Reserve Variability – Session I:
Where Are We Today?
Mark R. Shapland, FCAS, ASA, MAAA
Casualty Actuarial Society Spring Meeting
San Juan, Puerto Rico
May 7-10, 2006
Overview

Definitions of Terms

Ranges vs. Distributions

Methods vs. Models

Types of Methods/Models

What is “Reasonable?”
Definitions of Terms
Statements of Statutory Accounting Principles (SSAP):



Management’s Best Estimate – Management’s best estimate
of its liabilities is to be recorded. This amount may or may not
equal the actuary’s best estimate.
Ranges of Reserve Estimates – When management believes
no estimate is better than any other within the range,
management should accrue the midpoint. If a range can’t be
determined, management should accrue the best estimate.
Management’s range may or may not equal the actuary’s
range.
Best Estimate by Line – Management should accrue its best
estimate by line of business and in the aggregate. Recognized
redundancies in one line of business cannot be used to offset
recognized deficiencies in another line of business.
Definitions of Terms
Actuarial Statement of Principles No. 36 (ASOP 36):
 Risk Margin – An amount that recognizes
uncertainty; also known as a provision for
uncertainty.

Determination of Reasonable Provision – When
the stated reserve amount is within the actuary’s
range of reasonable reserve estimates, the
actuary should issue a statement of actuarial
opinion that the stated reserve amount makes a
reasonable provision for the liabilities.
Definitions of Terms
Actuarial Statement of Principles No. 36 (ASOP 36):

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. The actuary may include risk margins in a range
of reasonable estimates, but is not required to do so. A range
of reasonable reserves, however, usually does not represent
the range of all possible outcomes.
Definitions of Terms
Current ASB Exposure Draft:

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.

Reasonableness – The actuary should assess the
reasonableness of the unpaid claim estimate, using appropriate
indicators or tests that, in the actuary’s professional judgment,
provide a validation that the unpaid claim estimate is
reasonable.
Definitions of Terms
Current Principles Exposure Draft:

Actuarially Reasonable Loss Reserve – An actuarially
reasonable loss reserve for a defined group of claims is an
estimate, derived from reasonable assumptions and
appropriate methods.

Probability Distribution Representation – The unpaid
amounts required to settle a defined group of claims can be
represented as a probability distribution for which neither the
form nor the parameters are necessarily known.

Range of Reserves – The uncertainty in the estimate of the
unpaid amounts required to settle a defined group of claims
implies that a range of loss reserves can be actuarially
reasonable.
Definitions of Terms
Other Definitions :




Reserve – an amount carried in the liability section
of a risk-bearing entity’s balance sheet for claims
incurred prior to a given accounting date.
Liability – the actual amount that is owed and will
ultimately be paid by a risk-bearing entity for claims
incurred prior to a given accounting date.
Loss Liability – the expected value of all estimated
future claim payments.
Risk (from the “risk-bearers” point of view) – the
uncertainty (deviations from expected) in both timing
and amount of the future claim payment stream.
Definitions of Terms
Other Definitions :

Process Risk – the randomness of future outcomes
given a known distribution of possible outcomes.

Parameter Risk – the potential error in the
estimated parameters used to describe the
distribution of possible outcomes, assuming the
process generating the outcomes is known.

Model Risk – the chance that the model (“process”)
used to estimate the distribution of possible
outcomes is incorrect or incomplete.
Definitions of Terms
Measures of Risk from Statistics:

Variance, standard deviation, kurtosis, average
absolute deviation, Value at Risk, Tail Value at Risk,
etc. which are measures of dispersion.

Other measures useful in determining
“reasonableness” could include: mean, mode,
median, pain function, etc.

The choice for measure of risk will also be
important when considering the “reasonableness”
and “materiality” of the reserves in relation to the
capital position.
Ranges vs. Distributions

A “Range” is not the same as a “Distribution”

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.

A Distribution is a statistical function that
attempts to quantify probabilities of all
possible outcomes.
Ranges vs. Distributions
A Range, by itself, creates problems:
 A range can be misleading to the
layperson – it can give the impression
that any number in that range is equally
likely.

A range can give the impression that as
long as the carried reserve is “within the
range” anything is reasonable.
Ranges vs. Distributions
A Range, by itself, creates problems:
 There is currently no specific guidance
on how to consistently determine a
range within the actuarial community
(e.g., +/- X%, +/- $X, using various
estimates, etc.).

