properly pricing the catastrophe exposure

04/19/231

Properly Pricing the Properly Pricing the Catastrophe ExposureCatastrophe Exposure

David Chernick, FCAS, MAAAMichael Devine, FCAS, MAAASara Drexler, FCAS, MAAACAS Ratemaking SeminarMarch, 2004

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AgendaAgenda Introduction History of Methods

– Terminology– Exposure Base– Capping

Discussion of Several Alternative Approaches– Traditional– Methods Using Cat/AIY – State Based– Methods Using Cat/AIY – Countrywide Based

Wrap-up

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IntroductionIntroduction

The PanelistsThe DataThe Issue

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The Panelists

IntroductionsSara DrexlerMichael DevineDavid Chernick

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IntroductionIntroduction The Data

The base data we are using in this presentation is included in the handouts.Real dataLarge Catastrophe in 1998

343.2% loss ratio9.67 Ratio of Cat/AIY

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The Issue– Operating results

$19.1 Million profit prior to 1998$41.4 Million loss in 1998

– StatisticsMean:1998 was 22.7 Standard Deviations from mean.

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The Issue– A rate should include all costs associated with

the transfer of risk.– 20 or 30 or even 40 years of data is not

sufficient to properly quantify the tail of the distribution

– What is the true prospective average (mean) catastrophe provision?

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The Issue– Perspective of this presentation is from large

insurers without reinsurance coverage.– Reinsurance covering some portion of the

catastrophe exposure would most likely be an upper bound of the true mean.

– What is the true prospective average (mean) catastrophe provision?

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TerminologyTerminology

AIY – Amount of Insurance years 1AIY=$1,000 of dwelling coverage

Losses/AIY – Damage Ratios or Cat/AIY

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History of MethodsHistory of Methods

Category Data

Averaging MethodsAdjustment for Extreme Cats

Category # 1ISO Excess Wind &Variations

A. Ratio of wind losses to ex-wind losses

B. Ratio of cat losses to ex-cat losses

Base: Non-wind incurred lossesCat Data: Annual wind incurred losses over threshold

Base: Non-cat incurred lossesCat Data: Incurred event losses over fixed external threshold

Arithmetic average of excess wind factors and average wind to ex-wind ratios. For each additional year, give 95% weight for long-term average, 5% weight for additional year.

Can give less weight to unusual years

Can adjust catastrophes for unusual years

Category #2Cat/AIY Methods

A. Utilize state Cat/AIY with no direct use of countrywide cat experience

B. Utilize state Cat/AIY with direct use of countrywide cat experience

Base: AIYCat Data: Incurred event losses over fixed external threshold

Straight long-term average Confidence interval

approach Average adjusted catastro-

phes and add increment for extreme

Trended exponential smoothing

Balance state provisions to countrywide expectation

May or may not include adjustments for extreme catastrophes

Category #3Modeled Cat Losses

Base: AIYCat Data: Computer application generated losses

Average annual losses are based on the stochastic event set

Extreme events are given their appropriate statistical weight

Category #4Reinsurance Cost Based

Cat cover reinsurance cost passed through primary pricing

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What Base to Relate Catastrophes To?What Base to Relate Catastrophes To?

Premium:Cat provisions impacted by rate changesTrends in non-catastrophe loss & expense dictate cat provision

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Non-Cat Loss/Ex-Wind Loss:Still heavily dictated by trends in Crime, Liability, etc. lossEx-wind losses can include catastrophic losses

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Non-Cat Loss/Ex-Wind Loss:Still heavily dictated by trends in Crime, Liability, etc. lossEx-wind losses can include catastrophic losses

AIY or Amount of Insurance Years Definition: $1000 of Building Coverage in force for one year

Inflation sensitiveDirect measurement of exposure – incorporates policy growth and changes in building costs

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Should Individual CatastrophesBe Capped?

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Should Individual CatastrophesBe Capped?

Stabilizes provision Can serve to more appropriately match experience period used with event return periods Potentially more accurate estimate of expected value results

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What Are Some Problems WithCapping Individual Catastrophes?

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What Are Some Problems WithCapping Individual Catastrophes?

What criteria should be used? The “unthinkable” is happening every year somewhere. Is the result systematic underestimation of loss costs? How do we really know appropriate event return periods?

