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A Hybrid Framework for Reinsurance Optimization: Integrating Generative Models and Reinforcement Learning
Keywords: Reinsurance Optimization, Generative Models, Reinforcement Learning, Variational Autoencoders (VAEs), Proximal Policy Optimization (PPO), \textcolor{black}{Capital Management}, Catastrophe Risk, \textcolor{black}{Dynamic Treaty Design}, \textcolor{black}{Hybrid AI Framework}
The insurance and reinsurance industries play a pivotal role in managing financial risks and ensuring economic stability. Reinsurance, which involves the transfer of risk from insurers to reinsurers, is a cornerstone of risk management strategies aimed at maintaining solvency and optimizing financial performance. However, designing effective reinsurance strategies remains a highly complex challenge due to the stochastic nature of claims, multi-dimensional constraints, and the dynamic interplay between risk retention, profitability, and regulatory compliance Asmussen_and_Albrecher2010, Schmidli2008.
\textcolor{black}{A central difficulty lies in managing tail risk: catastrophic but rare claims can dominate portfolio outcomes, and classical actuarial models often underestimate such events. This motivates the need for adaptive, data-driven frameworks that explicitly account for tail behavior and align with evolving solvency regulations.}\textcolor{black}{This emphasis on catastrophe risk and model-based stress testing aligns with recent RMIR perspectives on data science for catastrophe modeling and disaster risk reduction Steptoe2022_RMIR,Surminski2018_RMIR.}
Traditional approaches to reinsurance optimization, such as the classical Cram\'{e}r---Lundberg model, have provided foundational insights into surplus dynamics and ruin probabilities. These models, while mathematically rigorous, rely on static assumptions about premium rates and claim distributions, limiting their applicability to modern reinsurance practices. Extensions to these models, including proportional and layered reinsurance structures, address some of these limitations but often remain computationally intensive and insufficiently adaptable to high-dimensional, real-world scenarios Gerber1970, Albrecher_et_al2017. \textcolor{black}{Recent surveys highlight that static optimization struggles in the presence of heavy-tailed claims and under regulatory frameworks such as Solvency II and IFRS 17 Mikosch2009, Embrechts_et_al2014.}
\textcolor{black}{Building on this, a growing literature explores optimization of reinsurance treaties in dynamic and stochastic settings Schmidli2008, Albrecher_et_al2017. Parallel advances in machine learning open new opportunities: generative models such as Variational Autoencoders (VAEs) capture complex data distributions and generate synthetic samples, helping address data scarcity and the underrepresentation of catastrophic claims Kingma_and_Welling2014, Goodfellow_et_al2014, Wuthrich2020. Reinforcement learning (RL), particularly Proximal Policy Optimization (PPO), has demonstrated strong performance in sequential decision-making under uncertainty Schulman_et_al2017, Kolm_et_al2020, with recent applications highlighting its promise in insurance risk management Buehler2019, Buehler2022.} \textcolor{black}{Related RMIR contributions underscore the role of advanced analytics and organizational learning for risk management, reinforcing the relevance of our hybrid approach TextMining2024_RMIR,Resilience2023_RMIR.}
\textcolor{black}{Despite these advances, little work has combined generative modeling of claim processes with dynamic optimization of reinsurance strategies. Our contribution is to close this gap by proposing a hybrid framework that explicitly targets tail-risk robustness in reinsurance optimization.}
This paper introduces a novel hybrid framework that integrates generative AI with reinforcement learning to optimize reinsurance strategies dynamically and adaptively. By leveraging VAEs to model claim distributions and generate synthetic scenarios, the framework overcomes challenges associated with data scarcity and variability. The PPO algorithm, on the other hand, dynamically adjusts reinsurance parameters---such as retention rates and layer boundaries---based on evolving claim distributions, market conditions, and regulatory constraints. This synergy enables the framework to evaluate and optimize complex reinsurance strategies in real time, addressing high-dimensional uncertainties and ensuring financial stability Albrecher_et_al2017, Cheng_et_al2020.
\textcolor{black}{The framework is empirically evaluated on three representative claim distributions---lognormal, Pareto, and a lognormal---Pareto mixture---selected for their relevance to insurance modeling and their contrasting tail behaviors Embrechts_et_al2014. Results demonstrate improved robustness in the upper tail and closer alignment with solvency requirements compared to classical optimization approaches.}
Key contributions of this work include:
{\textcolor{black}{The remainder of this paper is organized as follows. Section (ref) presents the mathematical foundations of the surplus process, reinsurance structures, and optimization objectives. Section (ref) introduces the proposed hybrid framework, describing the integration of generative modeling with reinforcement learning. Section (ref) details the experimental design, results, and benchmarking against established baselines. Section (ref) examines practical implications and limitations, while Section (ref) summarizes key findings and outlines directions for future research. By uniting actuarial rigor with modern AI techniques, this study advances a new paradigm for reinsurance optimization---enhancing financial resilience, strengthening decision-making under uncertainty, and laying the groundwork for next-generation risk management strategies.}}
\textcolor{black}{This section presents the mathematical foundations of our framework, which is designed to capture the operations of an insurer over a finite planning horizon $T$}. \textcolor{black}{The framework combines a discrete-time formulation Asmussen_and_Albrecher2010, a generalized surplus process Schmidli2008, and flexible reinsurance mechanisms Gerber1970, Albrecher_et_al2017 to address the dual challenges of financial stability and risk management under uncertainty.} \textcolor{black}{By explicitly structuring the surplus process to accommodate both proportional and layered reinsurance treaties, as well as dynamic treaty adjustments, the model provides a tractable yet versatile basis for optimization Mikosch2009, Embrechts_et_al2014.}
\textcolor{black}{This mathematical foundation supports the integration of learning-based approaches in later sections. In particular, reinforcement learning agents optimize reinsurance decisions by adapting retention rates and layer structures to evolving claims and market conditions, extending classical actuarial models toward adaptive, data-driven frameworks Buehler2019, Wuthrich2020.}
