Assumptions for multiplicative tariff in Non-Life Insurance My question is: what assumptions are made (and why are they sensible), when working with a multiplicative tariff structure in non-life insurance.
That is, we study E(S), the expectation value of the total portfolio claim S.
With 


*

*$S=\sum_{i=1}^NY_i$ is composite Poisson distributed (thus $E(S)=\lambda v E(Y_1)$ with $\lambda$ being the expected number of claims),

*$S_l=\sum_{i=1}^{N^l}Y_i^l$ being all claims for policy $l=1,...,v$, 

*$v_l=1$, that is, the number of policies in a portfolio of one policy is equal to one, 

*$\mu=E(S)/v=\lambda E(Y_1)$ is the expected claim amount of the whole portfolio


we can rewrite
$$
E(S)=\sum_{l=1}^vE(S_l)=\sum_{l=1}^v\lambda_l v_l E(Y_1^l)=\sum_{l=1}^v\lambda_l E(Y_1^l)=\mu\sum_{l=1}^v\frac{\lambda_l E(Y_1^l)}{\mu}=:\mu\sum_{l=1}^v\chi^l.
$$
Where $\chi^l$ is called the risk characteristic of policy $l$.
At this point, the usual way is to go from working with individual policies $l$ to working with (2 or more) risk classes. Working with only two $(i,j)$, e.g. for a motor insurance, we might take something like $i=$ horse powers of the engine, $j=$ age of vehicle. This changes the above to
$$
E(S)=\mu\sum_{i,j}v_{i,j}\,\,\chi^{(i,j)},
$$
with $v_{i,j}$ being the number of policies in the risk class $(i,j)$.
Now, the above risk characteristic $\chi^{(i,j)}$ is assumed to be multiplicative:
$$
\chi^{(i,j)}=\chi_{1,i}\,\,\chi_{2,j}.
$$
Why are we allowed to make this assumption? Why does it make sense? What is the motivation of this assumption? Does this multiplicative ansatz assume that the two risk factors, i,j are not correlated?
 A: I believe the underlying motivation for any assumption is to avoid having $v$ parameters in:
$$E[S]=\mu\sum_{l=1}^{v}\chi^{(l)}$$
Thus, a parametric form for $\chi^{(l)}$ is assumed, which is a function of some ratings variables. As for why a multiplicative structure is assumed, I think it's just for simplicity. The goal with tarrification is to account for an amount of heterogeneity in a group so that premiums can reflect different risks within the group. In order to account for this, we have to assume that there is some structure (e.g. multiplicative) that is dependent on some ratings variables. In this sense, we are allowed to make this assumption because it is practical to do so.
For example, let's take a simple example in GI. Let's say we have $K=2$ tariff criteria:


*

*$\chi_{1,i}$: Kilometres driven yearly

*$\chi_{2,j}$: Years driven without an accident


We can construct the tarrification scheme as follows:
$$\begin{array}{cc|cccc}
\text{} &  \textbf{Years} & \text{0 years} & \text{1 year} & \text{2 years} & \text{3+ years}\\
 &  \chi_{2,\cdot} & 1.2 & 1.1 & 1.0 & 0.9\\
\textbf{Kilometers} & \chi_{1,\cdot} &  &  &  & \\
\hline
\text{0-15,000} & 0.8 &  &  &  & \\
\text{15,000-25,000} & 0.9 &  &  & \chi^{2,3} & \\
\text{25,000+} & 1 &  &  &  & \\
\end{array}$$
As you can see, the tariff is, for $(i=2,j=3)$:
$$\chi^{2,3}=\chi_{1,2}\,\chi_{2,3}$$
In terms of the question you ask about whether there is no correlation between two risk factors. It is important to conduct multivariate statistical analysis to avoid an "over-correction" for dependent factors. For example, for each $k=1,\ldots,K$ risk factors, you need to ensure each risk factor isn't strongly related to another risk factor as this would lead to too much correction. In terms of the example above, there may be some correlation between kilometers driven and years without accident which could lead to some issues unless corrected for.
