Insurance is an economic institution that allows the transfer of ﬁnancial risk from an individual to a pooled group of risks by means of a two-party contract. The insured party obtains a speciﬁed amount of coverage against an uncertain event for a smaller but certain payment. Insurers may offer ﬁxed, speciﬁed coverage or replacement coverage, which takes into account the increased cost of putting the structure back to its original condition.

Most insurance policies have some form of deductible, which means that the insured party must cover the ﬁrst portion of their loss. For example, a 10 percent deductible on a $100,000 earthquake policy means that the insurer is responsible for property damage that exceeds $10,000 up to some prespeciﬁed maximum amount, the coverage limit.

Losses and claims

A policyholder is a person who has purchased insurance. The term loss is used to denote the payment that the insurer makes to the policyholder for the damage covered under the policy. It is also used to mean the aggregate of all payments in one event. Thus, we can say that there was a “loss” under the policy, meaning that the policyholder received a payment from the insurer. We may also say that the industry “lost” $12.5bn in the Northridge earthquake.

A claim means that the policyholder is seeking to recover payments from the insurer for damage under the policy. A claim does not result in a loss if the amount of damage is below the deductible, or subject to a policy exclusion, but there still are expenses in investigating the claim. Even though there is a distinction between a claim and a loss, the terms are often used interchangeably to mean that an insured event occurred, or with reference to the prospect of having to pay out money.

The law of large numbers

Insurance markets can exist because of the law of large numbers which states that for a series of independent and identically distributed random variables, the variance of the average amount of a claim payment decreases as the number of claims increases. If you go to Las Vegas and place a bet on roulette, you are expected to lose a little more than five cents every time you bet $1. But each time you bet, you either win or lose whole dollars. If you bet ten times, your average return is your net winnings and losses divided by ten. According to the law of large numbers, the average return converges to a loss of five cents per bet. The larger the number of bets, the closer the average loss per bet is to five cents.

Fire is an example of a risk that satisﬁes the law of large numbers since its losses are normally independent of one another. To illustrate this, suppose that an insurer wants to determine the accuracy of the ﬁre loss for a group of identical homes valued at $100,000, each of which has a 1/1,000 annual chance of being completely destroyed by ﬁre. If only one ﬁre occurs in each home, the expected annual loss for each home would be $100 (ie, 1/1,000 × $100,000).

If the insurer issued only a single policy, then a variance of approximately $100 would be associated with its expected annual loss.

As the number of issued policies, n, increases the variance of the expected annual loss or mean decreases in proportion to n. Thus, if n = 10, the variance of the mean is approximately $10. When n = 100 the variance decreases to $1, and with n = 1000 the variance is $0.10. It should thus be clear that it is not necessary to issue a large number of policies to reduce the variability of expected annual losses to a very small number if the risks are independent.

However, natural hazards – such as earthquakes, ﬂoods, hurricanes, and conﬂagrations such as the Oakland ﬁre of 1991 – create problems for insurers because the risks affected by these events are not independent. They are thus classiﬁed as catastrophic risks. If a severe earthquake occurs in Los Angeles, there is a high probability that many structures would be damaged or destroyed at the same time. Therefore, the variance associated with an individual loss is actually the variance of all of the losses that occur from the speciﬁc disaster. Because of this high variance, it takes an extraordinarily long history of past disasters to estimate the average loss with any degree of predictability. This is why seismologists and risk assessors would like to have databases of earthquakes, hurricanes, or other similar disasters over 100-to 500-year periods. With the relatively short period of recorded history, the average loss cannot be estimated with any reasonable degree of accuracy.

One way that insurers reduce the magnitude of their catastrophic losses is by employing high deductibles, where the policyholder pays a ﬁxed amount of the loss (eg, the ﬁrst $1,000) or a percentage of the total coverage (eg, the ﬁrst 10 percent of a $100,000 policy). The use of coinsurance, whereby the insurer pays a fraction of any loss that occurs, produces an effect similar to a deductible. Another way of limiting potential losses is for the insurer to place caps on the maximum amount of coverage on any given piece of property.

An additional option is for the insurer to buy reinsurance. For example, a company might purchase a reinsurance contract that covers any aggregate insured losses from a single disaster that exceeds $50m up to a maximum of $100m. Such an excess-of-loss contract could be translated as follows: the insurer would pay for the ﬁrst $50m of losses, the reinsurer the next $50m, and the insurer the remaining amount if total insured losses exceeding $100m. An alternative contract would be for the insurer and reinsurer to share the loss above $50m, prorated according to some predetermined percentage.

This article is an edited version of an entry in the “Encyclopedia of Quantitative Risk Analysis and Assessment”, Copyright © 2008 John Wiley & Sons Ltd. Used by permission.

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