I have a quick question with regards to the calculation of σ^2(θi) = Vi * var(Xi | θi) as given in the Buhlmann Straub formulae. You will notice that in the Sept 2017Q2 paper, the way the parameters of the Gamma distribution are specified will allow for cancellation of the Vi's once this stage of the calculation is reached (i.e. calculation of σ^2(θi)). However in Apr2019, I can't seem to find any discernable way the same cancellation will apply. Am I missing something here? Looking at the solution in the examiner's report it isn't obvious whether this cancellation has occurred or not or whether it was even considered.
In April 2019 Q6, we are given the distribution for the number of claims per year for an individual within a risk class, whereas when the Bühlmann-Straub model is discussed in the Core Reading, we instead begin with the distribution of the claim ratio for a risk, which depends on the volume associated with that risk. As we are working with the distribution for the number of claims for an individual within a risk class, there is no volume factor to apply in the formula for σ^2(θi).
From the definition of BS model, the same volume measure V, is defined to be in the formulas for both σ^2(θi) and the credibility factor z. How come here we still include it (V=37) in the latter to calculate z = 0.965? Whereas for σ^2(θi) we use V=1.
The formulation used in this question does not quite match the formulation given for the Bühlmann-Straub model in the Core Reading. In the question we are working with the distribution for the number of claims for an individual within a risk class, so there is no volume factor to apply in the formula for σ^2(θi) (or equivalently you could argue V=1 as there is one individual). The formulation in the Core Reading relates to the distribution of the claim ratio for a risk, which depends on the volume associated with that risk. When we calculate the Bühlmann-Straub credibility factor we are deciding how much weight to give to our past experience. In this case V=37 because we have 37 individuals in the past data from 2016 to 2018.
