3.5 Risk Aversion and Price of Hedging Risk

Learning Objectives

  • In this section we focus on risk aversion and the price of hedging risk. We discuss the actuarially fair premium (AFP) and the risk premium.
  • Students will learn how these principles are applied to pricing of insurance (one mechanism to hedge individual risks) and the decision to purchase insurance.

From now on, we will restrict ourselves to the E(U) theory since we can predict behavior with it. We are interested in the predictions about human behavior, rather than just a description of it.

The risk averter’s utility function (as we had seen earlier in Figure 3.2 "A Utility Function for a Risk-Averse Individual") is concave to the origin. Such a person will never play a lottery at its actuarially fair premium, that is, the expected loss in wealth to the individual. Conversely, such a person will always pay at least an actuarially fair premium to get rid of the entire risk.

Suppose Ty is a student who gets a monthly allowance of $200 (initial wealth W0) from his parents. He might lose $100 on any given day with a probability 0.5 or not lose any amount with 50 percent chance. Consequently, the expected loss (E[L]) to Ty equals 0.5($0) + 0.5($100) = $50. In other words, Ty’s expected final wealth E (FW) = 0.5($200 − $0) + 0.5($200 − $100) = W0 − E(L) = $150. The question is how much Ty would be willing to pay to hedge his expected loss of $50. We will assume that Ty’s utility function is given by U( W )= W —a risk averter’s utility function.

To apply the expected utility theory to answer the question above, we solve the problem in stages. In the first step, we find out Ty’s expected utility when he does not purchase insurance and show it on Figure 3.6 "Risk Aversion" (a). In the second step, we figure out if he will buy insurance at actuarially fair prices and use Figure 3.6 "Risk Aversion" (b) to show it. Finally, we compute Ty’s utility when he pays a premium P to get rid of the risk of a loss. P represents the maximum premium Ty is willing to pay. This is featured in Figure 3.6 "Risk Aversion" (c). At this premium, Ty is exactly indifferent between buying insurance or remaining uninsured. What is P?

Figure 3.6 Risk Aversion

Key Takeaways

  • In this section, students learned that risk aversion is the key to understanding why insurance and other risk hedges exist.
  • The student should be able to express the demand for hedging and the conditions under which a risk-averse individual might refuse to transfer risk.

Discussion Questions

  1. What shape does a risk-averse person’s utility curve take? What role does risk aversion play in market demand for insurance products?
  2. Distinguish between risk premium and AFP. Show the two on a graph.
  3. Under what conditions will a risk-averse person refuse an insurance offer?