The CCDM Procedure

As an illustration of the simulation process, consider a very simple example of analyzing the distribution of an aggregate loss that is incurred by a set of policyholders of an automobile insurance company in a period of one year. It is postulated that the frequency and severity distributions depend on three variables: Age, Gender (1: female, 2: male), and CarType (1: sedan, 2: sport utility vehicle). So these variables are used as regressors while you fit the count model and severity scale regression model by using the CNTSELECT and SEVSELECT procedures, respectively. Now, consider that you want to use the fitted frequency and severity models to estimate the distribution of the aggregate loss that is incurred by a set of five policyholders. The characteristics of the five policyholders are encoded in a data table named mycas.Scenario that has the following contents:

Obs     age     gender     carType
1       30      2          1
2       25      1          2
3       45      2          2
4       33      1          1
5       50      1          1

The column Obs contains the observation number. It is shown only for the purpose of illustration. It does not need to be present in the data table. The following PROC CCDM step simulates the scenario in the data table mycas.Scenario:

proc ccdm data=mycas.scenario
        severityest=<severity parameter estimates data table>
        countstore=<count model store> nreplicates=<sample size>;
    severitymodel <severity distribution name(s)>;
run;

The following process generates a sample from the aggregate loss distribution for the scenario in the data table mycas.Scenario:

To illustrate the simulation process, consider the following simple example. In this example, your severity model does not contain any regressors. An example that uses a severity scale regression model is illustrated later. Assume that you have made 10 random draws from an external count model and recorded them in the variable ExtCount of a data table named mycas.Counts1, as follows:

Obs     extCount
1       3
2       2
3       0
4       1
5       3
6       4
7       1
8       2
9       0
10      5

Because the data table does not contain an ID= variable, the observation number that is shown in the Obs column acts as the replicate identifier. The following PROC CCDM step simulates an aggregate loss sample by using the data table mycas.Counts1:

proc ccdm data=mycas.counts1 nreplicates=5
        severityest=<severity parameter estimates data table>;
    severitymodel <severity distribution name(s)>;
    externalcounts count=extCount;
run;

The simulation process works as follows:

The process results in an aggregate loss sample of size 50, which is equal to the number of replications in the data table (10) multiplied by the value of the NREPLICATES= option (5).

Now, consider an example in which the severity models in the SEVERITYEST= data table are scale regression models. In this case, the severity distribution that is used for drawing the severity value is determined by the values of regressors in the observation that is being processed. Consider that you want to simulate the aggregate loss that one policyholder incurs and you have recorded, in the variable ExtCount, the results of 10 random draws from an external count model. The DATA= data table has the following contents:

Obs     age     gender     carType  extCount
1       30      2          1        5
2       30      2          1        2
3       30      2          1        0
4       30      2          1        1
5       30      2          1        3
6       30      2          1        4
7       30      2          1        1
8       30      2          1        2
9       30      2          1        0
10      30      2          1        5

The simulation process in this case is the same as the process in the previous case of no regressors, except that the severity distribution that PROC CCDM uses for drawing the severity values has a scale parameter that is determined by the values of the regressors Age, Gender, and CarType in the observation that is being processed. In this particular example, all observations have the same value for all regressors, indicating that you are modeling a scenario in which the characteristics of the policyholder do not change during the time for which you have simulated the number of events. You can also model a scenario in which the characteristics of the policyholder change by recording those changes in the values of the appropriate regressors.

Extending this example further, consider that you want to analyze the distribution of the aggregate loss that is incurred by a group of policyholders, as in the example in the section Illustration of Aggregate Loss Simulation Process. Let the data table mycas.Counts2 record multiple replications of the number of losses that each policyholder might generate. The contents of the data table mycas.Counts2 are as follows:

Obs     replicateId age     gender     carType  extCount
1       1           30      2          1        2
2       1           25      1          2        1
3       1           45      2          2        3
4       1           33      1          1        5
5       1           50      1          1        1

6       2           30      2          1        3
7       2           25      1          2        2
8       1           45      2          2        0
9       2           33      1          1        4
10      2           50      1          1        1

The variable ReplicateId records the identifier for the replication. Each replication contains multiple observations, such that each observation represents one of the policyholders that you are analyzing. For simplicity, only the first two replications are shown here.

The following PROC CCDM step simulates an aggregate loss sample by using the data table mycas.Counts2:

proc ccdm data=mycas.counts2 nreplicates=3
        severityest=<severity parameter estimates data table>;
    severitymodel <severity distribution name(s)>;
    externalcounts count=extCount id=replicateId;
    output out=aggloss samplevar=totalLoss;
run;

When you specify an ID= variable in the EXTERNALCOUNTS statement, the DATA= data table must be partitioned by the ID= variable as noted in the description of the ID= option.

The simulation process works as follows:

If your data table mycas.Counts2 contains 10,000 distinct values of ReplicateId, then 30,000 observations are written to the data table mycas.AggLoss that you specify in the OUTPUT statement of the preceding PROC CCDM step. Because you specify SAMPLEVAR=TotalLoss in the OUTPUT statement, the aggregate loss sample is available in the TotalLoss column of the data table mycas.AggLoss.

PROC CCDM assumes that the severity distribution that you specify in the SEVERITYEST= data table or the SEVERITYSTORE= item store is an untruncated distribution—that is, a distribution that has the full support. For severity distributions that have a scale parameter, full support is , and for severity distributions that have a log-scale parameter, full support is . If you want PROC CCDM to make draws from a truncated distribution, which is especially useful in insurance applications, then you can specify the LEFTTRUNCATION= or RIGHTTRUNCATION= options in the PROC CCDM statement.

Let L and R denote the left-truncation threshold and right-truncation limit that you specify in the LEFTTRUNCATION= and RIGHTTRUNCATION= options, respectively. Let F and denote the cumulative distribution functions (CDF) of the untruncated and truncated severity distributions, respectively. Not specifying the LEFTTRUNCATION= option is equivalent to specifying —that is, . Not specifying the RIGHTTRUNCATION= option is equivalent to specifying —that is, .

PROC CCDM makes a random draw from the truncated severity distribution by using the following steps: