catastrophe modeling

1. Catastrophe Modeling Jim Maher, FCAS MAAA Chief Risk Officer Platinum Re US

3. Hazard Module Seismology Meteorology Terrorism Non random frequency Non random severity

4. Non-modeled perils Tsunami Meteor strike Est. RP of 1,000 years for 10 megaton event Most recent Siberia (1908) River Flood Wildfire Winterstorm

5. Non-modeled coverages Life/Health Personal Accident Group Life Disability Marine Yachts Offshore Oil Rigs Cargo

6. Earthquake Major Types of Earthquake Location of Earthquake Hazard Major Historical US Earthquakes Recent US Earthquakes Vulnerability and Financial Models Earthquake prediction (?)

7. Major Types of Earthquakes Strike-Slip Rock on one side of fault slides horizontally San Andreas Fault Dip-Slip (subduction) Fault is at an angle to the surface of the earth Movement of the rock is up or down Great Kanto Earthquake (Japan 1923)

8. Location of Earthquakes Plate Boundaries 90% of worlds earthquakes occur here Seven Major Crustal Plates on the Earth Rocks usually weaker, yield more to stress than Examples: California, Japan, etc. Ring of Fire Intra-plate Earthquakes New Madrid (1812) Newcastle, Australia (1989) Charleston (1886)

9. Plate Boundaries & “Ring of Fire”

11. Modified Mercalli Scale IV Felt by many indoors but by few outdoors. Moderate V Felt by almost all. Many awakened. Unstable objects moved. VI Felt by all. Heavy objects moved. Alarm. Strong. VII General alarm. Weak buildings considerably damaged. Very strong. VIII Damage general except in proofed buildings. Heavy objects overturned.

12. Modified Mercalli ctd. IX Buildings shifted from foundations, collapse, ground cracks. Highly destructive. X Masonry buildings destroyed, rails bent, serious ground fissures. Devastating. XI Few if any structures left standing. Bridges down. Rails twisted. Catastrophic. XII Damage total. Vibrations distort vision. Objects thrown in air. Major catastrophe.

13. Major Historical US Quakes San Francisco (1906) Magnitude 7.8, 3000 deaths Significant fire following element Charleston (1886) Magnitude 7.3, 100 deaths New Madrid (1811/12) 12/16/1811 Northeast Arkansas 1/23/1812 & 2/7/1812 New Madrid, Missouri Estimated Magnitude 8.0 Destroyed New Madrid, severe damage in St. Louis, rang church bells in Boston

15. Recent US Earthquakes Loma Prieta (1989) Northridge (1994) Nisqually/ (Seattle) (2001)

16. Loma Prieta (1989) Magnitude 6.9 on San Andreas Fault Largest since 1906 earthquake 63 deaths, 3,757 injuries, $6 BN economic damage, $1.0 BN insured damage Severe property damage in Oakland and San Francisco Collapse of Highways, viaducts

17. Loma Prieta ctd. Liquefaction San Francisco’s Marina district loosely consolidated, water saturated soils. Loosely consolidated soils tend to amplify shaking and increase structural damage. Water saturated soils compound the problem due to their susceptibility to liquefaction and corresponding loss of bearing strength. Unreinforced masonry construction Engineered buildings performed well Earthquake liquefaction, often referred to simply as liquefaction, is the process by which saturated, unconsolidated soil or sand is converted into a suspension during an earthquake. The effect on structures and buildings can be devastating, and is a major contributor to urban seismic risk. Sand Compaction Liquefaction essentially means that the soil is turned into a liquid. The key ingredient is a formation of loose, saturated sand. As seen in the figure, uniform sand grains can be packed either in a loose or a compact (dense) formation. Loose sand has usually been deposited gently underwater, either naturally, or sluiced into what is called hydraulic fill. The loose grains can support considerable weight, with the help of the water, which forms a good portion of the mass. Once strong earthquake shaking begins, the grains are sheared into the more compact arrangement. The water, however, interferes, and the grains float in a liquid slurry. The excess water is squeezed out which causes the quicksand condition at the surface. If there is a soil crust or impermeable cap, then the sand boils out in the form of sand volcanoes.Earthquake liquefaction, often referred to simply as liquefaction, is the process by which saturated, unconsolidated soil or sand is converted into a suspension during an earthquake. The effect on structures and buildings can be devastating, and is a major contributor to urban seismic risk. Sand Compaction Liquefaction essentially means that the soil is turned into a liquid. The key ingredient is a formation of loose, saturated sand. As seen in the figure, uniform sand grains can be packed either in a loose or a compact (dense) formation. Loose sand has usually been deposited gently underwater, either naturally, or sluiced into what is called hydraulic fill. The loose grains can support considerable weight, with the help of the water, which forms a good portion of the mass. Once strong earthquake shaking begins, the grains are sheared into the more compact arrangement. The water, however, interferes, and the grains float in a liquid slurry. The excess water is squeezed out which causes the quicksand condition at the surface. If there is a soil crust or impermeable cap, then the sand boils out in the form of sand volcanoes.

