Search Results (65)

This paper investigates a robust optimal reinsurance and investment problem for an insurance company operating in a Markov-modulated financial market. The insurer’s surplus process is modeled by a diffusion process with jumps, which is correlated with financial risky assets through a common shock structure. The economic regime switches according to a continuous-time Markov chain. To address model uncertainty concerning both diffusion and jump components, we formulate the problem within a robust optimal control framework. By applying the Girsanov theorem for semimartingales, we derive the dynamics of the wealth process under an equivalent martingale measure. We then establish the associated Hamilton–Jacobi–Bellman (HJB) equation, which constitutes a coupled system of nonlinear second-order integro-differential equations. An explicit form of the relative entropy penalty function is provided to quantify the cost of deviating from the reference model. The theoretical results furnish a foundation for numerical solutions using actor–critic reinforcement learning algorithms. Full article

This paper investigates the role of proportional reinsurance as a practical and flexible tool for managing demographic risk in life insurance, with a focus on its impact on both the Solvency Capital Requirement (SCR) and expected profitability. While much of the existing literature focuses on mortality modeling or longevity-linked reinsurance instruments, this paper proposes a novel framework for analyzing traditional proportional reinsurance structures within the Solvency II market-consistent valuation environment. The framework integrates proportional reinsurance into the valuation of liabilities and the calculation of Solvency Capital Requirement, beginning with an outline of cash flow structures and their valuation under Solvency II principles. A key contribution is the introduction and decomposition of the net of reinsurance Claims Development Result (CDR), which allows us to assess the dual impact of reinsurance on risk mitigation and profit transfer. Through numerical analysis, we show how proportional reinsurance can effectively reduce capital requirements while quantifying the trade-off in expected profit transferred to the reinsurance company, with insights into how different reinsurance treaties affect capital efficiency and profitability. Full article

This article provides an overview of the current state-of-the-art in cyber risk and cyber risk management, focusing on the mathematical models that have been created to help with risk quantification and insurance pricing. We discuss the main ways that cyber risk is measured, starting with vulnerability functions that show how systems react to threats and going all the way up to more complex stochastic and dynamic models that show how cyber attacks change over time. Next, we examine cyber insurance, including the structure and main features of the cyber insurance market, as well as the growing role of cyber reinsurance in strategies for transferring risk. Finally, we review the mathematical models that have been proposed in the literature for setting the prices of cyber insurance premiums and structuring reinsurance contracts, analysing their advantages, limitations, and potential applications for more effective risk management. The aim of this article is to provide researchers and professionals with a clear picture of the main quantitative tools available and to point out areas that need further research by summarising these contributions. Full article

The recent losses and damages due to climate change have destabilized the insurance industry. As global warming is one of the most critical aspects of climate change, it is essential to investigate to what extent greenhouse gas emissions affect the financial stability of insurers. Insurers typically do not emit substantial greenhouse gases directly, while their underwriting and investment activities play a substantial role in enabling companies that do. This article uses panel data regressions to analyze companies in all insurance segments and in all geographic regions of the world from 2004 to 2023. The main finding is that insurers that increase their greenhouse gas emissions become financially unstable. This result is consistent in all three scopes (scope 1, scope 2, and scope 3) of emissions. Furthermore, the findings reveal that this impact is related to reserves and reinsurance. Specifically, reserves increase with greenhouse gas emissions, while premiums ceded to reinsurers decline. Thus, high-emissions insurers retain a significant share of carbon risk and eventually become financially weak. The results encourage several policy recommendations, highlighting the need for instruments that improve the assessment and disclosure of insurers’ carbon footprints. This is crucial to achieving environmental targets and improving the stability of both the insurance market and the economic system. Full article

Insurance companies need to calculate solvency capital requirements in order to ensure that they can meet their future obligations to policyholders and beneficiaries. The solvency capital requirement is a risk management tool essential for addressing extreme catastrophic events that result in a high number of possibly interdependent claims. This paper studies the problem of aggregating the risks coming from several insurance business lines and analyses the effect of reinsurance on the level of risk. Our starting point is to use a hierarchical risk aggregation method which was initially based on two-dimensional elliptical copulas. We then propose the use of copulas from the Archimedean family and a mixture of different copulas. Our results show that a mixture of copulas can provide a better fit to the data than an individual copula and consequently avoid over- or underestimation of the capital requirement of an insurance company. We also investigate the significance of reinsurance in reducing the insurance company’s business risk and its effect on diversification. The results show that reinsurance does not always reduce the level of risk, but can also reduce the effect of diversification for insurance companies with multiple business lines. Full article

