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Increased Limits, Excess & Deductible Ratemaking Joseph M. Palmer, FCAS, MAAA, CPCU Assistant Vice President Increased Limits & Rating Plans Division Insurance Services Office, Inc.

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Increased Limits Ratemaking is the process of developing charges for expected losses at higher limits of liability.

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Increased Limits Ratemaking is the process of developing charges for expected losses at higher limits of liability. Expressed as a factor --- an Increased Limit Factor --- to be applied to basic limits loss costs

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Calculation Method Expected Costs at the desired policy limit _____________________________________________________________________________________________________________ Expected Costs at the Basic Limit

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KEY ASSUMPTION: Claim Frequency is independent of Claim Severity

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This allows for ILFs to be developed by an examination of the relative severities ONLY

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Limited Average Severity (LAS) Defined as the average size of loss, where all losses are limited to a particular value. Thus, the ILF can be defined as the ratio of two limited average severities. ILF (k) = LAS (k) / LAS (B)

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Example 1,250,000 250,000 150,000 75,000 50,000 @1 Mill Limit@100,000 LimitLosses

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Example (cont’d) 100,0001,250,000 100,000250,000 100,000150,000 75,000 50,000 @1 Mill Limit@100,000 LimitLosses

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Example (cont’d) 1,000,000100,0001,250,000 250,000100,000250,000 150,000100,000150,000 75,000 50,000 @1 Mill Limit@100,000 LimitLosses

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Example – Calculation of ILF 3.588Increased Limits Factor (ILF) $1,525,000Limited to $1,000,000 $425,000Limited to $100,000 (Basic Limit) $1,775,000Total Losses

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Why Take This Approach? Loss Severity Distributions are Skewed Many Small Losses/Fewer Larger Losses Yet Larger Losses, though fewer in number, are a significant amount of total dollars of loss.

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Basic Limits vs. Increased Limits Use large volume of losses capped at basic limit for detailed, experience-based analysis. Use a broader experience base to develop ILFs to price higher limits

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Loss Distribution - PDF 0 Loss Size

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Loss Distribution - CDF 0 1 Claims

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Claims vs. Cumulative Paid $ $ $ 0 0 0 1 Liability Property Claims

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A novel approach to understanding Increased Limits Factors was presented by Lee in the paper --- “The Mathematics of Excess of Loss Coverages and Retrospective Rating - A Graphical Approach”

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Lee (Figure 1)

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Limited Average Severity Size method; ‘vertical’ Layer method; ‘horizontal’

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Size Method k 01 Loss Size

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Layer Method k 10 Loss Size

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Empirical Data - ILFs 10 50 200 500 1,000 Occs. -15,000,000- 1 Million 30,000,000 1 Million 500,001 60,000,000500,000250,001 75,000,000250,000100,001 25,000,000100,0001 LASLossesUpperLower

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Empirical Data - ILFs LAS @ 100,000 (25,000,000 + 760 * 100,000) / 1760 = 57,386 LAS @ 1,000,000 ( 190,000,000 + 10 * 1,000,000 ) / 1760 = 113,636 Empirical ILF = 1.98

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Issues with Constructing ILF Tables Policy Limit Censorship Excess and Deductible Data Data is from several accident years Trend Loss Development Data is Sparse at Higher Limits

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Use of Fitted Distributions Enables calculation of ILFs for all possible limits Smoothes the empirical data Examples: Truncated Pareto Mixed Exponential

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“Consistency” of ILFs As Policy Limit Increases ILFs should increase But at a decreasing rate Expected Costs per unit of coverage should not increase in successively higher layers.

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Illustration: Consistency 01 k3k3 k2k2 k1k1 Loss Size

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“Consistency” of ILFs - Example 1.40250,000 5.505 Million 4.302 Million 2.751 Million 1.80500,000 ---1.00100,000 MarginalDiff. ILFDiff. Lim.ILFLimit

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“Consistency” of ILFs - Example 0.401501.40250,000 1.203,0005.505 Million 1.551,0004.302 Million 0.955002.751 Million 0.402501.80500,000 ---1.00100,000 MarginalDiff. ILFDiff. Lim.ILFLimit

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“Consistency” of ILFs - Example.00270.401501.40250,000.00041.203,0005.505 Million.001551.551,0004.302 Million.00190.955002.751 Million.00160.402501.80500,000 ---1.00100,000 MarginalDiff. ILFDiff. Lim.ILFLimit