18. Northridge (1994) Magnitude 6.8 earthquake Occurred on previously unknown fault 60 killed, 7,000 injured, 20,000 homeless, 40,000 buildings damaged $15 BN insured damage, $44 BN economic Fires caused damage in San Fernando Valley, Malibu, Venice Liquefaction at Simi Valley

20. Northridge-PCS Estimates

21. Nisqually/(Seattle) (2001) Magnitude 6.8, 400 people injured Major damage in Seattle-Tacoma area Insured Damage $305 Million Max. intensity VIII in Pioneer Square area Landslides in the Tacoma area Liquefaction and sand blows

22. Earthquake vulnerability factors Building construction Unreinforced masonry vs. seismic designed Building height Taller buildings vulnerable to long-period waves Soft story (hotel lobby) increases vulnerability Building location Soil type is critical Fire following losses can be very significant

23. Financial model factors CEA mini-policy Earthquake sublimits on commercial Per policy Per location Regional sublimits (e.g. CA only) Interlocking clause Reduces event loss across multiple treaty years Hard to model

24. Differences between models Detailed vs. Aggregate Detailed models better capture these vulnerability and financial considerations Fire Following Significant difference in modelers New Madrid Significant difference in return period

25. Earthquake prediction Earthquakes not a Poisson process Poisson implies inter-arrival times are exponentially distributed (memory-less) 1999 Izmit (Turkey) Earthquake Increased risk for a quake in Istanbul San Andreas Fault Is an earthquake due? Where on fault?

27. Izmit Quake ctd. 60% chance of Istanbul earthquake in next 30 years - Thomas Parsons, USGS Researchers took into account the stress transfer from a magnitude 7.4 earthquake in Izmit, Turkey in August 1999.

28. San Andreas Fault Over the past 1,500 years large earthquakes have occurred at about 150-year intervals on the southern San Andreas fault. As the last large earthquake on the southern San Andreas occurred in 1857, that section of the fault is considered a likely location for an earthquake within the next few decades The San Francisco Bay area has a slightly lower potential for a great earthquake, as less than 100 years have passed since the great 1906 earthquake

29. Cat Models and Earthquake Pred. At least one cat modeling firm has variable earthquake rate (changes with calendar date) Annual model updates allow for changing earthquake rate with time.

30. Hurricanes Meteorology of Hurricanes Frequency of Hurricanes by category Recent Hurricane Activity Hurricane prediction Vulnerability and Financial Models

31. Meteorology of Hurricanes Occur in both Northern and Southern Hemispheres Don’t occur on the equator Factor in the 2004 Tsunami tragedy Coriolis Force spin clockwise in southern hemisphere spin counter-clockwise in northern hemisphere Need warm sea surface temperatures Always travel from east to west

33. Safir-Simpson Scale

34. Atlantic Basin Hurricanes

35. US Landfalling Hurricanes

36. Average & Recent Hurricane Activity To sum up: Average Hurricane numbers per year:

38. 12 tropical storms/hurricanes. Few hurricanes. Activity focused on the US coast. (red is hurricane)12 tropical storms/hurricanes. Few hurricanes. Activity focused on the US coast. (red is hurricane)

39. More activity (16) but focus moved away from the coast. (red is hurricane)More activity (16) but focus moved away from the coast. (red is hurricane)

40. Similar amounts of activity to ’03 season but now focus did swing back to the US Coast. 4 Hurricane Landfalls in FL.Similar amounts of activity to ’03 season but now focus did swing back to the US Coast. 4 Hurricane Landfalls in FL.

