Transcript Session II: Reserve Ranges Who Does What Presented by Roger M. Hayne, FCAS, MAAA May 8, 2006

Session II: Reserve Ranges
Who Does What
Presented by
Roger M. Hayne, FCAS, MAAA
May 8, 2006
Reserve – A Loaded Term
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“Reserve” is an accounting term
A “reserve” to be a “reasonable estimate” of the
unpaid claim costs
Definition is vague at best
Only requires a “reasonable estimate” (whatever that
is)
Seems to imply an amount that will happen
Even if we knew entire process this is not much help
in what to record
One number is not enough – the “reserve” is really a
set of outcomes with related probabilities, a
distribution
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Why A Distribution?
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Future payments are uncertain
A single number cannot convey that
uncertainty
If we had a distribution then concepts like
mean, mode, median, probability level, value
at risk, or any other statistic have specific
meaning
If accountants insist on a single number
embodying all that “reserve” means then we
can talk about which of (infinitely) many
numbers best convey that message
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Our Solution
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Our solution – punt
We will try not to talk about “reserves”
We will try to focus on the “distribution of
outcomes” under a policy or group of policies
(or for an insurer)
The “distribution of outcomes” will inform the
reserve to be recorded
Better having the “what to book” discussion
with this knowledge than without it
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How Did We Get Here?
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Accounting definition seems to be
intentionally vague
Current CAS Statement of Principles on
reserves also somewhat vague
– Reasonable estimates
– Ranges of reasonable estimates
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No mathematical/statistical precision
To quote Tevye – “Tradition”
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Tradition
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Traditional actuarial forecast methods are
deterministic
Do not have an underlying model (more on
this later)
They produce “estimates”
– Not estimates of the expected (mean)
– Not estimates of the most likely (mode)
– Not estimates of the middle-of-the-road
(median)
– Just “estimates” without quantifying variability
or uncertainty
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Measuring Uncertainty In Times Past
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Traditional methods do not give underlying
distributions
Our “Fore-parents” knew this and tradition
included the application of a variety of
methods
Bunching up of methods gave a sense of
variability or uncertainty
Methods gave similar answers => little
uncertainty
Methods gave disperse answers => much
uncertainty
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Reasonable Ranges and Ranges of
Reasonable Results
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Need for “Range of Reasonable Estimates”
– Still important to discuss uncertainty
– Still a need to quantify how uncertain an
estimate is
– Lack of statistical qualities in traditional
forecasts
– Solution “Range of Reasonable Estimates”
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Definition still “soft” without any statistical
meaning
Depends on nebulous term “reasonable”
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A Hint of the Future?
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Consider a more reasoned approach
Assume that a “reserve” still needs to be a
single number but that (big assumption here)
we all agree on which statistic to use
(Presenter’s digression – I like Rodney
Kreps’ “Least Pain” statistic)
What statistical sense do terms like
“reasonable estimate” and “range of
reasonable estimates” convey?
To help let’s define a few terms
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Talking About Uncertainty
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Ultimate future payments on insurance
claims are generally unknown
Theoretically, for a given amount there is a
probability that future payments will not
exceed that amount
Problem, we usually need to estimate those
probabilities
The way we do this can (should?) involve
several steps
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Simple Example
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Write policy 1/1/2006, roll fair die and hide result
Reserves as of 12/31/2006
Claim to be settled 1/1/2007 with immediate payment
of $1 million times the number already rolled
All results equally likely so some accounting
guidance says book low end ($1 million), others
midpoint ($3.5 million)
Mean and median are $3.5 million, there is no mode
What would you book as a reserve?
Note here there is no model or parameter uncertainty
If only one statistic is “reasonable” then “range of
reasonable estimates” is a single point
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Almost Simple Example
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Claim process as before
This time die is not fair:
– Prob(x=1)=Prob(x=6)=1/4
– Prob(x=2)=Prob(x=5)=1/6
– Prob(x=3)=Prob(x=4)=1/12
Mean and median still $3.5 million
“Most likely” is either $1 million or $6 million
What do you book now?
The means are the same but is the reality?