A range, in and of itself, needs some
other context to help define it (e.g., how
to you calculate a risk margin?)
Ranges vs. Distributions
A Distribution provides:
 Information about “all” possible
outcomes.

Context for defining a variety of other
measures (e.g., risk margin, materiality,
risk based capital, etc.)
Ranges vs. Distributions
Should we use the same:
 criterion for judging the quality of a
range vs. a distribution?

basis for determining materiality? risk
margins?

selection process for which numbers
are “reasonable” to chose from?
Methods vs. Models

A Method is an algorithm or recipe – a
series of steps that are followed to
give an estimate of future payments.

The well known chain ladder (CL) and
Bornhuetter-Ferguson (BF) methods
are examples
Methods vs. Models

A Model specifies statistical
assumptions about the loss process,
usually leaving some parameters to
be estimated.

Then estimating the parameters gives
an estimate of the ultimate losses and
some statistical properties of that
estimate.
Methods vs. Models

Many good probability models have been
built using “Collective Risk Theory”

Each of these models make assumptions
about the processes that are driving claims
and their settlement values

None of them can ever completely eliminate
“model risk”
Methods vs. Models

Processes used to calculate liability ranges
can be grouped into four general categories:
1) Multiple Projection Methods,
2) Statistics from Link Ratio Models,
3) Incremental Models, and
4) Simulation Models
Multiple Projection Methods
Description:

Uses multiple methods, data, assumptions

Assume various estimates are a good proxy
for the variation of the expected outcomes
Primary Advantages:
 Better than no range at all
 Better than +/- X%
Multiple Projection Methods
Problems:
 It does not provide a measure of the density of the
distribution for the purpose of producing a probability
function
 The “distribution” of the estimates is a distribution of
the methods and assumptions used, not a distribution
of the expected future claim payments.
 Link ratio methods only produce a single point
estimate and there is no statistical process for
determining if this point estimate is close to the
expected value of the distribution of possible
outcomes or not.
Multiple Projection Methods
Problems:
 Since there are no statistical measures for these
models, any overall distribution for all lines of
business combined will be based on the addition of
the individual ranges by line of business with
judgmental adjustments for covariance, if any.
Multiple Projection Methods
Uses:
 Data limitations may prevent the use of more
advanced models.
 A strict interpretation of the guidelines in ASOP No.
36 seems to imply the use of this “method” to
create a “reasonable” range
Statistics from Link Ratio Models
Description:
 Calculate standard error for link ratios to calculate
distribution of outcomes / range
 Typically assume normality and use logs to get a
skewed distribution
 Examples: Mack, Murphy, Bootstrapping and others
Primary Advantages:
 Significant improvement over multiple projections
 Focused on a distribution of possible outcomes
Statistics from Link Ratio Models
Problems:
 The expected value often based on multiple
methods
 Often assume link ratio errors are normally
distributed and constant by (accident) year – this
violates three criterion
 Provides a process for calculating an overall
probability distribution for all lines of business
combined, still requires assumptions about the
covariances between lines
Statistics from Link Ratio Models
Uses:
 If data limitations prevent the use of more
sophisticated models
Caveats:
 Need to make sure statistical tests are satisfied.
 ASOP No. 36 still applies to the expected value
portion of the calculations
Incremental Models
Description:
 Directly model distribution of incremental claims
 Typically assume lognormal or other skewed
distribution
 Examples: Finger, Hachmeister, Zehnwirth, England,
Verrall and others
Primary Advantages:
 Overcome the “limitations” of using cumulative values
 Modeling of calendar year inflation (along the
diagonal)
Incremental Models
Problems:

Actual distribution of incremental payments may not
be lognormal, but other skewed distributions
generally add complexity to the formulations

Correlations between lines will need to be
considered when they are combined (but can
usually be directly estimated)

