Notes on Using Property Catastrophe Model Results

David Homer and Ming Li

Abstract

This article will discuss the use of results from popular Property Catastrophe mod-

els. It will explain common terms like Occurrence Exceedance Probability (OEP) and

Aggregate Exceedance Probability (AEP) and show how these are related to event

count and event size ideas. Simulation and the use of multiple models (blending) will

also be discussed.

Keywords. Catastrophe Modeling, Occurrence Exceedance Probability, OEP, Ag-

gregate Exceedance Probability, AEP, Probable Maximum Loss, PML.

1 Introduction

A reinsurance or insurance actuary will frequently need to work with the results of com-

mercial Property Catastrophe models. This work may include incorporating results such as

an average annual loss (AAL) into a pricing exercise or making additional calculations such

as simulating catastrophes in a capital model. This article will provide an introduction to

some of the simpler uses of commercial catastrophe models, including common terms, basic

calculations and simulation. Combining or blending of models will also be discussed.

1.1 Popular Cat Models

Two models will be reviewed along with their standard formats. These are Risk Management

Solution’s RMS platform and Verisk’s AIR platform.

1.1.1 AIR

The AIR output is provided in the form of sample data and some capital models refer to

this as pre-simulated data. Table 1 provides an example. The table values are illustrative

and don’t represent any particular exposures to actual losses. Columns are provided for

simulation number or year, event id, and claim size. This format is relatively easy to use

because it looks like a historical loss listing. This table is sometimes referred to as a year-

event loss table (YELT) because it provides loss detail by year and event. Note that Table

1 is missing year 2 and that year 3 has multiple events.

The mean and standard deviation of the annual loss from an AIR YELT created from n

simulated years with n

events in year y (which could be zero) are

µ = (

y=1

e=1

loss

y,e

)/n (1)

σ =

y=1



e=1

loss

y,e



− µ

(2)

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Table 1: AIR-style Year Event Loss Table (4 Years)

Year Event ID Loss

1 1 100

3 2 500

3 3 300

4 4 100

The events within each year are summed by year before computing the annual mean and

standard deviation. For the annual mean, this is equivalent to a straight sum of all the

event-year data divided by the number of years n. For Table 1 we have

µ = 250 = (100 + 500 + 300 + 100)/4 and (3)

σ = 320 =

(100)

+ (500 + 300)

+ (100)

)/4 − 250

. (4)

1.1.2 RMS

The RMS output is provided in the form of a list of parameters for each event. Table 2

provides a brief description of each column. There are two columns that need additional

comment, Sdi and Sdc. These two columns represent an approximation that RMS uses to

represent the standard deviation of the loss for a given event. The standard deviation for the

event loss is the sum of an independent component, Sdi, and a correlated component, Sdc.

This split facilitates RMS calculations as the event loss is built up from components whose

individual losses are partially dependent on one another. Later on we will discuss events

that are split into subcomponents like Personal lines and Commercial lines. The Sdi-Sdc

split will be important then, but for now we can just think of their sum as the standard

deviation of the event loss.

Table 2: RMS-style Event Loss Table Parameters

Column Name Description

Event ID Unique identiﬁer of the event

Rate Annual event frequency

Mean Average Loss if the event occurs

Sdi Independent component of the spread of the loss if the event occurs.

Sdc Correlated component of the spread of the loss if the event occurs.

Exposure Total amount of limits exposed to the event (Maximum loss)

Table 3 provides an example of RMS output. This table is often referred to as an event

loss table (ELT) because it provides event details. As before, the table values are illustrative

and don’t represent any particular exposures to actual losses.

