Actuarial Concept: Projected Case Estimates ( PCE ) Model
Subtopics: Case Estimates Development Factor, Payout factor, Now basis claims triangles and transitions
Use: Used in valuations
Purpose: Just to explain as best as I can what I’ve learnt, to my current understanding, as an actuarial analyst. Hopefully will be good documentation to refer back to, and easy enough to help anyone new to the concept to understand it.
Method
PCE model, uses two triangles. A payment and a case estimate triangle.
Payment triangles are straightforward enough – it’s just the amount you pay out for the claim, be it lawyer fees, physio or medical fees, past economic loss (loss incurred due to not being able to work because of injury), future economic loss (FEL, same as past econ loss but into the future, e.g. someone disabled), rehabilitation cost, claims handling expenses etc.
Case Estimate – what is it? It is the reserves $$ set aside for amounts of future payments we will likely need to pay out for this claim. Most of the time, how much to set aside depends on the type of damage (physio, FEL etc as above) and the injury codes and injury severity score (ISV score). At least for QLD Australia in 2012 this is the case. It includes known payment amounts that we will definitely be paying out, but there can also be a lot of uncertainty surrounding case estimates, where someone just decides to put an approximate amount of reserves when not enough information has been provided yet e.g. medical reports have not arrived for us to put into database yet.
Triangles used like this –
· Payments, inflated, now basis, incremental, at data date (e.g. 201203 = 31 march 2012 month end). By accident period and development period. E.g. AccQuarter (the year and quarter of dateOccurred) and Development Quarter (Dev quarter as number of quarters since dateOccurred quarter). (e.g. pmts_Lg_MCWCnow_1203)
· Case Estimates:
o case_Lg_MCWCnow_1203, inflated, cumulative snapshot as at data date (*NOT* incremental movements!!),
o Now basis too (as in, the claim is, right NOW, in that particular segment, e.g. if the claim is currently a large claim although it was in the small claims segment 3 months ago, then it is still split into the large segment).
Now basis also means this – if a claim is currently in Large segment, then all the past payments and current snapshot of case estimates are ALL put into large triangle. Even when the claim was small last quarter, all last quarter’s payments and case are attributed to the Large triangle appropriately. So the entire history of the claim moves with which segment the claim is NOW in. That is what Now basis means.
So basically if you compared the Large snapshot case estimates triangle from last time to this time (ie. 201203 triangle minus 201109 triangle), and a claim transitioned from Large segment to small segment in devQtr 1, this is a sample of what case estimates difference triangle you get (*note: fake data used):
Note: Negatives values means transition out of. Positive values mean transition into.
Note that because case estimates is a snapshot, you see the one –ve amount’s effects carried across through to every development quarter between the transition quarter and the current quarter. The current quarter is given by the lastest diagonal of cells. Current quarter is 2012Q1 i.e. 201203 yearmth which is much further down the triangle so you can’t see the latest diagonal here, that’s why this looks more like a rectangle of data rather than a triangle!!
Below are the sample actual payments triangle as at 201203. Note that in reality all triangles used for PCE model are inflated payments to current dollars. The reconciliations are between uninflated payments and case amounts for the same triangles. The definition of large has been simplified to define a large claim as 500,000 dollars. But in reality it is more like an inflated 500,000 from a particular past date/year.
But if you compare the payments triangle in 201203 to 201109 by taking a difference of the two, this is more like what you see:
So the rationale behind this is: If a claim that has total incurred (payments + case) greater than 500,000 then it is called a large claim.
Using this definition then, we see that because of an increase of $1001 in one of the large claims in 201203 whose accident quarter is 199801 and development quarter is 3 (i.e. this -1000 transition applies to a change during 199804 quarter, and we realised there was this change in current 201203 quarter), the total incurred is now is now above the 500,000 mark, so it transitions out of small represented claims segment, into large claims segment.
That is, a sample total incurred for this time (201203) = 80,000 + 421000 = 501,000
And a sample total incurred for last time in (201109) = 80,000 + (421000-1001) = 499,999
One more example.
In 199803 dev 1, because of a 50,200 reduction in realised payments this quarter when compared to 1109, total incurred has dropped below the $500,000 mark, and as a result has pushed a claim from large segment away and into the small segment.
