CAS Individual Claim SimulatorMethodology Documentation

ReservePrism

April 2018

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Table of Contents1. Background.............................................................................................................................................3

2. Simulation Process..................................................................................................................................3

3. Model Fitting...........................................................................................................................................6

3.1 Claim Data.........................................................................................................................................6

3.2 Distribution Fitting.............................................................................................................................6

3.3 Copula Fitting...................................................................................................................................10

3.4 Fitting Report...................................................................................................................................12

4. Simulation.............................................................................................................................................12

4.1 Open Claim (IBNER).........................................................................................................................12

4.2 IBNR.................................................................................................................................................14

4.3 Future Claim (UPR)..........................................................................................................................16

4.4 Claim Reopen...................................................................................................................................17

4.5 Copula..............................................................................................................................................18

5. Reporting..............................................................................................................................................18

Reference..................................................................................................................................................20

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1. BackgroundThis document describes the simulation and fitting methods used in the CAS Individual Claim Simulator. The simulator provides claim level simulation functions (a triangle-free approach) to accommodate individual differences. It can simulate open claim development, closed claim reopenness, IBNR and UPR for multiple business lines in a consistent way.

Most of the methods used in simulation follows Triangle-free reserving: a non-traditional framework for estimating reserves and reserve uncertainty (Parodi, 2013). In addition, suggested additions including deductible, limit, LAE, correlation/copula and claim reopenness have been incorporated as well. The simulator can also fit distribution and copula assumptions from claim data. Standard methods such as MLE, moment matching, and percentile matching are made available for distribution fitting and copula fitting.

2. Simulation ProcessThe simulator includes three modules: fitting, simulation, and reporting. It encompasses the entire process from claim experience to reserving analysis. Figure 1 illustrates the process.

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Figure 1. CAS Claim Simulator Architecture

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Fitting

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Based on the claim file, the fitting process tries to generate the following eight assumptions for each business line/type.

1. Report Lag2. Settlement Lag3. Monthly frequency4. Ultimate loss severity5. The correlation (copula) among severity, report lag and settlement lag6. The correlation (copula) of monthly frequencies of multiple lines of business.7. Deductible empirical distribution.8. Limit empirical distribution.

Assumptions that are not fitted but need to be provided by users include exposure data (business volume by time), severity index (severity trend), open claim development factors, closed claim reopen probability, closed claim reopen lag, closed claim reopen development factors, and loss adjustment expense (LAE) functions. Quantification of these assumptions requires historical development information of individual claims instead of static claim data.

Simulation

Four types of claims can be simulated for each business line/type.

1. Open Claim. Given the current incurred loss, a settlement date is simulated and then the ultimate loss is simulated based on the development factors and the simulated settlement date. Ultimate loss may also be simulated based on conditional severity distribution.

2. Closed Claim Reopenness. For each closed claim, the program simulates to determine whether the claim will reopen or not. If it will reopen, a reopen date and a resettlement date will be simulated. The ultimate loss will then be simulated based on reopen development factors.

3. Incurred but not Reported (IBNR). Monthly frequencies of IBNR are simulated first. For each IBNR claim, occurrence date, report date, settlement date, and ultimate loss are simulated.

4. Future claim (UPR). Monthly frequencies of future claims are simulated first. For each UPR claim, occurrence date, report date, settlement date, and ultimate loss are simulated.

Based on the LAE assumptions, the simulator can also simulate the LAE for each individual claim.

Reporting

Simulated individual claims are output to a csv file. They are then summarized to provide descriptive statistics (mean, standard deviation, and percentiles) for major measures such as count, total ultimate loss, average ultimate loss, etc. A html report containing main results and assumptions is also generated.

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3. Model Fitting3.1 Claim DataThe model fitting starts with claim data. Table 1 lists the required information in the claim file.

Table 1. Claim Data Fields

Data Field Type DescriptionClaimID Mandatory Claim unique IDLoB Mandatory Line of Business LineType Mandatory Claim type to differentiate claims within a business line. For

example, normal claims, big claims, jump claims, short-term claims, long-term claims, etc.

