Risk aversion is a trait common to most individuals. This is why we buy insurance. Most people must be compensated for accepting additional risk. This is why investors demand higher returns on riskier investments. A risk averse person must be compensated for accepting additional risk even if the expected value is the same. A risk neutral person only cares about expected value.
An example: A person is give the choice between two scenarios. In scenario one they can flip a coin and win $10 for heads or lose $10 for tails. In scenario two they can flip a coin and win $10,000 for heads or lose $10,000 for tails. The expected value of both is $0.
Remember that expected value is frequency x amount. The frequency (probability) of a coin flip is 50% (.5).
So, (.5 x $10) + (.5 x -$10) = $0 and (.5 x $10,000) + (.5 x -$10,000) = $0
The expected value of each is $0. They are identical in expected value, but a risk averse person would need to be compensated for the additional risk. This is because the loss of $10,000 would hurt them more than the gain of $10,000 would benefit them. The required compensation could come in the form of better odds, higher winnings or lower losses. The amount of compensation depends on the level of risk aversion associated with the individual. This also works in the other direction as well, the more risk averse a person the more they are willing to pay to reduce risk (uncertainty).
When we pay insurance premiums, we pay more than our expected cost (ignoring adverse selection). To put it simply, in addition to our expected cost we pay the insurance company other fees including administrative and processing fees, loadings, costs of adverse selection, investor returns and for residual uncertainties. We pay more than our expected cost for insurance because the cost of paying extra money over time hurts us less than the possibility of an uninsured accident, which could demand large amounts of available capital. The effects of such an event could be financially devastating. This additional cost above our expected cost is worth the ability for us to convert the uncertainty of not being able to financially manage the situation into the certainty of being able to manage it.