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Reserve Uncertainty Reserve Uncertainty 1999 CLRS 1999 CLRS by by Roger M. Hayne, FCAS, MAAA Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc. Milliman & Robertson, Inc.

Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

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Page 1: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Reserve UncertaintyReserve Uncertainty1999 CLRS1999 CLRS

byby

Roger M. Hayne, FCAS, MAAARoger M. Hayne, FCAS, MAAA

Milliman & Robertson, Inc.Milliman & Robertson, Inc.

Page 2: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Reserves Are Uncertain?Reserves Are Uncertain?

Reserves are just numbers in a financial statementReserves are just numbers in a financial statement What do we mean by “reserves are uncertain?”What do we mean by “reserves are uncertain?”

– Numbers are Numbers are estimatesestimates of future payments of future payments» Not estimates of the averageNot estimates of the average

» Not estimates of the modeNot estimates of the mode

» Not estimates of the medianNot estimates of the median

– Not really much guidance in guidelinesNot really much guidance in guidelines

Rodney’s presentation will deal with this moreRodney’s presentation will deal with this more

Page 3: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Let’s Move Off the PhilosophyLet’s Move Off the Philosophy

Should be more guidance in accounting/actuarial Should be more guidance in accounting/actuarial literatureliterature

Not clear what number should be bookedNot clear what number should be booked Less clear if we do not know the distribution of Less clear if we do not know the distribution of

that numberthat number There may be an argument that the more uncertain There may be an argument that the more uncertain

the estimate the greater the “margin”the estimate the greater the “margin” Need to know distribution firstNeed to know distribution first

Page 4: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

““Traditional” MethodsTraditional” Methods

Many “traditional” reserve methods are somewhat Many “traditional” reserve methods are somewhat ad-hocad-hoc

Oldest, probably development factorOldest, probably development factor– Fairly easy to explainFairly easy to explain

– Subject of much literatureSubject of much literature

– Not originally grounded in theory, though some have Not originally grounded in theory, though some have tried recentlytried recently

– Known to be quite volatile for less mature exposure Known to be quite volatile for less mature exposure periodsperiods

Page 5: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

““Traditional” MethodsTraditional” Methods

Bornhuetter-FergusonBornhuetter-Ferguson– Overcomes volatility of development factor method for Overcomes volatility of development factor method for

immature periodsimmature periods

– Needs both development and estimate of the final Needs both development and estimate of the final answer (expected losses)answer (expected losses)

– No statistical foundationNo statistical foundation

Frequency/Severity (Berquist, Sherman)Frequency/Severity (Berquist, Sherman)– Also ad-hocAlso ad-hoc

– Volatility in selection of trends & averagesVolatility in selection of trends & averages

Page 6: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

““Traditional” MethodsTraditional” Methods

Not usually grounded in statistical theoryNot usually grounded in statistical theory Fundamental assumptions not always clearly Fundamental assumptions not always clearly

statedstated Often not amenable to directly estimate variabilityOften not amenable to directly estimate variability ““Traditional” approach usually uses various Traditional” approach usually uses various

methods, with different underlying assumptions, methods, with different underlying assumptions, to give the actuary a “sense” of variabilityto give the actuary a “sense” of variability

Page 7: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Basic AssumptionBasic Assumption

When talking about reserve variability When talking about reserve variability primary assumption is:primary assumption is:

Given current knowledge there is a Given current knowledge there is a distribution of possible future payments distribution of possible future payments (possible reserve numbers)(possible reserve numbers)

Keep this in mind whenever answering the Keep this in mind whenever answering the question “How uncertain are reserves?”question “How uncertain are reserves?”

Page 8: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Some ConceptsSome Concepts

Baby steps first, estimate a distributionBaby steps first, estimate a distribution Sources of uncertainty:Sources of uncertainty:

– Process (purely random)Process (purely random)

– Parameter (distributions are correct but parameters Parameter (distributions are correct but parameters unknown)unknown)

– Specification/Model (distribution or model not exactly Specification/Model (distribution or model not exactly correct)correct)

Keep in mind whenever looking at methods that Keep in mind whenever looking at methods that purport to quantify reserve uncertaintypurport to quantify reserve uncertainty

Page 9: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Why Is This Important?Why Is This Important?

