Step 10
Is there a way to show the following formulas are equivalent?
BasicPrem.=(ExpectedPer-occurrenceexcessloss+netins.charge)⋅(c+UWProfit%)+Fixedexp
b = e – E(c – 1) + cI
Is there a way to show the following formulas are equivalent?
BasicPrem.=(ExpectedPer-occurrenceexcessloss+netins.charge)⋅(c+UWProfit%)+Fixedexp
b = e – E(c – 1) + cI
Comments
Yes, the net insurance charge is I, so you can write B = cI + (remaining terms), where cI is in dollars. Dividing by the standard premium (SP) to get b (and cI in terms of standard premium) then means (remaining terms)/SP equals e - (c-1)*E.
Here, e is the expense ratio which includes UW profit, fixed expenses and LAE so in this case e = (fixed expense)/SP +UW Profit % *(Expected loss [per-occurrence and aggregate] + net insurance charge)/SP + LAE % * Expected loss [per-occurrence and aggregate]/SP.
However, the basic premium ratio should only account for fixed loss adjustment expenses so it's necessary to subtract out the variable LAE which is (c-1)*E. When we do this, we're left with c*Expected loss [per-occurrence and aggregate]/SP which, when multiplied by SP, is the last term needed in the equation for B above.
I don't follow. Is B = b? What is mean by remaining terms? What I was trying to do is go from
b = e - E(c-1)+cI
to
BasicPrem.=(ExpectedPer-occurrenceexcessloss+netins.charge)⋅(c+UWProfit%)+Fixedexp
Essentially I am trying to solve 2012 Fall Q 23 using BasicPrem.=(ExpectedPer-occurrenceexcessloss+netins.charge)⋅(c+UWProfit%)+Fixedexp
This is the formula given in the lesson.
This question is an example of one of the challenges presented by Exam 8 and the CAS not releasing newer exam questions in general. The CAS model solution(s) refers to 'b = e - E(c-1) + cI' whereas the Fisher text refers to B = e - (c-1)E + cI. These are the same equation for the dollar amount of basic premium - it's just the syllabus readings have changed over time and the notation along with it.
In general, on the current syllabus (Fall 2021), B=b*SP where B is the basic premium in dollars, SP is the standard premium, and b is the basic premium ratio. Further, B= e -(c-1)E + cI, where e is the expected expense dollars, E is the expected loss, c is the loss conversion factor and I is the net insurance charge (in dollars). You can go back and forth between working in terms of dollars or in terms of ratios to standard premium by dividing or multiplying the equations by SP appropriately.
When we know more specifically about what we're pricing, such as whether there is a per-occurrence deductible/limit, or an aggregate deductible/limit, or both, we can more easily break this equation down into individual components.
In the case when there's only a per-occurrence deductible/limit plus a maximum and/or minimum rateable loss, the basic premium is expressed as the expected loss (in excess of the per-occurrence limit) loaded for ULAE (via the loss conversion factor) and UW profit + an insurance charge to account for the max/min rateable loss, plus a provision for fixed expenses. These are all dollar amounts which are fixed at policy inception, i.e. do not vary with the actual rateable loss observed.
In our post above, in dollars, the "remaining terms" are:
(ExpectedPer-occurrenceexcessloss)⋅(c+UWProfit%) + net insurance charge * UW Profit% + Fixedexp, that is the part that is not the net insurance charge in dollars. This has to equal e - (c-1)E.
The basic premium only accounts for expenses that do not vary with the actual losses. Since ULAE is (c-1)*E we then have e = UW Profit% (Expected per-occurrence excess loss + net insurance charge) + fixed expenses. So the equations are equivalent.
However, translating back and forth between the two equations likely isn't the best use of your resources during the exam. Instead, focus on identifying the information you're given and which basic premium framework it fits best. In the case of Fall 2012, Q23 we're given dollar amounts for the expense provision and expected total loss (implicit this is what the insurer is responsible for). This is better suited to the B = e - (c-1)E + cI approach. If we were given detailed percentages for the UW profit, general expenses, ULAE, etc. then the "word" approach is likely more approachable.
I think I got it. Thanks.
I still got confused by this whole topic.
You mentioned that "the basic premium ratio should only account for fixed loss adjustment expenses so it's necessary to subtract out the variable LAE which is (c-1)*E. When we do this, we're left with c*Expected loss [per-occurrence and aggregate]/SP".
I don't understand why variable LAE is (c-1)*E, in the note of Fisher. Risk Sharing lesson, it stated that c accounts for expenses which vary with the actual loss experience. Does that mean variable LAE should be c*E?
And I also don't get it why c*Expected loss [per-occurrence and aggregate]/SP was left after subtracting (c-1)*E from LAE % * Expected loss [per-occurrence and aggregate]/SP. Does that mean LAE% equal to (2c-1) ?
I think you're confusing the idea of a variable expense percentage with a risk multiplier.
In Fisher.RiskSharing we see the retrospective premium is R = (B+cL)T. If c is the variable expense percentage then the retro premium fails to account for the ratable loss dollars (those not in excess of either the per-occurrence or aggregate limits). So in this context, c is a multiplier. That is, c = 1+d% where d% is the variable expense percentage. This is why the variable LAE is then (c-1)*E.
The basic premium should account for fixed expenses plus those expected expenses and other provisions (UW profit, credit risk etc. if applicable) associated with losses in excess of either the per-occurrence and/or aggregate limits.
The expense ratio, e, is a ratio of the expenses to the standard premium, SP. I'm multiplying through by SP here to make things look cleaner. The expenses include the fixed expenses, UW profit, and LAE. UW profit is only charged on the expected excess loss because the ratable loss is passed through to the insured.
e*SP = fixed expense + UW profit% *(expected excess loss) + LAE% * E
LAE% is the variable expense, it's given by (c-1)E.
However, we don't want to include the variable expense associated with the ratable loss in the basic premium as we'd double count it then (once in B and once via cL).
So we subtract off the entire variable expense and add on only the expected expenses associated with losses in excess of the per-occurrence and aggregate limits which is cI
Hence B = e - (c-1)E + cI
Thanks for your explanation, I think I followed it. However, I still don't get it where is c*Expected loss [per-occurrence and aggregate]/SP after we subtract (c-1)E from e ?
I understand this equation:
e*SP = fixed expense + UW profit% *(expected excess loss) + LAE% * E
and now I also understand LAE% *E is (c-1)E. And after we subtract LAE%*E,
e*SP - (c-1)E = fixed expense + UW profit% *(expected excess loss) = BasicPremium - cI, (1)
But at the beginning of this discussion, we mentioned that there is formula for basic premium:
Basic Premium =(ExpectedPer-occurrenceexcessloss+netins.charge)⋅(c+UWProfit%)+FixedExp
then we subtract cI from equation above:
Basic Premium - cI =(Expected per-occurence excess loss) * c +(Expected per-occurence excess loss) *UW profit% + Fixed Expense ,(2)
The first item in equation (2) above is an extra item compared to equation (1).
I think the difficulty here lies in converting between word equations and symbols when there is inconsistent notation throughout the industry. On page 40 of the Fisher text, footnote 19 says:
"If a retro policy also has a per claim loss-limit, the charge for that is sometimes considered part of the insurance charge, and sometimes considered a separate charge. The terminology is not entirely consistent across the industry, and the actuary should be careful to understand what is being measured or estimated."
The I in B = e - (c-1)E +cI must implicitly include the expected per-occurrence excess loss as the expense ratio, e, does not count this loss as an expense.
Thank you! Now it makes sense to me.