The Lee family doesn’t know how big its medical bills will be next year. If all goes well, it won’t have any medical expenses at all. Let’s assume that there’s a 50% chance of that happening. But if family members require hospitalization or expensive drugs, they will face medical expenses of $10,000. Let’s assume that there’s also a 50% chance that these high medical expenses will materialize.

In this example—which is designed to illustrate a point, rather than to be realistic—the Lees’ medical expenses for the coming year are a random variable, a variable that has an uncertain future value. No one can predict which of its possible values, or outcomes, a random variable will take. But that doesn’t mean we can say nothing about the Lees’ future medical expenses. On the contrary, an actuary (a person trained in evaluating uncertain future events) could calculate the expected value of expenses next year—the weighted average of all possible values, where the weights on each possible value correspond to the probability of that value occurring. In this example, the expected value of the Lees’ medical expenses is (0.5 × $0) + (0.5 × $10,000) = $5,000.

To derive the general formula for the expected value of a random variable, we imagine that there are a number of different states of the world, possible future events. Each state is associated with a different realized value—the value that actually occurs—of the random variable. You don’t know which state of the world will actually occur, but you can assign probabilities, one for each state of the world.

Let’s assume that P₁ is the probability of state 1, P₂ the probability of state 2, and so on. And you know the realized value of the random value in each state of the world: S₁ in state 1, S₂ in state 2, and so on. Let’s also assume that there are N possible states. Then the expected value of a random variable is:

In the case of the Lee family, there are only two possible states of the world, each with a probability of 0.5.

Notice, however, that the Lee family doesn’t actually expect to pay $5,000 in medical bills next year. That’s because in this example there is no state of the world in which the family pays exactly $5,000. Either the family pays nothing, or it pays $10,000. So the Lees face considerable uncertainty about their future medical expenses.

But what if the Lee family can buy health insurance that will cover its medical expenses, whatever they turn out to be? Suppose, in particular, that the family can pay $5,000 up front in return for full coverage of whatever medical expenses actually arise during the coming year. Then the Lees’ future medical expenses are no longer uncertain for them: in return for $5,000—an amount equal to the expected value of the medical expenses—the insurance company assumes all responsibility for paying those medical expenses. Would this be a good deal from the Lees’ point of view?

Yes, it would—or at least most families would think so. Most people prefer, other things equal, to reduce risk—uncertainty about future outcomes. (We’ll focus here on financial risk, in which the uncertainty is about monetary outcomes, as opposed to uncertainty about outcomes that can’t be assigned a monetary value.) In fact, most people are willing to pay a substantial price to reduce their risk; that’s why we have an insurance industry.

But before we study the market for insurance, we need to understand why people feel that risk is a bad thing, an attitude that economists call risk aversion. The source of risk aversion lies in a concept we first encountered in our analysis of consumer demand, back in Chapter 10: diminishing marginal utility.