copula-POT Model, excelling in pricing efficiency with a Sharpe Ratio of 0.9542 and robust distribution fit, as confirmed by its successful KS test results. Other models, such as the Unified Bayesian Framework and Regression Approach, showed limitations in predictive ability due to parameter sensitivities and incomplete data, while the Signal-Processing Approach underperformed owing to uniform event states in the CISSM dataset. In portfolio optimization, incorporating cyber cat bonds under a modified MPT framework enhances diversification, offering a distinct risk-return profile that outperforms the balanced portfolio and traditional HY bonds in stability across varying market conditions, as shown in the comparative and sensitivity analyses.\u003c/p\u003e\u003cp\u003eThese contributions advance the literature by providing a robust framework for pricing cyber cat bonds and empirical evidence of their role as an alternative asset class, addressing three primary objectives outlined and filling research gaps identified regarding model comparisons and practical applications. Empirically, the study assessed and compared the performance of multiple pricing models using a standardized CISSM dataset supplemented with public loss amount data, systematically evaluating their predictive validity and real-world applicability through metrics like accuracy, robustness, and KS tests. Moreover, the identification of the highest-performing model for loss prediction was also achieved. Theoretically, the application of these results to MPT maximized risk-adjusted returns, demonstrating through baseline optimizations and comparisons that adding cyber cat bonds into a balanced portfolio can enhance the risk-adjusted return, and they can serve as a viable substitute for HY bonds, providing investors with diversification benefits.\u003c/p\u003e\u003cp\u003ePractically, the portfolio optimization results demonstrate tangible benefits, such as elevating the baseline portfolio's Sharpe Ratio from 0.176 to 0.447 with a 10% allocation priced via Copula-POT Model, and outperforming HY bonds with a Sharpe Ratio of 2.7348 in a 30% HY bonds and 70% cyber cat bonds allocation. Sensitivity analyses confirm robustness across trigger amounts, correlation and allocation, emphasizing the models' resilience under varying assumptions. While limitations such as data inconsistencies and model complexities tempered the results, the objectives were met by offering actionable insights that bridge theoretical modeling with real-world investment implications.\u003c/p\u003e\u003cp\u003eIn final remarks, cyber cat bonds may represent a transformative tool in investment portfolios, particularly in a time when cyber risks are projected to cost the global economy \u003cspan\u003e$\u003c/span\u003e10.5 trillion annually by 2025 (Cybersecurity Ventures 2020). Their inclusion not only mitigates systemic risks through capital market transfer but also enhances investment efficiency, as evidenced by the 154% Sharpe Ratio improvement in optimized portfolios. As markets evolve, with cyber cat bonds comprising only 1.4% of the \u003cspan\u003e$\u003c/span\u003e56.1\u0026nbsp;billion cat bond market (Artemis.bm 2025), their untapped potential underscores the need for greater adoption. This study demonstrated that the Copula-POT Model is the most suitable choice among five selected pricing models for valuing cyber cat bonds and cyber cat bonds offer diversification beyond traditional assets, appealing to institutional investors seeking HY alternatives with low market linkage. However, unlocking their full potential requires resolution of data and modeling issues as posited in future research initiatives. In summary, this study affirms the strategic role of cyber cat bonds in encouraging innovative and resilient risk management and consequently greater and safer acceptance in the financial world.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eI have no statement\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAnderson, David. 2018. Desmond Higham, and Yuyuan Sun. Computational complexity analysis for Monte Carlo approximations of classically scaled population processes. arXiv preprint arXiv:1512.01588v3 [math.NA]. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://arxiv.org/abs/1512.01588v3\u003c/span\u003e\u003cspan address=\"https://arxiv.org/abs/1512.01588v3\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eArtemis.bm. 2025. Catastrophe bonds \u0026amp; ILS outstanding by risk or peril. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.artemis.bm/dashboard/cat-bonds-ils-by-risk-or-peril/\u003c/span\u003e\u003cspan address=\"https://www.artemis.bm/dashboard/cat-bonds-ils-by-risk-or-peril/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eArtemis.bm. 2025. Catastrophe bonds \u0026amp; ILS Market Dashboard. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.arte- mis.bm/dashboard/\u003c/span\u003e\u003cspan address=\"https://www.arte- mis.bm/dashboard/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eArtemis.bm. 2025. Catastrophe Bond \u0026amp; Insurance-Linked Securities Deal Directory. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.artemis.bm/deal-directory/\u003c/span\u003e\u003cspan address=\"https://www.artemis.bm/deal-directory/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBodie, Zvi, and Alex Kane, Alan Marcus. 2014. \u003cem\u003eInvestments\u003c/em\u003e. 10th ed. New York: McGraw Hill Higher Education.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBraun, Alexander, Martin Eling, and Christian Jaenicke. 2023. Cyber insurance-linked se- curities. \u003cem\u003eASTIN Bulletin\u003c/em\u003e 53(3):684\u0026ndash;705. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1017/asb.2023.22\u003c/span\u003e\u003cspan address=\"10.1017/asb.2023.22\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCarter, Steve. 2018. \u003cem\u003eCyber-Catastrophe Insurance-Linked Securities\u003c/em\u003e. on Smart Ledgers. Long Finance.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChigada, Joel Rujeko Madzinga. 2021. Cyberattacks and threats during COVID-19: A systematic literature review. \u003cem\u003eSA Journal of Information Management\u003c/em\u003e 23. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.4102/sajim.v23i1.1277\u003c/span\u003e\u003cspan address=\"10.4102/sajim.v23i1.1277\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChimamiwa, George. 2024. Managing cyber risks in the face of AI- and ML-Driven Ad- versarial Attacks. 71\u0026ndash;79. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.70301/CONF.SBS-JABR.2024.1/1.6\u003c/span\u003e\u003cspan address=\"10.70301/CONF.SBS-JABR.2024.1/1.6\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCummins, J. David. 2013. and Pauline Barrieu. Innovations in Insurance Markets: Hybrid and Securitized Risk-Transfer Solutions. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1007/978-1-4614-0155-1_20\u003c/span\u003e\u003cspan address=\"10.1007/978-1-4614-0155-1_20\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCummins, J. David, Philippe Trainar. 2009. Securitization, Insurance, and Reinsurance. \u003cem\u003eThe Journal of Risk and Insurance\u003c/em\u003e 76(3):463\u0026ndash;492. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1111/j.1539-6975.2009.01313.x\u003c/span\u003e\u003cspan address=\"10.1111/j.1539-6975.2009.01313.x\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCummins, J., Daivd, and Mary A. Weiss. 2009. Convergence of Insurance and Financial Markets: Hybrid and Securitized Risk-Transfer Solutions. \u003cem\u003eJournal of Risk and Insurance\u003c/em\u003e 76(3):493\u0026ndash;545. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2139/ssrn.1260399\u003c/span\u003e\u003cspan address=\"10.2139/ssrn.1260399\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCyberCube. 2023. Digital Ties and Natural Divides: Correlation and Diversification in Cyber Catastrophe Bonds. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://insights.cybcube.com/correlation-and-diversification-in- cyber-catastrophe-bonds\u003c/span\u003e\u003cspan address=\"https://insights.cybcube.com/correlation-and-diversification-in- cyber-catastrophe-bonds\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCybersecurity Ventures. 2020. Cybercrime To Cost The World \u003cspan\u003e$\u003c/span\u003e10.5 Trillion Annually By 2025. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://cybersecurityventures.com/hackerpocalypse-cybercrime-report-2016/\u003c/span\u003e\u003cspan address=\"https://cybersecurityventures.com/hackerpocalypse-cybercrime-report-2016/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Ac- cessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eDemers-B\u0026eacute;langer, and Kim, Can Sin Lai. 2020. Diversification benefits of cat bonds: An in‐depth examination. \u003cem\u003eFinancial Markets Institutions \u0026amp; Instruments\u003c/em\u003e 29(5):165\u0026ndash;228. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1111/fmii.12132\u003c/span\u003e\u003cspan address=\"10.1111/fmii.12132\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eDomfeh, Daniel, Anirban Chatterjee, and Matthew Dixon. 2022. A Unified Bayesian Framework for Pricing Catastrophe Bond Derivatives. arXiv preprint arXiv:2205.04520. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://arxiv.org/abs/2205.04520\u003c/span\u003e\u003cspan address=\"https://arxiv.org/abs/2205.04520\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eDrobetz, Wolfgang, and Henning Schr\u0026ouml;der, Lars Tegtmeier. 2019. The Role of CAT Bonds in an International Multi-Asset Portfolio: Diversifier, Hedge, or Safe Haven? \u003cem\u003eSSRN Elec- tronic Journal\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2139/ssrn.3359277\u003c/span\u003e\u003cspan address=\"10.2139/ssrn.3359277\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eDubois, Elisabeth V., F. Omer, and Keskin. 2022. and Unal Tatar. Cyber Risk Modeling Meth- ods and Data Sets: A Systematic Interdisciplinary Literature Review for Actuaries. SOA Research Institute.