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Title: Loss Reserve Estimates: A Statistical Approach for Determining


1
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
2
Scope of the Paper
  • Definition of Terms
  • Measures of Risk
  • What is Reasonable?
  • Risk Concepts, Assumptions Considerations
  • Methods for Calculating Ranges
  • Practical Considerations
  • Conclusions

3
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.

4
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.

5
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.

6
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)

7
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.

8
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.

9
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.

10
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.

11
What is Reasonable?
11M
16M
12
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.

13
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.

14
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.

15
What is Reasonable?
  • Comparison of Reasonable Reserve Ranges
  • by Method


16
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.

17
What is Reasonable?
Comparison of Normal vs. Skewed Liability
Distributions
18
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
19
What is Reasonable?
Comparison of Aggregate Liability Distributions
20
What is Reasonable?
Comparison of Aggregate Liability Distributions
21
What is Reasonable?
Comparison of Aggregate Liability Distributions
Capital 1,000M
Capital 600M
22
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.

23
What is Reasonable?
Others can Define Reasonability
Reasonable Prudent Margin
Reasonable Conservative Margin
24
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.

25
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
26
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
27
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
28
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.

29
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.

30
What is Reasonable?
31
What is Reasonable?
32
What is Reasonable?
33
What is Reasonable?
34
What is Reasonable?
35
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.

36
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.

37
Risk Concepts, AssumptionsAnd Considerations
38
Risk Concepts, AssumptionsAnd Considerations
39
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.

40
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.

41
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.

42
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.

43
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?

44
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?

45
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?

46
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.

47
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

48
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

49
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.

50
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.

51
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

52
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

53
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

54
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

55
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)

56
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

57
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.

58
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

59
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.

60
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.

61
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?

62
Practical Considerations
63
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.

64
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.

65
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?

66
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,

67
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.

68
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.

69
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.

70
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.

71
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.

72
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.

73
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?

74
Questions?
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