Consider an exampleConsider an example ““Usual” development factor projection methodUsual” development factor projection method Assume:Assume:

– Reserves can be estimated by development factor Reserves can be estimated by development factor methodmethod

– Age-to-age factors lognormalAge-to-age factors lognormal

– Age-to-age factors independentAge-to-age factors independent

– You can estimate age-to-age parameters using observed You can estimate age-to-age parameters using observed factorsfactors

Page 10: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

ConclusionsConclusions

Use “customary” parameterization of Use “customary” parameterization of lognormal (based on transformed normal)lognormal (based on transformed normal)

Parameters for distribution of age-to-age Parameters for distribution of age-to-age factors can be estimated using:factors can be estimated using: ii = Average of logs of observed age-to-age = Average of logs of observed age-to-age

factorsfactors ii

22 = (Sample corrected) variance of logs of = (Sample corrected) variance of logs of

observed age-to-age factorsobserved age-to-age factors

Page 11: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

ConclusionsConclusions

Given assumptions distributions of age-to-ultimate Given assumptions distributions of age-to-ultimate factors are lognormal with parameters:factors are lognormal with parameters: ii

ii22

Given amounts to date one derives a distribution Given amounts to date one derives a distribution of possible future payments for one exposure yearof possible future payments for one exposure year

Convolute years to get distribution of total Convolute years to get distribution of total reservesreserves

Page 12: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Sounds Good -- Huh?Sounds Good -- Huh?

Relatively straightforwardRelatively straightforward Easy to implementEasy to implement Gets distributions of future paymentsGets distributions of future payments Job done -- yes?Job done -- yes? Not quiteNot quite Why not?Why not?

Page 13: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

An ExampleAn Example

Apply method to paid and incurred Apply method to paid and incurred development separatelydevelopment separately

Consider resulting distributionsConsider resulting distributions What does this say about the distribution of What does this say about the distribution of

reserves?reserves? Which is correct?Which is correct?

Page 14: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

A “Real Life” ExampleA “Real Life” Example

0

0.05

0.1

0 50,000 100,000 150,000 200,000

Paid Incurred

Page 15: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

What Happened?What Happened?

Conclusions follow unavoidably from Conclusions follow unavoidably from assumptionsassumptions

Conclusions contradictoryConclusions contradictory Thus assumptions must be wrongThus assumptions must be wrong Independence of factors? Not really (there Independence of factors? Not really (there

are ways to include that in the method) are ways to include that in the method) What else?What else?

Page 16: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

What Happened?What Happened?

Obviously the two data sets are telling different Obviously the two data sets are telling different storiesstories

What is the range of the reserves?What is the range of the reserves?– Paid method?Paid method?– Incurred method?Incurred method?– Extreme from both?Extreme from both?– Something else?Something else?

Main problem -- the method addresses only one Main problem -- the method addresses only one method under specific assumptionsmethod under specific assumptions

Page 17: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

What Happened?What Happened?

Not process (that is measured by the Not process (that is measured by the distributions themselves)distributions themselves)

Is this because of parameter uncertainty?Is this because of parameter uncertainty? No, can test this statistically (from normal No, can test this statistically (from normal

distribution theory)distribution theory) If not parameter, what? What else?If not parameter, what? What else? Model/specification uncertaintyModel/specification uncertainty

Page 18: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Why Talk About This?Why Talk About This?

Almost every paper in reserve distributions Almost every paper in reserve distributions considersconsiders– Only one methodOnly one method– Applied to one data setApplied to one data set

Only conclusion: distribution of results Only conclusion: distribution of results from a single methodfrom a single method

NotNot distribution of reserves distribution of reserves

Page 19: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

DiscussionDiscussion

Some proponents of some statistically-Some proponents of some statistically-based methods argue analysis of residuals based methods argue analysis of residuals the answerthe answer

Still does not address fundamental issue; Still does not address fundamental issue; model and specification uncertaintymodel and specification uncertainty

At this point there does not appear much (if At this point there does not appear much (if anything) in the literature with methods anything) in the literature with methods addressing multiple data setsaddressing multiple data sets

Page 20: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Moral of StoryMoral of Story

Before using a method, understand Before using a method, understand underlying assumptionsunderlying assumptions

Make sure what it measures what you want Make sure what it measures what you want it toit to

The definitive work may not have been The definitive work may not have been written yetwritten yet

Casualty liabilities very complex, not Casualty liabilities very complex, not readily amenable to simple modelsreadily amenable to simple models