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eFederal Bureau of Investigation. 2023. Internet Crime Complaint Center Report. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.ic3.gov/Media/PDF/AnnualReport/2023_IC3Report.pdf\u003c/span\u003e\u003cspan address=\"https://www.ic3.gov/Media/PDF/AnnualReport/2023_IC3Report.pdf\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eFederal Bureau of Investigation. 2024. Internet Crime Complaint Center Report. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.ic3.gov/Media/PDF/AnnualReport/2024_IC3Report.pdf\u003c/span\u003e\u003cspan address=\"https://www.ic3.gov/Media/PDF/AnnualReport/2024_IC3Report.pdf\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eFRED. 2025. ICE BofA US High Yield Index Effective Yield. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://fred.stlou- isfed.org/series/BAMLH0A0HYM2EY\u003c/span\u003e\u003cspan address=\"https://fred.stlou- isfed.org/series/BAMLH0A0HYM2EY\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eFRED. 2025. Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity, Quoted on an Investment Basis. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://fred.stlouisfed.org/series/DGS10\u003c/span\u003e\u003cspan address=\"https://fred.stlouisfed.org/series/DGS10\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed Sep- tember 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eG\u0026ouml;tze, Thomas, and Marc G\u0026uuml;rtler, Eileen Witowski. 2020. Improving CAT bond pricing models via machine learning. \u003cem\u003eJournal of Asset Management\u003c/em\u003e 21(5):428\u0026ndash;446. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1057/s41260-020-00167-0\u003c/span\u003e\u003cspan address=\"10.1057/s41260-020-00167-0\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHarry, Charles, Nancy Gallagher. 2018. Classifying Cyber Events. \u003cem\u003eJournal of Infor- mation Warfare\u003c/em\u003e 17(3):17\u0026ndash;31.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHofer, Lorenz, Patrizio Gardoni, and Michele A. Zanini. 2019. Risk-Based CAT Bond Pricing Considering Parameter Uncertainties. \u003cem\u003eSustainable and Resilient Infrastructure\u003c/em\u003e 6(5\u0026ndash;6):315\u0026ndash;331. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1080/23789689.2019.1667116\u003c/span\u003e\u003cspan address=\"10.1080/23789689.2019.1667116\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eIBM. 2024. Cost of a Data Breach Report. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.ibm.com/reports/data-breach\u003c/span\u003e\u003cspan address=\"https://www.ibm.com/reports/data-breach\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Ac- cessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eICE. 2025. Index Platform. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://indices.ice.com\u003c/span\u003e\u003cspan address=\"https://indices.ice.com\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKish, Richard. 2023. Catastrophe (CAT) bonds: risk offsets with diversification and high returns. \u003cem\u003eFinancial Services Review\u003c/em\u003e 25(3):303\u0026ndash;329. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.61190/fsr.v25i3.3281\u003c/span\u003e\u003cspan address=\"10.61190/fsr.v25i3.3281\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKshetri, Nir. 2020. The evolution of cyber-insurance industry and market: An institutional analysis. \u003cem\u003eTelecommunications Policy\u003c/em\u003e 44 (8). 102007. \u003cdiv class=\"ExternalRefDOI\"\u003e10.1016\u003c/div\u003e/j.telpol.2020.102007 Kolesnikov, Oleg, Aleksei Markov, Dmitry Smagulov, and Sergejs Solovjovs. 2022. Cyber.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLoss Distribution Fitting. 2022. A General Framework towards Cyber Bonds and Their Pricing Models. \u003cem\u003eInternational Journal of Mathematics and Mathematical Sciences\u003c/em\u003e 1\u0026ndash;20. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1155/2022/7689828\u003c/span\u003e\u003cspan address=\"10.1155/2022/7689828\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLane, Marton. 2000. Pricing Risk Transfer Transactions. \u003cem\u003eASTIN Bulletin\u003c/em\u003e 30(2):259\u0026ndash;293. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2143/AST.30.2.504635\u003c/span\u003e\u003cspan address=\"10.2143/AST.30.2.504635\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLi, Yifei, Richard Mamon. 2023. The Price Tag of Cyber Risk: A Signal-Processing Approach. \u003cem\u003eIeee Access : Practical Innovations, Open Solutions\u003c/em\u003e 11:44294\u0026ndash;44318. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1109/ACCESS.2023.3272572\u003c/span\u003e\u003cspan address=\"10.1109/ACCESS.2023.3272572\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMarkowitz, Harry. 1952. Portfolio Selection. \u003cem\u003eThe Journal of Finance\u003c/em\u003e 7(1):77\u0026ndash;91. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2307/2975974\u003c/span\u003e\u003cspan address=\"10.2307/2975974\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMastroeni, Loretta, and Alessandro Mazzoccoli, Maurizio Naldi. 