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Credible Loss Ratio Claims Reserves - Hurlimann

Synopsis

Be able to calculate \(R^{\text{ind}}\) and \(R^{\text{coll}}\) using \(m_{k}\) knowing which assumptions apply. Learn the basic formulae for the 3 credibility factors. Be able to calculate MSE of ultimate losses too.

Study Strategy

  • Understand what \(R^{\text{ind}}\) and \(R^{\text{coll}}\) mean by following the example in the material.
  • Learn how to solve for \(m_{k}\) and get the intuition.
  • Learn how to calculate the Individual and collective reserves, \(R^{\boxed{.}}\)
  • Once done, just know the \(Z^{\text{GB}}\), \(Z^{\text{WN}}\) and how to calculate them.

  • Don't worry about the big formula for optimal \(Z\). Just know that when \(Var(U_{i})=Var(U_{i}^{BC})\)

  • Don't worry about \(E(\alpha_{i}^{2}(U_{i}))\) because it comes from the distribution of the Ultimate losses. For the first pass, just do it the TIA way. #later go to the source text or Rising Fellow for details.

My Notes

  • There are two methods (know what to call them):
    • Collective Loss Ratio method
    • Individual Loss Ratio method
  • If the premiums and exposures are constant and not given assume a certain exposure or premium (otherwise your calculations will be conk and wrong. Unless you perfectly understand the logic of averaging the incremental loss ratios for calculating \(m_{k}\)). Be on the safe side, assume the exposure to be a reasonable number.

Know that you will always have to do the first step of calculating the \(m_{k}\) so muscle memorize this.

12 24 36 Total
m 0.32 0.14 0.10 0.55
### Remember
  • For \(mse(R_{i})\) in general, write \(p_{i}, q_{i}, t_{i}\) in the denominators first. Then write \(Z_{i}^{2}\), 1 and \((1-Z_{i})^{2}\). And don't forget to multiply the \(q_{i}^{2}\)!

Muffs

  • Be careful about cumulative triangles, you NEED TO CONVERT THEM TO INCREMENTAL.
  • Multiplying \(Z\) to \(R^{\text{coll}}\) instead of \(R^{\text{ind}}\)
  • They asked which is a better choice (\(R_{c}\) vs other options). I should have just told that \(R_{c}\) since it is optimal (optimal \(\implies\) least variance \(\implies\) better from a statistical POV) but I calculated the MSE for each. Dummy!
  • Read the question and pay attention to
    • which maturity period is being talked about: It was told 12-24, but I assumed and started looking at 0-12.