《保险经济学》期末考试试题

1. An individual has the following utility function:

Her initial wealth is 10 and she faces the lottery

a) Compute the exact value of the certainty equivalent and of the risk premium.

b) Apply Pratt’s formula to obtain the approximation of the risk premium.

c) Show that with such a utility function absolute risk aversion is decreasing in wealth while relative risk aversion is constant.

d) If the risk becomes , compute the new risk premium as approximated by Pratt’s formula. Why is the approximated risk premium four times smaller than for?

2. Consider lottery distributed as (-10,1/3;0,1/3;+10,1/3) and a “white noise”distributed as (-5,1/2;+5,1/2). First generate lottery by attaching the white noise to the worst outcome of (i.e. –10) and then generate lottery by attaching the white noise instead to the best outcome of (i.e. +10).

a) Compute , .

b) Draw the cumulative distributions of and. Can you say that one is riskier than the other? Why or why not?

c) If a decision maker has a quadratic utility such as u(w)＝w－0.001w2, compute and

. Are you surprised by the fact that =?

d) Choose a utility function such that u"'＞0 and then show that<

3. An economy consists of two individuals, Sempronius and Jacobus, each with logarithmic utility. There are two equally likely states of nature. Sempronius has an endowment of 10-ducats worth of consumption in both states. Jacobus has an endowment of a contingent claim for 15-ducats worth of consumption in state 1, but only 5-ducats worth of consumption in state 2. Jacobus offers Sempronius the following trade: Jacobus will give Sempronius 3-ducats worth of consumption in the event that state 1 occurs in exchange for a promise from Sempronius to provide Jacobus with 2-ducats worth of consumption in the event that state 2 occurs.

(a) Are both individuals made better off by the proposed trade?

(b) Is the final contingent-consumption allocation resulting from this trade Pareto efficient?

(c) Describe the set of all Pareto-efficient allocations in this economy.

(d) How would your answer to part (c) above change if the probability of state 1 was p = 0.4, instead of one-half?

4. An individual owns assets of value W0＝10, which may suffer a random loss described by a discrete random variable:

this exercise).

b) What is the actuarially fair premium when a deductible D = 3 is selected? What happens to the premium when D = 6? Why doesn’t the premium fall by 50%?

c) For each deductible compute the coinsurance rate that yields the same premium.

d) Draw the cumulative distribution of final wealth first if D = 6 and then if the policy is characterized by the coinsurance rate that yields the same premium as D = 6. Reference to the integral condition shows that the policy with a coinsurance rate induces a riskier distribution of final wealth.