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When Indemnity Insurance Fails: Parametric Coverage under Binding Budget and Risk Constraints

Benjamin Avanzi, Debbie Kusch Falden, Mogens Steffensen

TL;DR

This paper shows that in high-risk environments where indemnity insurance is unaffordable or impractical, parametric insurance can yield higher welfare for risk-averse individuals under a mean-variance objective. By explicitly modeling frequency via a compound Poisson process and severity with censored exponential tails, the authors derive closed-form optimal contracts and premiums, highlighting a dual relation where E[Y_i] = d^* + k^* under Poisson frequency and equal loadings. The analysis demonstrates that parametric coverage remains welfare-enhancing particularly under binding premium budgets, with a non-monotonic pattern: parametric dominates at low budgets but loses its edge as budgets widen and indemnity becomes feasible. Policy implications emphasize recognizing parametric instruments as complementary risk-transfer tools that can speed payouts, support resilience, and complement public disaster relief, especially when traditional indemnity markets fail the affordability test.

Abstract

In high-risk environments, traditional indemnity insurance is often unaffordable or ineffective, despite its well-known optimality under expected utility. We compare excess-of-loss indemnity insurance with parametric insurance within a common mean-variance framework, allowing for fixed costs, heterogeneous premium loadings, and binding budget constraints. We show that, once these realistic frictions are introduced, parametric insurance can yield higher welfare for risk-averse individuals, even under the same utility objective. The welfare advantage arises precisely when indemnity insurance becomes impractical, and disappears once both contracts are unconstrained. Our results help reconcile classical insurance theory with the growing use of parametric risk transfer in high-risk settings.

When Indemnity Insurance Fails: Parametric Coverage under Binding Budget and Risk Constraints

TL;DR

This paper shows that in high-risk environments where indemnity insurance is unaffordable or impractical, parametric insurance can yield higher welfare for risk-averse individuals under a mean-variance objective. By explicitly modeling frequency via a compound Poisson process and severity with censored exponential tails, the authors derive closed-form optimal contracts and premiums, highlighting a dual relation where E[Y_i] = d^* + k^* under Poisson frequency and equal loadings. The analysis demonstrates that parametric coverage remains welfare-enhancing particularly under binding premium budgets, with a non-monotonic pattern: parametric dominates at low budgets but loses its edge as budgets widen and indemnity becomes feasible. Policy implications emphasize recognizing parametric instruments as complementary risk-transfer tools that can speed payouts, support resilience, and complement public disaster relief, especially when traditional indemnity markets fail the affordability test.

Abstract

In high-risk environments, traditional indemnity insurance is often unaffordable or ineffective, despite its well-known optimality under expected utility. We compare excess-of-loss indemnity insurance with parametric insurance within a common mean-variance framework, allowing for fixed costs, heterogeneous premium loadings, and binding budget constraints. We show that, once these realistic frictions are introduced, parametric insurance can yield higher welfare for risk-averse individuals, even under the same utility objective. The welfare advantage arises precisely when indemnity insurance becomes impractical, and disappears once both contracts are unconstrained. Our results help reconcile classical insurance theory with the growing use of parametric risk transfer in high-risk settings.
Paper Structure (29 sections, 59 equations, 3 figures)

This paper contains 29 sections, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical framework for traditional indemnity and parametric insurance
  3. Losses and benefits
  4. Premia
  5. Decision making criteria
  6. Research question: indemnity vs parametric insurance
  7. Dual relationship between $d^*$ and $k^*$ in the compound Poisson case
  8. Optimal results under censored exponential losses
  9. Robustness and modeling choices
  10. Numerical illustrations
  11. Baseline calibration
  12. Unconstrained optima and one-dimensional comparative statics
  13. Budget-constrained choice and two-dimensional budget surfaces
  14. Policy implications
  15. Insurance failure in high-risk areas
  16. ...and 14 more sections

Figures (3)

  • Figure 1: Comparison of indemnity and parametric insurance under premium matching. Panel (a) shows one-dimensional comparative statics in the indemnity fixed cost $\gamma_d$, while panel (b) extends the comparison to the two-dimensional $(d,\gamma_d)$ space and highlights the region in which parametric insurance yields higher mean--variance utility.
  • Figure 2: Premium matching in the space of indemnity pricing parameters. Panel (a) shows how premiums and mean--variance utilities vary with the indemnity loading $\theta_d$ in one dimension. Panel (b) extends the comparison to the two-dimensional $(\theta_d,\gamma_d)$ space and identifies regions in which the premium-matched parametric contract delivers higher mean--variance utility than the optimal indemnity contract.
  • Figure 3: Budget-constrained comparison of indemnity and parametric insurance. Panel (a) illustrates how optimal contract choice and welfare evolve as the available premium budget $\bar{P}$ increases for a fixed cost $\gamma_d$. Panel (b) extends the analysis to the two-dimensional $(\bar{P},\gamma_d)$ space and highlights the region in which parametric insurance strictly dominates the best budget-feasible alternative.

Theorems & Definitions (5)

  • Remark 2.1: Event-level versus claim-level deductibles
  • Remark 2.2: Basis risk and richer parametric designs
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3