Question

Problem #3 Thousands of surfers spend their holidays every year on the Selection island. They are exposed to the risk of getting bitten by a shark. The probability of such event varies for different groups (depending on the frequency of entering the water) - it has a uniform distribution over the interval [0;0.24] (informally: in a large population every risk of getting bitten level between 0 and 24% occurs equally often). Individual probability of getting bitten is known to each surfer but to noone else. A bitten person must bear the costs of medical treatment amounting to $10,000. Because of risk aversion each surfer is willing to pay for a full insurance covering the entire treatment costs) a maximum amount of two times the individual expected value of the treatment costs (.e. 2*individual probability of getting bitten* 10,000). During the first year the local insurer offers insurance at a price (premium) which brings zero expected profits assuming that everyone buys insurance. In each of the subsequent years the premium is set so as to guarantee zero profits assuming that the average number of injuries per insuree will be the same as it were in the preceding year. a) What is the probability of an injury within the entire population? Hence, what will the premium be in the first year? Which surfers will be willing to buy insurance at such price? What will be the probability of a injury among those insured? Will the insurance company indeed achieve zero profits? b) What will happen with the shark-bite insurance market in the second year: what premium (adjusted to the number of injuries in the preceding year) will be set by the insurance company, who will buy insurance, how much will the company gain/lose? c) What will be the premium and who will be insured in the long run? The government of Selection island introduces mandatory insurance against shark bites with a premium guranteeing the insurer zero expected profits. What proportion of the surfers will be happy about this solution in comparison with point c)? Assuming that there are 15,000 surfers, what will be the social gain/loss resulting from this solution in comparison with no insurance at all? (Add up the willingness to pay for insurance and substract all premiums.) Will the situation improve or deteriorate in comparison with the one described in point c)?

Answer 1

Solution Page No - 1 their holidays s spend Liven the thousands of susters spend their every year on the selection island. They are expe to the risk of getting bitten by a shark. Th Probability of such event varies too differen groups and it has a unitorm distribution au The interval [0, 0.24]. The individual probability on getting bitten is known to each surter but to noone else. person bear must costs the H bitten of to medical -ment Amounting to $ 10,000 > Fall Insurence = 2 x individual probability of x10,000 getting bitten -24 0.24 X 10,000 = 2x 24 x 101000 100 = 2 x 24x10 = 4800 Qusing birst year a price which The everyone bugs the local Insurey offers insurance at brings zero expected protits assuming Insurance.

Page No - 2 a 7 we in over know that different groups [0, 0.24] the and probability it has of insusy uniform a intury varies unitorm distribution 1 insury with in tue Hence, entire the probability population is of 0.24 Hence, the premium in first year of getting bitten = $(2x individual probability x 10,000 ) (2x 0.24 x 10000) 8 $ 4800 = As -bility au The at such premium all the of getting surtuces a price Gurantee sustaces know the individual proba bitten. They it is expected that will be willing to buy insurance as it is also noticable That zero pootit. 3 Hence is are insured The Lokole probability of injury among those who is 0.244 of The whole popalation nag unitorm distribution of being insured. &

Page No -3 ting injured is nes of probability AS The probability of getting insured is o2 That is the probability of not getting injus 0.76 which is more than three times of poot of getting injured it all curtesy bay insurance on some of them will even not buy insurance then The insurance company will not have probit Insurance company will always have a good protit. also The (6) in each of the subsament years the premium is Set 30 as to guarantee zero protits assuming That The average number of injuries per insurce coin be some of it in the preceeding year. Hence, in the second year The company will Insure some amount of Insurance To The sustans who are bitten shork. Company will gain = $(2 x 0.26 x 10000) = $ 1.200 per year will be and the susters who insured in long - premium is 8 4800 are bitten by shark dun > The governament of - datory insurance against - teeing the insurer zero The grecsance selection shark expected island introduced man. bits with a premium garan protits.

page not-H about tai = Gence solution 76% in of the suster will be happy composision with point (cl. I There will must Assume be gain with no that there are 181000 susters. there will resulting from this solution in comparist insurance at all Cocoson is alsoudy discused mount of gain & (15000 x 240.76 x 10,000) & (15000 X 4000) =$ 15000 X (15200 -1800) $ 156000 000. over all 15000 suster. Situation will Improve or deteriorate according of The consistaness of the suster. but here situation, will Improve in comparison with the point (c) og Insurance is mandatory.