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“Consistency” of ILFs - Example.00270.401501.40250,000.00041.203,0005.505 Million.001551.551,0004.302 Million.0019*0.955002.751 Million.00160.402501.80500,000 ---1.00100,000 MarginalDiff. ILFDiff. Lim.ILFLimit

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Components of ILFs

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Expected Loss Allocated Loss Adjustment Expense (ALAE) Unallocated Loss Adjustment Expense (ULAE) Parameter Risk Load Process Risk Load

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ALAE Claim Settlement Expense that can be assigned to a given claim --- primarily Defense Costs Loaded into Basic Limit Consistent with Duty to Defend Insured Consistent Provision in All Limits

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Unallocated LAE – (ULAE) Average Claims Processing Overhead Costs e.g. Salaries of Claims Adjusters Percentage Loading into ILFs for All Limits Average ULAE as a percentage of Losses plus ALAE Loading Based on Financial Data

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Process Risk Load Process Risk --- the inherent variability of the insurance process, reflected in the difference between actual losses and expected losses. Charge varies by limit

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Parameter Risk Load Parameter Risk --- the inherent variability of the estimation process, reflected in the difference between theoretical (true but unknown) expected losses and the estimated expected losses. Charge varies by limit

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Increased Limits Factors (ILFs) ILF @ Policy Limit (k) is equal to: LAS(k) + ALAE(k) + ULAE(k) + RL(k) ____________________________________________________________________________________________________________ LAS(B) + ALAE(B) + ULAE(B) + RL(B)

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Components of ILFs 1.741351,43297467812,3082,000 1.5512380390567811,3921,000 1.3710841982167810,265500 1.19941937236788,956250 1.0079766136787,494100 ILFPaRLPrRLULAEALAELASLimit

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Deductibles

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Types of Deductibles Loss Elimination Ratio Expense Considerations

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Types of Deductibles Reduction of Damages Impairment of Limits

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Types of Deductibles Reduction of Damages Insurer is responsible for losses in excess of the deductible, up to the point where an insurer pays an amount equal to the policy limit An insurer may pay for losses in layers above the policy limit (But, total amount paid will not exceed the limit) Impairment of Limits The maximum amount paid is the policy limit minus the deductible

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Deductibles (example 1) Reduction of DamagesImpairment of Limits Example 1: Policy Limit: $100,000 Deductible: $25,000 Occurrence of Loss: $75,000 Payment is $50,000 Reduction due to Deductible is $25,000 Payment is $50,000 Reduction due to Deductible is $25,000 Loss – Deductible =75,000 – 25,000=50,000 (Payment up to Policy Limit) Loss does not exceed Policy Limit, so: Loss - Deductible =75,000 – 25,000=50,000

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Deductibles (example 2) Reduction of DamagesImpairment of Limits Example 2: Policy Limit: $100,000 Deductible: $25,000 Occurrence of Loss: $125,000 Payment is $100,000 Reduction due to Deductible is $0 Payment is $75,000 Reduction due to Deductible is $25,000 Loss – Deductible =125,000 – 25,000=100,000 (Payment up to Policy Limit) Loss exceeds Policy Limit, so: Policy Limit - Deductible =100,000 – 25,000=75,000

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Deductibles (example 3) Reduction of DamagesImpairment of Limits Example 3: Policy Limit: $100,000 Deductible: $25,000 Occurrence of Loss: $150,000 Loss – Deductible =150,000 – 25,000=125,000 (But pays only up to Policy Limit) Loss exceeds Policy Limit, so: Policy Limit - Deductible =100,000 – 25,000=75,000 Payment is $100,000 Reduction due to Deductible is $0 Payment is $75,000 Reduction due to Deductible is $25,000

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Liability Deductibles Reduction of Damages Basis Apply to third party insurance Insurer handles all claims Loss Savings No Loss Adjustment Expense Savings Deductible Reimbursement Risk of Non-Reimbursement Discount Factor

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Deductible Discount Factor Two Components Loss Elimination Ratio (LER) Combined Effect of Variable & Fixed Expenses This is referred to as the Fixed Expense Adjustment Factor (FEAF)