Status Mandatory Open/ClosedoccurrenceDate Mandatory The date that the accidence happenedreportDate Mandatory The date that the accidence was reportedsettlementDate Mandatory The date that the claim was closed. It is only required for

closed claimsincurredLoss Mandatory The sum of paid and outstanding amount at the valuation

date.osRatio Mandatory Outstanding ratio, which is outstanding amount divided by

Incurred Loss. It will be 1 for closed claims.Paid Optional Total paid loss. It should be the same as (1 – OS Ratio) ×

Incurred Loss.totalLoss Optional Total loss before deductible and limit.Deductible Optional Deductible for each claimLimit Optional Loss limit for each claim. It is the limit on payment after

deductible.LAE Optional Loss adjustment expense. It can be omitted if expense is not

modeled separately.claimLiability Optional Indicating whether it is a zero-payment claim (FALSE) or not

(TRUE). Small claims with loss amount less than deductibles are still considered as valid claim (TRUE).

3.2 Distribution FittingThe program will try to fit four distributions for each business line/type: frequency, severity, report lag, and settlement lag. Empirical distributions of deductible and limit are also constructed based on claim data if available. Probability of zero-payment claim by development year (p0) is also calculated based on claim data if invalid claims are included. Figure 2 illustrates the distribution fitting process.

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Claim Data

Report Lag

Settlement Lag

Monthly Frequency

Severity

Discrete Distribution

Pool

Continuous Distribution

Pool

Visualization

Statistical Test

Select

Deductible Limit

EmpiricalDistribution

Figure 2. Distribution Fitting Process

Data Preparation

The data used for distribution fitting can be calculated from the claim data file.

1. Report Lag = Report Date – Occurrence Date. 2. Settlement Lag = Settlement Date – Report Date3. Severity is detrended before distribution fitting. In simulation, the trend will be applied back.

Severity= Total LossSeverity Index

The severity index reflects the trend of severity by time, such as claim inflation rate, cyclical patterns, etc. If total loss before deductible and limit is not available, incurred loss together with deductible and limit will be used to fit severity distribution. For claims with zero loss due to deductible, log(prob(deductible)) is used in loglikelihood calculation. For claims with loss equal to the limit, log(prob(deductible + limit)) is used in loglikelihood calculation. Invalid claims (claimLiability == FALSE) are excluded when calibrating severity distribution. Users can also choose whether closed claim data or closed + open claim data is used for severity distribution fitting. Using closed + open claim data may be useful for new business lines with less historical data.

4. Because reported claims may not cover all the accidents (IBNR is not observed in the claim data), monthly frequency data needs to be adjusted before distribution fitting. The data is processed using the following steps:

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Counting the number of claims that occurred in each month according to occurrence date. Adjusting the monthly claim counts to reflect the IBNR. For example, assuming the

valuation date is Dec. 31, 2016, which is also the date by which the claims has been reported. If the count of claims during June 2016 is 100, it means that 100 claims happened in June 2016 have been reported by Dec. 31, 2016. However, IBNR claims need to be accounted for as well. Claims happened during June 2016 that may still be reported after the valuation date. The adjustment can be made based on the report lag distribution.

Expected ClaimCount= ReportedClaimCountF(report lag< (ValuationDate−MidMonth Date ))

On average, 6.5 months (approximately 195 days) had elapsed from Mid June to the end of December. If the report lag follows an Exponential distribution with an average lag of 100 days, 100 claims will be adjusted as follows.

ExpectedClaimCount= 1001−e−0.01×195 =117

To achieve this step, report lag distribution needs to be constructed before monthly frequency distribution.

Adjusting monthly claim counts further to remove the impact of business volume changes over time. In simulation, the exposure index will be applied back.

AdjustedClaimCount= ExpectedClaimCountExposure Index

Adjusted monthly claim counts are then used for frequency distribution fitting.

5. Deductible and Limit. Experience data from the claim data is used directly.

Distribution Pool

For the first four variables (report lag, frequency, settlement lag and severity), the program will automatically iterate through a pool of distributions trying to find the best fit. The pool of distribution for frequency fitting consists of the following:

Poisson Distribution Negative Binomial Distribution Geometric Distribution

The pool of distributions for severity, report lag, and settlement lag consists of the following:

Exponential Distribution Normal Distribution Lognormal Distribution Weibull Distribution Pareto Distribution Gamma Distribution Uniform Distribution

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If the user is not satisfied with any of the fitting results, empirical distribution can be chosen so that it is all data driven.

For limit and deductible, only empirical distribution is used to describe the data.

Distribution Fitting

Three methods can be used for distribution fitting: maximum likelihood estimation (MLE), moment matching and percentile matching.