The planning horizon $T$ is partitioned into $n$ discrete intervals, denoted $t_1, \ldots, t_n$, where $t_1 = 0$ and $t_n = T$. \textcolor{black}{Each interval serves as a decision epoch at which the insurer updates its risk portfolio, collects premiums, and settles claims. This setup mirrors industry practice, where financial positions are reviewed and adjusted at regular reporting periods, such as quarterly or annual solvency assessments Asmussen_and_Albrecher2010, Kaas2008.}
\textcolor{black}{Formulating the model in discrete time enables fine-grained analysis of both risk exposure and financial stability. In particular, it facilitates the incorporation of stochastic variability in claims and premiums, while preserving tractability for optimization Embrechts_et_al2014. Unlike continuous-time surplus models, which are analytically elegant but often less practical, the discrete-time approach aligns naturally with regulatory reporting cycles (e.g., Solvency II, NAIC) and supports implementation in computational frameworks for dynamic decision-making Mikosch2009, Wuthrich2020.}
Such granularity is essential for evaluating the dynamic interplay between claims, premium flows, and reinsurance decisions over the horizon $T$. \textcolor{black}{Moreover, by embedding the decision epochs in a stochastic control setting, the framework accommodates adaptive strategies: retention levels, layer structures, and capital allocations can be adjusted at each $t_i$ in response to realized experience. This flexibility is critical for designing robust policies that mitigate solvency risk while maintaining profitability under uncertainty Schmidli2008, Albrecher_et_al2017.}
The insurer’s financial surplus, defined as the difference between assets and liabilities, evolves as premiums are collected and claims are paid. \textcolor{black}{We adopt an enhanced Cram\'er---Lundberg framework in discrete time, which provides a tractable yet flexible foundation for capturing surplus dynamics while remaining consistent with actuarial practice. This discrete-time adaptation is particularly well-suited for incorporating decision epochs and reinsurance adjustments at regular intervals, in line with reporting and regulatory cycles Kaas2008, Asmussen_and_Albrecher2010.}
Let $N_i$ denote the number of claims in interval $[t_{i-1}, t_i)$, modeled as a Poisson random variable with intensity $\lambda \Delta t_i$, where $\Delta t_i = t_i - t_{i-1}$. Each claim amount $X_{ij}$ is assumed to be independent and identically distributed (i.i.d.). The surplus recursion is then given by:
where:
\textcolor{black}{For clarity, the claims within period $i$ may also be represented in vector form as $\vec{X}_i = (X_{i1}, \ldots, X_{iN_i})$, denoting the collection of realized claim severities. This compact representation emphasizes that the cumulative loss term $\sum_{j=1}^{N_i} X_{ij}$ depends jointly on the random frequency $N_i$ and the distribution of severities in $\vec{X}_i$ Gerber1970, Kaas2008.}
\textcolor{black}{The recursive surplus process in Eq. (ref) directly links financial health to stochastic claim arrivals and premium inflows, providing a dynamic and probabilistic perspective on solvency. Its tractability allows for rigorous study of ruin probabilities, capital adequacy, and the evaluation of alternative reinsurance strategies. In particular, this formulation serves as the baseline upon which proportional and layered reinsurance mechanisms (Section (ref)) are introduced and optimized.}
Reinsurance is a fundamental risk-transfer tool that enables insurers to share liabilities with reinsurers and stabilize surplus trajectories. \textcolor{black}{In our framework, we incorporate proportional, layered, and dynamically adjustable reinsurance structures, ensuring that both traditional static contracts and adaptive, market-responsive strategies are represented. This taxonomy reflects both classical actuarial theory, where proportional and excess-of-loss treaties form the two canonical families Kaas2008,Schmidli2008, and modern practice, where hybrid and adaptive contracts are increasingly common in response to capital market conditions Albrecher_et_al2017,Wuthrich2020.}
\textcolor{black}{We introduce a retention parameter $\alpha \in [0,1]$ to capture proportional reinsurance. The premium rate $c$ remains defined as in ((ref)), while the insurer retains only a fraction $\alpha$ of each claim. The resulting surplus recursion is:}
\textcolor{black}{with $(1-\alpha)$ of each claim transferred to the reinsurer. This specification preserves consistency in premium definition while making the insurer’s retained liability explicit Schmidli2008. In practice, the choice of $\alpha$ is driven by solvency considerations, volatility targets, and market reinsurance pricing. A higher $\alpha$ increases retained earnings in benign years but leaves the insurer more vulnerable to capital depletion under stress scenarios Daykin1994,Embrechts_et_al2014.}
Layered treaties partition claims into coverage bands with distinct retention rates. For a claim $X_{ij}$, the retained loss is:
where:
\textcolor{black}{This formulation enables insurers to allocate risk exposure strategically across severity levels, balancing affordability with protection against extreme events. For instance, ordinary attritional claims may be retained entirely, mid-sized claims partially ceded, and catastrophic losses passed upwards to reinsurers. Such structures are particularly relevant in natural catastrophe and long-tail liability lines, where tail protection stabilizes solvency capital requirements Mikosch2009,Embrechts_et_al2014.}
\textcolor{black}{In practice, treaties are rarely static. Retentions and layer boundaries evolve in response to capital positions, reinsurance pricing, regulatory constraints, and updated risk assessments. To capture this adaptive behavior, we model treaty parameters as time-varying decision variables ${\alpha}(t_i), {a}(t_i), {b}(t_i)$.}
For the $k$-th layer at decision time $t_i$:
where baseline values $({\alpha}^{\text{base}}, {a}^{\text{base}}, {b}^{\text{base}})$ are modified by adjustments $({\delta}(t_i), \Delta {a}(t_i), \Delta {b}(t_i))$. \textcolor{black}{These adjustments are independent control actions determined at each decision epoch, not sequential increments, thereby allowing flexibility in responding to changing market or regulatory conditions.}
\textcolor{black}{From a practical perspective, $\delta_k(t_i)$ represents the degree of additional retention an insurer is prepared to assume in layer $k$. It is typically computed by capital models or solvency tests (e.g., 99.5% VaR under Solvency II or TailVaR under NAIC RBC), ensuring that equity-at-risk remains within tolerable levels Sandstrom2010. The adjustments $\Delta a_k(t_i)$ and $\Delta b_k(t_i)$ correspond to shifts in attachment and detachment points, respectively. These are often derived from stress-test exercises (e.g., simulated catastrophe years) or market benchmarks: increasing $\Delta a_k$ raises the deductible and reduces ceded premium, while increasing $\Delta b_k$ extends coverage upward, often motivated by affordability of retrocession markets Cummins2008.}
