Strategic Risk Reduction: Self-Protection and Self-Insurance

This paper studies how a risk holder should combine self-protection and self-insurance strategies when market insurance is absent. Self-protection reduces loss frequency, while self-insurance reduces loss severity. The risk holder incurs a joint risk-reduction cost that allows technological interaction between the two strategies and evaluates residual risk using either Value-at-Risk or Tail Value-at-Risk. In a Bernoulli model, we show that Value-at-Risk leads to a threshold-driven solution in which the optimal strategy is either no risk reduction, pure self-protection, or pure self-insurance, thereby exhibiting a substitution-type structure between the two risk-reduction strategies. By contrast, although Tail Value-at-Risk also admits a left-region/right-region decomposition, its left-region problem creates a direct residual frequency-severity interaction, making the local problem non-convex even in the Bernoulli setting. We solve this problem using an isoquant geometry method based on the marginal-balance curves for self-protection and self-insurance. The analysis identifies boundary, extreme constrained, touching, and crossing candidates, and shows how the confidence level and the cost technology determine whether self-protection and self-insurance behave as substitutes or complements. Illustrative examples compare the Value-at-Risk and Tail Value-at-Risk strategies, show how the confidence level changes the relevant isoquant geometry, and demonstrate that multiple crossings may generate non-unique optimal joint risk-reduction strategies.