MLE: A parameter estimation method that maximizes the likelihood of making the observations given the parameters.

Moment Matching: A parameter estimation method that matches the distribution moments (mean, variance, skewness, kurtosis, etc.) to the data moments. The number of moments used for matching equals the number of distribution parameters. For example, only mean is used for Exponential distribution fitting. Both mean and variance is used for Normal distribution fitting.

Percentile Matching: A parameter estimation method that matches some chosen percentiles of the data and the distribution. For example, we may choose to match the 50th and the 90th percentiles of a Lognormal distribution to the data. The number of percentiles that are matched equals the number of distribution parameters.

MLE is used as the default method given its ability to capture individual differences instead of focusing on some key statistics such as mean and variance.

For each tried distribution, the user is able to visualize the goodness of fit, as shown in Figure 3.

Figure 3. Frequency Fitting with Negative Binomial Distribution (Example)

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The program also provides a summary of the fitting showing some test statistics, as shown in Table 2.

Table 2. Distribution Fitting Summary (Example)

LoBType Fit

Distribution

Method Parameter SD1

DoF2 KS3

pks4 AIC BIC

Auto N

severity Normal mle

mean:3272.218; sd:17567.545;

381.0755; 235.2049;

2230 0.5 0

50229.23

50240.66

Auto N

severity Lognormal mle

meanlog:2.038; sdlog:2.97;

0.0629; 0.0444;

2230 0.7 0

20293.37

20304.79

Auto N

severity Pareto mle NA NA NA NA NA NA NA

Auto N

severity Weibull mle

shape:0.336; scale:34.055;

0.005; 2.2801;

2230 0.7 0

20667.61

20679.04

Auto N

severity Gamma mle

shape:0.165; scale:4668.656;

0.0038; 270.378;

2230 1 0

22269.96

22281.38

Auto N

severity Uniform mle

min:0.01; max:850690.16; NA;

2230

0.98 0 NA NA

Auto N

severity

Exponential mle NA NA NA NA NA NA NA

1. Standard deviation of the parameter estimation. In case with MLE based on incurred loss after deductible and limit, optimization convergence information is recorded in this data field.

2. DoF: degree of freedom3. KS: Kolmogorov–Smirnov (K-S) test statistic. For discrete distributions, Chi-squared test is performed

instead.4. pks: p-value of K-S test.

Based on the test statistics, users can choose a preferred distribution or could select the empirical distribution constructed from the data.

3.3 Copula FittingCorrelation modeling is used in two places in CAS Claim Simulator. The process is similar to distribution fitting, as shown in Figure 4.

Figure 4. Copula Fitting Process

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Claim Data

Severity, Report Lag and Settlement Lag Within Each

Business Line

Monthly Frequency Among Multiple Business Lines

CopulaPool

Visualization

Statistical Test

Select

Data Preparation

The data used for copula fitting is the same as used for distribution fitting.

Severity, report Lag and settlement lag within each business line: Severity, report lag and settlement lag data.

Monthly frequency among multiple business lines: adjusted monthly claim counts for each business line.

Following the methods described in Kojadinovic and Yan (2010) and options available in R package “Copula”, the data is then transformed into percentiles for copula fitting. For example, an observation of 5 following a Normal distribution N(5,2) will be transformed into 0.5. Therefore, all the data will be transformed into the range between 0 and 1. After the copula fitting, the percentiles can be then transformed back into original variable according to the inverse of marginal distribution. This way, marginal distribution fitting and copula fitting can be separated and monitored separately.

Copula Pool

For each relationship, the program will automatically iterate through a pool of copulas trying to find the best fit. The pool of copulas consists of the following:

Guassian t Clayton Frank Gumbel Joe

Copula Fitting

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MLE is used for copula fitting. For each tried copula, the user is able to visualize the goodness of fit, as shown in Figure 5. Three lines’ monthly frequency data are used to drive the Gaussian copula parameters, i.e., the correlation matrix.

Figure 5. Frequency Copula Fitting with Gaussian Copula (Example)

In addition, statistical testing is performed to assess the goodness of fit, as shown in Table 3. Users can then choose the best fit and use it for simulation.