\textcolor{black}{In many insurers’ workflows, such adjustments are proposed by risk managers, validated against internal capital adequacy frameworks, and then negotiated with reinsurers during renewal. They are bounded by simple governance constraints: retentions must remain between 0 and 1, layers must not overlap ($a_{k+1}\geq b_k$), and premium budgets must be respected. These operational safeguards ensure that even dynamically adjusted treaties remain consistent with solvency regulation (e.g., Solvency II, IFRS 17) and industry best practice Wuthrich2020,Buehler2019.}
\textcolor{black}{By integrating these adaptive controls, the framework captures how real-world reinsurance evolves under uncertainty. While optimization tools such as reinforcement learning Sutton_and_Barto2018,schulman2017proximal may be used to automate the selection of adjustments, the levers themselves are firmly actuarial in nature: $\delta_k$ mirrors choices about risk appetite, $\Delta a_k$ reflects deductible calibration, and $\Delta b_k$ represents capital allocation to tail protection. This dual perspective---classical treaty structures with adaptive parameter shifts---offers a tractable yet realistic bridge between theory and practice in risk management.}
\textcolor{black}{We now formalize the insurer’s decision-making process as a stochastic control problem governed by a reinforcement learning (RL) policy. At each decision epoch $t_i$, the insurer observes its current financial and risk state $s_i$ and selects an action $a_i$ that adjusts treaty parameters. The policy $\pi_\theta(a|s)$, parameterized by $\theta$, specifies a probability distribution over actions given the state Sutton2018, Barto2003.}
\textcolor{black}{The state $s_i$ may include the current surplus $S_i$, realized claim history $\vec{X}_{1:i}$, current reinsurance parameters $(\alpha_k(t_i), a_k(t_i), b_k(t_i))$, and external signals such as premium levels or catastrophe indicators Asmussen2010, Embrechts1997. The action $a_i$ encodes adjustments to treaty parameters, represented as:}
\textcolor{black}{Here, $\delta_k(t_i)$ adjusts the retention rate in layer $k$, while $\Delta a_k(t_i)$ and $\Delta b_k(t_i)$ shift the attachment and detachment points, respectively. These controls correspond to operational levers available to risk managers during treaty renewal or intra-year adjustments Frees2010, Boucher2016.}
\paragraph{Integration into the Surplus Process.} \textcolor{black}{With policy-driven treaty adjustments, the retained loss for claim $X_{ij}$ under the dynamically updated parameters becomes:}
\textcolor{black}{The surplus recursion incorporating these decisions is therefore:}
\textcolor{black}{where $L_{ij}(t_i)$ reflects the state-dependent, action-modified retention profile at epoch $t_i$. The dependence of $L_{ij}(t_i)$ on $(\delta_k, \Delta a_k, \Delta b_k)$ makes explicit how RL policy decisions $\pi_\theta$ alter capital trajectories McNeil2015, Krvavych2014.}
\paragraph{Policy Optimization.} \textcolor{black}{The policy parameters $\theta$ are optimized to maximize the expected return over the horizon $T$:}
\textcolor{black}{where the reward $R(s_i,a_i)$ encodes the scalarized trade-off between surplus growth, solvency protection, and premium costs, as specified in Section (ref). In practice, this optimization is performed using Proximal Policy Optimization (PPO), which ensures stable updates to $\pi_\theta$ while handling high-dimensional, continuous action spaces Schulman2017, Espeholt2018.}
\textcolor{black}{The insurer’s central objective is to design reinsurance strategies that maximize long-run financial stability while respecting regulatory and capital constraints. In our framework, this is formalized as the maximization of the expected utility of terminal surplus $S_n$:}
\textcolor{black}{where the decision vectors $({\alpha}, {a}, {b})$ represent the retention rates and layer boundaries across all layers. The utility-based formulation balances profit-seeking and risk-aversion, in line with actuarial practice.}
This objective is subject to the following constraints:
\textcolor{black}{Together, these constraints ensure that reinsurance strategies remain economically viable, legally compliant, and practically implementable. By enforcing solvency protection, cost control, and structural validity, the framework provides a disciplined foundation for decision-making. In Section (ref), these optimization objectives are embedded into the hybrid learning architecture, guiding policy design under uncertainty.}
\textcolor{black}{While our primary formulation relies on expected utility, alternative objectives are widely used in both actuarial literature and regulatory applications. One important class involves coherent risk measures such as Conditional Value-at-Risk (CVaR), which directly target the tail of the loss distribution and are embedded in Solvency II and IFRS 17 capital frameworks Rockafellar2000, Pflug2007, Embrechts_et_al2014. Another approach emphasizes risk-adjusted return metrics such as Return on Risk-Adjusted Capital (RORAC), which balance profitability and capital efficiency Cummins2008, Wuthrich2020. Our framework can accommodate these formulations by substituting the terminal utility objective in ((ref)) with a risk measure or performance ratio, without altering the structural constraints.}
\textcolor{black}{\paragraph{Practical scalarization of surplus---ruin trade-offs.} In implementation, we operationalize the bi-objective problem (maximize expected surplus; bound ruin probability) via a scalarized objective in the RL reward: $R_t=\Delta S_t-\lambda_{\text{ruin}}\mathbf{1}\{S_t<0\}-\eta\,\text{Premium}_t$ with a small terminal bonus for solvency. The weights $(\lambda_{\text{ruin}},\eta)$ are calibrated so that the learned policy satisfies $\mathbb{P}(\text{ruin})\le \psi_{\text{target}}$ across Monte Carlo evaluation paths, thus preserving the constrained formulation of Section (ref) while enabling efficient learning Asmussen_and_Albrecher2010, Glasserman2003.}
{\color{black}{Reinsurance optimization requires methods that can both model complex claim distributions and adaptively adjust treaty structures under uncertainty. Traditional actuarial approaches, while mathematically elegant, often assume simple parametric distributions and static treaty parameters, limiting their applicability in modern, high-dimensional settings with systemic risks. Recent advances in machine learning provide complementary tools that address these gaps.
In this section we introduce the two core components of our framework--- Variational Autoencoders (VAEs) for generative modeling of claims and Proximal Policy Optimization (PPO) for sequential decision-making---and explain how they integrate into a unified approach to reinsurance optimization. The VAE enriches the claims environment by generating synthetic scenarios, including rare catastrophic events, thereby mitigating data scarcity and capturing dependencies across lines of business. PPO then operates within this enriched environment to learn adaptive treaty strategies, balancing profitability with solvency constraints.
We first outline the structure and training objectives of VAEs, emphasizing their ability to model high-dimensional claims data and generate coherent joint loss scenarios. We then describe the PPO algorithm, its policy formulation, and its adaptation to the insurer’s surplus optimization problem with explicit ruin constraints. Finally, we present the integrated workflow that combines these components into a single hybrid optimization engine.}}
Variational autoencoders (VAEs) provide the generative backbone of our framework. They learn a probabilistic representation of observed claims data and use it to generate synthetic samples that are both realistic and statistically consistent with historical experience kingma2014auto,rezende2014stochastic. Structurally, a VAE consists of three components:
\paragraph{Multivariate nature of claims portfolios.} \textcolor{black}{We use the term “high-dimensional” in an economic and statistical sense, not as raw feature vectors with hundreds of coordinates.} Although an individual claim amount is a scalar outcome, insurance data are not purely one-dimensional. Each record typically includes attributes such as line of business, policyholder characteristics, geographical exposure, event type, and accident or development year, often supplemented with macroeconomic or catastrophe indices. When modeling across multiple lines or accident years, the joint distribution of severities introduces dependencies that further increase effective dimensionality. Thus, the term high-dimensional refers to the multivariate structure of the claims dataset used to train the VAE, not to the univariate representation of a single claim severity. VAEs are well suited to capture these cross-feature and cross-line dependencies, which traditional univariate severity models cannot. \textcolor{black}{In particular, the VAE does not replace marginal severity models, but augments them by learning a joint representation across heterogeneous features and lines of business. This allows the simulation of coherent portfolios of losses, rather than isolated claim draws Buehler2019, Wuthrich2020.} \textcolor{black}{This use of representation learning is consistent with RMIR’s recent analytics agenda, where text- and data-driven methods complement traditional actuarial techniques TextMining2024_RMIR,Steptoe2022_RMIR.}
\paragraph{Training objective.} The VAE is trained by minimizing the evidence lower bound (ELBO), which balances reconstruction accuracy and regularization of the latent space:
where $x$ denotes observed claims, $z$ is the latent representation, $q_\phi(z|x)$ is the encoder distribution, $p_\theta(x|z)$ the decoder likelihood, $p(z)$ a prior on the latent variables, and $\beta$ controls the strength of regularization. The reconstruction term encourages fidelity to historical data, while the KL divergence enforces smoothness and diversity in the latent space. \textcolor{black}{Equivalently, the objective can be written in maximization form as}
\textcolor{black}{which is algebraically identical to the minimization of $\mathcal{L}_{\text{VAE}}$.}
\textcolor{black}{From a practical standpoint, this objective allows the VAE to interpolate smoothly between observed outcomes and to extrapolate towards rare but plausible extremes, which is particularly valuable for stress testing and capital modeling Frey2019.}
\textcolor{black}{\paragraph{Reconstruction choices and tail emphasis.} In our implementation, the reconstruction term $\mathbb{E}_{q_\phi(z|x)}[-\log p_\theta(x|z)]$ is instantiated as a Gaussian negative log-likelihood for (log) claim severities and a Poisson (or negative binomial, when overdispersion is present) likelihood for counts, following standard actuarial practice Kaas2008, Embrechts_et_al2014. To reflect the importance of the upper tail in reinsurance, we adopt a simple tail-weighting scheme that upweights large losses:}
\textcolor{black}{where $\text{q}_{\tau}(X)$ is the $\tau$-quantile of the empirical severity distribution (e.g., $\tau=0.95$) and $\omega>0$ controls the strength of tail emphasis. The full training loss then becomes}
\textcolor{black}{with $\beta$ optionally annealed from small to larger values over training to stabilize optimization Kingma_and_Welling2014. This formulation preserves the probabilistic semantics of the VAE while explicitly improving fidelity in ranges that matter most for solvency analysis.}
\paragraph{Comparison with classical severity models.} Traditional severity models such as lognormal, Pareto, or Burr assume fixed parametric forms and typically treat claims as independent klugman2012loss,embrechts1997modelling. VAEs, in contrast, are non-parametric and can learn complex dependencies, including joint tail behavior, across lines of business. While a parametric model may offer superior fit for a single marginal distribution, the VAE’s ability to produce correlated scenarios makes it particularly valuable for stress testing and reinsurance optimization. \textcolor{black}{This portfolio perspective is essential: classical models can fit single marginals well, but they cannot capture correlated extremes across business lines (e.g., catastrophe and non-catastrophe losses occurring jointly). The VAE therefore complements classical tools by generating coherent multi-line scenarios that are directly relevant for stress testing, solvency, and capital adequacy. In this sense, parametric models remain useful for interpretability and calibration, while the VAE supplies a flexible scenario generator that enhances the robustness of optimization exercises.}
\paragraph{Integration into the hybrid framework.} In our framework, the trained VAE generates synthetic claims that populate the reinforcement learning environment. By enriching the simulation with extreme but realistic loss scenarios, the VAE provides the foundation upon which the PPO agent learns adaptive treaty strategies. \textcolor{black}{This integration ensures that the optimization agent is not trained solely on average-case scenarios, but also learns from rare, correlated, and high-severity events that are critical for solvency and capital adequacy. In this sense, the VAE acts as a bridge between actuarial modeling traditions and modern machine learning techniques, combining statistical soundness with generative flexibility.}
Whereas the VAE enriches the claims environment, the decision-making engine of our framework is Proximal Policy Optimization (PPO), a reinforcement learning algorithm designed for stable policy updates in sequential settings schulman2017proximal. PPO is particularly well-suited for reinsurance because treaty design involves repeated, path-dependent choices under uncertainty. In contrast to static optimization approaches such as stochastic programming or credibility-based reserving Daykin1994,Sandstrom2010, PPO adapts dynamically, updating strategies as new information unfolds over time.
\paragraph{Policy definition.} We define the insurer’s decision policy as \[ \pi_\theta({a}_t \mid {s}_t), \] which specifies the probability of selecting an action vector ${a}_t$ given the current state ${s}_t$. The state ${s}_t \in \mathbb{R}^d$ summarizes key information such as current surplus, historical losses, line-of-business exposures, and relevant macroeconomic or catastrophe indices. The action vector ${a}_t \in \mathbb{R}^{3K}$ encodes adjustments to treaty parameters across $K$ layers: \[ {a}_t =
, \] where $\delta_k(t)$ adjusts the retention rate of layer $k$, and $\Delta a_k(t), \Delta b_k(t)$ adjust its attachment and detachment points. These adjustments are interpreted as incremental, sequential treaty tweaks that are economically meaningful and practically implementable for insurers and brokers.
