The HPCDM Procedure (Experimental)

Analyzing the Effect of Parameter Uncertainty on the Compound Distribution

Continuing with the previous example, note that you have fitted the frequency and severity models by using the historical data. Even if you choose the best-fitting models, the true underlying models are not known exactly. This fact is reflected in the uncertainty that is associated with the parameters of your models. Any compound distribution estimate that is computed by using these uncertain parameter estimates is inherently uncertain. You can request that PROC HPCDM conduct parameter perturbation analysis, which assesses the effect of the parameter uncertainty on the estimates of the compound distribution by simulating multiple samples, each with parameters that are randomly perturbed from their mean estimates.

The following PROC HPCDM step adds the NPERTURBEDSAMPLES= option to the PROC HPCDM statement to request that perturbation analysis be conducted and the PRINT=PERTURBSUMMARY option to request that a summary of the perturbation analysis be displayed:

/* Conduct parameter perturbation analysis of
   the Poisson-gamma compound distribution model */
proc hpcdm countstore=countStorePoisson severityest=sevest
           seed=13579 nreplicates=10000 nperturbedsamples=30
           print(only)=(perturbsummary) plots=none;
   severitymodel gamma;
   output out=aggregateLossSample samplevar=aggloss;
   outsum out=aggregateLossSummary mean stddev skewness kurtosis
          p01 p05 p95 p995=var pctlpts=90 97.5;
run;

The Work.AggregateLossSummary data set contains the specified summary statistics and percentiles for all 30 perturbed samples. You can identify a perturbed sample by the value of the _DRAWID_ variable. The first few observations of the Work.AggregateLossSummary data set are shown in Figure 4.3. For the first observation, the value of the _DRAWID_ variable is 0, which represents an unperturbed sample—that is, the aggregate sample that is simulated without perturbing the parameters from their means.

Figure 4.3: Summary Statistics and Percentiles of the Perturbed Samples

_SEVERITYMODEL_ _COUNTMODEL_ _DRAWID_ _SAMPLEVAR_ N MEAN STDDEV SKEWNESS KURTOSIS P01 P05 P90 P95 P97_5 var
Gamma Poisson 0 aggloss 10000 4062.76 3429.57 1.14604 1.76466 0 0 8792.64 10672.49 12391.70 15877.89
Gamma Poisson 1 aggloss 10000 4008.04 3406.22 1.10747 1.43304 0 0 8658.62 10521.82 12279.33 16152.05
Gamma Poisson 2 aggloss 10000 4426.67 3719.94 1.14337 1.66525 0 0 9484.05 11522.70 13523.54 17575.20
Gamma Poisson 3 aggloss 10000 3991.87 3480.10 1.23233 2.07634 0 0 8672.80 10568.25 12472.90 16969.77
Gamma Poisson 4 aggloss 10000 3807.58 3303.61 1.08965 1.15633 0 0 8375.09 10319.59 11884.11 15255.16
Gamma Poisson 5 aggloss 10000 4083.70 3429.83 1.08043 1.31018 0 0 8836.78 10707.19 12399.09 16236.24
Gamma Poisson 6 aggloss 10000 4185.82 3525.20 1.12642 1.49282 0 0 9095.46 11056.46 12752.18 16519.99
Gamma Poisson 7 aggloss 10000 3882.99 3372.81 1.22931 1.95615 0 0 8515.35 10371.84 12245.23 16153.91
Gamma Poisson 8 aggloss 10000 4092.94 3483.60 1.10040 1.47077 0 0 8923.13 10757.13 12522.34 16275.95
Gamma Poisson 9 aggloss 10000 4039.82 3454.69 1.17185 1.84608 0 0 8696.09 10679.34 12611.43 16350.84
Gamma Poisson 10 aggloss 10000 3851.17 3287.52 1.12302 1.60240 0 0 8383.29 10129.41 11725.89 15303.35



The PRINT=PERTURBSUMMARY option in the preceding PROC HPCDM step produces the "Sample Perturbation Analysis" and "Sample Percentile Perturbation Analysis" tables that are shown in Figure 4.4. The tables show that you can expect a mean aggregate loss of about 4,049.1 and the standard error of the mean is 193.6. If you want to use the VaR estimate to determine the amount of reserves that you need to maintain to cover the worst-case loss, then you should consider not only the mean estimate of the 99.5th percentile, which is about 16,339.1, but also the standard error of 692.8 to account for the effect of uncertainty in your frequency and severity parameter estimates.

Figure 4.4: Summary of Perturbation Analysis of the Poisson-Gamma Compound Distribution

The HPCDM Procedure
Severity Model: Gamma
Count Model: Poisson

Sample Perturbation Analysis
Statistic Estimate Standard
Error
Mean 4049.1 193.55480
Standard Deviation 3448.5 132.43375
Variance 11909479 919586.4
Skewness 1.14075 0.04610
Kurtosis 1.64953 0.27146
Number of Perturbed Samples = 30
Size of Each Sample = 10000

Sample Percentile Perturbation Analysis
Percentile Estimate Standard
Error
0 0 0
1 0 0
5 0 0
25 1386.8 114.41389
50 3368.2 185.13314
75 5944.8 265.53061
90 8756.0 365.86765
95 10663.6 441.16381
97.5 12454.8 519.67311
99 14685.6 620.49261
99.5 16339.1 692.79352
Number of Perturbed Samples = 30
Size of Each Sample = 10000