2023. Cyber Insurance Premium Setting for Multi-Site Companies under Risk Correlation. \u003cem\u003eRisks\u003c/em\u003e 11(10):167. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.3390/risks11100167\u003c/span\u003e\u003cspan address=\"10.3390/risks11100167\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMouelhi, Chaker. 2021. The Relationship Between Cat Bond Market and Other Financial Asset Markets: Evidence from Cointegration Tests. \u003cem\u003eEuropean Journal of Business and Management Research\u003c/em\u003e 6(2):78\u0026ndash;85. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.24018/ejbmr.2021.6.2.790\u003c/span\u003e\u003cspan address=\"10.24018/ejbmr.2021.6.2.790\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMunich Re. 2025. Cyber Insurance: Risks and Trends 2025. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.munichre.com/en/insights/cyber/cyber-insurance-risks-and-trends-2025.html\u003c/span\u003e\u003cspan address=\"https://www.munichre.com/en/insights/cyber/cyber-insurance-risks-and-trends-2025.html\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNiehaus, Greg. 2002. The Allocation of Catastrophe Risk. \u003cem\u003eJournal of Banking and Fi- nance\u003c/em\u003e 26(2\u0026ndash;3):585\u0026ndash;596. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1016/S0378-4266(01)00235-7\u003c/span\u003e\u003cspan address=\"10.1016/S0378-4266(01)00235-7\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eOrlando, Albina, Maria Francesca Carfora, Fabio Martinelli, and Francesco Mercaldo. 2018. and Art- siom Yautsiukhin. Cyber Risk Management: A New Challenge for Actuarial Mathe- matics: MAF 2018. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1007/978-3-319-89824-7_36\u003c/span\u003e\u003cspan address=\"10.1007/978-3-319-89824-7_36\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePain, David. 2024. Catalysing Cyber Risk Transfer to Capital Markets: Catastrophe bonds and beyond. \u003cem\u003eGeneva Association\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.genevaassociation.org/publica- tion/cyber/catalysing-cyber-risk-transfer-capital-markets-catastrophe-bonds-and-beyond\u003c/span\u003e\u003cspan address=\"https://www.genevaassociation.org/publica- tion/cyber/catalysing-cyber-risk-transfer-capital-markets-catastrophe-bonds-and-beyond\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePeters, Gareth, Pavel Shevchenko, Ruben Cohen, and Diane Maurice. 2017. Understand- ing Cyber Risk and Cyber Insurance. \u003cem\u003eSSRN Electronic Journal\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2139/ssrn.3065635\u003c/span\u003e\u003cspan address=\"10.2139/ssrn.3065635\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePham, Nhan, and Bing Cui. 2025. and Umar Ruthbah. The Performance of the 60/40 Portfolio: A Historical Perspective. CFA Institute. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://rpc.cfainstitute.org/research/re- ports/2025/performance-of-the-60-40-portfolio\u003c/span\u003e\u003cspan address=\"https://rpc.cfainstitute.org/research/re- ports/2025/performance-of-the-60-40-portfolio\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePolacek, Andy. 2018. Catastrophe bonds: A primer and retrospective. \u003cem\u003eChicago Fed Letter\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.chicagofed.org/publications/chicago-fed-letter/2018/405\u003c/span\u003e\u003cspan address=\"https://www.chicagofed.org/publications/chicago-fed-letter/2018/405\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eS\u0026amp;P Global Rating. 2025. Cyber Risk Insights: Cyber Catastrophe Bonds Offer Greater Scope for Risk Mitigation. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.spglobal.com/ratings/en/regulatory/article/250317-cyber-risk-insights-cyber- catastrophe-bonds-offer-greater-scope-for-risk-mitigation-s13438912\u003c/span\u003e\u003cspan address=\"https://www.spglobal.com/ratings/en/regulatory/article/250317-cyber-risk-insights-cyber- catastrophe-bonds-offer-greater-scope-for-risk-mitigation-s13438912\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSharpe, William. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Con- ditions of Risk. \u003cem\u003eThe Journal of Finance\u003c/em\u003e 19(3):425\u0026ndash;442. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2307/2977928\u003c/span\u003e\u003cspan address=\"10.2307/2977928\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSkeoch, Hannah. 2023. and Christos Ioannidis. The barriers to sustainable risk transfer in the cyber-insurance market. arXiv preprint arXiv:2303.02061. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://arxiv.org/abs/2303.02061\u003c/span\u003e\u003cspan address=\"https://arxiv.org/abs/2303.02061\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eStatista. 2025. Estimated Cost of Cybercrime Worldwide 2018\u0026ndash;2029. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.sta- tista.com/forecasts/1280009/cost-cybercrime-worldwide\u003c/span\u003e\u003cspan address=\"https://www.sta- tista.com/forecasts/1280009/cost-cybercrime-worldwide\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eStrupczewski, Grzegorz. 