In our study, we investigate reinsurance issues and optimal investment related to derivatives trading for a mean-variance insurer, employing game theory. Our primary objective is to identify strategies that are time-consistent. In particular, the insurer has the flexibility to purchase insurance in proportion to its needs, explore new business, and engage in capital market investments. This is under the assumption that insurance companies’surplus capital adheres to the classical Cramér-Lundberg model. The capital market is made up of risk-free bonds, equities, and derivatives, with pricing dependent on the underlying stock’s basic price and volatility. To obtain the most profitable expressions and functions for the associated investment strategies and time guarantees, we solve a system of expanded Hamilton–Jacobi–Bellman equations. In addition, we delve into scenarios involving optimal investment and reinsurance issues with no derivatives trading. In the end, we present a few numerical instances to display our findings, demonstrating that the efficient frontier in the case of derivative trading surpasses that in scenarios where derivative trading is absent. Full article

Climate change has brought several natural disasters to South Africa in the form of floods, heat waves, and droughts. Neighbouring countries are also experiencing tropical cyclones, almost on a yearly basis. The insurance sector is faced with an increased level of climate change risk with individuals, corporates, and even the government approaching it for financial cover. However, with an increased level of competition in the insurance sector, (re)insurers must engage in massive product research and development. Therefore, this paper looks at the possibility of the insurance industry developing new products in the form of catastrophe resilience bonds (CAT R Bonds). A qualitative approach is used following content analysis of (re)insurers’ product development policies, marketing documents, company reports, and risk management reports as well as the Conference of Parties 27 and 28 resolution papers. The findings reveal that (re)insurers’ underwriting capacity, reinsurance protection, and innovative and creative product development increase because of CAT R Bonds. CAT R Bonds enhance the interaction between the capital market and money market, thereby giving speculative investors another investment option. Increased investment into new product development such as CAT R Bonds must continue in South Africa in pursuit of climate change resilience goals. Full article

Estimating the expected capital and its variability is a crucial objective for a non-life insurance company, which enables the firm to develop effective management strategies. Many studies have been devoted to this topic, with simulative approaches being especially employed for solving the complexity of the interacting risks, not manageable through closed-form solutions. In this paper, we present a realistic framework based on Solvency II for the definition of next-year capital of a non-life insurer, including reinsurance treaties and counterparty default risk, in a multi-line of business setting. We determine the mean and variance of the stochastic capital considering both quota share and excess-of-loss reinsurance. We show how these closed-form results enable the analysis of many different real-world strategies, granting the insurer the possibility of choosing the optimal policy without the computational resources and time constraints required by simulative approaches. Full article

This study explores the insurance pricing domain in the motor insurance industry, focusing on the creation of “technical models” which are essentially obtained after combining the frequency model (the expected number of claims per unit of exposure) and the severity model (the expected amount per claim). Technical models are designed to predict the loss costs (the product of frequency and severity, i.e., the expected claim amount per unit of exposure) and this is a main factor that is taken into account for pricing insurance policies. Other factors for pricing include the company expenses, investments, reinsurance, underwriting, and other regulatory restrictions. Different machine learning methodologies, including the Generalised Linear Model (GLM), Gradient Boosting Machine (GBM), Artificial Neural Networks (ANN), and a unique hybrid model that combines GLM and ANN, were explored for creating the technical models. This study was conducted on the French Motor Third Party Liability datasets, “freMTPL2freq” and “freMTPL2sev” included in the R package CASdatasets. After building the aforementioned models, they were evaluated and it was observed that the hybrid model which combines GLM and ANN outperformed all other models. ANN also demonstrated better predictions closely aligning with the performance of the hybrid model. The better performance of neural network models points to the need for actuarial science and the insurance industry to look beyond traditional modelling methodologies like GLM. Full article