Total incurred for this time 201203 = 80,000 + 421000 = 450,000
And a sample total incurred for last time in 201109 = 80,000 + (421000-1001) = 500,200
***Note, there can be cases where case est is more than the difference triangles’ values we see above, such that their differences are 80,000. In fact this is more often the case. However, for understanding’s sake, I just simplified it saying that the entire case estimate is what is seen in the difference, and made both the transition in and out case be due to a difference in payments between 1109 and 1203. But in fact it is equilikely if not more likely that case estimates change as case estimates are reserves that have not been actualise and thus more subject to change than a payment that you actually paid out (one case where a change in payments applies is a reversal or recovery of some sort, then you get into complications with timing issues etc. Also, sometimes, data just changes, and it might take a lot of investigation to find out exactly why).
And similar concept for the other transitions in and out of segments.
o by the same accident period and development period as payments triangle. E.g. AccQuarter and Development Quarter
Once the triangles are imported to the spreadsheet, here is what we do with it: --
· We start off with generating a new triangle of development factors for the case estimate. This is, to no one’s surprise, called the Case Estimate Development Factor. For the purposes of this doco, we will use quarters as our period – accident quarters, development is also in quarters.
The Case Estimate Development Factor is defined as:
(this quarter’s case estimates + this quarter’s payments)
last quarter’s case estimates
that is, it is a measure of how the case estimates from last time has progressed since last quarter. Start at the first quarter. We are assuming that by the next quarter, some of those case estimates would have actualised into payments. So if there has been no new case estimate increases or changes then for one particular claim, first quarter’s case estimates = next quarter’s case estimate balance + amount paid out.
However, in reality, case estimates will change as we gain more data to a certain point. And new case estimates for other claims will be added to the mix each quarter. So when things are more complicated, the CED is a measure of how case estimates for accidents which occurred in that quarter develop, increase, get paid out, get adjusted, quarter upon quarter. So one would expect high CED factors (i.e. high proportion of change) in the first say 3 years of a long tail insurance business such as CTP, and then as we progress towards development quarter 80 (we choose to cut off at dev80) i.e. 20years, the closer we get to 80 we see the CED change less and less, with many quarters’ CED factor = 1.
Tail behaviour: Due to lumpy payments especially for large claims in the later development periods, we see patterns where CED is = 1 for several quarters in a row, and then suddenly a large number like 1.3, and then back to 1 again next quarter and 1’s onwards, and then another large CED. CED’s can also be < 1, but greater than 0 – e.g. if the claimant died in the process and suddenly for a large claim (serious injury) we didn’t have to pay out a lot of money anymore, and because it is very late development quarter, say dev30, no new case estimates have been added to the mix for that accident quarter because no new claims have been reported after such a long time.
· Looking at the equation for CED, you might be wondering – then how do I know which portion of the case estimates has been paid out from last time, and which is change in case estimates due to new claims and case est revisions?
Well, that’s when we come to the Payout (PO) Factor, which is the next step.
We develop a payout factor, given by:
This quarter’s payments
Last quarter’s case est
that is, the PO factor is the proportion of payments made in the quarter as a fraction of last quarter’s view of the total remaining amount of money we expect to pay out in the future for that particular accident quarter.
· Then finally, we do two things.
A) We project the amount of payments to be paid out next quarter
B) We project the projected amount of remaining case estimates we expect to see for that accident quarter using A).
A) Expected Projected Payments for the next quarter is found like so:
this time’s (quarter’s) case estimates * next quarter’s case estimate development factor
So if it is the first quarter of projection for the first row of the case est triangle, then if the triangle ended at development quarter 67, then the 68th quarter case estimates would be a projected case estimates achieved by 67th quarter’s case*selected dev68’s PO factor.
Now, because we are dealing with more than one accident year, and it is easier to spot trends using more than just a single accident quarter (because by having a larger dataset we reduce volatility in experience), the selected PO factors is usually an averaged/weighted PO factor curve of multiple accident quarters’ experiences. We will cover the multi acc quarter case later in this doco, and how to derive them.
Using the same formula, we keep projecting the 69th payments using 68th case estimates and 69th selected PO factor... and so on until development quarter 80 which is when we decided to cut the portfolio’s projections off.