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Insurance Services Office (ISO)Insurance Services Office (ISO)Excess Wind ProcedureExcess Wind ProcedureBasic Approach Separate wind & non-wind losses Examine wind/non-wind ratios Years where wind/non-wind exceed 1.5 times

median are “excess” Average factor for excess wind Factor developed for excess wind applied to non-wind,

non-excess losses

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Insurance Services Office (ISO)Insurance Services Office (ISO)Excess Wind ProcedureExcess Wind ProcedureBasic Approach Separate wind & non-wind losses Examine wind/non-wind ratios Years where wind/non-wind exceed 1.5 times

median are “excess” Average factor for excess wind Factor developed for excess wind applied to non-wind,

non-excess losses

Characteristics Straightforward application Definition of “excess wind” can change as median changes Assumes stable relationship between wind & non-wind losses Doesn’t consider variability of wind losses Doesn’t consider non-wind catastrophes

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The Fix ‘Em Up Insurance GroupThe Fix ‘Em Up Insurance GroupHomeownersHomeowners

The State of Mich-con-otaThe State of Mich-con-ota20-Year Average Approach20-Year Average Approach

Year

Amount ofInsurance Years

(AIY)Cat Incurred

Loss CAT/AIYRunningProvision

1993199419951996199719981999200020012002

4,276,1354,306,8154,540,9134,774,7835,001,1645,193,1905,367,5665,574,5065,745,3446,223,199

$ 307,946 259,784 4,378 1,529,513 2,736,486 50,241,886 8,141,594 6,676,296 12,086,512 6,091,053

0.07200.06030.00100.32030.54729.67461.51681.19762.10370.9788

0.21480.21020.17190.18620.21280.69480.71140.75660.84650.8929

Provision (20-Year Average ) 0.8929

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Confidence Interval Approach

Factors include risk tolerance, surplus position/ availability of capital, reinsurance Determine confidence demands for long-term companywide cat provision Calculate companywide mean cat/aiy Calculate standard deviation of mean cat/aiy

Step 1 – Establish Company Objective:

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Company has established that it would like to be 90% certain it has an adequate catastrophe provision over the long-term The following have been calculated from the companywide catastrophe history: Mean Cat/AIY = .3151

Standard Deviation of Mean Cat/AIY = .0372

Step 1 – Establish Company Objective (Cont.):

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The long-run companywide benchmark cat provision is established as follows: Provision (Cat/AIY) = Mean + (t) x (Standard Deviation) = .3151 + (1.323) x (.0372) = .3643 Where : Mean = average cat/aiy companywide 1.323 = t – stat for 90% and (N-1) degrees of freedom .0372 = standard deviation of the mean

Step 1 – Establish Company Objective (Cont.):

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Goal period becomes interval rates are in effect Need to be reasonably certain provision is adequate Desire to use cap on individual cats to limit volatility Largest 5% of companywide cats exceeded .65/AIY Establish required confidence for state capped average

Step 2 – Establish State Level Objective:

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Non-Hurricane Catastrophe ProvisionsNon-Hurricane Catastrophe Provisions

ConfidenceIntervals

Sum of StatesUncapped Capped

CompanywideUncapped

50% 55 60 65 70 75 80 85 90 95

0.3334 0.2682 0.3825 0.2998 0.4328 0.3323 0.4848 0.3657 0.5392 0.4008 0.5988 0.4392 0.6657 0.4823 0.7447 0.5332 0.8452 0.5981 0.9999 0.6973

0.31510.31980.32470.32970.33490.34060.34710.35460.36430.3791

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It’s determined that 65% confidence is required Calculate state mean cat/aiy (capped) Calculate state standard deviation of cat/aiy (capped)

Step 2 – Establish State Level Objective (Cont.):

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The following was calculated from the capped state level catastrophe history:

Mean Cat/AIY = .3912Standard Deviation of Cat/AIY = .5450

The short-run state cat provision is established as follows: Provision (Cat/AIY) = Mean + (t) x (standard deviation) = (.3912) + (.389) x (.5450)

= .6032 Where: Mean = average capped cat/aiy for Mich-con-ota .389 = t – stat for 65% and (N-1) degrees of freedom .5450 = standard deviation of the annual capped cat/aiy

Step 3 – Calculate State Provision:

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The State of Mich-con-otaThe State of Mich-con-otaConfidence Interval ApproachConfidence Interval Approach

Year

Amount ofIns. Years

(AIY)Cat Incurred

Loss CAT/AIYCapped

CAT/AIYRunningProvision

1993199419951996199719981999200020012002

4,276,1354,306,8154,540,9134,774,7835,001,1645,193,1905,367,5665,574,5065,745,3446,223,199

$ 307,946 259,784 4,378 1,529,513 2,736,486 50,241,886 8,141,594 6,676,296 12,086,512 6,091,053

0.07200.06030.00100.32030.54729.67461.51681.19762.10370.9788

0.07200.06030.00100.32030.54721.61751.51681.19761.95370.9788

0. 29830.29110.28250.28650.30160.39340.46120.50040.58350.6032

Provision (Confidence Interval Approach) 0.6032

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Issues With Confidence Interval ApproachIssues With Confidence Interval Approach

Pluses Recognizes individual state variability Stable provision Provides means to assure companywide sufficiency

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Issues With Confidence Interval ApproachIssues With Confidence Interval Approach

Pluses Recognizes individual state variability Stable provision Provides means to assure companywide sufficiency

Drawbacks Not particularly responsive to distributional changes,

coverage changes, etc. (data back to 1971) Capping can result in less responsiveness Recognition of variability interpreted as risk margin

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The State of Mich-con-otaThe State of Mich-con-otaExtreme Events AdjustmentExtreme Events Adjustment

Year

Amount ofIns. Years

(AIY)Cat Inc.

