Risk model

Hi Yuvraj

Both are models for the aggregate number of claims on a portfolio. However, they approach the problem differently.

Under the collective risk model, we take what you may call a 'top-down' approach. We have:

S = X1 + X2 + ... + XN
S = 0 if N = 0

where Xi is the RV that represents the aggregate claim amount for the ith claim and N is the RV representing the number of claims. We assume that the Xi and N are independent and that the Xi are iid (ie we have the same distribution for each claim amount and that each claim amount is independent to the others).

Under the individual risk model, we take more of what you may call a 'bottom-up' approach. Instead of considering the random number of claims we receive on a particular portfolio, we break the portfolio down into the individual risks such that each of these risks can have one claim or zero claims.

We then have:

S = X1 + X2 + ... + Xn

Where n is the (fixed, not random) number of individual risks and Xi is the amount claimed on the ith risk. Xi is either 0 (if there is no claim on the ith risk) or some value (if there is one claim on the ith risk). The value of the claim on the ith risk (should there be a claim) is also a random amount. We can denote this by Yi. So we have:

Xi = 0 if 0 claims
Xi = Yi if 1 claim

This is one potential source of confusion as here we use the collective risk model within the individual risk model. Each of these risks has a random number of claims (either 0 or 1) and we have a random claim amount associated with each risk. This sounds like the set up for the collective risk model. So we can treat the aggregate claims on each of the individual i risks as being modeled by the collective risk model with the following set up:

Xi = Yi,1 + Yi,2 + ... + Yi,Ni
Xi = 0 if Ni = 0

Where Ni ~ Bin(1, qi). As Ni is a binomial with one trial (otherwise known as a Bernoulli trial), the only two outcomes are either 0 claims or 1 claim, ie simplifying the notation back to what we had before, we have:

Xi = 0 if 0 claims
Xi = Yi if 1 claim

One other difference to note between the models, in the collective risk model we assumed that the Xi's were iid. In the individual risk model, we allow for the Yi RVs to potentially have different distributions, so the claim amounts on individual risks are not necessarily identically distributed.

Hope this helps!

Andy