If we name it the "Patrik/Weissner" method, it may not be descriptive but
at least credit will fall where credit is due, i.e., to American casualty
No one on CASnet has walked thru the derivation for Mike who passed Part
7 before Cape Cod was there. Let me make an attempt, admitting it was not
on my Part 7 either. Please excuse me for not being able to put subscripts
below the line (or umlauts above the vowels, for that matter). I hope my S
comes out as sigma.
1. We need to know an exposure measure for each year i of underwriting
experience. Call each Pi, which does not have to be premium, but it will
be easier to remember as such.
2. It is best if this is on-leveled/adjusted so that we believe all years
have about the same unknown expected loss ratio, E=Ei. This is one of the
keys to the method. E is something we can solve for using the known
3. Next, use the classic Bornhuetter/Ferguson assumption that losses to
emerge will be a function of expected by year. Loss dollars by year are
4. In any reasonable way, calculate factors for development to ultimate,
respective of year, LDFi. In the usual chain-ladder methodology, these
would be applied directly to known emerged losses by year, Ai, to produce
ultimate losses by year.
5. Instead, assume the ultimate losses Ui for year i are given by:
Note that PixEi/LDFi is losses expected to have emerged as of the current
evaluation.. We multiply this amount by LDFi-1 to obtain losses expected
to emerge in the future. Spend some time with this formula, which is due
6. This is better written
Ui = Ai + Pi*E(1 -1/LDFi)
I am now using the assumed single expected loss ratio E.
7. We aggregate this expression for all the experience years, calculating
the aggregate ultimate losses and equate them to E times the aggregate
SAi + SPi*E*(1 -1/LDFi) = ESPi.
8. Solve this for E (easy exercise) to obtain:
E = (SAi)/(SPi/LDFi)
9. Some call Pi/LDFi the "reported exposure", and Ai are commonly known
as reported losses. The formula makes sense with these names.
10. Now ultimate losses for each year are given by the formula in 7. Note
that the derived ultimate loss ratio for each year will usually not be E.
This is Bornhuetter/Ferguson with a systematic estimate of the expected
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