SUMMARY

1. The expected value of a random variable is the weighted average of all possible values, where the weight corresponds to the probability of a given value occurring.

2. Risk is uncertainty about future events or states of the world. It is financial risk when the uncertainty is about monetary outcomes.

3. Under uncertainty, people maximize expected utility. A risk-averse person will choose to reduce risk when that reduction leaves the expected value of his or her income or wealth unchanged. A fair insurance policy has that feature: the premium is equal to the expected value of the claim. A risk-neutral person is completely insensitive to risk and therefore unwilling to pay any premium to avoid it.

4. Risk aversion arises from diminishing marginal utility: an additional dollar of income generates higher marginal utility in low-income states than in high-income states. A fair insurance policy increases a risk-averse person’s utility because it transfers a dollar from a high-income state (a state when no loss occurs) to a low-income state (a state when a loss occurs).

5. Differences in preferences and income or wealth lead to differences in risk aversion. Depending on the size of the premium, a risk-averse person is willing to purchase “unfair” insurance, a policy for which the premium exceeds the expected value of the claim. The greater your risk aversion, the higher the premium you are willing to pay.

6. There are gains from trade in risk, leading to an efficient allocation of risk: those who are most willing to bear risk put their capital at risk to cover the losses of those least willing to bear risk.

7. Risk can also be reduced through diversification, investing in several different things that correspond to independent events. The stock market, where shares in companies are traded, offers one way to diversify. Insurance companies can engage in pooling, insuring many independent events so as to eliminate almost all risk. But when the underlying events are positively correlated, all risk cannot be diversified away.

8. Private information can cause inefficiency in the allocation of risk. One problem is adverse selection, private information about the way things are. It creates the “lemons problem” in used-car markets, where sellers of high-quality cars drop out of the market. Adverse selection can be limited in several ways—through screening of individuals, through the signaling that people use to reveal their private information, and through the building of a reputation.

9. A related problem is moral hazard: individuals have private information about their actions, which distorts their incentives to exert effort or care when someone else bears the costs of that lack of effort or care. It limits the ability of markets to allocate risk efficiently. Insurance companies try to limit moral hazard by imposing deductibles, placing more risk on the insured.