CAS Spring Meeting 2006
Reserve Variability Session 1
Where Are We Today?
Rodney Kreps
A Fundamental Truth
In order to be meaningful and useful,
any measurement or estimate must
also have a sense of the size of its
uncertainty.
5 yards
Guy Carpenter

15 feet
 180 inches
2
We know this
We use this automatically, often without doing
a conscious calculation.
It is so automatic we don’t even think about it
having done it, for example in crossing the
street.
Why are we not explicit about this in our work?
Guy Carpenter
3
Corollaries
The statement of the estimate
frequently implies the size of the
uncertainty, correctly or not.
When the uncertainty gets too big,
the estimate loses all meaning.
The size of the desired uncertainty
depends on the situation.
Guy Carpenter
4
Confusion Source
on Reserve Uncertainty and Range

The term “reserve” is used in many contexts: As the
– information actually booked.
– estimate of the dollars still to be paid for claims.
– actual outcome of paying off claims.
– distribution of possible outcomes of paying off claims.

Each of these has its own notion of uncertainty.

The only meaning relevant to the actual underlying current
economic value of the company is the last, even though
analysts and stockholders react strongly to the first.

Wouldn’t it be nice if the first reflected the last, even if it
required sophisticated interpretation?
Guy Carpenter
5
Uncertainty in Booked

The number, as booked, is known (at least to the nearest
thousand)

The range of numbers all of which were possible
management estimates and could have been booked is
known to management, and usually not elsewhere.

The choice from this range of estimates depends on many
factors, such as what they think competitors will do, what
the current cash flows may look like, what reserve
changes they have taken recently, and so on.
Guy Carpenter
6
Uncertainty in the Actuarial Estimate

Usually there are several ways to get the estimate.

The Actuarial Standards of Practice are meant to provide guidance,
and ASOP 36 speaks only to this uncertainty, not that of the outcomes.

The guidance suggests strongly that the several ways should all be
different ways of calculating a mean value.

This may suggest a range of possible permissible estimates, but says
nothing about the variability in what you are going to pay.

It does, at least, allow possible other statistics. See, for example,
“Management’s Best Estimate” for a discussion of an economically
rational way to choose a statistic.
Guy Carpenter
7
ASOP 36
Definition 2.6


“Expected Value Estimate - An estimate of the mean value of an
unknown quantity where the mean value represents a probabilityweighted average of the quantity over the range of all possible
values.”
Please note for later reference the use of the phrases “all possible
values” and “probability-weighted.”
Guy Carpenter
8
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. Err on
the high side. We know these are skewed distributions.
Guy Carpenter
9
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.
Wait a minute – what happened to the definition?
Question: So, if I don’t need a distribution, what is my expected
value? What I expect to happen?
Question: If I have a distribution, why should I not use my
judgment as to the appropriate statistic? Why is the mean
enshrined?
Guy Carpenter
10
Uncertainty in Actual Outcome
Why can’t you actuaries get the reserves right?
Guy Carpenter
Feel like a target?
11
Uncertainty in Actual Outcome
= Distribution of Outcomes

Now we are acknowledging there is a distribution of possible outcomes.

There is always some underlying sense of the distribution of outcomes.

Although, formulation of a formal statistical distribution may be difficult
or impossible. The standards of practice go out of their way not to
require one.

Nevertheless, the practitioner should and usually does have some
sense for it.
– “Well, I estimated 100 million and would be surprised if it ended up
being over 115 million or under 95 million.”
– This is not a statement that all values in the range are equally good.
Guy Carpenter
12
Uncertainty in the Distribution

Some techniques such as predictive modeling and some classes of
models will give actual distributions.

Chain ladder estimates have actual underlying statistical models, but
they are seldom invoked.
– In fact, the data is seldom systematically tested for the applicability
of the underlying models in using a technique.

Even when there is distribution, there is usually considerable
parameter uncertainty in it.


Nevertheless, one can arrive at one’s best estimate of the predictive
distribution.
This is the fundamental basis for actuarial estimates, whether explicitly
or implicitly – and more often the latter.
Guy Carpenter
13
Where Are We Now?

ASB Exposure Draft

Property/Casualty Unpaid Claim and Claim Adjustment Expense
Estimates

Comment Deadline June 30, 2006. There is still time to comment.

