Question

Suppose you work for an insurance company. You know that there are equal numbers of individuals who will get in an accident with probability 0.2 and 0.3 and that the loss from getting in an accident is $4,000. Suppose you wish to screen individuals by offering full- coverage insurance as well as 30 percent coinsurance (i.e., the insurance pays 30 percent of the loss amount, or $1,200, in the event of an accident). You wish to include a 5 percent premium over the actuarially fair values for these policies to help cover overhead and pro- vide a profit margin for your company.

Will this pair of policies screen individuals into risk classes if we assume that all insuredindividuals have utility function U(X) ? 2000 · (X/4000)0.5 where X is income? Hint: Calculate utility (or expected utility) for each of the three options (no insurance, full- coverage insurance, and coinsurance) for both types of individuals and compare ex- pected utility levels to see what each will choose.

Background on Expected Utility: In this conceptualization of utility, we can compare uncertain outcomes with certain outcomes. For example, if an individual does not have insurance and does not get into an accident, he or she has a utility level of 2,000 (because X ? $4,000) or zero if he or she gets into an accident (because then X ? $0). The expected utility (EU) is based on the likelihood of each event occurring. A per- son facing a 10 percent chance of having an accident would have an expected utility of 1,800 using this utility function because EU ? 0.9 · 2,000 ? 0.1 · 0. This person would be indifferent between facing this “lottery” and having $3,240 with certainty (because U(3240) ? 1,800). This certainty equivalent value was obtained by solving for X in the equation 1800 ? 2000 · (X/4000)0.5. The lottery would be preferred to amounts less than $3,240 offered with certainty.

Answer 1

Answer:

The actuarially reasonable premium is equivalent to the normal incentive from the misfortune. The Coinsurance is a protection wherein the organization pays a part of the all out misfortune accured to the shopper. The full protection is where the organization pays the all out misfortune accures to the shopper.

Let the underlying abundance of each kind of customer is $4000. At that point with no protection they will have $0 pay on the off chance that of misfortune and $4000 if there should arise an occurrence of no misfortune. The utility from $x salary is given as

U(x)=2000*(x/4000)0.5

Thusly, with no protection the utility if there is no misfortune is

U(x)=2000*(4000/4000)0.5=2000 *1=2000

The utility if there is the occasion that produces misfortune

U(x)=2000*(0/40000)0.5=2000*0=0

The normal utility of riches is the likelihood weighted normal of all out abundance of misfortune and no misfortune. The normal utility of a high hazard individual if there should arise an occurrence of no protection is  EU=0.3*0+0.7*2000=0+1400=1400

The normal utility of a generally safe individual is EU=0.2*0+0.8*2000=0+1600=1600

For coinsurance the purchaser follows through on the cost of $247. At that point his pay if there should be an occurrence of misfortune is ($1200-$247)=$953 also, if there should arise an occurrence of no misfortune ($4000-$247)=$3753. The utility grom $x salary is given as

U(x)=2000*(x/4000)0.5

In this way, with coinsurance the utility if there is no misfortune is

U(x)=2000*(3753/4000)0.5=1937.27

The utility if there is the eent that creates misfortune is

U(x)=2000*(953/4000)0.5=976.22

The normal utility of riches is the likelihood weighted normal of complete abundance of misfortune and no misfortune. The normal utility of a high hazard individual in the event of coinsurane is EU=0.*976.22+0.7*1937.27=1648.95

The normal utility of okay individual is, EU=0.2*976.22+0.8*1937.27=1745.6

Presently with full protection every purchaser will have $400 salary in the event of misfortune and no misfortune as full protection spread for the all out misfortune. For full protection the purchaser pays the proce of $1260. At that point his salary in the event of misfortune and no misfortune is ($4000-$1260)=$2740.

The utility from $x salary is given as U(x)=2000*(x/4000)0.5

In this manner, with coinsurnace the utility if there is no misfortune or no misfortune is U(x)=2000*(2740/4000)0.5=1655.29

The normal utility of riches is the probablity weighted normal of absolute abundance of misfortune and no misfortune. The normal utility of a high hazard person if there should be an occurrence of full protection is EU=0.3*1655.29+0.7*1655.29=1655.29

The normal utility of a generally safe individual is, EU=0.2*1655.29+0.8*1655.29=1655.29.

The normal utiity for both the protection and no protection for both the buyer is abridged in the table beneath:

Expected utility No insurance Coinsurance Full insurance

Low Risk 1600 1745.6 1655.29

High Risk 1400 1648.95 1655.29

The confined numbers the table gives the most noteworthy anticipated utility for the two kinds of shopper. It is apparent from the table that an okay individual will have most noteworthy expected utility in the event that he purchases the coinsurance and a high hazard person will have most noteworthy expected utility on the off chance that he purchases the full insurnace. Therfore, the estimating structure decided above wil effectively seperate two gatherings of person.