This paper examines an insurer’s optimal asset allocation and reinsurance policies. The financial market framework includes one risk-free and one risky asset. The insurer has two business lines, where the ordinary claim process is modeled by a compound Poisson process and catastrophic claims follow a compound dynamic contagion process. The dynamic contagion process, which is a generalization of the externally exciting Cox process with shot-noise intensity and the self-exciting Hawkes process, is enhanced by accommodating the dependency structure between the magnitude of contribution to intensity after initial events for catastrophic insurance products and its claim/loss size. We also consider the dependency structure between the positive effect on the intensity and the negative crashes on the risky financial asset when initial events occur. Our objective is to maximize the insurer’s expected utility of terminal surplus. We construct the extended Hamilton–Jacobi–Bellman (HJB) equation using dynamic programming principles to derive an explicit optimal reinsurance policy for ordinary claims. We further develop an iterative scheme for solving the value function and the optimal asset allocation policy and the reinsurance policy for catastrophic claims numerically, providing a rigorous convergence proof. Finally, we present numerical examples to demonstrate the impact of key parameters.

A company with n geographically widely dispersed sites seeks an insurance policy that pays off if m out of the n sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study compares three strategies for an insurance company wishing to offer such an m-out-of-n policy, assuming the existence of markets for insurance on the individual sites with coverage periods of various lengths of a year or less. Strategy A is static: at the beginning of the year it buys a reinsurance policy on each individual site covering the entire year and makes no later adjustments. By contrast, Strategies S and C are dynamic and adaptive, exploiting the availability of individual-site policies for shorter periods than a year to make changes in the coverage on individual sites as quakes occur during the year. Strategy S uses the payoff from reinsurance when a quake occurs at a particular site to increase coverage for the remainder of the year on the sites that have not yet had quakes. Strategy C buys individual-site policies covering successive time periods of fixed length, observing the system at the beginning of each period and using cash on hand plus cash obtained from a reinsurance payoff (if any) during the previous period to decide how much cash to retain and how much reinsurance to purchase for the current period. The study relies on expected utility to determine indifference premiums and compare the premiums and loss probabilities for the three strategies.

The rise of large language models (LLMs) has marked a substantial leap toward artificial general intelligence. However, the utilization of LLMs in (re)insurance sector remains a challenging problem because of the gap between general capabilities and domain-specific requirements. Two prevalent methods for domain specialization of LLMs involve prompt engineering and fine-tuning. In this study, we aim to evaluate the efficacy of LLMs, enhanced with prompt engineering and fine-tuning techniques, on quantitative reasoning tasks within the (re)insurance domain. It is found that (1) compared to prompt engineering, fine-tuning with task-specific calculation dataset provides a remarkable leap in performance, even exceeding the performance of larger pre-trained LLMs; (2) when acquired task-specific calculation data are limited, supplementing LLMs with domain-specific knowledge dataset is an effective alternative; and (3) enhanced reasoning capabilities should be the primary focus for LLMs when tackling quantitative tasks, surpassing mere computational skills. Moreover, the fine-tuned models demonstrate a consistent aptitude for common-sense reasoning and factual knowledge, as evidenced by their performance on public benchmarks. Overall, this study demonstrates the potential of LLMs to be utilized as powerful tools to serve as AI assistants and solve quantitative reasoning tasks in (re)insurance sector.

We develop explicit bounds for the tail of the distribution of the all-time supremum of a random walk with negative drift, where the increments have a truncated heavy-tailed distribution. As an application, we consider a ruin problem in the presence of reinsurance.

The systemic nature of climate risk is well established, but the extent may be more severe than previously understood, particularly with regard to cyber risk and economic security. Cyber security relies on the availability of insurance capital to mitigate economic security sector risks and support the reversibility of attacks. However, the cyber insurance industry is still in its infancy. Pressure on insurance capital from increasing natural disaster activity could consume the resources necessary for economic security in the cyber domain in the near term and create long-term conditions that increase the scarcity of capital to support cyber security risks. This article makes an original contribution by exploring the under-researched connection between the nexus of cyber and economic security and the climate change threat. Although the immediate pressure on economic resources for cyber security is limited, recent natural disaster activity has clearly shown that access to capital for cyber risks could come under significant pressure in the future.

