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Reserve Reserve UncertaintyUncertainty

byby

Roger M. Hayne, FCAS, MAAARoger M. Hayne, FCAS, MAAAMilliman USAMilliman USA

Casualty Loss Reserve SeminarCasualty Loss Reserve SeminarSeptember 10-11, 2001September 10-11, 2001

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Reserves Are Uncertain?Reserves Are Uncertain?

Reserves are just numbers in a financial Reserves are just numbers in a financial statementstatement

What do we mean by “reserves are What do we mean by “reserves are uncertain?”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 Kreps has more to say on this Rodney Kreps has more to say on this

subjectsubject

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Let’s Move Off the Let’s Move Off the PhilosophyPhilosophy

Should be more guidance in Should be more guidance in accounting/actuarial literatureaccounting/actuarial literature

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

of that numberof that number There may be an argument that the more There may be an argument that the more

uncertain the estimate the greater the uncertain the estimate the greater the “margin”“margin”

Need to know distribution firstNeed to know distribution first

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““Traditional” MethodsTraditional” Methods

Many “traditional” reserve methods Many “traditional” reserve methods are somewhat ad-hocare somewhat ad-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 Not originally grounded in theory, though

some have tried recentlysome have tried recently– Known to be quite volatile for less mature Known to be quite volatile for less mature

exposure periodsexposure periods

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““Traditional” MethodsTraditional” Methods

Bornhuetter-FergusonBornhuetter-Ferguson– Overcomes volatility of development Overcomes volatility of development

factor method for immature periodsfactor method for immature periods– Needs both development and estimate of Needs both development and estimate of

the final answer (expected losses)the final answer (expected losses)– No statistical foundationNo statistical foundation

Frequency/Severity (Berquist, Frequency/Severity (Berquist, Sherman)Sherman)– Also ad-hocAlso ad-hoc– Volatility in selection of trends & averagesVolatility in selection of trends & averages

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““Traditional” MethodsTraditional” Methods

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

clearly statedclearly stated Often not amenable to directly estimate Often not amenable to directly estimate

variabilityvariability ““Traditional” approach usually uses Traditional” approach usually uses

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

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

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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 Parameter (distributions are correct but

parameters unknown)parameters unknown)– Specification/Model (distribution or model not Specification/Model (distribution or model not

exactly correct)exactly correct)

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

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Why Is This Important?Why Is This Important?

Consider “usual” development factor Consider “usual” development factor projection method, projection method, CCikik accident year accident year i, i, paid paid

by ageby age k k Assume:Assume:

– There are development factors There are development factors ffii such thatsuch that

E(E(CCi,k+1i,k+1||CCi1i1, C, Ci2i2,…, C,…, Cikik)= )= ffkk CCikik

– {{CCi1i1, C, Ci2i2,…, C,…, CiIiI}, {}, {CCj1j1, C, Cj2j2,…, C,…, CjIjI} independent for } independent for i i

jj– There are constants There are constants kk such thatsuch that

Var(Var(CCi,k+1i,k+1||CCi1i1, C, Ci2i2,…, C,…, Cikik)= )= CCik ik kk22

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ConclusionsConclusions

Following Mack (Following Mack (ASTIN Bulletin, ASTIN Bulletin, v. 23, No. 2, pp. v. 23, No. 2, pp. 213-225)213-225)

, 1 ,1 1

ˆ

I k I k

k j k j kj j

f C C

are unbiased estimates for the development are unbiased estimates for the development factors factors ffii

Can also estimate standard error of reserveCan also estimate standard error of reserve

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ConclusionsConclusions

Estimate of mean squared error (Estimate of mean squared error (msemse) ) of reserve forecast for one accident of reserve forecast for one accident year:year:

212

21

1

ˆ 1 1ˆˆˆ ˆ

Ik

i iI I kk I i ikk

jkj

mse R CCf C

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ConclusionsConclusions

Estimate of mean squared error (Estimate of mean squared error (msemse) ) of the total reserve forecast:of the total reserve forecast:

2 212

12 1 1

1

ˆˆ2ˆ ˆˆ ˆs.e.

