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Loss Reserve EstimatesA Statistical Approach

for Determining Reasonableness

- Mark R. Shapland, FCAS, ASA, MAAA

Casualty Actuarial Society Spring Meeting Disney

Contemporary Resort Orlando, Florida June 20, 2007

Scope of the Paper

- Definition of Terms
- Measures of Risk
- What is Reasonable?
- Risk Concepts, Assumptions Considerations
- Methods for Calculating Ranges
- Practical Considerations
- Conclusions

Definition of Terms

- From the Statements of Statutory Accounting

Principles (SSAP)

- Managements Best Estimate Managements best

estimate of its liabilities is to be recorded.

This amount may or may not equal the actuarys

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 cant be determined,

management should accrue the best estimate.

Managements range may or may not equal the

actuarys 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.

Definition of Terms

- From the 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 actuarys

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.

Definition of Terms

- From the 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.

Definition of Terms

- From the Proposed Unpaid Claim Estimates ASOP

- Actuarial Central Estimate An estimate that

represents a mean excluding remote or speculative

outcomes that, in the actuarys 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. - (First Draft)
- An estimate that represents an expected value

over the range of reasonably possible outcomes. - (Second Draft)

Definition of Terms

- Other Definitions Offered in the Paper

- Reserve an amount carried in the liability

section of a risk-bearing entitys 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.

Measures of Risk

- From Statistics

- Variance, standard deviation, skewness, average

absolute deviation, Value at Risk, Tail Value at

Risk, etc. 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.

Measures of Risk

- Types of Risk

- 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.

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. - There is currently no specific guidance 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 could 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

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

What is Reasonable?

Comparison of Aggregate Liability Distributions

LOB A

Aggregate Distribution with 100

Correlation (Added)

LOB B

Aggregate Distribution with 0 Correlation (Indepe

ndent)

LOB C

What is Reasonable?

Comparison of Aggregate Liability Distributions

What is Reasonable?

Comparison of Aggregate Liability Distributions

What is Reasonable?

Comparison of Aggregate Liability Distributions

Capital 1,000M

Capital 600M

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?

Others can Define Reasonability

Reasonable Prudent Margin

Reasonable Conservative Margin

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

Situation C

Situation B

Situation A

Loss Reserves

Prob. Of Ins.

Amount

Prob. Of Ins.

Amount

Prob. Of Ins.

Amount

Prob.

Amount

1

160

15

120

40

80

50

100

What is Reasonable?

- Comparison of Reasonable Reserve Ranges
- with Probabilities of Insolvency

Medium Reserve Risk

Corresponding Surplus Depending on Situation

Situation C

Situation B

Situation A

Loss Reserves

Prob. Of Ins.

Amount

Prob. Of Ins.

Amount

Prob. Of Ins.

Amount

Prob.

Amount

10

160

40

120

60

80

50

100

What is Reasonable?

- Comparison of Reasonable Reserve Ranges
- with Probabilities of Insolvency

High Reserve Risk

Corresponding Surplus Depending on Situation

Situation C

Situation B

Situation A

Loss Reserves

Prob. Of Ins.

Amount

Prob. Of Ins.

Amount

Prob. Of Ins.

Amount

Prob.

Amount

20

160

50

120

80

80

50

100

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?

What is Reasonable?

What is Reasonable?

What is Reasonable?

What is Reasonable?

Risk Concepts, AssumptionsAnd Considerations

- Concept 1 For each accident year, the

coefficient of variation should be the largest

for the oldest (earliest) year and will,

generally, get smaller when compared to more and

more recent years. - Concept 2 For each accident year, the standard

error (on a dollar basis) should be the smallest

for the oldest (earliest) year and will,

generally, get larger when compared to more and

more recent years.

Risk Concepts, AssumptionsAnd Considerations

- Concept 3 The coefficient of variation should

be smaller for all accident years combined than

for any individual year. - Concept 4 The standard error (on a dollar

basis) should be larger for all accident years

combined than for any individual year.

Risk Concepts, AssumptionsAnd Considerations

Risk Concepts, AssumptionsAnd Considerations

Risk Concepts, AssumptionsAnd Considerations

- Concept 5 The standard error should be smaller

for all lines of business combined than the sum

of the individual lines of business on both a

dollar basis and as a percentage of total

liabilities (i.e., coefficient of variation). - Concept 6 In theory, it seems reasonable to

allocate any overall reserve margin (selected

by management) based on the standard error by

line after adjusting for covariances between

lines.

Risk Concepts, AssumptionsAnd Considerations

- Concept 7 Whenever simulated data is created,

it should exhibit the same statistical properties

as the real data. In other words, the simulated

data should be statistically indistinguishable

from real data.

Risk Concepts, AssumptionsAnd Considerations

- Assumption 1 For lines of business with small

payment sizes (e.g., Auto Physical Damage)

Normality might be a reasonable simplifying

assumption. - Assumption 2 For most lines of business, the

distribution of individual payments, or payments

grouped by incremental period, is skewed toward

larger values. Thus, it would be better to model

the claim payment stream using a Lognormal,

Gamma, Pareto, Burr or some other skewed

distribution function that seems to fit the

observed values.

