Moral hazard in insurance markets with distortion risk measures

Non-concave Corporate Management with Option Incentives under Value-at-Risk Constraint

This article studies a dynamic corporate risk management problem by considering the decision-making of risk-averse managers who exert costly effort and select project risk. We study how a Value-at-Risk (VaR) constraint affects managerial decisions and the distribution of firm value when the manager's objective is non-concave with a fixed salary and options. By the concavification technique, we analyze the optimal terminal firm value on the concave envelope of the objective function. Applying the quantile formulation and the martingale approach, we can derive explicit solutions for optimal effort, terminal firm value, and project choice. The optimal terminal firm value can be divided into nine cases by carefully discussing the choices of VaR floor and tail probability. Compared with the benchmark case, we find that a VaR manager will smooth terminal firm value across states, reducing it in good states while supporting it in adverse states. Moreover, a VaR requirement generally improves downside protection and reduces bankruptcy probability when the VaR floor is low or moderate. However, when the VaR floor is sufficiently high, it can increase bankruptcy probability and induce gambling-for-recovery behavior in adverse states. Our sensitivity analysis indicates that greater managerial effort uniformly improves firm value. Moreover, more incentive options make managers more responsible, leading to a smoother terminal firm value across states. In contrast, a high fixed salary makes the manager less responsible and ultimately causes a more dispersed firm value.

Research Division Federal Reserve Bank of St. Louis Working Paper Series Dynamic Optimal Insurance and Lack of Commitment

Risk management in the insurance sector: theory and practical implications

This paper provides an overview of risk management in the insurance sector, combining theoretical principles with practical and regulatory perspectives. Starting with a simplified model of risk pooling, we demonstrate how diversification creates benefits for risk-averse policyholders. In a next step, we present a simple model to illustrate the main goal of quantitative risk management: The optimization of performance subject to a variety of constraints rather than a pure minimization of risks. We continue by reviewing the main fields of application of risk management and elaborate on the rationality of risk management due to market frictions. Finally, our discussion of solvency regulation and alternative policyholder protection mechanisms highlights the trade-offs between financial resilience, costs, and market efficiency. Overall, the paper demonstrates that, in addition to being a regulatory requirement, risk management in insurance is a strategic instrument for balancing policyholder protection, economic efficiency, and long-term sustainability.

Risk Pricing Under Gain–Loss Asymmetry

We propose a novel gain–loss asymmetric utility model—losses relative to a reference point incur discontinuously more disutility than comparable gains—under which we derive closed-form asset pricing solutions within the workhorse “long-run risk” asset pricing model. Our formal analysis reveals gain–loss asymmetry has a dual impact on risk prices. First, a level effect: the expected excess returns of risky assets are made higher and the risk-free rate lower by the kink in the preferences. Second, a cross-sectional effect: the increase in expected returns for an additional unit of risk is higher (lower) for safer (riskier) assets, so expected returns increase nonlinearly with risk exposures. This second effect, absent in standard smooth utility models, can help rationalize, both qualitatively and quantitatively, key puzzles in empirical finance—the fit of the security market line and the downward-sloping term structure of equity risk premia—without compromising on the model’s ability to match the equity premium in the data. This paper was accepted by Tomasz Piskorski, finance. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2025.02286 .

Optimal Life Insurance Decision in Mean-Variance DC Management with Mortality Improvements

This paper studies the investment and insurance strategies of defined-contribution (DC) pension plans under the mean-variance framework. We consider a stochastic environment with time-varying interest rates, contributions, and mortality risk. The DC plan members are allowed to decide their bond and stock allocations, as well as their life insurance coverage. Adopting the martingale approach, we derive the closed-form optimal strategies and the mean-variance efficient frontier. Further numerical analysis investigates how mortality improvements affect investment and insurance decisions, as well as the sensitivity of the optimal decision to market parameters. Our analysis suggests that longevity raises expectations of future contributions, allowing pension members to adopt a less risky investment strategy. Meanwhile, insurance strategy shifts toward early adulthood to protect the high value of future income and decreases significantly at later ages. Moreover, we conduct sensitivity analyses on the target expected wealth, market price of risk, and contribution growth. These findings provide practical guidance for pension members on investment and offer insights for the design of DC pension plans.

Behavioral Participating Insurance: Optimal Investment under Probability Distortion and Aspiration Constraints

We study optimal investment for insurers managing participating (profit-sharing) contracts under probability distortion and probability benchmark (aspiration) constraints. The problem combines three theoretical complexities: (i) nonconcave effective utilities induced by embedded guarantees and surplus-sharing rules, (ii) probability weighting capturing behavioral aspects of long-horizon decisions, and (iii) aspiration-type constraints formalizing solvency requirements. Using quantile formulations and concavification techniques, we derive explicit closed-form solutions for optimal terminal wealth and trading strategies in both complete and incomplete Black-Scholes markets. Our utility class accommodates the piecewise hyperbolic absolute risk aversion (PHARA) family and covers nonconcavities arising naturally in insurance contexts. The framework reveals how probability distortion weakens lock-in behavior and induces time inconsistency: under inverse S-shaped distortions, insurers overestimate upside probabilities and increase risky investment relative to undistorted benchmarks. Asymptotic analysis and numerical illustrations demonstrate regime switches in optimal policies driven by regulatory thresholds and capital constraints. Our results extend the hope-fear-aspirations framework of He and Zhou (2016) and provide practical insights for managing insurance balance sheets under behavioral preferences and solvency constraints.