value-at-risk optimization

This study addresses how risk holders optimally coordinate self-protection (reducing loss probability) and self-insurance (mitigating loss severity) in the absence of market insurance. Within a Bernoulli loss framework, the authors develop a technical model incorporating interaction costs under Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) criteria, and propose an analytical approach combining marginal trade-off curves with isoquant geometry. The analysis reveals that VaR yields threshold-type corner solutions, whereas TVaR leads to non-convex optimization problems. For the first time, the optimal combination under TVaR is solved using isoquant geometry, with optimal strategies shown to occur at boundaries, extreme points, or tangency/intersection locations. The study further clarifies how confidence levels and cost structures determine whether self-protection and self-insurance act as substitutes or complements.

This paper addresses the challenge of optimizing insurance reserves under macroeconomic volatility by proposing the first reinforcement learning framework integrating tail-risk control, regulatory compliance, and macro-state awareness. Methodologically, it formulates reserve adjustment as a CVaR-constrained finite-horizon MDP, incorporating a hidden Markov model for macro-state identification, PPO-based constrained policy optimization, Solvency II/ORSA-aligned composite reward design, and progressive volatility exposure training. The key contribution lies in enabling multi-objective co-optimization of CVaR sensitivity, capital efficiency, and solvency floor adherence, while supporting stress testing and macro-state attribution analysis. Evaluated on real-world workers’ compensation and general liability datasets, the framework reduces CVaR₀.₉₅ by 18.7%, improves capital utilization efficiency by 23.4%, lowers regulatory violation rate to 0.3%, and demonstrates robustness under fixed-shock stress tests.

This work addresses the challenge that when the loss function depends on decision variables, the regularity properties—such as continuity and differentiability—of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are generally not guaranteed, thereby hindering theoretical and algorithmic advances in risk-aware optimization. Focusing on such decision-dependent losses, the paper establishes simple yet rigorous sufficient conditions under which CVaR is proven for the first time to be continuously differentiable, and provides an explicit expression for its gradient. By integrating tools from perturbation analysis of probability measures, path-differentiability, and real analysis, the study develops a unified theoretical framework that ensures the continuity of VaR and the continuous differentiability of CVaR, thereby furnishing reliable gradient information and convergence guarantees for optimization problems involving tail risk.

This work addresses the problem of safe policy optimization under Value-at-Risk (VaR) constraints in reinforcement learning. To prevent constraint violations during training, the authors propose a sample-efficient conservative policy optimization method that directly incorporates VaR constraints into the policy optimization framework. By leveraging a one-sided Chebyshev inequality, they construct a differentiable surrogate constraint based on the first and second moments of the cost return, which is then integrated with an extended trust region mechanism. This approach provides theoretical guarantees for both policy improvement and worst-case constraint satisfaction. Empirical results demonstrate that the method achieves zero constraint violations throughout training across multiple environments, significantly outperforming existing baselines while maintaining high sample efficiency.

This paper addresses the optimal insurance contract design problem under the Lambda-Value-at-Risk (ΛVaR) framework, jointly accommodating heterogeneous risk sensitivity among policyholders and insurers’ robustness requirements. Methodologically, it pioneers the integration of ΛVaR into actuarial optimization—overcoming the rigid tail-risk assumptions inherent in traditional Value-at-Risk (VaR) and Conditional VaR (CVaR) by enabling continuous, preference-based tail-risk calibration. Leveraging convex analysis, distributionally robust optimization, and calculus of variations, the study constructs a structured solution framework incorporating monotonicity constraints and incentive compatibility. It derives explicit optimal reinsurance forms—e.g., stop-loss–stop-loss contracts—and establishes their existence and uniqueness. Numerical experiments demonstrate that the proposed ΛVaR-based contracts improve the joint expected utility of policyholders and insurers by 12%–19% relative to conventional VaR/CVaR benchmarks.

This work addresses the challenge in safety-sensitive stochastic optimization where tail risk exhibits high sensitivity to decision variables, making it difficult for conventional methods to simultaneously optimize average performance and control extreme risks. The authors propose a novel framework that integrates a “safe-start” mechanism with variance reduction techniques. By employing simulation-guided safe initialization, the approach overcomes convergence failures in stochastic gradient descent caused by step-size selection, achieving provably improved sample complexity. Empirical evaluations on portfolio optimization and robust neural network classification demonstrate that the method substantially enhances decision safety under extreme risk scenarios while improving algorithmic efficiency.

This study investigates how risk-averse managers optimally choose effort levels and project risk under Value-at-Risk (VaR) constraints to shape the firm’s terminal value distribution, given a compensation structure combining fixed salary and stock options. By integrating concavification techniques, quantile representations, martingale methods, and dynamic optimization, the paper provides the first analytical solution for optimal terminal wealth, effort provision, and project selection in a non-concave setting. It systematically characterizes nine distinct regimes through which VaR constraints affect the value distribution. The analysis reveals that moderate VaR limits enhance downside protection and reduce bankruptcy risk, whereas excessively stringent thresholds can induce gambling behavior in distress. Moreover, greater option-based incentives lead managers to act more prudently, while higher fixed pay results in a more dispersed value distribution.

This study addresses the limitations of traditional volatility control methods, which often rely on fixed rules or static estimators and struggle to adapt to dynamic market conditions. The authors propose reframing volatility control as a market-state-dependent policy routing problem and introduce a three-stage architecture—comprising state inference, gating review, and policy selection—that dynamically switches among multiple estimator–controller pairs. The framework flexibly accommodates rule-based systems, learnable modules, and large language models as decision mechanisms and incorporates relative policy evaluation to enable continual optimization. Empirical evaluations across three real-world scenarios, including the S&P 500, demonstrate that the approach significantly enhances risk-adjusted returns, achieving a Sharpe ratio as high as 1.222, while simultaneously reducing maximum drawdown and Conditional Value-at-Risk (CVaR), thereby validating the efficacy and novelty of state-aware routing in adaptive risk management.