Why buy insurance?

Question

I don't understand why people buy insurances when they know that odds are is not their favor. Is it not better bet to stay uninsured, and save money? I am not talking about house insurance. About things of which cost can be covered by one. Laptop, phone etc.

Answer 1

Because people are Risk Averse.

Suppose that you own an asset worth $10,000 to you.

Suppose that each year, the asset has 1% chance of being stolen (or completely broken).

The expected value is 99% x 10,000 + 1% x $0 = $9,900. This is the average outcome if you do not buy insurance.

Now consider two mutually exclusive outcomes:

1. 99% chance of keeping $10,000 and 1% chance of losing everything
   (expected value: $9,900) 
2. 100% chance of keeping $9,900 (expected value: $9,900) 

Everyone would choose option 2, even though the expected values are the same.

Option 2 is an insurance that cost $100 (Actuarially fair, aka the odds are fair).

Now suppose the insurance costs $150 instead of $100 (despite that the bad probability is still 1%). You are faced with

1. 99% chance of keeping $10,000 and 1% chance of losing everything
   (expected value: $9,900) 
2. 100% chance of keeping $9,850 (expected value: $9,850) 

Some people would still choose option 2, even though the expected value is actually lower.

The $50 is called Risk Premium, which people are willing to pay in order to avoid uncertainty. The odds are unfair, but the Risk Premium has its value.

That being said, competition between insurance companies would drive down the premium until the insurance is close to actuarially fair, but they have cost to cover (sales, administration, etc), making the odds "unfair".