Main limitation to these models seems to be only
when some data issues are present
Incremental Models
Uses:
 Usually, they allow the actuary to tailor the model
parameters to fit the characteristics of the data.
 An added bonus is that some of these models allow
the actuary to thoroughly test the model parameters
and assumptions to see if they are supported by the
data.
 They also allow the actuary to compare various
goodness of fit statistics to evaluate the
reasonableness of different models and/or different
model parameters.
Simulation Models
Description:
 Dynamic risk model of the complex interactions
between claims, reinsurance, surplus, etc.,
 Models from other groups can be used to create
such a risk model
Primary Advantage:
 Can generate a robust estimate of the distribution of
possible outcomes
Simulation Models
Problems:
 Models based on link ratios often exhibit statistical
properties not found in the real data being modeled.
 Usually overcome with models based on incremental
values or with ground-up simulations using separate
parameters for claim frequency, severity, closure
rates, etc.
 As with any model, the key is to make sure the
model and model parameters are a close reflection
of reality.
What is “Reasonable”?
A Range, by itself, creates problems:

A range (arbitrary or otherwise) can be misleading
to the layperson – it can give the impression that
any number in that range is equally likely.

A range can also give a false sense of security to
the layperson – it gives the impression that as long
as the carried reserve is “within the range” anything
is reasonable (and therefore in compliance) as long
as it can be justified by other means.
What is “Reasonable”?
A Range, by itself, creates problems:

There is currently no specific guidance on how to
consistently determine a range within the actuarial
community (e.g., +/- X%, +/- $X, using various
estimates, etc.).

A Range, in and of itself, has insufficient meaning
without some other context to help define it.
What is “Reasonable”?
$11M
$16M
What is “Reasonable”?
Premise:

We should define a “reasonable” range based
on probabilities of the distribution of possible
outcomes.

This can be translated into a range of liabilities
that correspond to those probabilities.
What is “Reasonable”?
A probability range has several advantages:

The “risk” in the data defines the range.

Adds context to other statistical measures.

A “reserve margin” can be defined more precisely.

Can be related to risk of insolvency and materiality
issues.

Others can define what is reasonable for them.
What is “Reasonable”?
A probability range has several advantages:

The “risk” in the data defines the range.

Adds context to other statistical measures.

A “reserve margin” can be defined more precisely.

Can be related to risk of insolvency and materiality
issues.

Others can define what is reasonable for them.
What is “Reasonable”?
Comparison of “Reasonable” Reserve Ranges
by Method
Relatively Stable LOB
Method
More Volatile LOB
Low
EV
High
Low
EV
High
Expected +/- 20%
80
100
120
80
100
120
50th to 75th Percentile
97
100
115
90
100
150
What is “Reasonable”?
A probability range has several advantages:

The “risk” in the data defines the range.

Adds context to other statistical measures.

A “reserve margin” can be defined more precisely.

Can be related to risk of insolvency and materiality
issues.

Others can define what is reasonable for them.
What is “Reasonable”?
Comparison of “Normal” vs. “Skewed” Liability Distributions
“Normally” Distributed Liabilities
“Skewed” Liability Distribution
Probability
Probability
50th Percentile
Expected Value, Mode & Median
Value
50th Percentile
Mode
Median
Expected Value
Value
What is “Reasonable”?
Comparison of Aggregate Liability Distributions
Probability
Liability Distribution for Line A
50 th Percentile
Mode
Median
Value
Expected Value
Probability
Aggregate Liability Distribution with 100% Correlation
(Added)
50 th Percentile
Liability Distribution for Line B
Probability
Mode
Median
Value
Expected Value
50 th Percentile
Aggregate Liability Distribution With No Correlation
(Independent)
Value
Expected Value
50 th Percentile
Probability
Liability Distribution for Line C
50 th Percentile
Mode
Median
Value
Expected Value
Probability
Mode
Median
Mode
Median
Value
Expected Value
What is “Reasonable”?
A probability range has several advantages:

The “risk” in the data defines the range.

Adds context to other statistical measures.

A “reserve margin” can be defined more precisely.

Can be related to risk of insolvency and materiality
issues.

Others can define what is reasonable for them.
What is “Reasonable”?
50 th Percentile
Probability
Reasonable & Prudent Margin
75th Percentile
Reasonable & Conservative Margin
Mode
Median
Expected Value
Liability Estimate
What is “Reasonable”?
A probability range has several advantages:

The “risk” in the data defines the range.

Adds context to other statistical measures.

A “reserve margin” can be defined more precisely.

Can be related to risk of insolvency and materiality
issues.