In the field of finance and insurance, addressing the optimal investment and reinsurance issue is a focal point for researchers. This paper contemplates the optimal strategy for insurance companies within a model where wealth dynamics adhere to a jump–diffusion process. The fractional structure of the diffusion term is extremely interpretative. This model encompasses elements of risky assets, risk-free assets, and proportional reinsurance. Based on this model and grounded in the principles of stochastic control, the corresponding HJB equation is derived and solved. Consequently, explicit expressions for the optimal investment and reinsurance ratios are obtained, and the solution’s verification theorem is proven. Finally, through a numerical analysis with varying parameters, results consistent with real-world scenarios are achieved. Full article

In this study, we design an algorithm to work on gate-based quantum computers. Based on the algorithm, we construct a quantum circuit that represents the surplus process of a cedant under a reinsurance agreement. This circuit takes into account a variety of factors: initial reserve, insurance premium, reinsurance premium, and specific amounts related to claims, retention, and deductibles for two different non-proportional reinsurance contracts. Additionally, we demonstrate how to perturb the actuarial stochastic process using Hadamard gates to account for unpredictable damage. We conclude by presenting graphs and numerical results to validate our capital modelling approach. Full article

This paper studies insurance companies’ optimal reinsurance–investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer’s surplus process and the variance premium principle to calculate premiums. In this paper, we assume that insurance companies can invest in risk-free assets, risky assets, and zero-coupon bonds, where the Cox–Ingersoll–Ross model describes the dynamic change in stochastic interest rates and the Heston model describes the price process of risky assets. The analytic solution of the optimal reinsurance–investment strategy is deduced by employing related methods from the stochastic optimal control theory, the stochastic analysis theory, and the dynamic programming principle. Finally, the influence of model parameters on the optimal reinsurance–investment strategy is illustrated using numerical examples. Full article

In univariate data, there exist standard procedures for identifying dominating features that produce the largest number of observations. However, in the multivariate setting, the situation is quite different. This paper aims to provide tools and methods for detecting dominating directional components in multivariate data. We study general heavy-tailed multivariate random vectors in dimension d ≥ 2 and present procedures that can be used to explain why the data are heavy-tailed. This is achieved by identifying the set of the riskiest directional components. The results are of particular interest in insurance when setting reinsurance policies, and in finance when hedging a portfolio of multiple assets. Full article

Optimal reinsurance problems under the risk measures, such as Value-at-Risk ($\mathrm{VaR}$) and Tail-Value-at-Risk ($\mathrm{TVaR}$), have been studied in recent literature. However, losses based on $\mathrm{VaR}$ may be underestimated and $\mathrm{TVaR}$ allows us to account better for catastrophic losses. In this paper, we propose a new family of flexible risk measures denoted by $\mathrm{LVaR}$, which is a weighted combination of $\mathrm{VaR}$ and $\mathrm{TVaR}$. Based on the new risk measures, we deal with the optimal reinsurance problem by minimizing the $\mathrm{LVaR}$ of the total risks of an insurer when two types of constraints for reinsurer’s risk exposure are considered. The results indicate that the two-layer reinsurance is always an optimal reinsurance policy with both types of constraints. Also, we find that the optimal reinsurance policy depends on the confidence level, the weight coefficient, the safety loading, the tolerance level, as well as the relations between them. Finally, we illustrate the results by numerical examples and compare them with the results in Lu et al. Full article

In this paper, we solve an optimal reinsurance problem in the mathematical finance area. We assume that the surplus process of the insurance company follows a controlled diffusion process and the constant interest rate is involved in the financial model. During the whole optimization period, the company has a choice to buy reinsurance contract and decide the reinsurance retention level. Meanwhile, the bankruptcy at the terminal time is not allowed. The aim of the optimization problem is to minimize the distance between the terminal wealth and a given goal by controlling the reinsurance proportion. Using the stochastic control theory, we derive the Hamilton-Jacobi-Bellman equation for the optimization problem. Via adopting the technique of changing variable as well as the dual transformation, an explicit solution of the value function and the optimal policy are shown. Finally, several numerical examples are shown, from which we find several main factors that affect the optimal reinsurance policy. Full article