2021. Defining Cyber Risk. \u003cem\u003eSafety Science\u003c/em\u003e 135:105143. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1016/j.ssci.2020.105143\u003c/span\u003e\u003cspan address=\"10.1016/j.ssci.2020.105143\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eTang, Yuchen, Chang Wen, Chenggang Ling, and Yujun Zhang. 2023. Pricing Multi- Event-Triggered Catastrophe Bonds Based on a Copula\u0026ndash;POT Model. \u003cem\u003eRisks\u003c/em\u003e 11(8):151. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.3390/risks11080151\u003c/span\u003e\u003cspan address=\"10.3390/risks11080151\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eTobin, James. 1958. Liquidity Preference as Behavior Towards Risk. \u003cem\u003eThe Review of Eco- nomic Studies\u003c/em\u003e 25(2):65\u0026ndash;86. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2307/2296205\u003c/span\u003e\u003cspan address=\"10.2307/2296205\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eTrottier, Denis-Alexandre, Van Son Lai, and Anne-Sophie Charest. 2017. CAT Bond Spreads via HARA Utility and Nonparametric Tests. \u003cem\u003eJournal of Fixed Income\u003c/em\u003e 28(2):52\u0026ndash;69. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.3905/jfi.2018.1.062\u003c/span\u003e\u003cspan address=\"10.3905/jfi.2018.1.062\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eVaugirard, Victor E. 2003. Pricing Catastrophe Bonds by an Arbitrage Approach. \u003cem\u003eQuar- terly Review of Economics and Finance\u003c/em\u003e 43(1):119\u0026ndash;132. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1016/S1062-\u003c/span\u003e\u003cspan address=\"10.1016/S1062-\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eVerizon. 2025. 2025 Data Breach Investigations Report. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.verizon.com/busi- ness/resources/reports/dbir/\u003c/span\u003e\u003cspan address=\"https://www.verizon.com/busi- ness/resources/reports/dbir/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWillard, John. 2025. Cyber ILS market poised for growth, but must overcome key risk challenges: S\u0026amp;P. \u003cem\u003eArtemis.bm\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.artemis.bm/news/cyber-ils-market-poised-for- growth-but-must-overcome-key-risk-challenges-sp/\u003c/span\u003e\u003cspan address=\"https://www.artemis.bm/news/cyber-ils-market-poised-for- growth-but-must-overcome-key-risk-challenges-sp/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Accessed September 24, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWoods, Daniel, Jessica Wolff. 2023. A History of Cyber Risk Transfer. \u003cem\u003eSSRN Elec- tronic Journal\u003c/em\u003e. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.2139/ssrn.4493171\u003c/span\u003e\u003cspan address=\"10.2139/ssrn.4493171\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Cyber cat bonds, Copula-POT Model, Modern Portfolio Theory (MPT), portfolio optimization, tail risk, pricing models","lastPublishedDoi":"10.21203/rs.3.rs-7923044/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7923044/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This study evaluates and compares five pricing models for cyber catastrophe (cat) bonds—the Loss Distribution Framework, Unified Bayesian Framework, Signal-Processing Approach, Regression Approach, and Copula-POT Model—to identify the most effective method for loss prediction and portfolio optimization under Modern Portfolio Theory (MPT). Using CISSM data, the Copula-POT Model shows the highest predictive accuracy and robustness, making it the preferred framework despite a zero trigger probability due to limited extreme value data. Integrating cyber cat bonds priced by this model into an MPT-optimized portfolio improves diversification and risk-adjusted returns, outperforming traditional high-yield bonds. The study highlights challenges including data scarcity, parameter sensitivity, and model uncertainty, and proposes hybrid modeling and data enrichment as directions for future research. Overall, these findings emphasize the potential of cyber cat bonds as an innovative asset class and an effective tool for cyber risk transfer in capital markets.","manuscriptTitle":"Valuation of Cyber Catastrophe Bonds and Their Role in Portfolio Efficiency: An Analysis of Model Selection and Investment Implications","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-17 08:33:56","doi":"10.21203/rs.3.rs-7923044/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"42a33539-b87a-4de4-9303-e6be24d5f1a2","owner":[],"postedDate":"November 17th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-02-04T10:58:22+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-17 08:33:56","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7923044","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7923044","identity":"rs-7923044","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}