How Insurance Works: The Math of Sharing Risk

1.Sharing Risk and the Law of Large Numbers

No one can know in advance who will have an accident, but if you gather a large group of people, you can predict fairly accurately roughly how many of them will. Insurance relies on this fact. People who face similar risks each put in a small amount of money ahead of time, and that money is used to cover the losses of those who actually suffer an accident. A large loss that would be hard for one person to bear turns into a small cost that everyone shares. This is called risk pooling.

The tendency for the actual number of accidents to come closer to the expected rate as the group grows is called the law of large numbers. You can see it with the standard deviation from Math Lesson 8. If the number of people grows 100 times, the swing in the number of accidents (the standard deviation) grows only 10 times, so the swing relative to the average shrinks to one tenth. That is why the more policyholders there are, the more steadily premiums can be set.

2.How Premiums Are Built: An Expected-Value View

Expected value is the sum of "each possible outcome × its probability": the average you would get if the same situation were repeated many times. The core of a premium is the amount the insurer expects to pay out to one person on average, in other words the expected value of the loss. This part is called the pure premium (risk premium).

But the insurer also has to pay staff, cover sales costs, manage contracts, and keep a cushion in case there are more accidents than expected. The part added for these costs is called the loading (expense charge). So the premiums policyholders pay are larger than the claims they receive on average, and from the policyholder's side the expected value of insurance is usually a loss (negative). This is not a trick; it follows from the way insurance is built.

3.When Insurance Makes Sense Despite a Negative Expected Value

If the expected value is negative, is insurance always a bad deal? Expected value is only an average; you also have to think about how hard a single outcome would hit your life. In the example above, paying an extra 50,000 won a year is a small cost you can handle, but a 20,000,000 won loss arriving all at once could wreck your finances or force you into debt. Turning a large loss you could not absorb into a small cost you can afford: that is why insurance can be sensible.

On the other hand, if a loss is small enough to cover from the emergency fund you built in Lesson 3, then on average you come out ahead by bearing it yourself instead of paying the same expected-value loss every year. This is sometimes called self-insurance. A deductible works on the same principle. If the policyholder agrees to pay the first part of any loss, small claims drop and the premium goes down. For example, with a 1,000,000 won loss and a 200,000 won deductible, the insurer pays 800,000 won.

4.Protection vs. Savings Policies, and Surrender Values

Broadly, insurance can be divided into protection policies, which guard against risks, and savings policies, which add a savings feature on top of the coverage. A savings policy's premium includes both the cost of coverage and expenses, so not all of the money you pay is saved. When you compare one with the compound-interest calculations from Lesson 1, base the comparison on the amount actually set aside.

If you cancel a policy partway through, you receive a surrender value. Many policies charge a large share of expenses early in the contract, so if you cancel in the early years, the surrender value may be far less than the premiums you paid, or almost nothing. Some protection policies reduce or eliminate the surrender value in exchange for a lower premium. Before signing up, check the table of surrender values by cancellation date and think about whether you can keep paying the premiums for a long time.

5.Renewable vs. Level-Premium Policies, and the Duty of Disclosure

A renewable policy resets the premium at set intervals to reflect your age and risk level at the time. The premium starts low, but because the chance of an accident rises with age, it can go up at renewal. A level-premium (non-renewable) policy charges the same premium throughout the payment period, but because you pay in advance for the higher risks of later years, the starting premium tends to be higher. There is no single right answer; it comes down to weighing the total amount paid against your future ability to pay.

The duty of disclosure is the obligation, when you apply, to answer truthfully about important matters the insurer asks about, such as your health and occupation. Premiums are set based on each applicant's level of risk, so if you give false information or leave something out, the contract may later be canceled or a claim may be denied. Even if a salesperson tells you "you don't need to mention that," the rule is to answer the questions on the form truthfully yourself.

6.Questions for Working Out the Coverage You Need

What coverage you need differs from person to person, and this lesson does not recommend any particular coverage. Instead, try answering the questions below yourself; they will show what you need and what overlaps. If you already have insurance, list its coverage in a table to find overlaps and gaps.

• If this loss happened, could I cover it with my emergency fund and income, or would my finances collapse?
• Is there family that depends on my income?
• Does it overlap with coverage I already have (through work, public programs, or existing policies)?
• Can I keep paying the premium for years or even decades? How much would I lose if I canceled early?
• Have I checked the scope of coverage and the cases it does not cover (exclusions) in the policy terms?
• Do I need to bundle protection and savings in one product? Have I compared it with doing them separately?