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 in a given 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.
a. 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 in a given 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]".
b. 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.
c. 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)
d. 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 Simtools 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. If you use @Risk, you can superimpose the cumulative probability of one output overlaid on the active graph. 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?)
| Exhibit 1 Eastern Airlines Hull Insurance-Comparison of Costs Based on Catastrophic Losses |
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Losses |
Loss Conversion Plan |
Profit Commission Plan |
| 1. |
$1,000,000 |
$5,000,000 |
$3,500,000 |
| 2. |
1,000,000 |
5,000,000 |
3,500,000 |
| 3. |
1,000,000 |
5,000,000 |
3,500,000 |
| Total |
$3,000,000 |
$15,000,000 |
$10,500,000 |
| Incentive credit |
|
1,200,000 |
1,500,000 |
| Net cost |
|
$13,800,000 |
$9,000,000 |
| 1. |
$1,000,000 |
$5,000,000 |
$3,500,000 |
| 2. |
1,000,000 |
5,000,000 |
3,500,000 |
| 3. |
6,000,000 |
8,100,000 |
8,500,000 |
| Total |
$8,000,000 |
$18,100,000 |
$15,500,000 |
| Incentive credit |
|
1,010,000 |
1,500,000 |
| Net cost |
|
$17,090,000 |
$14,000,000 |
| 1. |
$1,000,000 |
$5,000,000 |
$3,500,000 |
| 2. |
6,000,000 |
8,100,000 |
8,500,000 |
| 3. |
6,000,000 |
8,100,000 |
8,500,000 |
| Total |
$13,000,000 |
$21,200,000 |
$20,500,000 |
| Incentive credit |
|
820,000 |
1,500,000 |
| Net cost |
|
$20,380,000 |
$19,000,000 |
| 1. |
$6,000,000 |
$8,100,000 |
$8,500,000 |
| 2. |
6,000,000 |
8,100,000 |
8,500,000 |
| 3. |
6,000,000 |
8,100,000 |
8,500,000 |
| Total |
$18,000,000 |
$24,300,000 |
$25,500,000 |
| Incentive credit |
|
630,000 |
1,500,000 |
| Net cost |
|
$23,670,000 |
$24,000,000 |
| 1. |
$1,000,000 |
$5,000,000 |
$3,500,000 |
| 2. |
1,000,000 |
5,000,000 |
3,500,000 |
| 3. |
11,000,000 |
10,500,000 |
10,500,000 |
| Total |
$13,000,000 |
$20,500,000 |
$17,000,000 |
| Incentive credit |
|
750,000 |
0 |
| Net cost |
|
$19,750,000 |
$17,000,000 |
| 1. |
$1,000,000 |
$5,000,000 |
$3,500,000 |
| 2. |
6,000,000 |
8,100,000 |
8,500,000 |
| 3. |
11,000,000 |
10,500,000 |
10,000,000 |
| Total |
$18,000,000 |
$23,600,000 |
$22,000,000 |
| Incentive credit |
|
560,000 |
0 |
| Net cost |
|
$23,040,000 |
$22,000,000 |
| 1. |
$1,000,000 |
$5,000,000 |
$3,500,000 |
| 2. |
11,000,000 |
10,500,000 |
10,000,000 |
| 3. |
11,000,000 |
10,500,000 |
10,000,000 |
| Total |
$23,000,000 |
$26,000,000 |
$23,500,000 |
| Incentive credit |
|
300,000 |
0 |
| Net cost |
|
$25,700,000 |
$23,500,000 |
| 1. |
$6,000,000 |
$8,100,000 |
$8,500,000 |
| 2. |
6,000,000 |
8,100,000 |
8,500,000 |
| 3. |
11,000,000 |
10,500,000 |
10,000,000 |
| Total |
$23,000,000 |
$26,700,000 |
$27,000,000 |
| Incentive credit |
|
370,000 |
0 |
| Net cost |
|
$26,330,000 |
$27,000,000 |
| 1. |
$6,000,000 |
$8,100,000 |
$8,500,000 |
| 2. |
11,000,000 |
10,500,000 |
10,000,000 |
| 3. |
11,000,000 |
10,500,000 |
10,000,000 |
| Total |
$28,000,000 |
$29,100,000 |
$28,500,000 |
| Incentive credit |
|
110,000 |
0 |
| Net cost |
|
$28,990,000 |
$28,500,000 |
| 1. |
$11,000,000 |
$10,500,000 |
$10,000,000 |
| 2. |
11,000,000 |
10,500,000 |
10,000,000 |
| 3. |
11,000,000 |
10,500,000 |
10,000,000 |
| Total |
$33,000,000 |
$31,500,000 |
$30,000,000 |
| Incentive credit |
|
0 |
0 |
| Net cost |
|
$31,500,000 |
$30,000,000 |
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