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Econometrics Procedures
 

CCDM Procedure

Simulation Processes for Pure Premium and Custom Simulation Modes

This section describes the simulation processes for pure premium mode (when you specify SIMULATIONMODE=PP) and custom mode (when you specify SIMULATIONMODE=CUSTOM).

Like the simulation processes for the collective risk mode, which are described in the section Simulation Processes for Collective Risk Simulation Mode, the simulation processes for pure premium and custom modes also depend on whether you specify external counts or you request that PROC CCDM simulate the counts and on whether or not the severity or frequency models contain regression effects. The following sections describe the processes for the different possibilities.

Simulation with No Regressors and No External Counts

If you specify severity and frequency models that contain no regression effects and if you do not specify externally simulated counts in the EXTERNALCOUNTS statement, then PROC CCDM uses the simulation process that this section describes.

The process is described for one severity distribution, dist. If you specify multiple severity distributions in the SEVERITYMODEL statement, then the process is repeated for each specified distribution. Let D denote the number of adjusted severity symbols that you specify in the ADJUSTEDSEVERITY= option.

The following steps are repeated M times to generate a compound distribution sample of size M, where M is the value of the NREPLICATES= option:

  1. Use the frequency model that you specify in the COUNTSTORE= item store or the COUNTMODEL statement to draw a value N from the count distribution.

  2. Use the parameter estimates of the severity distribution dist (which are read from the SEVERITYEST= data table or the SEVERITYSTORE= item store that you specify) to draw a severity value X. (For more information about how PROC CCDM draws an individual severity value, see the section Making a Random Draw from a Severity Distribution.)

  3. Compute one sample point of different aggregate loss samples as follows:

    • For SIMULATIONMODE=CUSTOM, do not compute the unadjusted aggregate loss sample point. Because D must be nonzero for this mode, compute D sample points upper S Superscript d of the D adjusted aggregate loss samples as follows:

      1. Set the _FREQ_ and _SEV_ placeholder symbols equal to N and X, respectively. (For more information about the placeholder symbols, see the section Simulation of Adjusted Compound Distribution Sample.)

      2. Execute your severity adjustment program.

      3. Retrieve the adjusted severity values upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D) that your adjustment program computes.

      4. Set the sample point upper S Superscript d of the dth adjusted aggregate loss sample equal to upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D).

    • For SIMULATIONMODE=PUREPREMIUM, multiply the frequency and severity values to compute the sample point S of the unadjusted aggregate loss sample as upper S equals upper N upper X. If upper D greater-than 0, then follow the same steps as the custom mode to compute D sample points upper S Superscript d of the D adjusted aggregate loss samples, except for the following two differences:

      • Do not fill the _FREQ_ placeholder symbol, because it is valid only in the custom mode.

      • Compute upper S Superscript d as upper S Superscript d Baseline equals upper N upper X Superscript d (d equals 1 comma ellipsis comma upper D).

Simulation with Regressors and No External Counts

If the severity or frequency models contain regression effects and if you do not specify externally simulated counts in the EXTERNALCOUNTS statement, then you must specify a DATA= data table to provide values of the regression variables, which together represent a scenario for which you want to simulate the CDM. In this case, PROC CCDM uses the simulation process that this section describes.

The process is described for one severity distribution. If you specify multiple severity distributions in the SEVERITYMODEL statement, then the process is repeated for each specified distribution. Let D denote the number of adjusted severity symbols that you specify in the ADJUSTEDSEVERITY= option.

Note that you are performing scenario analysis when regression effects are present. Let K denote the number of observations that form the scenario. This is the number of observations either in the current BY group or in the entire DATA= data table if you do not specify the BY statement. If upper K greater-than 1, then you are modeling the scenario for a group of entities. If K = 1, then you are modeling the scenario for one entity.

The following steps are repeated M times to generate a compound distribution sample of size M, where M is the value of the NREPLICATES= option:

  1. For each observation k (k equals 1 comma ellipsis comma upper K), draw a value upper N Subscript k from the frequency model. If you specify the COUNTSTORE= item store, then the parameters of the frequency model are determined by the frequency regressors in observation k. If you specify the COUNTMODEL statement, the frequency model does not depend on the regressors, so upper N Subscript k is drawn by using the information that you specify in the COUNTMODEL statement.

  2. For each observation k (k equals 1 comma ellipsis comma upper K), draw a severity value upper X Subscript k from the severity model. If you specify a scale regression model for the severity distribution, then the scale parameter of the severity distribution is determined by the values of the severity regressors in the kth observation. (For more information about how PROC CCDM makes a random severity draw, see the section Making a Random Draw from a Severity Distribution.)

