Marsh & McLennan (A) A Marsh & McLennan are international insurance brokers' and employee benefit consultants. The firm is an independent contractor remunerated on a commission or fee basis. Its principal function is to assist its clients in placing risks in the insurance market at conditions and premiums equitable to both insured and insurer, as well as to provide consulting, actuarial, and communication services for its clients' employee benefit plans. Marsh & McLennan's services are primarily purchased by corporate and institutional organizations which require specialized and professional counsel to assess their exposures to risk and to fulfill their insurance and employee benefit requirements. Aircraft Insurance Early in 1969, in preparation for the renewal of coverage for a three-year policy term, Marsh & McLennan began reviewing the insurance program for one of its clients, Eastern Airlines, Inc. Aircraft insurance received little attention until after World War I. There were no special forms to cover aviation risks, and the few policies that were issued made use of the ordinary fire and automobile forms. Initially, underwriting was characterized by a considerable element of trial and error. The volume of business was small, and values in a single risk were great. Experience upon which to predict rates was lacking. Early insurers charged high premiums, imposed heavy deductibles, and used some complicated policy conditions. For a considerable period, insurance companies were organized into underwriting syndicates to handle the growing volume of aircraft business. Such syndicates still account for a substantial part of the business, especially the protection for the large airline transportation and aircraft manufacturing companies. At first glance, the aircraft coverages seem to parallel the familiar automobile coverages. Each is divided into two classifications: direct loss and liability coverages. However, compared with 1 From the Marsh & McLennan Story: "The responsibility of the insurance] broker is to provide for his client, through quality service by technical, administrative, and executive personnel, an expert appraisal of his insurance requirements, development of custom-made policy conditions to provide proper insurance protection, and the purchase of such coverage in the insurance markets of the world. Additionally, the broker provides during the policy term technical advice and assistance on a continuing basis to assist the client and all related problems." automobile risks, aircraft insurance involves huge sums for each accident; and direct loss insurance, depreciation, and obsolescence become important factors. Hull Insurance The specific insurance policy under study by Marsh & McLennan was hull insurance for Eastern's entire jet fleet (Exhibit 1). Premiums for aircraft hull insurance are frequently determined in a retrospective fashion and are based on the amount of losses during the coverage period. There is an upper limit on the premium amount which transfers some risk to the insurer, but generally the insurer is directly reimbursed for losses paid under the contract. There are many alternative reimbursement formulas, giving the insurer varying margins for expense and risk premiums, but Eastern was considering only two hull insurance proposals submitted by Associated Aviation Underwriters. A third alternative, self-insurance, was not being considered by Eastern at this time. Loss Conversion Plan Eastern had been utilizing a Loss Conversion formula. Under this plan, annual premiums are equal to 135% of all losses incurred within that year, subject to a maximum annual rate of $1.05 and a minimum rate of $.50 per $100 insured value. With this method of premium calculation, an incentive credit equaling 10% of total three-year premiums in excess of total three-year losses is given the insured if the plan runs for the full three-year term. Profit Commission Plan John Lawton, Marsh & McLennan's account executive, suggested that Eastern modify its insurance program to utilize a cumulative three-year Profit Commission method of calculation. Annual premiums under this plan equal losses plus $.25 per $100 insured, subject to a maximum annual premium of 1% of insured value. The incentive credit refunded at the end of the policy was equivalent to the excess, if any, of (a) premiums paid over (b) total losses plus $.20 per $100 value per year. When the Profit Commission plan was suggested to Peter Mullen, Eastern's director-insurance, he expressed some concern that it could be more expensive than the Loss Conversion plan. He pointed out that Eastern had 206 jet aircraft with a total fleet value of nearly $1 billion and an average value of $4.6 million per aircraft . If the assumption were made that one "average aircraft" was lost each year, the Loss Conversion plan would produce significant savings when compared to the Profit Commission plan. In addition, of the 206 aircraft in the Eastern jet fleet, all but two were presently valued at amounts which, in the event of loss, would produce lower costs under the Loss Conversion plan. John Lawton thought there were other factors that should be considered. Peter Mullen was correct in pointing out that Eastern's annual insurance premium could be less under the Loss Conversion plan provided that losses were "average." However, the alternative Profit Commission plan would cost less if there were no losses or if two or more losses occurred in a given year. Considering various possible combinations of losses over a three-year term, the Profit Commission plan produced a lower cost in 8 out of 10 sample adjustments. See Exhibit 1 for the sample adjustments submitted to Eastern In order to prepare for an upcoming conference with Mullen and other Eastern executives, Lawton asked Marsh & McLennan actuary, Charles Porter, to evaluate the relative merits of the two alternative plans and to prepare his recommendations prior to the meeting. Additional Information A variety of industrywide statistics were available. Losses could be studied in detail as to type of aircraft and cause of loss, or related to the number of flights, revenue miles, flying hours, etc. In an attempt to estimate the probability that an aircraft would be lost in the course of a calendar year, Charles Porter had found a research report of the FAA. This statistical survey had shown that the accident rate per cruise hour was essentially identical for all types of aircraft. To adjust for increased exposure during takeoff and landing, 3.7 hours of cruising were added for each flight. Thus, a flight between New York and San Francisco (flight time 6 hours) would be equivalent to 9.7 hours of cruising; whereas a flight between New York and Boston (flight time 45 minutes), although only one- eighth as long in terms of flight time, would be equivalent to 4.45 hours of cruising To determine the equivalent hours" of exposure for all jets flown by domestic trunk carriers, Porter found detailed data for both 1967 and 1968. During this period of time, 15,660,000 "equivalent hours" were recorded, with total losses of jet aircraft numbering 10. While Eastern's recent experience had been much better, data relating solely to Eastern would be too scanty to provide a meaningful basis. In addition to total losses, partial hull damage (e.g., minor aircraft damage on a landing or a takeoff) might be estimated from Eastern's experience as $500,000 to $1,000,000 per year. Information on future fleet expansion and flight schedules was not firm; but for the purposes of his analysis, Porter was told to base his evaluation of alternative plans upon the current fleet size (Exhibit 2) and an annual estimate of 10,060 "equivalent hours" per jet. The recent congestion along the eastern seaboard would eliminate any possibility of significantly higher utilization. Armed with these data, Charles Porter prepared to make his evaluation of the two plans. Exhibit 1 Eastern Airlines Hull Insurance Comparison of Costs Based on Catastrophic Losses Losses 1. 2. 3. Total Incentive credit Net cost S 1,000,000 1,000,000 1,000,000 S 3,000,000 Loss Conversion Plan $5,000,000 5,000,000 5,000,000 $15,000,000 1.200.000 $13,800,000 Profit Commission Plan $ 3,500,000 3.500.000 3.500.000 $10.500,000 1.500.000 9,000,000 1. 2 3 Total Incentive credit Net cost $ 1,000,000 1,000,000 6.000.000 S 8.000.000 $5,000,000 5,000,000 8.100.000 $18,100,000 1,010,000 $17,090,000 $ 3,500,000 3,500,000 8.500.000 $15.500.000 1.500.000 $14,000,000 1. 2 3 Total Incentive credit Net cost $ 1,000,000 6.000.000 6.000.000 $13,000,000 $5,000,000 8,100,000 8.100.000 $21,200,000 820,000 $20,380,000 $ 3,500,000 8.500.000 8.500.000 $20.500.000 1.500.000 $19.000.000 1. 2. 3. Total Incentive credit Net cost $ 6,000,000 6,000,000 6.000.000 $18,000,000 $ 8,100,000 8,100,000 8.100.000 $24,300.000 630,000 $23,670,000 $ 8,500,000 8,500,000 8.500.000 $25,500,000 1.500.000 $24,000,000 $ 3,500,000 3.500.000 10.500.000 $17,000,000 $17.000.000 1. 2. 3. Total Incentive credit Nel cost $1,000,000 1,000,000 11.000.000 $13,000,000 $5,000,000 5,000,000 10.500.000 $20,500,000 750,000 $19,750,000 1 2. 3 Total Incentive credit Net cost $ 1,000,000 6.000.000 11.000.000 $18,000,000 $5,000,000 8,100,000 10.500.000 $23,600,000 560.000 $23,040,000 $ 3,500,000 8.500.000 10.000.000 $22.000.000 $22,000,000 1 2 3. Total Incentive credit Net cost $1,000,000 11,000,000 11.000.000 $23,000,000 1 2. 3. Total Incentive credit Nel cost $ 6,000,000 6,000,000 11.000.000 $23,000,000 $5,000,000 10,500,000 10.500.000 $26.000.000 300.000 $25,700,000 $ 8,100,000 8,100,000 10.500.000 $26,700,000 370.000 $26,330,000 $ 8,100,000 10,500,000 10.500.000 $29,100,000 110.000 $28,990,000 $10,500,000 10,500,000 10.500.000 $31,500,000 $31,500,000 $ 3,500,000 10,000,000 10.000.000 $23,500,000 $23,500,000 $ 8,500,000 8,500,000 10.000.000 $27,000,000 $27,000,000 $ 8,500,000 10,000,000 10.000.000 $28,500,000 1 2 3. Total Incentive credit Net cost $ 6,000,000 11,000,000 11.000.000 $28,000,000 1 2. 3. Total Incentive credit Net cost $11,000,000 11,000,000 11.000.000 $33,000,000 $28,500,000 $10,000,000 10,000,000 10.000.000 $30,000,000 $30,000,000 Exhibit 2 Eastern Airlines Jet Fleet (January 1, 1969) Aircraft Book Value Plane Value Fleet Value No. DC-8-61 DC-8-63 17 2 19 $ 8,709,000 11,100,000 $148,053,000 22.200.000 $170,253,000 Insured Value Plane Value Fleet Value $ 9,000,000 $153,000,000 11,300,000 22.600.000 $175,600,000 15 14 DC-9-14 DC-9-21 DC-9-3 DC-9-31 $ 3,310,000 2,772,000 3,825,000 3,825,000 $ 49,650,000 38,808,000 237,150,000 19.125.000 $344,733,000 $3,400,000 3,000,000 3,900,000 3,900,000 $ 51,000,000 42,000,000 241,800,000 19.500.000 $354,300,000 96 720 15 15 $ 2,455,000 $3,200,000 $ 36,825.000 $ 36,825,000 $ 48.000.000 $ 48,000,000 727 727-225 727-QC 727-QC 50 1 17 $ 3,854,000 6,092,225 5,710,000 5,710,000 $192,700,000 6,092,225 97,070,000 45.680.000 $341,542,225 $4,200,000 6,200,000 5,900,000 5,900,000 $210,000,000 6,200,000 100,300,000 47.200.000 $363,700,000 76 TOTALS 206 $893,353,225 $941,600,000 As with all insurance policies, the insurance premium charged depends on the insurance company's cost experience over the life of the policy, which in this case is three years. In each year, the total cost to the insurer consists of two parts: total loss of airplanes due to crashes and a partial loss to the fleet due to minor damage. Each year, the number of airplanes lost due to crashes depends upon the total number of airplanes in the fleet and the probability of each airplane's crashing. The probability that an airplane will crash each year depends on the number of "equivalent cruise hours" that it flies. The number of airplane crashes together with the average insured value of an aircraft determines the annual loss due to aircraft crashes. In addition, there are losses due to partial hull damage to the fleet (such as during landings and takeoffs). Adding the two losses in a year then provides the total annual loss that the insurer experiences. The insured party's premium is then based on the loss experience over three years. (Note the typo in Exhibit 1 in the data file for Exhibit 1. D30 should be $10 million not $10,500,000) 1. Given the total number of equivalent hours flown, the total number of crashes recorded, and the equivalent hours each plane flies, estimate the probability that an aircraft will crash each year. Use an appropriate probability distribution to simulate the number of airplane crashes and the corresponding loss in a year. Also simulate additional uncertain loss due to partial damage to the fleet. Repeat the simulation of the total annual loss three times to yield the loss experience during one policy period of three years. The total three-year cost, together with any incentive credit then yields the net cost to the insured in a three-year policy period under each of the two plans being considered. Under Profit Commission plan, interpret the incentive credit as "premiums paid in excess (if any) of [total losses plus $0.20 per $100 insured value per year]". 