Fall 2014 17b
Is the solution in the examiner's report missing the expected per-occurrence excess losses in the basic premium calculation? I.e. shouldn't there be a "+0.214*1.1" in the formula?
Is the solution in the examiner's report missing the expected per-occurrence excess losses in the basic premium calculation? I.e. shouldn't there be a "+0.214*1.1" in the formula?
Comments
This is another great example of the CAS regularly changing the syllabus readings so they no longer match the past exam papers they've released.
According to the Fisher reading in the 2021 syllabus, the basic premium includes both the expected per-occurrence excess losses and the expected aggregate excess losses. So you are correct, there should be a +1.1*0.214 in the formula to incorporate the expected per-occurrence excess losses.
At the end of the examiner's report for part b they note at the time they didn't penalize candidates who included this term as it's a standard industry practice but isn't done so in the source material at the time.
I am a bit confused on basic premium formula. Could I please confirm when we should use unlimited E and when we should use limited E?
When attempting this question please keep in mind the notation in the examiner's report is that used in a previous reading which is no longer on the syllabus.
In this problem we have both a per-occurrence limit and an aggregate limit (implied by the min and max premiums). As such, you can price the policy either using a Table L approach or through pricing the per-occurrence piece using a Table M_D approach and then adding on the price for the aggregate excess. The question doesn't make it particularly clear what sort of table you're given but since there are expected loss groups you should take this as a hint to use the ICRLL procedure, taking the Limited Table M approach to price each piece separately.
Using Fisher's notation, the basic premium formula is b = e - (c-1)E + cI. The basic premium formula covers expenses which do not vary with the losses. Working with ratios to standard premium we have e is the total expense ratio and (c-1)E is the portion of expenses which are related to losses. So e - (c-1)E is the fixed expense ratio (general expenses, acquisition costs etc.).
We now add back on an expense ratio which accounts for capping the losses at the aggregate limit. This is c\hat{I} where \hat{I} is calculated as E[A_D]*(phi(r_G)-psi(r_H)) because the aggregate limit is calculated using loss data which has been capped first by the per-occurrence limit, and through using the ICRLL procedure we're able to use a Table M (hence phi rather than phi_D etc.). This is the third term in the equation you posted above.
However, according to the 2022 syllabus we may not be done yet. This is because we've not included any expense provision for the portion of losses above the per-occurrence limit. Fisher (p. 19) says the basic premium may or may not include a provision for losses in excess of the per-occurrence deductible. The expense we need to add on is c*(excess loss factor). The examiner's report says they accepted solutions regardless of whether this last step was taken or not.
Thank you! I have a follow-up question.
In the Fisher.Visualization, there are a few formulas for basic premium. For limited table M, the formula is
In this question, shouldnt we use the formula above where the second term is limited E?
Thanks for calling this out as there are a couple of things going on here.
Firstly, there is a small typo in that formula, the second term should be (c-1)E[A] to reflect we're removing all loss adjustment expenses associated with the retro plan. This is because we're now pricing both the losses excess of the aggregate limit and excess of the per-occurrence limit within the basic premium. If you use the version without the kE[A] at the end then it should be (c-1)E[A_D] to recognize you're ignoring the pricing of the per-occurrence limit. We've cleaned up the wiki to hopefully make this much more explicit. See here: https://battleacts8.ca/8/wiki/index.php?title=Fisher.Visualization#Limited_Table_M_Balance_Equations
Secondly, in the context of Fall 2014 Q17 we don't need to apply the Limited Table M basic premium formula because we've used the ICRLL procedure to adjust the expected losses to reflect the presence of the per-occurrence limit so we can use a Table M approach. This means we should be using the Table M version of the basic premium formula.
Hope this helps
On your second point, if we use the Table M version of basic premium formula, it would be the following which uses I, that is calculated using E(A).
However, it is different than the solution where I is calculated using E(A_D), not E(A). The balance equations in the solution are also using the one from Limited Table M, not Table M.
Could you please confirm the following?
when policy has both aggregate limit and per-occurrence limit,
Thanks!
All three of your points are correct.
The syllabus has changed materially since 2014 and the current Fisher reading says it is optional whether you include the cost of the per-occurrence limit within the basic premium.
My second point in the prior post needs correcting. The ICRLL procedure is used to avoid needing a Limited Table M when pricing in this situation. It does so by translating the expected losses to those of a larger, more stable risk to reflect the stability introduced by the per-occurrence limit to the ratable losses. Then we can use a Table M to produce the net aggregate loss factor. Once we have that, we should resume pricing using the actual information associated with the risk. Namely, the net insurance charge is priced by applying the net aggregate loss factor to the expected losses limited by the per-occurrence limit, E[A_D].
Thanks for the clarification!!
The problem says that the Excess Loss Factor is 0.241. Wouldn't the Excess Loss Factor be a number such that if you multiply the Unlimited Expected Losses by the ELF then you get the Excess Losses? Ie: (Unlimited Expected Loss) * ELF = E - E[A_D], so ELF = (E - E[A_D])/E? Which would match our definition of k. Why then, do they divide the ELF by E to get k?
This is a terminology thing I'm afraid you have to get used to and also hope the CAS uses it in the right context. k is the loss elimination ratio which is a ratio to losses, whereas the excess loss factor is a ratio to standard premium.
See Alice's note below the applying the ICRLL procedure example here:
By dividing the excess loss factor by the expected loss ratio, the standard premium cancels out to leave you with k.
Loss elimination ratios can be confusing because they depend on the context. When we have a per-occurrence limit and an agg limit the LER is the loss above the per-occurrence limit because this is "eliminated" when pricing the agg limit. However, if you were working on say a Bahnemann problem pricing a deductible then the LER is E[AD]/E[A] because the piece that gets eliminated by the deductible is the bit below the deductible.
Alice's best advice is to drink coffee and focus understanding and keeping track of what you're trying to count/eliminate rather than memorizing too many formulas.
Hi. I have an additional question. Fisher says we may include the cost of the per-occurrence limit within the basic premium. However, the discussion above (especially tracyguo8's #3 point above about the ICRLL procedure) gives me the impression that this is a situation where we must include it. Am I correct? How was I supposed to know to include it in this problem, and how can I know for future problems? Thanks!
Also, since the updated method is not in the examiner's report, do you mind confirming that my approach is correct here?
In our opinion there isn't really a good answer to your question because the Fisher text currently leaves things too open-ended by saying the basic premium may or may not include the cost of the per-occurrence limit.
If you're asked, as is the case here, for the basic premium without going on to do anything further with it then you should state whether or not you're choosing to include the cost of the per-occurrence limit in the basic premium. The CAS should accept either answer unless they gave explicit instructions for its treatment.
If you're asked to price the policy then you need to use the basic premium and account for the cost of both the aggregate and per-occurrence limits. Then it doesn't really matter what you're calling the basic premium provided you include the cost of both parts in the price. This is what they're trying to show in the case study.
Your answer is almost right. You have the hard part correct but the expense dollars are wrong. e is a ratio to standard premium, not expected losses so the expense dollars are $200,000 instead of $130,000. Once you fix this, you'll get the same answer as adding c*k*E[A] to the examiner's report answer.