I I Ik k

i iI jI Ii j i k I i

nkn

fmse R R C C

C

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

Relatively straightforwardRelatively straightforwardEasy to implementEasy to implementGets distributions of future Gets distributions of future

paymentspaymentsJob done -- yes?Job done -- yes?Not quiteNot quiteWhy not?Why not?

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An ExampleAn Example

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

Consider resulting estimates and Consider resulting estimates and errorserrors

What does this say about the What does this say about the distribution of reserves?distribution of reserves?

Which is correct?Which is correct?

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““Real Life” ExampleReal Life” Example

Paid and Incurred as in handouts Paid and Incurred as in handouts (too large for slide)(too large for slide)

ResultsResults

PaidPaid IncurredIncurred

Case ReserveCase Reserve $96,917$96,917

Reserve Est.Reserve Est. $358,453$358,453 90,58090,580

s.e.(Est.)s.e.(Est.) 41,63941,639 13,52413,524

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A “Real Life” ExampleA “Real Life” Example

100,000 150,000 200,000 250,000 300,000 350,000 400,000 450,000 500,000

Paid Incurred

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A “Real Life” ExampleA “Real Life” Example

100,000 150,000 200,000 250,000 300,000 350,000 400,000 450,000 500,000

Paid Incurred Actual

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What Happened?What Happened?

Conclusions follow unavoidably Conclusions follow unavoidably from assumptionsfrom assumptions

Conclusions contradictoryConclusions contradictoryThus assumptions must be wrongThus assumptions must be wrong Independence of factors? Not Independence of factors? Not

really (there are ways to include really (there are ways to include that in the method) that in the method)

What else?What else?

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What Happened?What Happened?

Obviously the two data sets are telling Obviously the two data sets are telling different storiesdifferent stories

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

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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 Is this because of parameter uncertainty?uncertainty?

No, can test this statistically (from No, can test this statistically (from normal distribution theory)normal distribution theory)

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

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Why Talk About This?Why Talk About This?

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

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

NotNot distribution of reserves distribution of reserves

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DiscussionDiscussion

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

Still does not address fundamental Still does not address fundamental issue; model and specification issue; model and specification uncertaintyuncertainty

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

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Moral of StoryMoral of Story

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

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

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

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

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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 Approach does not necessarily have

“single model” problems in others “single model” problems in others described so fardescribed so far

Keeps some flavor of “traditional” Keeps some flavor of “traditional” approachesapproaches

Some theory already developed by the Some theory already developed by the CAS (Committee on Theory of Risk, Phil CAS (Committee on Theory of Risk, Phil Heckman, Chairman)Heckman, Chairman)

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Collective Risk ModelCollective Risk Model

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

count distribution (often Poisson, but not count distribution (often Poisson, but not necessary)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 Can get closed form expressions for

moments (under suitable assumptions)moments (under suitable assumptions)

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Adding Parameter Adding Parameter UncertaintyUncertainty

Heckman & Meyers added parameter Heckman & Meyers added parameter uncertainty to both count and severity uncertainty to both count and severity distributionsdistributions

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

1 and variance 1 and variance cc (“contagion” parameter) (“contagion” parameter)– Select claim counts Select claim counts NN from a Poisson from a Poisson

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

binomialbinomial

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Adding Parameter Adding Parameter UncertaintyUncertainty

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 from a distribution with mean 1

and variance and 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

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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 will have a decreasing variance

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

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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 remainsVariance of average tends to Variance of average tends to

((bb++cc++bcbc)E)E22((XX))Not zeroNot zeroOtherwise 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

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

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

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

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

Problem with potential dependence Problem with potential dependence among years when combining for final among years when combining for final reservesreserves

