# Discussing Insurance, Risk Tolerance and Game Theory

• Leo Liu
In summary, people buy insurance to mitigate potential financial losses. They do this because the costs of the potential losses are often much higher than the amount of money they would make from the insurance.
Leo Liu
The title is my question, and this post is meant to be a place for discussions. I think it is somewhat related to game theory and microeconomics, but I am not sure how to accurately categorize it.

Let me give you an example. Let us suppose that Mr. Goose, an ideal person with a national average probability of getting serious chronic diseases, lives in a country which doesn't have universal healthcare. A salesperson is trying to sell him an insurance that covers the costs incurred by the treatments for serious chronic diseases. Assuming that the chances of an average person's developing a serious chronic disease is 10% in that nation, which is the same as the value for an average insured, that the average cost of treatments is $100,000 and that the life-long insurance is priced at$11,000. We shall also assume that Mr. Goose can only afford to pay $50,000 max for the possible treatments. In this case, the insurance company is likely to make$1,000 from Mr. Goose while Mr. Goose's expectation is -$1000. However, most people who are in similar situations as Mr. Goose still choose to let their insurance companies rob them of their money. Why? I managed to come up with a theory myself though I know it is imperfect. I think people like Mr. Goose still buy insurances because their risk tolerance is lower than insurance companies, and the latter are more likely to obtain a better model for their business due to the law of large numbers. If Mr. Goose didn't buy the insurance, he would have difficulty in paying the extra$50,000 for the treatments although the chances are relatively low. However if the payer were the insurance company, while it would lose $89,000, the insurance company still makes money from other insured. As a result, when the pool of the insured people is large enough, the insurance company will almost certainly gain profit because the math works, and due to the large sum of money the company has, it possesses higher tolerance against financial risk. Above is my theory and reasonings. How do they sound? Can you tell the reasons why people buy insurances? Last edited: PeroK Your math is wrong for Mr. Goose. The$50k he cannot afford will have additional costs associated with it (loan interest, life disruption, etc.). Some large, wealthy organizations self insure using your logic because they can afford the $100k. Leo Liu caz said: Your math is wrong for Mr. Goose. The$50k he cannot afford will have additional costs associated with it (loan interest, life disruption, etc.). Some large, wealthy organizations self insure using your logic because they can afford the $100k. Yeah I am making a highly simplified model for my question, as unnecessary complexities will do no good to our understanding of it. It's like an economic model in a college textbook or a model for a physics problem which requires you to neglect air friction and Coriolis effect. If you measure loss as the expectation value, that defines your objective function for minimalization, and covariances between events to measure the risk, then you get a well-defined optimization problem. The clue is, that the covariances can be used for minimization. It is called risk diversification. You get a lower expectation value the more participants share the fond. Now you can simply calculate whether the costs for managing the fond are higher or lower than the yield by covariances. This defines the price for insurance policies up to which participants are willing to join the fond. This is the same mechanism that is used by fond managers for maximizing the expectation value of yields. What also has to be taken into account is, that the sum of many payings for insurances is still far below the costs of a possible hazardous event. This is the same reason why people take part in lotteries: They have basically no chance to win, but the costs are so low, that they are paying them to buy a very small chance, despite the fact that the expectation value is negative. Astronuc, Leo Liu and BWV I think you have oversimplified things. I have insurance, but I do not buy extended warranties because I know what I can afford to lose. There is a large literature on how people perceive risk, expectation values and future vs current liabilities. It is known that people avoid potential losses more than potential gains with identical expectation values. Last edited: Leo Liu fresh_42 said: What also has to be taken into account is, that the sum of many payings for insurances is still far below the costs of a possible hazardous event. They have also priced in limited liability, government bailouts and skipping town. In economics utility also enters it. If the odds of my$300K house burning down (say I have no mortgage) in anyone year period is 1/1000, so the expectation value is $300. Would you pay more than that to protect yourself from the loss?(forget about all the other damage possible- the house is either OK or a total loss) Most people would pay more than the expectation value to protect themselves from a catastrophic loss Why should the insurance company underwrite policies at loss? If they sold you the policy for$300, they would go out of business as they have costs - employees, office space etc.

the offsetting factor will be the return on the premiums received - as in a diversified pool in this example would have a loss rate of 0.10% per year. The company would invest the premiums received and most years earn a positive rate of return. So the pool earns a market rate of return less expected losses and overhead costs.

But who owns insurance companies? largely retirement vehicles, college endowments and charitable foundations. The insurance companies have a fiduciary duty to these shareholders to provide the, with a return on their savings. The capital for insurance companies is not free. They must either borrow it by issuing bonds or offer ownership by issuing stock. Why would anyone invest their savings in an insurance company to subsidize their overhead, borrowing costs and expected losses? The weighted average of their borrowing cost and the rate of return demanded by equity investors is the cost of the insurance company’s capital

so in reality, the insurance company will underwrite policies at premium rates sufficiently above the expectation value plus costs to cover their cost of capital

Last edited:
PeroK and Leo Liu
@Leo Liu in the UK people buy car insurance because it's illegal to drive a car while uninsured. Go figure that one!

Leo Liu
Why should we buy insurance policies when insurance companies profit from them?

Why should we buy apples when grocery stores profit from them?

russ_watters and BillTre
Why should we buy apples when grocery stores profit from them?
Because they are delicious and are rich in nutrients. I can't see how health insurances are healthy though .

PeroK
Seems we have come to a dead-end, which had to be expected.

There is an easy answer, that sounds a bit more professional than the grocery store but is of the same quality: We do not need insurances. However, there is obviously a demand. Hence, there are companies that attempt to satisfy this demand. That is how our economies work. First course, first lecture.

Many answers have been given, and except for the concept of a solidarity community, which is another possible reason, all have likely been mentioned.

For further studies, I recommend reading the detailed Wikipedia entry
https://en.wikipedia.org/wiki/Insurance
and the 87 references in that article alone.

Although economics isn't our core competence, we might be able to address specific questions, but we cannot cover what normally is a course at a university.

Leo Liu

## 1. What is the purpose of discussing insurance?

The purpose of discussing insurance is to understand the concept of risk and how it can be managed through various insurance policies. Insurance helps individuals and businesses mitigate financial losses by transferring the risk to an insurance company in exchange for a premium.

## 2. What is risk tolerance and why is it important in insurance?

Risk tolerance refers to an individual's or organization's ability and willingness to take on risk. It is important in insurance because it helps determine the level of coverage and premiums that are suitable for a particular person or business. Those with a higher risk tolerance may opt for less coverage and lower premiums, while those with a lower risk tolerance may choose more coverage and higher premiums.

## 3. How does game theory apply to insurance?

Game theory is the study of strategic decision-making in situations where the outcome of one's choices depends on the choices of others. In insurance, game theory can be applied to analyze the behavior of insurance companies and policyholders. It helps determine the most advantageous strategies for both parties, such as setting premiums and filing claims.

## 4. What are some common types of insurance policies?

Some common types of insurance policies include life insurance, health insurance, property insurance, and liability insurance. Life insurance provides financial protection in the event of a person's death, health insurance covers medical expenses, property insurance protects against damage to physical assets, and liability insurance covers legal costs and damages in the event of a lawsuit.

## 5. How can individuals and businesses manage their insurance needs?

Individuals and businesses can manage their insurance needs by assessing their risk tolerance, determining the types and levels of coverage they need, comparing policies and premiums from different insurance companies, and regularly reviewing and updating their policies as their needs and circumstances change.

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