\paragraph{PPO surrogate objective.} PPO maximizes a clipped surrogate objective schulman2017proximal:
where $r_t(\theta) = \pi_\theta(a_t|s_t)/\pi_{\theta_{\text{old}}}(a_t|s_t)$ is the policy likelihood ratio, $\hat{A}_t$ the advantage estimator, and $\epsilon$ a trust-region parameter. The clipping acts as a guardrail that prevents the algorithm from overreacting to rare but extreme claims, a desirable property in reinsurance where tail events dominate risk assessment Mnih2016,Sutton2018.
\paragraph{Reward design in insurance.} To align PPO’s per-period objective with the insurer’s terminal-surplus goal, we define
where $\Delta S_t$ is the net surplus increment (including claims and ceded premiums), $\text{Premium}_t$ is the cost of reinsurance, the indicator penalizes insolvency, and $\widehat{\text{Tail}}_t$ is a CVaR-type penalty Rockafellar2000,Asmussen2010,Buehler2019. With $\gamma \approx 1$ and a terminal bonus $b_{\text{term}}=\rho\,\mathbf{1}\{S_n \ge 0\}$ at horizon $n$, the cumulative reward closely tracks $\mathbb{E}[U(S_n)]$, enforcing solvency discipline alongside surplus maximization.
\paragraph{Reconciling objectives.} Thus PPO’s optimization \[ J(\pi_\theta) = \mathbb{E}_{\pi_\theta}\!\left[\sum_{t=0}^{n} \gamma^t r(s_t,a_t)\right] \] is consistent with the insurer’s actuarial problem \[ \max_{\pi_\theta}\;\; \mathbb{E}[U(S_n)]. \] This reconciliation ensures that the RL agent avoids short-term strategies (e.g., overly aggressive retentions) that jeopardize solvency, a common criticism in financial applications of machine learning Buehler2019.
\paragraph{Surplus---ruin trade-offs.} The formulation explicitly encodes the fundamental trade-off of reinsurance:
By balancing rewards and penalties, PPO converges to treaty strategies that maximize long-run surplus while respecting solvency constraints. This balance mirrors actuarial practice and embeds regulatory capital requirements (e.g., Solvency II 99.5% or NAIC RBC thresholds) directly into the learning environment Cummins2008,Sandstrom2010. Methodologically, this approach formalizes surplus---ruin management as a sequential, data-driven optimization problem scalable to high-dimensional treaty portfolios.
To illustrate the methodological implications of our design, Table (ref) provides a side---by---side comparison of the proposed VAE---PPO framework and traditional actuarial optimization techniques Asmussen2010,McNeil2015,Buehler2019.
As summarized in Table (ref), the hybrid framework supports dynamic, data-driven decision making and enhanced tail-risk management, whereas classical methods rely on fixed distributional assumptions and require frequent manual recalibration Asmussen2010,McNeil2015. These contrasts explain the superior adaptability of our approach to high-dimensional treaty portfolios and changing market environments.
The hybrid framework integrates the generative capabilities of the VAE with the adaptive decision-making of PPO to optimize reinsurance strategies under uncertainty. The process unfolds in four steps:
\paragraph{Division of roles.} The two components address complementary challenges. The VAE mitigates data scarcity by augmenting the environment with realistic catastrophic scenarios and captures cross-line correlations that classical univariate models cannot. PPO, in turn, provides adaptive optimization by learning treaty strategies that evolve in response to stochastic claim dynamics and capital constraints. \textcolor{black}{This separation of tasks reflects a broader principle in hybrid AI---actuarial systems: generative modeling enhances the data landscape, while reinforcement learning adapts strategy in real time. Together, they bridge the gap between actuarial simulation and operational decision-making Buehler2019, Kaas2008.}
\paragraph{Practical implications.} \textcolor{black}{The integrated workflow can be deployed in an iterative cycle aligned with insurers’ planning horizons (e.g., quarterly or annually). Synthetic claims generated by the VAE provide forward-looking distributions, while the PPO agent translates these into adaptive treaty adjustments. This ensures that reinsurance strategies remain both profitable and resilient under Solvency II and NAIC regulatory frameworks Daykin1994, Cummins2008.}
The integration of generative modeling and reinforcement learning yields a framework that is, to our knowledge, the first to jointly address two longstanding challenges in reinsurance optimization:
By combining these components, the hybrid framework provides a tractable computational approach to reinsurance design that is both robust to tail risks and responsive to evolving claim dynamics. {\color{black}This synergy between generative modeling and reinforcement learning is, to our knowledge, unique in the actuarial and financial literature, and represents a step toward AI-native risk management tools. In this sense, the framework contributes both a methodological novelty and a bridge between modern machine learning and classical actuarial science Buehler2019,Wuthrich2020.} This methodological novelty sets the stage for the empirical evaluation in Section (ref), where we benchmark performance against traditional actuarial and computational methods.
\textcolor{black}{In this section, we conduct a systematic evaluation of the proposed hybrid reinsurance optimization framework. The analysis is organized around three key components: (i) simulation setup and training metrics used to assess learning stability, (ii) surplus trajectory dynamics under the learned policies, and (iii) benchmark comparisons against established optimization methods.} \textcolor{black}{By structuring the evaluation in this way, we make clear how our approach performs both in absolute terms and relative to traditional actuarial and computational techniques.}
\textcolor{black}{To ensure reproducibility and clarity, we first specify the simulation environment used to evaluate the hybrid framework. The configuration is designed to balance tractability with sufficient realism, providing a testbed that captures the essential features of reinsurance operations.} Table (ref) summarizes the initial parameters.