2. Develop appropriate formulas for determining the total premium, net of any incentive credit in one policy period of three years under each of the two suggested plans. Verify and show the accuracy of your formulas by applying them to the net cost calculations shown in Exhibit 1 for ten policy periods. (This exhibit assumes the total fleet value of $1 billion, and the average value of an aircraft to be $4.6 million.) Also compute the total cost to the airline with self-insurance (i.e., without insurance). This completes one simulation of the net cost calculation over a three-year policy period for each of the three plans. 3. Repeat this simulation of the three net costs 1000 times, find the average values and standard deviations of the net costs to Eastern to see which of the three plans Eastern should choose. (Partial Answer: The average loss with self-insurance should be about $20 million) 4. Finally, chart two cumulative probability distributions of the net costs to Eastern under the two insurance plans being considered. If you use data tables or Simulation Supplement to simulate the net costs under the two plans, sort these costs in an ascending order, and plot their inverse cumulative distributions on the same chart. The x-axis values will then show cumulative probabilities, and the y-axis values are net costs under the two plans. Compare the two graphs of cumulative distributions of costs to the insured and interpret the result in probabilistic terms. (Hint: Do they cross each other? What does that mean?) Marsh & McLennan (A) A Marsh & McLennan are international insurance brokers' and employee benefit consultants. The firm is an independent contractor remunerated on a commission or fee basis. Its principal function is to assist its clients in placing risks in the insurance market at conditions and premiums equitable to both insured and insurer, as well as to provide consulting, actuarial, and communication services for its clients' employee benefit plans. Marsh & McLennan's services are primarily purchased by corporate and institutional organizations which require specialized and professional counsel to assess their exposures to risk and to fulfill their insurance and employee benefit requirements. Aircraft Insurance Early in 1969, in preparation for the renewal of coverage for a three-year policy term, Marsh & McLennan began reviewing the insurance program for one of its clients, Eastern Airlines, Inc. Aircraft insurance received little attention until after World War I. There were no special forms to cover aviation risks, and the few policies that were issued made use of the ordinary fire and automobile forms. Initially, underwriting was characterized by a considerable element of trial and error. The volume of business was small, and values in a single risk were great. Experience upon which to predict rates was lacking. Early insurers charged high premiums, imposed heavy deductibles, and used some complicated policy conditions. For a considerable period, insurance companies were organized into underwriting syndicates to handle the growing volume of aircraft business. Such syndicates still account for a substantial part of the business, especially the protection for the large airline transportation and aircraft manufacturing companies. At first glance, the aircraft coverages seem to parallel the familiar automobile coverages. Each is divided into two classifications: direct loss and liability coverages. However, compared with 1 From the Marsh & McLennan Story: "The responsibility of the insurance] broker is to provide for his client, through quality service