Reserve Ranges
Roger M. Hayne, FCAS, MAAA
C.K. “Stan” Khury, FCAS, MAAA
Robert F. Wolf, FCAS, MAAA
2005 CAS Spring Meeting
Changing Scene

Changes:
– Changes in the 2005 NAIC reporting
requirements (best estimate, ranges, etc.)
– SEC pending rule changes about disclosures with
respect to items involving uncertainty
– Pending changes in the reserving principles
– Pending changes in the ASOP

Unifying theme driving all of these changes:
– A reserve is really a probability statement
consisting of an amount x plus the probability
that the final settlement will not exceed x
A Range – Gas or Electric?
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Start simple – a range around what?
Accountants say it is to be a
“reasonable estimate” of the unpaid
claim costs
CAS says that “an actuarially sound
loss reserve … is a provision, based on
estimates derived from reasonable
assumptions …”
First Question – An
Estimate of What?
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An “estimate” of amount unpaid.
Is it an estimate of the average
amount to be paid? No
It is an estimate of the most likely
amount to be paid? No
It is an estimate of the amount to be
paid
Simple Example
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Reserves as of 12/31/2005
Claim to be settled 1/1/2006 with immediate
payment of $1 million times roll of fair die
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
An Almost-Simple
Example
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
Reserves as of 12/31/2005
Claim to be settled 1/1/2006 as $1 million
times toss of loaded die:
– Prob(x=1)=Prob(x=6)=1/4
– Prob(x=2)=Prob(x=5)=1/6
– Prob(x=3)=Prob(x=4)=1/12
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What do you book now?
Mean and median still $3.5 million
“Most likely” is either $1 million or $6 million
Traditional Approach
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Traditional actuarial methods:
– “Deestribution? We don’ need no steenkin’
deestribution.”
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Traditional methods give “an estimate”
No assumptions, thus no conclusions on
distributions
There are stochastic versions of some
methods (chain ladder, BornhuetterFerguson)
Traditional Estimates

Traditional methods give “estimates”
–
–
–
–
–
–

Not estimates of
Not estimates of
Not estimates of
Not estimates of
Not estimates of
Just “estimates”
the mean
the median
the mode
a percentile
any statistic of the distribution
Distributions come only after added
assumptions
Range of Reasonable
Results
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Designed for traditional analysis
Does not address or even talk about
distributions
Definition is “soft” and talks about
results of “reasonable” methods under
“reasonable” assumptions
Does not refer to the distribution of
potential outcomes
Reasonable?