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An ExampleAn Example

Consider the data set included in Consider the data set included in the handoutsthe handouts

This is hypothetical data but This is hypothetical data but based on a real situationbased on a real situation

It is residual bodily injury liability It is residual bodily injury liability under no-faultunder no-fault

Rather homogeneous insured Rather homogeneous insured populationpopulation

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An ExampleAn Example(Continued)(Continued)

Applied several “traditional” Applied several “traditional” actuarial methodsactuarial methods– Usual development factorUsual development factor– Berquist/ShermanBerquist/Sherman– Hindsight reserve methodHindsight reserve method– Adjustments forAdjustments for

Relative case reserve adequacyRelative case reserve adequacy Changes in closing patternsChanges in closing patterns

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An ExampleAn Example(Continued)(Continued)

Reserve Estimates by Method

Accident Paid Adjusted CD Adjusted Paid

Year Incurred Devel. Sev. Pure Prem. Hindsight Incurred Devel. Sev. Pure Prem. Hindsight

1986 744 2,143 1,760 1,909 1,687 394 1,936 1,842 1,950 675

1987 2,335 6,847 5,583 5,128 5,128 2,348 6,000 5,790 5,220 2,301

1988 8,371 19,768 16,246 13,451 14,428 10,391 17,352 16,433 13,399 8,001

1989 25,787 44,631 36,887 29,232 32,199 26,048 39,241 36,431 28,512 19,174

1990 60,211 83,760 73,987 61,846 62,974 55,734 79,667 70,246 57,192 43,286

1991 83,093 130,907 95,283 95,185 78,616 79,573 154,268 87,625 84,688 72,157

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An ExampleAn Example(Continued)(Continued)

Now review underlying claim Now review underlying claim informationinformation

Make selections regarding the Make selections regarding the distribution of size of open claims for distribution of size of open claims for each accident yeareach accident year– Based on actual claim size distributionsBased on actual claim size distributions– RatemakingRatemaking– OtherOther

Use this to estimate contagion (c) Use this to estimate contagion (c) valuevalue

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An ExampleAn Example(Continued)(Continued)

Accident Reserve Unpaid Single Claim Implied

Year Selected Std. Dev. Counts Average Std. Dev. c Value

1986 1,357 637 106 12,802 18,913 0.190

1987 4,260 1,620 330 12,909 19,072 0.135

1988 12,866 3,525 926 13,894 20,527 0.072

1989 30,212 6,428 1,894 15,951 23,566 0.044

1990 62,516 10,198 3,347 18,678 27,595 0.026

1991 90,014 19,166 4,071 22,111 32,666 0.045

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An ExampleAn Example(Continued)(Continued)

Thus variation among various Thus variation among various forecasts helps identify parameter forecasts helps identify parameter uncertainty for a yearuncertainty for a year

Still “global” uncertainty that Still “global” uncertainty that affects all yearsaffects all years

Measure this by “noise” in Measure this by “noise” in underlying severityunderlying severity

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An ExampleAn Example(Continued)(Continued)

Accident Severity Estimate Year Selected Fitted of 1/

1986 7,723 7,780 0.9931987 8,501 8,196 1.0371988 9,577 8,634 1.1091989 9,919 9,095 1.0911990 10,739 9,581 1.1211991 12,194 10,093 1.208

Variance 0.019

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An ExampleAn Example(Continued)(Continued)

100,000 150,000 200,000 250,000 300,000

With Uncertainty Without Uncertainty Actual

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CAS To The RescueCAS To The Rescue

Still assumed independenceStill assumed independence CAS Committee on Theory of Risk CAS Committee on Theory of Risk

commissioned research intocommissioned research into– Aggregate distributions without independence Aggregate distributions without independence

assumptionsassumptions– Aging of distributions over life of an exposure Aging of distributions over life of an exposure

yearyear

Will help in reserve variabilityWill help in reserve variability Sorry, do not have all the answers yetSorry, do not have all the answers yet