estimating the cost of commercial airlines catastrophes

Estimating the Cost of Commercial Airlines Catastrophes

A Stochastic Simulation Approachby Romel Salam, FCAS, MAAA

March 2003

Why a stochastic Model?Simulation Model Better reflects current

environment in terms of exposures, frequency, fleet composition, liability and hull costs, passenger loads.

Provides results that are statistically stable even for layers exposed to rare events.

Allows one to better understand all the components in the loss process.

More conducive to pricing covers with a lot of bells and whistles.

Traditional Experience Rating

May not reflect current environment

Results not statistically stable, especially for layers exposed to rare events.

No attempt to piece together loss components.

Not very good for pricing covers with lots of contingent features.

Generate # of events over exposureperiod for given airline

For each event

Determine exact aircraft modelinvolved

For given aircraft model

Obtain Seating capacity

Generate % of capacity fiilled

Calculate # of passengers onboard

Generate % of fatalities/injuries

Generate flight itinerary(Domestic/International)

Calculate average passengerawards for given itineray for

fatalities/injuries

Calculate Passenger liability

Calculate # of fatalities/injuries

Projecting the # of Airline Catastrophes

Choosing a frequency model Poisson Negative Binomial Non-parametric


Picking an exposure base:a) Departuresb) Miles/Kilometers FlownC) Hours Flown All three measures almost perfectly correlated. If using different sources, make sure definitions

are consistent. Public Sources include: International Civil

Aviation Organization (ICAO), International Air Transport Association (IATA), National Transportation Safety Board (NTSB).

Keep in mind these statistics were not produced with the actuary in mind.


Classification May need to account for possible differences in

expected frequency of catastrophic accidents amongst airlines.

US vs Rest of the World is a typical line of demarcation. Does it really make sense as far as frequency is concerned?

Rating variables could include: airline flag country, airline size, average age of fleet, fleet make up (i.e. western built vs. other).

A rating scheme is presented in Appendix A of this paper based on methodology introduced in prior writing.


Accounting for Trend in Frequency Has the rate of accident changed over time? How do we project accident rates 1, 2 or

several years hence? Use extrapolation carefully. Choose trend curve carefully. A linear model

may not be appropriate. Simple linear regression may not be

appropriate as some assumptions are violated (i.e. equal variance).

Be mindful of error of statistical estimates.

Accounting for Trend in Frequency

Major Accident per million Departures

(0.40)(0.20)0.000.200.400.600.801.001.201.40

Year

Fre

qu

en

cy

Actual Linear Model Exponential Decay Model


Modeling the number of aircrafts involved in an accident.

Need to account for the possibility of collision involving several aircrafts.

Cost of such accidents may be prohibitive. Fortunately, these types of events are

relatively rare. Hence, modeler needs to use judgment in establishing probabilities.

Projecting the Cost of Catastrophes

Hull Cost Need to know Airline fleet, utilization

schedule and insured values as pre-agreed in contract.

If insured values are not known, find way to approximate these values.

Probability of any given aircraft involved in an accident may be based on its percentage utilization.

Others may use factors such as age and type of aircraft in figuring probability.


Passenger Liability Cost Need to know airline fleet, utilization schedule,

approximate capacity of each aircraft, passenger load factors, survival ratios, destination profile.

Need to come up with average passenger award. Award may vary by jurisdiction/country. May focus on ratio of average passenger award

to, say, income per capita. May use a Classification scheme to group

jurisdictions.


Third Party Liability Cost Highly volatile. Not a lot of history. One approach may be to lump

Third Party Liability cost with Passenger Liability cost.

Build scenarios through judgment.


Products Liability Aircraft and parts manufacturers are often

named in lawsuits resulting from airline accidents.

Need to allocate liability between operators and manufacturers. Specific allocation depends on determined cause of loss.

For given manufacturer, need to aggregate exposure over the universe of airline operators.

Much judgment may be needed.

Validation Does the model work? Are the assumptions

realistic? Need to validate results. Some results are easier to validate, i.e. # of

accidents, # of passengers, # of fatalities. Others are harder to validate, i.e. Passenger

or Third Party Liability Costs. One approach is to project latest ten years

based on data available in all preceding years and compare with actual results.

ValidationData Year

ProjectedProjected Distributio

n

Define pth Confidence Interval

Actual Results Bernoulli Distributed

Variables w/ Prob p

80 – 89

90 F90 [L90(p),U90(p)] r90 s90 = 0, 1

80 – 90

91 F91 [L91(p),U91(p)] r91 s91 = 0, 1

80 – 91

92 F92 [L92(p),U92(p)] r92 s92 = 0, 1

80 – 92

93 F93 [L93(p),U93(p)] r93 s93 = 0, 1

80 – 93

94 F94 [L94(p),U94(p)] r94 s94 = 0, 1

80 – 94

95 F95 [L95(p),U95(p)] r95 s95 = 0, 1

80 – 95

96 F96 [L96(p),U96(p)] r96 s96 = 0, 1

80 – 96

97 F97 [L97(p),U97(p)] r97 s97 = 0, 1

80 – 97

98 F98 [L98(p),U98(p)] r98 s98 = 0, 1

80 – 98

99 F99 [L99(p),U99(p)] r99 s99 = 0, 1

80 – 99

00 F00 [L00(p),U00(p)] r00 s00 = 0, 1

80 – 00

01 F01 [L01(p),U01(p)] r01 s01 = 0, 1

Validation Example - Projected Distribution of Claims Count vs Actual

Year Projected 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Actual 21 24 26 22 21 19 24 24 25 23 23 27

