actuarial pricing methods

This study addresses the critical limitation of traditional catastrophe risk pricing models, which ignore the non-stationarity in disaster frequency induced by climate change, leading to biased reinsurance and catastrophe bond valuations and a systematic underestimation of capital reserves. To rectify this, the authors propose a climate-aware pricing framework that, for the first time, incorporates a temperature-driven stochastic intensity into a Cox process: the catastrophe arrival intensity depends on a temperature index modeled by an Ornstein–Uhlenbeck process with a time trend, coupled with a compound Poisson loss structure. Valuation is performed under a risk-adjusted measure via Monte Carlo simulation. Empirical results demonstrate that the model significantly increases excess-of-loss reinsurance premiums and reduces catastrophe bond prices; compared to a stationary benchmark, the 99.5% TVaR reveals that economic capital requirements are underestimated by approximately 13.7%, underscoring the essential role of integrating climate dynamics into risk management.

This paper addresses the incentive deficiency and risk imbalance arising from static reinsurance premium structures by proposing a performance-based variable premium mechanism: premiums are initially set stochastically and subsequently adjusted ex post—via bonuses or penalties—based on realized claims, thereby linking premiums to both the loss distribution and actual losses. Methodologically, we integrate risk measures, stochastic optimization, and Bowley bilateral bargaining theory to formulate an equilibrium model wherein the insurer selects an optimal reinsurance strategy subject to the reinsurer’s risk constraints, extending the Meyers–Chen actuarial pricing framework. Numerical experiments demonstrate that, compared with the conventional expected-value principle, our mechanism significantly reduces the reinsurer’s aggregate risk exposure while enhancing contract efficiency and incentive compatibility. To the best of our knowledge, this is the first systematic design embedding performance feedback directly into variable reinsurance pricing.

This study investigates strategic interactions and competitive equilibrium within a multi-layer reinsurance chain comprising m insurers and n reinsurers. A stochastic differential game framework is developed, wherein Stackelberg games model hierarchical reinsurance relationships across layers, non-zero-sum games capture inter-insurer competition, and investment in both risky and risk-free assets is incorporated. Under the mean-variance criterion, explicit equilibrium strategies are derived for the first time for both proportional and excess-of-loss reinsurance contracts, revealing how intensified market competition exerts downward pressure on safety loading rates. By solving an extended system of Hamilton–Jacobi–Bellman (HJB) equations and conducting numerical experiments, the analysis demonstrates that heightened competition among insurers significantly reduces safety loadings across all reinsurance layers.

This study investigates how insurers jointly optimize underwriting, investment, refinancing, and dividend policies to maximize shareholder value in the presence of model uncertainty and financial frictions. The authors develop the first dynamic equilibrium model of liquidity management that incorporates model uncertainty, characterizing firms’ robust decision-making under their subjectively worst-case scenarios. Key contributions include establishing the existence of equilibrium, uncovering a liquidity-driven underwriting cycle and associated risk-hedging mechanisms, and identifying a counterintuitive phenomenon—negative risk loadings—when insurance risks are positively correlated with financial markets, along with a clear explanation of its underlying mechanism.

Under heavy-tailed loss distributions—such as those arising from extreme weather events—the expected loss may be infinite, rendering conventional indemnity-based insurance inadequate for risk transfer. Method: This paper proposes a hybrid insurance mechanism integrating traditional capped indemnity insurance (front-end) with index-based parametric insurance (back-end) for excess losses. We develop a novel optimization criterion specifically tailored to Pareto-type extreme-value distributions and rigorously establish its convergence under heavy-tailed conditions. The framework combines extreme value theory (EVT) modeling, Monte Carlo simulation, and empirical calibration using U.S. tornado loss data. Contribution/Results: The calibrated hybrid contract significantly outperforms conventional capped indemnity contracts in both protection efficacy and cost efficiency. Notably, it enhances payout speed, accuracy, and robustness under extreme-loss scenarios—demonstrating superior risk-transfer performance while maintaining actuarial feasibility.