What about the next accident quarter? Well, you use the same selected PO factor curve, but pick the appropriate dev quarter to use. I.e. for second row of triangle which is one accqtr later in time than the first row, the final cell would be 66th dev quarter since the dateOccurred accQuarter. So to project the 67th devqtr’s payment amount, you go:
66th quarter’s actual case est*dev67’s PO factor. By ‘curve’ I really mean the one single series of 80 development factors (dev1 to dev80) we decided to use for the whole triangle (btw, case estimates triangle starts from dev0. Applying the formula, you realise that for dev0, the PO factor is 0 because you don’t have a ‘last time’ to base anything on. So you start at the next quarter which is dev1). You get the point.
Thus we develop a lower-right triangle full of projected payments 
where the upper-left is the actual case estimates experience/data to date 
. Your next quarter total paid for the entire segment is then the sum of the 67th diagonal down the triangle.
We usually push each row of this lower triangle to the left all the way, so that we group all the payments to be made in the next quarter in the same column so that the lower triangle now looks like this: 
.
B) Expected Projected Case Estimates for the next quarter is found like so:
Next quarter case = This quarter’s case * next quarter’s selected CED factor – projected amount paid next quarter from A)
It’s basically a reverse engineer of how we came up with the CED factor.
So this is saying that at the end of next quarter, the pure/true case estimates next quarter is the total amount of case estimates we project to remain minus any payments we might make during the quarter.
I.e. the projected case est amount is the pure case est sourced from any remaining money we know we are going to pay but haven’t – plus—changes to the original case estimates either from new case estimates coming in for old/new claims OR revisions to the old case estimates.
· Then we usually inflate the projected triangles by Average Weekly Earnings inflation and Super Imposed Inflation up by however many quarters to reach their respective payment dates. Sum of this inflated projected pay triangle will then give us the total undiscounted liability (OSC) for that segment.
And that is the PCE in a nut shell (pretty big nutshell). ^_^
<<THE END>>
Catering for multiple years’ worth of experience in the PO and CED
Now I did mention before that we are going to look at coming up with a series of selected PO and CED curves using combined data from multiple accyears. So here goes:
· Because the first row of our triangle is a prior accQtr row (i.e all the prior data with accQtr more than 67qtrs away from current valuation quarter), we ignore that. So the useable number of rows for us is 66 in a triangle with 67 rows.
So CED is the entire colum of this quarter’s case + proportion of last quarter’s case that we paid out.
I.e. CED = {sum(of This quarter’s case column) + sum(of this quarter’s pay column)}/sum(last quarter’s case estimates omitting the most recent accQtr, i.e. minus bottom row which is the latest diagonal’s case)
The denominator’s accQtrs summed need to be of the same length as the numerator’s accQtr so that we are getting a CED factor that compares fairly the experience in accordance to the number of quarters. See below table, CED factor for all periods (ie. All accqtrs’ worth of experience combined) is: Blue shaded in dev 1 of case triangle plus same blue shaded dev 1 in pay triangle (i.e. this time’s case plus this time’s case), divided in total by the orange shaded area in dev0 of case triangle (last time’s case). Note there is one more quarter’s worth of useful experience denoted in dark purple, but to make sure the lengths of the orange and blue areas are the same, it is omitted. Similarly for dev4’s CED factor.
In the case that we want not all periods, but the last 3 quarter’s worth, we then use:
Notice that for the last 3 qtr combo of CED factors the last value we have is at dev4. Because after that, there is not enough data to have three-quarters’-worth of experience. From dev5 onwards, the CED factors are blank. We sometimes make it = 1 which means no development.
· A similar concept applies for multi accyear PO factors
Again, because the first row of our triangle is a prior accQtr row (i.e all the prior data with accQtr more than 67qtrs away from current valuation quarter), we ignore that. So the useable number of rows for us is 66 in a triangle with 67 rows.
So PO is the entire colum of this quarter’s payments / last quarter case column, for the same number of accident quarters.
I.e. PO = sum(of This quarter’s payment column)/sum(last quarter’s case estimates omitting the most recent accQtr, i.e. minus bottom row which is the latest diagonal’s case)
The denominator’s accQtrs summed need to be of the same length as the numerator’s accQtr so that we are getting a PO factor that compares the same length of payments and case experience.
For PO factor for period of ‘all years’, to dev1 PO factor = blue 1 in payments triangle divided by orange 1 in case triangle. Note that deep purple cell of additional experience in case triangle is omitted so that the total number of rows we are comparing remains the same across pay and case triangles.
Similar concept for Last 3 qtrs, and you will only get up to dev4 since there is not enough experience to go any further than that.