We consider a robust optimal investment–reinsurance problem to minimize the goal-reaching probability that the value of the wealth process reaches a low barrier before a high goal for an ambiguity-averse insurer. The insurer invests its surplus in a constrained financial market, where the proportion of borrowed amount to the current wealth level is no more than a given constant, and short-selling is prohibited. We assume that the insurer purchases per-claim reinsurance to transfer its risk exposure to a reinsurer whose premium is computed according to the mean–variance premium principle. Using the stochastic control approach based on the Hamilton–Jacobi–Bellman equation, we derive robust optimal investment–reinsurance strategies and the corresponding value functions. We conclude that the behavior of borrowing typically occurs with a lower wealth level. Finally, numerical examples are given to illustrate our results.

Insurers draw on sophisticated models for the probability distributions over losses associated with catastrophic events that are required to price insurance policies. But prevailing pricing methods don’t factor in the ambiguity around model-based projections that derive from the relative paucity of data about extreme events. I argue however that most current theories of decision making under ambiguity only partially support a solution to the challenge that insurance decision makers face and propose an alternative approach that allows for decision making that is responsive to both the evidential situation of the insurance decision maker and their attitude to ambiguity.

We study the optimal investment-reinsurance problem in the context of equity-linked insurance products. Such products often have a capital guarantee, which can motivate insurers to purchase reinsurance. Since a reinsurance contract implies an interaction between the insurer and the reinsurer, we model the optimization problem as a Stackelberg game. The reinsurer is the leader in the game and maximizes its expected utility by selecting its optimal investment strategy and a safety loading in the reinsurance contract it offers to the insurer. The reinsurer can assess how the insurer will rationally react on each action of the reinsurer. The insurance company is the follower and maximizes its expected utility by choosing its investment strategy and the amount of reinsurance the company purchases at the price offered by the reinsurer. In this game, we derive the Stackelberg equilibrium for general utility functions. For power utility functions, we calculate the equilibrium explicitly and find that the reinsurer selects the largest reinsurance premium such that the insurer may still buy the maximal amount of reinsurance. Since in the equilibrium the insurer is indifferent in the amount of reinsurance, in practice, the reinsurer should consider charging a smaller reinsurance premium than the equilibrium one. Therefore, we propose several criteria for choosing such a discount rate and investigate its wealth-equivalent impact on the expected utility of each party.

Due to the presence of reporting and settlement delay, claim data sets collected by non-life insurance companies are typically incomplete, facing right censored claim count and claim severity observations. Current practice in non-life insurance pricing tackles these right censored data via a two-step procedure. First, best estimates are computed for the number of claims that occurred in past exposure periods and the ultimate claim severities, using the incomplete, historical claim data. Second, pricing actuaries build predictive models to estimate technical, pure premiums for new contracts by treating these best estimates as actual observed outcomes, hereby neglecting their inherent uncertainty. We propose an alternative approach that brings valuable insights for both non-life pricing and reserving. As such, we effectively bridge these two key actuarial tasks that have traditionally been discussed in silos. Hereto, we develop a granular occurrence and development model for non-life claims that tackles reserving and at the same time resolves the inconsistency in traditional pricing techniques between actual observations and imputed best estimates. We illustrate our proposed model on an insurance as well as a reinsurance portfolio. The advantages of our proposed strategy are most compelling in the reinsurance illustration where large uncertainties in the best estimates originate from long reporting and settlement delays, low claim frequencies and heavy (even extreme) claim sizes.