The essential element is the notion of Actuarial Central Estimate,
which replaces the old mean estimate.
Guy Carpenter
14
ASB Exposure Draft
Background Section


“The subcommittee discussed whether or not the proposed standard
should specify the measurement the actuary should produce when
estimating unpaid claims, such as specifying that the estimate must be
a mean or median. In the end, the subcommittee decided the
proposed standard should instruct the actuary to describe the estimate
produced, without mandating the specifics of that estimate. “
“The intent of using the term actuarial central estimate is to allow the
actuary to make an estimate without determining the complete
probability distribution or using a more sophisticated statistical
analysis. “
The intent is good: we don’t have to use the mean, and we don’t
have to have a formal distribution. Let’s see what came out of it.
Guy Carpenter
15
ASB Exposure Draft
Definition Section

“Actuarial Central Estimate—An estimate that represents a mean
excluding remote or speculative outcomes that, in the actuary’s
professional judgment, is neither optimistic nor pessimistic. An
actuarial central estimate may or may not be the result of the use of a
probability distribution or a statistical analysis. This definition is
intended to clarify the concept rather than assign a precise statistical
measure, as commonly used actuarial methods typically do not result
in a statistical mean.“
So if I have a distribution, use the mean but throw away parts of
the distribution? How remote, how speculative?
And if I don’t, make up what I thought the mean would be if I
had one?
And what happened to those other statistical estimates?
Guy Carpenter
16
ASB Exposure Draft
Definition Section

“Actuarial Central Estimate—An estimate that represents a mean
excluding remote or speculative outcomes that, in the actuary’s
professional judgment, is neither optimistic nor pessimistic. An
actuarial central estimate may or may not be the result of the use of a
probability distribution or a statistical analysis. “
Optimistic and pessimistic mean what? That I have been
unprofessionally bending numbers in the process to lean some
direction? If I feel guilty, I should re-think my estimate?
Or, that there is too small a chance of the estimate being close to
the outcome? And I evaluate that how, without a distribution?
With what criteria?
Guy Carpenter
17
ASB Exposure Draft
Definition Section

“Process Risk - Risk associated with the projection of future
contingencies, which are inherently variable. For example, even if the
mean frequency and the mean severity are properly evaluated, the
actual observed results will generally vary from the underlying means.“
Hey, good. We are talking outcomes. But, not only generally
vary, but almost surely. Never play dice with someone who can
tell you exactly how they will come up.
However, it would be nice to know on average how far away
they will be, or some other measure of variability.
Guy Carpenter
18
ASB Exposure Draft
Definition Section

“Parameter Risk - The risk that the parameters within the models are
misestimated.“
Well, sort of. It may well be not a problem in the estimation
but that that there is intrinsic variability in the parameters
themselves and we are looking at a mixed process.
Guy Carpenter
19
ASB Exposure Draft
Analysis of Issues and Recommended Practices

“Methods—The actuary should consider methods for estimating
unpaid claims that, in the actuary’s professional judgment, are
appropriate. …For any particular component of the unpaid claim
estimate, the actuary should use multiple applicable methods, unless,
in the actuary’s professional judgment, a particular method is clearly
superior to other methods. ”
Are we back to “how many ways can we find a mean,
without even a distribution?”
And is using multiple methods supposed to be a surrogate
for having a more real estimate of the actual variability of
the outcomes?
Guy Carpenter
20
ASB Exposure Draft
Analysis of Issues and Recommended Practices

“Uncertainty - The actuary is not required to measure uncertainty, but
if the actuary is measuring uncertainty, the actuary should consider the
types and sources of uncertainty being measured and choose the
methods and assumptions that, in the actuary’s professional judgment,
are appropriate for the measurement of such uncertainty. Such types
and sources of uncertainty surrounding unpaid claim estimates may
include uncertainty due to model risk, parameter risk, and process
risk. ”
It would be nice if there were SOME estimate of the
uncertainty of the outcomes were present.
In fact, for meaningfulness it is really ESSENTIAL. See
original soapbox.
Guy Carpenter
21
A Glimmer of Hope
ASB Exposure Draft Section 4.2 – Additional Disclosures
4.2.a says
“in the case when the actuary specifies a range of estimates, the actuary
should disclose the basis of the range provided, for example, a range
of actuarial central estimates; a range representing a confidence
interval within the range of outcomes produced by a particular model
or models; or a range representing a confidence interval reflecting
certain risks, such as process risk and parameter risk ..”
At least an estimate of the uncertainty of the outcomes is allowed.
Guy Carpenter
22
Summary

Historically, we started with ad-hoc methods to estimate a reserve.
Originally the focus was on getting a number for the annual statement.

Then came the realization that variability in the outcomes which that
number represented could be quite significant, and not necessarily a
tiny fraction of the current cash flow.

This encourages the statement of a range, preferably with probabilities
but at least with some sense of what would be surprising. The
estimate is really not meaningful without it.

And now, with all good intentions and sophisticated recognition of the
random nature of the claims process and parameter estimation, we
seem to have gone back to only thinking of a number. And that from a
modified mean.

I do hope I am wrong.
Guy Carpenter
23