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Why buy insurance?
Discussions around expected values and risk premiums are very useful, but there's another thing to consider: cash flow. Some individuals have high value assets that are vital to them, such as  transportation or housing.  The cost of replacing these assets is prohibitive to them: their cashflow means that their rate of saving is too low to accrue a fund large enough to cover the asset's loss. However, their cashflow is such that they can afford insurance. While it may be true that, over time, they would be "better off" saving that money in an asset replacement fund, until that fund reaches a certain level, they are unprotected. Thus, it's not just about being risk averse; there are some very pragmatic reasons why individuals with low disposable income might elect to pay for insurance when they would be financially better off without it.