Adverse Selection With Frequency and Severity Risk: Alternative Risk‐Sharing Provisions

ABSTRACT The analysis considers an insurance market with adverse selection where individuals' loss distributions may differ with respect to both the frequency and severity of loss. We show that the combination of deductibles and coinsurance can be used to sort rationed policyholders. Because of their screening properties, coinsurance and deductibles may both be equilibrium forms of risk sharing for a particular insurer facing asymmetric information, with different rationed consumers choosing different risk-sharing provisions. INTRODUCTION In practice, a single insurer will frequently offer some policies with deductibles, other policies with coinsurance (i.e., proportional coverage), and still others with a combination of these provisions. For example, these types of risk-sharing provisions are common in health insurance policies. Many HMO policies require a copayment, which operates like a per visit deductible, whereas non-HMO group and individual health insurance polices frequently include both an annual deductible and proportional coinsurance for amounts above the deductible. Deductibles (or HMO copayments) and coinsurance rates are typically higher for some kinds of treatment (e.g., dental care, mental health treatments). However, the economic rationale for the particular form of these risk-sharing provisions is not clear. In the absence of information asymmetries, full coverage is optimal for risk-averse consumers if insurance is available at an actuarially fair price. With a positive insurance cost, less than full insurance is optimal. Raviv's (1979) work suggests that with a risk-neutral insurer and costs that are linear in the payoff to the policyholder, full coverage above a deductible is the optimal form of risk sharing. (1) Insurer risk aversion or convexity of insurer costs is necessary for coinsurance to be an optimal risk-sharing form. These results suggest that all contracts written by a particular insurer should feature the same form of risk sharing. The purpose of this study is to determine whether deductibles and coinsurance can be used to solve problems of adverse selection when there is more than one rationed risk type. Our purpose is not to determine the optimal risk-sharing arrangements in a multiple-risk-type adverse selection environment but rather to determine whether the simultaneous use of deductibles and coinsurance represents an improvement over environments where only one of these forms of cost sharing is available. In general, equilibrium risk sharing might involve some combination of deductibles, coinsurance, and upper limits and could involve ex post conditioning of benefit payments. However, before one can address the multitude of possible equilibrium configurations arising from various policy form combinations or ex post adjustments to benefits, one needs to fully understand the interactions between consumer risk characteristics and basic policy forms. Accordingly, in order to focus the analysis, attention is restricted to the choice between policies with different combinations of coinsurance and deductibles. In insurance markets with adverse selection, risk sharing has been suggested as a screening device to solve the adverse selection problem (Rothschild and Stiglitz, 1976; Wilson, 1977; Miyazaki, 1977; Spence, 1978). Most insurance models of adverse selection specify a two-point loss distribution. A loss of some fixed amount either occurs or does not occur, and high- and low-risk individuals simply differ with regard to the probability that a loss will occur. These models have proven to be quite useful for showing that self-selection methods can provide a solution to the adverse selection problem and, indeed, have been extended to provide insight into issues like insurer organizational form (Smith and Stutzer, 1990; Ligon and Thistle, 2005). Individuals with high frequency risk choose contracts with relatively higher levels of coverage (full coverage, if the price is actuarially fair) or, more generally, less risk sharing. …