Table 3. Copula Fitting Summary (Example)

Fit Copula Method Parameter SD1 DoF2 Sn3 p4

freqCorrelation normal mpl 0.197;0.4748;0.2027 0.1469;0.0973;0.1466 3 0.0252 0.6095freqCorrelation clayton mpl 0.4283 0.1472 3 0.0436 0.1915freqCorrelation gumbel mpl 1.1778 0.0729 3 0.0683 0.0174freqCorrelation frank mpl 1.5971 0.5129 3 0.0442 0.1517freqCorrelation joe mpl 1.1641 0.1086 3 0.1301 0.0075freqCorrelation t mpl 0.1906;0.4924;0.2053 0.1556;0.1081;0.154 9.77959 NA NA

1. Standard deviation of the parameter estimation2. DoF: degree of freedom3. Sn: Cramer-von Mises Statistic introduced by Genest, Remillard and Beaudoin (2009).4. p: p-value of Sn test using the multiplier method introduced by Kojadinovic and Yan (2011).5. In the API, only Normal copula is fitted for fast processing. Running fitting for all copulas could take hours

to finish.

3.4 Fitting ReportIn addition, a html fitting report is generated by the R program listing the fitting results for each business line/type. A sample report is attached below.

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4. SimulationOpen claim development, claim reopenness, IBNR and future claims (UPR) can be simulated by the CAS Claim Simulator.

4.1 Open Claim (IBNER)For open claims, occurrence date, report date and incurred loss are known information from the claim data file. The only three variables to simulate are settlement date, ultimate loss and LAE.

1. Settlement Lag. It is simulated from a truncated distribution of settlement lag that has the following CDF.

F ( x<X )=FSL ( x<X )−FSL ( x<T )

1−F SL ( x<T )if X>T

0 if X≤TWhere

T = Valuation Date – Report DateFSL ( x<X ) is the cumulative distribution function of the settlement lag.

Here, the simulation is done based on the condition that the settlement lag is greater than T.

2. Ultimate Loss. Four methods are available for simulating ultimate loss.a. Developing the ultimate loss from incurred loss using year-to-year development factors.

The year-to-year development factor assumptions contain both the mean value and the standard deviation, if individual claim experience data is sufficient for a credible estimation. Standard deviation controls the random part of the simulation. Table 4 shows an example of the assumption input.

Table 4. Year-to-Year Development Factor Assumption (Example)

Current Development Year

Year-to-Year Development FactorMean Standard Deviation

1 1.200 0.0592 1.150 0.0433 1.100 0.0884 1.050 0.1005 1 0

Assume an open claim has an incurred loss of 10,000 with a report lag of 1 day. At the valuation date, the claim is in the second development year. The simulated settlement lag is 4 years, which means the claim will be closed in the fourth development year. The ultimate loss is simulated as

10,000 × (1.15+0.043×) × (1.1+0.088×)

Here and are random numbers generated from standard Normal distribution.

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Lognormal and Gamma distributions can also be used with meanlog and sdlog parameter inputs for Lognormal distribution, and shape and scale parameter inputs for Gamma distribution.

Year-to-year development is used like the triangle method, expect here it is done at the individual claim level with volatility.

b. Developing the ultimate loss from incurred loss with an estimated cumulative development factor based on a link function which can be identity, log, inverse, and exponential.

identity(Linear Function): Cumulative Development Factor=α+β1d+β2l+β3os+…+ϵ

exponential(Loglinear Function): Cumulative Development Factor=log (α+ β1d+β2l+β3os+…+ϵ)

log(Exponential Function): Cumulative Development Factor=eα+ β1d+β2 l+β3os+…+ ϵ

inverse(Reciprocal Linear Function):

Cumulative Development Factor= 1α+β1d+β2l+β3os+…+ϵ

Where

d: development year

l: incurred loss

os: outstanding ratio

ϵ : random variable following Normal distribution

Simulated Ultimate Loss = Incurred Loss × Simulated Cumulative Development Factor

For example, if we have the cumulative development factor function as below:

Cumulative Dev. Factor = log(0.1 + 0.01×d + 0.005×l + 0.002×o/s ratio + 5

follows the standard Normal distribution.