Risk Concepts, AssumptionsAnd Considerations

- Assumption 3 Estimating the distribution of

loss liabilities assuming normality could produce

misleading results. - Assumption 4 Estimating the distribution of

loss liabilities assuming normality, but

simulating the loss distribution using a

lognormal distribution (or some other skewed

distribution) is marginally better.

Risk Concepts, AssumptionsAnd Considerations

- Consideration 1 The extra information in the

case reserves is generally believed to add value

by giving a better estimate of the expected

mean. However, does this extra information

really change the estimate of the expected value

of the payment stream (by year), or does it give

a better credibility adjusted estimate of the

likely outcome (by year) as the additional

information comes to light and leave the expected

value of the payments unchanged?

Risk Concepts, AssumptionsAnd Considerations

- Consideration 2 Consider two identical books of

business with two different insurance companies.

They are identical except that one company sets

up case reserves on the claims and the other does

not. The estimates of the total liabilities

(IBNR vs. case plus IBNR) are identical. Will

the deviations of actual from the expected value

of the future claim payments be any different?

Risk Concepts, AssumptionsAnd Considerations

- Consideration 3 Since measuring the variations

in the incurred claims does not directly measure

the variations in the payment stream, should risk

measures based on incurred claims be used to

quantify risk for management?

Models For Calculating Ranges

- 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 - All models are wrong. Some models are useful.

Models For Calculating Ranges

- 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 Bootstrapping, 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.

Practical Considerations

- Are Reasonable Assumptions Enough?

- Some may not agree with the statement a

reasonable range is meaningless without some

other context. - Context is provided by the ASOP No 36 phrase,

that could be produced by appropriate actuarial

models or alternative sets of assumptions that

the actuary judges to be reasonable. - In other words, The reasonable range is from A

to B must make sense in light of reasonable

statements about the history of cost drivers and

about the history of loss development.

Practical Considerations

- Are Reasonable Assumptions Enough?

- What makes selecting A as the final reserve any

more or less reasonable than B or any other

number in between? - Without any further guidance do we, as a

profession, have any basis for selecting one

number in the range over another?

Practical Considerations

Practical Considerations

- Are Reasonable Assumptions Enough?

- What makes selecting A as the final reserve any

more or less reasonable than B or any other

number in between? - Without any further guidance do we, as a

profession, have any basis for selecting one

number in the range over another?

- All of the subjectiveness cannot be removed, so

setting an absolute percentile may not be a good

idea. - But theoretically at least, the expected value

seems to be a logical minimum for a

reasonableness standard.

Practical Considerations

- Are Reasonable Assumptions Enough?

- A standard that is less than the expected value

would be akin to recommending to a casino that

they set the odds at something less than in their

favor. - Some constituents may consider a percentage lower

than the expected value to be a reasonable lower

bound - However, the principle of greatest common

interest would suggest that other interested

parties would likely insist on at least an

expected value standard as the minimum for the

reasonable probability range.

Practical Considerations

- Are Reasonable Assumptions Enough?

- Current guidelines seem to say that if you can

document the reasonableness of the models and

assumptions used to arrive at a possible

outcome then, ipso facto, that possible

outcome is reasonable. - Shouldnt we look at the reasonableness of that

possible outcome in relation to all other

possible outcomes? - No matter how reasonable a given model and

assumptions are, is that possible outcome

reasonable if it is less than the expected value

given a reasonable distribution of possible

outcomes?

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

to ASOP No. 36 - Whenever the actuary can produce a reasonable

distribution of possible outcomes, a lower bound

for the reasonable range within that distribution

should not be less than the expected value of

that distribution.

Conclusions

- The ASOP definitions of Expected Value could be

improved by adding wording similar to the

following - The expected value from a distribution should

include a statistically calculated amount to

reflect both process and parameter risk and

it could also include a judgmental amount to

reflect model risk.

Conclusions

- The ASOP definition of Risk Margin could be

improved by adding wording similar to the

following - The actuary can recommend adding a risk margin

to judgmentally reflect model risk if not

already included with the expected value.

Alternatively, the actuary can recommend

selecting a percentile above the expected value

in order to create a risk margin.

Conclusions

- Since distributions are not always possible,

required or desirable, adding wording similar to

the following to the ASOPs would be consistent

with the SSAPs - Whenever a range of estimates is produced and

the actuary has no further means of producing a

reasonable distribution of possible outcomes, or

is not obligated to produce a distribution, then

the midpoint of the range should be used as the

minimum reasonable reserve.

Conclusions

- Other issues that should be addressed in our

standards include - 1) the need to consider language to more directly

require testing of the assumptions for different

models, - 2) a more definitive solution for how to

consistently disclose the relative reserve risk,

and - 3) a more precise definition of material change

as it relates to reserve risk.

Closing

- WHAT IF you knew the EXACT distribution of

possible outcomes? - 1) Would you feel comfortable giving a clean

opinion to a company that wanted to carry less

than the expected value on their books? - 2) Will your answer give the public added

confidence in the profession? - 3) Doesnt it make sense to strengthen our

standards in order to increase public confidence?

Questions?