Page 21: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

All May Not Be LostAll May Not Be Lost

Not presenting the definitive answerNot presenting the definitive answer More an approach that may be fruitfulMore an approach that may be fruitful Approach does not necessarily have “single Approach does not necessarily have “single

model” problems in others described so farmodel” problems in others described so far Keeps some flavor of “traditional” approachesKeeps some flavor of “traditional” approaches Some theory already developed by the CAS Some theory already developed by the CAS

(Committee on Theory of Risk, Rodney Kreps, (Committee on Theory of Risk, Rodney Kreps, Chairman)Chairman)

Page 22: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Collective Risk ModelCollective Risk Model

Basic collective risk model:Basic collective risk model:– Randomly select Randomly select N, N, number of claims from claim count number of claims from claim count

distribution (often Poisson, but not necessary)distribution (often Poisson, but not necessary)– Randomly select Randomly select NN individual claims, individual claims, XX11, X, X22, …, X, …, XNN

– Calculate total loss as Calculate total loss as TT = = XXii Only necessary to estimate distributions for Only necessary to estimate distributions for

number and size of claimsnumber and size of claims Can get closed form expressions for moments Can get closed form expressions for moments

(under suitable assumptions)(under suitable assumptions)

Page 23: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Adding Parameter UncertaintyAdding Parameter Uncertainty

Heckman & Meyers added parameter uncertainty Heckman & Meyers added parameter uncertainty to both count and severity distributionsto both count and severity distributions

Modified algorithm for counts:Modified algorithm for counts:– Select Select from a Gamma distribution with mean 1 and from a Gamma distribution with mean 1 and

variance variance cc (“contagion” parameter) (“contagion” parameter)

– Select claim counts Select claim counts NN from a Poisson distribution with from a Poisson distribution with mean mean

– If If cc < 0, < 0, NN is binomial, if is binomial, if cc > 0, > 0, NN is negative binomial is negative binomial

Page 24: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Adding Parameter UncertaintyAdding Parameter Uncertainty

Heckman & Meyers also incorporated a Heckman & Meyers also incorporated a “global” uncertainty parameter“global” uncertainty parameter

Modified traditional collective risk modelModified traditional collective risk model– Select Select from a distribution with mean 1 and from a distribution with mean 1 and

variance variance bb– Select Select NN and and XX11, X, X22, …, X, …, XNN as before as before

– Calculate total as Calculate total as TT = = XXii

Note Note affects affects allall claims uniformly claims uniformly

Page 25: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Why Does This Matter?Why Does This Matter?

Under suitable assumptions the Heckman & Under suitable assumptions the Heckman & Meyers algorithm gives the following:Meyers algorithm gives the following:– E(E(TT) = E() = E(NN)E()E(XX))– Var(Var(TT)= )= (1(1+b+b)E()E(XX22)+)+22((bb++cc++bcbc)E)E22((XX))

Notice if Notice if bb==cc=0 then =0 then – Var(Var(TT)= )= E(E(XX22))– Average, Average, TT//NN will have a decreasing variance as will have a decreasing variance as

E(E(NN)=)= is large (law of large numbers) is large (law of large numbers)

Page 26: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

Why Does This Matter?Why Does This Matter?

If If bb 0 or 0 or cc 0 the second term remains 0 the second term remains Variance of average tends to (Variance of average tends to (bb++cc++bcbc)E)E22((XX)) Not zeroNot zero Otherwise said: No matter how much data Otherwise said: No matter how much data

you have you still have uncertainty about you have you still have uncertainty about the meanthe mean

Key to alternative approach -- Use of Key to alternative approach -- Use of bb and and cc parameters to build in uncertainty parameters to build in uncertainty

Page 27: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

If It Were That Easy …If It Were That Easy …

Still need to estimate the distributionsStill need to estimate the distributions Even if we have distributions, still need to Even if we have distributions, still need to

estimate parameters (like estimating estimate parameters (like estimating reserves)reserves)

Typically estimate parameters for each Typically estimate parameters for each exposure periodexposure period

Problem with potential dependence among Problem with potential dependence among years when combining for final reservesyears when combining for final reserves

Page 28: Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc

CAS To The RescueCAS To The Rescue

CAS Committee on Theory of Risk commissioned CAS Committee on Theory of Risk commissioned research intoresearch into– Aggregate distributions without independence Aggregate distributions without independence

assumptionsassumptions

– Aging of distributions over life of an exposure yearAging of distributions over life of an exposure year

Paper on the first finished, second nearly soPaper on the first finished, second nearly so Will help in reserve variabilityWill help in reserve variability Sorry, do not have all the answers yetSorry, do not have all the answers yet