The mean and standard deviation of the annual loss described by an RMS ELT with m

event rows are

µ =

e=1

(Rate

)(Mean

) (5)

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Table 3: RMS-style Event Loss Table

Event ID Rate Mean Sdi Sdc Exposure

1 .10 500 500 500 10,000

2 .10 300 400 800 5,000

3 .50 200 300 400 4,000

σ =

e=1

(Rate

)((Sdi

+ Sdc

)

+ Mean

). (6)

For Table 3 we have

µ = 180 = (.1)(500) + (.1)(300) + (.5)(200) (7)

σ = 737 = [(.1)((500 + 500)

+ 500

) +

(.1)((400 + 800)

+ 300

) +

(.5)((300 + 400)

+ 200

)]

1/2

. (8)

These formulae can be derived by assuming each event is an independent collective risk

model (CRM)

with Poisson mean Rate

and a severity distribution with mean Mean

and

standard deviation Sdi

+ Sdc

A common use for these tables is to apply reinsurance terms and then estimate prices or

distributions net of reinsurance. This is straightforward with AIR-style data (Table 1) and

a bit more diﬃcult with RMS-style data (Table 3). So it is common to use the parameters

from RMS-style data to simulate individual events and then work with the simulated data

directly. Simulating from RMS-style ELTs will be discussed in section 3.2.

1.2 OEP, Return Period, AEP and PML

The terms Occurrence Exceedance Probability (OEP), Return Period, Aggregate Exceedance

Probability (AEP), and Probable Maximum Loss (PML) are commonly used and commonly

confused. Sometimes OEP and AEP will be abbreviated as EP (Exceedance Probability)

[GK05]. We will step through each term and explain it. To begin, it is useful to think of these

deﬁnitions in the context of a collective risk model with an annual event count distribution

and an event size distribution. Imagine simulating a set of losses from these distributions

and presenting the results in a form similar to Table 1. The statistics that we want, like

OEP and AEP, can then be computed from that table.

1.2.1 OEP

The Occurrence Exceedance Probability(OEP) curve O(x) describes the distribution of the

largest event in a year. In particular, O(x) is the probability that the largest event in a year

exceeds x.

A collective risk model assumes a claim count N and claim sizes X

, i = 1, ..., N with each X

independent

and identically distributed and each X

indepdendent from N .

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The distribution of the largest event in the year is not the same as the distribution of the

event size.

Consider our AIR-style (Table 1) losses. The empirical claim size distribution F

(x)

shown in Table 4 reﬂects all event losses. In contrast, the OEP curve is derived from the

Table 4: Empirical Claim Count and Severity Distribution Derived from Table 1

x Pr(X = x) n Pr(N = n)

100 50% 0 25%

300 25% 1 50%

500 25% 2 25%

largest event in each year, which is shown in Table 5. Table 6 presents our empirical OEP

Table 5: Largest Events by Year Derived from Table 1

Year Largest Event

1 100

2 0

3 500

4 100

curve.

Table 6: Empirical OEP Curve Derived from Table 1

PML

occ

OEP Return Period

x O(x) r = 1/O(x)

0 75% 1.33

100 25% 4.00

500 0% ∞

The OEP is often used by primary insurers to help select their catastrophe reinsurance

program limits and retentions.

1.2.2 Return Period

It is common to talk about a return period r instead of the OEP, where r = 1/O(x). It is

the expected number of years between events that exceed x.

1.2.3 PML

The dollar amount of loss x is often called the Occurrence Probable Maximum Loss (PML)

at return period r, or simply the PML for the return period r. Thus,

1/r = O(x) = O(PML

occ

) (9)

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PML

occ

(r) = O

−1

(1/r), (10)

where O

−1

(x) is the inverse OEP function. The OEP and the PML are linked. Sometimes

actuaries will refer to an OEP curve or a PML curve; they refer to the same thing. Table

6 represents an OEP or PML curve. The PML column shows dollars and the OEP column

shows probabilities, though often OEP is supplemented or replaced with its reciprocal, the

return period. As part of their rating process, AM Best asks companies for their Occurrence

PML losses for the 100-year return period for wind and for the 250-year return period for

earthquake. [Irw16]

1.2.4 AEP

The Aggregate Exceedance Probability(AEP) curve A(x) describes the distribution of the

sum of the events in a year. In particular, A(x) is the probability that the sum of the events

in a year exceeds x.

The AEP is not the same thing as the OEP, but is often confused with it.

The AEP can be very diﬀerent from the OEP when the probability of two or more events

is signiﬁcant. The AEP and OEP can be similar when the probability of two or more events

is very small; they are identical when there is zero probability of two or more events. (See

appendix A.) The AEP is used to help consider the total volume of catastrophe events in a

year. The total losses by year from Table 1 are shown in Table 7 and used to compute an

empirical AEP in Table 8.

Table 7: Total Losses by Year Derived from Table 1

Year Total Losses

1 100

2 0

3 800

4 100

Table 8: Empirical AEP Curve Derived from Table 1

PML

agg

AEP Return Period

x A(x) r = 1/A(x)

0 75% 1.33

100 25% 4.00

800 0% ∞

Note that our empirical OEP and AEP curves are not the same. However, our OEP and

AEP curves would be identical if year 3 did not have a second event.

Sometimes modelers will use the term aggregate PML which is deﬁned in a manner

similar to the occurrence PML but with the aggregate distribution. The aggregate PML is

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essentially the inverse function of the AEP.

PML

agg

(r) = A

−1

(1/r). (11)

It should be noted that PML is often used informally and its meaning is not always clear.

Usually PML used by itself is understood to mean Occurrence PML, but it can also refer to

an Aggregate PML. It may simply refer to an intuitive notion of a large loss without a well

deﬁned statistical meaning.

2 OEP and the Collective Risk Model

Sometimes a reinsurance actuary will receive an OEP table or even part of one and be asked

to apply reinsurance terms for pricing. In these situations it is helpful to be able to reverse

engineer a claim count distribution F

(n) and a severity distribution F

(x) from the OEP

curve. Using the claim count and severity distributions one can then simulate individual

losses and apply reinsurance terms to the simulated data. It is easy to start with detailed

event loss data and compute the OEP curve as we did with Table 1 and Table 6, and just a

bit harder to go the other way.

Conversely, there may be situations where an actuary starts with the claim count dis-

tribution and claim size distribution and it may be convenient to compute the OEP curve

directly, without simulating.

These tasks are relatively easy if we assume that the vendor models can be represented

by a collective risk model with independent claim counts and independent and identically

distributed claim sizes. This is probably an oversimplication, but it provides a convenient

and useful framework.

2.1 Converting OEP Curves to Claim Count/Severity Curves

There is substantial information contained in the OEP and it is tightly connected to the dis-

tribution of the number of events in a year and the distribution of the size of an event. Given

the cumulative distribution function (cdf) F

(x) for the claim size X and the probability

function P

(n) for the claim count N, O(x) can be written explicitly.

O(x) = Pr(M > x) where M = max(X

, ..., X

) (12)

= 1 − Pr(X

≤ x for i = 1, ..., N) (13)

= 1 − E

(x)

) = 1 − PGF(F

(x)) (14)

where PGF(x)

is the probability generating function for N. The claim size cdf F

(x) may

then be derived from this equation. For some claim count distributions PGF(t)

−1

is easily

expressed and we obtain

(x) = PGF

−1

(1 − O(x)). (15)

This process does not generally produce a unique size distribution F

(x) because we need

to select the claim count distribution F

(n) and its parameters. A diﬀerent F

(n) will yield

a diﬀerent F

(x). However, the size distributions computed this way will be consistent with

the starting OEPs and the claim count assumption.

The PGF of a discrete distribution is deﬁned as PGF(t) = E(t

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2.1.1 OEP Conversion Example

Let’s take our empirical OEP (Table 6) and using the empirical PGF

PGF(t) = 0.25 + .5t + .25t

, (16)

estimate F

(100). In practice, we usually use the Poisson claim count assumption, but it

is convenient here to stick with the empirical ﬁgures. We don’t have a closed form for the

inverse function of the empirical PGF, but since it is a quadratic we can solve for the roots.

O(100) = .25 = 1 − (.25 + .5F

(100) + .25(F

(100))

). (17)

0 = −.25(F

(100))

− .5F

(100) + (1 − .25 − .25) (18)

The negative root yields

F (100) = .73 =

.5 −

− (4)(−.25)(.5)

(2)(−.25)

(19)

Table 9 completes this process. The inverted claim size curve is a coarse approximation (we

are only working with three points), but it is entirely consistent with the starting OEP.

Table 9: Claim Size Distribution from OEP

x O(x) Inverted F

(x) Table 4 F

(x)

0 75% 0% 0%

100 25% 73% 50%

500 0% 100% 100%

2.1.2 OEP and the Poisson Distribution

For the Poisson claim count distribution we have

PGF(t) = exp(λ(t − 1)) (20)

O(x) = 1 − exp(λ(F (x) − 1)) (21)

F (x) = 1 +

log(1 − O(x))

. (22)

Equation 22 can be used to convert an OEP to an event size distribution if an estimate of λ

is available. This is very convenient.

In theory, λ may be estimated directly from O(x).

exp(−λ) = Pr(0) = 1 − O(0) (23)

=⇒ λ = − log(1 − O(0)). (24)

The OEP can “pack” both claim count and severity information if the count distribution is Poisson and

(0) = 0. See appendix B

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This may be diﬃcult to apply in practice since it is common to receive only a partial OEP

curve without an entry for zero or Pr(0) will be very nearly zero when the catastrophe

distribution includes frequent losses.

A common practice is to take the smallest claim size entry x

min

of interest and compute

λ = − log(1 − O(x

min

)). (25)

In this case, λ represents the Poisson mean for claims greater than x

min

and equation 22 is

applied to produce a claim size distribution for claims greater than x

min

. Note, F (x

min

) = 0.

2.1.3 OEP and the Negative Binomial Distribution

Using the mean-contagion form [HM83] for the Negative Binomial claim count distribution

we have

PGF(t) = (1 − cλ(t − 1))

−1/c

(26)

O(x) = 1 − (1 − cλ(F (x) − 1))−1/c (27)

F (x) = 1 +

1 − (1 − O(x))

−c

cλ

. (28)

3 Aggregation and Simulation of Cat Losses

A common use for catastrophe modeling output is to feed it into capital models to be

mixed with other sources of loss. Randomly drawn catastrophe losses are combined with

randomly drawn losses from other sources. In order to do this the capital model has to have

a mechanism for using the catastrophe output. In the case of AIR-style output it is often

as simple as randomly drawing a year and then looking that year up in a table like Table

1. In the case of RMS-style output, the capital model needs to use the parameter table to

perform its own simulation.

3.1 AIR

The YELT produced by AIR is essentially pre-simulated data and can be used directly

or re-sampled. One should be careful with re-sampling if the results are to be combined

with other AIR model results (for example, merging two companies’ cat results) because

the dependencies among events can be destroyed by independently sampling two separate

YELTs that share perils. Two AIR-style tables should be joined by common years and

events. Alternatively, one can draw a single random year and use the same year to extract

losses from both tables.

3.2 RMS

The RMS ELT contains parameters for both the number of events and the size of each event.

3.2.1 Number of Events

The number of events N can be simulated from a Poisson with mean λ set to the sum of the

ELT rates.

λ =

Rate

(29)

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N ∼ Poisson(λ). (30)

3.2.2 Size of Events

A claim size can be simulated for each claim in two steps. First, we determine which event

occurs, that is, which ELT row are we using. This is done by drawing a random row/event

R from the ELT with each row/event having a probability in proportion to its rate.

U ∼ Uniform(0, 1) (31)

R = min{r : U ≤

i=1

Rate

/λ} (32)

Second, now that we know which row or event we are using, we use the event parameters

to draw a random claim size X from a scaled Beta distribution. The Beta distribution

parameters a and b are computed as follows:



Mean

Sdi

+ Sdc



1 −

Mean

Exposure

−

Mean

Exposure

(33)

= a



Exposure

Mean

− 1



(34)

X ∼ (Exposure

)(Beta(a

, b

)). (35)

The cdf for the Beta distribution is the incomplete Beta function

F (x) = β(a, b; x) =

Γ(a + b)

Γ(a)Γ(b)

a−1

(1 − t)

b−1

dt. (36)

In EXCEL one can generate a scaled beta variate with

= BETA.INV(RAND(),a

, b

)∗Exposure

. (37)

Table 10 illustrates RMS-style simulation using the parameters from our RMS-style ELT

(Table 3).

Table 10: RMS-Style Simulation using Table 3-ELT

Poisson Uniform Beta Beta Scaled

Trial Count Draw Row Parameter Parameter Beta

N U R a b X

1 1 0.70 3 11.5875 220.1625 204

2 0 – – – – –

3 2 0.15 2 14.9800 234.6867 272

3 – 0.40 3 11.5875 220.1625 168

4 1 0.05 1 3.3750 23.6250 268

This procedure works if the ELT does not have event parameters sub-divided by region or

line of business. When each event is sub-divided by region or line of business the simulation

process requires additional steps to preserve dependencies between sub-divisions. Table 11

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Table 11: RMS-Style ELT with Two Sub-categories

Personal Lines Commercial Lines

Event ID Rate Mean Sdi Stc Exposure Mean Sdi Sdc Exposure

1 0.1 300 400 300 3000 200 300 200 1000

2 0.1 100 371 267 1000 200 150 533 4000

3 0.5 100 224 200 2000 100 200 200 2000

provides an example of an ELT with sub-divisions. In order to simulate from Table 11 we

need to aggregate it to make it look like Table 3. The following approximation has worked

well for the authors.

1. Aggregate the event parameters as follows

Mean

R,k

(38)

Exposure

R,k

(39)

Sdi

R,k

(40)

Sdc

R,k

. (41)

2. Apply equations 33–35 to the results of step 1 (equations 38–41) to simulate the total

event loss X.

3. Allocate X to the sub-divisions X

= X

Mean

R,k

Mean

. (42)

This allocation is not perfect but it assures that the sub-categories sum to the simulated

total and preserves much of the component dependencies. The values in Table 11 can be

aggregated across Personal and Commercial Lines using equations 38–41 to reproduce our

simpler RMS-style ELT (Table 3).

4 Model Blending

The catastrophe models available in the market can produce a wide range of loss results.

Companies that use these models need to understand the model diﬀerences and determine

the best model(s) to manage their catastrophe risk. A common practice of using multiple

models is to blend the models together. For example, Florida Hurricane Catastrophe Fund

uses a weighted average of ﬁve models (RMS, AIR, EQE, ARA, FPM) in their ratemaking

[Inc16]. The beneﬁt of model blending is that it reﬂects elements of a range of models,

stabilizes changes in individual models across time and yields a single set of results.

It is beyond the scope of this paper to discuss how to determine the weights that should

be used to blend the models. We will focus on the technical approaches that can be used to

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blend the models, assuming the weights have already been determined. The model blending

approaches can become quite complicated if we consider breaking down the models into

various components and blending them at the component level. However, those approaches

need to be supported by tremendous amount of independent research and the practical

diﬃculties have limited their applications. Below we will discuss two more straightforward

and much more widely used approaches for model blending: ELT/YELT blending and OEP

blending.

4.1 ELT/YELT Blending

When the YELT results are provided, we can blend them together by following the steps

below. For the sake of simplicity, we assume that the RMS (ELT) and AIR (YELT) loss

results are provided and the blending weights to be used are ω RMS and (1-ω) AIR, which

can be easily generalized to other cases if necessary.

1. Convert RMS ELT to YELT format using Monte Carlo simulation described in section

3.2.

2. Sample from a uniform distribution. For a given year, if the sampled value is less than

ω, take the losses from the RMS YELT, otherwise take the losses from the AIR YELT.