Others can define what is reasonable for them.
What is “Reasonable”?
Comparison of “Reasonable” Reserve Ranges
with Probabilities of Insolvency
“Low” Reserve Risk
Corresponding Surplus Depending on Situation
Loss Reserves
Situation A
Prob.
Amount Of Ins.
Situation B
Situation C
Amount
Prob.
Of Ins.
Amount
Prob.
Of Ins.
Amount
Prob.
100
50%
80
40%
120
15%
160
1%
110
75%
70
40%
110
15%
150
1%
120
90%
60
40%
100
15%
140
1%
What is “Reasonable”?
Comparison of “Reasonable” Reserve Ranges
with Probabilities of Insolvency
“Medium” Reserve Risk
Corresponding Surplus Depending on Situation
Loss Reserves
Situation A
Situation B
Prob.
Amount Of Ins.
Prob.
Amount Of Ins.
Situation C
Amount
Prob.
Of Ins.
40%
160
10%
100
40%
140
10%
80
40%
120
10%
Amount
Prob.
100
50%
80
60%
120
120
75%
60
60%
140
90%
40
60%
What is “Reasonable”?
Comparison of “Reasonable” Reserve Ranges
with Probabilities of Insolvency
“High” Reserve Risk
Corresponding Surplus Depending on Situation
Loss Reserves
Situation A
Prob.
Amount Of Ins.
Situation B
Situation C
Amount
Prob.
Of Ins.
Amount
Prob.
Of Ins.
Amount
Prob.
100
50%
80
80%
120
50%
160
20%
150
75%
30
80%
70
50%
110
20%
200
90%
-20
80%
20
50%
60
20%
What is “Reasonable”?
A probability range has several advantages:

The “risk” in the data defines the range.

Adds context to other statistical measures.

A “reserve margin” can be defined more precisely.

Can be related to risk of insolvency and materiality
issues.

Others can define what is reasonable for them.
What is “Reasonable”?
Satisfying Different Constituents:

Principle of Greatest Common Interest – the
“largest amount” considered “reasonable” when a
variety of constituents share a common goal or
interest, such that all common goals or interests are
met; and the

Principle of Least Common Interest – the
“smallest amount” considered “reasonable” when a
variety of constituents share a common goal or
interest, such that all common goals or interests are
met.
What is “Reasonable”?
$11M
$16M
What is “Reasonable”?
Probability
50 th Percentile
75th Percentile
$16M
$11M
Mode
Median
Expected Value
Liability Estimate
What is “Reasonable”?
$11M
Probability
50 th Percentile
75th Percentile
$16M
Mode
Median
Expected Value
Liability Estimate
What is “Reasonable”?
Probability
50 th Percentile
75th Percentile
$16M
$11M
Mode
Median
Expected Value
Liability Estimate
What is “Reasonable”?
Probability
50 th Percentile
75th Percentile
$11M
Mode
Median
$16M
Expected Value
Liability Estimate
Conclusions




Users of actuarial liability estimates based on
probability ranges will get much more information
for risk evaluation and decision-making,
The width of the dollar range will be directly related
to the potential volatility (uncertainty) of the actual
data,
The concept of materiality can be more directly
related to the uncertainty of the estimates,
Risk-Based Capital calculations could be related to
the probability “level” of the reserves,
Conclusions



Both ends of the “reasonable” range of reserves
will be related to the probability distribution of
possible outcomes in addition to the
“reasonableness” of the underlying assumptions,
The concept of a “prudent reserve margin” could
be related to a portion of the probability range and
will then be directly related to the uncertainty of the
estimates, and
The users of actuarial liability estimates would
have the opportunity to give more specific input on
what they consider “reasonable.”
Conclusions

To implement the advantages of the statistical
approach, the actuarial profession should consider
adding wording similar to the following:
“Whenever the actuary can produce a reasonable
distribution of possible outcomes, a reasonable
unpaid claim estimate should not be less than the
expected value of that distribution.”
Conclusions

To implement the advantages of the statistical
approach, the actuarial profession should consider
adding wording similar to the following:
“Whenever the actuary uses multiple methods to
determine a range of central estimates, if no one
estimate is any better than the others, then a
reasonable unpaid claim estimate should not be
less than the midpoint of the range.”
Questions?