41. Lots of hurricanes and activity shifted even further westward into the gulfLots of hurricanes and activity shifted even further westward into the gulf

42. Very little hurricane activity and focus is away from coast. Good parts of 02 and 03 seasons combinedVery little hurricane activity and focus is away from coast. Good parts of 02 and 03 seasons combined

43. 2004 Hurricanes cost

44. 2005 Hurricanes cost

45. Hurricane Prediction

46. Hurricane Prediction, ctd.

47. Hurricane Prediction 2007

48. Vulnerability model factors Construction Concrete bunkers vs. mobile homes Location Properties near ocean very vulnerable to storm surge Secondary modifiers E.g. Roof tie downs

49. Financial model factors percentage deductibles can be very significant New season deductible in FL What is a risk? Issue for per-risk treaties For hurricanes, widely dispersed buildings on one policy often considered one “risk” E.g. school district

50. Differences between models Detailed vs. Aggregate models Location (distance to coast) is critical Need detailed model to properly assess Northeast Hurricane Significant difference between modelers Caribbean clash Not all modelers facilitate this analysis

51. Modeling Issues raised by ‘04/05 storms Storm Surge Demand Surge Frequency Distribution of Hurricanes Offshore oil rig losses Caribbean Clash modeling

52. Data/Modeling Issues Need for completeness Reinsurers need compensation for all risks being accepted Model all exposures Model all perils Run multiple models

53. Missing exposures Sometimes only get tier 1 wind counties Sometimes only certain states E.g. CA, Pacific NW, New Madrid only Other shake exposure ignored (e.g. East Coast) Fire following exposures ignored Sometimes entire books of business are missing Must cross-check cat model exposure data Premium often n.a. , policy counts (?)

54. Modeling Tricks Failing to load for LAE Failing to consider demand surge Abuse of secondary modifiers “Really, all my policyholders have roof tie-downs!” Running all the models and providing the lowest different modeling firms Aggregate vs. detailed models

55. Portfolio Management Event Set framework is a powerful tool for portfolio management Ability to model portfolio’s risk vs. return Determine portfolio capital and allocate to individual deals The last topic is portfolio management- how cat models can provide a powerful tool to assess a portfolio’s risk vs. return and how each contract in the portfolio contributes to the overall risk vs. return. The last topic is portfolio management- how cat models can provide a powerful tool to assess a portfolio’s risk vs. return and how each contract in the portfolio contributes to the overall risk vs. return.

56. Portfolio Framework Example Consider two countries Oceania and Eurasia 5 possible events for each country Industry losses specified Goal-determine risk vs. return for various reinsurance portfolios To keep things simple, consider a world with just two countries with cat exposure (George Orwell fans may recognize these from 1984). There are 5 possible events in each country with specified industry losses. Our goal is to determine how to assess the risk vs. return of reinsurance portfolios and how the contracts within the portfolio contribute to that risk/return. To keep things simple, consider a world with just two countries with cat exposure (George Orwell fans may recognize these from 1984). There are 5 possible events in each country with specified industry losses. Our goal is to determine how to assess the risk vs. return of reinsurance portfolios and how the contracts within the portfolio contribute to that risk/return.

57. Event Sets Here are the event sets for Oceania and Eurasia. Standard stuff.Here are the event sets for Oceania and Eurasia. Standard stuff.

58. Create a set of Simulation Years The next step is to come up with an annual aggregate framework. Again to keep things simple I have simulated 40 years worth of events. I have assumed that all 10 events are poisson distributed. In year 1, event O_4 occurred. No events occurred in years 2, 3, or 4. In year 5 event E_1 occurred, etc. In year 39, 3 events occurred O_4, E_3 and E_5. No event occurred in year 40. Note this event set has been carefully constructed to replicate the rates stated above-e.g. Event 0_4 has an annual rate of 10% and occurs exactly 4 times in the 40 year period, etc.The next step is to come up with an annual aggregate framework. Again to keep things simple I have simulated 40 years worth of events. I have assumed that all 10 events are poisson distributed. In year 1, event O_4 occurred. No events occurred in years 2, 3, or 4. In year 5 event E_1 occurred, etc. In year 39, 3 events occurred O_4, E_3 and E_5. No event occurred in year 40. Note this event set has been carefully constructed to replicate the rates stated above-e.g. Event 0_4 has an annual rate of 10% and occurs exactly 4 times in the 40 year period, etc.

59. Check against Poisson As a check we can compare this simulation set against the poisson distribution. Since each of the 10 events has been assumed to be poisson, the total number of events is also poisson distributed with mean 47.5%. According to the poisson then, we should expect that 11 of the 11.8 of the 40 years would have exactly one event- our simulation et has 12, etc.As a check we can compare this simulation set against the poisson distribution. Since each of the 10 events has been assumed to be poisson, the total number of events is also poisson distributed with mean 47.5%. According to the poisson then, we should expect that 11 of the 11.8 of the 40 years would have exactly one event- our simulation et has 12, etc.

60. Contracts Now let’s look at some contracts that are available in the market- for simplicity we will consider only 3. These are industry trigger contracts- one exposed in Oceania, one in Eurasia and one in both.Now let’s look at some contracts that are available in the market- for simplicity we will consider only 3. These are industry trigger contracts- one exposed in Oceania, one in Eurasia and one in both.

