How Insurance Works: The Law of Large Numbers
Insurance is a financial product that helps individuals and businesses protect themselves from financial losses. At its core, insurance works by pooling risks. By spreading the risk among many, the insurance company can provide financial protection to each individual without any one person bearing the full burden of a catastrophic loss.
The Law of Large Numbers
The foundation of insurance is built on a key principle in statistics: the Law of Large Numbers. This law states that as the size of a group of insured people (the “pool”) increases, the actual results will become more predictable and stable, aligning closely with the expected outcome.
Let’s assume Greg is concerned about the financial impact of his death on his family. Greg is 35, healthy and wants to know his family will be cared for financially. But there is no way of knowing whether Greg will die during this time. Fortunately, the odds of him dying are fairly low but it is still possible.
Greg goes out and buys insurance to cover this risk.
Greg
What the Insurance Company Knows
The insurance company works with more people than just Greg. They work with thousands, tens of thousands or hundreds of thousands of people just like Greg. They take these people and group them together by characteristics and identify how many people in that group have died in the past. So, while they also don’t know if Greg will die during this timeframe, they do know how many people in Greg’s group will die with a fair degree of accuracy based on history and population data. The more individuals an insurance company insures, the more accurately it can predict the likelihood of events occurring, such as deaths or accidents, based on historical data.
They then use this data to set premiums to be able to cover the estimated claim payouts, their own expenses, and produce a profit.
A Simple Example
Greg has been grouped with 49 other 35-year-old in good health (a cohort). The insurance company knows that 1 person will die each year for the next 5 years (Ok, I admit that’s a terrible death rate but I’m dealing with small numbers so work with me here).
Assuming a $100,000 death benefit for each person, the life insurance company will need to have enough money to pay for a death benefit of $100,000 each year for the next 5 years.
For the first year, they can take the expected claim of $100,000 divided by the premium pool (50) to get the premium for that year to cover that claim which is $2,000.
However, the second year, they are assuming one person has died so the pool is only 49 people paying premiums but the claim is still $100,000. So the second year premium is $2,040.82 to pay the same benefit. This continues each of the years until the end of the period.
The insurance company then takes the total premiums for each year, adds on some profitability, and then applies that to each policy holder.
Fortunately, Greg lives on.
The Initial Cohort
The Final Cohort
Two Sides of the Same Coin
The previous example demonstrates how life insurance works. With life insurance, the coverage protects against the risk of death within a specified timeframe, whether it’s a fixed term, such as 10 years, or over a lifetime.
Similarly, ANNUITIES rely on the same Law of Large Numbers but focus on the opposite outcome—estimating the total payments they will need to make based on how many people remain alive.
In essence, life insurance provides the greatest benefit if you pass away early, whereas annuities offer the most value if you live to an advanced age, such as 105.
Summary
The Law of Large Numbers is the mathematical backbone of the insurance industry. It allows insurers to predict the probability of events, such as death or longevity, with a high degree of accuracy when dealing with large numbers of people. In life insurance, it helps insurers estimate how many policyholders will die in a given year, ensuring they collect enough in premiums to cover claims. In annuities, it enables insurers to predict how long they will need to make payments, ensuring the company can meet its obligations while remaining financially stable.
Through the power of risk pooling and statistical predictability, insurance companies can offer vital financial protection to individuals, providing peace of mind in exchange for a manageable premium.
The Law of Large Numbers applies to life insurance, annuities, social security, pensions, property and casualty insurance, and many more.