Fall 2008 Version Professor Dan C. Jones FINA 4355 Class Problem

Risk Management and Insurance: Perspectives in a Global Economy 19. The Economic Foundations of Insurance Professor Dan C. Jones FINA 4355 Class Problem

3 Study Points Expected utility and the demand for insurance Insurance supply: characteristics of ideal insurable exposures

Expected Utility and the Demand for Insurance This section extends the discussion in Chapter 2.

5 Insurance Demand with Premium Loadings (Figure 19.1)

6 Insurance Demand in Markets with Moral Hazard Ex-ante moral hazard Insurance fraud Ex-post moral hazard Insurer responses to moral hazard Controlling the marginal benefit of being careful or the marginal cost of being careless Loss sharing through deductible and coinsurance Insight 19.2 Rewarding insureds who undertake loss preventing activities Retrospective or experience rating Discussion in page 482

7 Deductible and Coinsurance (Insight 19.2) Deductible Coinsurance Deductible Stop loss Insurer’s Share Insured’s Share

8 Insurance Demand in Markets with Adverse Selection The effect of adverse selection on insurance markets Insurer responses Eliciting more information about applicants and insureds Designing insurance contracts that encourage insureds with differing risk types to self-select into the most appropriate risk class Table 19.1 Discussion in pages

9 Substitutes for Insurance Substitutes Higher insurance prices tend to decrease the amount of market insurance purchased by risk-averse individuals and increase the amount of loss reduction “bought.” Complements Loss prevention and market insurance are complements, not substitutes. An investment in loss prevention may actually raise the amount of risk that a risk-averse person faces and therefore raises the demand for market insurance.

10 Why Corporations Purchase Insurance Already covered are: Managerial self-interest Corporate taxation Cost of financial distress Capital market imperfections Other reasons include: Insurers may offer real service efficiencies. Regulated industries have a higher demand for insurance. The purchase of some types of insurance is required by government. Discussion continues from Chapter 2

Insurance Supply: Ideal Insurable Exposures

12 “Ideal” Insurable Exposures Presence of numerous independent and identically distributed (IID) units Unintentional losses Easily determinable losses as to time, amount, and type Economically feasible premium

13 Numerous IID Exposure Units Each exposure unit in an insurance pool represents a possible liability for the insurer. In the ideal case, these exposure units should be IID. Two random variables (e.g., exposures units) are independent if the occurrence of an event affecting one of the variables has no affect on the other variable. The independence property is important because it affects how well insurers can diversify the systematic risk of their insurance pools. Random variables are identically distributed if the probability distributions of two random variables prescribe the same probability to each potential occurrence.

14 Numerous IID Exposure Units The law of large numbers Variance and standard deviation as measures of dispersion Effects of pooling IID exposures units – A fire insurer would be interested in the following four statistics: The total amount of losses expected to be paid during the year; The standard deviation of the total loss distribution (to understand the riskiness inherent in providing this insurance) The average loss (to determine the premium to be charged); The standard deviation of the average loss distribution (to determine the risk each exposure unit contributes to the risk class) Discussion in pages

15 Average Loss Distribution of an Insurance Pool (Figure 19.2)

16 Probability of Ruin (Figure 19.3)

17 Accidental Losses Losses should be accidental or unintentional We made this point earlier in the context of moral hazard From a societal viewpoint, it clearly is not good public policy to allow policyholders to collect insurance proceeds for internationally causing losses. Some losses occur naturally over time. It is usually less expensive to budget for possible repair or replacement of the property than to purchase insurance.

18 Determinable Losses The details of the insured loss – time, place, and amount – must be verified and the payment amount agreed upon by the insured and the insurer. The costs of verifying loss details should be relatively low for insurance to be offered at an economically feasible premium.

19 Economically Feasible Premiums On the one hand, rational risk-averse individuals will pay a maximum premium equal to the expected value of the loss plus the risk premium. On the other, the owners of private insurance companies require that insurance rates be enough to give them a competitive return on their investments. Factors affecting this range Competition in the market Threats of new entrances Price Threat of alternative products and substitutes The bargaining power of consumers The degrees of risk attitudes of consumers

Discussion Questions

21 Discussion Question 1 Hannah owns a home worth US$50,000, which is subject to the risk of fire. The probability of a fire is 25 percent and the amount of damage due to the fire would be US$40,000. Assume Hannah’s utility function is the square root of wealth. Hannah has been offered full insurance at a cost of US$13,000. Will she buy the insurance? Why or why not?

22 Discussion Question 1

23 Discussion Question 2 A frequency distribution shows the number of accidents that an insurer can expect from each exposure unit in its insurance pool during the year. Use the information provided below to answer the following questions: Calculate the expected number of accidents a single exposure unit could expect during the next year. Calculate the standard deviation of the number of accidents a single exposure unit could expect during the next year. Calculate the standard deviation of the number of accidents.

24 Discussion Question 4 Consider the following lotteries, x, y and z: Calculate the expected value of each gamble. Assuming a risk-averter’s utility function of wealth is given below. Calculate the expected utility of each gamble for a person who has an initial wealth level of 10. Which gamble does this person prefer? Why?