by technical, administrative, and executive personnel, an expert appraisal of his insurance requirements, development of custom-made policy conditions to provide proper insurance protection, and the purchase of such coverage in the insurance markets of the world. Additionally, the broker provides during the policy term technical advice and assistance on a continuing basis to assist the client and all related problems." automobile risks, aircraft insurance involves huge sums for each accident; and direct loss insurance, depreciation, and obsolescence become important factors. Hull Insurance The specific insurance policy under study by Marsh & McLennan was hull insurance for Eastern's entire jet fleet (Exhibit 1). Premiums for aircraft hull insurance are frequently determined in a retrospective fashion and are based on the amount of losses during the coverage period. There is an upper limit on the premium amount which transfers some risk to the insurer, but generally the insurer is directly reimbursed for losses paid under the contract. There are many alternative reimbursement formulas, giving the insurer varying margins for expense and risk premiums, but Eastern was considering only two hull insurance proposals submitted by Associated Aviation Underwriters. A third alternative, self-insurance, was not being considered by Eastern at this time. Loss Conversion Plan Eastern had been utilizing a Loss Conversion formula. Under this plan, annual premiums are equal to 135% of all losses incurred within that year, subject to a maximum annual rate of $1.05 and a minimum rate of $.50 per $100 insured value. With this method of premium calculation, an incentive credit equaling 10% of total three-year premiums in excess of total three-year losses is given the insured if the plan runs for the full three-year term. Profit Commission Plan John Lawton, Marsh & McLennan's account executive, suggested that Eastern modify its insurance program to utilize a cumulative three-year Profit Commission method of calculation. Annual premiums under this plan equal losses plus $.25 per $100 insured, subject to a maximum annual premium of 1% of insured value. The incentive credit refunded at the end of the policy was equivalent to the excess, if any, of (a) premiums paid over (b) total losses plus $.20 per $100 value per year. When the Profit Commission plan was suggested to Peter Mullen, Eastern's director-insurance, he expressed some concern that it could be more expensive than the Loss Conversion plan. He pointed out that Eastern had 206 jet aircraft with a total fleet value of nearly $1 billion and an average value of $4.6 million per aircraft . If the assumption were made that one "average aircraft" was lost each year, the Loss Conversion plan would produce significant savings when compared to the Profit Commission plan. In addition, of the 206 aircraft in the Eastern jet fleet, all but two were presently valued at amounts which, in the event of loss, would produce lower costs under the Loss Conversion plan. John Lawton thought there were other factors that should be considered. Peter Mullen was correct in pointing out that Eastern's annual insurance premium could be less under the Loss Conversion plan provided that losses were "average." However, the alternative Profit Commission plan would cost less if there were no losses or if two or more losses occurred in a given year. Considering various possible combinations of losses over a three-year term, the Profit Commission plan produced a lower cost in 8 out of 10 sample adjustments. See Exhibit 1 for the sample adjustments submitted to Eastern In order to prepare for an upcoming conference with Mullen and other Eastern executives, Lawton asked Marsh & McLennan actuary, Charles Porter, to evaluate the relative merits of the two alternative plans and to prepare his recommendations prior to the meeting. Additional Information A variety of industrywide statistics were available. Losses could be studied in detail as to type of aircraft and cause of loss, or related to the number of flights, revenue miles, flying hours, etc. In an attempt to estimate the probability that an aircraft would