Question

assumptions: (1) underwriting costs is 30% of pure premium; (2) interest rate is 5%; (3) fair profit is 10% of pure premium. 3. Suppose that Annie's loss distribution is as follows: $ 5.000 with probability 0.004 $1,000 with probability 0.006 Loss = $ 250 with probability 0.055 $ 0 with probability 0.935 Assume that the only administrative cost is the cost of processing a claim, which equals $500 regardless of the claim size. Ignoring moral hazard, adverse selection, the time value of money. and capital costs, the cost of providing insurance is the sum of expected claim costs and expected claim processing costs. Fill in the table below for a policy with a $250 deductible and a policy without a deductible. What does the table illustrate about the marginal cost of insuring small losses and the desirability of deductibles?

Answer 1

		Policy with Full Coverage	Policy with $250 deductible	Implicit cost of coverage for $250
		(1)	(2)	(1)-(2)
a	Expected Claim Costs	39.75	23.5	16.25
b	Expected Claim Processing Costs	32.5	5	27.5
(a)+(b)	Fair Premium	72.25	28.5	43.75

Working Note 1(Expected Claim Cost for Full Coverage)

(a)Loss Distribution (In $)	(b)Probability	(c) Expected Claim Costs=(a)*(b)
5000	0.004	20
1000	0.006	6
250	0.055	13.75
Total		39.75

Working Note 2(Expected Claim Processing cost for full coverage)

Expected claim processing cost=Claim Processing Cost (1-probability for nil loss)

=500*(1-0.935)

= 32.5

Working Note 3(Expected Claim Cost for $250 deductible)

(a)Loss Distribution(In $)	(b)Deduction	(c)Net Loss Distribution	(d)Probability	(e) Expected Claim Costs=(c)*(d)
5000	250	4750	0.004	19
1000	250	750	0.006	4.5
250	250	0	0.055	0
Total				23.5

Working Note 4(Expected Claim Processing cost for $250 deductible)

There will not be any claim processing for $250 distribution as it became nil after having deducted $250

=$500*(0.004+0.006)

=5

Conclusion:

Percentage Loading of implicit processing cost on implicit claim cost

=27.5/16.25%

=169.2%

The marginal cost of insuring small losses is extremely high.