TY - JOUR
TI - Deductible insurance and the transfer of risk
DO - https://doi.org/doi:10.7282/T3TB15CC
PY - 2014
AB - This thesis develops a deepened understanding of insurance and its benefits, focusing on practical aspects of insurance coverage and risk reduction. It is easy to see that the purchase of insurance increases the expected loss suffered by the insured, otherwise the insurer’s expected profit would be negative. In view of this, we show that insurance is a variance-reducing mechanism. We first prove that the customer’s variance is less than the variance that would be experienced if insurance was not purchased, and further show that the variance of an insured loss X is equal to the sum of the covariances of the insured and insurer losses with X. As insurance increases the insured’s expected loss and decreases the variance, we develop a mean-variance model of insurance demand, showing how the relationship between the premium and the insured’s risk preference defines the demand for insurance. We verify Arrow’s classical (utility-based) result that the optimal policy has full coverage above a non-zero deductible and consider the insurer’s perspective, showing that the customer can be induced to purchase the insurer’s optimal policy. Next, we consider different forms of coinsurance. We show that the optimal straight coinsurance policy is inferior to the optimal deductible policy, while coinsurance combined with either a stop-loss limit or a deductible is equivalent to a deductible policy (in the optimum). We also show that, in each of the coinsurance cases, the optimal policy involves partial coverage if the premium exceeds the expected reimbursement. Finally, we consider the system composed of the insured and insurer, discussing how certain customers may receive discounted premiums that are subsidized by other customers and showing how the customer and the company share in the variance. We then discuss a benefit of insurance; in the case of a single insurer and a single customer, the sum of their individual variances is less than the variance in the uninsured case, and in the case of a single insurer and multiple insured, the variance of this system is smaller than the sum of the individual uninsured variances if the insurer reimbursements are sufficiently uncorrelated.
KW - Operations Research
KW - Insurance--United States
KW - Deductibles (Insurance)
LA - eng
ER -