61. Calc. Contract Losses by year The 40 year simulation set allows us to calculate contract losses easily. For example in year 19, 2 events occur: O_3 and E_5, causing industry losses of 5,000 and 1,500 respectively. Contracts A is totalled by event O_3. Contract B is caused 750 of loss by event E_5. For contract C, event O_3 causes 4,000 of loss whereas E_5 causes 500 of loss for a total of 4,500 of loss. For the years not shown, the contract losses are obviously zero as no events have occurred.The 40 year simulation set allows us to calculate contract losses easily. For example in year 19, 2 events occur: O_3 and E_5, causing industry losses of 5,000 and 1,500 respectively. Contracts A is totalled by event O_3. Contract B is caused 750 of loss by event E_5. For contract C, event O_3 causes 4,000 of loss whereas E_5 causes 500 of loss for a total of 4,500 of loss. For the years not shown, the contract losses are obviously zero as no events have occurred.

62. Compute AAL and expected profit for each contract Next we can compute the AAL and expected profit for each contraact. Note the AAL is just the average of the losses for each contract over the 40 years. E[profit] = premium-AAL. To calculate the sd profit we need the distribution of profit by year for each of the 40 years…Next we can compute the AAL and expected profit for each contraact. Note the AAL is just the average of the losses for each contract over the 40 years. E[profit] = premium-AAL. To calculate the sd profit we need the distribution of profit by year for each of the 40 years…

63. Distribution of profit/(loss) For example contract C has premium of 2,000 and had 5,000 of losses in year 39 so the result is a loss (3,000).For example contract C has premium of 2,000 and had 5,000 of losses in year 39 so the result is a loss (3,000).

64. Calculate return on capital Next we want to calculate each contracts return on capital. To do this we first have to have a capital rule- how we will define capital. There are many such rules that can be used. For this example we will define capital as 2x the standard deviation of profit. Later on we will discuss a more tail-oriented capital metric. Based on our sd(profit) capital metric we see that contract A appears to be the most profitable followed by C and B.Next we want to calculate each contracts return on capital. To do this we first have to have a capital rule- how we will define capital. There are many such rules that can be used. For this example we will define capital as 2x the standard deviation of profit. Later on we will discuss a more tail-oriented capital metric. Based on our sd(profit) capital metric we see that contract A appears to be the most profitable followed by C and B.

65. Portfolio Effects Now assume that the reinsurer’s portfolio consists of certain shares of these 3 contracts Want to calculate the overall portfolio capital and Each contract’s share of this portfolio capital Now let’s consider a portfolio of certain shares of these 3 contracts. We want to calculated the overall portfolio capital and then how to allocate this capital to the individual contracts that make up the portfolio.Now let’s consider a portfolio of certain shares of these 3 contracts. We want to calculated the overall portfolio capital and then how to allocate this capital to the individual contracts that make up the portfolio.

66. Portfolio Consider the following portfolio: P = 20% A + 10% B + 5% C Then consider 3 other portfolios P+0.1% A P+0.1% B P+0.1% C Let’s consider a portfolio with 20% of contract A, 10% of contract B, and 5% of contract C. Then consider 3 other portfolios, where we add 0.1% of A, B, and C respectively to this base portfolio.Let’s consider a portfolio with 20% of contract A, 10% of contract B, and 5% of contract C. Then consider 3 other portfolios, where we add 0.1% of A, B, and C respectively to this base portfolio.

67. Portfolio ctd. Obviously the Portfolio Premium, Expected Loss (AAL) and E[Profit] are just weighted averages of those of the individual contracts using the weights 20%, 10% and 5%. For example, for the expected profit, 83.1 = 20%*225 + 10%*56 + 5%*650 However since the contracts are not perfectly correlated the standard deviations and capitals do not add in the same way. In other words the portfolio capital 422 is not equal to 20%*1037+10%*944+5%*4033 (which equals 503). Obviously the Portfolio Premium, Expected Loss (AAL) and E[Profit] are just weighted averages of those of the individual contracts using the weights 20%, 10% and 5%. For example, for the expected profit, 83.1 = 20%*225 + 10%*56 + 5%*650 However since the contracts are not perfectly correlated the standard deviations and capitals do not add in the same way. In other words the portfolio capital 422 is not equal to 20%*1037+10%*944+5%*4033 (which equals 503).