Loss

ExtremeCat Inc.

Loss

NonExtremeCat Inc.

LossExtreme

CAT/AIY

Contrib.From

Extreme

Contrib.FromNon

ExtremeRunningProvision

1993199419951996199719981999200020012002

4,276,1354,306,8154,540,9134,774,7835,001,1645,193,1905,367,5665,574,5065,745,3446,223,199

$ 307,946 259,784 4,378 1,529,513 2,736,486 50,241,886 8,141,594 6,676,296 12,086,512 6,091,053

$

45,217,697

$ 307,946 259,784 4,378 1,529,513 2,736,486 5,024,189 8,141,594 6,676,296 12,086,512 6,091,053

8.7071 1.7414

0.07200.06030.00100.32030.54720.96751.51681.19762.10370.9788

0.21480.21020.17190.18620.21280.34650.36310.40830.49820.5446

20 Year Average 0.0871 0.4576

Provision (Extreme plus Non-Extreme) 0.5446

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Extreme Events AdjustmentExtreme Events Adjustment

PlusesRelatively stableAs opposed to censoring, reflects events fully

DrawbacksAccurate determination of event return period

difficultCan be viewed as arbitrary and difficult to

explain

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95% / 5% Trended Approach:

All years used Exponential smoothing Trend factor applied – recognizes static cat definition 10% annual cap to change in provision

Methodology:

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The State of Mich-con-otaThe State of Mich-con-ota95/5 Trended95/5 Trended

Year

Amount ofInsurance Years

(AIY)Cat Incurred

Loss CAT/AIYRunningProvision

1993199419951996199719981999200020012002

4,276,1354,306,8154,540,9134,774,7835,001,1645,193,1905,367,5665,574,5065,745,3446,223,199

$ 307,946 259,784 4,378 1,529,513 2,736,486 50,241,886 8,141,594 6,676,296 12,086,512 6,091,053

0.07200.06030.00100.32030.54729.67461.51681.19762.10370.9788

0. 19720.19300.20540.22590.24850.27330.30070.33070.36380.4002

Provision (95/5 Trended) 0.4002

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95% / 5% Trended Approach:

Not volatile, yet responsive for non-extreme events Simple to understand Trend factor to compensate for static definition of cats Reduced data complications

Advantages:

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Summary of Results So Far:Summary of Results So Far:

Approach

Mich-con-otaCat/AIY

StabilityRank

Cat/Ex-Cat20-Year AverageConfidence Interval ApproachExtreme Events95% / 5% TrendedAll Years Weighted Average

.7292

.8929

.6032

.5446

.4002

.9940

452316

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Pivotal Question: Can Countrywide or Regional Data Help Quantify the True ProspectiveMean Catastrophe Loss in a GivenState?

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Pivotal Question:Can Countrywide or Regional DataHelp Quantify the True ProspectiveMean Catastrophe Loss in a GivenState?

Provisions need to reflect adequacy and stability All company surplus is generally available and at risk Are regional or sub state provisions appropriate? Perceived cost sharing will be scrutinized

Issues:

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Goals of Relativity MethodGoals of Relativity Method

Develop accurate, stable results by state that results in an appropriate provision on a countrywide basis

Systematic approach to handle extreme events so a single outlying year does not drive the cat provision for a state

Appropriate application of credibility procedure Provide result that is responsive to recent

demographic and cat definition shifts

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Issues AddressedIssues Addressed

How to be responsive to changes in risk due to population shifts or cat definition changes while still including an appropriate number of years

How does one define an outlying event– Individual state vs. countrywide

How to incorporate credibility

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State Relativity Weighted with Countrywide State Relativity Weighted with Countrywide Complement – General OutlineComplement – General Outline

I. Develop State Damage RatiosII. Calculate Countywide Damage RatiosIII. Calculate State RelativitiesIV. Cap State RelativitiesV. Average Capped RelativitiesVI. Credibility Weight with CW Average of 1.000VII. Balance Back to CW Average of 1.000VIII. Calculate Statewide Catastrophe Provision

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State Relativity Weighted with State Relativity Weighted with Countrywide ComplementCountrywide Complement Develop each state’s damage ratios for years 1981-2000

– State Damage Ratios – Losses/AIY

– Only use years 1981 forward. Data for years 1971 through 1980

is sparse as evidenced by yearly variance.

