Limited Table M vs. Table L
I believe the argument goes that Table L streamlines the process by allowing for a single insurance charge to be stored in a single Table that can be applied to a single expected loss estimate (E). We do this by changing the aggregate insurance charge basis to be divided by total unlimited losses (E) instead of limited losses, so that it is now on the same basis as the per-occurrence excess ratio, k. Because they are both proportional to E, they can be summed and stored together as table L.
Limited Table M, on the other hand, expresses the aggregate insurance charge on the basis of limited expected losses in the denominator, and then (like Table L) adds the per-occurrence charge k, which is proportional to unlimited expected losses (E). Because of this basis difference, we cannot add the two ratios and store them together in one table since they must each be applied to a different claim exposure base. I think this is why we call it a 2-step method even though Table L also must derive k as a second step then add it to the aggregate charge estimate.
To help verify my understanding of the linkage between the two methods, I tried estimating the Limited Table M insurance charge estimate for Source text: Chapter 3 Q14. Compared to the Table L estimate we get r_D = 1.5 x 250/200 = 1.875, and then integrating with a Uniform(0,2) limited entry ratio distribution (instead of U(0,1.6) for Table L), we get phi_D(1.875) = 1.25 x (phi*(1.5)-k) = E/E_D * (phi(1.5) - k), exactly the basis the change I was expecting. This then shows that both methods add the value of k on-top of this aggregate charge estimate, except that Limited Table M cannot add the 2 ratios together to store in a common table due to the basis difference.
So this is where my question begins. Table L overcame the basis difference between the aggregate and per-occurrence charge by expressing the aggregate charge on the basis of E to match k. Is there good reason why Limited Table M could not have alternatively re-expressed k on the basis of limited expected losses to match the base of its aggregate insurance charge? Then the 2 could be summed, and in theory applied to an estimate of limited expected losses instead of vs. unlimited claims like for Table L.
I'm trying to reason why this is not a considered alternative; I don't think I've seen justification for this in the reading, but this seems like an interesting comparison to draw. Would the problem with this latter alternative be that excess per-occurrence claims relative to limited claims would not normalize for policy size or severity skewness as well, and thus contain too much over-dispersion to yield reliable estimates? I'm not totally certain on this reasoning yet, just trying to think my way through it now so I won't have to on the exam if needed. Curious to hear what others' thoughts are, or if I've maybe gone astray while thinking my way through any of the above details.
Thanks!
Comments
This is an interesting question thanks. From what I can tell your approach to re-basing in terms of the expected limited distribution works. My question is why would you want to do this given we already have the Table L approach - does this add any additional insight or simplify solving the problem? In both the Limited Table M and Table L approaches we require the limited loss distribution and the excess ratio with respect to a basis - so we're not reducing the volume of information needed. It would certainly be a unique take on the material in the exam - exactly the sort of thing the CAS likes to try from time to time!
From a practical standpoint, remember we may have sufficient data to price the per-occurrence limit, but lack data to price the aggregate limit. A solution to this is to use say countrywide data to price the aggregate limit instead of state/hazard group or even class data. If we use a Table L approach or your (Table P?) approach then we're building the table using data from a single source which introduces additional uncertainty.
It is also perhaps easier to build an empirical Table L rather than use the approach you outline as with your approach you would need to limit the losses by the per-occurrence limit first. Not a barrier but if unlimited loss data is already available then this is an extra set of calculations to do.
Thanks for the reply. I wouldn't say there is particular reason why this alternate approach would be voluntarily used in favour of Table L and M_D. It was more the output of trying to understand the linkage/comparison of table L and M_D, which I found was more a memorization exercise at first until I made time to think about it more carefully.
In theory, the CAS could restrict the information they give us on the exam to force the candidate to draw the connection, and adapt what we know in a slightly different way than formally presented in the source. Unlikely, sure... though a compare and contrast sort of problem could easily be on the table. I feel this exercise has at least helped improve my understanding of what we're doing, which should make it easier to explain and distinguish the two methods if and where needed.
In practice, we'll routinely have to identify and weigh alternative approaches to solve new problems. For this reason, deconstructing known methods to see how the authors approached and resolved their own problems can be educational in guiding our own ways of thinking.
We're glad you found the exercise helpful; it's certainly a great way to strengthen your understanding in readiness for the IQ problems and situations you'll encounter in your day to day work.
I have a related question. In the pencil-mark exercise
Since it's under the section limited Table M, so we should know the insurance charge rate should apply to the expected limited loss. However, if we just see the question without knowing it's limited Table M question, how we can know we should apply to unlimited loss or limited loss? Did I miss some indication in the question?
Thanks!
Under the "Policy Characteristics" we're told there is a per-occurrence limit and an aggregate deductible. This means we need to price using either the Limited Table M approach or Table L approach.
Generally speaking you can choose which method to apply but in practice it depends on the table(s) you have to hand. In this case we're implicitly given a Limited Table M as there's no mention of the aggregate limit associated with the table which is what we'd need to have a Table L.
Once you know you're working with a Limited Table M then you know to apply the limited loss.