As I understand it, the original question was whether loss development factors
for losses limited to $50,000 should be greater than the LDF's for losses
limited to $25,000. A more general version of this issue would be to ask how
the development patterns for total limits and limited losses compare.
This question requires a distinction between expected values and actual
Part I -- Actual values:
It's easy to construct an example for which the actual LDF for total limits
losses is less than the actual LDF for limited losses.
For example, suppose several years of development have passed and that every
claim but one has already closed. Also suppose that the reserve on the one
open claim is more than is needed to settle the claim (this settlement value
will be more than $25,000). When this claim closes, the actual LDF for total
limits losses at that age will be less than 1.000 but the LDF for losses
limited to $25,000 will equal 1.000. In this case, the total limits LDF is
less than the LDF for limited losses at this age.
Other situations in which our prior expectations on loss development aren't
met include (1) when development is downward rather than upward due to company
overreserving, (2) salvage and subrogation recoveries, and (3) recovery of the
per claim deductible from the insured after the loss has settled. This is
just a listing of situations that have simple explanations. There are other
situations for which a simple explanation can't be provided.
Part II -- Expected values and expectations:
With that said, now consider the real issue, which is whether your
expectations regarding loss development patterns mean anything and whether you
should rely on them. The examples given above as well as some of the other
replies you received could be characterized as being special exceptions.
However, special exceptions don't tell you whether your original expectation
was valid or not.
To get to the nub of the matter, I think that your original expectation that
the LDF's didn't make sense (I hope I'm not reading too much into your
message) was a reasonable conclusion, but you need to review the data and the
analysis to confirm that.
Part III -- Comments about the analysis
You can take one of two approaches to a loss development analysis. One way is
to rely solely on the data. The other way is to go with your feeling about
how LDF's ought to behave. Both approaches can be subject to criticism, but I
think the second approach is criticized more since it doesn't seem scientific
or actuarial. Here are some thoughts on these approaches:
First, you would ordinarily think that it would be difficult to argue with
conclusions drawn from the data. However, the data may be misleading. You
should consider gathering additional information. This can include previous
LDF evaluations which may be inconsistent with the current analysis,
countrywide data to supplement state data, industry data, etc.
Second, there may be a more fundamental flaw. Most of us like to believe that
we have a special ability to select the proper age-to-age factors from the
loss development triangle. This belief isn't limited to actuaries either -- a
lot of people will second guess the actuarial selections. However, the
process of selecting values from the data is more unreliable than many people
This can be observed by comparing the selections two different actuaries make.
One actuary may simply select the average of the last five points while
another may do something more complex such as excluding certain points because
of a feeling that the value is outside the normal range. Assuming that the
two sets of selections are very different, which one is right? On the other
hand, if the two actuaries should agree to take the average of their
selections, I would think that this would reflect on the quality of their work
Third, isn't the feeling that one person may have about how LDF's ought to
behave just as valid from an actuarial standpoint as the feelings that another
person may have about including and excluding certain points from the
selection of age-to-age factors? Why should one feeling be considered valid
and the other invalid? That feeling about the LDF's may be the only thing
that prevents an unfortunate pricing decision.
Actually, there really is some justification for having an expectation about
how LDF's ought to behave. However, since a complete explanation would be
lengthy, I won't give it here (if Fermat could get away with this, why can't
all of us?). So, if you have prior knowledge (or expectations), then it
would seem reasonable to use that knowledge to help you make selections.
Part IV -- Advice, take it or leave it
If the volume of your data is small, then there may not be enough data to
establish reliable loss development patterns. In that case, you shouldn't
have any prior expectation about the relationship of total vs. limited LDF's.
On the other hand, suppose that that the volume of your data is sufficient to
determine development factors. Also assume that there are no influences that
are significantly changing the loss development patterns. That is, you expect
the patterns to be fairly stable. In a situation like this, a common reason
for errors in the projection of the ultimate losses is the presence or absence
of large losses in the data. Don't be surprised if the development patterns
you found for Workers Compensation were due to the effect of large claims on
the selections of age-to-age factors.
Here's how you might test for large losses in the data: First, divide the
total limits loss triangle by the limited loss triangle to create a triangle
of ratios. Review this triangle to determine whether the development of large
losses is changing. Second, compute the average ratio at each age of
development. This tells you how large losses develop as compared to limited
losses. Third, compare the average ratios to the values on the latest
diagonal. This should tell you whether the latest point has more or less
large losses than normal. If the latest point has a normal amount of large
losses, then the historical patterns may be appropriate. On the other hand,
if the latest point has an unusual amount of large losses, then the historical
development patterns may not be appropriate.
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