Strategic Risk Reduction: Self-Protection and Self-Insurance

2026-06-29

AI Summary

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.

Abstract

This paper studies how a risk holder should combine self-protection and self-insurance when market insurance is absent. In a Bernoulli loss model, self-protection reduces the residual loss probability, while self-insurance reduces the residual loss severity. The risk holder evaluates residual risk using either Value-at-Risk or Tail Value-at-Risk and incurs a joint risk-reduction cost that allows technological interaction between the two activities. We show that Value-at-Risk leads to a threshold-driven solution that the optimal strategy is either no risk reduction, pure self-protection, or pure self-insurance. By contrast, Tail Value-at-Risk creates a direct interaction between residual frequency and residual severity, making the problem non-convex even in the Bernoulli setting. We solve it using an isoquant geometry method based on the marginal-balance curves for self-protection and self-insurance. The analysis identifies when optimal strategies lie on boundaries, extreme constrained candidates, touching components, or crossing components, and shows how the confidence level and the cost technology determine whether self-protection and self-insurance behave as substitutes or complements.

Problem

• self-protection
• self-insurance
• risk reduction
• Value-at-Risk
• Tail Value-at-Risk

Innovation

• self-protection
• self-insurance
• Tail Value-at-Risk
• non-convex optimization
• isoquant geometry