3. Repeat the above for year 1 to year 10k to create a 10k blended YELT.

This process is illustrated in Table 12 for a 50/50 weighting. In terms of the OEPs of the

Table 12: 50/50 Blending of Models Using AIR-Style Table 1 and RMS-Simulation Table 10

Model Model Event

Trial Uniform Selected Count Loss

1 0.599 AIR 1 100

2 0.041 RMS 0 –

3 0.401 RMS 2 168

3 – – – 268

4 0.925 AIR 1 100

component models, the theoretical OEP derived from blending the simulations is

mix

(x) = ωO

rms

(x) + (1 − ω)O

air

(x), (43)

where ω is the weight given to the RMS model. The advantage of this approach is that

it produces a blended set of results comprised of speciﬁc modeled events, it is simple to

implement, and it can be used to model dependencies with other portfolio results as long

as the same uniform draw and technique is used to blend the other portfolio results. Its

disadvantage is that the blended results with this approach could be diﬀerent from the

blended OEPs that are usually presented to the management teams, regulatory bodies and

rating agencies.

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4.2 OEP Blending

In practice, the modeling results are often presented to various interested parties as sum-

maries like the OEPs instead of the underlying ELTs or YELTs. The weighting factors are

usually directly applied to the dollar amounts x for a ﬁxed return period r or exceedance

probability 1/r. Table 13 shows a 50/50 blend of RMS and AIR results derived from Tables

1 and 10. The Table 13 AIR PML for return period 2 was interpolated from Table 6 and

Table 13: 50/50 OEP Blending Using AIR-OEP Table 6 and RMS-Simulation Table 10

Return AIR RMS 50/50

Period PML PML PML

1.33 0 0 0

2 50 204 127

4 100 268 186

∞ 500 272 384

the RMS PML column was constructed from Table 10 RMS simulated losses.

The theoretical OEP for this approach, in terms of the occurrence PMLs, is

PML

mix

(r) = ωPML

rms

(r) + (1 − ω)PML

air

(r) or (44)

blend

(PML

mix

) = 1/r. (45)

This is not equivalent to ELT/YELT Blending. The ELT/YELT blending essentially weights

probabilities at ﬁxed amounts while the OEP blending weights dollars amounts at ﬁxed

probabilties. A better name for OEP blending might be PML blending.

The OEP blending is certainly very intuitive and it has become a common practice

to present results this way. However, this only provides a high level summary of blended

results. For some calculations, actuaries need the underlying loss details by event. The

OEP conversion technique introduced in section 2.1 can be applied to the blended OEPs to

produce claim count and severity distributions that can be used in simulation models and

will yield the “blended” OEP curve.

5 Conclusion

It is helpful to understand the various terms used by consumers of catastrophe modeling and

their relationship with the traditional claim count/severity collective risk model (CRM).

In particular, one can avoid common areas of confusion:

1. The OEP and AEP are not the same.

The OEP and AEP keep track of diﬀerent random variables, respectively, the largest

event each year versus the total of each year’s events.

2. The probable maximum loss (PML) can be associated with the OEP or the AEP.

PML is often used informally and usually refers to the dollar amount x associated with

a particular return period r or exceedance probability 1/r, that is, the inverse OEP

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function,

PML(r) = x = O

−1

(1/r). (46)

Sometimes PML refers to the AEP, where it might be called the aggregate PML, and

it becomes the inverse AEP function (A

−1

PML(r) = x = A

−1

(1/r). (47)

3. Blending OEPs is not the same as blending simulated results.

It is a relatively simple experiment to take two cat models and compare a 50/50 weight-

ing of their Occurrence PML curves (OEP blending) with the OEP curve produced

by simulating from one model half the time and the other model the rest of the time

(ELT/YELT Blending). These are not the same. The former weights dollar amounts

for a ﬁxed exceedance probability or return period, while the latter weights probabili-

ties for a ﬁxed dollar amount. It is an unfortunate use of language that OEP blending

actually refers to weighting PMLs or dollar amounts while ELT/YELT blending actu-

ally refers to weighting probabilities (or OEPs).

