# Premium Ratios with Capital Costs Included

Insurance economics often assumes that the equilibrium price of a policy should closely approximate the expected losses on the policy plus some sort of "loading" to handle fixed costs and the costs of claims processing. In fact, however, it is probably appropriate to include the cost of risky capital in determining equilibrium premiums. Investors and regulators may demand that insurers' surplus exceed expected losses in order that the insurer is able to cope with higher than expected losses.
This Demonstration examines the magnitude of the effect of capital costs on equilibrium premiums. You determine a conventional load by setting the ratio of fixed costs,
for example, the cost of underwriting, to the expected losses on the policy. You also set the capital-premium ratio to the level demanded either by regulators or investors. And you determine the rate of return the insurer is able to make on funds it invests. The Demonstration produces a graph showing the relationship between the return of return demanded by investors on equity invested in the insurer and the corresponding ratio between premiums and expected losses. You can set how you want this relationship to be displayed with either the premium/expected loss ratio or return on equity placed on the axis.

### DETAILS

Snapshot 1: with cost of capital and loading excluded, the ratio between premium and expected loss is one
Snapshot 2: if the insurer cannot make high returns on safe investments and the capital-premium ratio required by regulators or investors is likewise high (perhaps due to the insurer's inability to diversify risk away), then insurers who face high costs of capital will charge premiums that can be double, triple, or even higher multiples of expected losses
Snapshot 3: the premium/expected loss ratio is placed on the axis in a situation where regulators do not demand high levels of capital relative to premiums
The equation underlying this Demonstration may be found in various publications on catastrophe insurance, notably those of professor Neil Doherty and others of the Wharton Business School.
Notice that for some settings available in the Demonstration, the ratio between premium and expected loss is quite high. If the insured is not particularly risk averse, it may sometimes make more sense in these circumstances for the insured to "self insure" against losses in this setting and pay for damages with some sort of borrowing after a disaster occurs.
This Demonstration may clarify the circumstances under which insurance premiums appear extraordinarily high relative to the expected value of the risk transferred. Excess insurance (or derivative reinsurance) often covers risks (such as large hurricanes, powerful earthquakes, widespread default on debt) that are extremely unlikely to materialize but extraordinarily expensive if they do materialize. They may thus have low expected value but high standard deviation. If these risks cannot be diversified away, regulators and investors should require the insurer to keep large amounts of capital either on hand or readily available. Since these investors will likely want a high rate of return on their risky investment, the cost of capital will be high and the insurer will need to charge premiums significantly above the expected value of losses in order to avoid losses.

### PERMANENT CITATION

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