Full Insurance Theorem Risks to Wealth

Motives  Practical risk management  Analysis: base case  First example of general principles

Fair pricing I  Fair price = expected value of loss  Law of large numbers says a firm can survive with nearly fair pricing.  Competition of insurers says it must do so.

Fair pricing II  Finance theory says investors diversify risk.  They compete.  Therefore they are not rewarded for bearing insurance risks. They get only the expected value.  Details: insurance risks are not (very much) correlated with the stock market.

Ignore interest for now  Separate it from risk.  Absent risk, clients pay insurers first, receive it back later.  Insurers should pay some interest, just like banks.  They do, in fact.

The consumer  Suppose that a consumer  is a risk-averse, expected utility maximizer  with utility independent of state  and the same subjective probabilities as the insurer  That means

Two states  The pi’s are the same as for the insurer.  The U’s are not of the form  as might happen if state s involved illness or death.

Risk aversion  Risk aversion means the second derivative of the vN-M utility function is negative. That means the first derivative is decreasing, which is the same as decreasing marginal utility of wealth.

Note on the use of risk aversion  Since marginal utility is always decreasing, each value of marginal utility corresponds to exactly one value of wealth. MU w

Insurers price fairly  i.e.., price of state-s wealth is the probability of state-s Suppose state-s has probability =.5, then a dollar for state s costs fifty cents.

the world in the model Time zeroTime one s=1 s=S Make insurance contracts, i.e., trade state-contingent claims Execute the contracts and consume.

Full insurance theorem  Suppose that the consumer is a risk- averse, expected utility maximizer with utility independent of state and having the same subjective probabilities as the insurer. Suppose further that the insurer prices fairly. Then the optimum insurance for the consumer is full insurance

= endowed risk non loss state loss state equation of the budget constraint:

non loss state loss state certainty line slope = MRS indifference curve

Proof: At the consumer optimum, MRS = price ratio. Recall that MRS is the ratio of marginal utilities. Specifically The price ratio is the ratio of probabilities, implying

proof continued Divide by the probability ratio on both sides. Result: Then which implies which means full insurance. Q.E.D.