CCDM Procedure

In many loss modeling applications, you analyze the loss events by modeling the severity (magnitude) of loss and the frequency (count) of loss separately. The primary goal of creating these models is to estimate the aggregate loss—that is, the total loss that occurs over a period of time for which the frequency model is applicable. For example, an insurance company might want to assess the expected and worst-case losses for a particular business line, such as automobile insurance, over an entire year by using the models for the number of losses in a year and the severity of each loss. A bank might want to assess the value-at-risk (VaR), a measure of the worst-case loss, for a portfolio of assets by using the frequency and severity models for each asset type.

Severity of loss and frequency of loss are random variables, so the aggregate loss is also a random variable. Instead of preparing a point estimate of the expected aggregate loss, it is better to estimate its probability distribution, because this enables you to infer various aspects of the aggregate loss, such as measures of location, scale (variability), and shape, in addition to percentiles. For example, the VaR that banks or insurance companies use to compute regulatory capital requirements is usually the estimate of the 97.5th or 99th percentile from the aggregate loss distribution.

Let N represent the frequency random variable for the number of loss events that occur in the time period of interest. Let X represent the severity random variable for the magnitude of one loss event. Then, the aggregate loss S is defined as

The goal is to estimate the probability distribution of S. Let denote the cumulative distribution function (CDF) of X, let denote the n-fold convolution of the CDF of X, and let denote the probability of seeing n losses as per the frequency distribution. The CDF of S is theoretically computable as

This probability distribution model of S, characterized by the CDF , is referred to as a compound distribution model (CDM). The CCDM procedure computes an estimate of the CDM by using the distribution models of X and N.

Direct computation of is usually a difficult task because of the need to compute the n-fold convolution. Klugman, Panjer, and Willmot (1998, Ch. 4) suggest some relatively efficient recursion and inversion methods for certain combinations of severity and frequency distributions. However, those methods assume that distributions of N and X are fixed and all Xs are identically distributed. When the distributions of X and N are conditional on external factors (regressors), each set of regressor values results in a different distribution. So you must repeat the recursion and inversion methods for each combination of regressor values, and this repetition makes these methods prohibitively expensive. PROC CCDM instead estimates the compound distribution by using a Monte Carlo simulation method, which can use all available computational resources to generate a sufficiently large, representative sample of the compound distribution while accommodating the dependence of distributions of X and N on external factors. Conceptually, the simulation method works as follows:

Steps 1 through 3 are repeated M times, where M is a number that you specify, to obtain the representative sample of the CDM. PROC CCDM analyzes this sample to compute empirical estimates of various summary statistics of the compound distribution such as the mean, variance, skewness, and kurtosis in addition to percentiles such as the median, the 95th percentile, the 99th percentile, and so on. You can also use PROC CCDM to write the entire simulated sample to an output data table.

The simulation process gets more complicated when the frequency and severity models contain regression effects. The CDM is then conditional on the values of regressors. The simulation process essentially becomes a scenario analysis, because you need to specify the expected values of the regressors that together represent the scenario for which you want to estimate the CDM. PROC CCDM enables you to specify an input data table that contains the scenario. If you are modeling a group of entities together (such as a portfolio of multiple assets or a group of insurance policies), and each entity has a different set of characteristics, then the scenario consists of more than one observation, and each observation corresponds to a different entity. PROC CCDM enables you to specify such a group scenario in the input data table and creates a realistic simulation of loss events that each entity can generate.

PROC CCDM accepts the severity model of X as estimated by the SEVSELECT procedure, and it accepts the frequency model of N as estimated by the CNTSELECT procedure. Both the SEVSELECT and CNTSELECT procedures are part of SAS Econometrics software. Both procedures allow models of X and N to be conditional on external factors (regressors). In particular, you can model the severity distribution such that its scale parameter depends on severity regressors, and you can model the frequency distribution such that its mean depends on frequency regressors. The frequency model can also be a zero-inflated model. PROC CCDM uses the estimates of model parameters and the values of severity and frequency regressors to estimate the compound distribution model.

PROC CCDM also enables you to specify externally simulated counts. This is useful if you have an empirical frequency model or if you estimate the frequency model by using a method other than using PROC CNTSELECT. To specify an external count model, you create a data table that contains M replications of counts that are randomly drawn from your frequency model. For each of the replications, in step 1 of the simulation, PROC CCDM uses the count N that is associated with that replication in the count data table. If the severity model contains regression effects, then you can specify the scenario to analyze by specifying the severity regressor data for each of the M replications in the same count data table.

If the estimates of the parameters of your severity and frequency models have uncertainty associated with them, and they usually do, then you can use PROC CCDM to conduct parameter perturbation analysis to assess the effect of the uncertainty in parameter estimates on the estimates of CDM. If you request that P perturbed samples be generated, then the parameter set is perturbed P times, and each time PROC CCDM makes a random draw from either the univariate normal distribution of each parameter or the multivariate normal distribution over all parameters. For each of the P perturbed parameter sets, a full compound distribution sample is simulated and summarized. This process yields P estimates for each summary statistic and percentile, which are then used to provide you with estimates of the location and variability of each summary statistic and percentile.

You can also use PROC CCDM to compute the distribution of an aggregate adjusted loss. For example, in insurance applications, you might want to compute the distribution of the amount paid in a particular time period after applying adjustments such as the deductible and the policy limit to each individual loss. PROC CCDM enables you to specify SAS programming statements to adjust each severity value. If represents the dth adjusted severity value, then PROC CCDM computes , the dth aggregate adjusted loss, as

You can define and analyze as many adjusted losses as you want in one PROC CCDM step. All the analyses that PROC CCDM conducts for the aggregate unadjusted loss, including scenario analysis and parameter perturbation analysis, are also conducted for each aggregate adjusted loss. This gives you a comprehensive picture of the compound distribution model for each adjusted loss.

PROC CCDM is the next-generation version of PROC HPCDM. It requires SAS Cloud Analytic Services (CAS) in order to run. Because PROC CCDM is a next-generation high-performance analytical procedure, it also does the following: