On the Impact of Insurance on Households Susceptible to Random Proportional Losses: An Analysis of Poverty Trapping

The trapping probability, $\psi$, as defined in Kovacevic and Pflug (2011), is modelled by assuming proportional capital losses, both in the case where there is no insurance and in the case where insurance is purchased by the household. Insurance coverage is likewise proportional, mirroring the structure of quota-share contracts, which are both prevalent in practice and analytically convenient. New closed formulae for $\psi$ are obtained in the case of no insurance when the distribution of the remaining proportion of capital is a power law, extending the results in Kovacevic and Pflug (2011). When proportional insurance is acquired and the remaining proportion of capital is uniformly distributed on $[0,1]$, $\psi$ satisfies a non-local differential equation whose analysis is based on the properties of diffusion processes. The non-local nature of the equation can be addressed using iterative solution methods, leading to a constructive determination of the trapping probability. Constraints on the parameters governing the capital process are derived in both the uninsured and insured cases to prevent the certainty of trapping. Numerical calculations are used to determine the trapping probability for the insured process and to illustrate the impact of different parameters. Consequences on the trapping probability for vulnerable non-poor populations with initial capital slightly above the poverty line are discussed.