be lost in the course of a calendar year, Charles Porter had found a research report of the FAA. This statistical survey had shown that the accident rate per cruise hour was essentially identical for all types of aircraft. To adjust for increased exposure during takeoff and landing, 3.7 hours of cruising were added for each flight. Thus, a flight between New York and San Francisco (flight time 6 hours) would be equivalent to 9.7 hours of cruising; whereas a flight between New York and Boston (flight time 45 minutes), although only one- eighth as long in terms of flight time, would be equivalent to 4.45 hours of cruising To determine the equivalent hours" of exposure for all jets flown by domestic trunk carriers, Porter found detailed data for both 1967 and 1968. During this period of time, 15,660,000 "equivalent hours" were recorded, with total losses of jet aircraft numbering 10. While Eastern's recent experience had been much better, data relating solely to Eastern would be too scanty to provide a meaningful basis. In addition to total losses, partial hull damage (e.g., minor aircraft damage on a landing or a takeoff) might be estimated from Eastern's experience as $500,000 to $1,000,000 per year. Information on future fleet expansion and flight schedules was not firm; but for the purposes of his analysis, Porter was told to base his evaluation of alternative plans upon the current fleet size (Exhibit 2) and an annual estimate of 10,060 "equivalent hours" per jet. The recent congestion along the eastern seaboard would eliminate any possibility of significantly higher utilization. Armed with these data, Charles Porter prepared to make his evaluation of the two plans. Exhibit 1 Eastern Airlines Hull Insurance Comparison of Costs Based on Catastrophic Losses Losses 1. 2. 3. Total Incentive credit Net cost S 1,000,000 1,000,000 1,000,000 S 3,000,000 Loss Conversion Plan $5,000,000 5,000,000 5,000,000 $15,000,000 1.200.000 $13,800,000 Profit Commission Plan $ 3,500,000 3.500.000 3.500.000 $10.500,000 1.500.000 9,000,000 1. 2 3 Total Incentive credit Net cost $ 1,000,000 1,000,000 6.000.000 S 8.000.000 $5,000,000 5,000,000 8.100.000 $18,100,000 1,010,000 $17,090,000 $ 3,500,000 3,500,000 8.500.000 $15.500.000 1.500.000 $14,000,000 1. 2 3 Total Incentive credit Net cost $ 1,000,000 6.000.000 6.000.000 $13,000,000 $5,000,000 8,100,000 8.100.000 $21,200,000 820,000 $20,380,000 $ 3,500,000 8.500.000 8.500.000 $20.500.000 1.500.000 $19.000.000 1. 2. 3. Total Incentive credit Net cost $ 6,000,000 6,000,000 6.000.000 $18,000,000 $ 8,100,000 8,100,000 8.100.000 $24,300.000 630,000 $23,670,000 $ 8,500,000 8,500,000 8.500.000 $25,500,000 1.500.000 $24,000,000 $ 3,500,000 3.500.000 10.500.000 $17,000,000 $17.000.000 1. 2. 3. Total Incentive credit Nel cost $1,000,000 1,000,000 11.000.000 $13,000,000 $5,000,000 5,000,000 10.500.000 $20,500,000 750,000 $19,750,000 1 2. 3 Total Incentive credit Net cost $ 1,000,000 6.000.000 11.000.000 $18,000,000 $5,000,000 8,100,000 10.500.000 $23,600,000 560.000 $23,040,000 $ 3,500,000 8.500.000 10.000.000 $22.000.000 $22,000,000 1 2 3. Total Incentive credit Net cost $1,000,000 11,000,000 11.000.000 $23,000,000 1 2. 3. Total Incentive credit Nel cost $ 6,000,000 6,000,000 11.000.000 $23,000,000 $5,000,000 10,500,000 10.500.000 $26.000.000 300.000 $25,700,000 $ 8,100,000 8,100,000 10.500.000 $26,700,000 370.000 $26,330,000 $ 8,100,000 10,500,000 10.500.000 $29,100,000 110.000 $28,990,000 $10,500,000 10,500,000 10.500.000 $31,500,000 $31,500,000 $ 3,500,000 10,000,000 10.000.000 $23,500,000 $23,500,000 $ 8,500,000 8,500,000 10.000.000 $27,000,000 $27,000,000 $ 8,500,000 10,000,000 10.000.000 $28,500,000 1 2 3. Total Incentive credit Net cost $ 6,000,000 11,000,000 11.000.000 $28,000,000 1 2. 