If the claim during the second development year has an incurred loss of 10,000 and an outstanding ratio of 0.1, then the ultimate loss is simulated as

10,000 × log(0.1+ 0.01 × 2 + 0.005 × 10,000 + 0.002 × 0.1 + 5 ×

Users can include additional variables in the functions as long as they are available in the claim data.

c. Generating the total loss directly from the severity distribution. The ultimate loss is total loss after deductible and limit and floored by the incurred losses. For example, if the severity follows a Lognormal distribution (=3,=2.5), we can simulate a random number from that distribution. After applying deductible and limit, if it is larger than 10,000 (current incurred loss), say 12,000, we will use it as the simulated ultimate loss. If it is smaller than 10,000, say 9,000, we will use 10,000 as the ultimate loss.

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d. Generating the total loss based on conditional severity distribution where total loss is no less than paid loss. The ultimate loss is total loss after deductible and limit and floored by the incurred losses. For example, if the severity follows a Lognormal distribution (=3,=2.5) and paid loss is 500, we can simulate a random number from the conditional Lognormal distribution where X > 500. Deductible and limit are applied to the random number to get the simulated ultimate loss. If settlement date and severity are assumed to be correlated, copula will be used to simulate settlement date and severity together.

The developed ultimate loss is then adjusted by severity index based on settlement date. If p0, the probability of zero-payment invalid claim by development year is non-zero, zero-payment claims will be simulated as follows.

a. For each open claim, generating a random number from uniform distribution in range [0,1];b. If the generated random number is no greater than p0, the open claim is assumed to be a

zero-payment claim with ultimate loss equals zero. 3. Loss adjustment expense (LAE). If development factors are used to develop the loss, LAE is

assumed to develop by the same percentage. If loss is simulated based on the severity distribution, LAE can be modeled separately from the indemnity. Like loss development factor, LAE development factor can be simulated using a regression function or a factor table.

Simulated LAE = Current LAE × Simulated Cumulative Development Factor

4.2 IBNRFigure 6 shows the process of simulating IBNR claims.

Figure 6. IBNR Claim Simulation

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Monthly Frequency

Distribution

Report Lag

Distribution

IBNR Count

Report Date

Uniform in Accident Month

Severity, Deductible,

Limit

Occurrence Date

Settlement Date

Ultimate Loss

LAE

Truncated Report Lag

Settlement Lag

LAE Assumption

Exposure Index

Monthly IBNR claim counts are simulated first based on monthly frequency distribution and report lag distribution. For example, if the monthly frequency follows a Poisson distribution with a mean of 100 and the report lag follows an exponential distribution with a mean of 150 days, monthly counts are simulated for the entire study period: from the start date (Jan. 1, 2016) to the valuation date (Dec. 31, 2016).

Table 5. IBNR Count Simulation Example

Accident Month

Simulated Monthly Count1

Probability of Reporting after Dec. 31, 20162

Exposure Index

IBNR Count3

Jan 2016 105 0.032 1 3Feb 2016 98 0.043 1 4Mar 2016 101 0.058 1 6Apr 2016 86 0.078 1 7

May 2016 117 0.105 1 12Jun 2016 109 0.142 1 16Jul 2016 101 0.192 1 19

Aug 2016 108 0.259 1 28Sep 2016 101 0.350 1 35Oct 2016 117 0.472 1 55Nov 2016 113 0.638 1 72Dec 2016 94 0.861 1 811. Simulated from Poisson Distribution with =1002. Calculated based on report lag distribution as an Exponential distribution with =0.01. For example, for

Nov. 2016, the probability is calculated as e−0.01× 45=0.638 where 45 is 45 days representing 1.5 months till the valuation date.

3. IBNR Count = Simulated Monthly Count × Probability of Reporting after Valuation Date × Exposure Index

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With the simulated IBNR count for each accident month, each IBNR claim is then simulated using the following steps:

1. Simulating the occurrence date in a specific accident month using uniform distribution. For example, Nov. 2016 has 72 IBNR claims occurred which means that on average 2.4 claims happened each day during that month.

2. Simulating the report lag using a truncated report lag distribution.

F ( x<X )=FRL ( x<X )−FRL ( x<T )

1−FRL ( x<T )if X>T

0if X ≤T

Where

T = Valuation Date – Occurrence Date

FRL ( x<X ) is the cumulative distribution function of the report lag.

Here, the simulation is done based on the condition that the settlement lag is greater than T so that the claims will be reported after the valuation date. In order to speed up simulation, for claims occurred in the same accident month, T is set as the difference between valuation date and the beginning of the accident month. This enables bulk simulation instead of individual simulation. However, report lag may be biased up a little bit.

Report Date = Occurrence Date + Report Lag

3. Simulating settlement lag according to the settlement lag distribution assumption.Settlement Date = Report Date + Settlement Lag

4. Ultimate loss is simulated based on severity, deductible and limit distributions. Total loss is first simulated according to the severity distribution adjusted by severity index. Deductible and limit are then simulated and applied to the total loss to get ultimate loss. Each claim is also simulated if it is a zero-payment invalid claim.

5. Simulating LAE according to LAE assumption. LAE is assumed to be a function of ultimate loss and other variables such as development year at valuation date. The assumption can take the form as a linear, loglinear, exponential and rlinear (reciprocal linear) function. It can also be defined as a table that defines LAE as a percentage of ultimate loss. The percentage in the table varies by the development year at valuation date, with an example shown in Table 6. The setting is the same as for development factors except that the parameters will be different.

Table 6. LAE Assumption (Example)

Current Development Year

LAE (% of Ultimate Loss)Mean Standard Deviation

1 1% 0.1%2 1% 0.1%

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Monthly Frequency

DistributionFuture Claim Count

Report Date

Uniform in Accident Month

Severity, Deductible,

Limit

Occurrence Date

Settlement Date

Ultimate Loss

LAE

Report Lag

Settlement Lag

LAE Assumption

Exposure Index

3 1.5% 0.2%4 2% 0.5%5 2% 0.5%

4.3 Future Claim (UPR)Future claims are simulated in a similar way to IBNR simulation, as shown in Figure 7.

Figure 7. UPR Claim Simulation

Future claim simulation is different from IBNR simulation in two places.

1. Monthly claim count is directly simulated from frequency distribution, without the need to reflect the impact of report lag.

2. Report lag is simulated from the report lag distribution instead of the truncated report lag distribution.

4.4 Claim ReopenFor closed claims, claim reopenness can be simulated as well, with the process shown in Figure 8.

Figure 8. Claim Reopen Simulation

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Reopen?Closed Claims

Resettlement Date

Reopen Lag

Reopen Loss Development

Factors

Reopen Date

Ultimate Loss

LAE

Resettlement Lag

LAE Assumption

Reopen Assumptions

For each closed claim, it is simulated to be reopened or not in the future. The reopen assumption can be based on a regression model, taking the form of a linear, loglinear or exponential function. It can also be defined in a table with the reopen probability changing with the development year, with an example shown in Table 7.

Table 7. Claim Reopen Assumption (Example)

Current Development Year

ProbabilityMean Standard Deviation

1 0.02 0.0052 0.015 0.0023 0.01 0.0014 0.005 0.0015 0 0

If a closed claim is simulated to be reopened, details of the reopened case will be simulated in the following steps.

1. Simulating the reopen lag using a truncated reopen lag distribution.

F ( x<X )=FRO ( x<X )−F RO ( x<T )

1−FRO ( x<T )if X>T

0 if X≤T

Where

T = Valuation Date – Previous Settlement Date

FRO ( x<X ) is the cumulative distribution function of the reopen lag, the difference between the reopen date and the previous settlement date.

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Here, the simulation is done based on the condition that the reopen lag is greater than T so that the claims will be reopened after the valuation date.

Reopen Date = Settlement Date + Report Lag

2. Simulating resettlement lag according to the resettlement lag distribution assumption.Resettlement Date = Reopen Date + Settlement Lag

3. Simulating ultimate loss with the same methods used for open claim simulation: year-to-year development factors, cumulative development factors based on regression models and truncated severity distribution. The ultimate loss is then adjusted by severity index based on resettlement date.

4. Simulating LAE with the same methods used for open claim LAE simulation: year-to-year development factors and cumulative development factors based on regression models.

4.5 CopulaCopula may be used for two places during the simulation, based on user’s choice.

1. Copula among severity, report lag and settlement lag. For IBNR and UPR claims, severity, report lag and settlement lag are simulated together if a copula assumption is used. For open claims, if conditional severity distribution (ultimate loss > paid loss) is used to simulate the loss amount, settlement lag and severity are simulated together according to the copula assumption.

2. Copula among monthly frequencies of multiple business lines. For IBNR and UPR, if a copula is chosen, monthly claim counts are generated for all business lines at the same time to retain the relationship.

5. ReportingAfter the simulation, four files will be generated including information on simulated individual claims, statistical summary, triangles, and a html report, respectively.