The insurer's starting surplus was set at \$20,000, with claims modeled using a Poisson process with an average frequency of \( \lambda = 10 \) claims per year Ross2014. Claim sizes were sampled from a lognormal distribution with parameters \( \mu = 3.5 \) and \( \sigma = 1.0 \) Aitchison_and_Brown1957. \textcolor{black}{These choices reflect standard actuarial assumptions that capture both the frequency and skewness of insurance losses, while remaining simple enough for benchmark comparability.}
\textcolor{black}{The synthetic claims generated under these assumptions were used to train the Variational Autoencoder (VAE), which in turn produced enriched and correlated scenarios for reinforcement learning. This integration ensures that the PPO agent is trained not only on stylized losses but also on high-dimensional, realistic claim dynamics.}
\textcolor{black}{To evaluate the effectiveness of the PPO agent, we analyze both training metrics and the resulting surplus trajectories. This dual perspective captures how well the policy converges during learning and whether the optimized treaties translate into financially stable outcomes.} Table (ref) outlines key metrics, and Figure (ref) illustrates the surplus trajectory across 6,144 timesteps.
Figure (ref) highlights the PPO agent's learning process. Early fluctuations reflect exploration, while stabilization over time underscores convergence to effective policies. \textcolor{black}{The overall trajectory remains consistently above the ruin threshold, underscoring the role of reward shaping in aligning learning with solvency objectives.}
The metrics in Table (ref) provide deeper insights:
\textcolor{black}{To rigorously evaluate the contribution of the proposed framework, we benchmarked it against four widely recognized optimization methods, each chosen to represent a distinct class of actuarial and computational techniques. These baselines were implemented according to canonical formulations to ensure reproducibility and fairness of comparison:}
\textcolor{black}{In contrast, our Hybrid RL with Generative Models embeds PPO within a reinforcement learning environment seeded by VAE-simulated claims. This allows adaptive reinsurance decisions to be learned directly under both budget and ruin constraints, overcoming limitations of the static or sample-intensive baselines.}
\textcolor{black}{\paragraph{Baseline implementation details.} All baselines were implemented to reflect canonical actuarial practice and to ensure fair comparison:}
\textcolor{black}{All methods use identical claim generators and premium budgets; evaluation uses common random numbers for variance reduction Glasserman2003.}
Table (ref) summarizes the comparative outcomes. \textcolor{black}{From a performance-measurement perspective, these results echo RMIR studies of efficiency and value creation in insurance operations Efficiency2022_RMIR.}
\paragraph{Analysis of Results:}
\textcolor{black}{Importantly, the budget utilization column is reported as N/A for DP, MC, HDMC, and MOO because those methods were implemented in their canonical forms, which constrain ruin probability but not cost. Only the Hybrid RL approach integrates budget directly into policy learning, making this metric explicit and economically meaningful.}
\textcolor{black}{Overall, the results highlight the advantage of combining generative modeling with adaptive reinforcement learning: the framework not only achieves superior financial outcomes but also demonstrates scalability and robustness that classical methods lack.}
\textcolor{black}{Reinsurance optimization remains a challenging problem because the underlying risk environment is inherently uncertain, heavy-tailed, and highly dynamic. To be practically useful, any optimization framework must demonstrate not only strong in-sample performance but also robustness across a range of realistic stressors.}
\textcolor{black}{In this section, we assess the applicability of the proposed hybrid framework along four applied dimensions: (i) performance across alternative claim distributions to test generalizability, (ii) out-of-sample and sensitivity analyses to evaluate stability under parameter shifts, (iii) stress-testing against catastrophic scenarios and tail events to probe resilience, and (iv) scalability assessments to understand feasibility for large, multi-line portfolios.}
\textcolor{black}{The results illustrate that the hybrid approach maintains surplus stability and low ruin probability under a wide variety of operational settings, while also adapting to distributional shifts and extreme shocks. At the same time, several limitations emerge---particularly the need for more accurate tail modeling in rare-event regimes and the computational burden associated with very large portfolios.}
\textcolor{black}{Together, these findings provide a balanced perspective: the framework is demonstrably applicable to real-world reinsurance problems, but also highlights important areas for refinement to ensure robustness, scalability, and industry adoption.}
The performance of the generative claim model was evaluated across Lognormal, Pareto, and combined Lognormal-Pareto distributions, focusing on its ability to replicate key statistical properties. Using the Kolmogorov-Smirnov (KS) test and visual comparisons, we highlight the model's strengths in capturing central tendencies and its limitations in modeling tail behavior. Accurate tail modeling is critical for reinsurance applications due to the disproportionate impact of extreme claims Embrechts_et_al2014.
\textcolor{black}{In reinsurance practice, the choice of reference distribution is not merely a statistical exercise but directly informs capital adequacy, solvency testing, and pricing decisions. Hence, understanding where the generative model succeeds and where it falls short provides practical guidance for both model refinement and actuarial application McNeil2015.}
The KS test results indicate significant discrepancies between the training and generated datasets, with a KS statistic of \(0.6264\) and a \(p\)-value of \(0.0000\). The maximum difference location (\(D\)) of \(14.7174\) highlights the model’s difficulty in capturing extreme claims, which dominate risk assessments in reinsurance. \textcolor{black}{As shown in Figure (ref), the empirical CDFs reveal systematic underestimation of large claims, underscoring the model’s current limitations in the upper tail. This finding is consistent with prior evidence that generative neural networks often excel at fitting central distributions but struggle in replicating heavy-tail behavior without explicit regularization Kuo2022,Goodfellow2016.}
For the Lognormal distribution, the KS statistic of \(0.5896\) and \(p\)-value of \(0.0000\) reveal significant differences between the training and generated datasets, particularly in the tail regions. The maximum difference location (\(D\)) of \(12.0666\) emphasizes the model’s challenges in replicating the distribution’s long-tailed nature. \textcolor{black}{Figure (ref) confirms this: while the body of the distribution is well-captured, the right tail is consistently underestimated. This underestimation suggests that the VAE tends to over-regularize extreme outcomes in order to preserve reconstruction accuracy in the bulk of the data. In actuarial contexts, such behavior would lead to systematic underpricing of high-excess layers, where profitability depends critically on accurate tail risk estimates Daykin1994.} Strategies such as tail-prioritized loss functions and data augmentation focusing on rare events may enhance performance Bengio_et_al2013.