Expected 20 20 22 25 27 28 28 29 28 28 28 29

5% 13 13 15 17 19 20 20 21 19 20 20 21

10% 14 14 17 18 20 22 22 22 21 22 22 23

15% 15 15 17 20 22 23 23 24 22 23 23 24

20% 16 16 18 20 22 24 24 25 23 24 24 25

25% 17 17 19 21 23 25 25 25 24 25 25 26

30% 17 17 20 22 24 26 25 26 25 25 26 26

35% 18 18 20 23 24 26 26 27 26 26 26 27

40% 19 18 21 23 25 27 27 28 26 27 27 28

45% 19 19 22 24 26 28 27 28 27 27 28 28

50% 20 19 22 25 26 28 28 29 27 28 28 29

55% 20 20 23 25 27 29 29 29 28 29 29 30

60% 21 20 24 26 27 30 29 30 29 29 29 31

65% 22 21 24 27 28 30 30 31 29 30 30 31

70% 22 22 25 27 29 31 31 32 30 31 31 32

75% 23 22 26 28 30 32 31 32 31 32 32 33

80% 24 23 27 29 31 33 32 33 32 33 32 34

85% 24 24 28 30 32 34 33 34 33 34 34 35

90% 26 25 29 31 33 35 35 36 35 35 35 37

95% 28 27 31 33 35 38 37 38 36 37 37 39

This example shows that the model has an upward bias in the more recent years, as the actual is lower than the 50th percentile of the projected distribution in the last consecutive nine years. This is comparable to getting nine consecutive "heads" on tosses of a fair coin, which has less than a 1% odd!

ValidationOur Hypothesis:The r’s are random draws from the F’s.Let the s’s = 1 when the r’s fall in the confidenceinterval, 0 otherwise.If our Hypothesis is true, then The s’s are Bernoulli distributed w/ parameter p. The sum of the s’s has a Binomial distribution

with parameters (p,n) where n is the number of observations, 12 in this example.

Use our knowledge of the Binomial distribution to test our hypothesis.

Use same process for various values of p.

Terrorism Actuary has to work with other

experts to make proper assessment. Potential acts of terrorism include:

Hijackings. Forced collision w/ other aircraft. Surface to air missiles. Sabotaging engine, electrical system,

navigation system, or other vital equipment.

Tampering with food, water, or air. Damaging garaged planes and equipment.

Terrorism

Unlike most pundits, actuary has to actually try to quantify the risk of terrorism.

Past history may not be a good guide.

Risk of terrorism is highly fluid. Invariably, assessment will be very

subjective.

A Simple Application Cover for a hypothetical group of

airlines for accidents occurring in the 2003 year that pays: for the full insured value of a destroyed or

damaged aircraft $50,000 per passenger fatality $100,000 per injured passenger

Cover excludes acts of war and terrorism

A Simple Application

Information and Assumptions Fleet, utilization profile, and seating capacity. Projected departures for 2003. Projected average passenger load. Expected frequency of accidents per million

departures. Distribution of passenger survival ratios. Conditional probability for the number of

aircrafts involved in an accident.

Aircraft Type Count SeatsInsured Value

(MM)

# of Depart

ures Prob

Airbus Industrie A300-600 79 298 118 58,390 0.69%

Airbus Industrie A300B2/B4 19 298 118 6,165 0.07%

Airbus Industrie A310 44 249 92 22,253 0.26%




Avro RJ Avroliner 36 70 26 71,171 0.85%

BAE SYSTEMS (HS) 146 18 94 40 47,420 0.56%

Boeing (McDonnell-Douglas) DC-10 239 264 110 123,748 1.47%



Boeing (McDonnell-Douglas) MD-11 66 325 150 41,923 0.50%

Boeing (McDonnell-Douglas) MD-80 670 155 60 1,056,052 12.57%

Boeing (McDonnell-Douglas) MD-90 21 163 60 34,382 0.41%

Boeing 717 31 106 40 37,779 0.45%

Boeing 727 729 167 60 715,326 8.51%

Boeing 737 (CFMI) 779 149 60 1,672,505 19.90%

Total 6,245 1,108,002 450,332 8,403,831 100.00%

A Simple Application

Fleet, Utilization Profile, and Seating Capacity

Variables Accident Count

Aircraft Count Passenger Count

Fatal Count

Injured Count

Hull Cost

Passenger Cost

Total Cost

Best Case 0 0 0 0 0 0 0 0

Worst Case 13 14 1790 1015 1012 1131 1376 2507

Expected 3.59 3.69 375 197 179 233 286 519

Standard Deviation

1.91 1.99 230 154 144 139 182 315

5% 1 1 58 0 0 50 42 95

10% 1 1 109 18 8 60 76 149

15% 2 2 144 43 30 95 105 204

20% 2 2 176 65 51 115 128 248

25% 2 2 204 81 70 120 150 283

30% 2 3 231 97 86 145 172 322

35% 3 3 259 114 103 170 194 363

40% 3 3 287 131 121 180 214 398

45% 3 3 317 148 135 195 236 434

50% 3 4 347 167 150 216 258 474

55% 4 4 377 184 168 230 282 518

60% 4 4 407 209 190 252 305 559

65% 4 4 440 230 212 270 333 604

70% 4 5 474 254 233 290 364 652

75% 5 5 509 281 258 315 396 714

80% 5 5 554 314 287 345 430 772

85% 6 6 614 355 325 375 473 845

90% 6 6 681 407 371 415 531 932

95% 7 7 792 488 447 485 627 1089

Final Thoughts

Similarly to the use of simulation in property catastrophe analysis, for commercial aviation, simulation may:

enhance the comprehensibility of prices.

reduce information risk. promote stable pricing.

Final Thoughts

Some areas in need of more work How to make realistic projections

for Third Party and Products Liability.

Multi-aircraft collisions Terrorism.

estimating the cost of commercial airlines catastrophes

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