How does the expected number of claims impact the insurance charge?
As the expected number of claims increases the insurance charge decreases for entry ratios above 1.
Could you please explain why it only applies to entry ratios above 1?
As the expected number of claims increases the insurance charge decreases for entry ratios above 1.
Could you please explain why it only applies to entry ratios above 1?
Comments
Update 9/1: Cleaning up this answer to keep everything in one place.
The Fisher text says (top p.90) "All other things being equal, the higher the expected number of claims, the lower the variance on the distribution of entry ratios, and the smaller the Table M charge is for entry ratios above 1."
How you choose to interpret "all other things being equal" changes how you draw a Lee diagram and view the conclusions.
Plotting a Lee diagram with a small number of expected claims results in a steep inverted S shape curve, call this curve F_1. Plotting one with a large number of expected claims results in a much flatter curve as the variance of the claims is reduced, call this curve F_2. If both sets of claims share the same claim severity distribution, they have the same maximum entry ratio. If they only shared the same expected severity then they need not meet at a common point in the top right part of the graph.
For entry ratios greater than 1, curve 1 lies above curve 2, so for any entry ratio, r, greater than 1, increasing the number of expected claims means you reduce the area corresponding to the insurance charge because area A is no longer included. Conversely, for entry ratios below 1 you increase the insurance charge if and only if area E is greater than area A+C.
We spent a lot of time trying to figure out why F_1(r) = F_2(r) means r = 1. To be honest, the only way it made sense to us is if we assume both claim severity distributions have the same expected severity and the same proportion of claims below the expected severity. Note this doesn't have to mean the two claim severity distributions are the same. Fisher likely intended them to have the same severity distribution and the difference between curves F_1 and F_2 is a result of the sample size for the number of claims.
Do you mind creating a graph? it is not too clear describing them in words.
We've added a graph and cleaned up our original answer. Hopefully this helps.
Thank you for the graph and the detailed explanation!