Transcript reserve estimator

Determining Reserve Ranges and
the Variability of Loss Reserves
CLRS 2001
by
Rodney Kreps
Guy Carpenter Instrat
Motivation: ASOP 36, 3.6.3
Expected Value Estimate – In evaluating the reasonableness of
reserves, the actuary should consider one or more expected
value estimates of the reserves, except when such estimates
cannot be made based on available data and reasonable
assumptions.
Other statistical values such as the mode (most likely value) or
the median (50th percentile) may not be appropriate measures
for evaluating loss and loss adjustment expense reserves, such
as when the expected value estimates can be significantly
greater than these other estimates.
Translation: use the mean. We don’t want low reserves.
ASOP 36, 3.6.3 (cont.)
The actuary may use various methods or assumptions to arrive
at expected value estimates. In arriving at such expected value
estimates, it is not necessary to estimate or determine the
range of all possible values, nor the probabilities associated
with any particular values.
Translation: distribution? I think I saw one once.
Expected? This is what I expect will do me the most good.
ASOP 36, 3.6.4
Range of Reasonable Reserve Estimates – The actuary may
determine a range of reasonable reserve estimates that reflects
the uncertainties associated with analyzing the reserves. A
range of reasonable estimates is a range of estimates that
could be produced by appropriate actuarial methods or
alternative sets of assumptions that the actuary judges to be
reasonable.
Translation: how many ways can you get a mean?
Without even introducing a distribution?
The real title of this talk is:
Why the mean?
OR
Why the mean is almost surely not the best
reserve measure.
OR
What is an economically rational basis for
reserve measures?
To answer this we need to go back to the beginning and
consider fundamentals —
Why can’t you actuaries
get the reserves right?
Feel like a target?
What are Reserves?
1 Actual Dollars To Be Paid.
2 Distribution of Potential Actual Dollars To
Be Paid.
3 Estimator of the Distribution of Potential
Actual Dollars To Be Paid.
4 An esoteric mystery dependent on the
whims of the CFO/CEO.
And the Right Answer -
ALL of the above.
What are Reserves?
1 Actual Dollars To Be Paid.
Actual Dollars
• Are only known after runoff.
• Give a hindsight view.
• Lie behind the question
“Why can’t you get it right?”
A changing estimate does not imply a mistake.
What are Reserves?
1 Actual Dollars To Be Paid.
2 Distribution of Potential Actual Dollars To
Be Paid.
Distribution of Potential
Actual Dollars To Be Paid
• All planning quantities are distributions.
• ALL planning quantities are distributions.
• ALL planning quantities are
DISTRIBUTIONS.
• Basically, anything interesting in the future
is a distribution.
Distributions are frequently
characterized by spread and
estimator
• 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.
• A stochastic variable is essentially not
known to within its 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%.
What are Reserves?
1 Actual Dollars To Be Paid.
2 Distribution of Potential Actual Dollars To
Be Paid.
3 Estimator of the Distribution of Potential
Actual Dollars To Be Paid.
Estimator of the Distribution of
Potential Actual Dollars Paid
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Can’t book a distribution.
Need a estimator 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 and middle of the road.
Some Possible estimators
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Mean
Mode
Median
Fixed percentile
Other ?!!
How to choose a relevant
estimator?
• 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
estimator, and 1 when it is not.
• This corresponds to doing a simple bet.
• Minimize the pain over the distribution:
• This leads to choosing the estimator as the most
probable single value.
Generalization to continuous
variables
• Define an appropriate pain function.
– Depends on business meaning of distribution.
– Function of estimator and stochastic variable.
• Choose the estimator 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
Mathematical representation
• f(x) – the distribution density function
• p(m,x) – the relative pain if x ≠ m
• P(m) = ∫ p(m,x) f(x) dx – the average of the
pain over the distribution
• Choose the pain to represent business
reality
• Choose m so as to minimize the average
pain.
Claim: All the usual estimators
can be framed this way.
Further claim: this gives us a
way to see the relevance of
different estimators in the given
business context.
Example: Mean
• Pain function is quadratic in x with
minimum at the estimator:
• p(m,X) = (X- m)^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 estimator:
• P(m) = V + (M- m)^2
• Clearly a minimum at m = 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 estimator, and 1 elsewhere.
• The estimator is the most likely result.
• Could generalize to any finite interval (and
get differing results)
• Corresponds to simple bet, no degrees of
pain.
Example: Median
• Pain function is the absolute difference of x
and the estimator:
• p(m,X) = Abs(m -L)
• Equally bad on upside and downside, but
linear: two dollars off is only twice as bad
as one dollar off.
• The estimator is the 50th percentile of the
distribution.
Example: Arbitrary Percentile
• Pain function is linear but asymmetric with
different slope above and below the estimator:
• p(m,X) = (m -X) for X< m and S*(X- m) for X> m
• If S>1, then coming in high (above the estimator)
is worse than coming in low.
• The estimator is the S/(S+1) percentile. E.g., S=3
gives the 75th percentile.
What are Reserves?
1 Actual Dollars To Be Paid.
2 Distribution of Potential Actual Dollars To
Be Paid.
3 Estimator of the Distribution of Potential
Actual Dollars To Be Paid.
4 An esoteric mystery dependent on the
whims of the CFO/CEO.
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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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.
• Problems 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:
• 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 and reinsurance problematical.
Reserving Pain function
• Climbs much more steeply on the high side
than on the low.
• Probably has steps as critical values are
exceeded.
• Is probably non-linear on the high side.
• Has weak dependence on the low side
Reserving Pain function (cont.)
• The pain function for the mean is quadratic
and therefore symmetric.
• It gives too much weight to the low side
• Consequently, the estimate is almost surely
too low.
Reserving Pain function (cont.)
• Simplest form is linear on the low side and
quadratic on the high:
• p(m,X) = S*(m -X) for X< m and (X- m)^2
for X> m
• S an “appropriate” constant
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% low is the same pain as 10% high, 16% high
is the same pain as 25% low, and 25% high is the same
pain as 63% low.
• Estimator 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 MUCH LARGER
THAN the uncertainty in the reserve estimator.
• The appropriate reserve estimate depends
on the pain function.
– The mean is almost surely not the correct
estimator, since it comes from a symmetric pain
function.
– It is probably too low.