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Determining Reserve Ranges
CLRS 1999
by
Rodney Kreps
Guy Carpenter Instrat
Why can’t you actuaries
get the reserves right?
Feel like a target?
What are Reserves?
1 Actual Dollars Paid.
2 Distribution of Potential Actual Dollars
Paid.
3 Locator of the Distribution of Potential
Actual Dollars Paid.
4 An esoteric mystery dependent on the
whims of the CFO.
And the Right Answer -
ALL of the above.
Actual Dollars Paid
• Only true after runoff.
• Gives a hindsight view.
• Lies behind the question
“Why can’t you get it right?”
Distribution of Potential
Actual Dollars Paid
• All planning estimates are distributions.
• ALL planning estimates are distributions.
• ALL planning estimates are
DISTRIBUTIONS.
• Basically, anything interesting on a goingforward basis is a distribution
Distributions frequently
characterized by locator and
spread
• However, the choice of these is basically a
subjective matter.
• Mathematical convenience of calculation is
not necessarily a good criterion for choice.
• Neither is “Gramps did it this way.”
Measures of spread
• Standard deviation
• Usual confidence interval
• Minimum uncertainty
Standard deviation
• Simple formula.
• Other spread measures often expressed as
plus or minus so many standard deviations.
• Familiar from (ab)normal distribution.
Usual confidence interval
• Sense is, “How large an interval do I need
to be reasonably comfortable that the value
is in it?”
• E.g., 90% confidence interval. Why 90%?
• Why not 95%? 99%? 99.9%?
• Statisticians’ canonical comfort level seems
to be 95%.
• Choice depends on situation and individual.
Minimum uncertainty
• AKA “Intrinsic uncertainty,” Softness,” or
“Slop.”
• All estimates and most measurements have
intrinsic uncertainty.
• The stochastic variable is essentially not
known to within the intrinsic uncertainty.
• Sense is, “What is the smallest interval
containing the value?”
Minimum uncertainty (2)
• “How little can I include and not be too
uncomfortable pretending that the value is
inside the interval?”
• Plausible choice: Middle 50%.
• Personal choice: Middle third.
• Clearly it depends on situation and
individual.
E.g. Catastrophe PML
• David Miller paper at May 1999 CAS
meeting.
• Treated only parameter uncertainty from
limited data.
• 95% confidence interval was factor of 2.
• Minimum uncertainty was 30%.
Locator of the Distribution of
Potential Actual Dollars Paid
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•
•
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Can’t book a distribution.
Need a locator for the distribution.
Actuaries have traditionally used the mean.
WHY THE MEAN?
WHY THE MEAN?
• It is simple to calculate.
• It is encouraged by the CAS statement of
principles.
• It is safe - “Nobody ever got fired for
buying IBM.”
Some Possible locators
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Mean
Mode
Median
Fixed percentile
Other ?!!
How to choose a relevant
locator?
• Example: bet on one throw of a die whose
sides are weighted proportionally to their
values.
• Obvious choice is 6.
• This is the mode.
• Why not the mean of 4.333?
• Even rounded to 4?
What happened there?
• Frame situation by a “pain” function.
• Take pain as zero when the throw is our chosen
locator, and 1 when it is not.
• This corresponds to doing a single bet.
• Minimize the pain over the distribution:
• Choose as locator as the most probable
value.
Generalization to continuous
variables
• Define an appropriate pain function.
– Depends on business meaning of distribution.
– Function of locator and stochastic variable.
• Choose the locator so as to minimize the average
pain over the distribution.
• “Statistical Decision Theory”
– Can be generalized many directions
• Parallel to Hamiltonian Principle of Least Work
Claim: All the usual locators
can be framed this way
Further claim: this gives us a
way to see the relevance of
different locators in the given
business context.
Example: Mean
• Pain function is quadratic in x with
minimum at the locator:
• P(L,X) = (X-L)^2
• Note that it is equally bad to come in high
or low, and two dollars off is four times as
bad as one dollar off.
Squigglies: Proof for Mean
• Integrate the pain function over the
distribution, and express the result in terms
of the mean M and variance V of x. This
gives Pain as a function of the Locator:
• P(L) = V + (M-L)^2
• Clearly a minimum at L = M
Why the Mean?
• Is there some reason why this symmetric
quadratic pain function makes sense in the
context of reserves?
• Perhaps unfairly: ever try to spend a
squared dollar?
Example: Mode
• Pain function is zero in a small interval
around the locator, and 1 elsewhere.
• Generates the most likely result.
• Could generalize to any finite interval (and
get a different result)
• Corresponds to simple bet, no degrees of
pain.
Example: Median
• Pain function is the absolute difference of x
and the locator:
• P(L,X) = Abs(X-L)
• Equally bad on upside and downside, but
linear: two dollars off is only twice as bad
as one dollar off.
• Generates the X corresponding to the 50th
percentile.
Example: Arbitrary Percentile
• Pain function is linear but asymmetric with
different slope above and below the locator:
• P(L,X) = (L-X) for X<L and S*(X-L) for
X>L
• If S>1, then coming in high (above the
locator) is worse than coming in low.
• Generates the X corresponding to the
S/(S+1) percentile. E.g., S=3 gives the 75th
percentile.
An esoteric mystery dependent
on the whims of the CFO
• What shape would we expect for the pain
function?
• Assume a CFO who is in it for the long
term and has no perverse incentives.
• Assume a stable underwriting environment.
• Take the context, for example, of one-year
reserve runoff.
Suggestion for pain function:
The decrease in net economic worth
of the company as a result of the
reserve changes.
Some interested parties who
affect the pain function:
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•
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•
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policyholders
stockholders
agents
regulators
rating agencies
investment analysts
lending institutions
If the Losses come in lower than
the stated reserves:
• Analysts perceive company as strongly
reserved.
• Not much problem from the IRS.
• Dividends could have been larger.
• Slightly uncompetitive if underwriters talk
to pricing actuaries and pricing actuaries
talk to reserving actuaries.
If the Losses come in higher than
the stated reserves:
• If only slightly higher, same as industry.
• Otherwise, increasing problems from the
regulators.
– Start to trigger IRIS tests.
• Credit rating suffers.
• Analysts perceive company as weak.
– Possible troubles in collecting Reinsurance, etc.
• Renewals problematical.
Generic Reserving Pain function
• Climbs much more steeply on high side
than low.
• Probably has steps as critical values are
exceeded.
• Probably non-linear on high side.
• Weak dependence on low side
Generic Reserving Pain function (2)
• Simplest form is linear on the low side and
quadratic on the high:
• P(L,X) = S*(L-X) for X<L and (X-L)^2 for
X>L
Made-up example:
• Company has lognormally distributed reserves,
with coefficient of variation of 10%.
• Mean reserves are 3.5 and S = 0.1 (units of
surplus).
• Then 10% high is as bad as 10% low, 16% high
is as bad as 25% low, and 25% high is as bad as
63% low.
• Locator is 5.1% above the mean, at the 71st
percentile level.
...
ESSENTIALS
...
• All estimates are soft.
– Sometimes shockingly so.
– The uncertainty in the reserves is NOT the
uncertainty in the reserve estimator.
• The appropriate reserve estimate
depends on the pain function.
– The mean is unlikely to be the correct
estimator.