Sometimes we can assess the probability of a state of nature by determining the frequency with which it has occurred in the past. This concept is known as objective probability. At other times, we may assess the probability of a state of nature by using subjective judgment, a concept known as subjective probability.
If we know the probability and payoff associated with each state of nature, we can find the probability distribution of the payoffs. We can also calculate the expected payoff, as well as the standard deviation and the variance, two measures of variability.
Risk preferences
To analyze decisions involving risk, we can apply the theory developed in Chapters 4 through 6, thinking of a consumption bundle as a list of the amount of each good consumed in each state of nature. Indifference curves represent the consumer's preferences for consumption in different states of nature.
With only two possible states of nature, a risk-averse consumer's preferred point on any constant expected consumption line lies on the guaranteed consumption line.
With only two possible states of nature, we can find the certainty equivalent of a risky consumption bundle by identifying the indifference curve that runs through the bundle and determining the level of consumption that corresponds to the point where the curve crosses the guaranteed consumption line.
The difference between a bundle's expected level of consumption and its certainty equivalent, known as the risk premium, reflects the psychological cost of exposure to risk.
At greater levels of risk aversion, indifference curves bend more sharply where they cross the guaranteed consumption line. The certainty equivalent of any risky bundle is lower and the risk premium higher with greater levels of risk aversion.
With only two possible states of nature, a risk-loving consumer's least preferred point on any constant expected consumption line lies on the guaranteed consumption line. For risk-neutral consumers, indifference curves coincide with the constant expected consumption lines.
Under some conditions, we can use an expected utility function to describe a consumer's risk preferences.
For expected utility, a concave benefit function implies risk aversion; a convex function implies riskloving preferences. A linear benefit function implies risk neutrality.
Insurance
If insurance is actuarially fair, a risk-averse consumer will purchase full insurance.
If insurance is less than actuarially fair, a risk-averse consumer will purchase partial insurance or no insurance at all. The amount of insurance purchased will depend on the degree of risk aversion. Those who are not very risk averse will purchase no insurance.
The value of insurance equals the difference between the certainty equivalent of the consumer's consumption bundle after purchasing insurance and the certainty equivalent of the bundle before purchasing insurance. The greater the risk aversion, the higher the value of the insurance.
Other methods of managing risks
One way to make a risky investment more attractive is to share the risk by dividing it among several people. Companies can expand the opportunities for risk sharing by issuing equity shares. As long as an investment's expected payoff is positive, even an extremely risk-averse person will benefit from taking a small share of it.
Hedging reduces risk, because when the payoffs from two activities are negatively correlated, the gains offset the losses.
Diversification reduces risk because it creates opportunities for gains to offset losses, raising the likelihood of intermediate outcomes. The risk-reducing effects of diversification are smaller when the payoffs are more positively correlated, making offsetting gains and losses less likely. One of the most important functions of the stock market is to allow people to diversify their risky investments by purchasing small interests in many companies, instead of betting everything on the performance of a single business.
Better information about probable events leads to better decisions, reducing the likelihood of a loss. The value of information equals the difference between the certainty equivalent of the risky outcome when an individual is informed and the certainty equivalent of the risky outcome when he is uninformed.
