Insurance companies rely on the law of large numbers to help estimate the value and frequency of future claims they will have to pay out to policyholders. When it works perfectly, insurance companies run a stable business, consumers pay a fair and accurate premium, and the entire financial system avoids serious disruption. However, the theoretical benefits from the law of large numbers do not always hold up in the real world.

### How the Law of Large Numbers Works

The law of large numbers stems from probability theory in statistics. It proposes that when the sample of observations increases, variation around the mean observation declines. In other words, the average value gains predictive power.

For example, consider a simple trial in which someone flips a quarter. Every time the quarter lands on heads, the person records one point. No points are recorded when it lands as tails. The expected value of a coin flip in this trial is 0.5 points because there is only a 50% chance that the quarter will land as heads.

If you only flip the coin only twice the average value could end up far from the expected value. Consecutive heads produce an average value of one point while two tails have an average value of zero points. Increasing the number of observations is more likely to yield an average value closer to the expected value. If there are 53 heads and 47 tails during 100 flips, the average value would be 0.53, which is very close to the 0.5 expected value.

This is how the law of large numbers works.

### How Insurers Use the Law

In the insurance industry, the law of large numbers produces its own axiom. As the number of exposure units, or policyholders, increases, the probability is higher that the actual loss per exposure unit will equal the expected loss per exposure unit.

To put it in economic language, there are returns to scale in insurance production.

In practical terms, this means that it is easier to establish the correct premium and thereby reduce risk exposure for the insurer as more policies are issued within a given insurance class. An insurance company is better off issuing 500 rather than 150 fire insurance policies, assuming a stable and independent probability distribution for loss exposure.

To see it another way, suppose that a health insurance company discovers that five out of 150 people will suffer a serious and expensive injury during a given year. If the company insures only 10 or 25 people, it faces far greater risks than if it is able to ensure all 150 people. The company can be more confident that 150 policyholders will collectively pay sufficient premiums to cover the claims from five customers who suffer serious injuries.

### When It Does Not Work

There were nearly 6,000 insurance carriers in the United States as of 2016, according to the National Association of Insurance Commissioners. Some carriers are more successful than others who provide the same or similar types of coverage. If there are increasing returns to scale in insurance, thanks to the law of large numbers, then why are there so many insurance companies rather than a few giants dominating the industry?

First, all insurance companies are not equally adept at the business of providing insurance. This includes maintaining operational efficiency, calculating effective premiums and mitigating loss exposure after a claim is filed. Most of these features do not impact the law of large numbers.

However, the law of large numbers becomes less effective when risk-bearing policyholders are independent of one another. This is most easily seen in the health and fire insurance industries because diseases and fire can spread from one policyholder to another if not properly contained. This problem is known as contagion.

There are also potentially insurable risks for which the law of large numbers theoretically could be useful, but there are not enough potential customers to make it work. Consider trying to insure a city against the risk of nuclear or biological warfare. It would take thousands or millions of major cities paying premiums to offset the cost of one realized risk. There aren't enough cities in the world to make it work.

Finally, each insurance consumer has an individual risk preference, time preference and price point for insurance. As the variety in demands increases, the potential benefit from the law of large numbers decreases because fewer people want similar types of coverage.