The number of claims of each Type arising from an occurrence is modeled via a multinomial distribution

 

The number of claims of each Type arising from an occurrence is modeled via a multinomial distribution. This allows for the possibility of multiple claims from the same occurrence, either of the same Type or of multiple Types.  It typically introduces correlations between frequencies across pairs of Types within the same Line.

 

Each occurrence k = 1 to N can generate multiple numbers of “claims” of various “types.”  Presumably the number of claims is another random variable N2(k).  Apparently, these N2 claims get distributed to various types based on a multinomial distribution.  In the uncorrelated situation, one could use the function RMULTINOM in “R” to generate the number of each type.  If correlation applies, then one can generate an uncorrelated sample and apply the matrix L to it, where L is the “square root” of the correlation matrix.

 

Example: 

 

In one simulation definition, 60% of claims involve a single Physical Damage claim, 20% involve a single Physical Damage claim and a single Property Damage claim, 10% involve asingle Bodily Injury claim, 8% involve one Property Damage and one Bodily Injury claim, and 2% involve one Property Damage and two BodilyInjury claims. The system lets him enter this model as:

Physical Damage Property Damage  Bodily Injury Proportion

Normalized Probability

1 0 0 60 0.60
1 1 0 20 0.20
0 0 1 10 0.10
0 1 1 8 0.08
0 1 2 2 0.02

Step 1: System will generate the number of occurrences N. N is a random variable based on the amount of exposure, seasonality adjustments, trends,  and the parameters specified for the frequency distribution. For example, we have N=530

Step2:  For this N=530 occurrences, based upon the proportions entered above, system will do a multinomial  simulation with sample size 1000: rmultinom(1000, 530, c(60, 20, 10, 8, 2)) , and get the occurrence sample as:

> x<-rmultinom(1000, 530, c(60, 20, 10, 8, 2))
> x[,189]             #pick a random
[1] 316 101  61  38  14
> mean(x[1,])        #check the random mean
[1] 317.804

Physical Damage Property Damage  Bodily Injury Occurrences
1 0 0 316
1 1 0 101
0 0 1 61
0 1 1 38
0 1 2 14

Step3: The number of claims of each Type arising from the 530 occurrence is:

Physical Damage Property Damage  Bodily Injury Total
Number of Claims 316+101=417 101+38+14=153 61+38+2x14=127 697