Simulated individual claims are stored in a csv file containing the data fields listed in Table 8.

Table 8. Simulated Individual Claim Data Fields

Data Field DescriptionClaimID Claim unique IDLoB Line of Business LineType Claim type to differentiate claims within a business line. For

example, normal claims, big claims, jump claims, short-term claims, long-term claims, etc.

status OPEN/CLOSED/IBNR/UPR. Closed claims are claims that were closed at the valuation date but are simulated as reopened cases.

occurrenceDate The date that the accidence happenedreportDate The date that the accidence was reportedsettlementDate The date that the claim was closed. It is only required for closed

claimsincurredLoss The sum of paid and outstanding amount at the valuation date.

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osRatio Outstanding ratio, which is outstanding amount divided by Incurred Loss.

ultimateLoss Simulated ultimate loss.expectedLoss Expected ultimate loss based on severity assumption (for IBNR and

UPR) or loss development factors (for open claims and reopened claims)

Paid Total paid loss. It should be the same as (1 – OS Ratio) × Incurred Loss.

totalLoss Total loss before deductible and limit.Deductible Deductible. Simulated for IBNR and UPR.Limit Limit. Simulated for IBNR and UPR.LAE Loss adjustment expense at the valuation date.claimLiability Whether it is a zero-payment invalid claim.Ultimate LAE Simulated ultimate LAE.Expected LAE Expected LAE based on LAE assumptions.reopenDate Date that a closed claim is reopened.resettleDate Date that a reopened claim is resettled.reopenLoss Ultimate loss for a reopened claim.Sim Simulation Number.

Statistical summary is also stored in a csv file, showing the statistics of the data fields in Table 9.

Table 9. Simulation Summary Data Fields

Data Field DescriptionLoB Line of Business Line, or the total portfolioType Claim type to differentiate claims within a business line, or the total

of a business line.class IBNER/ROPEN/IBNR/UPR/TOTAL.

IBNER claims are claims that are open at the valuation date.ROPEN claims are claims that were closed at the valuation date but are simulated as reopened cases.In addition to IBNER, ROPEN, IBNR and UPR, Total includes closed claims that are not reopened as well.

Accident Year Calendar year when the claim occurred. “Total” contains the results of all accident years.

Measure Average, standard deviation, min, max, percentiles (1-99,99.5,99.9, and 99.95th)

Frequency Claim countAvg. Incurred Loss Incurred Loss per claim in a single portfolio simulation. Multiple

portfolio simulations allow the construction of the distribution of avg. incurred loss.

Total Incurred Loss Total Incurred Loss of all claims in a single portfolio simulation. Multiple portfolio simulations allow the construction of the distribution of total incurred loss.

Avg. Ultimate Loss Simulated ultimate loss per claim in a single portfolio simulation. Multiple portfolio simulations allow the construction of the

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distribution of avg. ultimate loss.Total Ultimate Loss Simulated total ultimate loss of all claims in a single portfolio

simulation. Multiple portfolio simulations allow the construction of the distribution of total incurred loss.

Total Paid Total paid amount at the valuation date.Total LAE Simulated total ultimate LAE.Paid LAE Total paid LAE at the valuation date.Reopen Probability Simulated reopen probability.

Triangles and rectangles are constructed for reported count, closed count, incurred loss, and paid loss (assuming single payment) based on the claim data and the simulation data including the average value or major percentiles (10th, 20th, …, 90th, 99.5th and 99.95th). Table 10 and Table 11 shows examples of constructed triangles from the simulated data.

Table 10. Closed Count Triangle based on Claim Data (Example)

Accident Year M12 M24 M36 M48 M602012 345 355 355 355 3552013 393 406 406 4062014 490 512 5122015 455 4812016 499

Table 11. Average Closed Count Rectangle based on Simulated Data (Example)

Accident Year M12 M24 M36 M48 M60 M72 M842012 345 355 355 355 355 355 3552013 393 406 406 406 406 406 4062014 490 512 512 512 512 512 5122015 455 481 481 481 481 481 4812016 499 528.5 528.5 528.5 528.5 528.5 528.52017 532 552.2 552.2 552.2 552.2 552.2 552.2

In addition, a html fitting report is generated by the R program listing the simulation summary results for each business line/type and the total portfolio, and the assumptions used for simulation. A sample report is attached below.

ReferenceGenest, C., Rémillard, B., and Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics 44, 199–214.

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