Range of reasonable results an
attempt to quantify an actuary’s “gut
feel”
Typically you do a lot of methods
– If they “bunch up” you feel “good”
– If they are “spread out” you feel
“uncomfortable”
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In the end – quite subjective
Model and Method
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A method is a general approach
– Chain ladder
– Bornhuetter-Ferguson
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A model specifies an underlying process or
distribution and the focus is in
parameterizing the model
Many traditional actuarial forecasting
approaches are methods and not models
Stochastic Methods
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Stochastic methods have assumptions about
underlying models
Nearly all focus on a single data set (paid
loss triangle, incurred loss triangle, etc.)
Do not directly model multiple sources of
information (e.g. counts, paid, and incurred
at the same time)
Mack/Quarg method not yet stochastic
Some Vocabulary
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Components of uncertainty:
– Process
– Parameter
– Model/Specification
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Any true estimate of the distribution of
outcomes should recognize all three
Process
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Uncertainty that cannot be avoided
Inherent in the process
Example – the throw of a fair die
– You completely know the process
– You cannot predict the result with certainty
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Usually the smallest component of insurance
distribuitons (law of large numbers)
Parameter
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Uncertainty about the parameters of
models
The underlying process is know
Just the position of some “knobs” are
not
Example – flip of a weighted coin
– Uncertainty regarding the expected
proportion of heads
Model/Specification
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The uncertainty that you have the
right model to begin with
Not just what distributions, but what
form the model should take
Most difficult to estimate
Arguably unestimable for P&C
insurance situations
Distribution of Outcomes
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Combines all sources of uncertainty
Gives potential future payments at
point in time along with associated
likelihood
Must be estimated
Estimation is itself subject to
uncertainty, so we are not away from
“reasonable” issues
Example Distribution I
Distribution of IBNR Outcomes
Using Chain Ladder Method - UnWeighted Analysis
0.55%
0.50%
0.45%
0.35%
0.30%
0.25%
0.20%
0.15%
0.10%
0.05%
0.00%
1,
05
4
1,
20
2
1,
35
0
1,
49
8
1,
64
6
1,
79
4
1,
94
2
2,
09
0
2,
23
8
2,
38
6
2,
53
4
2,
68
2
2,
83
0
2,
97
8
3,
12
6
3,
27
4
3,
42
2
3,
57
0
3,
71
8
3,
86
6
4,
01
4
4,
16
2
4,
31
0
4,
45
9
4,
60
7
4,
75
5
16
5
31
3
46
1
61
0
75
8
90
6
Frequency
0.40%
IBNR Amounts
BassAndKhury.com
Example Distribution II
Distribution of IBNR Outcomes
Using Chain Ladder Method - Incurred - UnWeighted Analysis
0.65%
0.60%
0.55%
0.50%
Frequency
0.45%
0.40%
0.35%
0.30%
0.25%
0.20%
0.15%
0.10%
0.05%
0.00%
-3.1 -1.7 -0.3 1.2 2.6 4.0 5.5 6.9 8.4 9.8 11.2 12.7 14.1 15.6 17.0 18.4 19.9 21.3 22.7 24.2 25.6 27.1 28.5 29.9 31.4
IBNR Amounts
BassAndKhury.com
What is Reasonable?
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I use a series of methods
My “range of reasonable estimates” is the
range of forecasts from the various methods
Is this reasonable?
What if one or more of the methods is really
“unreasonable”?
Is something outside this range
“reasonable”?
A Range Idea
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Take largest and smallest forecast by
accident year
Add these together
Is this a “reasonable range”
Example:
– Roll of single fair die, 2/3 confidence interval is
between 2 an 5 inclusive
– Roll of a pair of fair dice, 2/3 confidence interval
is between 5 and 9 inclusive, not 4 to 10 (5/6).
You Missed Again!!
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Your best estimate is $x
Actual future payments is $y (>$x)
Conclusion – you were “wrong”
Why? The myth that the estimate
actually will happen
Problem – a reserve is a distribution
not a single point, any treatment
otherwise is doomed to failure
Why Can’t the Actuaries
Get it Right?
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Actually, why can’t the accountants get it
right?
The accountants need to deal with the fact
rather than the myth that the actual
payments will equal the reserve estimate
Need to
–
–
–
–
Be able to book a distribution
Recognize the entire distribution
Recognize context (company environment)
Realize that future payments = reserves is an
accident with a nearly 0% chance of happening
An Economically Rational
Reserve
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Why not set reserves so that the loss in
company value when actual payments turn
out different is the least expected
Note expectation taken over all possible
reserve outcomes (along with their
probabilities)
Economically rational – focuses on company
worth
Least Pain
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Since any number will be “wrong” let me
submit a reasonable estimate of reserves
(complements of Rodney Kreps)
Suppose
– (a really BIG suppose) we know the probability
density function of future claim payments and
expenses is f(x)
– For simplicity one year time horizon
– g(x,μ) denotes the decrease in shareholder
(policyholder) value of the company if reserves
are booked at μ but payments are actually x.
Least Pain (Cont.)

A rational reserve (i.e. “estimate of future
payments”) is that value of μ that minimizes

P      g  x,   f  x dx
0
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i.e. the expected penalty for setting
reserves at μ over all reserve outcomes
A Reasonable g
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Not likely symmetric
Likely flat in a region “near” μ
Increases faster when x is above μ
than when x is below
Likely increases at an increasing rate
when x is above μ
Such a function generally gives an
estimate above the mean