68. Allocating Portfolio Capital The portfolio capital can be allocated as follows: Cap[20%A]= 20%/0.1% * (422.89-422.02)=174 Cap[10%B]= 10%/0.1% * (422.56-422.02)= 54 Cap[5%C] = 5%/0.1% * (425.90-422.02)=194 -------------- -------- Cap[Portfolio] = 422 Here is the crucial step. To allocate the 422 in portfolio capital to the component contracts, we start by looking at how much capital each contract A, B, C contributes on the margin to portfolio capital. We ask the question, what if I wrote 0.1% more of contract A- how would that change the portfolio capital. It turns out that the capital would increase from 422.02 to 422.89 or a 0.87 increase- this can be simply calculated from the data on page 14. The allocated capital for the 20%A part of the portfolio is then calculated as above. Taking this 0.87 dividing by the 0.1% marginal share and multiplying by the actual 20% share. Same steps for B, C. The allocated capital exactly equals the portfolio capital!!!Here is the crucial step. To allocate the 422 in portfolio capital to the component contracts, we start by looking at how much capital each contract A, B, C contributes on the margin to portfolio capital. We ask the question, what if I wrote 0.1% more of contract A- how would that change the portfolio capital. It turns out that the capital would increase from 422.02 to 422.89 or a 0.87 increase- this can be simply calculated from the data on page 14. The allocated capital for the 20%A part of the portfolio is then calculated as above. Taking this 0.87 dividing by the 0.1% marginal share and multiplying by the actual 20% share. Same steps for B, C. The allocated capital exactly equals the portfolio capital!!!

69. Return on Allocated Capital Now that we have allocated capital to contract we can calculate the return on allocated capital for each contract in the portfolio. In this example, tt turns out that each contract has a higher return on allocated capital than it does on a stand alone basis. However the portfolio ROC is 19.7% which is lower than the stand-alone ROC for contract A. Based upon this, our current portfolio is clearly not optimal- it would be better just to write contract A alone. One could run this portfolio through an optimizer- I did this and found that the optimal portfolio had 22%, 8% and –2% of A, B, C respectively. The overall portolio ROC was 22.4%. The ROACs for each contract in the portfolio was also 22.4%.Now that we have allocated capital to contract we can calculate the return on allocated capital for each contract in the portfolio. In this example, tt turns out that each contract has a higher return on allocated capital than it does on a stand alone basis. However the portfolio ROC is 19.7% which is lower than the stand-alone ROC for contract A. Based upon this, our current portfolio is clearly not optimal- it would be better just to write contract A alone. One could run this portfolio through an optimizer- I did this and found that the optimal portfolio had 22%, 8% and –2% of A, B, C respectively. The overall portolio ROC was 22.4%. The ROACs for each contract in the portfolio was also 22.4%.

70. Tail oriented Capital Metrics Approach also works for tail oriented capital metrics- e.g. TVAR Define capital = 3 x TVAR (80%) Now let’s consider a more tail oriented capital metric- TVAR. In real life reinsurers often look at 90% TVAR or 95% TVAR. Since we only have 40 years, let’s look at 80% TVAR and define capital as 3x this.Now let’s consider a more tail oriented capital metric- TVAR. In real life reinsurers often look at 90% TVAR or 95% TVAR. Since we only have 40 years, let’s look at 80% TVAR and define capital as 3x this.

71. Tail oriented ROAC To calculate 80% TVAR we need to look at the 20% worst years in the simulation set- I.e. the 8 worst years. It is important to point out that since the contracts are not 100% correlated that these years will in general be different for each contract. TVAR 80% is just the average over these worst 8 years. Using this metric, the portfolio ROC is 18.9%. To calculate 80% TVAR we need to look at the 20% worst years in the simulation set- I.e. the 8 worst years. It is important to point out that since the contracts are not 100% correlated that these years will in general be different for each contract. TVAR 80% is just the average over these worst 8 years. Using this metric, the portfolio ROC is 18.9%.

72. Allocated Capital Calcs As before, alloc. capital based on marginal For example, for the 20%A contract: 450 = (793.5-791.25)/0.1% * 20% Portfolio Cap = Sum of Alloc. Capitals N.B. according to this capital metric, 10%B has the highest ROAC in the portfolio and contract C has the highest return on allocated capital in the portfolio and A the lowest –remember A had the highest ROAC under the sd metric. and contract C has the highest return on allocated capital in the portfolio and A the lowest –remember A had the highest ROAC under the sd metric.

73. Summary CAT Models provide a powerful tool for portfolio management Can be used to derive capital for a contract within a portfolio and ROC There is no “contract order” issue as is sometimes thought Portfolio can then be optimized to maximize ROC