I. Develop State Damage Ratios

II. Calculate Countrywide Damage Ratios

III. Calculate State Relativities

IV. Cap State Relativities

V. Average Capped Relativities

VI. Credibility Weight with CW Average of 1.000

VII. Balance Back to CW Average of 1.000

VIII. Calculate Statewide Catastrophe Provision

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State Relativity Weighted with State Relativity Weighted with Countrywide ComplementCountrywide Complement Each year’s Countrywide damage ratio is

calculated as the weighted average of state damage ratios using latest year AIYs as weights– Eliminates distortion of state distributional shifts over

time Countrywide catastrophe provision is the

arithmetic average of the most recent 10 years of damage ratios









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CW Cat damage ratios including 2000

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

1970 1975 1980 1985 1990 1995 2000 2005

Year

Da

ma

ge

ra

tio

Years Linear trend

1971-1978 0.006

1979-1989 0.000

1990-1999 -0.019

1990-2000 -0.010

Figure 1Figure 1

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State Relativity Weighted with State Relativity Weighted with Countrywide ComplementCountrywide Complement

Calculate state relativities as the ratio of state damage ratios to countrywide damage ratios

Relativities should be more stable than damage ratios Trend should not be a problem so we can use more years

of data than the Countrywide Catastrophe Provision









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State Relativity Weighted with State Relativity Weighted with Countrywide ComplementCountrywide Complement Any relativity greater than the mean plus three

standard deviations is capped to the next lowest relativity (not the cap number)– Intuitively we are replacing a once in a hundred year

event with a once in 20 Benefit of capping process

– Represents a systematic approach to dealing with extreme events

– Cap is dynamic and is allowed to shift if exposure in a state is changing over time

– Censoring at the cap would not have much impact and therefore would not result in increased stability









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Calculate arithmetic average of 1981-2000 capped relativities– Using a arithmetic average is simple

– No benefit of weighting relativities has been shown since relationship of variability to exposure level is unclear

– Arithmetic average relativity does not differ significantly from an AIY weighted average









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Uses Buhlmann credibility factor: n/(n+k)– n = number of years of relativities in average

We use number of years rather than exposures because exposures not

independent, especially past a certain threshold where exposure

concentration increases

– k = average process variance/variance of hypothetical means The process variance and variance of hypothetical means are

calculated using all available years of capped relativities across all states.

Complement of credibility of 1.000 is not appropriate when there is a wide spread of average relativities– Solution lies in balancing process described on next slide









04/19/2349

State Relativity Weighted with State Relativity Weighted with Countrywide ComplementCountrywide Complement At this point, the individual state relativities result in a

countrywide relativity of less than 1.000. Relativities are adjusted to achieve an overall adequate level as follows:– Determined on a countrywide basis what our expected losses

would be based on the countrywide selected catastrophe factor– Sum the pre-balanced expected losses across all states – We distribute the difference between 1 and 2 in proportion to

each state’s standard deviation measured in latest year expected losses.

Using this approach has several benefits:– Results in an appropriate provision countrywide – It compensates for high (low) relativity states being

underestimated (overestimated) by the use of a 1.000 complement of credibility.

– Each state’s resulting cat load is a function of its own size and variability









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Statewide catastrophe provision is calculated by multiplying capped, credibility weighted, balanced relativity by the countrywide catastrophe provision

Benefits of method:– Allows use of long term data to determine relativity

while using more responsive data for countrywide provision

– Adjustments to data are determined objectively with each state’s characteristics used to determine both capping and balancing









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Issues AddressedIssues Addressed How to be responsive to changes in risk due to population

shifts or cat definition changes while still including an appropriate number of years– We use as many years of relativities as possible while using only the

latest 10 years of Countrywide to determine the Countrywide load. How do you define an outlying event

– Greater than the mean plus 3 standard deviation in any given state. On a countrywide basis these “outlying” events occur fairly regularly (about 2% of relativities have been capped)

How to incorporate credibility– Uses Buhlmann credibility to account for variability in relativities.

Used a relativity of 1.000 as complement. Balancing method adjusts for bias in complement for when a 1.000 may not be an appropriate complement.

04/19/2352

Summary of Results:Summary of Results:

Approach

Mich-con-otaCat/AIY

Cat/Ex-Cat20-Year AverageConfidence Interval ApproachExtreme Events95% / 5% TrendedAll Years Weighted AverageRelativity Method

.7292

.8929

.6032

.5446

.4002

.9940

.6270

04/19/2353

ConclusionConclusion

We have made significant progress as a profession in quantifying the catastrophe exposure.

Do our current methods capture the true mean?

My purpose will be achieved if this session helps to keep focus on this issue

04/19/2354

ConclusionConclusion

Comments and Discussion

properly pricing the catastrophe exposure

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