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AIR Currents

October 24, 2012


Editor's Note: The AIR Institute, AIR's catastrophe risk management training and education program, is launching a series of AIRCurrents articles that explore the fundamentals of catastrophe modeling. This first one, by Greg Sly, Manager, Client and Consulting Services, and Nan Ma, Senior Editor, explains the do's and don'ts of combining exceedance probability metrics.

AIR software provides a wide range of modeled loss output to suit the diverse needs of catastrophe model users. One of the most commonly used types of output is a cumulative distribution of potential losses, along with their associated probabilities of exceedance, called an exceedance probability (EP) curve. Often, model users expect that losses at a particular exceedance probability can be simply added across perils, across regions, or across portfolios to calculate the combined loss. To explore why this is not the case, this article will provide a primer on AIR stochastic catalogs and what exceedance probabilities actually mean, along with some examples of when losses can and cannot be added.

The AIR Catalog

To develop a catastrophe model's stochastic catalog of simulated events, AIR scientists gather information from various sources including data about past events in order to infer what can happen in the future. Based on this combination of historical data and scientific expertise, AIR's stochastic catalog answers the questions of where and how frequently certain types of events are likely to occur and how large or severe they are likely to be.

Justin Pierce  

By: Greg Sly, CCM, and Nan Ma, CCM

A 10,000-year U.S. hurricane catalog, for example, contains 10,000 potential realizations of what an upcoming year of hurricane activity may look like. As in nature, each simulated year may have no U.S. hurricane landfalls, one, or multiple landfalls, as shown in Table 1. While the simulated events have their basis in historical data, they extend beyond the scope of recorded experience and do not mirror actual events exactly. And as no event will repeat itself in exactly the same way, events in the catalog do not have associated rates of occurrence. While we can estimate what the losses would be if Hurricane Andrew were to recur today, the probability that an actual hurricane will recur exactly as it did in 1992 is virtually zero.

Thus, rather than simulating what has happened in the past, AIR's catalogs simulate thousands of years of activity to provide a comprehensive view of risk that represents the entire spectrum of potential future catastrophe experience—not just events of average probability but also the most extreme and rare events that make up the tail of the statistical distribution. Because AIR's catalogs for different models are developed independently, they can be used in combination to provide a holistic view of risk across regions and perils.

Table 1. AIR catalogs, illustrated here for U.S. hurricane, include date of landfall, simulated storm parameters, and landfall location
Year Day Landfall Number Central Pressure Max Wind Speed Landfall Latitude Landfall Longitude Radius of Maximum Winds Forward Speed
1 265 1 956 110.3 34.78 -76.6 39.4 15.4
2 160 1 980.4 82.9 29.85 -84.17 23 8.5
2 272 1 978 91.1 28.89 -95.43 23.7 6.8
3 216 1 958 113.9 28.83 -95.51 30.2 11
3 216 1 989.2 79.1 32.27 -80.5 39.8 20.4
3 280 1 971.9 106.3 25.87 -80.15 14.8 11.6
4 198 1 911.6 126.1 30.36 -88.38 31.4 26.2
7 220 1 962.8 113 30.37 -88.2 25.1 16.1
8 205 1 977.3 90.3 34.56 -77.13 25.4 10.9
8 261 1 967.5 99.9 34.23 -77.76 27.2 9
11 259 1 973.6 110.5 29.19 -90.04 43.7 21.8
11 260 1 968.8 106.4 32.69 -79.96 9.7 19
11 260 2 986.8 86.9 41.48 -71.03 18.2 33.6
12 243 1 986.1 74.5 34.65 -76.93 39.5 8.5

Understanding Loss Exceedance Probabilities

When the catalog is run against a portfolio of exposure, the software can output the loss for each event and for all simulated years. This can then be used to calculate the probability of exceedance of various levels of loss, on either an annual occurrence basis or an annual aggregate basis.

To do this, losses are ranked, or sorted, from highest to lowest, based on the largest event loss within each simulated year (the occurrence loss) or based on the sum of all event losses within each simulated year (the aggregate loss). The exceedance probability corresponding to each loss is equal to its rank divided by the number of years in the catalog. For a 10,000-year catalog, for example, the 40th largest loss corresponds to the 40/10,000 or 0.40% exceedance probability (also known as the 250-year loss when expressed as a return period). As shown in Table 2 for a contrived portfolio, the correct interpretation of this metric is that there is a 0.40% probability that the portfolio will experience an aggregate loss of 153,560,977 or greater in a given year.

Table 2. Ranked annual losses to a hypothetical portfolio based on a 10,000 year U.S. hurricane catalog
Rank Loss Exceedance Probability
(Return Period)
Simulation Year Company Aggregate Loss
... ... ... ...
36 0.36% (277 years) 7059 161,869,892
37 0.37% (270 years) 3760 161,302,846
38 0.38% (263 years) 4548 160,577,066
39 0.39% (256 years) 6450 156,414,682
40 0.40% (250 years) 698 153,560,977
41 0.41% (244 years) 9604 147,410,429
42 0.42% (238 years) 7511 144,528,943
43 0.43% (233 years) 6974 143,893,274
44 0.44% (227 years) 1591 141,225,611
... ... ... ...

Combining Loss Metrics

A common error in interpreting model results is to add exceedance probability metrics across multiple perils, regions, or lines of business to compute the combined loss. For example, it is tempting to assume that the 1% exceedance probability loss for a portfolio exposed to both the hurricane and earthquake perils is simply the sum of the 1% EP loss for hurricane and the 1% EP loss for earthquake. This is not the case.

In the case of aggregate losses, each point (loss) on the aggregate EP curve corresponds to a specific simulation year of catastrophe activity. The likelihood that the 1% EP hurricane loss occurs in the same year as the 1% EP earthquake loss is close to zero, and it would be meaningless to add the losses from events that occur in different years (and even if they were to occur in the same year, it cannot be assumed that their combined loss would correspond to the 1% EP loss on the combined EP curve). The same objection holds for the occurrence EP curve, on which each point corresponds to the largest event loss in a specific year.

The examples below illustrate this point in further detail.

Combining Losses Across Perils

As is suggested by the discussion above, to obtain the aggregate EP loss across multiple perils, the losses must first be added by year and only then ranked to compute the combined peril loss. This is easily illustrated in the following much simplified example, which looks at losses to a hypothetical portfolio exposed to hurricanes and earthquakes. In this example, we'll use a much more manageable 10-year catalog rather than the usual 10,000-year catalog to illustrate how this calculation is done.

Suppose you want to find the 20% EP (five-year return period) loss for your portfolio from hurricanes and earthquakes combined. You run separate loss analyses using 10-year hurricane and earthquake catalogs to produce the annual aggregate losses shown in Table 3. Note that in only two of the 10 years do earthquake losses occur. (Of course, in the AIR software, you can run combined-peril loss analyses to aggregate losses across perils by simulation year automatically; this exercise is for illustrative purposes only.)

Table 3. Losses to a hypothetical portfolio from hurricanes and earthquakes, by simulated year
Hurricane Earthquake
Simulation Year Loss (USD millions) Simulation Year Loss (USD millions)
1 45 1 0
2 9 2 0
3 1,200 3 0
4 34 4 0
5 544 5 215
6 39 6 0
7 199 7 0
8 379 8 0
9 14 9 750
10 888 10 0

Table 4 shows the losses ranked separately for each peril, from largest to smallest. At the 20% exceedance probability, the hurricane loss is USD 888 million and the earthquake loss is USD 215 million. Simply adding these two numbers produces a combined loss of USD 1,103 million, but this would not be a meaningful value because these losses did not occur in the same simulation year.

Table 4. Separately ranked losses (X)
Hurricane Earthquake
Rank (Exceedance Probability) Simulation Year Loss (USD millions) Rank (Exceedance Probability) Simulation Year Loss (USD millions)
1 (10%) 3 1,200 1 (10%) 9 750
2 (20%) 10 888 2 (20%) 5 215
3 (30%) 5 544 3 (30%) 3 0
4 (40%) 8 379 4 (40%) 4 0
5 (50%) 7 199 5 (50%) 6 0
6 (60%) 1 45 6 (60%) 7 0
7 (70%) 6 39 7 (70%) 8 0
8 (80%) 4 34 8 (80%) 10 0
9 (90%) 9 14 9 (90%) 1 0
10 (100%) 2 9 10 (100%) 2 0

To correctly calculate the 20% EP loss to this portfolio from both perils combined, the losses for each peril must first be added by year, and then ranked, as shown in Table 5. In this case, the second highest combined loss (corresponding to the 20% exceedance probability) is USD 888 million. This loss occurred in Year 10 and is attributed to hurricane losses only (earthquake losses were 0 that year).

Table 5. Losses are combined first, then ranked (✓)
Simulation Year Combined Loss (USD millions) Rank (Exceedance Probability) Simulation Year Ranked Combined Loss (USD millions)
1 45 1 (10%) 3 1,200
2 9 2 (20%) 10 888
3 1,200 3 (30%) 9 764
4 34 4 (40%) 5 759
5 759 5 (50%) 8 379
6 39 6 (60%) 7 199
7 199 7 (70%) 1 45
8 379 8 (80%) 6 39
9 764 9 (90%) 6 39
10 888 10 (100%) 2 9

To calculate an occurrence EP curve for multiple perils, only the largest single event loss from each year (either hurricane or earthquake, in this example) is used. Losses from all other events that year, from any of the perils under consideration, do not have a bearing on the occurrence EP curve.

Combining Losses Across Regions or Zones

Similar to the example above for combining losses for different perils, EP metrics for different regions (or zones) cannot simply be added to calculate the combined loss metric. The error in this approach stems from the assumption that the simulated event that corresponds to a given EP loss to one region produces the same EP loss to another region. For a variety of reasons—including dissimilar geography, construction, and replacement values—this is usually not the case.

For example, suppose that a company divides their national portfolio into regional zones. Zone A contains exposure located entirely in Florida, while Zone B is concentrated in Texas and Louisiana. The largest loss-causing event for Zone A—for example a Category 4 hurricane making landfall near West Palm Beach—may cause only minimal losses for Zone B. Simply adding the losses at a given exceedance probability would not make sense.

Again, it is necessary to first combine losses by year (or event, if calculating occurrence losses for a particular peril) before ranking them (see Table 6). Note that at low exceedance probabilities, the combined loss is usually less than the sum of losses of the individual zones at that EP because of the effects of diversification (i.e., the different regions are less likely to suffer catastrophic losses in the same year or as a result of the same event).

Table 6. Losses at given exceedance levels from multiple zones cannot be added together to determine the combined loss
  AAL 5.0% 2.0% 1.0% 0.4% 0.2% 0.1%
Zone A 37,695,233 85,460,377 419,274,877 870,502,218 1,777,705,108 2,944,937,856 4,687,918,483
Zone B 9,753,215 39,039,688 71,478,640 110,627,388 173,434,169 300,949,785 442,842,103
Simple Addition 47,448,448 124,500,065 490,753,517 981,129,606 1,951,139,277 3,245,887,641 5,130,760,586
Actual Combined Loss 47,448,448 125,998,539 450,173,474 901,654,579 1,798,968,674 3,019,618,503 4,692,832,286

While this example involves combining losses from different regions, the same concept applies for combining other types of books or portfolios—for example, ones characterized by different lines of business, building/construction types, policy conditions, etc.

In AIR software, combined EP curves can be calculated by running the analysis at a higher level, for example at the business unit level instead of at the book level. Another option is to export the event losses from each analysis, sum the losses by event for that peril, and then either take the largest event loss per year if, calculating occurrence EP ,or sum all event losses for each year if calculating the aggregate EP. The losses can then be ranked to determine the combined EP metrics.

Combining Average Annual Losses

While simple addition does not work for losses at a given exceedance probability, it does work for the aggregate average annual loss (AAL). This is because the aggregate AAL is calculated by averaging losses across all modeled years, and is therefore independent of any given year. More precisely, the aggregate AAL is calculated by summing the annual loss across all years in the catalog and dividing the result by the number of years in the catalog (e.g., 10,000 years).

Occurrence AALs, however, cannot be added across perils, regions, or books and portfolios because it is calculated by determining the largest event loss for each year and averaging across all the event years in the catalog. It is unlikely that the largest occurrence loss in each simulated year is caused by the same event for different regions. Across different perils, the occurrence AAL is calculated by determining the largest event loss for each year (either earthquake or hurricane, for example) and averaging across all the event years in the catalog. The actual combined occurrence AAL is likely to be larger than each of the individual occurrence AALs for the separate perils, but most certainly smaller than the sum of the occurrence AALs for the separate perils. Note that the occurrence AAL does not provide a very robust measure of overall risk; is using the aggregate AAL is recommended.

Closing Thoughts

While AIR models output probabilities of loss, not of events, it is important to remember that the loss associated with each point on the exceedance probability curve corresponds either to a particular year in the catalog (for an aggregate EP) or a particular event in a particular year (in the case of an occurrence EP). Accordingly, losses at a given exceedance probability cannot simply be added across perils, portfolios, or regions. Exceedance probability metrics provide invaluable information about the risk profile of a portfolio, but care must be taken to ensure that metrics from different loss analyses are treated appropriately to yield meaningful results.

The authors would like to acknowledge Peter Baltatzidis and Shane Latchman for their valuable contributions.




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