HMRC internal manual

General Insurance Manual

Economic basis of insurance: law of large numbers

The majority of insurers, however, will be unwilling or unable to go back to their policyholders for additional payment if losses turn out to be greater than expected. They must rely on a cushion of working capital (provided by the shareholders in anticipation of an investment return in a proprietary company) to meet such losses. One of the main aims of insurance regulators is to ensure that companies always have a sufficient margin of assets over estimated liabilities appropriate to the business that they conduct.

The sharing and pooling of risk is still, however, vitally important. In the real world the pattern of losses (cars stolen or houses burning down) is unstable. Suppose that, on average, one car in ten is stolen each year. If the thefts are independent of one another an insurer who had only insured ten cars would find that there was a one in four chance that two or more would be stolen, which would double the expected outlay on claims. It would not be possible to do business on that basis.

But if 100,000 cars are insured, the probability that more than 10,200 (or less than 9,800) will be stolen is only about 1%. This is an example of the operation of the ‘law of large numbers’, which may be expressed as follows:

“The observed frequency of an event more nearly approaches the underlying probability of the population as the number of trials approaches infinity.”

In other words, the more cars insured, the more accurately can be predicted the percentage of cars likely to be stolen. It is this aspect of probability theory that enables the insurer to cope with variations in the pattern of actual losses. Underwriters and actuaries may also consider various measures of dispersion, that is the difference between the actual losses and average losses, when setting premiums or assessing liabilities.