This paper addresses the robust Pareto-optimal reinsurance design problem between multiple heterogeneous risk-averse primary insurers and a single reinsurer in a monopoly market, where the joint loss distribution is unknown and only marginal distributions are available. Methodologically, it introduces a novel robust optimization framework based on the Range Value-at-Risk (RVaR) family of risk measures. It fully characterizes the worst-case dependence structure and derives the optimal indemnity function in closed form, reducing the infinite-dimensional optimization to a finite-dimensional problem with only two or three parameters. For i.i.d. risks, it constructs analytically tractable, two-tier asymptotically optimal reinsurance contracts. The theoretical analysis reveals how dependence uncertainty and insurer heterogeneity critically influence reinsurance allocation and systemic risk assessment. Numerical experiments demonstrate the method’s effectiveness and practical applicability.

This study addresses the design of reinsurance contracts in peer-to-peer (P2P) insurance, aiming to align the strategic interactions between plan managers and reinsurers to optimize risk sharing and overall welfare. By formulating a game-theoretic model that incorporates a risk-sharing pool and a reinsurer, the authors propose two contract types: a Pareto-optimal cooperative contract grounded in coalition stability, and a Stackelberg-type Bowley contract satisfying pointwise individual rationality constraints, both admitting closed-form solutions. Theoretical analysis reveals that the Bowley contract is never Pareto optimal and consistently yields lower aggregate welfare. Numerical experiments further demonstrate that introducing reinsurance within the Pareto framework significantly enhances welfare, whereas a single premium rate mechanism disproportionately disadvantages high-risk members.

This paper addresses the inadequate modeling of multiregional loss dependence structures in cross-regional catastrophe (CAT) bond pricing. We develop a unified multi-regional pricing framework that systematically characterizes independent, proportional, and general bivariate extreme-value dependence structures, and integrates the Wang transform to explicitly incorporate market risk preferences. Using historical PCS loss data, we empirically assess how alternative dependence assumptions affect CAT bond prices and derive a closed-form normal approximation solution. Results demonstrate that the choice of dependence structure significantly impacts pricing outcomes, while the normal approximation maintains high accuracy under real-world data. This study provides a theoretically consistent, computationally efficient, and operationally practical valuation tool for the design and risk management of multi-regional CAT bonds.

In high-risk environments, indemnity-based insurance often fails due to budget constraints, fixed administrative costs, and heterogeneous premium loadings. Method: Within a mean–variance framework, we introduce joint budget and risk constraints and systematically compare the welfare implications of indemnity-based versus parametric insurance. Contribution/Results: We prove theoretically that when indemnity insurance is infeasible—due to tight budgets or high premium loadings—parametric insurance strictly increases the expected utility of risk-averse agents. Crucially, this welfare advantage is bounded: it holds only when indemnity mechanisms break down. This is the first formal refutation—in a normative model incorporating multiple real-world frictions—of the classical claim that indemnity insurance is universally optimal. Our results provide a rigorous theoretical foundation for promoting parametric insurance in climate-vulnerable regions and other high-risk settings.

This study investigates the endogenous determination of reinsurance pricing in a competitive insurance market, characterizing the strategic interactions between a reinsurer and a heterogeneous population of insurers, along with the feedback effects arising from their common risk exposure. Within a Stackelberg game framework, the reinsurer acts as the leader by setting a uniform premium rate and investment strategy, while insurers endogenously choose their risk retention levels based on individual performance, relative performance concerns, and shared noise. The work innovatively uncovers a threshold structure and spillover mechanism in risk retention driven by relative performance motives, establishes for the first time the convergence of finite-player equilibria under non-unique mean-field equilibria, and introduces an efficient threshold continuation algorithm. Numerical experiments demonstrate that relative performance concerns significantly amplify systemic spillovers and can induce multiple Stackelberg equilibria, while also revealing a three-stage behavioral pattern—from full reinsurance to full retention—as premiums vary.