Insurance

What is insurance? Consider a club 100 members About the same age, about the same lifestyles About once a year one of the members gets sick and incurs expenses of $1,000. Club collects $10 from each member each year. Invests it somewhere to maintain or increase its value. Pays it out to members who file claims.

What has happened? Insured pay $10 per year, guaranteed, to avoid the possibility of having to pay $1,000. Although outlays for an individual may be highly variable, Outlays for a group are generally rather predictable. The “Law of Large Numbers” suggests that as group size increases, the distribution of the average rate of illness will collapse around the “true” probability of the illness.

Insurance Terms Premium - $X premium for $Y of coverage. Coinsurance and Copayment - The insured person must pay the loss. –% paid is the coinsurance rate (varies from 0 to 100%). –amount paid is the copayment. Deductible - Some amount may be deducted from the payment to the insured person, irrespective of coinsurance. Why coinsurance and deductibles? Discuss.

Risk and Insurance What is “expected value?” What’s the expected value of a coin flip that pays $1 for heads and 0 for tails. A> Probability of heads * $1 + Probability of tails * 0. E = 0.5 * * $0 = $0.5 How much would you pay to play this kind of game? Why do we care? A> Because insurance is based on expected losses!

Marginal Utility of Wealth and Risk Aversion Go back to earlier example. Would you bet $50 on a coin flip that would give you either 0 or $100. Would you bet $5000 on a coin flip that would give you either 0 or $10000? Why? Consider a utility function with wealth. Wealth Utility

Marginal Utility of Wealth and Risk Aversion Suppose wealth is 10 (thousand). It gives him U = 140 Wealth Utility Suppose wealth is 20 (thousand). It gives him U =

Expected wealth is due to risk Wealth Utility E (U) = (Prob. Healthy * Utility of Wealth if Healthy) + (Prob. Ill * Utility of Wealth if Ill) E (U) = Prob. Healthy * Utility of 20 + Prob. Ill * Utility of 10 We’re on the maroon line rather than the blue curve BECAUSE of RISK

Expected wealth is due to risk Wealth Utility We see that total utility is increasing, but what about marginal utility We see that it is decreasing. Why? In this example, losses bother us more than gains, help us, so we lose utility due to risk. Wealth Mgl. Utility

Insurance puts us on the Blue Line Wealth Utility Suppose the probability of illness is E (U) = Prob. Healthy * Utility of 20 + Prob. Ill * Utility of 10 E (U) = 0.95 * * 140. E (U) = 197. BUT, without risk, Utility would be 199. Wealth Mgl. Utility Insurance  risk (you end up at wealth = 19.5 and U=199 whether you’re sick or not). 19.5

Does everyone buy insurance? Depends on marginal utility of wealth. Some people “love” risk. How would we draw this? Wealth Utility Wealth Mgl. Utility How would we draw curves for someone who was “risk neutral.”

How much insurance? E (U) = (Prob. Ill * Utility of Wealth if Ill) + (Prob. Healthy * Utility of Wealth if Well) (11.3) If insured, new wealth = Wealth - insurance premium new wealth = W - aq q is the value of the insurance purchased a is the fraction paid, called the premium Wealth if ill = [(W - L) + q] - aq = W - L + (1-a)q Wealth if well = W - aq

How much insurance? E (U) = (Prob. Ill * Utility of Wealth if Ill) + (Prob. Healthy * Utility of Wealth if Well) (11.3) Wealth if ill = [(W - L) + q] - aq = W - L + (1-a)q Wealth if well = W - aq So we put the maroon stuff into the blue equation to get. E (U) = [Prob. Ill * Utility (W - L + (1-a)q] + [Prob. Healthy * Utility (W - aq)] [1] [2]

How much insurance? E (U) = [Prob. Ill * Utility (W - L + (1-a)q] + [Prob. Healthy * Utility (W - aq)] [1] [2] A purchase of 1 extra dollar of insurance increases utility in [1]. When he is ill, it will increase wealth by (1-a). The extra utility is (1-a) * MU (wealth when ill) * p. (Since wealth is lower, this is a relatively large marginal benefit). When he is health, it will decrease wealth by a. “Loss” of utility is a * MU (wealth when well) * (1-p).

Amount of Insurance A purchase of 1 extra dollar of insurance increases utility in [1]. When he is ill, it will increase wealth by (1-a). The extra utility is (1-a) * MU (wealth when ill) * p. (Since wealth is lower, this is a relatively large marginal benefit). When he is health, it will decrease wealth by a. “Loss” of utility is a * MU (wealth when well) * (1-p). Amount of insurance Mgl. Benefits, Mgl. Costs MB MC q*