For the Pareto distribution, the KS test results show a statistic of \(0.6230\) with a \(p\)-value of \(0.0000\), highlighting the model's inability to adequately represent the heavy-tailed characteristics of the data. \textcolor{black}{As illustrated in Figure (ref), the generated distribution systematically underestimates catastrophic losses, a critical shortcoming for reinsurance risk modeling. This is particularly concerning given that Pareto-type tails are widely used in catastrophe and operational risk modeling because of their theoretical grounding in regular variation Embrechts1997.} Incorporating custom-tailored loss functions and oversampling tail regions could mitigate these deficiencies. \textcolor{black}{Another promising avenue involves hybrid modeling: coupling generative models for the bulk of the data with parametric Pareto fits for the extreme tail, thereby combining data-driven flexibility with theoretical rigor Frey2019.} \textcolor{black}{While parametric fits achieve lower KS statistics for marginal distributions, they cannot capture joint dependencies across lines, which are central to PPO-based optimization.}
The combined Lognormal-Pareto distribution provides further insights into the model’s limitations. The KS statistic of \(0.4438\) and a \(p\)-value of \(0.0000\) confirm discrepancies in the tails, as shown in Figure (ref). \textcolor{black}{While the hybrid distribution reproduces the central body effectively, the generated data consistently fails to capture the probability of rare catastrophic claims. This underrepresentation implies that capital requirements estimated using such a model may be biased downward, potentially leading to solvency shortfalls if used without adjustment.} Increasing latent dimensionality and introducing loss functions that explicitly weight tail events could enhance robustness Bengio_et_al2013. \textcolor{black}{This refinement is especially important in reinsurance, where solvency and capital requirements are disproportionately driven by extreme losses. Practical implementations could draw on existing actuarial approaches to extreme value theory (EVT) and tail risk measures, such as CVaR, to guide loss weighting in training McNeil2015,Rockafellar2000.}
The generative claim model's performance was evaluated through out-of-sample testing, sensitivity analysis, and visualization of results. This assessment highlights both the model's robustness in generalizing to unseen data and its limitations when confronted with real-world variability.
The out-of-sample testing revealed a mean surplus of \(16,686.73\) with a ruin probability of \(0.00\%\), demonstrating the model's capability to effectively manage surplus within a simulated insurance environment. \textcolor{black}{Figure (ref) illustrates these dynamics: the surplus distribution remains stable, while ruin events are entirely absent, indicating that the model successfully balances premium income against claims volatility. In actuarial contexts, this is equivalent to maintaining solvency margins under regulatory standards such as Solvency II or NAIC risk-based capital regimes McNeil2015, Sandstrom2010.} This stability suggests strong potential for operational deployment, particularly in settings where solvency must be guaranteed under stochastic conditions.
\textcolor{black}{For claim-size comparisons, cumulative distribution functions (CDFs) are used instead of histograms, following best practice for detecting subtle differences across datasets Frees2010.} The claim size distribution, shown in Figure (ref), demonstrates that the model replicates central tendencies of the training data. \textcolor{black}{However, tail discrepancies are more visible in the CDF view, with the model underestimating the frequency of extreme claims---an important weakness for reinsurance contexts where rare, high-severity events dominate solvency calculations Embrechts1997, McNeil2015.}
Sensitivity analysis further evaluated robustness under distributional shifts. When claim parameters were altered (\(\mu=3.6, \sigma=1.1\)), the mean surplus declined modestly to \(16,009.44\) while maintaining zero ruin probability. With further variations (\(\mu=3.7, \sigma=1.2\)), the model adapted effectively, producing a higher mean surplus of \(17,052.60\). \textcolor{black}{These findings, summarized in Table (ref), confirm that the framework preserves solvency across parameter regimes, though performance levels vary with distributional assumptions. Importantly, such stress-testing resembles regulatory Own Risk and Solvency Assessment (ORSA) practices, where insurers must demonstrate resilience to parameter drift and distributional ambiguity Daykin1994, Cummins2008.} This robustness is particularly valuable in practice, where claim severity parameters are uncertain and may drift over time.
Stress testing provides insight into the framework’s robustness under clustered catastrophic events, where multiple large claims occur in rapid succession. Such clustering often overwhelms traditional reinsurance optimization methods, as extreme events disproportionately affect surplus and solvency. \textcolor{black}{These clustered shocks mimic real-world features such as natural catastrophes, pandemics, or financial crises, where dependence and temporal correlation among losses lead to compounding effects that standard actuarial models often underestimate Embrechts1997, McNeil2015, Asmussen2010. Stress-testing is therefore a key regulatory and risk management tool under Solvency II and NAIC ORSA regimes Sandstrom2010, Cummins2008.} \textcolor{black}{This focus on clustered extremes parallels RMIR discussions on disaster risk reduction and catastrophe-model usage in risk management Surminski2018_RMIR,Steptoe2022_RMIR.}
To evaluate resilience, clustered shocks were simulated by introducing bursts of heavy-tailed claims. Across 1,000 simulation runs, the Hybrid RL with Generative Models maintained a mean surplus of \textcolor{black}{\$9{,}741.52} while limiting ruin probability to \textcolor{black}{2.36%}. In contrast, non-hybrid baselines exceeded \textcolor{black}{5% ruin probability under identical stress conditions and produced lower average surplus}, underscoring the hybrid model’s superior adaptability to catastrophic clustering. \textcolor{black}{This performance gap highlights the value of adaptive control policies that dynamically adjust retention and layering strategies in response to loss shocks, as opposed to static optimization frameworks that assume independence across events Boucher2016, Lopez2020.}
\textcolor{black}{These results reinforce Section (ref)’s benchmark analysis: while DP, MC, HDMC, and MOO perform adequately in static or i.i.d.\ claim settings, their resilience erodes under clustered catastrophic regimes. By directly learning adaptive retention and layering policies through PPO Schulman2017, and leveraging VAE-based tail generation Kingma2014, Hybrid RL preserves solvency more effectively and sustains higher surplus even in highly adverse environments.}
\textcolor{black}{Nonetheless, as discussed in Section (ref), the framework still faces challenges in fully capturing the far tail of claim distributions, which remains a critical limitation despite its comparative advantages. Future work may benefit from extreme value theory (EVT)-based augmentations or copula-driven dependence modeling to further enhance tail fidelity Embrechts2013, ChavezDemoulin2005.}
While the hybrid RL framework demonstrates notable improvements in surplus preservation and ruin reduction relative to established baselines, several limitations remain. The most prominent challenge is \textcolor{black}{tail fidelity}. As shown in Section (ref), the VAE struggles to fully capture the extreme upper quantiles of loss distributions, leading to underestimation of rare but catastrophic claims. Although stress testing indicates that the hybrid approach mitigates ruin more effectively than non-hybrid baselines, \textcolor{black}{its performance in the far tail remains imperfect and warrants further methodological advances. This limitation is well-recognized in actuarial science, where extreme value theory (EVT) and generalized Pareto distribution (GPD) models are standard tools for modeling catastrophic risks Embrechts1997, McNeil2015. Empirical studies confirm that misspecification in the far tail can lead to severe underestimation of capital requirements Embrechts2013, ChavezDemoulin2005, directly impacting solvency assessments. Future work may therefore integrate tail-focused loss functions Frey2019, EVT-based augmentation, or adversarial training strategies to improve tail fidelity in generative models.}
A second limitation lies in \textcolor{black}{scalability}. The framework has been validated primarily on simulated portfolios with manageable dimensionality. Scaling to real-world reinsurance portfolios---which may involve thousands of treaties, clauses, and cedents---poses computational challenges. High-dimensional state-action spaces increase training times and may require substantial infrastructure. \textcolor{black}{In the RL literature, scalability bottlenecks are well documented Sutton2018, Mnih2016. Distributed RL approaches, such as IMPALA Espeholt2018, demonstrate how parallelized rollouts and asynchronous optimization can accelerate convergence. Portfolio-level strategies, including clustering cedents by exposure profiles Kuo2022, or applying hierarchical RL Barto2003 to break down treaty optimization into subproblems, offer promising pathways for extending hybrid frameworks to realistic market settings. These techniques would allow hybrid RL to handle combinatorial complexity while maintaining tractable training times.}
Finally, \textcolor{black}{model interpretability} deserves attention. While the framework provides quantitative improvements, transparency in decision-making (e.g., why a specific retention or limit was chosen) is critical for regulatory adoption under regimes such as Solvency II or NAIC. \textcolor{black}{Regulatory frameworks increasingly emphasize explainability and governance, with IFRS 17 and NAIC requiring justification of capital adequacy assumptions Lopez2020. Black-box RL policies may face resistance if they cannot provide clear rationales for treaty structures DoshiVelez2017. Techniques such as attention-based explanations, rule-extraction from policies, or surrogate interpretable models Lundberg2017 could improve trustworthiness and bridge the gap between technical performance and supervisory acceptance.}
In sum, \textcolor{black}{the hybrid RL framework represents a significant advance in reinsurance optimization but requires enhancements in tail modeling, scalability, and interpretability before achieving broad real-world adoption. These limitations directly motivate the directions outlined in Section (ref), where we chart opportunities for advancing hybrid approaches toward practical, large-scale deployment.}
{\color{black}{This paper introduced a hybrid framework that integrates generative modeling and reinforcement learning for reinsurance optimization. By combining Variational Autoencoders (VAEs) Kingma2014 to simulate complex, heavy-tailed loss distributions with Proximal Policy Optimization (PPO) Schulman2017 to adaptively manage treaty structures, the framework addresses the twin challenges of modeling systemic risk and dynamically allocating capital under uncertainty. Across simulation experiments, the approach demonstrated improved surplus stability, reduced ruin probability, and adaptability to evolving claim environments relative to dynamic programming (DP), Monte Carlo (MC), hybrid deep Monte Carlo (HDMC), and multi-objective optimization (MOO) baselines. These results build on recent advances in actuarial machine learning Frees2010,Boucher2016 and reinforcement learning in operations research Barto2003,Espeholt2018, demonstrating their relevance to solvency and reinsurance design.}} \textcolor{black}{In line with RMIR’s recent emphasis on data-driven risk management and resilience Steptoe2022_RMIR,Resilience2023_RMIR,TextMining2024_RMIR, our findings highlight the practicality of hybrid analytics for solvency-aware treaty design.}
{\color{black}{Evaluation under out-of-sample and sensitivity scenarios confirmed that the hybrid method generalizes effectively beyond training distributions, a critical property for risk management where model misspecification is common McNeil2015,Kuo2022. Stress-testing further revealed that the framework sustains lower ruin probabilities under clustering and catastrophic shocks, though performance still deteriorates in the extreme upper tail. The surplus preservation advantage over baselines (e.g., \(2.36\%\) ruin vs. \(5\%+\) for non-hybrid methods) underscores robustness, but also highlights that catastrophic persistence and “black swan” dynamics Embrechts1997,ChavezDemoulin2005 remain open challenges for next-generation actuarial AI models.}}
{\color{black}{A recurring methodological concern is the use of VAEs instead of direct parametric severity fitting. While traditional severity models (e.g., lognormal, Pareto, Burr) provide interpretable and statistically tractable tail estimates Asmussen2010,Embrechts2013, VAEs were chosen not for marginal optimality but for their ability to generate correlated, high-dimensional stress scenarios. This trade-off sacrifices marginal fit to gain richer joint-loss structures more aligned with capital adequacy testing and treaty-layer optimization. Future work could explore hybrid approaches that combine parametric marginals with VAE-learned dependence, paralleling recent developments in copula-based and GAN-based generative risk models patton2009copula,goodfellow2016deep,Lopez2020.}}
{\color{black}{Several research avenues emerge from these findings. First, methodological advances are needed to improve tail fidelity, such as loss functions weighted toward extreme quantiles, adversarial augmentation for rare-event synthesis, or embedding extreme value theory (EVT) priors directly into generative networks Krvavych2014. Second, distributed RL training and hierarchical policy architectures could address scalability constraints, enabling application to portfolios with thousands of treaties and cedents. Third, interpretability remains essential for regulatory adoption: explainable ML techniques such as SHAP Lundberg2017 or interpretable surrogate policies DoshiVelez2017 could help bridge technical performance with transparency requirements under Solvency II and NAIC frameworks. Finally, validating the framework on real-world, multi-line datasets---and incorporating external factors such as macroeconomic volatility, regulatory shocks, and climate-driven catastrophe risk Glasserman2003,Ross2014---will be critical steps toward practical deployment.}}
{\color{black}{In summary, the hybrid RL framework represents a significant advance in reinsurance optimization: it improves financial resilience under stochastic claims, highlights the importance of tail-aware modeling, and establishes a roadmap for future work spanning methodological rigor, computational scalability, and regulatory alignment. As such, it contributes to the broader vision of AI-augmented actuarial science---one in which advanced generative models and reinforcement learning jointly enable reinsurance strategies that are adaptive, transparent, and resilient under systemic uncertainty.}}
The author thanks James R. Finlay for reading the manuscript and providing helpful suggestions on presentation and language.