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Loss Elimination Ratio Net Indemnity Costs Saved – divided by Total Basic Limit/Full Coverage Indemnity & LAE Costs Denominator is Expected Basic Limit Loss Costs

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Loss Elimination Ratio (cont’d) Deductible (i) Policy Limit (j) Consider [ LAS(i+j) – LAS(i) ] / LAS(j) This represents costs under deductible as a fraction of costs without a deductible. One minus this quantity is the (indemnity) LER Equal to [ LAS(j) – LAS(i+j) + LAS(i) ] / LAS(j)

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Loss Elimination Ratio (cont’d) LAS(j) – LAS(i+j) + LAS(i) represents the Gross Savings from the deductible. Need to multiply by the Business Failure Rate Accounts for risk that insurer will not be reimbursed The result is the Net Indemnity Savings from the deductible

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Fixed Expense Adjustment Factor Deductible Savings yields Variable Expense Savings Does not yield Fixed Expense Savings Variable Expense Ratio represents Variable Expenses as a percentage of Premium So: Total Costs Saved from deductible equals Net Indemnity Savings divided by (1-VER)

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FEAF (cont’d) Now: Basic Limit Premium equals Basic Limit Loss Costs divided by the Expected Loss Ratio (ELR) We are looking for: Total Costs Saved / Basic Limit Premium

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FEAF (cont’d) Total Costs Saved / Basic Limit Premium Is Equivalent to: Net Indemnity Savings / (1-VER) _________________________________________________________________________________________ Basic Limit Loss Costs / ELR Which equals: LER * [ ELR / (1-VER) ] So: FEAF = ELR / ( 1 – VER )

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Deductibles - Summary Deductible Discount Factor Expected Net Indemnity Total Expected B.L. Indemnity + ALAE + ULAE LER= Fixed Expense Adjustment Factor (FEAF) Loss Elimination Ratio (LER) X= Expected Loss Ratio 1 – Variable Expense Ratio FEAF=

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A Numerical Example New Premium = ? Deductible Discount Factor = ?.10Net LER ?FEAF.30VER ?ELR5Fixed Expenses ?Premium65Expected Losses

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A Numerical Example New Premium = 90.71 Deductible Discount Factor =.0929.10Net LER.929FEAF.30VER.65ELR5Fixed Expenses 100Premium65Expected Losses

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Numerical Example (cont’d) New Net Expected Losses = ( 1 -.10 ) * 65 = 58.5 Add Fixed Expenses => 58.5 + 5 = 63.5 Divide by ( 1 – VER ) => 63.5 /.70 = 90.71 Which agrees with our previous calculation

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Layers of Loss

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Expected Loss ALAE ULAE Risk Load

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Limited Average Severity (Ded.) Size method; ‘vertical’ Layer method; ‘horizontal’

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Size Method & LAS is equal to

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Size Method – Layer of Loss 10 Loss Size k2k2 k1k1

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“Layer Method” – Layer of Loss Loss Size 10 k1k1 k2k2

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Layers of Loss Expected Loss ALAE ULAE Risk Load

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Effect of Inflation

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Inflation – Leveraged Effect Generally, trends for higher limits will be higher than basic limit trends. Also, Excess Layer trends will generally exceed total limits trends. Requires steadily increasing trend.

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k2k2 Effect of Inflation k1k1 01

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An Example of Trends at Various Limits – Comm. Auto BI Bodily Injury Severity Low Statewide Credibility All Commercial Auto Data used in analyses Only 4 States with Greater than 15% Credibility Limited Variation between States after Trends are Credibility-Weighted

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Commercial Automobile Policy Limit Distribution ISO Database Composition: 70% at $1 Million Limit 5% at $300,000 Limit 15% at $500,000 Limit 5% at $2 Million Limit Varies by Table

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Commercial Automobile Bodily Injury Data Through 6/30/2004 Paid Loss Data --- $100,000 Limit 12-point:+ 4.0% 24-point:+ 5.0%

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Commercial Automobile Bodily Injury Data Through 6/30/2004 Paid Loss Data --- $1 Million Limit 12-point:+ 6.1% 24-point:+ 7.3%

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Commercial Automobile Bodily Injury Data Through 6/30/2004 Paid Loss Data --- Total Limits 12-point:+ 6.3% 24-point:+ 7.6%

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Commercial Automobile Bodily Injury Excess Trends Data Through 6/30/2004 Paid Loss Data --- Excess Limits $400,000 xs $100,000 12-point:+ 8.5% 24-point:+ 9.9%

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Commercial Automobile Bodily Injury Excess Trends Data Through 6/30/2004 Paid Loss Data --- Excess Limits $500,000 xs $500,000 12-point:+ 9.7% 24-point: + 12.6%

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General Liability Policy Limit Distribution ISO Database Composition: 85% at $1 Million Limit 4% at $500,000 Limit 5-6% at $2 Million Limit 1-2% at $5 Million Limit

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Mixed Exponential Methodology

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Issues with Constructing ILF Tables Policy Limit Censorship Excess and Deductible Data Data is from several accident years Trend Loss Development Data is Sparse at Higher Limits

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Use of Fitted Distributions May address these concerns Enables calculation of ILFs for all possible limits Smoothes the empirical data Examples: Truncated Pareto Mixed Exponential

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Mixed Exponential Distribution Trend Construction of Empirical Survival Distributions Payment Lag Process Tail of the Distribution Fitting a Mixed Exponential Distribution Final Limited Average Severities

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Trend Multiple Accident Years are Used Each Occurrence is trended from the average date of its accident year to one year beyond the assumed effective date.

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Empirical Survival Distributions Paid Settled Occurrences are Organized by Accident Year and Payment Lag. After trending, a survival distribution is constructed for each payment lag, using discrete loss size layers. Conditional Survival Probabilities (CSPs) are calculated for each layer. Successive CSPs are multiplied to create ground- up survival distribution.

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Conditional Survival Probabilities The probability that an occurrence exceeds the upper bound of a discrete layer, given that it exceeds the lower bound of the layer is a CSP. Attachment Point must be less than or equal to lower bound. Policy Limit + Attachment Point must be greater than or equal to upper bound.

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Empirical Survival Distributions Successive CSPs are multiplied to create ground-up survival distribution. Done separately for each payment lag. Uses 52 discrete size layers. Allows for easy inclusion of excess and deductible loss occurrences.

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Payment Lag Process Payment Lag = (Payment Year – Accident Year) + 1 Loss Size tends to increase at higher lags Payment Lag Distribution is Constructed Used to Combine By-Lag Empirical Loss Distributions to generate an overall Distribution Implicitly Accounts for Loss Development

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Payment Lag Process Payment Lag Distribution uses three parameters -- - R1, R2, R3 (Note that lags 5 and higher are combined – C. Auto) R3= Expected % of Occ. Paid in lag (n+1) Expected % of Occ. Paid in lag (n) R2= Expected % of Occ. Paid in lag 3 Expected % of Occ. Paid in lag 2 R1= Expected % of Occ. Paid in lag 2 Expected % of Occ. Paid in lag 1

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Lag Weights Lag 1 wt. = 1 / k Lag 2 wt. = R1 / k Lag 3 wt. = R1 * R2 / k Lag 4 wt. = R1 * R2 * R3 / k Lag 5 wt. = R1 * R2 * [R3*R3/(1-R3)] / k Where k = 1 + R1 + [ R1 * R2 ] / [ 1 – R3 ]

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Lag Weights Represent % of ground-up occurrences in each lag Umbrella/Excess policies not included R1, R2, R3 estimated via maximum likelihood.

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Tail of the Distribution Data is sparse at high loss sizes An appropriate curve is selected to model the tail (e.g. a Truncated Pareto). Fit to model the behavior of the data in the highest credible intervals – then extrapolate. Smoothes the tail of the distribution. A Mixed Exponential is now fit to the resulting Survival Distribution Function

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Μean parameter: μ Policy Limit: PL Simple Exponential

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Mixed Exponential Weighted Average of Exponentials Each Exponential has Two Parameters mean (μ i ) and weight (w i ) Weights sum to unity *PL: Policy Limit

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Mixed Exponential Number of individual exponentials can vary Generally between four and six Highest mean limited to 10,000,000

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Sample of Actual Fitted Distribution 0.00057110,000,000 0.0044121,835,193 0.023622367,341 0.16859132,363 0.8028044,100 WeightMean

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Calculation of LAS *PL: Policy Limit

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Joe Palmer Assistant Vice President Insurance Services Office, Inc. 201-469-2599 Jpalmer@iso.com With many thanks to Brian Ko for his assistance

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