3. Total Incentive credit Net cost $11,000,000 11,000,000 11.000.000 $33,000,000 $28,500,000 $10,000,000 10,000,000 10.000.000 $30,000,000 $30,000,000 Exhibit 2 Eastern Airlines Jet Fleet (January 1, 1969) Aircraft Book Value Plane Value Fleet Value No. DC-8-61 DC-8-63 17 2 19 $ 8,709,000 11,100,000 $148,053,000 22.200.000 $170,253,000 Insured Value Plane Value Fleet Value $ 9,000,000 $153,000,000 11,300,000 22.600.000 $175,600,000 15 14 DC-9-14 DC-9-21 DC-9-3 DC-9-31 $ 3,310,000 2,772,000 3,825,000 3,825,000 $ 49,650,000 38,808,000 237,150,000 19.125.000 $344,733,000 $3,400,000 3,000,000 3,900,000 3,900,000 $ 51,000,000 42,000,000 241,800,000 19.500.000 $354,300,000 96 720 15 15 $ 2,455,000 $3,200,000 $ 36,825.000 $ 36,825,000 $ 48.000.000 $ 48,000,000 727 727-225 727-QC 727-QC 50 1 17 $ 3,854,000 6,092,225 5,710,000 5,710,000 $192,700,000 6,092,225 97,070,000 45.680.000 $341,542,225 $4,200,000 6,200,000 5,900,000 5,900,000 $210,000,000 6,200,000 100,300,000 47.200.000 $363,700,000 76 TOTALS 206 $893,353,225 $941,600,000 As with all insurance policies, the insurance premium charged depends on the insurance company's cost experience over the life of the policy, which in this case is three years. In each year, the total cost to the insurer consists of two parts: total loss of airplanes due to crashes and a partial loss to the fleet due to minor damage. Each year, the number of airplanes lost due to crashes depends upon the total number of airplanes in the fleet and the probability of each airplane's crashing. The probability that an airplane will crash each year depends on the number of "equivalent cruise hours" that it flies. The number of airplane crashes together with the average insured value of an aircraft determines the annual loss due to aircraft crashes. In addition, there are losses due to partial hull damage to the fleet (such as during landings and takeoffs). Adding the two losses in a year then provides the total annual loss that the insurer experiences. The insured party's premium is then based on the loss experience over three years. (Note the typo in Exhibit 1 in the data file for Exhibit 1. D30 should be $10 million not $10,500,000) 1. Given the total number of equivalent hours flown, the total number of crashes recorded, and the equivalent hours each plane flies, estimate the probability that an aircraft will crash each year. Use an appropriate probability distribution to simulate the number of airplane crashes and the corresponding loss in a year. Also simulate additional uncertain loss due to partial damage to the fleet. Repeat the simulation of the total annual loss three times to yield the loss experience during one policy period of three years. The total three-year cost, together with any incentive credit then yields the net cost to the insured in a three-year policy period under each of the two plans being considered. Under Profit Commission plan, interpret the incentive credit as "premiums paid in excess (if any) of [total losses plus $0.20 per $100 insured value per year]". 2. Develop appropriate formulas for determining the total premium, net of any incentive credit in one policy period of three years under each of the two suggested plans. Verify and show the accuracy of your formulas by applying them to the net cost calculations shown in Exhibit 1 for ten policy periods. (This exhibit assumes the total fleet value of $1 billion, and the average value of an aircraft to be $4.6 million.) Also compute the total cost to the airline with self-insurance (i.e., without insurance). This completes one simulation of the net cost calculation over a three-year policy period for each of the three plans. 3. Repeat this simulation of the three net costs 1000 times, find the average values and standard deviations of the net costs to Eastern to see which of the three plans Eastern should choose. (Partial Answer: The average loss with self-insurance should be about $20 million) 4. Finally, chart two cumulative probability distributions of the net costs to Eastern under the two insurance plans being considered. If you use data tables or Simulation Supplement to simulate the net costs under the two plans, sort these costs in an ascending order, and plot their inverse cumulative distributions on the same chart. The x-axis values will then show cumulative probabilities, and the y-axis values are net costs under the two plans. Compare the two graphs of cumulative distributions of costs to the insured and interpret the result in probabilistic terms. (Hint: Do they cross each other? What does that mean?)