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## Abstract

Restrictions on insurance risk classification can lead to troublesome adverse selection. A simple version of the usual argument is as follows. If insurers cannot charge risk-differentiated premiums, more insurance is bought by higher risks and less insurance is bought by lower risks. This raises the equilibrium pooled price of insurance above a population-weighted average of true risk premiums. Also, since the number of higher risks is usually smaller than the number of lower risks, the total number of risks insured usually falls. This combination of a rise in price and fall in demand is usually portrayed as a bad outcome, both for insurers and for society.

However, some restrictions on insurance risk classification are common in practice. For example, since 2012 insurers in the European Union has been barred from using gender in underwriting; and many countries have placed some limits on insurers' use of genetic test results. We can observe that policymakers often appear to perceive some merit in such restrictions. This observation motivates a careful re-examination of the usual adverse selection argument.

In this talk, we study the implications of insurers not being allowed to use risk-differentiated premiums. We model the insurance purchasing behaviour of individuals based on their degrees of risk aversion and utility of wealth. We assume that an equilibrium has been reached, where insurers break even by charging the same pooled' premium to both high and low risks. We characterise this equilibrium by two quantities: adverse selection, defined as the correlation of insurance coverage and losses; and loss coverage', defined as the expected losses compensated by insurance.

We find that adverse selection is always higher under pooling than under risk-differentiated premiums, as expected. However, loss coverage can be higher or lower under pooling than under risk-differentiated premiums. Loss coverage is higher under pooling if the shift in coverage towards higher risks more than compensates for the fall in number of risks insured. In other words, loss coverage is higher under pooling if adverse selection at the equilibrium is modest, but lower under pooling if adverse selection at the equilibrium is severe.

Loss coverage represents the expected losses compensated by insurance for the whole population. We argue that this is a good metric for the social efficacy of insurance, and hence one which public policymakers may reasonably wish to maximise. If this argument is accepted, modest adverse selection under pooling can be a good thing, because it leads to higher loss