Optimal Insurance to Maximize Exponential Utility when Premium is Computed by a Convex Functional

Paper: arXiv 2401.08094

Authors: Jingyi Cao, Dongchen Li, Virginia R. Young, Bin Zou

Abstract

We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity; then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents strong theoretical rigorousness with advanced mathematical formulations and convergence proofs for optimal insurance under convex premium principles. However, it lacks empirical validation using real-world market or actuarial data, relying instead on stylized numerical examples.

Research Flowchart

  flowchart TD
    A["Research Goal: Maximize Exponential Utility of Terminal Wealth"] --> B["Inputs: Exponential Utility & Convex Premium Functional"]
    B --> C["Derive Necessary Condition for Optimal Indemnity"]
    C --> D["Formulate Implicit Candidate Optimal Indemnity"]
    D --> E["Develop Iterative Numerical Algorithm"]
    E --> F["Prove Convergence to Unique Optimal Policy"]
    F --> G["Outcomes: Verified Optimal Indemnity & Numerical Examples"]