The OEP contains substantial information and it can be used to infer information about

all events. Catastrophe modeling can be viewed in the context of a collective risk model

(CRM) with a claim count distribution and a severity distribution. Understanding the

connection between the OEP and an underlying CRM allows one to go back and forth

between the two forms.

One can convert OEP output from a vendor model into claim count and severity distri-

butions that can then be included in a capital model that uses claim count/severity inputs.

Conversely, one can compute OEPs from a custom model, built from a traditional claim

count/claim size approach, and compare the custom results with vendor models.

The trick is to understand what the components are and how they are diﬀerent.

Acknowle dgment

The authors would like to thank Bradley Andrekus, James Chang, Kara Kemsley, and

Mohsen Rahnama for numerous thoughtful and helpful suggestions.

A When are the AEP and the OEP alike?

The AEP is generally not the same as the OEP. However, the two can be similar when the

probability of 2 or more claim counts is very small. They are identical when the probability

of 2 or more claim counts is zero. To see this, recall that for a collective risk model with size

cdf F

(x) and count probabilities P

(n) the aggregate distribution for

Z = X

+ ... + X

(48)

(x) =

(n)F

(n)

(x), (49)

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where F

(n)

(x) is the nth convolution of F

with itself. Therefore, the AEP which is the

probability of annual losses Z exceeding a given amount x is 1 − F

(x) or

A(x) = 1 −

(n)F

(n)

(x). (50)

Compare this to the OEP

O(x) = 1 −

(n)(F

(x))

. (51)

When P

(n) = 0 for n > 1 then A(x) = O(x) because F

(1)

= F

. Similarly, A(x) ≈ O(x) if

A(x) − O(x) =

∞

n=2

(n)(F

(n)

(x) − (F

(x))

) (52)

is suﬃciently small.

B OEP packing

Generally speaking, we need to add information about the claim count distribution to our

knowledge about the OEP in order to compute a size distribution consistent with the OEP.

However, if we can make two speciﬁc assumptions then we can compute a size distribution

solely from the OEP.

1. There is no point mass at zero in the claim size distribution, F

(0) = 0.

2. The claim count distribution is Poisson.

We can imagine all the claim count and severity information as packed into the OEP when

these two conditions are met. The claims size distribution F (x) is extracted as described

in section 2.1.2 and the Poisson mean is extracted from O(0). Recall from equation 22 for

Poisson counts that

F (x) = 1 +

log(1 − O(x))

. (53)

Adding that F (0) = 0 implies

λ = − log(1 − O(0)). (54)

References

[GK05] Patricia Grossi and Howard Kunreuther, editors. Catastrophe Modeling: A New

Approach to Managing Risk (Huebner International Series on Risk, Insurance and

Economic Security). Springer, 2005th edition, 2005.

[HM83] Philip E. Heckman and Glenn G. Meyers. The calculation of aggregate loss distribu-

tions from claim severity and claim count distributions. Proceedings of the Casualty

Actuarial Society, LXX:36, 1983.

[Inc16] Paragon Strategic Solutions Inc. Florida hurricane catastrophe fund (fhcf)

2016 ratemaking formula report. page 7, 2016. https://www.sbafla.

com/fhcf/Portals/FHCF/Content/AdvisoryCouncil/2016/0315/2016_

RatemakingReportFINAL.pdf?ver=2016-06-08-094808-847.

Notes on Using Property Catastrophe Model Results

Casualty Actuarial Society E-Forum, Spring 2017-Volume 2

[Irw16] Stephen Irwin. Understanding BCAR for u.s. property/casualty insurers. page 18,

2016. http://www3.ambest.com/ambv/ratingmethodology/OpenPDF.aspx?rc=

197686.

Notes on Using Property Catastrophe Model Results

Casualty Actuarial Society E-Forum, Spring 2017-Volume 2