Insurance

Expected Value and Expected Utility

Suppose you are confronted with an uncertain prospect. During the next year, you believe that there is a small likelihood (probability p that you will have an automobile accident. To make our lives easy, assume that your wealth is w₁, and that will also be your wealth at the end of the year if you do not have an accident. If you do have an accident, you believe that your wealth will fall to w₀<w₁, where the difference, w₁-w₀, consists of the cost of repairing your car and the amount that the legal system requires you pay to compensate your victims.

Ignorring discounting and timing issues and taking all of the above values as known, your expected wealth today is the weighted average of the two possible outcomes. The weight applied to the accident-reduced wealth, w₀, is the probability of an accident, p, and the weight assigned to w₁ is the probability of no accident, (1-p), so that the expected value of your wealth, which we will denote w^*, is given by

w^* = pw₀ + (1-p)w₁.

These values are illustrated graphically in figure 1. The figure suggests a relatively high likelihood of an accident (a low likelihood would place w^* close to w₁ ).

Figure 1: Expected Wealth

Notice that if you were not bothered by risk, you would be just indifferent between having w^* with certainty and the prospect of w₀ with probability p and w₁ with probability (1 -p). This means that you would be willing to pay an insurance premium of up to w₁ - w^* for a policy that paid off w₁ -w₀ in the event of an accident; with such a policy you are guaranteed w^* (w₁ - your premium) no matter what happens.

An insurance company is certainly not willing to accept less than w₁ - w^* for insuring you and people like you. If it insures enough people with the same prospect as yours, and that all of the prospects are independent, the company will end up paying out all its premiums in the form of compensation to those who do incur losses.

For example, if the prospect in question is that of totalling a $20,000 car, and if probability of a wreck in a year is 0.01, then w₁ is $20,000, w₀ is zero, and w^* is $19,800. If the the insurance company insures a large number of drivers, it can be reasonably confident that its $20,000 payouts will be made to close to one in one hundred of its policy holders, for a cost of $20,000 per 100 policyholders. It receives $200 from each policyholder and therefore just breaks even, if it has no costs other than payments for claims.

Real insurance companies do have costs other than claims payouts. This means that the premiums they charge must exceed the amount they pay out. Consumers, not surprisingly, pay something for insurance. The expected wealth of the consumer who does not insure exceeds the certain wealth of an ensured person. It must be, therefore, that given a choice between our uncertain prospect with expected value of w^* and, alternatively, w^* with certainty, consumers would opt for the certain alternative—they get more utility from the certain prospect.

Figure 2 graphs consumer utility for the two points w₀ and w₁. That is, u(w₀) is the utility you receive if you have suffered a clamity and u(w₁) is the utility you receive if you do not have to worry about that clamity. Each of these represents the utility of having a certain level of wealth with certainty. We can average these to levels of utility, using p and (1-p), as shown in the diagram. But we know that this expected utility is less that we would get from having w^* with certainty—the utility function for certain prospects that passes through (w₀,u(w₀)) and (w₁,u(w₁)) must lie above the line connecting the two points.

Figure 2: Expected Utility

The solid red line in Figure 2 illustrates such a utility functiion. Note first that the utility of receiving w^* with certainty, u(w^*), exceeds the probability-weighted average of u(w₀) and u(w₁). Note that this means that the utility function must be getting "flatter" in the sense that as wealth rises, each additional dollar of wealth raises utility by a smaller amount—the marginal utility of wealth is declining.

Figure 3: Expected Utility and Insurance Premiums

If the red line represents your utility, how much are you willing to pay for an insurance policy? Notice from the figure that the consumer is indifferent between the uncertain prospect the with expected value w^* and w´ with certainty. That is, a consumer starting with w₁ would be willing to pay up to w₁ - w´ to avoid the uncertain prospect.