In my first post on insurance principles I introduced the inequality

Expected payouts that has to hold for the industry as a whole and also for any one insurance company to stay solvent (yes you can collect interest on premiums, so consider this appropriately adjusted for timing of cash flows already).But what about the individual person or company seeking insurance? It follows immediately that this inequality must also bet true for many of those seeking insurance. Why? Well if the opposite were true for every insured, i.e. Expected payouts >= Expected premiums, then clearly the insurance scheme wouldn’t meet the initial inequality in aggregate.So that raises an interesting question. Why do people or companies buy insurance if generally the expected payout is less than the expected premiums they have to pay? It must be the case that there are insurance buyers for whomExpected payouts and yetExpected benefit >= Expected costNow that seems like a direct contradiction at first but not once we understand that the first inequality is in dollar terms and the second one is in terms of utility for individuals and equity value for companies. So we can rewrite the second inequality asExpected utility(payouts) >= Expected utility(premiums)Or substitute equity(payouts) and equity(premiums) for companies. Put differently the first inequality is an expectation and the second is an expectation over a function of these values.This where we can use one of the truly beautiful mathematical results, known as Jensen’s inequality. It tells us thatE[f(x)] if f(x) is a concave function. This is a generalization of the idea that if you pick two points on a concave curve and connect them with each other by a straight line (a “secant line”) then the function lies everywhere above the line.So in order for an insurance purchase to make sense even when the expected payouts are less than the expected premiums it must be true that the utility function or equity value function is concave.Why is that often the case? Well for individuals tend to generally have a diminishing marginal utility of money which means they have a concave utility function. For corporations this seems less obvious. It starts to make sense if you consider that when faced with too big a negative event they may not have the liquidity to continue operating as normal and may forfeit business or in the extreme face bankruptcy (wiping out all equity value). So for large enough negative events the equity value function is definitely concave.The existence of successful insurance schemes thus depends on the concavity of the utility / equity value function. This concavity is also sometimes referred to as risk aversion. This is critical to understand both from the perspective of insurance providers and insurance buyers.To see just how important this is let’s recognize that all insurance companies have administrative cost and also have marketing expenses. In a low returns on capital environment these are likely to outweigh any investment gains the company can make on premiums that are collected before payouts occur. That means almost certainly that most insurance schemes are not “fair” where a fair scheme would be one in which Expected payouts = Expected premiums, instead Expected payouts < Expected premiums (also: for publicly traded or otherwise for profit instead of mutual insurance carriers lets not forget about profits).Conversely from an insurance buyer perspective the concavity requirement means that it is easy to be over insured. A great example of that are the various gadget insurances out there such as mobile phone insurance. These make sense only if you really are risk averse over the relevant amounts. Sadly for many Americans that is the case because they could not come up with the cash needed to buy a replacement on the spot and hence their utility function is extremely concave. But if you have sufficient cash so that purchasing a replacement device is not a big deal at all then you should probably decline to buy this kind of insurance (learning this may take a bit of emotional discipline — since if you will be really upset then your utility function is in fact made concave by your emotional response).So the key takeaway: Without risk aversion insurance math fundamentally would not add up. The concavity of the utility or equity value function is the central reason for the existence of insurance.