Still no parameter or model uncertainty
Again, if only one statistic is “reasonable” then “range
of reasonable estimates” is a single point
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Steps In Estimating
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Define one or more models of the future payment
process
Estimate the parameters underlying the model(s)
Assess the volatility of the process under the
assumption that the model(s) and parameters are all
correct
Aggregate the uncertainty from each of these steps
Particular contributions called respectively
model/specification, parameter and process
uncertainty
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Some Context
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The aggregate distribution you get in the end
is useful in talking about the “range of
potential outcomes”
The “range of reasonable results” is not this
range of potential outcomes
If one defines a particular statistic (mean,
“least pain,” value at risk, etc.) as a
“reasonable” reserve estimate then it makes
sense to look at the distribution of that
statistic under different selections of models
and parameters
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Simple Inclusion of Parameter
Uncertainty
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Adding parameter uncertainty is not that
difficult
Very simple example
– Losses have lognormal distribution,
parameters m (unknown) and σ2 (known),
respectively the mean and variance of the
related normal
– The parameter m itself has a normal
distribution with mean μ and variance τ2
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Simple Example Continued
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Expected (“reasonable estimate”) is lognormal
– Parameters μ+ σ2/2 and τ2
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– c.v.2 of expected is exp(τ2)-1
“What will happen” (“possible outcome”) is lognormal
– Parameters μ and σ2+τ2
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– c.v.2 is exp(σ2+τ2)-1
c.v. = standard deviation/mean, measure of relative
dispersion
Note expected is much more certain (smaller c.v.)
than “what will happen”
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Carry the Same Thought Further
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Suppose that judgmentally or otherwise we
can quantify the likelihood of various models
Think of each of them as different possible
future states of the world
Why not use this information similar to the
way the normal distribution was used in the
example to quantify parameter/model
uncertainty
Simplifies matters
– Quantifies relative weights
– Provides for a way to evolve those weights
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An Evolutionary (Bayesian) Model
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Again take a very simple example
Use the die example
For simplicity assume we book the mean
This time there are three different dice that can be
thrown and we do not know which one it is
Currently no information favors one die over others
The dice have the following chances of outcomes:
1
1/6
1/21
6/21
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2
1/6
2/21
5/21
3
1/6
3/21
4/21
4
1/6
4/21
3/21
5
1/6
5/21
2/21
6
1/6
6/21
1/21
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Evolutionary Approach
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“What will happen” is the same as the first
die, equal chances of 1 through 6
The expected has equally likely chances of
being 2.67, 3.50, or 4.33
If you set your reserve at the “average” both
have the same average, 3.5, the true
average is within 0.83 of this amount with
100% confidence
There is a 1/3 chance the outcome will be
2.5 away from this pick.
We now “observe” a 2 – what do we do?
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How Likely Is It?
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Likelihood of observing a 2:
– Distribution 1
– Distribution 2
– Distribution 3
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1/6
2/21
5/21
Given our distributions it seems more likely that
the true state of the world is 3 (having observed
a 2) than the others
Use Bayes Theorem to estimate posterior
likelihoods
Posterior(model|data)likelihood(data|model)prior(model)
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Evolutionary Approach
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Revised prior is now:
– Distribution 1
– Distribution 2
– Distribution 3
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0.33
0.19
0.48
Revised posterior distribution is now:
1
2
3
4
5
6
0.20
0.19
0.17
0.16
0.15
0.13
Overall mean is 3.3
 The expected still takes on the values 2.67, 3.50, and
4.33 but with probabilities 0.48, 0.33, and 0.19
respectively (our “range”)
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Next Iteration
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Second observation of 1
Revised prior is now (based on observing a 2 and a
1):
– Distribution 1
– Distribution 2
– Distribution 3
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0.28
0.05
0.67
Revised posterior distribution is now:
1
2
3
4
5
6
0.25
0.21
0.18
0.15
0.12
0.09
Now the mean is 3.0
 The expected can be 2.67, 3.50, or 4.33 with
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probability 0.67, 0.28, and 0.05 respectively
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Not-So-Conclusive Example
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Observations 3, 4, 3, 4
Revised prior is now :
– Distribution 1
– Distribution 2
– Distribution 3
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0.34
0.33
0.33
Revised posterior distribution is unchanged from the
start:
1
2
3
4
5
6
1/6
1/6
1/6
1/6
1/6
1/6
As are the overall mean and chances for various
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states
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Summary
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Though our publics seem to want certainty future
payments are uncertain
It is virtually certain actual future payments will differ
from any estimate
Quantifying the distribution of future payments will
inform discussion
Keep process, parameter and model/specification
uncertainty in mind
Models contain more information than methods
“Ranges of reasonable estimates” are different than
“ranges of possible outcomes”
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