Insurance serves as a potent tool for transferring undesirable risks to third parties. However, insurance contracts often fall short of providing perfect indemnification for all potential losses in any given scenario. This deficiency has been acknowledged in the field of insurance economics, tracing back to the work of Doherty and Schlesinger in 1983.
In this study, our focus lies in examining the demand for insurance when confronted with an uninsurable risk that is contingent on the magnitude of the loss. We assume that an individual’s loss can be divided into two parts: the insurable component, denoted as [insurable component], and the uninsurable component, denoted as [uninsurable component]. The total loss experienced by the individual is determined by the sum of these two components, expressed as [total loss] = [insurable component] + [uninsurable component]. However, the insurance coverage only pertains to the insurable component, as the uninsurable component cannot be addressed through insurance mechanisms. We assume that the uninsurable component has a zero mean, thereby eliminating any wealth effects stemming from this particular component. This interpretation aligns with the concept of “approximate insurance” introduced by Gollier in 1996, where [uninsurable component] can be understood as the discrepancy between the actual loss magnitude and the insured loss amount. Essentially, we consider a scenario in which only estimated losses are insurable, while actual losses remain beyond the scope of insurance coverage.
There are several scenarios where an uninsurable component can emerge in various applications:
⦁ Limited Observability: The insurer faces challenges in accurately observing the extent of the loss. This situation often arises when the loss develops gradually over extended periods, but the compensation needs to be provided immediately. This exposes the decision-maker to price risk.
⦁ Simplified Indemnification Process: The insurer employs a simplified method for determining the compensation, relying on estimates rather than actual losses. In such cases, the insurance policies are intentionally designed not to depend on the precise magnitude of the loss. For instance, fixed indemnity insurance plans like hospital cash benefits exemplify this scenario.
⦁ Uninsurable Exchange Rate Risk: The insured loss is susceptible to exchange rate fluctuations that cannot be covered by insurance. In these instances, the unpredictable changes in currency values pose a risk that insurance providers are unable to underwrite.
Changes in initial wealth
One important finding in insurance economics is that the slope of absolute risk aversion plays a significant role in determining the impact of wealth on optimal insurance demand. When there is no background risk, an increase in initial wealth has no effect on the demand for fair insurance. However, for actuarially unfair premiums, insurance behaves as an inferior good when absolute risk aversion is decreasing. This holds true even when there is an independent background risk, as the derived utility function inherits the property of decreasing absolute risk aversion (Kihlstrom et al., 1981; Nachman, 1982; Pratt, 1988).
In this section, we examine preferences where the intuitive inverse relationship between wealth and insurance demand remains valid when the decision-maker faces loss-dependent background risk. Our first finding emphasizes that the results observed in the presence of independent background risk may not hold when confronted with loss-dependent background risk. Proposition 4 demonstrates that we can identify conditions under which the decision-maker increases their coinsurance demand in response to a wealth increase. In the case of independent background risk, no risk-averse decision-maker would alter their insurance demand as a reaction to increased wealth.
Insurance contracts often fail to fully compensate for actual losses due to various factors, making them imperfect tools for risk management. This study focuses on a specific scenario where insurable losses are accompanied by an additional background risk that cannot be insured against. This additional risk, known as loss-dependent background risk, increases as the size of the loss grows. One example of such a risk occurs when an insurance company employs a simplified method to determine indemnification based on standardized assumptions about the size of the loss. Consequently, insurance payouts are not based on the actual loss incurred but rather on an estimation of the true loss size.