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Variability in a heterogeneous/homogeneous portfolio

D

Delvesy888

Member
Hi. I am a bit stuck as to a section of the Risk Models (2) chapter. The notes state:

"Consider a portfolio consisting of $n$ independent policies. The aggregate claims from the $i$-th policy are denoted by the random variable $S_{i}$, where $S_{i}$ has a compound Poisson distribution with parameters $\lambda_{i}$ (all i.i.d), $\textbf{not known}$, and the CDF of the individual claim amounts distribution is $F(x)$, known."

The notes then go on to say that this implies that all of the $S_{i}$s are i.i.d. This makes intuitive sense.

However, the next section has the same set-up, but now the Poisson distribution parameters are all $\lambda$. The notes state "If the value of $\lambda$ were known, then the $S_{i}$ are i.i.d". I.e. the $S_{i}|\lambda$ are i.i.d. Implying that the $S_{i}$ themselves (i.e. with $\lambda$ $\textbf{not known}$) are dependent. This seems to contradict the first section.

If you can help me get my head around this, intuitively, that would be a great help.

Thanks very much.
 
Hi Delvsey888,

Sorry, I'm struggling to understand your question because something funny seems to have happened with the font. I've tried looking at the paragraph you are talking about but I can't see the apparent contradiction.

In particular, why would i.i.d. imply dependence?

" I.e. the $S_{i}|\lambda$ are i.i.d. Implying that the $S_{i}$ themselves (i.e. with $\lambda$ $\textbf{not known}$) are dependent "

John
 
Hi John,
Thanks for the reply. Yes, sorry about the font. I will try to be a little more clear..

The notes state that for a heterogeneous portfolio, the S_i are independent. if the lambda_i are unknown. This makes sense to me, since I cannot make an informed decision as to the value of one S_i, given another.

The notes then state that in a homogeneous portfolio, the S_i are independent if lambda (same for all S_i) are known. And therefore dependent if lambda is unknown.

I do not understand the reason for this last paragraph - i.e. why do we need to know lambda for the S_i to be independent???

I hope that makes more sense.

Thank you.
 
The Poisson parameters are the same for all policies in the portfolio. So, before we know the value of lambda, each policy is not independent of each other, in the sense that whatever the mean is for one policy, the other policies will have the same mean. However, once the lambda has been set, the policies then have nothing to do with each other, you can think of it as a coincidence that they have the same lambda.

If X~Po(lamda) and Y~Po(lamda) then X is NOT independent of Y because the value of lamda for X will be the same for Y

If X~Po(5) and Y~Po(5) then X IS independent of Y.

Hope this helps,
John
 
Hi John,
Thank you for the reply and the good explanation, that has certainly cleared things up.
 
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