Question

Problems (see the end of each question for its respective point value) 1. Bob's utility preferences with respect to wealth (W) can be described by the utility function U-0.5. Assume that Bob's current wealth level equals $400. Also assume that Bob faces a 20 percent chance of incurring a fire in the next year that would totally destroy his home (valued at $300) and an 80 percent chance of suffering no fire loss during the year. If Bob were charged the actuarially fair premium (AFP) to cover this fire exposure by his insurer, how much money would Bob pay for insurance? What will Bob's utility level be if he chooses to buy insurance at a price equal to the APP? c. What will Bob's utility level be if he chooses to remain uninsured against the fire exposure? d. What would the price of fire insurance be it Bob were indifferent between buying insurance versus remaining uninsured against the fire exposure? (6 points)

Answer 1

a)

Acutuarially fair premium=Probability of loss*Amount of loss=0.2*300=$60

b)

If he goes for premium,

Wealth in case of damage=400-300-60+300=$340

Wealth in case of no damage=400-60=$340

We are given that U=W^0.5

Expected Utility=p*U(340)+(1-p)U(340)=U(340)=340^0.5=18.4391

c)

Wealth in case of damage=400-300=$100

Utility in case of damage=U(100)=100^0.5=10 utils

Probability of damage=p=0.20

Wealth in case of no damage=$400

Utility in case of no damage=U(400)=400^0.5=20 utils

Probability of no damage=1-p=0.80

Expected Utility=p*U(100)+(1-p)U(400)=0.2*10+0.8*20=18 utils

d)

Let the price of insurance be X. We are given

U(400-X)=18

(400-X)^0.5=18

400-X=324

X=400-324=$76