  3. Compute one sample point of different aggregate loss samples as follows:

    • For SIMULATIONMODE=CUSTOM, do not compute the unadjusted aggregate loss sample point. Because D must be nonzero for this mode, compute D sample points upper S Superscript d of the D adjusted aggregate loss samples as follows:

      1. Execute your severity adjustment program K times, once per observation. To execute the program for the kth observation, set the _FREQ_ and _SEV_ placeholder symbols equal to upper N Subscript k and upper X Subscript k, respectively. (For more information about the placeholder symbols, see the section Simulation of Adjusted Compound Distribution Sample.)

      2. Retrieve the adjusted severity values upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D) that your adjustment program computes.

      3. After all K observations are processed, compute the sample point upper S Superscript d of the dth adjusted aggregate loss sample as upper S Superscript d Baseline equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D).

    • For SIMULATIONMODE=PUREPREMIUM, compute the sample point S of the unadjusted aggregate loss sample as upper S equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper N Subscript k Baseline upper X Subscript k. If upper D greater-than 0, then follow the same steps as the custom mode to compute D sample points upper S Superscript d of the D adjusted aggregate loss samples, except for the following two differences:

      • Do not fill the _FREQ_ placeholder symbol, because it is valid only in the custom mode.

      • Compute upper S Superscript d as upper S Superscript d Baseline equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper N Subscript k Baseline upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D).

If you specify the BY statement, then a separate sample of size M is created for each BY group in the DATA= data table.

Simulation with External Counts

If you specify externally simulated counts by using the EXTERNALCOUNTS statement, then each replication in the input data table represents the loss events that are generated by an entity. An entity can be an individual or organization for which you want to estimate the compound distribution. If an entity has any characteristics that are used as external factors (regressors) in developing the severity scale regression model, then you must specify the values of those factors in the DATA= data table. If you specify the ID= variable, then multiple observations for the same replication ID represent different entities in a group for which you are simulating the CDM.

This section describes the simulation process that PROC CCDM uses in the presence of externally simulated counts.

The process is described for one severity distribution. If you specify multiple severity distributions in the SEVERITYMODEL statement, then the process is repeated for each specified distribution. Let D denote the number of adjusted severity symbols that you specify in the ADJUSTEDSEVERITY= option.

Suppose there are M distinct replications in the current BY group of the DATA= data table or in the entire DATA= data table if you do not specify the BY statement. A replication is identified either by the observation number or by the value of the ID= variable that you specify in the EXTERNALCOUNTS statement.

For each of the M values of the replication identifier, the following steps are executed R times, where R is the value of the NREPLICATES= option:

  1. Read the value of the COUNT= variable, upper N Subscript k, for each observation k (k equals 1 comma ellipsis comma upper K), where K denotes the number of observations in the current replication—that is, the number of observations in which the replication identifier has the same value.

  2. For each observation k (k equals 1 comma ellipsis comma upper K), draw a severity value upper X Subscript k from the severity model. If you specify a scale regression model for the severity distribution, then the scale parameter of the severity distribution is determined by the values of the severity regressors in the kth observation. (For more information about how PROC CCDM makes a random severity draw, see the section Making a Random Draw from a Severity Distribution.)

  3. Compute one sample point of different aggregate loss samples as follows:

    • For SIMULATIONMODE=CUSTOM, do not compute the unadjusted aggregate loss sample point. Because D must be nonzero for this mode, compute D sample points upper S Superscript d of the D adjusted aggregate loss samples as follows:

      1. Execute your severity adjustment program K times, once per observation. To execute the program for the kth observation, set the _FREQ_ and _SEV_ placeholder symbols equal to upper N Subscript k and upper X Subscript k, respectively. (For more information about the placeholder symbols, see the section Simulation of Adjusted Compound Distribution Sample.)

      2. Retrieve the adjusted severity values upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D) that your adjustment program computes.

      3. After all K observations are processed, compute the sample point upper S Superscript d of the dth adjusted aggregate loss sample as upper S Superscript d Baseline equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D).

    • For SIMULATIONMODE=PUREPREMIUM, compute the sample point S of the unadjusted aggregate loss sample as upper S equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper N Subscript k Baseline upper X Subscript k. If upper D greater-than 0, then follow the same steps as the custom mode to compute D sample points upper S Superscript d of the D adjusted aggregate loss samples, except for the following two differences:

      • Do not fill the _FREQ_ placeholder symbol, because it is valid only in the custom mode.

      • Compute upper S Superscript d as upper S Superscript d Baseline equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper N Subscript k Baseline upper X Subscript k Superscript d (d equals 1 comma ellipsis comma upper D).

This process generates a compound distribution sample of size upper M times upper R. If you specify the BY statement, then a separate sample of size upper M times upper R is created for each BY group in the DATA= data table.

Last updated: November 24, 2025