This paper studies a Pareto-optimal reinsurance problem when the contract is subject to default of the reinsurer. We assume that the reinsurer can invest a share of its wealth in a risky asset and default occurs when the reinsurer's end-of-period wealth is insufficient to cover the indemnity. We show that without the solvency regulation, the optimal indemnity function is of excess-of-loss form, regardless of the investment decision. Under the solvency regulation constraint, by assuming the investment decision remains unchanged, the optimal indemnity function is characterized element-wisely. Partial results are derived when both the indemnity function and investment decision are impacted by the solvency regulation. Numerical examples are provided to illustrate the implications of our results and the sensitivity of solution to the model parameters.

This article examines the impact of the largest claims reinsurance treaties on loss reserve of the ceding company. The largest claims reinsurance, known as LCR, and ECOMOR reinsurance treaties are considered to be the two most appropriate reinsurance treaties for large or catastrophe claims. Then, it studies the impact of such treaties on loss reserves. Through a simulation study, it shown that, under a more general situation, the LCR treaty can be a more efficient (in some sense, see below) treaty than the ECOMOR treaty for the ceding company.

This paper studies the open-loop equilibrium strategies for a class of non-zero-sum reinsurance–investment stochastic differential games between two insurers with a state-dependent mean expectation in the incomplete market. Both insurers are able to purchase proportional reinsurance contracts and invest their wealth in a risk-free asset and a risky asset whose price is modeled by a general stochastic volatility model. The surplus processes of two insurers are driven by two standard Brownian motions. The objective for each insurer is to find the equilibrium investment and reinsurance strategies to balance the expected return and variance of relative terminal wealth. Incorporating the forward backward stochastic differential equations (FBSDEs), we derive the sufficient conditions and obtain the general solutions of equilibrium controls for two insurers. Furthermore, we apply our theoretical results to two special stochastic volatility models (Hull–White model and Heston model). Numerical examples are also provided to illustrate our results.

A company with $n$ geographically widely dispersed sites seeks insurance that pays off if $m$ out of the $n$ sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study describes an adaptive dynamic strategy that enables an insurance company to offer the policy with smaller loss probability than more conventional static policies induce, but at a comparable or lower premium. The strategy accomplishes this by periodically purchasing reinsurance on individual sites. Exploiting rarity, the policy induces zero loss with probability one if no more than one quake occurs during any review interval. The policy also may induce a profit if $m$ or more quakes occur in an interval if no quakes have occurred in previous intervals. The study also examines the benefit of more than one reinsurance policy per active site. The study relies on expected utility to determine indifference premiums and derives an upper bound on loss probability independent of premium.

Crop revenue insurance is unique, because it involves a guarantee subsuming yield risk and highly systematic price risk. This study examines whether crop insurers could use options instead of, or in addition to, assigning policies to the Commercial Funds of the USDA Federal Crop Insurance Corporation (FCIC) as per the Standard Reinsurance Agreement (SRA) to hedge the price risk of revenue insurance policies. The behavioral model examines the optimal hedge ratio for a crop insurer with a book of business consisting of corn Revenue Protection (RP) policies. Results show that a mix of put and call options can hedge the price risk of the RP policies. The higher optimal hedge ratios of call options as compared to put options imply that the risk of increased liability due to upside price risk can be hedged using options better than downside price risk. This study also analyzed the combination of options with the SRA at 35, 50, and 75% retention levels. The zero optimal hedge ratios at each retention level and the negative correlation between RP indemnities and the option returns when the crop insurer mixed options and SRA suggest that the purchasing of options provides no additional risk protection to crop insurers beyond what is provided by the SRA despite retention limits.

We establish a “top-down” approximation scheme to approximate loss distributions of reinsurance products and Insurance-Linked Securities based on three input parameters, namely the Attachment Probability, Expected Loss and Exhaustion Probability. Our method is rigorously derived by utilizing a classical result from Extreme-Value Theory, the Pickands–Balkema–de Haan theorem. The robustness of the scheme is demonstrated by proving sharp error-bounds for the approximated curves with respect to the supremum and L2 norms. The practical implications of our findings are examined by applying it to Industry Loss Warranties: the method performs very accurately for each transaction. Our approach can be used in a variety of applications such as vendor model blending, portfolio optimization and premium calculation.